Shape Memory Alloy Engineering: For Aerospace, Structural, and Biomedical Applications [2 ed.] 012819264X, 9780128192641

Table of contents :
Front-Matter_2021_Shape-Memory-Alloy-Engineering
Shape Memory Alloy Engineering
Copyright_2021_Shape-Memory-Alloy-Engineering
Copyright
Dedication_2021_Shape-Memory-Alloy-Engineering
Dedication
Contributors_2021_Shape-Memory-Alloy-Engineering
Contributors
About-the-editors-in-chief_2021_Shape-Memory-Alloy-Engineering
About the editors in chief
About-the-section-editors_2021_Shape-Memory-Alloy-Engineering
About the section editors
About-the-contributors_2021_Shape-Memory-Alloy-Engineering
About the contributors
Preface-to-the-second-edition_2021_Shape-Memory-Alloy-Engineering
Preface to the second edition
Preface-to-the-first-edition_2021_Shape-Memory-Alloy-Engineering
Preface to the first edition
Section-1---Introduction_2021_Shape-Memory-Alloy-Engineering
Introduction
Chapter-1---Historical-background-and-future-p_2021_Shape-Memory-Alloy-Engin
1. Historical background and future perspectives
1.1 Shape memory alloys
1.2 List of acronyms
1.3 Gold-based alloys
1.4 Nitinol
1.4.1 A story
1.4.2 Early commercial developments
1.4.2.1 Pipe coupling
1.4.2.2 Orthodontic wires
1.4.2.3 Other medical applications
1.4.3 A conclusion
1.5 Copper-based alloys
1.5.1 Copper–zinc–aluminum
1.5.2 Copper–aluminum–nickel
1.6 Iron-based alloys
1.7 Shape memory alloy community
1.8 Future perspectives
1.8.1 A status overview
1.8.2 A vision
1.8.3 Other shape memory materials
1.9 Summary tables
Bibliography
Chapter-2---Latest-attainments_2021_Shape-Memory-Alloy-Engineering
2. Latest attainments
2.1 Introduction
2.2 List of symbols and acronyms
2.3 Application and production technologies
2.3.1 Joint and fastener applications
2.3.2 Damping systems
2.3.3 Actuators
2.4 Technological process
2.4.1 Powder metallurgy
2.4.2 Additive manufacturing
2.5 Improvement of shape memory alloy properties
2.5.1 Improvement in thermomechanical performance
2.5.2 High-temperature shape memory alloy
2.5.3 Corrosion
2.5.3.1 NiTi-based shape memory alloy
2.5.3.2 Copper-based shape memory alloy
2.5.3.3 Ferrous shape memory alloy
2.6 Overview on modeling
2.7 Conclusions
Bibliography
Chapter-3---Standards-for-shape-memory-alloy-a_2021_Shape-Memory-Alloy-Engin
3. Standards for shape memory alloy applications
3.1 Introduction
3.2 List of symbols
3.3 International market interest and concern
3.4 American Society for Testing and Materials Standards
3.4.1 The Ni-Ti binary alloy, a catalyst of interest and attention
3.4.2 Shape memory alloy American Society for Testing and Materials references
3.5 Complementary recommendations
3.5.1 Free recovery thermal control method
3.5.1.1 Test setup
3.5.1.1.1 Thermostated bath
3.5.1.1.2 Test specimen and gripping system
3.5.1.1.3 Angle measurement
3.5.1.2 Test execution
3.5.1.3 Deformation analysis of specimen flaps
3.5.2 Stress-induced martensite measurement under strain control method
3.5.2.1 Method
3.5.2.2 Results
3.5.2.3 Results
3.6 Conclusions
Bibliography
Section-2---Material_2021_Shape-Memory-Alloy-Engineering
Section 2 Material
Chapter-4---Phenomenology-of-shape-memory-al_2021_Shape-Memory-Alloy-Enginee
4. Phenomenology of shape memory alloys
4.1 Introduction
4.2 General characteristics and martensitic transformations
4.3 Functional properties of shape memory alloys
4.3.1 Shape memory effect
4.3.2 Recovery stress generation
4.3.3 Superelasticity or pseudoelasticity
4.3.4 Damping capacity
4.3.5 Examples of shape memory alloys
4.4 Porous NiTi
4.5 Magnetic shape memory alloys
4.6 Conclusion
Bibliography
Chapter-5---Experimental-characterization-of-sha_2021_Shape-Memory-Alloy-Eng
5. Experimental characterization of shape memory alloys
5.1 Introduction
5.2 List of symbols
5.3 Calorimetric investigations
5.4 Thermomechanical characterization
5.4.1 Thermomechanical tests and parameters
5.5 Complete experimental characterization of thermal and mechanical properties
5.5.1 Test 1: differential scanning calorimetry
5.5.2 Test 2: T

Citation preview

Shape Memory Alloy

ENGINEERING

For Aerospace, Structural, and Biomedical Applications Second Edition

Edited by

ANTONIO CONCILIO Department of Adaptive Structures, CIRA e the Italian Aerospace Research Centre, Capua, Italy

VINCENZA ANTONUCCI CNR-IPCB Institute for Polymer, Composite, and Biomedical Materials, National Research Council, Portici, Italy

FERDINANDO AURICCHIO Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy & Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), CNR, Pavia, Italy

LEONARDO LECCE Novotech - Advanced Aerospace Technology s.r.l., Naples, Italy

ELIO SACCO Dipartimento di Strutture per l’Ingegneria e l’Architettura (DiSt), Universita di Napoli “Federico II”, Naples, Italy

Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-819264-1 For information on all Butterworth-Heinemann publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Christina Gifford Editorial Project Manager: Gabriela Capille Production Project Manager: Poulouse Joseph Cover Designer: Miles Hitchen Typeset by TNQ Technologies Cover credit (Left photo): “La Sapienza” University of Rome, DICMA-Metallurgy archives

to Pasquale, who reshaped his life

Contributors

Salvatore Ameduri Department of Adaptive Structures, CIRA e the Italian Aerospace Research Centre, Capua, Italy Vincenza Antonucci CNR-IPCB Institute for Polymer, Composite, and Biomedical Materials, National Research Council, Portici, Italy Maurizio Arena Department of Industrial Engineering, Smart Structures Lab, University of Napoli “Federico II”, Naples, Italy Edoardo Artioli Dipartimento di Ingegneria Civile e Ingegneria Informatica (DICII), Universita degli Studi di RomaeTor Vergata, Rome, Italy Domenico Asprone Dipartimento di Strutture per l’Ingegneria e l’Architettura (Dist), Universita degli Studi di Napoli Federico II, Naples, Italy Ferdinando Auricchio Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy Keyvan Safaei Baghbaderani The University of Toledo, Toledo, OH, United States Silvestro Barbarino Data Analytics, Joby Aviation Inc., Santa Cruz, CA, United States Paolo Bettini Politecnico di Milano, Department of Aerospace Science and Technology, Milan, Italy Elisa Boatti Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy; Istituto di Matematica Applicata ed Tecnologie Informatiche (IMATI), CNR, Pavia, Italy Matthew Bray Brayfoil Technologies, Johannesburg, Gauteng, South Africa Robert Bray Brayfoil Technologies, Johannesburg, Gauteng, South Africa Andrea Brotzu Department of Chemical Engineering Materials EnvironmenteSapienza, Rome University, Rome, Italy Lorenzo Casagrande Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy

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Contributors

Paolo Chiggiato European Organization for Nuclear Research (CERN), Geneva, Switzerland Antonio Concilio Department of Adaptive Structures, CIRA - the Italian Aerospace Research Centre, Capua, Italy Michele Conti Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy Giuseppe de Ceglia Technosprings Italia SRL, Besnate, Italy Vittorio Di Cocco Department of Civil and Mechanical Engineering, Universita di Cassino e del Lazio Meridionale, Cassino, Italy Ignazio Dimino Department of Adaptive Structures, CIRA - the Italian Aerospace Research Centre, Capua, Italy Eugenio Dragoni University of Modena and Reggio Emilia, Reggio Emilia, Italy Mohammad Elahinia The University of Toledo, Toledo, OH, United States Luca Esposito Department of Engineering, University of Campania Luigi Vanvitelli, Aversa (CE), Italy Massimiliano Ferraioli Dipartimento di Ingegneria, Universita degli studi della Campania Luigi Vanvitelli, Aversa, Italy Cedric Garion European Organization for Nuclear Research (CERN), Geneva, Switzerland Stefano Gualandris Technosprings Italia SRL, Besnate, Italy Francesco Iacoviello Department of Civil and Mechanical Engineering, Universita di Cassino e del Lazio Meridionale, Cassino, Italy Maryam Khoshlahjeh Flight Physics, Joby Aviation Inc., Santa Cruz, CA, United States Leonardo Lecce Novotech - Advanced Aerospace Technology s.r.l., Naples, Italy Carmine Maletta European Organization for Nuclear Research (CERN), Geneva, Switzerland; Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy Stefania Marconi Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy

Contributors

Sonia Marfia Dipartimento di Ingegneria, Universita di Roma Tre, Rome, Italy Alfonso Martone CNR-IPCB Institute for Polymer, Composite, and Biomedical Materials, National Research Council, Portici, Italy Costantino Menna Dipartimento di Strutture per l’Ingegneria e l’Architettura (Dist), Universita degli Studi di Napoli Federico II, Naples, Italy Simone Morganti Dipartimento di Ingegneria Industriale e dell’Informazione (DIII), Universita degli Studi di Pavia, Pavia, Italy Stefano Natali Department of Chemical Engineering Materials EnvironmenteSapienza, Rome University, Rome, Italy Mohammadreza Nematollahi The University of Toledo, Toledo, OH, United States Adelaide Nespoli CNR ICMATE Unit of Lecco, Lecco, Italy Fabrizio Niccoli European Organization for Nuclear Research (CERN), Geneva, Switzerland Rosario Pecora University of Naples “Federico II”, Industrial Engineering Department, Aerospace Division, Naples, Italy Maria Rosaria Ricciardi Institute for Polymer, Composite and Biomedical Materials, National Research Council, Portici, Naples, Italy Aniello Riccio Department of Engineering, University della Campania “Luigi Vanvitelli”, Aversa, Italy Daniela Rigamonti Politecnico di Milano, Department of Aerospace Science and Technology, Milan, Italy Elio Sacco Dipartimento di Strutture per l’Ingegneria e l’Architettura (DiSt), Universita di Napoli “Federico II”, Naples, Italy Giuseppe Sala Politecnico di Milano, Department of Aerospace Science and Technology, Milan, Italy Giulia Scalet Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy Franca Scocozza Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universita degli Studi di Pavia, Pavia, Italy

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Andrea Sellitto Department of Engineering, University della Campania “Luigi Vanvitelli”, Aversa, Italy Emanuele Sgambitterra DIMEG, University of Calabria, Rende, Italy Andrea Spaggiari University of Modena and Reggio Emilia, Reggio Emilia, Italy Francesco Stortiero GFM SpA, Mapello, Bergamo, Italy Cristian Vendittozzi Faculdade do Gama, Universidade de Brasìlia, Brasìlia, DF, Brazil Andrea Vigliotti Innovative Materials Laboratory, Centro Italiano Ricerche Aerospaziali, Capua, Italy Stefano Viscuso TSS InnovationsProjekte GmbH, Roveredo, Switzerland Valerio Visentin Technosprings Italia SRL, Besnate, Italy

About the editors in chief Antonio Concilio earned his degree in aeronautics engineering with honors at the University of Napoli “Federico II” (Italy) in 1989. There, he was awarded his PhD in aerospace engineering in 1995. In 2007, he completed the ECATA Master course in Aerospace Business Administration, at ISAE-Supaero, Toulouse (France). Since 1989, he has been a researcher at the Italian Aerospace Research Centre (CIRA), Italy, where he is currently head of the Adaptive Structures Division. Between 1995 and 2000, he was head of Methods within the Vibro-acoustics Department. In 2000, he was appointed head of the newborn Smart Structures Laboratory, later integrated with the Vibration and Acoustics Lab (2002). From 2012 to 2014, he was the national relationship manager. In 2014, he was appointed head of the newly founded Adaptive Structures Division, currently one of the 10 scientific areas at CIRA. He has represented the company in a number of specific liaisons with external firms. Among the founders, he was a longtime secretary of the ex-Alumni Association of the Aerospace Engineers at the University of Naples “Federico II” (AIAN) and a member of the Italian Association for Aeronautics and Astronautics (AIDAA). He is a member of MP4EVER, a nonprofit organization supporting people with eating disorders, and their families. He is member of the Spartacus Rugby Social Club and the Steel Bucks American Football Association. Since 2005, Dr. Antonio Concilio has been a lecturer on smart structures at the PhD School of the “Federico II” University. Since 2004, he has held courses on that subject for postgraduate students from national programs funded by the Italian Ministry for Education and Research and Regione Campania, and for other universities. In 2007, he held a course on smart structures within a “master’s in systems engineering,” jointly organized by the University of Missouri-Rolla and the “Federico II” University. His main research interests are vehicles’ noise and vibration control for increased comfort and protection of devices, smart structural systems design, integrated sensor networks, and the morphing of aircraft wing and control surfaces. He led CIRA activities within several European Union (EU) and nationally funded research contracts on smart structures and noise and vibration control. He was the European scientific coordinator of the Identification of an Aircraft Passenger Comfort Index project (1998e2001) and the national project manager of the Design and Realization Methods of Intelligent Systems for the Monitoring and Control of Aeronautic and Aerospace Structures project (2002e2006). He led an international team to develop, manufacture, and test an Adaptive Wing Trailing Edge within the EU Smart Aircraft Intelligent Structures project (2011e2015). He is the author of more than 220 scientific papers, published in specialized journals (58) or presented at conferences. In 2018, he earned the Italian National Qualification as Full Professor (ASN). According to Scopus, his H-index rank is 17. xxi

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Vincenza Antonucci was born in Caserta (Italy), on Sep., 28, 1972. She has been a permanent researcher at the Institute for Polymers, Composites, and Biomaterials (IPCB) of CNR since 2001, where she performs research in the area of composite and polymeric materials. She graduated cum laude in chemical engineering ( Jan. 1996) at the University of Naples “Federico II.” Since 1996, she has worked at the Department of Materials and Production Engineering, University of Naples “Federico II,” attaining a materials engineering PhD (XII cycle) with a thesis on “Polymer Infiltration Processes for the Production of Composite Materials” (Feb. 2000). Furthermore, she had a training period at the Mechanical Engineering Department of the University of Delaware (Newark, United States), collaborating with Prof. S.G. Advani and working on control methodologies of the polymerization stage in the Resin Transfer Moulding (RTM) process, and studying nanocomposites and the production of thermoset composites reinforced with carbon nanotubes. In general, her main research activities are related to polymer-based composite materials and their manufacturing technologies, in particular the: • Study and development of innovative out-of-autoclave technologies, based on dry reinforcement infiltration by resin; • Study of properties of composites obtained by innovative technologies; • Study of polymer matrix filled with flame retardants and carbon nanotubes; • Study of actuation systemsebased shape memory alloys to produce adaptive structures; and • Study of composite materials based on natural (hemp, flax, and cotton) or mineral (basalt) fibers for nonconventional applications. The main scientific results are summarized and diffused by publications of more than 65 papers in international technical journals, seven book chapters, and a national patent. Furthermore, as a scientific researcher for IPCB, Dr. Antonucci manages research activities within the frame of national and regional research projects, carrying out training activities through training courses and tutoring students for degrees and doctoral theses.

About the editors in chief

Ferdinando Auricchio is a professor of Solids and Structural Mechanics at the University of Pavia, Italy, with strong collaborations with the Department of Mathematics (he is also a research associate at IMATICNR Pavia) and with several medical institutions. He received the Fellow Award from the International Association for Computational Mechanics (IACM) in 2012, the San Siro Merit from Pavia Municipality in 2015, the Euler Medal from European Community of Computational Methods in Applied Sciences (ECCOMAS) in 2016, and the Theodore von Karman Fellowship for incoming scientists from RWTH Aachen University (Germany) in 2018. Since 2015, he has coordinated a strategic theme (out of five) for the University of Pavia, entitled “3D@UniPV: Virtual Modeling and Additive Manufacturing (3D printing) for Advanced Materials.” Since 2018, he has been a member of the Italian National Academy of Science, also known as Accademia dei XL. He served as the vice president of ECCOMAS from 2013 to 2019 and has been a member of the Executive Council for IACM since 2018. His major research interests are: • Three-dimensional (3D) printing: modeling of phenomena occurring during 3D printing at different scales and with different technologies (mainly, Fused Deposition Modeling (FDM) and Selective Laser Melting (SLM)), and activation of a 3D printing laboratory with different technologies. He has organized a 3D printing laboratory, exploring new materials, new printing technologies, and new uses for 3D printing ranging from civil engineering 3D-printed concrete beams to biomanufacturing. • Mixed finite elements: development and analysis of finite element methods for Reissner-Mindlin plates, laminates, shells, and locking problems in small and large deformation regimes. • Material constitutive modeling: static and dynamic response for low and high numbers of cycles (metals, polymers, and rubber), and advanced materials (shape memory alloys and self-diagnosing materials). • Biomechanics: constitutive laws for biological tissue, modeling, and investigation of minimally invasive procedures (stenting) as well as invasive cardiosurgery procedures and the generation of computational models from patient-specific medical images. • Isogeometric analysis: structural mechanics problems in small and large deformations. He has more than 280 publications in refereed international journals, an H-index of 46 on Scopus, and over 6000 citations on Scopus. He is the cofounder of two university spinoffs and has six patents.

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Leonardo Lecce graduated in 1971 with honors in aeronautical engineering at the University of Naples “Federico II” and has spent his academic career at that university. He retired from being Full Professor of Aerospace Structures on Sep. 1, 2016. From 2000 to 2006, he was the Aeronautical Engineering Department chair. He has supervised more than 250 graduation and 20 doctoral theses. He has been a member of the EU Expert Commission for the evaluation of research proposals many times. He has been appointed a member of the Scientific Committee at CIRA many times, as well. For many years until 2016, he was a member of the Board of the Italian branch of the Advisory Council for Aeronautics Research in Europe and of the Executive Committee of the European Association of Structural Health Monitoring. A founder of the AIAN, he was also its president for many years. From 2013 to 2017, he was president of the AIDAA, after having directed the Naples Chapter since 2010. From 2017 onward, he has been the CEO of the company Novotech, Advanced Aerospace Technology S.r.L., which he founded in 1992. In this position, he has continued to work in research and development, managing many research contracts with the EU’s national and regional institutions. Professor Lecce’s main research interests are the development of structures and systems integrating innovative smart materials; damage detection and health monitoring of structures; morphing of aircraft wing and control surfaces; and the prediction and control of noise and vibration in transportation systems. He led “Federico II” University activities within several EU- and nationally funded research contracts. He was the European scientific coordinator for the Magnetostrictive Actuators for Damage Analysis and Vibration Control, Fifth Framework Program (FP) and Magnetoelastic Energy Systems for Even More Electric Aircraft, Sixth FP projects. Within the Seventh FP (2007e2013), he was the coordinator of Airgreen (a company association made of nine partners), associate member of the Alenia-managed Green Regional Aircraft Consortium, part of the EU Joint Technology Initiative, Clean Sky Integrated Technological Demonstrator. In the new position of CEO of Novotech, he is the scientific coordinator of the EU Horizon 2020 project named NHYTE and the CS2 project TRINITI and a core partner within the CJ2 Integrated Airframe Technology Demonstrator-Green Regional Aircraft (IATD-GRA) with the Airgreen 2 Consortium. The results of his research activities are reported in more than 250 papers published in national and international journals and conference proceedings.

About the editors in chief

Elio Sacco (Researcher ID: G-5349-2017, ORCID: https://orcid.org/0000-0002-3948-4781) was born in Naples, Italy. He earned his degree (5-year MS) in civil engineering with honors at the University of Naples “Federico II” (Italy) in Jan. 1980. He was an assistant professor of solid and structural mechanics at the University of Rome “Tor Vergata” (1986e92), associate professor at the University of Cassino (1992e2000), and full professor at the University of Cassino (2000e17). He a full professor at the University of Naples “Federico II” (2007 to present). For research abroad, he was a visiting scientist (1988 and 1990) in Blacksburg, United States, a visiting professor (1991) in Morgantown, United States, professeur associee (1992e93) in Paris, France, enseignant invite (2008, 2009, and 2011) in Marseille France, and professeur invite (2012 and 2013) in Marseille, France. He is a member of the editorial advisory board of several international journals: International Journal of Mathematical Physics, Frattura ed Integrita Strutturale (Fracture and Structural Integrity), International Journal of Masonry Research and Innovation IJMRI, and European Journal of Computational Mechanics/Revue Europeenne de Mecanique Numerique, and an associate editor of Meccanica, Annals of Solid and Structural Mechanics, and International Journal for Computational Methods in Engineering Science and Mechanics. His major research interests are (1) material constitutive modeling: static and dynamic response for cohesive and ductile materials and advanced materials (shape memory alloys); (2) micromechanics and homogenization techniques: analysis of composite materials characterized by nonlinear behavior of the constituents; (3) multiscale analysis of heterogeneous structures: structural analyses developed considering different scales (i.e., the scale of the structure and the scale of the material); (4) mechanics of masonry materials and structures: development of specific constitutive laws and computational procedures for the analysis of masonry structures; and (5) analysis of plate and shells: development of models and finite elements.

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Salvatore Ameduri was born in Naples, Italy, on May 14, 1974. He graduated in aeronautic engineering in 1998 at the University of Naples “Federico II,” defending a thesis work in gas dynamics, with focus on a thermographic investigation of heat transfer caused by a jet in the cross-flow condition. In 2002, he received a doctorate in aerospace engineering, defending a thesis on the design of a piezo-based adaptive system to mitigate the dramatic effects caused by the interaction of a normal shock wave and the airfoil boundary layer in a transonic regime. In that year, he entered the Italian Aerospace Research Center (CIRA) as a senior researcher in the Smart Structures and Materials Department. Since 2015 and to date, he heads the research unit Sensors and Actuators for Smart Structures of CIRA. He is the author of about 80 journal and conference papers (on smart structures and materials, morphing, noise and vibration, acoustics, deicing, and smart materials such as shape memory alloys (SMAs), piezoelectric, magnetorheological fluids, and shape memory polymers), the author and coauthor of two national and international patents on a drop-nose Leading Edge (LE) and an SMA-based variable chamber Trailing Edge (TE), and a reviewer for the Journal of Intelligent Material Systems and Structures (JIMSS), Smart Structure and Systems (SSS), Smart Materials and Structures (IOP), and Actuators MDPI, and has had a scientific role in more than 15 projects funded by national and international programs in the general aviation and defense fields, with a focus on different classes and types of vehicles. Lorenzo Casagrande was born in Senigallia, Italy, on Sep. 19, 1988. He earned his degree in civil engineering with honor at the University of Pavia (Italy) in 2014; there, he was also awarded his PhD in civil engineering and architecture in 2020. In 2014, he devoted professional full-time activity at Modeling and Structural Analysis Konsulting (Italy), with the task of conducting advanced numerical simulation of precast steel and concrete structures to study their seismic vulnerability. In 2015, he worked as a researcher at the European Centre for Training and Research in Earthquake Engineering (Italy), numerically modeling precast concrete structural walls and their connections, in support of the experimental test conducted at the experimental laboratory. In 2017, he worked as a researcher at the National Research Council (Italy), with the task of conducting numerical modeling and experimental activity on envelope systems and structural and nonstructural elements, with a significant impact on safety. Since Mar. 2020, he has been a postdoctoral research fellow at the University of Pavia. His main research interests are in numerical modeling and simulations, three-dimensional (3D) concrete printing, SMAs, and structural and nonstructural systems. Together with Professor Ferdinando Auricchio of the University of Pavia, he led a national team to develop and manufacture xxvii

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3D-printed envelopes for home automation sensors, within the TPRO.sl project (Smart Living call from Lombardy Region, 2017e19). He is the author of more than 10 scientific papers presented at conferences or published in specialized journals. Michele Conti was born in Scicli (RG), Italy, on Apr. 8, 1982. Since 2019, Michele Conti has been an associate professor of industrial bioengineering at the University of Pavia. He is also a member of the PhD program in health technologies, bioengineering, and bioinformatics. His research activities cover (1) in vitro and in silico simulations for vascular biomechanics; (2) medical image analysis for computational mechanics; (3) cardiovascular constitutive modeling; and (4) 3D (bio)printing for biomedical applications. Moreover, he has close collaborations with national and international clinical institutions. He was the recipient of an ESC research grant in 2016 and Livanova donations and grants in 2019. He is the unit leader of three Ricerca Finalizzata projects funded by the Italian Ministry of Health. According to Scopus, he has published 63 papers in indexed international journals. In 2010, his PhD thesis was selected as the Italian candidate for the European Community on Computational Methods in Applied Sciences Award for the Best PhD Thesis. In 2014, he received the Kiefer Prize at The Sixth International Congress for Aortic Surgery. He is a reviewer for several journals in the field of biomechanics. He is a member of the Italian Association of Theoretical and Applied Mechanics, the Italian Society for Bioengineering, and Italian Digital Biomanufacturing Network. Eugenio Dragoni graduated in mechanical engineering from the University of Bologna, Italy, in 1982. He served as a researcher and lecturer of machine design at the Department of Mechanical Engineering of the University of Bologna from 1983 to 2000. At end of 2000, he became full professor of mechanical engineering design at the University of Modena and Reggio Emilia, where he was chair of the Department of Sciences and Methods for Engineering from 2003 to 2018. His teaching assignments cover topics ranging from machine design to product design and development for bachelor’s, master’s and PhD students. Professor Eugenio Dragoni leads the research group of Machine Design, established in 2001 at the Department of Sciences and Methods for Engineering, and manages research grants from private companies and public institutions in the range of V300e500,000 per year. He has authored or coauthored more than 100 papers published in peer-reviewed journals on a variety of subjects including computational mechanics, the mechanical behavior of adhesives and nonmetals, engineering applications of smart materials, and product design and development. Professor Dragoni is the international cochair of the International Conference on Smart Material, Adaptive Structures and Intelligent Systems, organized by the American Society of Mechanical Engineers (ASME). He serves on a regular basis as a reviewer for the scientific journals and proceedings published by the Institution of Mechanical Engineers (United Kingdom), by the ASME, and many other renowned institutions. He is an associate editor for the International Journal of Adhesion and Adhesives, Journal of Adhesion, Journal of Materials: Design and Applications, and Meccanica.

About the section editors

Carmine Maletta is a professor of mechanical engineering at the University of Calabria (Unical), Italy. He received a degree in mechanical engineering in 1999, and in 2006 he obtained a PhD in structural engineering at Unical. From 2000 to 2002, he was employed as a business consultant in the international consulting company Accenture. Since 2016, he has been a cooperation associate at the European Organization for Nuclear Research (CERN), within the context of a CERN-Unical research project on smart pipe coupling technologies based on SMAs, where he serves as a technical coordinator for the University of Calabria. He is the cofounder and CEO at 2SMArtEST S.r.l, an innovative startup on SMA-based smart solutions and technologies established in 2019. He is the scientific coordinator of the material testing and mechanical design laboratory at Unical and serves as a scientific coordinator of several national and European Union research and development projects. His research interests cover both numerical methods and experimental techniques in the fields of mechanical engineering and materials science. In particular, special focuses are devoted to the fatigue and fracture of engineering materials and the thermomechanical analysis of SMAs. So far, Dr. Maletta has authored more than 100 research papers. Sonia Marfia has been an associate professor at the Department of Engineering of Roma Tre University since 2019. In 1996, she earned a degree in civil engineering with honors at the University of Roma Tor Vergata, where she was also awarded a PhD in structural engineering in 2000. From 2001 to 2006, she was an assistant professor, and from 2006 to 2018, she was an associate professor at the University of Casino and Southern Lazio. Since 2019, she has been the coordinator of the Italian Group of Computational Mechanics, and since 2017, she has been a member of the General Assembly of the International Association for Computational Mechanics. She is a Member of the PhD Committee of Civil Engineering at the University Roma Tre. She won a grant (1998e99) at the Technical University of Denmark (DTU) in Lyngby, Copenhagen, Denmark, to spend 6 months at the DTU. She was invited as an academic guest  cole Polytechnique Federale de Lausanne in Lausanne (Switzerland). (2004) at the E She won a short mobility CNR grant (2005) to spend a month at the Czech Technical University of Prague. She is the author of more than 90 papers published in national and international journals and proceedings. Her research interests focus on theoretical and applied mechanics and computational mechanics. In particular, she developed research in the fields of fracture and damage mechanics, mechanics of composite materials, multiscale analysis of heterogeneous structures, finite element method, and the modeling of SMAs. She participated in more than 40 national and international conferences and organized several sessions or minisymposia in international and national conferences. She received funds for her research activities from the Italian Ministry of the University and Research.

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Maria Rosaria Ricciardi was born in Naples, Italy, on Mar. 26, 1973; she graduated in materials engineering in 2005 at the University of Naples, where she also received her PhD in materials and structures engineering (2010). After some postdoctoral positions, she has a fixed-term researcher position at the National Research CouncileInstitute for Polymers, Composites and Biomaterials, Naples (Italy). She taught materials science and technology at Universita degli Studi di Napoli, Department of Medicine and Surgery. Her scientific activity has been disseminated through the publication of more than 20 papers in international peer-reviewed journals, two book chapters, and more than 22 contributions to national and international conferences. Dr. Ricciardi’s scientific activity includes studies in the area of composite materials design and manufacturing, in particular the infusion process, the development of innovative composite materials manufacturing processes (Italian Patent IT2009NA0067, “Pulsed Infusion”), the physicochemical characterization of thermosetting and thermoplastics, the thermal characterization of flame retardants, the mechanical characterization of fiber-reinforced composites, the acquisition and analysis of experimental data on Thermogravimetric Analysis (TGA), Differential Scanning Calorimetry (DSC), Dynamic Mechanical Analysis (DMA), Thermomechanical Analysis (TMA), conductivity tests, flammability tests, a fire testing technology dual cone, and a fire testing technology microcalorimeter. Dr. Ricciardi is interested in the formation of nanohybrids through the coupling of low-dimensional, inorganic nanostructures (carbon nanofibers and nanotubes) and graphene nanosheets for application in the automotive, aeronautical, and aerospace fields. Cristian Vendittozzi has been an associate professor at the aerospace engineering course at the University of Brasilia (Brazil), since 2015. He holds a master’s degree in aerospace engineering and a PhD in materials and raw materials engineering, both from Sapienza, Rome University (Italy). In 2014, he completed the EMBA Master of Business Administration at the MIP Business School of the Polytechnic of Milan (Italy). His research focuses on activities related to aerospace, but with frequent initiatives in other sectors, such as civil engineering infrastructures, architectural heritage, particle physics, and mechanical engineering. Since the PhD, he has been interested in integrating sensory receptors within the structures; thus, he focused his research activity on embedding fiber-optic sensors into materials, trying to resolve issues related to the process of integration in the different families of materials, mainly polymers and metals. He is strongly interested in engineering biomimetic applications aimed at coupling what was previously learned with the idea of developing a metamaterial that can be printed (3D) and that can change shape when properly stimulated (þ1D): a 4D material that can feel changes taking place in the operating environment (by the embedded sensory system) and react by adapting itself to the new conditions. The study on shape memory materials mainly aims to develop adaptive systems for space applications, such as deorbiting systems, solar sails, and atmospheric reentry systems. In 2019, he was been invited to become a scholar at CIRA to deepen concepts related to adaptive structures. He is the author of two books on metallurgy, the coinventor of two Italian patents, and the author of scientific articles presented at conferences or published in specialized scientific journals.

About the contributors

Edoardo Artioli (Chapter 9) He received his degree in civil engineering from the University of Bologna in 2002, where he continued his studies and earned his PhD in structural mechanics in 2006. In 2007 he was a postdoctoral researcher at the Institute for Applied Mathematics and Computer Science of the National Research Council in Pavia and a visiting research scholar at the Civil and Environmental Engineering Department at the University of California, Berkeley. In 2008, he joined the faculty of the Civil Engineering Department at the University of RomeeTor Vergata as an assistant professor in Solid and Structural Mechanics. Since 2018, he has served as an associate professor. He has authored and coauthored more than 50 scientific papers published in peer-reviewed international journals. Domenico Asprone (Chapters 21e23) He received a PhD in Materials and Structures in 2010; he has been an associate professor of structural engineering at the University of Naples “Federico II,” since 2010 and an associate researcher at the Institute of Complex Systems at National Research Council since 2015. His research interests cover different topics related to technological innovations in structural engineering and construction, ranging from the integrated sustainable design and the implementation of Building Information Modelling (BIM) methodologies into the design, construction, and management processes of civil works to the resilience and robustness of civil structures against natural and man-made hazards, the sustainability of structural materials, components, and systems, additive manufacturing (AM), and three-dimensional (3D) printing techniques for the production of structural systems. He is the cofounder of two university spinoff companies and the author of more than 100 scientific papers published in international peer-reviewed journals. Keyvan Safaei Baghbaderani (Chapter 6) He earned his Bachelor of Science degree in Mechanical Engineering at the Isfahan University of Technology, Iran in 2013 and his Master of Science at Amirkabir University of Technology, Iran in 2015. He joined the Mechanical Department at the University of Toledo, United States in 2018 as a doctoral student. He is currently pursuing his PhD at the Dynamic and Smart System Laboratory at the University of Toledo, where his main research interest is focused on AM, shape memory alloys (SMAs), the 3D printing of polymers, and materials characterization. He is a coauthor of several scientific papers published or presented in journals and conferences. He also serves as a reviewer for several scientific journals.

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Silvestro Barbarino (Chapters 11 and 17) He is a lead data scientist at Joby Aviation, Inc. He received his MS in aerospace engineering from the University of Napoli “Federico II” (Italy) in 2006; there, he was also awarded his PhD in aerospace engineering in 2010. In 2018, he joined Joby Aviation, a rising leader in the Electrical Vertical TakeOFF and Landing (eVTOL) field. Previously, he worked at Sikorsky, a Lockheed Martin company, from 2014 to 2018 as a senior engineer, first in the Rotor Structures group and then as a data scientist analyst within the iNtelligent Technologies, Analytics and Sustainability branch. Between 2010 and 2013, he was a research associate and adjunct faculty at Rensselaer Polytechnic Institute, a postdoctoral scholar at Penn State University, and a research officer at Swansea University (United Kingdom). His expertise includes topics such as aerospace morphing structures, smart materials, fatigue analysis of metals, nonlinear structural dynamics, optimization, data analytics, machine learning, and deep learning. He is the author of more than 45 technical papers, reports, and book chapters, and has been awarded patents in the European Union (EU) and United States. He is member of the American Institute of Aeronautics and Astronautics (AIAA), ASME, and AHS. Paolo Bettini (Chapter 16) He received his PhD in aerospace engineering at Politecnico di Milano in 2008; he has been an assistant professor at the Department of Aerospace Science and Technology of that University since 2013. He is a lecturer of “Aerospace Technologies and Materials” in master’s courses of aeronautical engineering, space engineering, mechanical engineering, and nanomaterials and materials engineering. His main research activities are focused on smart materials and their integration into smart structures, especially in the development of health monitoring systems and composite structures with morphing capabilities. He is the author of more than 90 papers printed in international journals or presented at international conferences. He has participated in many European and Italian research projects and has worked with several Italian companies and research centers. Currently, he is the Work Package (WP) leader for advanced technologies for space applications within a 15-year agreement between the Italian Space Agency and Politecnico di Milano. Elisa Boatti (Chapter 19) She earned her master’s degree in mechanical engineering from Polytechnic University of Turin (Italy) in 2011 and her PhD in computational mechanics and advanced materials from the Institute of Advanced Studies, Pavia (Italy) in 2016. The focus of her PhD dissertation was the constitutive modeling of SMAs and polymers. Right after graduation, she was granted a postdoctoral fellowship at Harvard University (Cambridge, MA, United States) to work on metamaterials, from 2016 to 2017. Subsequently, she completed a second postdoctoral assignment, this time in the field of biomechanics, at Georgia Institute of Technology (Atlanta, GA, United States), where she studied the mechanics of the cochlea. Since 2018, she has been working as a research engineer in the steel manufacturing industry for ArcelorMittal Global R&D.

About the contributors

Matthew Bray (Chapter 15) He received his master’s degree in international finance at the S~ao Paulo School of Economics (Brazil) and Universidade Nova de Lisboa (Portugal) in 2016. He practiced finance in London and Hong Kong with a particular focus on corporate finance and venture capital. After experience with Goldman Sachs and BASF, his focus has been on the commercialization of morphing structures in aerospace, as CEO of Brayfoil Technologies. There, he has overseen fundraising and the project management of engineering development within renewable energy and aviation. Robert Bray (Chapter 15) He completed his degree in architecture at the University of Witwatersrand (South Africa) in 1980 and practiced architecture in South Africa with particular specialist design skills in retail malls. He led the design team at BAI Architects on over 250 projects, and the company became the largest architectural firm in the Southern Hemisphere. His personal interest in aviation led him to study this subject after retirement, with over 15 years of research and development into morphing wing technology. He was successful in developing an auto-setting morphing wing technology and started Brayfoil Technologies to commercialize this, underpinned by global patent portfolios, with a focus on aviation and renewable energy. Andrea Brotzu (Chapters 2 and 3) He graduated in chemical engineering in 1993 and obtained his PhD in metallurgical engineering in 1998. For several years, he worked in the laboratories of various industries operating in the metallurgical and aerospace field. Since 2004, he has been working in the Department of Chemical Engineering, Materials, and Environment of the University of Rome “La Sapienza,” where he handles various metallurgy laboratories and collaborates in several research projects of the metallurgy research group. His latest research is focused on SMAs, corrosion and electrochemistry, cast TiAl intermetallic alloys, materials selection for a new passive satellite (LARES2), a metal matrix composite, Fibre Bragg Gratings (FBG) sensor applications, and characterization of additive manufactured products. He is author of more than 70 scientific papers and is the coinventor of two patents in the field of fiber-optic sensors. Paolo Chiggiato (Chapter 25) He studied nuclear engineering and materials at the Politecnico di Milano. Since 1988, he has worked at the European Organization for Nuclear Research (CERN) in the field of vacuum technology. His strengths include outgassing of vacuum materials, surface modifications for particle accelerators, vacuum system operation, and ultrahigh vacuum instrumentation. Since 2014, he has led the vacuum, surfaces, and coating group at CERN, a polyvalent unit dealing with several tasks essential for particle accelerator operation and development.

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Giuseppe De Ceglia (Chapter 18) He earned his degree in materials engineering and nanotechnology with honor at Politecnico of Milano (Italy) in 2015. He integrated his studies with an international exchange at the Materials Engineering Department of Chalmers Technical University (Sweden). Since 2016, he has been the research and development manager at Technosprings Italia S.r.l., where he also coordinates nationally and internationally funded projects. His activities focus on the development of SMA-based devices and the definition of their relative production processes. The main applications of SMA actuators include ventilation systems, fire systems, automotive, and aerospace. Moreover, he has specialized in the quality management system for the production of medical products. Hence, he is in charge of developing and producing medical components in superelastic NiTi alloy, as well as conventional materials besides SMA-based actuators. Vittorio Di Cocco (Chapters 2 and 3) He graduated with a master’s degree in mechanical engineering, with honor, at the University of Cassino in 2000; there, he also earned a PhD in civil and mechanical engineering in 2004. From 2005 to 2014, he was a researcher at DIMsAT. Since 2014, he has been an associate professor of metallurgy at DICeM of University of Cassino and Southern Lazio. Since 2012, he has been the scientific director of LaMeFi (Laboratory of Metallurgy and Physics) of DICeM. His research on metallic SMAs focuses on microstructure transitions, the thermodynamics of structure transformations, and the modeling of mechanical behavior. He is the author and coauthor of more than 150 papers published in international journals and conference proceedings. Ignazio Dimino (Chapter 15) He graduated with honors in aeronautical engineering at the University of Palermo in 2004. He achieved his PhD in aeronautics at Imperial College of London, United Kingdom. He was also a visiting researcher at the Center of Acoustics and Vibration at Penn State University, State College (Pennsylvania, United States) in 2008. He holds the position of scientist at the Department of Adaptive Structures of the Italian Aerospace Research Center (CIRA) and is project manager in the field of smart structures. He is the author of over 70 peer-reviewed papers on adaptive structures, structural dynamics, and vibroacoustic control and coauthor of two books. Mohammad Elahinia (Chapter 6) He is a university distinguished professor in engineering and chair of the Mechanical, Industrial, and Manufacturing Engineering Department at the University of Toledo. He graduated with a doctorate in mechanical engineering from Virginia Polytechnic Institute and State University in Aug. 2004. At UToledo, he has served as an investigator on several funded projects with a total budget of more than $15 million. He is a Fellow of American Society of Mechanical Engineers. He has served as the major advisor for 45 graduate students (10 PhD and 35 MS). Seven of his former students are assistant and associate professors at other universities. To disseminate his research findings,

About the contributors

he and his students coauthored three books, seven book chapters, and more than 100 journal papers. He serves as an associate editor for four journals in his area of research: Smart Material Research, The Scientific World Journal, Journal of Intelligent Material Systems and Structures, as well as the Journal of Shock and Vibration. Luca Esposito (Chapter 11) He was awarded a PhD in structural engineering in 2013 at the University of Napoli “Federico II” (Italy). He was a 4-year postdoctoral research fellow at the Department of Structures for Engineering and Architecture at the University of Napoli “Federico II.” Since 2013, he has been an adjunct professor at the master’s degree level of materials engineering and teaching assistant at the master’s degree level of structural engineering and biomedical engineering at the University of Napoli “Federico II.” In 2018, he was an invited professor at the University of Reykjavik. He works as a researcher at the Department of Engineering of the University of Campania “Luigi Vanvitelli” (Italy). He participated in different national and international research projects and is the author of more than 50 scientific papers that were presented at conferences or published in specialized journals. Massimiliano Ferraioli (Chapters 21 and 23) He received a magna cum laude degree in civil (structural) engineering from the University of Salerno (Italy) in 1994 and a PhD in Seismic Design of Buildings: Analysis and Strengthening of Structures in 1998. He was a research assistant at the Department of Civil Engineering of the University of Salerno from 1994 to 1998 and a researcher and assistant professor at the Second University of Naples from 2001 to 2018. Since 2018, he has been a professor of structural engineering at the University of Campania “Luigi Vanvitelli.” Since 2001, he has had full responsibility for many courses on seismic design and rehabilitation of structures in master’s degree courses in civil engineering and for the PhD degree. He is the author of more than 120 papers and articles, including book chapters, scientific journals, and international and national conferences. Cedric Garion (Chapter 25) He is graduated from the Ecole Normale Superieure de Cachan in mechanical engineering sciences. He arrived at CERN in 2000 as a PhD student. His PhD thesis was entitled Material and Structural Mechanical Modeling and Reliability of Thin-Walled Bellows at Cryogenic Temperatures. Application to LHC Compensation System. He joined the CERN vacuum group in 2008, where he is currently is the head of a section in charge of the design of vacuum systems in the vacuum, surfaces, and coating group. He is involved in designing vacuum systems for new projects and studies as well as developing new technologies for ultrahigh vacuum applications. In particular, he is developing new leak-tight connections for vacuum chambers. Stefano Gualandris earned his degree in political science with honor at the University of Milano (Italy) in 2004. Since 2007, he has attended several courses in aerospace structural simulations and standards and regulations. Since 2015, he has attended

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advanced seminars and work groups organized by the Swiss Space Center and Swiss Space Office on European Space Agency contracts, microelectromechanical systems, space mechanisms, commercial off-the-shelf technologies for space, miniaturization, and earth observation. From 1999 to 2004, he was a small business owner with the company North Italy Web Promoter in the field of service and sales of hardware and software. Since 2004, he has been chief technical officer (CTO) and managing director of the company Technosprings Italy S.r.l. Since January 2012, he has been CEO at TSS Innovationsprojekte GmbH in Roveredo (Switzerland). From 2008 to 2012, he was a member of the board of directors of Societa per i Mercati di Varese Spa. From 2008 to 2013, he was vice president of Fondazione Museo dell’AeronauticaeVolandia. In 2018e19, he was a technical and commercial advisor at the Presidency of the Council of Ministers (Under Secretary of State G. Giorgetti) for the aerospace and defense sector. Francesco Iacoviello (Chapters 2 and 3) He received his degree in nuclear engineering with honors at the University of Roma “La Sapienza” (Italy) in 1989; he was also awarded a PhD in metallurgy-corrosion in 1997 at the Ecole Centrale Paris (France). He is a full professor of metallurgy at the “Universita di Cassino e del Lazio Meridionale.” He is the coordinator of the didactic activities of engineering courses, responsible for the Innovation in Didactic and President of the Information Technology Center at the “Universita di Cassino e del Lazio Meridionale.” He is president of the Italian Group of Fracture and the European Structural Integrity Society and is the vice president of the International Congress of Fracture. He is the editor in chief of the international journal Frattura ed Integrita Strutturale and of Procedia Structural Integrity. He is author of more than 200 scientific papers presented at conferences or published in specialized journals. Maryam Khoshlahjeh (Chapter 17) She is lead aeroelasticity and program manager at Joby Aviation, an eVTOL venturebacked startup based in Santa Cruz, California. Before joining Joby Aviation, she worked at Sikorsky Aircraft, a Lockheed Martin company, as a deputy program manager for the Turkish Utility Helicopter Program and dynamics lead for the Combat Rescue Helicopter Program. She was the recipient of the 2016 Marshall-Tan Engineering Leadership Award at Sikorsky and the President’s Award in 2015. She started her undergraduate education in physics at Sharif University of Technology in Iran. She moved to the United States when she was 20 years old, and received her BS in mechanical engineering from San Jose State University. She started working as an aerospace engineer at Advanced Rotorcraft Technology, Inc. in Sunnyvale, California. She went on to receive an MS in Aeronautics and Astronautics from Stanford University and a PhD in aerospace engineering from Penn State, a Vertical Lift Research Center of Excellence. She is a member of the AIAA and the Vertical Flight Society, where she serves on technical committees.

About the contributors

Stefania Marconi (Chapters 19 and 20) She received a master’s degree in biomedical engineering in 2011 and a PhD in experimental surgery and microsurgery in 2015 from the University of Pavia, Italy. She is currently an assistant professor at the University of Pavia and coordinates the research activity of 3D4Med (www.3d4med.eu), the Clinical 3D printing laboratory of IRCCS Policlinico San Matteo (Pavia, Italy). Her research activity focuses on 3D printing technologies and materials, especially for medical application. She is the author of about 50 works on refereed international journals and two patents. Alfonso Martone (Chapter 4) In 2005, he graduated in aerospace engineering at the University of Naples “Federico II,” Italy, where he also received his PhD in materials and structures engineering in 2009. He worked at National Research CouncileInstitute for Polymers, Composites, and Biomaterials (CNR-IPCB) as a research fellow and was involved in several EU and national research programs, including within the area of composite material characterization and modeling. He works as a researcher at the Institute for CNR-IPCB. His current research interests include the study of carbon nanotube composites, hybrid advanced composites, viscoelasticity of polymers and composites, advanced composites manufacturing, thermomechanical characterization, and structural health monitoring. He has published more than 10 scientific contributions in international journals and books. Costantino Menna (Chapters 21e23) He has been an assistant professor of structural engineering at the University of Naples “Federico II” since 2016. He received his PhD in materials engineering and structures in 2013, working on damage modeling of advanced composite materials. He is involved in several multidisciplinary research activities mainly focused on advanced materials for civil and industrial applications. During his academic career, he was a visiting research scholar  cole Polytechnique de Montreal, Department of in several foreign institutions: E Engineering Science and Mechanics of Penn State University, Laboratoire de Mecanique des Solides of Ecole Polytechnique, and University of Greenwich. He is chair of FIB Task Group 2.11: Structures Made by Digital Fabrication (Federation Internationale du Beton) and the holder of three national patents. He is also cofounder of two university spinoff companies and the author of more than 40 scientific papers published in international peer-reviewed journals. Simone Morganti (Chapter 20) He is a tenure-track assistant professor of solid and computational mechanics at the Department of Electrical, Computer, and Biomedical Engineering at the University of Pavia (Italy). An Italian representative member of the European Community on Computational Methods in Applied Sciences (ECCOMAS) Young Investigator Committee, from Oct. 2013, he was secretary and a member of the PhD Programme Board of the Department of Civil Engineering and Architecture of the University of Pavia. In 2014, he was a recipient of the Tissue Mechanics Prize awarded by the Centre

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for Mechanics of Biological Materials of the University of Padua. In 2012, he competed for the ECCOMAS PhD Olympiad and was awarded Best Thesis Presentation. His main research topics are finite element method and cardiovascular mechanics, isogeometric analysis, and 3D printing and innovative materials. He is the author of more than 40 scientific papers. Stefano Natali (Chapters 2 and 3) He earned his degree in mechanical engineering at the University of Rome “La Sapienza” in 1988, where he also obtained a PhD in materials engineering. From May 1992 to Oct. 2000, he was a researcher in the disciplinary scientific field ING-IND 21 (metallurgy) at the University of Rome “La Sapienza.” Since Nov. 2000, he has been an associate professor in that area. Since 2012, he has been a member of the board of the Electrical, Materials and Nanotechnology Engineering Ph. Doctorate, for the materials curriculum. He is the author of more than 150 publications presented at conferences or in scientific journals in the research topics of corrosion, aluminum coatings, coatings with zinc alloys, catalytic electrodeposition processes, archaeometallurgy, copper alloys, duplex steels, and smart alloys. Mohammadreza Nematollahi (Chapter 6) He is a PhD candidate in mechanical engineering at the University of Toledo, United States. In 2015, he completed his Master of science in mechanical engineering at Sharif University of Technology, Tehran (Iran), where he worked on a mastereslave surgical robotic system at the Research Center for Biomedical Technologies and Robotics. Since 2016, he has worked as an assistant graduate researcher at Dynamic and Smart Systems Lab, where he has researched the fabrication and development of AM techniques for smart materials. He is the author of several scientific papers presented at conferences or published in specialized journals. His major areas of expertise include material science and characterization, AM, and smart materials. Adelaide Nespoli (Chapters 6 and 13) She graduated in biomedical engineering (technological and industrial course of studies) at Politecnico di Milano (Italy), presenting a dissertation on a new calibration procedure of distortion product otoacoustic emissions (2003e04 academic year). Since 2004, she has worked as a researcher at the CNR-ICMATE, Lecco Unit, where she is involved in SMAs, their characterization, and their applications in sensors and actuators, particularly for the aeronautical, civil, and biomedical fields. She is also involved in technological transfer and collaborates with private companies in developing new smart devices. Since 2012, she has also been involved in AM with specialization in fused deposition modeling and selective laser melting processes. She is the author of more than 50 scientific papers. Fabrizio Niccoli (Chapter 25) He studied mechanical engineering at University of Calabria, Italy. He has been carrying out research in the field of SMAs since the beginning of his postgraduate career and has published several scientific articles on topics related to the characterization and modeling of SMA structures. During his doctorate, he spent about 3 years at CERN

About the contributors

in Geneva (Switzerland), where he developed a new generation of clamping systems thermally activated for chambers operating in ultrahigh vacuum conditions. Since 2017, he has held a position as researcher at CERN (senior fellow) and is responsible for developing and implementing such devices in particle accelerators. Rosario Pecora (Chapter 15) He earned a master’s degree in aeronautical engineering and was awarded a PhD in transport engineering by the University of Naples “Federico II” in 2002 and 2005, respectively. He has been an assistant professor of aircraft structure stability and a lecturer of advanced aircraft structures at that university since 2011. He has worked for aircraft manufacturing companies and research centers as a technical advisor for loads, aeroelasticity, aircraft structures design, and certification. His research activity mainly focuses on the aero-servo-elasticity of unconventional structural systems, structures dynamics, and smart structures while covering leading roles in major European and extraEuropean areas. He is the author of 75 indexed papers, the editor of a book on morphing wing technologies for large civil airplanes, and the design inventor of European and US patents on SMA-based architectures for the morphing wing trailing edge. Aniello Riccio (Chapter 24) He graduated in aeronautical engineering at the University of Naples in 1996. He received his PhD in 1999 at the Second University of Naples (SUN). In 2000, his PhD thesis was awarded first prize in the frame of the Pratt & Whitney-EREA Award. Between 2000 and 2010, he was engaged by CIRA. In 2003, he became a member (chairman in 2008e2011) of the Structures and Materials Group of Responsibles within the GARTEUR organization. Since 2004, he has been a reviewer for many scientific journals. Between 2006 and 2012, he acted as chairman of European Projects and Mod-funded projects. In 2010, he joined the SUN (then the University of Campania “Luigi Vanvitelli”). In 2018, he became full professor in aerospace structures. Professor Riccio is the author of several publications on advanced materials and structures in international journals and conference proceedings. He also acts as associate editor for several international scientific journals. Daniela Rigamonti (Chapter 16) She earned a master’s degree with a specialization in space engineering in 2012 at Politecnico di Milano, with a numerical and experimental thesis on the use of SMA in optical support systems, carried out in collaboration with the ICMATE Institute of the CNR. She then obtained a PhD in aerospace engineering, deepening the modeling of actuators based on NiTi alloy, thus obtaining a global view of both the thermodynamic mechanisms and the design aspects. She also spent a period at Intelligent Material Systems Laboratory in Germany, flanking a group with great expertise in the design and control of actuators based on SMA. In the following years at the Department of Aerospace Sciences and Technologies, she extended the research field to other classes of smart materials (in particular, fiber-optic sensors) and smart structures in general, deepening aspects related to the embedding of sensors and actuators in composite structures.

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About the contributors

Giuseppe Sala (Chapter 16) He earned a master’s degree in 1982 in aeronautical engineering and obtained a PhD in aerospace engineering in 1987. He became an assistant professor in 1990, an associate professor in 1998, and a full professor in 2002. He gives classes in aerospace technologies and materials at the Department of Aerospace Science and Technology of Politecnico di Milano and Science and the Engineering of Materials at the Ph. Doctoral School of that university. He heads the Department of Aerospace Science and Technologies. He is the author of more than 150 papers in the fields of materials and technologies (nanotechnologies and rapid prototyping), as well as of advanced materials (smart materials and biomimetic materials). He is the coordinator of EU research projects and Join Research Centers, acts as reviewer for EU Research Programs (Self-Healing Materials, CleanSky1, and GRC), and acts as a referee for the Journal of Advanced Manufacturing Technologies and Journal of Smart Materials and Structures. Giulia Scalet (Chapters 7 and 10) She has been an assistant professor at the Department of Civil Engineering and Architecture of the University of Pavia (Italy) since 2019. She received a BS (2008) and MS (2010) cum laude in civil engineering and a PhD (2014) in civil and environmental engineering from the University of Bologna (Italy). She has been a postdoctoral student at Ecole Polytechnique (France) and the University of Pavia. She won the prize for Best Graduated Women of the Engineering and Chemistry Faculties of the University of Bologna by Unindustria et al. in 2011 and the Best PhD Thesis in Computational Solid Mechanics in 2014 Award by the Italian Group of Computational Mechanics. She visited several international institutions, including Texas A&M University (United States). She has published over 30 scientific papers presented at conferences or published in peer-reviewed journals. Franca Scocozza (Chapter 20) She received a master’s degree in bioengineering at the University of Pavia in 2017. Right after that, she was awarded a research fellowship at the Department of Civil Engineering and Architecture of University of Pavia, and afterward at IRCCS San Matteo Hospital (Pavia). In 2019, she started her PhD, which is in progress, in Bioengineering, Bioinformatics, and Health Technology at the University of Pavia. Her research is focused on bioprinting: 3D printing biological components (e.g. cells, proteins, genes) for realizing in vitro models for tissue engineering and regenerative medicine applications. Andrea Sellitto (Chapter 24) He graduated in aerospace engineering at SUN (then the University of Campania Luigi Vanvitelli) in 2008. Between 2008 and 2009, he worked in Alenia Aeronautica (then Leonardo Company), in Pomigliano (Italy), on the Boeing 787 program. In 2012, he received a PhD in aerospace science and technology at the University of Campania Luigi Vanvitelli. Since 2012, he has been collaborating with the Aerospace Composite Structures group of the University of Campania Luigi Vanvitelli in research

About the contributors

and education activities. He actively participates in national and European research programs, focusing on developing numerical models to study the damage behavior of composite material structures. He is the author of several scientific papers and conference proceedings on aerospace structure. He has held the role of chairman since 2012. He is an associate editor and member of the scientific committee of different international journals and conferences. Emanuele Sgambitterra (Chapter 7) He earned a degree in mechanical engineering with honor at the University of Calabria, Rende (Italy) in 2010; there, he was also awarded a PhD in materials and structures engineering in 2014, defending a thesis entitled Fatigue and Fracture Behavior in NickeleTitanium Based Shape Memory Alloys. From 2013 to 2019, he worked as a postdoctoral student at the University of Calabria. Since 2019, he has worked as a research fellow at the University of Calabria. Since 2019, he has been a lecturer at the University of Calabria on the finite element method for the analysis of solids and structures, techniques and instruments for experimentation. He is the author of more than 35 scientific papers presented at conferences or published in specialized journals. Andrea Spaggiari (Chapter 12) He completed his degree with honor in 2006 at the University of Modena and Reggio Emilia as well as a PhD at the School of High Mechanics and Automotive Design and Technology in 2010. Since 2011, he has worked as a researcher at the University of Modena and Reggio Emilia, where he is a lecturer for three academic courses for the mechatronics engineering degree and innovation design master’s degree. His current research interests are threefold. First, he studies the properties and mechanical behavior of structural adhesives and their efficient modeling; second, he is interested in the magneto-mechanical modeling of magnetorheological fluids and elastomers, including fundamental research and industrial applications; and third, he is devoted to the design of smart and efficient actuators based on SMAs. He is the author of more than 60 scientific papers in international scientific journals. Francesco Stortiero (Chapters 6 and 13) He is the technical director at GFM SpA, a company involved in manufacturing components for gas and steam turbines. He earned an MSc in materials science. After spending 15 years in the research and development of SMA-based applications for the aerospace, HVAC, and biomedical sectors, carrying out activity in both a public institution (CNR) and private companies (Saes Getters and Technosprings Italia S.r.l.), he moved to PowerGen industry. As the technical director of GFM SpA, he is also responsible for introducing new processes such as AM in company capabilities for the manufacturing of innovative components. Since 2019, he has also been the coordinator of the Additive Technology Center for the promotion of additive technologies in different industrial sectors supporting new applications of AM such as the printing of smart devices based on SMAs. He is the inventor of five patented devices and the coauthor of papers on related subjects.

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Andrea Vigliotti (Chapter 8) He received a master’s degree in aerospace engineering from the University of Naples in 1996 and a PhD in mechanical engineering from McGill University in 2012. From 2013 to 2014, he was recipient of a Newton International Fellowship from the Royal Society. Since Sep. 1996, he has been a researcher at CIRA. His research interests are mainly focused on material science and mechanics of materials and include the study of materials with controlled microarchitecture, multifunctional materials, and metamaterials. Stefano Viscuso (Chapter 18) He has been the quality manager and SMA research and development manager at TSS InnovationsProjekte GmbH since 2015. In his career, he has deepened his knowledge of SMAs, in particular their applications in biomedical field and actuation, with numerous scientific publications. He attained experience in processing shape memory components and functional characterization. He is active in developing new SMA applications for customers in various fields (medical, automotive, and aeronautical). In space, he coordinated the development of a high-temperature SMA pin-puller in an ESA-founded project. Valerio Visentin (Chapter 18) He graduated in aeronautical construction from the high school ITIS Gallarate (Italy) in 2006. Since 2011, he has worked for Technosprings Italia S.r.l., where he has the working duties of realizing parts and tool drawings, compiling laboratory testing software, and developing custom simple mechanical or electrical components. In the company, he carried out the role of research assistant on several projects: INNOSMAD, ESA GSTP6.1 4,000,123,630/18/NL/BJ/gp (consultant), SAPERE STRONG, SHREK, DE-LIGHT, and ALMAS. In PV SMART2, he will assist the technical manager mainly in WP4 activities and WP2.

Preface to the second edition

It is always a matter of praise when a publisher asks for a novel edition of a previous work that, surprisingly and pleasantly, encountered the favor of the public. This request is also a signal that the original book and the implemented architecture have met the public’s needs. Of course, the fascinating field of the still-mysterious world of shape memory alloys (SMAs) may have exerted a significant role in the success achieved. The original work was structured to touch on the main aspects of the realization of SMA-based systems. Fabrication, material modeling, device development, and practical implementations were all aspects that were treated, enriched with a bibliography that would have enabled the interested reader to deepen any selected subject. With such a solid background, it might have been a mistake not to try to expand and complete the former edition. Indeed, many aspects should have been considered in the second edition to give an even more complete vision of the actual and potential impact of SMA for industry. Interaction with the authors of the first edition allowed us to identify additional items that would immediately have been worth including in a new edition. For the sake of truth, Elsevier and many anonymous commenters provided formidable support to make it better by transmitting their valuable and appreciated opinion. The preliminary introductory section was then expanded with chapters devoted to the latest attainments and applicable regulations for the use and implementation of SMAs. The part dedicated to material behavior was enlarged to include a chapter concerning fatigue issues, and the one addressing modeling considers some innovative approaches. In the same way, the actuators section has been augmented with a work dedicated to unconventional actuators. However, the application segment has deserved the most significant expansion. It has been articulated into four parts dedicated to aerospace, biomedical, civil, and other industrial uses, and further organized into different chapters concerning selected topics. The final result is impressive. The new edition of the book now contains eight sections and 25 chapters, almost doubling the previous one. The number of involved authors who have graciously made their expertise available has grown to 54. Most come from academia (33), with significant contributions given by research centers, small and medium enterprises, and industries (21). They represent 23 Italian and international institutions (including Brazil, South Africa, Switzerland, and the United States) and provide an exhaustive overview of the current SMA scenario as perceived by the worldwide scientific and technological community. The editorial team has been enlarged from the former section editors to manage the expected large amount of data and information

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exchange properly. They have enthusiastically agreed to spend time on completing the envisaged opus. That has been a necessary, but not a sufficient step. The final product was finally achieved because all of the editors, section editors, and participants have behaved like a trained team. They have worked synchronously for the common objective, continuously supported by the publisher’s personnel, who have followed the historical evolution of this adventure since the very first moment. Here, we acknowledge the efforts of Gabriela Capille, Christina Gifford, Joseph Poulouse, Narmatha Mohan, and all others working on this project within Elsevier. After more than 50 years since the discovery of their properties, many potentialities of SMAs remain a black box. After the first trials and preliminary static implementations, many devices have been introduced in different fields of engineering, often at the edge of knowledge. Current systems refer only partially to the most acknowledged characteristic of SMAs, the shape memory. Many others instead rely on their superelastic, damping, or high-strain characteristics, for instance. It seems that it finally invaded the market to a significant extent only after the technical community started appreciating the material as a whole. In aerospace, SMA elements were thought to extrapolate their shape characteristics to wing sections. Today, more realistically, they are mostly used as inner actuators, suitably designed to generate outline variations. In the biomedical field, adaptive stents and filters are evidence of the shape memory potentialities. In the large and mass-market scenario, their constant stress peculiarity has provided dentistry with a new, impressive tool for patients. In the civil sector, seismic engineers can exploit the damping characteristics of SMAs, combined with their superelastic behavior, to suppress or reduce earthquake effects. We hope this book will attract new scientists to SMAs and inspire some colleagues to imagine the further expansion of SMA applications. As that engagement results in the novel benefits of that technology for the worldwide community, we will have achieved our objective. Antonio Concilio Vincenza Antonucci Ferdinando Auricchio Leonardo Lecce Elio Sacco

Preface to the first edition

The book is addressed to doctoral students, postdegree researchers, and professionals in the industry who want to have a complete survey on the main characteristics of shape memory alloys (SMAs) and their different fields of use to proceed with practical implementations. The book provides readers with details for a wide understanding of SMA behavior, modeling, and potentialities in engineering. The superelastic and shape memory effects are considered, different types of evidence of a single microstructural behavior. This comprehensive approach makes the book unique in the current scenario of available publications. Aeronautics is specially addressed while, to give a complete view, a conclusive section is dedicated to relevant employment in biomedical and civil engineering. The latest advances in design, Finite element (FE) representations, and available materials are considered. Case studies, successful applications, and typical failures and limitations of SMA devices are reported, together with a description of current manufacturing technologies. The book is arranged in monographic sections, each focusing on a particular topic concerning SMAs, from a general overview to their use into real structural projects. Apart a general introduction (Section 1eHistorical survey and perspectives), the work is structured as follows: Section 2eThe material; Section 3eModeling; Section 4eAeronautics; and Section 5eBiomedical and civil engineering. A complete survey is performed at each step, leading up to a wide bibliography for specific and deeper studies. Sections are further organized into three chapters dealing with a specific argument on the related topic. Propaedeutic matters are some basic concepts related to thermodynamics, crystallography, the theory of elasticity, structural dynamics, and others, some of which are referred in the bibliography. The work starts with a general introduction to the evolution and perspective applications of SMAs (Chapter 1). It gives the reader a sort of time trip, illustrating the origins of SMAs and their status and a perspective on the future use of these materials. How they were discovered, how they are used, and how they could develop according to further needs and specifications may furnish readers with the necessary forma mentis with which to face the relatively new world of SMA engineering as part of the more general field of smart materials and structures. The book continues with the study of SMA as materials, moving from the fundamentals and highlighting their experimental appearance and main properties (Chapter 2), illustrating a strategy for characterizing their behavior (Chapter 3) and discussing some operations connected to their manufacture, including material design concepts (Chapter 4). NiTi compounds are mainly considered, currently the most diffused SMAs. Hints are also given about the recent development of new substances as porous SMAs. Finally, an

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overview of SMA macroscopic behavior, connected to corrosion, aging, and fatigue aspects, is afforded. The constitutive elastic models and the different philosophy with which they have been constructed by many researchers are then considered. Characteristics of different formulations (both good and bad) are outlined, together with critical discussions about different approaches, including one-dimensional (1D) (Chapter 5) and 3D linear and nonlinear modeling (Chapter 6). How nonlinear models are used to produce FE representations is shown, providing readers with the opportunity to develop their own simulation codes on the basis of the referred philosophy. A presentation of some available SMA modularizations in commercial codes concludes the section (Chapter 7). The book finally refers to the description and application of SMA structural systems design criteria. Most of all, it discusses how SMA devices are manufactured and what the basic design criteria are (Chapter 8). A logic flow diagram is then formalized, schematizing steps for outlining and optimizing SMA-based architectures. Selection criteria are developed and formulated (Chapter 9). The book then illustrates actual examples of integrated systems for aeronautics and the rise of new challenges with respect to standard solutions (Chapter 10). A wide survey of conceived or already operative devices in the aerospace field completes the section. A wide overview of current applications in other relevant technological sectors (biomedical and civil engineering) concludes the work. The former is perhaps the area in which the largest worldwide SMA use takes place, from orthodontics to general surgery and orthopedics, owing to the high biocompatibility level of these materials (Chapter 11). A single chapter is devoted to the increasing use of SMA-based systems in the cardiovascular field, perhaps the top scientific and commercial branch (Chapter 12). Finally, other civil engineering examples are recalled: SMA-related technology potentialities are opening up to a lot of relevant sectors, including seismic protection and the preservation of archaeological heritage and historical buildings (Chapter 13). The editors have discovered the fascinating field of the SMA during long activity dealing with other forms of smart materials such as piezoelectric and magnetostrictive ones. They were fascinated when they discovered the great capabilities and wide field of use of SMAs. The resulting work was incredibly stimulating, as were the rich, interesting results in the field of aerospace structure morphing. This book is the outcome of the editors’ fortunate chance to meet many other scientists and colleagues during this adventure, carrying out relevant activities in the same field of application or in the general field of the SMA. The editors warmly thank all of the contributing authors who with their strong engagement and professional dedication have made it possible to collect in this book so much knowledge and professional experience. In addition, the editors want to thank the publisher, Elsevier, in particular Dr. Stephen Merken and Dr. Jeffrey Freeland, who guided them through the difficult art of organizing, managing, and finalizing a work with so many relevant contributors. Leonardo Lecce Antonio Concilio

SECTION 1

Introduction Editor:

Cristian Vendittozzi Faculdade do Gama, Universidade de Brasìlia, Brasília, DF, Brazil

List of chapters 1. 2. 3.

Historical background and future perspectives Latest attainments Standards for shape memory alloy applications

CHAPTER 1

Historical background and future perspectives Antonio Concilio1, Leonardo Lecce2 1

Department of Adaptive Structures, CIRA - the Italian Aerospace Research Centre, Capua, Italy; 2Novotech - Advanced Aerospace Technology s.r.l., Naples, Italy

1.1 Shape memory alloys In the common view, shape memory is a specific property some materials have to restore their original shape after a thermal load is applied [1e5]. In these substances, a rise in temperature may cause the full recovery of residual strains after a mechanical loadinge unloading process. This is macroscopically perceived as a cancellation of the impressed deformation, often also referred to as the shape memory effect (SME). In the following chapter, metal alloys will be discussed, although this kind of phenomenology is also observed in other compounds such as ceramics [6] and polymers [7], albeit on different physical bases. This property emerges as the result of a phase shift in which the crystal structure is reorganized. Such an atomic rearrangement can also occur when a stress field is imposed: thermal and mechanical fields have a sound reciprocal influence, and the action of each of the two amends the characteristic values of the other. For the sake of simplicity, let us say that in the case of shape memory alloy (SMA), two phases exist and are stable at low and high temperatures, respectively. The term “phase” is intended here as a peculiar inner arrangement of the crystal structure, with stable and uniform properties in a certain thermodynamic region. The cold phase, martensite, is named after the German physicist Adolf Martens, and is characterized by an eccentric and highly crystalline state that is stable in two further different forms. Originally, it indicated a metastable nonequilibrium allotrope of steel that is hard and is formed by rapid cooling [8]. The hot phase, austenite, takes its name after the English physicist Charles Austen; it has a centered cubic structure geometry and originally denoted a nonmagnetic form of iron that is stable at high temperatures [9]. Ironically, none of those two scientists had a role in SMA research: both had lived at the turn of 19th and 20th centuries and are considered the fathers of the materials science. A schematic aspect of martensite and austenite crystals is reported in Fig. 1.1, whereas the main geometrical characteristics are shown in Table 1.1 [10,11]. SMA exhibit further interesting macroscopic peculiarities, namely pseudoelasticity and thermoelasticity. The former is the property of the complete recovery of deformation after the material exhibits a sort of apparent yield; for some materials, they can attain large values of 10% or more. For that reason, it is

Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00001-7

© 2021 Elsevier Ltd. All rights reserved.

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c

c

b

b

a

a

Figure 1.1 Geometric appearance of austenite (left) and martensite (right). Distribution of titanium (light grey) and nickel (dark grey) atoms. Table 1.1 Characteristic lattice parameters for NiTi alloys [10] for austenite a, b ¼ cO2. Lattice parameters

Material Martensite Austenite

Length [Å]

a 4678 4262

b 4067 4262

Angle [deg]

c 2933 3.014

ab 90.00 90.00

bc 90.00 90.00

ca 98.26 90.00

also referred as the superelastic effect. The term “pseudoelasticity” is used because it involves reversible phase transformation between austenite and martensite, and therefore cannot be said to be strictly elastic (i.e., something that is linked to a simple geometry variation with no crystal rearrangement). In other words, the reference equations that describe the phenomenon are different [12]. The latter instead indicates a functional, reciprocal dependence of the elastic response and properties on thermal conditions. Properly speaking, thermoelasticity deals with reversible mechanical strain, thermodynamic behavior, and related coupling [13,14]. In the case of SMA, it primarily manifests as a strong dependence of the material response on temperature. For instance, depending on that, a certain alloy can show SME or superelastic effects [15]. In SMA, shape memory, superelasticity, and thermoelasticity all have a common base at the point at which they can be reconducted at the same physics law. Phase change and its connection with strain memory shape memory were first observed by Chang and Read in 1951, in a study on the AuCd alloy. They were the first to report the term “shape recovery” [16], although related behaviors such as pseudoelasticity or superelasticity were previously seen by other scientists. In 1932, € Olander noticed pseudoelasticity in the same alloy and described this occurrence in his research as a “rubber like effect” [17]. In 1938, Greninger and Mooradian demonstrated clear thermoelastic effects in CuZn during their investigation of brass alloys (copperezinc and copperetin) [18], because a martensitic phase was found to form and vanish as a function of temperature. A detailed study of thermoelasticity in CuZn alloys was published in 1949 by Kurdyumov and Khandros [19]. A second alloy,

Historical background and future perspectives

€ indiumethallium, was discovered to show SME in 1954 [20]. As in the case of Olander, the authors referred to the material as displaying rubber-like elastic characteristics when strained at low temperatures, highlighting its thermoelastic features. However, none of these studies led to an outburst of scientific interest in such a material. Yet, another alloy did, made of nickel and titanium. In 1965, Buehler and Wiley of the US Naval Ordnance Laboratory (NOL) obtained a US patent [21] for a series of NiTi alloys marked by uncommon performance. Among the compounds for which this invention applied, the second was reported as “paramagnetic, presenting high damping at ordinary temperatures, and capable to reassume nondeformed condition after having been plastically strained at about room environmental conditions, if brought to higher temperatures.” In the patent cited, the authors claimed anteriority rights for two-component alloys containing about 50e70% nickel and a corresponding amount of titanium. Because of their origin, these materials are currently referred to as Nitinol, and are derived from the union of their constituents’ chemical symbols and the NOL acronym. Since then, SME has been systematically observed in many other alloys. Copper-based alloys such as CuZnAl and CuAlNi [22,23] and iron-based alloys such as FePt, FePd, and FeMnSi [24e26] have attracted considerable attention for specific applications. For instance, copper-based materials exhibit high transformation temperatures and are useful for avoiding uncontrolled activation, whereas iron-based systems may be cheap and attractive for largescale implementation. To date, Nitinol is the alloy with the best general shape memory characteristics, together with three other remarkable properties: excellent corrosion resistance, a stable configuration, and almost perfect biocompatibility. These make Nitinol the preferred choice for implantation in human bodies. Nevertheless, attention is still devoted to further improving its properties [27]. On the other hand, such materials are expensive and difficult to melt and process [28].

1.2 List of acronyms Af Austenite finish; the characteristic temperature at which the transformation to austenite is completed (unloaded case) As Austenite start; the characteristic temperature at which the transformation to austenite begins (unloaded case) Mf Martensite finish; the characteristic temperature at which the transformation to martensite is completed (unloaded case) Ms Martensite start; the characteristic temperature at which the transformation to martensite begins (unloaded case)

1.3 Gold-based alloys € The first SME documented is attributed to Olander in 1932 [17,29]. He used electrochemical techniques to identify AuCd cubic (austenite) and B19 orthorhombic (martensite) phases. Furthermore, he was the first to recognize the peculiar behavior of these

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materials, although he did not yet acknowledge shape memory. He spoke of a “rubberlike effect” that allowed the alloy to come back to its undeformed configuration after it experienced plastic deformation [30,31]. Later, Bystrom and Almin [32], performed an x-ray investigation and found different phases of AuCd alloys for different compositions. The work was inspired by Olander’s previous study; surprisingly, the authors did not mention the strange phenomenon that had been previously discovered. The shape memory properties of Au-47.5% Cd were finally recognized and identified by Chang and Read [16]. They examined the movement of the boundaries between the two phases during transformation by x-ray analysis, electrical resistivity measurements, and motion picture observations. Results from that experiment allowed the researchers to conclude that the analyzed compound underwent diffusionless transformation from a high-symmetry B2 cubic to a low-symmetry B19 orthorhombic structure when it was cooled to its martensite finish (Mf) temperature (about 60  C for the specific investigated compound). Reverse transformation was found to occur upon heating the material to 80  C (austenite finish [Af] temperature). A qualitative resistanceetemperature graph of gold-based SMA is reported in Fig. 1.2. When it is cooled, a sharp rise in resistance occurs at martensite start (Ms). As transformation is completed at Mf, resistance again falls in line with classical materials behavior. Upon heating this martensitic phase, reverse transformation starts at austenite start (As), this time associated with a sharp decrease in electrical resistance, and is completed at Af, when resistance starts growing again. This macroscopic phenomenon may be explained by considering that martensite is less conductive than austenite, so as phase grows in the A / M transformation, the material shows a higher resistance to electricity flow, whereas in the M / A transformation, the natural increase linked to the thermal effect is somehow compensated by the austenite appearance.

1 Heating Relative Resistance

6

As

Mf

Af

30°C

Ms

Cooling 0°C

Relative Temperature

Figure 1.2 Pictorial view of the transformation characteristics of a typical Au-based SMA; the changes of resistivity with respect to temperature give an index of the phase state.

Historical background and future perspectives

In 1973, a work was published on the potentiality of gold-based SMA [33]. Brook noted that these kinds of compounds could have a market space for uses in which color, corrosion and tarnish behavior, and low electrical resistance characteristics typical of gold were important. Therefore, he envisaged that such a material could be used in electronics, jewelry, and even biomedical applications including dentistry. Undoubtedly, this vision was correct to a certain extent. This work caused some interest to the point that the South African Chamber of Mines sponsored a major research effort based on gold-based SMA, as reported by Brook himself 2 years later, in 1975 [34]. In that publication, interest was focused on a combination of goldecopperezinc memory alloys, which led to some patents [35]. Nominal compositions of the investigated compounds varied between 60% and 40% for the gold, 15% and 30% for the copper, and the residual difference for the zinc. Workability, ductility, mechanical strength, and shape recovery properties were studied in detail by specimen testing. The study focused on the crystallography of a specially ordered type of the beta-phase, known as the Heusler alloy, occurring at approximately Au25Cu25Zn50 (or AuCuZn2), where the subscripts refer to atomic percentages [36,37]. In those alloys, shape memory properties demonstrated a wide range of compositions, allowing a certain flexibility for optimizing targeted properties at room temperature. The optimal blend was found in the region of 22% atomic Au, 33% Cu, and 45% Zn (or 46% weight gold, 22% copper, and 31% zinc), for which strain recovery resulted in values of around 2.15%, whereas tensile tests confirmed that deformations up to 5.75% could be attained before failure. Generally, copper-rich alloys were found to perform better; they also showed the possibility of tolerating a further content of zinc. However, because of the high cost, practical applications did not take off. Another chapter dedicated to gold-based alloys was written when scientists from the University of Minnesota investigated with remarkable success how it was possible to enhance the reversibility characteristics of SMA by governing martensiteeaustenite transformations. The roots of that research were based on previous stochastic studies aimed at tuning the intimate geometrical structure of some materials (so-called compatibility conditions) to increase their ability to invert internal changes. Those scientific activities were initially aimed at finding ways to couple apparently incompatible properties such as ferromagnetism and ferroelectricity, in a material that could present phases, each marked by one of those properties [38]. Hints of further studies were then easily generated. Starting from some assumptions, those authors applied developed concepts to martensitic transformations to find a relation among conditions of compatibility between two phases and the hysteresis exhibited during cyclic microstructure changes [39]. Stronger and new compatibility conditions were finally formulated and known as cofactor conditions [40]. A paper reporting a systematic presentation of the impressive results attained during this wide experience was finally elaborated and published in 2018 [41].

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In synthesis, the research established that by respecting certain geometric and physical features in the crystalline structure, SMA was able to extend the reversibility of the austeniteemartensite transformation for a huge number of cycles. The key factor was found to ensure a no-stress transition between the phases of austenite and martensite, and especially an energetic equivalence of the twin phases of martensite. It is remarkable that the selected compound that could respect the cofactor conditions was the same Heusler alloy previously cited in the works of Brook and colleagues. Moving from that configuration by small steps in many possible directions, Au30Cu25Zn45 (atomic percentages) was found to fit the specifications almost perfectly. The result was an alloy that could go for more than 16,000 thermal cycles with no significant degradation in properties, with reference to either strain recovery or the transformation temperature itself, which shifted by around half a Kelvin degree. The comparison with nickele titanium alloys whose activation temperature may vary by tens of Kelvin degrees along the first tens of cycles is impressive. Equivalently, hysteresis stays confined within a couple of Kelvin degrees whereas nickeletitanium approaches almost 100 K [42]. Finally, the assessed alloy exhibited an appreciable 8% transformation strain, definitely in line with other, performant SMA. From a microscopic point of view, the alloy revealed a characteristic riverine structure of martensite twin phases not observed in other materials [43]. This latter property may be the key to the incredible properties of such materials. Considering the zero-stress boundary among the phases, it was experimentally demonstrated that whereas the macroscopic qualities remain unchanged along cyclic transformation processes (recoverable strain, activation temperatures, damping, and so on), its microstructure changed drastically, almost casually. This fact may be simply interpreted as the stress of change distributing along the material instead of repeating itself in certain, fixed regions. This can help the material itself to survive repeated loadingeunloading sequences, either mechanical or thermal, because the creation of microcracks is less probable. Although the attained results are interesting, the cost of such alloys remains the most relevant problem in their further development and commercial exploitation. It may be imagined that by increasing the demand for technological performance and reducing the size of implementation of the associated devices (micro- to nanomechanics) and associated costs, such materials could finally find a specific slot in the market. Apart from the jewelry industry, the excellent conductivity of gold makes these alloys suitable as adaptive electrical connectors. Moreover, because of their pronounced thermoelectrical coupling, goldepalladium alloys find applications in thermal measurements.

1.4 Nitinol 1.4.1 A story William J. Buhler started his work experience at the NOL, currently the Naval Surface Warfare Center (NSWC), White Oak, Maryland, in 1951, when he was hired as a

Historical background and future perspectives

mechanical engineer. In 1956, he was appointed a supervisory physical metallurgist. In 1958, he was in charge of selecting metal alloys for the nose of the Subroc, UUM-44A, a submarine-launched nuclear reentry missile for submarine warfare [44]. This submarine rocket was a large weapon designed to be fired from standard submarine torpedo tubes and aimed at hitting submarine targets, after a watereairewater mission. To preserve its structure during the reentry phase, the nose material had to satisfy severe thermal and heating conditions. In general, a missile head should also exhibit high impact resistance and a high melting temperature to guarantee successful penetration inside the enemy defenses. In some works, it is cited that this activity regarded the Polaris missile, instead [45], a two-stage, nuclear-armed, submarine-launched reentry ballistic warhead. Unfortunately, a speech reported by Buehler himself at the periodical meeting of the White Oak Laboratory Alumni Association (WOLAA), does not clarify this point. He spoke generally of finding “a metallic alloy material to withstand the high temperature rigors of a missile re-entry nose cone” [46]. Nevertheless, this ambiguity does not change the substance of the statements and the logic of the process reported here. Among tens of different compounds coming from a first screening carried out on the basis of a bibliographical survey, Buehler and his assistants, Everly and Heintzelman, selected a dozen for further investigation. Impact tests, performed by hammering small disk-shaped specimens, showed NiTi alloys to have the best response in terms of impact resistance, combined with excellent general structural and workability properties. It seems that at this point in the story, the acronym Nitinol was introduced, associating the material with the laboratory name. Then, it was meant to indicate something quite different from the current meaning. The selected alloy was produced in different forms from various processes and released for further experiments. One day in 1959, Buehler and his melter assistant made six NiTi arc bars. They took such specimens from the furnace and let them cool on a transit topped table. Because the operation was performed one-by-one, a wide range of temperatures was then available; the first deployed was cooler than the last ones. Buehler decided to bring the chiller beams to the workshop for chamfering, trying to move irregularities away from the surfaces. At this point of the tale, two versions are available in the literature. According to one version, on the way, the colder object fell to the ground, emitting a leaden-like dull sound. A second version states that this happened on purpose and not by accident [44e46]. In any case, this was the key point of the discovery, and in the opinion of the authors of this chapter, they do not add or detract merit from the scientists who were working on that compound. According to Buehler’s words, this episode worried him because some internal cracks could have been produced within the alloy during the arc-casting process, and that would have been a bad index for its performance [45]. He therefore decided to make the other, warmer bars fall to the ground as well. However, these specimens produced a clear, shrill sound. Something was off. To verify what was happening, he literally ran to the first available fountain to chill all of the other bars and

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check what kind of acoustics they might have generated. The cold bars produced dull sounds like the earlier one. Intrigued, he returned to check the behavior of the other samples, confirming the original results. He then made the original specimen hot again, cooled it down once more, and verified the responses in the two states. Again, deadened and sharp sounds were respectively produced. The process was repeated over and over, always with the same results. In his chronicle, released at the WOLAA, Buehler talked about suspended warm and cold arc bars that were repeatedly hit [46]. The two different sounds led to an understanding of the coexistence of two different reversible phases in the same material at different temperatures. Diverse atomic arrangements, associated with various levels of damping, should have arisen as the material was heated or cooled. At that time, it was not yet understood that this inner material restructuring would have led to a much more interesting discovery. Nevertheless, investigations showed that something was extremely interesting and unusual. Among the most relevant evidence found in earlier studies, the following are worth recalling: • As polished alloy surfaces were heated slightly, irregularities appeared; some shearing signals were also manifest; • Small indentations applied at room temperature were stable but tended to vanish as the material was warmed slightly; • Surface finishing greatly depended on the process implemented, with extreme differences between classical abrasive and diamond polishing (i.e., it was strongly related to the induced stress); and • Treated plates and sheets showed some length reduction when undergoing a thermal (heating) process. Together with sound emission, damping, strain, and fine structure variations, these occurrences seemed unequivocally to link inner properties at the level of crystal arrangement with macroscopic transformations. However, no one was able to imagine what kind of further secrets the newly realized alloy could hide. Such a situation was not peculiar in the history of technology. Some centuries before, the link between temperature and the generation of electrostatic charges on the surface of a crystal, a phenomenon known as pyroelectricity, was considered a characteristic of some types of materials and captured the attention of scientists worldwide [47]. The first of them, tourmaline, was imported in Europe from Ceylon in 1703 and became famous for its property of attracting or repelling light substances (such as ash particles) when heated [48]. After a long time, Jacques and Pierre Curie recognized pyroelectricity as articular evidence of more general piezoelectricity (i.e., the link between imposed pressure and the generation of electrostatic charges, which had been neglected until then) [49,50]. Even in the case of SMA, there was a risk that science would settle for a long time on investigating materials with variable damping rather than strain recovery, or shape memory capability. This time, however, things went differently. Whatever it is, serendipity seems to rule smart materials and the affair of structures.

Historical background and future perspectives

After those first findings, great effort was invested to understand the behavior of the alloy with highly variable macroscopic properties, and it was also found to exhibit a certain resistance to fatigue. The research group expanded as two young metallurgists, Raymond C. Wiley and David Goldstein, joined the team. In 1962, Buehler and Wiley presented their results to management; there are different versions about who actually participated in that historical meeting [44e46]. They presented a strip some tenths of a millimeter thick and deployed in a sort of accordion-like arrangement, folded and refolded many times to demonstrate its resistance to cyclic loading. The committee members examined that specimen in turn, bending and wrapping them, verifying with interest that it was able to bear those repeated stresses well. But how would it have withstood heat? A pipe smoker on the commission, David S. Muzzey, asked the question, and perhaps not completely satisfied with the answers, he wanted to test it himself. This gentleman took the wrinkled wire, put it to the flame of his lighter, and to the great astonishment of all those present, almost immediately, the thread straightened to its original shape. The metal had a memory of its early state [44], activated by heat that converted into mechanical energy. In a short time, while the story diffused through the laboratory, Buehler had no difficulty in connecting this evidence to the acoustic behavior noticed earlier. Sound phenomena and variable damping could lead to only a limited number of applications. Strain recovery could instead result in a considerable number of uses [21]. The research team was completed in 1962 as Frederick E. Wang accepted the challenge of investigating the atomic behavior of the NiTi alloy and join efforts to characterize its many properties to point out possible practical uses. According to that point of view, the scientists handled a solution without yet having a problem. Under that circumstance, the outcomes are not certain. Relevant results of their preliminary, finalized research were published in the next years, focusing on phase change peculiarities and a description of crystal structures [51,52]. It is curious that those first publications inverted the alloy’s name: they referred to titanium-nickel (TiNi) instead of nickel-titanium (NiTi). The reason is not explained, but it would be nice to understand if there was one.

1.4.2 Early commercial developments After the discovery, many funded research efforts started, some with success and some with no result at all. Many patents were applied for as the studies spread to universities, research centers, and industries. Nevertheless, only few commercial products were launched in the 1960s and 1970s, and just one had relevant success: the pipe coupling system. Difficulty in manufacturing (spanning from melting accuracy, purity, forming to machining, and so on), high costs, and the initial lack of reliable suppliers easily explain such results. With these boundary conditions, a successful application had to be simple and allow discrete tolerances. Yet, in the following decades, many items entered the market, above all in the biomedical field.

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1.4.2.1 Pipe coupling The first Nitinol commercial implementation that was also the first industrialized SMA was used for fluid fitting couplings in the early 1970s [53,54], a product that is still on the cutting edge. This system resulted in wide use despite the many competitive conventional joining approaches already employed that forced prices to remain low. The proposed coupling process used the SME to connect pipes. In few words, a short SMA cylinder was employed whose internal diameter was slightly smaller than the external tubes to connect. The cylinder was cooled in liquid nitrogen to attain the martensite phase; then, it was mechanically stretched to fit over the components to be joined. The natural warming process took about 10e15 s, enough to permit the operation. As the SMA element was heated, it transformed into austenite, completely reducing its shape while trying to recover the initial shape, creating a tight connection. As the temperature tended to return to room conditions, the alloy stayed in the new phase (i.e., it was designed so that it did not overcome Ms). It is not difficult to find publications on the matter, even at that date [55e57]. From that experience, other minor devices were derived, such as fasteners [58] and even related fittings to enhance the fatigue behavior of holes in structural components [59]. The first couplings realized are also the most famous. Since 1969, they were installed on the US Navy F-14 Tomcat, an advanced fighter of the period. The company that produced such devices, Raychem Corporation, manufactured a special material derived from NiTi named Tinel, which had the characteristics of stainless steel. Such a nickele titaniumeiron compound was highly resistant to corrosion and had good fatigue behavior. The finished product was put on the market under the trademark Cryofit, clearly referring to the cryogenic process taking place before installation. They accomplished 3000 psi specifications (21 MPa) for that plane’s hydraulic system [60,61]. It is estimated that about 300,000 couplings were mounted until 1999 [62,63]. In 1973, a slightly more well-performing version (4000 psi, 28 MPa) was designed for surface ships of the British Royal Navy [60]. In 1975, it was applied to UK Trafalgar-class nuclear submarines (6000 psi, 42 MPa), and in 1977 to US Navy surface ships. Today, Cryofit is a common aircraft fitting acknowledged by the US Federal Aviation Administration [64], and many other regulatory and standard bodies such as the US National Fire Protection Association under the classification of Memory Metal Fittings [65e67]. In 1974, as a side product of that technology, Raychem started producing cryogenic electrical connectors under the commercial name Cryocon, initially devoted to the navigation system for the Trident missile, a reentry ballistic missile operated by submarines and developed by Lockheed Corporation. They displayed good signal transmission quality and had high resistance to shock and vibration environments [62,68]. A significant step forward was then achieved in the 1980s with the introduction of a new alloy integrating niobium, Tinel-Lock, whose name was transformed into UniLok in 1993. It guaranteed the same performance levels without needing low temperatures before

Historical background and future perspectives

installation and had success in military and aerospace applications, including satellite programs. Among the broad range of new applications, optical component clamping, pressure sensor cover sealing, aerospace valve joints, nuclear seals, and even valve stoppers on rocket motor shuttles may be recalled [69e72]. TiNb SMA has been proposed as a significant alternative to classical medical NiTi alloys for their intrinsic augmented biocompatibility levels [73]. Cryocon, Cryofit, Tinel, and Tinel-Lock are registered trademarks of the former Tyco Electronics, now TE Connectivity [74]. Such a survey would not be complete without citing some accidents that involved SMA piping connectors. The first one cited here regards the second flight of the F-14 prototype, on Dec. 30, 1970. The so-called first real flight ended with expulsion of the pilots’ and loss of the aircraft. Specialists found two titanium hydraulic lines destroyed, one for each niche of the main wheels. Further investigations demonstrated that the pipe material was not the main reason for the incident. During a simulation, hydraulic system breakage occurred in less than 2 min, but when stainless-steel pipes replaced the former architecture, damage happened on the 23rd min. The malfunction was eventually blamed on the absence of accumulators to dampen pressure fluctuations [75]. On Oct. 16, 2019, the US Nuclear Regulatory Commission published a report concerning an inspection at the Brunswick steam electrical plant and the follow-up assessment after a 1-inch Cryofit coupling placed on the steam lines underwent complete circumferential separation. In the letter, the cause of failure was confirmed to be the inappropriate selection of Tinel for long-term application in the reactor lines susceptible to hydrogen embrittlement. Repair considered a welded connection instead, and the process was extended to similar zones subjected to the same phenomenon. Brunswick piping specifications were finally revised to exclude the use of Tinel connections at those locations and to incorporate aging mechanisms concerning hydrogen embrittlement of those components into the Aging Management Program of the facility [76]. The tube connector, which is simple in concept and exceptional in performance, represented a real turning point in NiTi technology. Without a reference application, it could have remained a scientific curiosity for years or decades. The market explosion created the opportunity to produce large quantities of quality materials at lower prices, which in turn made the alloys available to many researchers worldwide. In this way, confidence with and knowledge about this alloy increased. Furthermore, because industrial engineers and designers were involved in this process, exploitation of other real products was facilitated. 1.4.2.2 Orthodontic wires In 1968, when he read some news about a new alloy with strange properties, George F. Andreasen asked Buehler for some samples. After an extensive test campaign, he considered that a peculiar composition behaved perfectly for use in orthodontics. He applied for a patent that is recognized as one of the most highly earning ones in history for the

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University of Iowa [77,78]. Andreasen’s work started after he directly asked Buehler for some specimens to perform the first investigations in his field. Such an episode was cited by the NOL researcher himself during his speech at the WOLAA in 2006 [46]. The medical scientist started his work soon after and published the first paper in 1971 [79]. In that article, he and his colleague, Tymothy B. Hillemann, tried to justify using cobalt as a substitute for some percentage of titanium to augment its properties in terms of the elastic limit and corrosion resistance, and to compare the new alloy’s behavior with classical single- and triple-strand (twist-flex) wires. Investigations continued in 1972, when a new article was published concerning the possibility of exploiting strain recovery as a way to fix the wire to the dental apparatus, almost automatically. However, despite the large recovery strains of the experienced wires (up to 8%), the exhibited elastic modulus was too low to imagine it could completely substitute for classical devices, but it was thought to complement ordinary arch-wires in producing an additional closing force [80]. The first article was proposed in new form in 1973 as an extensive and systematic comparison campaign was carried out, likening two kinds of cobalt-based with six other stainless-steel wires (usual and twist-flex) [81]. Research continued with the aim of ever-better characterization of the developed wire for clinical purposes [82]. This latter paper was the first to use terms such as “shorter treatment times,” “less discomfort” (light forces), and “fewer arch-wire changes,” explicitly referring to patients’ well-being and including doctors’ length of operation. In the meantime, the cited patent had been applied in 1972, with the only reference to Andreasen’s first job [79]. The wire was commercialized by Unitek Corporation (now, 3M Unitek), making it the second product on the market based on Nitinol SMA in history. In detail, the material was realized in a fairly equiatomic structure; it was composed of 55% weight nickel and 45% titanium. Cobalt was added to replace nickel for an atomic percentage equal to about 6.5% for applications aimed at facing flexural or torsional stresses. According to Andreasen’s presentation [77], the proposed compound promised to perform easier installation because of the reduced Young’s modulus and the possibility of using shape memory (called mechanical memory), to generate elastic forces trying to move teeth appropriately. Instead of a classical elastic wire, Nitinol wires would have exerted a constant nondecreasing force along the established path. Many passages of the invention clearly refer to “body temperature” as a suitable way to activate the alloy. Investigations continued in that direction even after the untimely death of Andreasen. In 1995, work was published dealing with an experimental campaign aimed at comparing the behavior of three different commercial wires manufactured by 3M Unitek, Ortho Arc Company, and GAC International Incorporated [83]. This simple fact gives an immediate idea of how much technology and the market had moved ahead in a few years, a clear signal of the economic impact of the invention. In that work, researchers reported that the tested compounds did not differ from each other with minor exception of standard deviations, an indicator of behavior repeatability and therefore wire quality.

Historical background and future perspectives

Strain recovery was confirmed to be in the range of 8e10%, whereas the addition of cobalt was confirmed as a way to adjust transformation temperatures, a key aspect for the targeted material, The publication proposes five points for the ideal alloy in orthodontics, in turn derived and summarized from previous work [84e86]: soft and passive at room temperature so as to be easily installed; able to be activated by body heat; safely sized to develop forces; suitable for clinical orthodontic applications; and stable with respect to the mouth’s environment. However, despite what was declared by the inventors and successive investigators, and what was reported earlier, the first NiTi materials for orthodontic applications, including those with cobalt substitutions, were SMA only by composition. The cold process implemented to stretch the material to 8e10% strain canceled the recovery effect [87e89]. Nevertheless, the new compound won over the market, which needed lighter installation forces and the availability of an increased working range, resulting in an overall larger springback with respect to what was provided by common solutions available at that time, such as stainless-steel and chromiumecobalt alloys [90]. In particular, forces for units of deactivation, which were five to six times less than the usual forces, better answered the need for lighter, more continuous orthodontic actions. In 1976, several brands of NiTi wires were put on the market, gathering wide acceptance because of their undiscussed benefits and expected perspectives [89]. Once that limitation was made clear, Andreasen started working on the thermal dynamic effects of Nitinol in 1985 [91], but he was unable to complete his investigations because he died only 4 years after that, when he was age 55. A comprehensive review of the orthodontic technologies available in 2014 is reported in Jyothikiran et al. [92], including an overall analysis of the major strengths and weaknesses of the many devices presented. The first use of NiTi alloys for human implants was reported in China in 1981 and referred to the implementation of compression staples [93,94]. This was followed by other records of bone applications for a large number of patients [95]. Eastern country capabilities in the sector had been revealed for the first time a few years earlier in a domestic conference on SMA held in 1986, when different papers were presented on a variety of topics [96]. The most famous product of those activities of NiTi technology for medical use was the realization of a new version of NiTi alloys especially suited for application in orthodontics, soon renamed Chinese NiTi [97]. Wire made of that material was the first biomedical element to show superelastic behavior; it generally exhibited outstanding mechanical characteristics with respect to equivalent contemporary products. For the sake of clarity, it is worth recalling that superelasticity in the biomedical field was not intended in those works as the ability to return to zero after having experienced a sort of yield along the load path and having attained remarkable value of deformation, but just the plateau segment of the diagram corresponding to the reversible martensiteeaustenite transformation stage. The utmost peculiarity was to display large and recoverable strains while maintaining a constant and limited force level corresponding to the characteristic

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plateau associated with martensiteeaustenite transformations in both directions. This particularity is important in orthodontics for at least two reasons. Along the installation phase, the technician should not worry about the possibility of applying a large force as a consequence of large displacements necessary to perform the arrangement correctly. During the operation, the patient would not experience excessive or ineffective forces if the rearrangement of teeth undergo significant displacements. A remarkable number of studies was then conducted worldwide aimed at producing wires with similar attributes. In this case, it seemed no one was really scared of trying to copy Chinese technology after all [98e100]. At the beginning of the 1980s, Fujio Miura and his staff conducted extensive research activities to establish the optimal path for tooth movement. To accomplish their objective, they needed a device that could impress a light and continuous action. The investigation then naturally moved to find something that could answer that necessity, finally approaching NiTi technology. In 1982, Tomy Incorporated, a manufacturer of orthodontic products, and Furukawa Electric Company, a supplier of wire materials for orthodontics, were approached with the target of realizing a new superelastic wire. The efforts resulted in the assessment of a filament capable of generating forces in the desired range, suitable for teeth implants, and around 8% pseudoelastic or superelastic deformation recovery. The product was traded in 1985 under the commercial name Sentalloy (superelastic NiTi alloy) [87,101,102]. Miura made explicit reference to Watanabe’s investigations in his papers [103,104]. In 1982, that scientist presented an NiTi compound with superelastic characteristics to assess its use in orthodontics and compare its behavior with stainless-steel cobaltechromium alloys and work-hardened NiTi wire. He found that the selected material presented an interesting curve with a plateau of 2e5% strain, and the ability to recover the imposed deformation naturally once the load was removed. Maximum recoverable deformation was evaluated to be around 11%; however, residual deformation up to 0.5% was recorded in cyclic experiments carried out up to 8% strain, adequate for use in orthodontics. The material was softer than CoCr or stainless steel; its stiffness was around 50% of those alloys. All of these results indicated the investigated wire to be a good candidate for orthodontic arches. The cantilever beam method used by Watanabe in his characterization research was the object of some criticism by Miura [101], who instead preferred three-point bending to characterize his specimens, because according to his view, superelastic behavior would not have been adequately distinguished. A further boost to orthodontics technology was linked to the exploitation of thermodynamic NiTi alloys in the early 1990s [105], on the bases of the precursors’ work. The new materials, attained by doping NiTi with a small amount of other substances such as copper, were enriched with the added attribute of being (finally!) activated by body temperature. Therefore, because the material would have received energy from the mouth, the reversal transformation into austenite would have started, contracting the

Historical background and future perspectives

wire and in turn exerting some force on the teeth. Many studies on the behavior of such wires were conducted, as reported in Kusy and Whitley [106]. A lower Af temperature allowed attaining a better and lower plateau. By properly modulating the dispersion of additives, the material properties could have been adequately varied to allow easier installation of the wires in the orthodontic structure or improve the patient’s comfort. Further evolutions of the concept were then proposed. For instance, some wires were designed with an Af well over the usual body temperature so that they could stay typically inactive, producing just a small force on the dental system. However, when the patient ate hot food, the orthodontic apparatus would have become active and applied a limited force for a confined time. That device was particularly suited for extremely sensitive people or people with periodontal problems [107]. 1.4.2.3 Other medical applications Orthodontics is not the only medical field in which SMA is used. In the 1990s, minimally invasive surgery gave a decisive boost to NiTi technology. The ability to provide large deformation and relevant forces is essential for those applications [108]. Shape recovery elements are valid alternatives to classical mechanical manipulation systems when available space is limited. If direct heating is possible for activation, SMA devices may be small; consequently, overall device dimensions may be minimized. Because NiTi is a nonferromagnetic alloy, it can also be used in magnetic resonance imaginge based surgery [109]. Apart widely employed orthodontic systems, early examples of NiTi alloys used for medical applications include augmentation of Harrington rodlike system capabilities for treating scoliosis in 1975 and the Simon filter in 1977. The Harrington rod is a medical system that aims to reduce spinal curvature and provide it with a certain stability. It was created as an alternative to uncomfortable and sometime unstable vertebrae fusions. In the first applications, SMA properties were used to adjust relative vertebrae position and reduce compression and extension loads [110]. The Simon filter is an SMA system that aimed to substitute for ordinary usual devices introduced in the vena cava to treat pulmonary embolism. Owing to the material’s property, the filter could be reduced to a thin wire that, after insertion, would expanded to its final shape, be activated by body temperature, and be enabled to trap thromboemboli further [111]. Those architectures were the bases for developing more famous stents, particularly aortic ones, starting from 1983 [112]. These are internal elements that keep the vessel open, achieving a sort of endoskeleton like an internal support structure. The basic idea was to realize an NiTi device that entered the vein in a compressed form like a wire, which is then forced to expand after activation by heating. In the first results, sufficient heating was induced by externally, whereas in the current versions, it relies simply on body temperature [113]. Self-expanding SMA stents are manufactured with a diameter larger than the target channel. Af is usually set slightly under such a value,

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so that the device may be easily worked under standard environmental conditions. To prevent premature release, it is constrained to a special rig that is removed after installation. The stent then assumes its final configuration in a few seconds [114e116]. Currently, it is estimated that about half of all implanted stents are made of NiTi alloy [117,118]. To give an idea of what the economical size of this application is and its perspectives, in 2018, Zion Market Research reported a value for the global vascular stent market of about US$8.13 billion in 2017, which is expected to rise to US$13.62 billion by the end of 2024 [119]. Apart those pioneering products, it was not until the 1990s that Nitinol started to have a serious impact on medicine with technological breakthroughs in design, modeling, and manufacturing. Since then, many other medical devices have been developed and have achieved great success, such as surgical devices for endoscopic and laparoscopic surgery, and orthopedic implants. It is believed that the first Nitinol low-cost certified medical device for sale on the Western mass market was the Mitek Homer Mammalok in 1990 [120,121], a limited intrusive and repositionable marker to help locate breast tumors and nodules, combined with classical radiology techniques. It is a two-piece device made of a superelastic NiTi wire ending in a half-circular hook, entering a hollowed needle. As this latter is inserted into the breast and carefully positioned to enter the supposed tumor zone, the needle is then extracted, assuming a characteristic hook configuration to tag the location for surgical intervention. If necessary, the wire can be retracted and then pulled again from the needle until it is correctly adjusted to the target site [122,123]. Because tumor material is usually denser than surrounding tissue, the locator tends to encircle it instead of passing through it. In its simplicity, the device offered some interesting design challenges: it should have been stiff enough to be placed tightly at the right position, yet soft enough to allow retraction in the needle. The optimal wire diameter was a tenth of a millimeter and the hook radius was several millimeters. Such an architecture would have been impossible to replicate with a stainless-steel material, for several reasons. For instance, having to respect the former geometrical and mechanical characteristics, it would have been too thin to secure the hook properly. For the same reason, there would have been a high risk for leaving fragments of wire along the deployment and release path [122]. Currently, this technology is widely implemented. The Homer breast localization needle is produced and commercialized by Argon Medical Devices [124], whereas other devices have been proposed, such as the Tumark Vision, by Somatex Medical Technologies GmbH [125]. Reason for the success of the NiTi alloy that caused it to be widely implemented for implants and medical devices were its excellent biocompatibility, mechanical characteristics, and aging performance, which made it particularly indicated for long-term installations. Early information on biocompatibility studies of Nitinol alloys may be found in Ryh€anen [126], who claimed approval of the device by the US Food and

Historical background and future perspectives

Drug Administration, and some references to analogous implantation that had occurred in Russia and China [127e130]. Along that line, another product entered was released in 1987, the Mitek bone anchor [131], which may be considered the first Western commercial permanent implant based on Nitinol [132]. That device consisted of a main body and superelastic wings able to anchor the system to the bone. The barbs could be made straight, and after installation it could be released to recover its initial shape to grab the bone, securing the device permanently. The insertion occurred through a small hole directly drilled onto the target part. Direct interventions (hole drilling) and collateral damage (securing) associated with the use of this device were minimal. In 1992, the original design was substituted by a new one, Mitek G2. In that release, the system assumed the shape of an arrow with the barbs aimed at securing the device to the bone once it was moved inside the bone. This is still the form currently commercialized under the brand DePuy Synthes Mitek GII Anchor, a Johnson & Johnson firm [133]. In a survey aimed at analyzing the results of such sealing structures [134], 66 interventions were reported, for a total of 166 installed anchors connecting soft tissues to the hand, wrist, and elbow bones. In that report, successive x-ray scanning verified that 65 implants stayed securely clamped to the bones. Common to other NiTi-based technologies, the technique proved to be easy to learn and use. An in-depth analysis of the Mitek anchor bone for orthodontic corrections is reported by the Keller Professional Group [135]. Today, bone implants, stents, valves, microvalves, and many other clinical systems have been developed [136e140], making NiTi alloy technology commonly used in many fields of medicine.

1.4.3 A conclusion Undoubtedly, since the early discovery of Nitinol properties, its name has run throughout the world and is currently a word that can be found in many English and international dictionaries. Many commercial applications still use that acronym, which brings in itself the reference to the place where its fantastic characteristics were revealed. But how much was it appreciated at home? It is worth citing Buehler’s words again, as reported in his speech at the WOLAA, in 2006. The text following reported is therefore extracted from that talk, [46]. In 1961, he was awarded the Meritorious Civilian Service Award, an acknowledgment for early Nitinol research and development activity. After exactly 10 years, in 1971, Buehler and Wang were proposed for a higher US Navy recognition level by NOL management. However, that application was rejected because in those times, the potential of the novel material was not yet fully proven. The Navy suggested waiting for a while and looking for further possible developments. After more than 20 years, in 1993, Buehler and Wang revisited the old recommendation and moved their request to the NSWC, the former NOL. Years passed and the institution changed

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its name. This time, however, the nomination did not need to arrive at US Navy Command to be rejected: it stopped at NSWC offices. The reason was written in an NSWC letter dated Nov. 24, 1993. In short, it stated that an award given for something achieved so far in the past would have somehow conflicted with standard US Navy practices. It appears that the letter concluded that there should have had to be very compelling reasons to make the administration justify such a late recognition. This chapter was conceived with the aim of presenting a historical perspective on SMA development. When the past is remembered, there is always some lesson to be learned. The authors are strongly convinced that this episode may well teach something to readers and young scientists, above all.

1.5 Copper-based alloys As with AuCd, the discovery of shape memory properties in CuZn goes back to the 1950s. The existence of a critical transformation temperature was initially found as a function of the composition and stress field [141]. In further work, the dependence of reversible martensite reaction characteristics, especially Ms and Mf, was preliminarily studied as a function of the composition of high-purity alloys [142]. Taking into account some additional results attained in work on the brass beta-phase [143], the study was expanded with a quantitative analysis of the tensile stress effects on modification of the material structure. In that circumstance, information on the thermoelastic behavior of the martensite in the first stages of the transformation was also obtained [144]. The SME in CuZn alloys was phenomenologically described in 1956 [145] and recalled in many other successive reports, f.i [146]. After the exploitation of NiTi, these compounds caught the interest of researchers because they were cheap, simply fabricated, and exhibited good shape memory characteristics. Such properties made them suitable, for instance, for applications to microelectromechanical systems, as shown for the CuZnAl formulation [147]. Brass is made of widely available gross materials and requires standard preparation and machining tools such as classical furnaces and the usual metal carving equipment [148,149]. Components of almost any shape and size can be produced even if they do not have excellent strength or corrosion resistance [150] and or good aging behavior that justifies a number of works on the theme [151,152]. The simple CuAl combination has well-defined shape memory characteristics but transformation temperatures that are too high for general use; furthermore, its polycrystalline form is generally brittle, preventing the envisioned applications because of manufacturing difficulties and unreliability of operation [153]. These drawbacks may be reduced by adding a third element. CuAlMn SMA exhibit excellent recovery response and superelastic features [154,155] while favorably altering ductility [156]. CuAlBe alloys are credited for appearing on the technology scene in 1982 [157] and soon created certain interest in their sound dissipation capacity in cyclical transformation. For this reason, they are particularly

Historical background and future perspectives

suited for ant-seismic devices, attaining around 5% equivalent damping in the whole frequency range of reference [158]. Because their mechanical properties are strictly connected to the grain size and the involved phases, some research has been devoted to their optimization [159]. Generally speaking, small fractions of Be are usually used, around 0.5%, which limits the impact of having introduced such an expensive material. Transformation temperatures are from 200 to 100  C and they exhibit excellent superelastic and damping properties [160]. However, the most studied mixes are CuZnAl and CuAlNi. The first ones have an outstanding advantage by integrating cheap metals through conventional processes, which makes them among the cheapest commercial SMA available. On other hand, martensitic phase stabilizes at room temperature whereas the material itself deteriorates when exposed to temperatures over 100  C. Maximum recoverable strain is around approximately 5% and attention should be paid during manufacturing or brittle architectures result [161]. The second ones were developed for high-temperature applications and display small hysteresis loops. Ordinary mechanical characteristics can be improved by adding further elements such as B, Be, and Zr, or simply changing the relative content of Al and Ni [162]. All of these compounds are available on the market, and many applications have been proposed [163,164]. The most common uses of these materials are in civil pipe couplings and hydraulic fittings, mechanical dampers, and thermal actuator and sensor systems. One of the first full-size uses of these alloys was to set up a heat engine [165], something that had already happened with NiTi alloys [166,167]. With the target of individuating a viable and cheap alternative to NiTi materials, many experiments were conducted on previously developed configurations. Many studies focused on the mechanical characterization of the new compound, as widely reported in the literature. For instance, low-frequency experiments were aimed at getting tension, bending, and twisting characteristics of CuZnAlNi and CuAlBe specimens [168]. A one-dimensional model was developed to predict materials’ long-term damping performance [169]. These and other research efforts were the bases for investigating the use of such alloys as energy dissipation devices for civil engineering structures [170,171]. This field seemed one of the most promising, for which massive uses are required and the cost of raw materials can make the ultimate difference. In another investigation, pretensioned CuAlBe cables were anchored to a scaled masonry wall model to verify the augmented structure seismic resistance [23]. Analogous devices were then inserted on the monumental aqueduct of Larnaca (Cyprus) to improve its dynamic characteristics, as verified after periodic measurements. Copper-alloy components have been also used in many industrial fields. Some were designed to work for many cycles in circuit board edge connectors, such as in Krumme [172,173], in which shape memory materials replaced an actuator lever system to control opening and closing of opposed pairs of contacts in cam-operated, multicontact,

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zero-insertion force connectors. Others were conceived for safety and protection devices as fire valves; for instance, an autonomous passive CuZnAl actuator shut off toxic or flammable gas flow as fire occurred [174]. Such a device was recalled from another patent on the subject, using instead NiTi alloys. In that case, an SMA disk designed to break the frangible body of a fire sprinkler at a temperature should have risen over a certain level to allow the passage of the extinguishing fluid [175]. Other uses extended to structural or quasistructural elements to improve their strength. For instance, CuAlNi microwires were proposed as additive elements for fiberreinforced composites and textiles. A suitable strategy to achieve high-ductility levels was defined, applied to obtain microwires 10e150 mm in diameter, characterized by a fine bamboo grain structure. The process enabled recovery strains up to about 7% to be reached [176]. Because of the possible impact of these alloys, and to prevent competition, leading companies were forced to enter this new market. Based on a CuZnAl compound, Raychem realized an economic variant of their famous pipe couplings for aluminum and copper tubing, devoted mainly to air-conditioning systems. The procedure was slightly more complex than the former. In this case, the SMA tube shrank a cylindrical liner on the pipes to be connected, whose tightness was enhanced by a sealant coating [177].

1.5.1 Copperezincealuminum Historically, copperezincealuminum (CuZnAl) was the first copper-based SMA to be commercially exploited [160]. It is a ternary compound commercially relevant and widely studied in the literature, starting from early investigations into their excellent damping properties [178], their critical fatigue characteristics [179,180], and some specific peculiarity such as the possibility of accessing two-way SME by specific training [181]. Copper binary alloys such as CuAl, CuZn, and CuSn have shape memory characteristics, but the associated criticalities are too severe to make them useful for real applications, at least currently. The transformation temperature may be too high (CuAl), workability may be poor (CuZn), or the martensite state too stable (CuSn) to allow common use. The addition of zinc, a low-cost and widely diffused element, may lead transition temperatures to decrease, attaining suitable values in the range of 100 to þ100  C, a function of both composition and thermomechanical treatments. Nevertheless, many compounds are reported with transformation characteristics (As / Mf) well over 200  C, such as Zn 25.4%, Al 3.3%, or Zn 18.1%, Al 5.35% with Cu in both cases enough for 100%. Respective ranges were 202e316  C, or 228e338  C [182]. Typical compositions contain 15e30% zinc, 3e7% aluminum, and copper to balance. Further additions of B, Ce, Co, Fe, Ti, V, and Zr permit the grain size to be controlled, and thus the brittleness of the alloy. Such quantities should generally be kept low (under 1%) because they can primarily affect the stability of the martensitic phase, and therefore shape memory properties.

Historical background and future perspectives

Among the major advantages of CuZnAl alloys are the basic material costs (they are made of relatively inexpensive metals) and production costs (it is possible to operate them through conventional processes such induction melting or powder metallurgy). Cold workability is feasible and results in a strong function of the percentage of Al. Memory properties are significant, with typical maximum recoverable strain attained at about 5%. Their major drawbacks are related to long-term cycling at room temperature, which stabilizes the martensitic phase, increases the transformation temperature, and degrades the shape memory performance. This results in its being almost independent on the considered alloy system, the long-range order degree, or temperature, whereas geometrical and chemical characteristics may have a relevant role in that [183,184]. Some compounds even decompose when exposed to temperatures above 100  C. Because such disadvantages are not usually compensated for by economical convenience, their commercial success has been strongly limited. Currently, copper aluminum zinc alloys are available in several forms such as bars, ribbons, wires, sheets, and foils. Ultrahigh purity and high-purity forms include metal powder, submicron powder, nanoscale targets for thin-film deposition, and pellets for chemical vapor deposition and physical vapor deposition applications. Compounds are available in many standard grades including Mil Spec (military grade), ACS, and reagent and technical grade, and follow applicable ASTM testing standards. Primary applications include bearing assemblies, ballast, casting, step soldering, and radiation shielding [185].

1.5.2 Copperealuminumenickel Copperealuminumenickel (CuAlNi) is usually preferred to the CuZnAl alloys; it replaces zinc with equally cheap nickel. They have atypical content of 11e14% Al, 3e5% Ni and Cu to balance. The most diffused commercially is perhaps Cu13%eAl4%eNi. Transformation temperatures are in the range 80e200  C, depending on their composition, particularly the aluminum content. Processing is more difficult because it can only be hot worked and heat treatment should be tightly controlled to attain the desired characteristics. Furthermore, workability levels seem to be independent of the variation of many parameters, including Al percentages. Nevertheless, in this case, standard methods may be implemented. As a consequence of hot working, controlled cooling processes are necessary to fix important shape memory attributes, and further treatments are usually required to stabilize those parameters. Despite these drawbacks that make them more expensive than the former copperbased system, they are still competitive with respect to nickeletitanium. Production has been tried via powder metallurgy [186]. As for CuZnAl, mechanical characteristics may be improved by using the same additives. For instance, small percentages of Mn replacing equal quantities of Al reduce brittleness, one of the most severe defects of CuAlNi alloys.

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It was proven that replacing 2% of Al content with Mn inhibited internal deterioration without decreasing transformation temperatures. In the same work, further addition of 1% Ti resulted in significant grain refinement, eliminating intergranular cracking [187]. For similar materials, significant aging was observed at 250 to 400  C [188], and studies were carried out to decrease brittleness in the high-temperature range [189]. Because extra elements affect the stability of the alloy, their use must be limited and carefully dosed. Damping properties are also relevant. A kind of metal matrix composite based on powders of CuAlNi SMA embedded in an indium matrix was shown to reach more than 50% structural damping at 0.01 Hz and 70  C. However, such a value exhibited strong dependency on frequency and temperature, decreasing at something like 7% at 1 Hz and the same temperature [190]. It is normal to find two-way alloys obtained using a simple technique. An alloy produced by classical melting was extruded in 4-mm-diameter wires bent around a cylinder mold that underwent a constrained thermal process [191]. Manufacturing technology underwent further progress with trials to produce thin films, combining magnetic techniques, high-temperature treatment, and hot quenching in ice water, or co-sputtering the component elements on an MgO surface [192]. In synthesis, the major advantages of CuAlNi include lower modification intervals; good stability even at high temperatures, making it the only one that can be used for applications above 100  C; and relatively low cost [160]. When the evident limitations are finally overcome [162], it can be imagined that such an alloy will soon effectively enter the market. CuAlNi alloys, whose applications are similar to CuZnAl, may be found in the same forms as either the macroscopic or powder ones, with the same standard grades, from military to medical and civil ones [185].

1.6 Iron-based alloys Traditional SMA have serious disadvantages regarding workability and costs, particularly for large applications such as civil engineering for bridge construction; in those instances, such drawbacks become real fiascos. This is the basic reason why research has moved toward ferrous-based compounds, specifically FeMnSi alloys, to establish easy-access materials that can be fabricated by classical methods and machines, in principle [193]. Tungsten inert gas, plasma, laser, and electron beam welding are applicable [194e196]. Nevertheless, real fabrication issues suggest that costs cannot be lowered to a significant target unless the needs reach those of usual steels. In other words, for plants sized to meet the common demand of that material, only an equally dimensioned use may make access convenient to those facilities. In general, ferrousbased SMA are manufactured by melting, casting, and final thermomechanical treatment [197]. Because FeMnSi alloy is similar to high-Mn stainless steel, it is almost trivial to use the same production facilities as for steel or stainless steel. However, it has been remarked that for a compound containing over 20% Mn, impurities contained in

Historical background and future perspectives

it could result in fatal faults in the shape memory properties. Of course, small percentages of Mn, yet maintaining above 13% [198], would reduce this problem and allow the use of mass production furnaces [199]. For instance, Fee17Mne5Sie10Cre4Nie1(V,C) was produced under normal atmospheric conditions and with usual tools [200]. An alternative production methodology was proposed by powder metallurgy and sintering processes, offering several advantages such as the realization of products in almost net shape that minimize additional machining operations to attain the final product. Produced alloys revealed shape memory properties similar to what was obtained by traditional casting [197]. With respect to NiTi alloys, the SME in Fe alloys is usually smaller, around 4% at most, and generally occurs at higher temperatures. However, there are many studies and much research is currently developed continuously raising the limit to almost 8% [201]. Moreover, unusual Fe alloys, using Ta and fractions of B have shown recovery strain up to about 14% [202]. Because the envisaged applications of Fe-based SMA for large infrastructures are mostly as joints to connect different architectural components easily and quickly, recovery stress is more important than recovery strain. The most appealing property is thus the maximum force exerted by the SMA element on the installed part. For Fe-based SMA, activation temperatures may start at around 50  C and end at several hundred degrees Celsius, depending on the alloy’s composition. Ideal stresses settle between 600 and 800 MPa [203], but ideal tensions would not be achieved because of both the presence of unavoidable tolerances at the connection area and the elasticity of the coupled structure. From a general point of view, the strength, shape of stressestrain curves, and other properties are similar to those of stainless steel. The research justifies such a vision; currently, many low-cost, iron-based alloys are on the market that exhibit high cold workability, good weldability, and excellent shape memory characteristics compared with NiTi systems [204]. SME in ferrous alloys was reportedly discovered in Japan in 1982 for FeMnSi alloys and then extended to their polycrystalline form [205]. In detail, stress-derived martensitic transformation was initially examined on Fee30Mne1Si alloy single crystals. Si additions were found to promote austenite / ε martensite phase change and to suppress the formation of alpha-martensite. In that way, the alloy did show almost complete strain recovery as heated over As. In these compounds, the SME is related to the nonthermoelastic martensitic transformation, but it exhibited interesting strain recovery levels. This fact confuted the former understanding that SME is basically correlated only with thermoelastic martensitic transformation [206]. Further research demonstrated that full shape memory recovery could also be obtained in polycrystalline alloys containing a significant amount of Si. Its presence was found to be beneficial for at least three reasons: austenite stability and the Neel temperature decreased; matrix strength increased, inhibiting the formation of permanent slips during memory deformation; and austenite stacking fault energy was reduced and martensite formation was fostered [207]. Addition of Cr and Ni increased corrosion resistance characteristics, and Fee28Mne6Sie5Cr alloy resulted

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in one of the most used in practical applications [208]. Many studies were carried out to characterize the fine behavior of this kind of alloy. For instance, optical, scanning electron, transmission electron, and scanning tunneling microscopy demonstrated that the pole mechanism was the only cause of martensite nucleation in Fee16Mne9Cre5Sie4 Ni [209]. A significant characterization study of Fee14Mne9Cre6Sie5Ni showed how these materials could attain about 4% recovery strain while presenting good strength (816 N/mm2), ductility (72% elongation), and corrosion resistance close to values for commercial construction steels. The addition of sulfur increased alloy workability without affecting shape memory characteristics [210]. Later, specific research established that NbC and VN precipitates in the microstructure could have enhanced shape memory characteristics and limits, eliminating the training phase [211,212]. The first activity led to a patent application regarding a method of processing and heat treating a classical ferrous alloy with the addition of NbC to improve its properties [213]. In the second one, the study aimed to develop high-stress, high-strain, cost-competitive SMA material for large civil transport engineering applications such as pipe joining in tunnel-making construction. In that case, it was estimated that over 400 MPa proof stress and 3.5% shape recovery strain were necessary. The objective was achieved by developing an Fe-based alloy (Fee28Mne6Sie5Cr) that contained a high density of coherent VN precipitates. As usual, the grain size was essential in assessing shape recovery properties. Its dimension may be conveniently reduced by proper thermal and mechanical treatment that upgrades the strain recovery level and diminishes the need for training. This last property is essential for large applications [214]. VC particles were also used to increase the ability of FeMnSibased alloys associated with sophisticated thermal, mechanical, and chemical treatment, reaching a grain size of around 50 mm [215]. Encouraging results were also attained with similar additions implemented during fabrication in traditional air-casting facilities. The produced material was then characterized in the hot forged and cold roll state, showing good shape recovery and shape memory stresses even when heated at 160  C [216]. Large joining pipes for tunnel construction and rail joint plates (known also as fishplates) are important examples of the successful employment of Fe-SMA [217]. A system used in tunnel construction consists of building a sort of whale skeleton from a smaller gallery by inserting segmented curved beams one after the other. These sections are soldered until a complete or almost complete ring is achieved. The resulting skeleton structure creates a sort of protected environment for further and larger excavations. The space of operation is narrow and work requires many hours. The use of SMA joints is a natural solution to this challenging problem, making the process simpler while ensuring the same levels of reliability. Generally, a further mechanical key (usually two C-rings) is also added to neglect any degree of freedom to the SMA joints. These connections show the same resistance characteristics as soldering or welding and require far fewer work hours to be implemented [208,218,219]. Fishplates for rail connections are another

Historical background and future perspectives

example of the large-scale use of Fe-SMA. In this case, the usual working process is a massive operation developed in open air and on-site. Many uncertainties then arise, eventually making two consequent rail elements imperfectly connected to each other. In other words, a gap remains among two successive parts that can be further dramatized by external environmental conditions. It keeps enlarging because of vibrations caused by trains, also producing bruises and lesions. Rail functions are then negatively affected and maintenance costs increase, not to mention critical safety issues. SMA fishplates can overcome this inconvenience. After installation and warming, the rail elements come increasingly closer, and at the end of the process, no visible gap is apparent [208,219,220]. During operations, SMA tensions keep the rail components together while ensuring good elastic properties. Both of these applications are possible because of two basic, unique features of Fe-based SMA: structural properties similar to steel and the relatively low cost, enabling extensive production. The ease of installation makes these solutions viable. Another widely used application has been suggested and tested, concerning the development of seismic dampers for increasing the safety of civil buildings and infrastructure [221]. In this case, the concept is to give the reference structure some added elements that can absorb and dissipate energy during the earthquake. Therefore, this time the application is dynamic instead of static and relies on hysteresis features of the Fe-based SMA that should be then maximized. The seismic damper may be in the form of a shear plate that squanders vibrational energy after elastoplastic deformation occurring between its top and bottom. Under a maximum deformation angle of about 2.3 degrees, the proposed device is estimated to be able to bear a load of around 400 tons, which makes it a first-class metallic damper [221]. Fe-based SMA may also enhance the strength of plaster, a building material that is commonly used for interior wall protection or decoration. For massive applications, it should usually be reinforced with internal fibers because it is not made to bear significant loads. Inexpensive FeMnSiCr SMA wires added to the reference compound may significantly increase the strength characteristics. In a literature example, such pretensioned additions are inserted in the plaster and treated at a hot temperature after curing. Apart from increased normal load resistance capabilities, laboratory tests demonstrated that bending characteristics improved by about 50% [222]. These figures resulted in an interesting function of SMA elements before strain. Because of the relatively low cost of the implemented alloy and perfect integrability with the existing production process of fiber-reinforced plaster, the proposed solution seems sustainable to obtain low-cost, market-appealing engineering products. An extensive review and experimental work analyzed the properties of ferrous SMA for strengthening civil structures. Related investigations allowed the authors to track some general statements correlated with implementing that kind of material. In particular, it showed good characteristics for implementation to be retrofitted to damaged structures or as a simple fortifier of existing constructions.

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A prestressing force of more than 400 MPa could be transferred to existing reinforced concrete to give it better ability to resist normal tension stress fields. Suitable boding strength were reached and improved through a dedicated process based on the proper choice of embedding lengths. These conclusions further paved the way to the extensive use of Fe-based SMA for civil structures [203]. Mechanical properties compared with steel alloys and low-cost are make iron-based SMA increasingly interesting for use in all fields of civil engineering, with effects on aspects such as assembly, damping, and on-site intervention for strengthening purposes.

1.7 Shape memory alloy community Early development of the SMA community coincided with the discovery of NiTi alloy properties. In a report published in 1978 and a first revision dated to 1980, a massive list of works and events associated with NiTi alloys was reported [223,224]. Among others, a study on the crystal structure of the alloy performed in Germany is cited, dated 1939 [225]. It is also clear that considerable attention was paid to Ti-based alloys as early as 1950 [226,227]. Of course, some other interesting studies may be found in the period between these dates. As often happens, if one looks just slightly beyond the usual, some interesting news appears, such as the lapidary sentence: The present data do not justify further investigation of binary titanium-germanium or titanium-nickel alloys [228]. The first symposium on NiTi alloys was held in 1967 [229]. About 18 articles were presented on NiTi and other alloys of titanium, dealing with theoretical modeling, the transformation process, the internal structure, and the macroscopic behavior, including damping, fatigue, and other mechanical characteristics. The primary scope of the meeting was to bring together scientists working on the novel discovery and to foster a better understanding of the unique properties disclosed [230]. In 1975, the first symposium entirely dedicated to SME (i.e., including other alloys apart from NiTi and compounds of titanium) was held in Toronto and organized by the Metallurgical Society of the American Institute of Mining, Metallurgical and Petroleum Engineers [231]. Thirtythree articles were presented including the seminar, dealing with many SMA, including copper-based, and the historical AuCd and TiNb. The works mainly focuses on characterizing the material behavior and its internal structure; just a few dealt with envisaged or experimental applications, including a general overview [232]: electrical connectors [54], integrated circuit packages [233], heat engines [234], vibration absorption [235], and the already cited interesting proposal for biomedical uses, concerning Harrington rods for scoliosis treatment in NiTi [110]. The NOL hosted a new conference on NiTi alloys in 1978, on their use for heat engines. The potentialities of that alloy were already clear, and increasing attention was devoted to the possibility of manufacturing engines powered through the transformation of thermal to mechanical energy. The proceedings report 12 of the 15 papers scheduled, eight of which introduce original prototype

Historical background and future perspectives

engines. A note by Jack Dixon, on behalf of the direction of the hosting structure, NSWC, states that several of these prototypes were subsequently shown on television in the US and Britain [236]. A simple consideration that can be made at this point is that no significant application was reasonably found at that time, so the scientists focused on a thematic of the highest concern in that period, dealing with energy independence and the oil crisis of the 1970s. With the objective of enhancing information sharing and discussion among scientists involved in the matter, a new symposium was announced, to be held in Leuven in 1982. Taking a small step backward to 1976, the Japan Institute of Metals organized the first international symposium on new aspects of martensitic transformation [237]. The event series was dedicated to general important metallurgical problems to promote the international exchange of information. Recent attainments on martensitic transformation and related attention given by many countries all over the world convinced the organizing team to choose that theme as the first to which the conference be dedicated. A total of 74 papers were submitted from 15 countries (including Japan, Belgium, Switzerland, the United States, and the USSR). Presentations were completed by 10 invited lectures. In the program, only one session is explicitly dedicated to SME, namely, Pseudoelasticity and Shape Memory Effect; it hosted eight papers, one of which was invited [238]. This conference was repeated 1 year later, changing its name to the Martensitic Transformations: International Conference (INCOMAT) [239]. Apart from the official release, there is also a side edition that summarizes the many papers presented in Russian, whose translation was not provided except for the titles [240]. That conference hosted 55 oral presentations and 82 posters for a total of 137 works; there were more than 200 attendees, more than double the number for the previous edition. However, just 24 of them came from outside the USSR, probably because of difficulties and delays in attaining visas at those times. It may be of some interest to report that, due to the respective small knowledge of Russian and English from both the parties, German was found the best compromise for communication.In short, the conference was mainly dedicated to analyses testing phenomenological theories of martensite crystallography, while presenting few works on applications. Apart from the proverbial capability of theoretical modeling, the domain of Eastern country scientists, it is presumed that the disclosure of actual technological implementation might somehow have been inhibited for national security reasons. In the same years, an International Committee on Martensite (INCOM) was established and ready to handle the organization of future symposiums on the matter. In 1979, finally realigning our tale, the conference reserved a session dedicated to the SME and once again changed its name and acronym to International Conference on Martensitic Transformation (ICOMAT), which is still used. It confirmed the participation of the previous meeting, counting almost 150 attendees from about 20

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countries [241]. By choice of the organizers, the event focused on a physical understanding of the martensitic transformation, trying to limit the presence of other works only to those dealing with insights into the nature of the material structure modification. No work on actual uses of SMA was presented. A remarkable number of presentations dealt with SME; a couple of them were dedicated to study of the NiTiCu alloy, a material that is not popular. In 1982, the ICOMAT conference was held in Leuven and incorporated the second originally planned symposium, as reported during the 1978 event at the NOL. More than 130 papers were presented, about 15 of which dealt with shape memory and an equal number with NiTi alloys. Many other dealt with copper-based compounds, although significant attention was apparently paid to ferrous systems. Again, no applications found space in that new conference. Significant participation of European and Japanese researchers was recorded, with inclusion of some from the United States and minor representatives from Russia and China [242]. With some irregularities, the event has been repeated on a generic 3-year basis. The most recent exception is the shift forward of ICOMAT 20 to 2021, because of the novel coronavirus-19 crisis (2020). It will be the 16th conference because the accepted numbering considers the symposium held in Kobe to be the first one, followed by the one in Kiev and the one in Leuven. When the conference came back to the United States in 1992, US Nitinol and US Office of Naval Research were among its sponsors, perhaps to mark continuity with the 1968 meeting. A great number of US and Japanese companies completed the supporters’ list, including just two European ones. More than 350 papers were presented, with growing attention to NiTi and Ti-based SMA. The event left significant space to industrial applications but maintained an important focus on material modeling and physical aspects. Production issues were also taken into consideration from the point of the view of both the alloy and device manufacturing [243]. With another step back into the past, the second international symposium completely devoted to SMA after the one held in Toronto took place in China under the initiative of the National Non-Ferrous Metals Society and the Japan Institute of Metals, in 1986 [244]. About 80 papers from 14 countries were published, almost 70 of which treated NiTi alloys, Cu-based alloys, and real implementation cases. A small number of works were dedicated to the relatively new Fe-based alloys. The opening lecture gave an almost complete overview of the status of research in China, with special emphasis on activities concerning the medical field. The symposium was then composed of five sections: general theory, NiTi alloys, Cu-based alloys, other SMA and materials, and manufacturing and experimental testing [245]. In 1988, another conference dedicated to the engineering and implementation of SMA was held in the United States [246] A structured collection of selected papers was published 2 years later [2]. The papers that were published and later rearranged were the occasion to state some important aspects of what still appeared to be new technology despite more than

Historical background and future perspectives

20 years passing since its discovery. In that moment, it became clear that the difficulty of finding actual applications in the real world was mainly because engineering lacked an actual science behind it. The editors and organizers of the event, and above all those who took care to select and arrange the papers in an ordered structure to give the reader the possibility of entering the world of SMA rationally from the material to possible implementation, classifies the uses into four main categories: free strain recovery, constrained strain recovery, actuation, and superelastic exploitation [2]. Without going into a discussion of those four domains, it is possible to state that at some extent, such a categorization may still be considered valid. In 1988, other conferences also took place on the largest topic of so-called advanced or smart or intelligent materials, sometimes hosting dedicated sessions or subconferences on shape memory materials [247]. A good example is the first meeting of the Material Research Society on Advanced materials [248]. Despite the relative novelty of the theme, it seemed already to be affirmed, because it was reported that more than 150,000 visitors were expected to visit the demonstration booths at a special event planned immediately before the upcoming symposium, Advanced Materials and Engineering Exhibition, 1988 May 27e30, at the Tokyo International Trade Fair Grounds (Harumi). The large conference was structured in 20 symposia plus a special one dedicated to the envisioned frontiers in materials science and engineering. Among those, there was one dedicated to shape memory materials, which would have counted about 100 among invited lectures, regular papers and posters [249]. In the same year, a special symposium was devoted to finding a formal definition of smart structures, organized by the US Army Research Office [250]. The potentiality of piezoelectric actuators was just discovered [251], and increasing numbers of scientists devoted interest to materials intrinsically able to transform several types of energy (electric, heat, magnetic, and so on) into mechanical energy. The opening lecture tried to give an answer to two formal questions essential for further scientific and technological developments: to come to a shared definition of smart or intelligent structures and materials, and to identify their characteristics. The result of the dissertation confined its attention on “smart structures” intended as a system . which has . sensor, actuator and control mechanism . [that makes it, editor’s note] . capable of sensing a stimulus, responding to it . in an . appropriate time and reverting to its original state as . the stimulus is removed [252]. With some additions and modifications, such a sentence is still currently used. In the second paper, some more details were given regarding how a smart structure could actually be realized, producing a wide overview on available applications and giving a clear vision of possible developments [253]. The first meeting on adaptive structures was held in 1989, with the aim of continuing to explore the path drawn by former work and exploring to what extent viable and reliable structural systems with variable capabilities could be produced [254]. Soon after that event was held was the first joint United StateseJapan conference on adaptive structures. It would have then have been classified as the first international conference on the

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topic and was the first of the series known as ICAST, currently held yearly [255]. Adaptive structures had become a scientific and technologic topic acknowledged worldwide. Meanwhile, the combined effect of stable growing interest in shape memory materials, the appearance on the scene of many other smart materials, and the explosion of application possibilities spanning virtually all engineering matters produced natural merging of different scientists into better-defined and specialist organizations. In 1992, the International Organization on Shape Memory and Superelastic Technologies (SMST) was founded alongside the ICOMAT conference series, following the recognized need for and possibility of having focused events on SMA engineering. The first conference of the new association was held in 1994 [256]. Since then, the event has been repeated almost yearly, whereas other symposia connected with SMA technology have been organized throughout the world. In 2004, SMST became an affiliate society of the American Society for Metals (ASM) International, a well-known organization committed to disseminating materials knowledge [257]. Among other worldwide institutions dealing with SMA, the Japanese Association for Shape Memory Alloys [258] is worth citing. It is linked to the Agne Gjiutsu Center, which is in turn an expression of the Agne Company Limited, a metallic materials development plant also involved with producing machinery for related physical and chemical research, which publishes the specialized scientific journal Metal (Kinzoku) [259].

1.8 Future perspectives 1.8.1 A status overview Before going toward future applications, it may be worth considering the 50-year period beginning with the discovery of Nitinol. The fascinating characteristics of SMA have made many scientists work on them. The possible applications of such alloys are almost infinite. Thousands and thousands of patents have been applied for and published. On the other hand, the devices that found a place in the market are considerably few and for precise aims. Currently, major applications are in medicine where, despite the small volume of materials dealt with, the economic value is impressive. In that case, orthodontics appliances and stents are the most famous products, as well as components in robotic actuators and micromanipulators. Pipe couplings are other devices that have received enormous success over the years for specific and costly systems (fighters, submarines, and ships) and are currently evolving for heavy civil applications. Here, the discovery of alternative and cheaper alloys has had a fundamental role. Side commercial areas have also developed with the introduction of well-known deformable eyeglass frames, women’s brassiere underwear, and cellular phone antennas, to name just some examples that have entered everyday life. The list continues by recalling the use of SMA for thermal valves, autonomous devices for opening and closing doors, a deep fryer actuator for moving the basket

Historical background and future perspectives

up and down according to the oil temperature, and many other items of that kind. It is also worth recalling the use of SMA for jewelry mounts and special bracelets, among the low masseimpact but high-cost value applications. At the same time, there are also a variety of uses that have not found their commercial outlet yet, despite their potentialities. For instance, SMA could be applied in vibration dampers because of their large energy dissipation characteristics and hysteresis properties. This concept is being applied to general transport vehicles as well as large buildings for augmented earthquake protection. In the same way, they could be useful for absorbing part of the cyclic movement a large bridge undergoes, or even to protect monuments and historical buildings. Although SMA cannot be used in a real dynamic control system because the response is slow, as any true for other thermally driven material, applications to modify the dynamic response of generic structures can easily be found in the literature. An example may be the concept of integrating SMA wires into composite panels to affect their stiffness by modifying the SMA crystal structure through temperature. It has been suggested to control large space truss architectures by elongating or stretching NiTi beams. The main reasons why of all these ideas have not yet become commercial products may be the associated cost of the proposed systems (dampers), the complexity of the solution (dynamic control), and the excessive scale (and cost) of the reference architecture (truss control). Speaking of marketed systems, the active technologies developed until recently are static or single actuationelife types. For instance, valve applications work alternatively on cold and hot water and have no particular dynamic characteristic needs or high-precision aims. The valve operates in one sense or the other, slowly and continuously, and accurate temperature versus motion control is not requested. SMA joints are devices that are applied, warmed, and left in that contracted state for the lifetime of the reference structure. The same consideration holds for medical stents: they have to deploy when immerged into the blood channels. On the other hand, active orthodontics wires are placed in the mouth and directly operated by the skill and force of the doctor. Even if they react to the mouth’s warmth, they can be removed by the operator’s action. All of them are sort of oneoff systems. Another relevant similarity among the cited examples and the almost totality of current shape memory actuator applications in the real world is the clear reference to a controlled temperature environment. In other words, SMA devices always lie within a clearly defined temperature range. The human body is an excellent example, but by definition, this statement also holds for the hot water valve or the fryer mover. Other uses are typically passive and refer instead to superelasticity. For this kind of use, SMA elements are employed for their remarkable strain recovery or energy dissipation capabilities. In turn, these are later associated with large hysteretic behavior and are not intended to be actuated.

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Therefore, an easy prediction is to say that these implementation constraints will also affect the future trends of SMA exploitation unless some breakthrough technology appears.

1.8.2 A vision In aeronautics, the frequency response limitations make SMA devices applicable only for maneuvers and not for attitude control. Within this frame, the main obstacle to the real use of SMA technology is temperature limitations. In particular, because typical operations of an aircraft occur at 50 to 80  C, any kind of actuator must be unresponsive to be safe from external disturbances. Besides, the activation interval should be expanded (currently, it is about 20  C for most alloys) to improve the controllability of the movement mechanism. Repeatability and reliability are two other parameters that should be considered, to ensure standard performance levels over a long period. SMA usually undergo strong aging phenomena that degrade their capabilities. Fatigue aspects must be also taken into account. They are strongly influenced by the requested strain recovery levels, which can be up to several unit percent, depending on the specific use. If neglected, aging and fatigue effects can easily lead to catastrophic failure of the system. Experiences with the design and characterization of structural mockups of adaptive flaps implementing SMA actuators have shown these critical aspects while confirming the wide potentiality of these devices [260e262]. Further investigations confirmed the preliminary impressions, and although the activation system was more sophisticated and overcame the need for an electrical current directly heating the SMA, basic limitations remained [263]. Nevertheless, there are special cases in which such major issues can be neglected, as in the case of devices mounted onboard small unmanned vehicles for specific operations. Lowaltitude missions, dealing mostly with usual environmental temperatures, say in the range 0e40  C, greatly relax basic specifications. Furthermore, in that case, an SMAbased, or generally a smart material-based system is no more than an improvement of an existing actuator, but it may be considered to enable the applications because it is the only solution that can fit the absolute requisites in terms of accessible room and weight [264,265]. An often overlooked element that can heavily affect the performance of a shape memoryebased actuation system is its strong dependence on the reference structure stiffness properties. Because they change according to the direction along which they are measured, the ability to transfer forces and displacements is strongly modified after its installation [266]. Adaptive twist for helicopter blades has been pursued for a long time. In principle, it is a well-confined problem with remarkable benefits in real cases. By considering that two major configurations for the rotorcraft flight are hovering and cruise, with different specifications that would lead to dissimilar rotor geometries, the possibility of attaining

Historical background and future perspectives

different shapes for the low- and high-speed phases undoubtedly attracts the interest of many investigators. One type of architecture that has been studied offers interesting perspectives on ongoing problems concerning installation, training, and operation. The system needs a specialized jig and tools for assembly, with extremely high forces to manage and high electrical power to be activated [267]. In the automotive field, many applications have been proposed for properly shaping air ducts or aerodynamic surfaces. An unusual system has been developed to improve the insulation characteristics of the sealant installed between the door and the main car body. The device tries to compensate for the pressure difference between the interior and exterior, which grows as the speed increases because of aerodynamic effects. The proposed concept could be installed within the existing components and has been demonstrated to lead to appreciable advantages in terms of cabin acoustic levels. The main drawbacks are once the mains supply, which is needed all during the cruise, and above all the costs, which highly affect the market appeal of the final product [268]. The following characteristics should be improved to allow SMA systems to be applied to real aircraft structures and undergo a certification process: • Stabilized aging over at least 20,000 cycles; • Fatigue life assessed to at least 200,000 cycles while ensuring the established recovery strain requirements; • Activation temperature over 100  C; and • Extended activation temperature range to enhance controllability. Of course, these are still minimal requirements that would not guarantee the actual and specific feasibility of the proposed devices. Further characteristics that should be improved or attained to increase the commercialization of SMA devices include: • Cost; • Energy requirements; • Supply architectures; and • Installation processes. Because Fe-based SMA are low in cost, in the future it is reasonable to expect the larger use of these kinds of SMA for building construction and other large civil structures such as tunnels and bridges. The same technology could be extended to other connection elements such as bridge beams, building pillars, and so on, which would result in cheaper, cleaner, and possibly safer systems, with the added potential to be extended to mass products such as cars and, in the future, urban air vehicles. In the more than 50 years since their introduction, SMA joints have proven their reliability in a variety of severe applications. The response of these devices to static loads, fatigue, high temperature, thermal cycles, and corrosion suggests they could also fit other uses such as bolting, riveting, sealing, clamping, and other types of connections whose large-scale technological development was historically inhibited by the high price of NiTi alloys.

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Concerning medical applications, the use of such compounds has become almost a standard and is continuously growing. Because of the limited material necessity (a few grams for orthodontic wires or a stent), the cost of the raw material is not a problem in this case. Instead, a major issue is the certification for biocompatibility and other aspects directly related to the use of medical implants. Research toward even more biocompatible materials such as TiNb thus finds fertile ground for development. Thus, on the hand side, there is an assessed technology with a wide and enlarging community; on the other hand, because of growing success, research for even more performing materials is ongoing, ensuring the continuity of SMA solutions. Apart from the wellestablished applications, many other concepts are being developed for SMA. The possibility of constructing small motors on the basis of their ability to transform heat in mechanical energy did not completely vanish. Energy harvesting ideas could easily take advantage of these materials in environments with natural thermal gradients. Rapid revolution satellites could be an interesting playground for this concept. Generally speaking, the possibility of using SMA components in complex and largely common mechanical systems such as car motor engines is wide, moving from springs to simple connection elements or windshields. Because of their sensibility to heat, they could be also naturally implemented to modulate air flow around the engine or other parts of a car, including the cabin, as a function of the inner and outer temperatures. Other potential applications of SMA are in the field of robotics, in the wake of what is already experienced for micromanipulators in medicine. Indeed, many further possibilities have been investigated, from classical mechanical hands to small moving systems, featuring insects, octopuses, and so on.

1.8.3 Other shape memory materials To complete this overview, it is necessary to recall the variety of materials that have shown some shape memory properties and are becoming further objects of investigation throughout the world: ceramics, polymers, and gels. Shape memory ceramics (SMC) [269,270], feature high-energy output and hightemperature use. Their most obvious inconvenience is represented by their inability to resist large strains for a large number of cycles. Research is currently focused on overcoming these limitations. Surprisingly, the expected strain recovery range is at the same level of classical SMA. In a patent, compounds of Zr, O, and Ce, plus many other alternatives were proposed. The authors claimed to have released a novel superelastic structure with absorption properties well over classical damping materials for its absolute characteristics of superior strength, recoverable strain, dissipation capacity, and temperature and chemical stability. Light weight is also mentioned. Envisaged applications regard armor, helmets, and many other kinds of protection, dissipation, and isolation systems [271].

Historical background and future perspectives

Shape memory polymers (SMP) [272,273] have basic macroscopic behavior close to that of alloys, with the significant difference of exhibiting larger strain recovery capabilities (even two magnitudes larger) but necessarily lower force levels (two magnitudes less) to work on the same amount of energy. A 2018 publication reports the perspective use of SMP in dynamic buildings for energy savings by proper incorporation in the external envelope, meeting general weather and temperature changes. In this way, better thermal efficiency is attained. Several examples are reported, but the paper focuses on possible configurations rather than giving insight into some specific realization. The envisaged concepts try to exploit shape variations to achieve different heat exchange capabilities or modify the light shielding or shading characteristics of a generic wall [274]. A special class of SMP is shape memory gels (SMG) [275e277]. Deformations may achieve several hundred percent, that means doubling or tripling the original length, or even more. The material stiffness itself may change a lot, from 0.1 to 10 MPa. Adaptive gels may be used to create an adaptive lens, as the shape change is used to modify the optical properties linked to geometry, or to create fabrics with variable properties, depending on the environmental temperature. Even other macroscopic features may be varied, such as transparency, to disclose exceptional possibilities in optics. There are also unusual but extremely effective applications. Smart medical bandages can be made, applicable to broken arms without exerting force causing potential pain, and which are self-removable. Moreover, a smart safety alarm button could be pushed under only certain environmental conditions, to avoid hoaxes [278]. Further examples may be found in the literature [279]. The shape memory properties in these materials rely on different physical bases compared with metallic alloys. SMC and SMP have several similarities: they undergo plastic deformation under the action of external loads at high temperatures, retain the acquired form after rapid cooling, and recover the original shape by heating them again. However, whereas in the first case the phenomenon is triggered by a combination of rigid and soft materials (mica and glass, for instance), the second of which tends to acquire the initial state after heating, in the second case, always relying on a combination of two different constituents as a polymer fabric and a bonding resin, the heating phase forces the long polymeric chain to return its natural outline. SMG is instead a class of materials with significant differences in mechanisms. Some hydrogels modify their mechanical characteristics by temperature and may be significantly stretched by applying external forces. As in the SMP and SMC, such a new form is maintained by rapid cooling. Successive heating will then restore the initial contour. Strains up to 50% may be reached. Other types of hydrogels may be manipulated by a structured sequence of chemical reactions, structuring molecular links by increasing the temperature and adding substances, and then

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cooling the product. In that case, strains up to 200% may be reached. The configuration is restored by wetting the gel with boiling water, for instance. Modulated gels are made of at least a control layer that is sensitive to temperature, and a substrate layer that is insensitive to temperature changes. In these architectures, shape variations up to 1000% may be reached through a complex mechanical and chemical process. By the proper arrangement of different layers, a wide range of shapes may be attained, even very complex ones [280]. As we look around, there are many innovative ideas for applying SMA and shape memory materials in general, while the number of assessed products is growing and worldwide research is trying to understand their behavior and their characteristics for better and larger use.

1.9 Summary tables Some reference SMA properties are reported, as extracted from available bibliography, and synthesized to give prompt support for rapid computations and considerations, while detailed design should access more accurate data. The properties listed here are indicative, because small changes in the composition, including the addition of further elements, can drastically modify the final properties of the alloy. Currently, much research is performed with that aim to reduce the drawbacks of each solution. Table 1.2 summarizes a few points on the ease of achieving desired products from raw material. Table 1.3 gives an overview of main mechanical data. A summary of some recurrent applications is reported in Table 1.4, to help focus attention on a certain type of SMA when approaching a specific design.

Table 1.2 Workability of shape memory alloys. Cold workability

Hot workability

Good

Fair to good

Poor Fair to good Good

Poor Good

Good to very good Poor to fair Very good Fair to good

Type

Forming

Aubased NiTi Cubased Febased

Poor to fair

Scale ¼ very poor, poor, fair, good, and very good.

Machinability

Transformation Hysteresis 8C (up to) 8C

Good

0 to 100

15

Poor Very Good

200 to 100 200 to 200

30 35

Fair

200 to 150

100

Historical background and future perspectives

Table 1.3 Mechanical properties of shape memory alloys. Alloy

Density (kg/m3)

AuCd

13.5

CuZnAl

7.7

CuAlNi

7.2

NiTi

6.4

FeMnSi

7.5

FeMnAlNi

6.9

Young’s modulus (GPa)

Austenite Martensite Austenite Martensite Austenite Martensite Austenite Martensite Austenite Martensite Austenite Martensite

100 96 72 70 85 80 83 30 210 140 100 50

Recoverable strain (%)

1.5 4.0 4.0 8.0 4.0 3.0

Table 1.4 Typical applications of shape memory alloys. Alloy

Main applications

Au-based

Dental Jewelry Pipe couplings Electrical connectors Actuators Orthodontics Stents Surgery implants Seismic attenuation Isolators and dampers in civil construction Reinforcing elements Connectors and couplings Safety valves and other safety devices Tube couplings Rail couplings Seismic attenuation Large-scale dampers

NiTi

Cu-based

Fe-based

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[225] F. Laves, H.J. Wallbaum, Die kristallstruktur von Ni3Ti und Si2Ti - zwei neue typen (The crystal structure of Ni3Ti and Si2Ti e two new types), Z. f€ ur Kristallogr. - Cryst. Mater. 101 (1) (1939) 78e93 [German]. [226] P. Duwez, J.L. Taylor, Structure of intermediate phases in alloys of titanium with iron, cobalt, and nickel, JOM 2 (1950) 1173e1176. [227] C.M. Craighead, W.O. Simmons, L.W. Eastwood, Titanium binary alloys, JOM 2 (1950) 485e513. [228] W.O. Simmons, C.T. Greenidge, C.M. Craighead, First Progress Report Covering the Period May 18 e September 18, 1949, on Research and Development on Titanium Alloys, Battelle Memorial Institute for AMC, Wright-Patterson Air Force Base, Columbus, 1949. Report N. ADB816506. [229] F.E. Wang (Ed.), Proceedings of Symposium on TiNi and Associated Compounds, 1967 Apr 3e4; Silver Spring, MD-USA, Oak Ridge: US Naval Ordnance Laboratory, 1967. Report N. AD0668696, TR-68-16. [230] USAFC Division of Technical Information, Nuclear science abstracts. Oak ridge: United States atomic energy commission, Div. Tech. Inf. 23 (7) (1969). E-book available from: https://play. google.com/books/reader?id¼qqhHAQAAMAAJ&hl¼en&pg¼GBS.PA1153 [cited 2020 Aug 02]. [231] Shape memory effects in alloys, in: J. Perkins (Ed.), Proceedings of the International Symposium on Shape Memory Effects and Applications; 1975 May 19e22, Springer Science þ Business Media, Toronto, ON-Canada. New York, 1975. [232] W.S. Owen, Shape memory effects and applications: an overview, in: J. Perkins (Ed.), Shape Memory Effects in Alloys, Proceedings of the International Symposium on Shape Memory Effects and Applications, Springer Science þ Business Media, Toronto, ON-Canada. New York, May 19e22, 1975. [233] H. Pops, Manufacture of an integrated circuit package, in: J. Perkins (Ed.), Shape Memory Effects in Alloys, Proceedings of the International Symposium on Shape Memory Effects and Applications, Springer Science þ Business Media, Toronto, ON, Canada. New York, May 19e22, 1975. [234] B. Ridgway, Nitinol heat engines, in: J. Perkins (Ed.), Shape Memory Effects in Alloys, Proceedings of the International Symposium on Shape Memory Effects and Applications, Springer Science þ Business Media, Toronto, ON, Canada. New York, May 19e22, 1975. [235] L. Kaufman, S.A. Kulin, P. Neshe, R. Salzbrenner, Internal vibration absorption in potential structural materials, in: J. Perkins (Ed.), Shape Memory Effects in Alloys, Proceedings of the International Symposium on Shape Memory Effects and Applications, Springer Science þ Business Media, Toronto, ON, Canada. New York, May 19e22, 1975. [236] D.M. Goldstein, L.J. McNamara (Eds.), Proceedings of the Nitinol Heat Engine Conference; 1978 Sep 26e27. Silver Spring, Oak Ridge: Naval Surface Weapons Centre, MD-USA, 1978. Report N. ADA108973. [237] H. Suzuki (Ed.), Proceedings of the First JIM International Symposium on New Aspects of Martensitic Transformation, Kobe, Japan, May 10e12, 1976. [238] I. Tamura, First JIM international symposium, new-aspects of martensitic transformation, Trans. ISIJ (17) (1997) 616e618. [239] V.N. Gridnev (Ed.), Proceedings of the International Conference on Martensitic Transformation in Metals and Alloys, INCOMAT 77; 1977 May 16e20, Naukova Dumka, Kiev, Ukraine-USSR. Kiev, 1978 [Russian and English]. [240] J. Perkins, in: INCOMAT 1977 International Conference on Martensitic Transformations, Kiev, USSR, 16e19 May 1977, Office of Naval Research, London, 1977. Report N. ADA045007.

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[259] Agne.co.jp [Internet]. Tokyo: Agne Technology Centre Company Limited; [released 2017 Jun 15; cited 2020 Aug 4]. Available from: https://www.agne.co.jp/kinzoku/index.htm. [260] S. Barbarino, R. Pecora, L. Lecce, A. Concilio, S. Ameduri, L. De Rosa, Airfoil structural morphing based on S.M.A. actuator series: numerical and experimental studies, J. Intell. Mater. Syst. Struct. 22 (10) (2011) 987e1004. [261] R. Pecora, S. Barbarino, A. Concilio, L. Lecce, S. Russo, Design and functional test of a morphing high-lift device for a regional aircraft, J. Intell. Mater. Syst. Struct. 22 (10) (2011) 1005e1023. [262] S. Ameduri, A. Brindisi, B. Tiseo, A. Concilio, R. Pecora, Optimization and integration of shape memory alloy (SMA)-based elastic actuators within a morphing flap architecture, J. Intell. Mater. Syst. Struct. 23 (4) (2012) 381e396. [263] S. Ameduri, A. Concilio, R. Pecora, D. Karagiannis, A single slotted morphing flap based on SMA technology, Smart Struct. Syst. 17 (5) (2016) 819e835. [264] S. Ameduri, A. Concilio, N. Favaloro, L. Pellone, A shape memory alloy application for compact unmanned aerial vehicles, Aerospace 3 (2) (2016) 16. [265] L. Pellone, S. Ameduri, N. Favaloro, A. Concilio, SMA-based system for environmental sensors released from an unmanned aerial vehicle, Aerospace 4 (4) (2017) 22. [266] A. Concilio, S. Ameduri, Influence of structural architecture on linear SMA actuators performance and morphing system layout optimisation, J. Intell. Mater. Syst. Struct. 25 (16) (2014) 2037e2051. [267] S. Ameduri, A. Brindisi, M. Ciminello, A. Concilio, V. Quaranta, M. Brandizzi, Car soundproof improvement through an SMA adaptive system, Actuators 7 (4) (2018) 23. [268] S. Ameduri, A. Concilio, A shape memory alloy torsion actuator for static blade twist, J. Intell. Mater. Syst. Struct. 30 (17) (2019) 2605e2636. [269] T. Matsumura, T. Nakamura, M. Tetsuka, K. Takashina, K. Tajima, Y. Nishi, Shape memory ceramics, MRS Proc. 604 (1999) 161e166. [270] E. De Guire, Shape Memory and Superelasticity in Zirconia, Ceramic Tech Today, 2013. Available from: http://ceramics.org/ceramic-tech-today/shape-memory-and-superelasticity-in-zirconia. [271] Z. Du, H. Yu, C.A. Schuh, C.L. Gan, Inventors, Nanyang Technological University Massachusetts Institute of Technology: Assignees. Shape Memory Ceramic Particles and Structures Formed Thereof, 2019. US Patent N. 20190039959. [272] C. Liu, H. Qinb, P.T. Mather, Review of progress in shape-memory polymers, J. Mater. Chem. 17 (2007) 1543e1558. [273] Y. Liu, H. Du, L. Liu, J. Leng, Shape memory polymers and their composites in aerospace applications: a review, Smart Struct. Syst. 23 (2) (2014), 023001. [274] J. Li, Q. Duan, E. Zhang, J. Wang, Applications of shape memory polymers in kinetic buildings. Special issue on smart buildings and construction materials, Ann. Mater. Sci. Eng. 2018 (2018) 13. [275] T. Kaneko, T. Miyazaki, K. Yamaoka, Y. Katayama, A. Matsuda, J. Gong, et al., Shape memory gels with multi-stimuli-responses, in: Proceedings of the SPIE Conference 1999 on Smart Structures and Materials þ Non-destructive Evaluation and Health Monitoring, March 1e5, 1999. Newport Beach, CA-USA. [276] S. Chaterji, K. Kwon, K. Park, Smart polymeric gels: redefining the limits of biomedical devices, Prog. Polym. Sci. 32 (8e9) (2007) 1083e1122. [277] C. Yuan, J. Guo, F. Yan, Shape memory poly(ionic liquid) gels controlled by host-guest interaction with B-cyclodextrin, Polymer 55 (16) (2014) 3431e3435. [278] M.H. Kabir, J. Gong, Y. Watanabe, F.H. Makino, The applications of shape memory gel as a smart material. JPS conf proc, in: Proceedings of the 12th Asia Pacific Physics Conference; 2013 Jul 14e19, vol. 1, Makuhari Messe Chiba, Japan, 2014, p. 012048. [279] J. Shang, X. Le, J. Zhang, T. Chen, P. Theato, Trends in polymeric shape memory hydrogels and hydrogel actuators, Polym. Chem. 10 (2019) 1036e1055. [280] A. Lendlein, S. Kelch, Shape-memory polymers, Angew. Chem. Int. Ed. 41 (12) (2002) 2034e2057.

CHAPTER 2

Latest attainments Andrea Brotzu1, Vittorio Di Cocco2, Francesco Iacoviello2, Stefano Natali1, Cristian Vendittozzi3 1

Department of Chemical Engineering Materials EnvironmenteSapienza, Rome University, Rome, Italy; 2Department of Civil and Mechanical Engineering, Universit
a di Cassino e del Lazio Meridionale, Cassino, Italy; 3Faculdade do Gama, Universidade de Brasìlia, Brasìlia, DF, Brazil

2.1 Introduction In recent decades, because of their distinctive characteristics, smart materials have been attracting increasing attention in the scientific and industrial fields. Among them, shape memory alloys (SMAs) are gaining importance in various fields: aerospace, biomedical, civil, automotive, and aeronautics. SMAs are characterized by two distinctive properties, the shape memory effect (SME) and the superelastic (or pseudoelastic) effect (SE). The SME is the ability to “memorize” a certain initial geometric shape, which, as a result of deformations, can then be restored (recalled) by heating. The SE is the possibility of undergoing great deformations (about 5e10%) and being able to recover them completely during the unloading phase without showing plastic phenomena. Both properties result because these materials can undergo a particular reversible crystallographic transformation called thermoelastic martensitic phase transition. This is usually induced by a temperature variation (SME) or by a variation of the stress state acting in the material (SE). Depending on the temperature or the exerted stress state, the material can show a martensitic crystallographic phase or an austenitic phase, with which different mechanical properties are associated. To provide a clearer understanding of this aspect, it is useful to consider the graphs shown in Fig. 2.1. Figure 2.1(a) shows a stressestrain diagram relative to a material that uses the SME. In its initial configuration, the product is in martensitic phase (T < Ms) and is subjected to a constant temperature loadingeunloading cycle. At the end of the loading cycle, the material will show residual deformations. At this stage, if the material has not been excessively deformed (ε < 5e10%), the SME activates, restoring the product’s initial configuration, just heating the material to a temperature above Af. Instead, for the SE, the material must work at a temperature between As and the temperature above which martensite formation is no longer possible, causing martensite formation directly from austenite owing to the applied loads that modify the material’s crystalline structure. In this temperature range, the martensite generated by applying stress, which is unstable, turns back into austenite as soon as the external load is removed. The diagram of Fig. 2.1(b) shows that the material has stressestrain curves concerning the SE; the upper plateau corresponds to the formation of martensite

Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00002-9

© 2021 Elsevier Ltd. All rights reserved.

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Figure 2.1 Different behaviors of shape memory alloys on stressestrain curve: (a) shape memory effect and (b) superelastic effect (at constant temperature).

under stress, whereas the lower plateau represents the reversion due to load removal. The amplitude of the hysteresis cycle depends on the composition of the material (it can be enlarged or reduced by introducing a third phase) and the load application speed (the higher the load application speed, the less symmetrical the curve will be). These dissipative effects are usually used to design damping systems. SMA properties have been exploited and applied in diverse fields including biomedical, automotive, robotics, and aerospace industries. In industrial applications, depending on what the target is, only the SE or SME is used. The chapter provides a bibliographic review summarizing recent advances in SMA applications, focusing attention on results of the past 5 years on improving fundamental SMA properties that should lead to their wider diffusion, and on recent innovations in the field of modeling. Section 2.3 provides an overview of possible applications of SMAs (joint, dampers, and actuators). Section 2.4 presents a brief survey of the latest proposals regarding production technologies, with an emphasis on powder metallurgy and additive manufacturing. Section 2.5 illustrates research trends aimed at deepening knowledge regarding the functioning principles of SMAs and enhancing their performance, to increase their use. A further limitation to the wider application of SMA devices is the lack of knowledge about their long-term operational behavior (deterioration and fatigue). The application of high stresses and high amplitude deformations can generate alterations in the crystalline structure; therefore, when a high number of cycles are required, it is necessary to operate with reduced levels of strain and stress. Although the working principle is easily explained, some characteristics of SMAs are still not sufficiently well known, especially those related to aging. Section 2.6 is dedicated to the latest news on SMA behavior modeling.

Latest attainments

2.2 List of symbols and acronyms Af Austenite finish As Austenite start EIS Electrochemical impedance spectroscopy HTSMA High-temperature shape memory alloy KIA Stress intensity factor related to austenite phase KIM Stress intensity factor related to martensite phase Mf Martensite finish Ms Martensite start Nitinol NickeleTitanium/Naval Ordnance Laboratory OCP Open circuit potential SIF Stress intensity factor SMA Shape memory alloy SME Shape memory effect TR Room temperature

2.3 Application and production technologies Although it can boast a good number of applications of reasonable commercial relevance, the technology of SMAs is still in a phase of rapid development, with a field of possible commercial applications destined to expand further in coming years. The main fields of application of SMAs are joints, fasteners, and connectors; actuators; smart materials; damping systems; biomedical devices; microelectromechanical devices; and miscellaneous.

2.3.1 Joint and fastener applications In SMA joint applications, the ability of shape memory materials to generate a force of considerable intensity is exploited when the return to high-temperature shape is mechanically prevented. The joint is made with a size so that when it is in the high-temperature shape, it exhibits adequate interference with the parts to be fixed. To allow assembly, the joint is brought into the martensitic phase by refrigeration and appropriately expanded. Once assembled, the joint is left to return to TR. In this way, phase transformation takes place and the joint tends to return to its shape at high temperature. Because this recovery is prevented by interference with the parts to be fixed, a stress is generated that carries out the fixing action. The main difference of shape memory joints compared with conventional ones is that the active clamping action persists even after installation. Obviously, the transition temperature of the used alloy must be lower than the minimum intended use, to prevent unwanted loosening of the joint at low temperature. This type of joint has been employed for a long time in the aerospace industry to couple pipes of hydraulic systems, [1]. Their diffusion is mainly limited by the high cost and the limits of the allowed thermal range (typically
20  C to þ200  C). There are numerous solutions in the literature that propose to replace traditional fastening systems with SMA devices for the aerospace sector. In particular, for space

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applications there are also several proposals for release systems, which belong to this category. Glucksberg et al. [2] show excellent results in attempting to replace traditional pyrotechnic bolts with a nonexplosive actuator. Among the many advantages that an SMA solution could provide instead of a pyrotechnic solution, including reducing high shock and vibration-induced levels, avoiding possible contamination of sensible instruments by dust released during explosion, and avoiding risks associated with the onboard storage of explosives, there is a key feature: the chance to carry out a complete functioning check of the release bolt that will actually be used on the flight model, which is obviously impossible with an explosive system.

2.3.2 Damping systems The damping capabilities demonstrated by SMAs, combined with good mechanical strength characteristics, have stimulated interest in applications aimed at noise reduction, vibration damping, and increased resistance to impulsive loads. The application areas of SMA damping systems range from sports equipment to aerospace systems components, and from ballistics to seismic protection. The latter sector is proving to be active by research projects dedicated to the study, experimentation, and validation of seismic protection devices based on the use of SMAs. In particular, the potential for exploiting SE for the absorption of seismic energy in structures has been explored in depth [3e6]. A further promising field of application of SMAs is that of ballistic protection. For this application, the ability of SMAs to dampen the shock wave quickly and provide good mechanical strength is advantageous [7]. Farsani and Khazaie [8] embedded SMA wires into basalt fiber metal laminates to evaluate their response to high-velocity impact in terms of the amount of absorbed energy, checking the limit-volume fraction of wires that improve energy absorption without affecting the mechanical resistance of the laminate. For Katsiropoulos et al. [9], who designed hybrid materials (in their case, aramideepoxy laminates) that combined the stiffness and strength of reinforcing fibers with the damping capability of SMA wires, this capability can be tuned by appropriately selecting the volume fraction of SMA wires (defined as a critical design parameter) and pretension level. As in the case of ballistic impact absorption systems, an interesting application involving combining SMAs (in particular, thin NiTI-based alloy wires) and composite materials that is not yet found in commercial applications is reported by Cohades and Michaud [10], who provide an in-depth review of the use of SMAs integrated into high-performance, fiber-reinforced polymer composite materials, providing a variety of potential applications such as a passive (as well as active) damping system to the structure. If the main interest is in allowing the shape morphing of active composites, the presence of thin SMA wires can improve the mechanical performance of laminates because of their high strength. For example, SMAs can reduce fiber breakage and

Latest attainments

puncture after an impact damage event. The characteristics of SMAs can turn laminates into intelligent materials, making a fundamental contribution to self-healing processes when they are properly positioned to close cracks after a damage event. The causes that currently limit commercial applications are mainly due to increases in cost and weight and difficulties in adapting the composite manufacturing process (the inertia in updating technological processes is a common difficulty that slows the implementation of new technologies).

2.3.3 Actuators During the shape recovery phase, which takes place at a high temperature, an SMA device is able to produce mechanical work. If an external action is applied to it that opposes shape recovery, the recovery force and the shape variation have the same direction and therefore result in the generation of (positive) mechanical work. The main interest of designers in various sectors (aerospace, automotive, biomedical, etc.) is the implementation of devices that fully exploit the properties of SMAs, mainly considering actuators able to replace traditional servomechanical systems. These devices made in SMA (SM-actuators) are supposed to simplify the products, improve their performance, and reduce mechanical complexity as well as with size and weight. In general, the objective is to improve reliability and therefore cut costs. The SM-actuator should be simpler because the devices can be designed to develop the required action directly, without requiring transmission and connection elements. Therefore, they do not require the use of auxiliary components based on friction phenomena (e.g., gears) and therefore do not generate dust and eliminate the risk for generating sparks (which is ideal for use in areas with a high explosive hazard). They also allow vibration-free and quiet operation. Among the main advantages recognized for SM-actuators is the high power-to-weight ratio, but this should be reserved exclusively for small size and weight (10 MPa), which hinders field-induced twin-boundary motion. On the other hand, the region of compositions of Ni-Mn-Ga Heusler alloys for which the coupled magnetostructural transition is observed, accompanied by a giant magnetocaloric effect [64], belongs to alloys that are extremely rich in Ni.

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Figure 4.15 Isothermal phase diagram. (Adapted from Entel et al. [64]).

4.6 Conclusion SMAs are metallic Ni-Tiebased alloys characterized by unique thermomechanical properties that are responsible for their ability to recover preimposed shapes and huge deformations completely upon a temperature change or a removal of load. If the material is deformed at a low temperature, its original shape can be recovered by heating it above a characteristic temperature so that the material exploits the SME. On the other hand, if the material is deformed at a high temperature, its original shape can be recovered simply by removing the applied load, because of the SE or pseudoelasticity characterizing the SMA at that temperature. These macroscopic properties of SMA are related to changes in the material microstructure. In particular, they depend on the presence in the alloy of two stable crystalline phases: a high-temperature and stronger phase, called austenite, and a low-temperature and weaker phase, called martensite. Together with heat-activatable alloys, there are also MSM alloys for which different orientations of the martensitic microstructure with an applied magnetic field are responsible for the macroscopic deformation. Compared with ordinary (temperature-driven) SMAs, magnetic control offers a faster response and the maximum deformation obtained from MSM alloys is larger than in ordinary magnetostrictive materials. The material can be actuated in a few milliseconds, allowing dynamic applications (100e1000 Hz). Because of these exceptional properties, SMAs are used as force and displacement actuators in many applications such as motors and lightweight actuators as well as biomedical devices and robotic parts.

Phenomenology of shape memory alloys

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[48] C. Greiner, S.M. Oppenheimer, D.C. Dunand, High strength, low stiffness, porous NiTi with superelastic properties, Acta Biomater. 1 (2005) 705e716. [49] Y.P. Zhang, B. Yuan, M.Q. Zeng, C.Y. Chung, X.P. Zhang, High porosity and large pore size shape memory alloys fabricated by using pore-forming agent (NH4HCO3) and capsule-free hot isostatic pressing, J. Mater. Process. Technol 192e193 (2007) 439e442. [50] S. Shabalovskaya, J. Ryh€anen, L. Yahia, Bioperformance of nitinol: surface tendencies, Mater. Sci. Forum 394e395 (2002) 131e138. [51] A. Bansiddhi, T.D. Sargeant, S.I. Stupp, D.C. Dunand, Porous NiTi for bone implants: a review, Acta Biomater. 4 (2008) 773e782. [52] B. Yuan, M. Zhu, C.Y. Chung, Biomedical porous shape memory alloys for hard-tissue replacement materials, Materials 11 (2018) 1716. [53] K. Ullakko, J.K. Huang, C. Kantner, R.C. O’Handley, V.V. Kokorin, Large magnetic-field-induced strains in Ni2 MnGa single crystals, Appl. Phys. Lett. 69 (1996) 1966e1968. [54] B. Kiefer, H.E. Karaca, D.C. Lagoudas, I. Karaman, Characterization and modeling of the magnetic field-induced strain and work output in magnetic shape memory alloys, J. Magn. Magn. Mater. 312 (2007) 164e175. [55] A. Sozinov, N. Lanska, A. Soroka, W. Zou, 12% magnetic field-induced strain in Ni-Mn-Ga-based non-modulated martensite, Appl. Phys. Lett. 102 (2013) 021902. [56] H. Luo, Q. Li, K. Sun, S. Liu, Z. Liang, Magnetic properties and site preference of Ru in Heusler alloys Fe2V1-xRuxSi (x¼0.25, 0.5, 0.75, 1), J. Magn. Magn. Mater. 496 (2020) 165908. [57] G. Durak Y€ uz€ uak, E. Y€ uz€ uak, A. H€ utten, Tailoring the structural and magnetic properties of epitaxially growth NieMneGa magnetic films by Fe addition, Philos. Mag. A 99 (2019) 2971e2983. [58] B. Emre, S. Yuce, N.M. Bruno, I. Karaman, Martensitic transformation and magnetocaloric properties of NiCoMnSn magnetic shape memory alloys, Intermetallics 106 (2019) 65e70. [59] X. Xu, et al., Magnetic-field-induced transition for reentrant martensitic transformation in Co-CrGa-Si shape memory alloys, J. Magn. Magn Mater. 466 (2018) 273e276. [60] E. Faran, D. Shilo, The kinetic relation for twin wall motion in NiMnGadpart 2, J. Mech. Phys. Solids 61 (2013) 726e741. [61] E. Faran, D. Shilo, A discrete twin-boundary approach for simulating the magneto-mechanical response of NieMneGa, Smart Mater. Struct. 25 (2016) 095020. [62] B. Kiefer, D.C. Lagoudas, Modeling the coupled strain and magnetization response of magnetic shape memory alloys under magnetomechanical loading, J. Intell. Mater. Syst. Struct. 20 (2009) 143e170. [63] A. Planes, L. Ma~ nosa, M. Acet, Magnetocaloric effect and its relation to shape-memory properties in ferromagnetic Heusler alloys, J. Phys. Condens. Matter 21 (2009) 233201. [64] L. Ma~ nosa, A. Planes, Materials with giant mechanocaloric effects: cooling by strength, Adv. Mater. 29 (2017) 1603607. [65] J.M. Gallardo Fuentes, P. G€ umpel, Strittmatter, J. Phase change behavior of nitinol shape memory alloys, Adv. Eng. Mater. 4 (2002) 437e452. [66] D. Lagoudas, et al., Shape memory alloys, Part II: modeling of polycrystals, Mech. Mater. 38 (2006) 430e462.

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CHAPTER 5

Experimental characterization of shape memory alloys Maria Rosaria Ricciardi1, Vincenza Antonucci2 1

Institute for Polymer, Composite and Biomedical Materials, National Research Council, Portici, Naples, Italy; 2CNR-IPCB Institute for Polymer, Composite, and Biomedical Materials, National Research Council, Portici, Italy

5.1 Introduction The techniques used to characterize the properties of alloys are reported and discussed for a better understanding of their material behavior as well as for the appropriate exploitation of their material properties for specific engineering applications and practical implementation. A proper determination of transformation temperatures is the preliminary characterization test that has to be performed on the alloys by calorimetric analysis. On the basis of calorimetric results, further investigations, such as thermomechanical tests, can be set up [1e9]. These tests are fundamentally important for the use of the alloys in structural or actuation devices, because for any kind of application, quantitative knowledge of the material properties is required. To identify the internal structure of the alloys and determine the values of the transformation temperatures, another useful experimental test is the electrical resistance (ER) measurement [10]. This technique is unconventional and has been investigated only for a few years. It is useful to assess the dependence of ER on the temperature and mechanical loads and tracking down the phase composition and microstructure changes induced by external thermomechanical conditions. In the next sections, a complete characterization of an SMA will be illustrated. The tests are described in the following and their results are analyzed by an engineering approach. The following experiments can be used as a sort of protocol for determining the basic properties of a new SMA; however, further characterization experiments can be carried out on the material to investigate specific properties. Several kinds of characterization techniques are used for the determination of the internal structure of shape memory alloys (SMAs), such as neutron diffraction technique and microscopic observations. These tests are used to detect the crystallographic changes upon phase transformation and can be carried out during heating or during the appliance of a magnetic field, to monitor the structural changes in situ. A description of these characterization techniques is not the aim of this chapter. For a more comprehensive illustration of all of the characterization methods used to identify SMA structures during phase transformations, neutron diffraction and other morphology techniques have been also described. Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00005-4

© 2021 Elsevier Ltd. All rights reserved.

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5.2 List of symbols Symbol

Parameter

sMs

Initiation stress for forward transformation into martensite Austenitic start temperature at zero stress Austenitic finish temperature at zero stress Martensitic start temperature at zero stress Martensitic finish temperature at zero stress Finish stress for forward transformation into martensite Start stress for transformation into austenite Finish stress for transformation into austenite Slope of austenite transformation lines in stressetemperature plane Slope of martensite transformation lines in stress etemperature plane Start stress for detwinning of martensite Finish stress for detwinning of martensite Maximum temperature at which austenite can be transformed into martensite by stress application

As Af Ms Mf sMf sAs sAf CA CM ss sf Md

5.3 Calorimetric investigations The current level of knowledge about the phenomenological behavior of the alloys has been made possible through an extensive investigation throughout the years as well as by the combination of experimental results and mechanical modeling. The phenomena observed in SMAs are related to microstructural changes occurring in the alloys upon varying the temperatures and loads. First studies on the microstructural properties of Nitinol as well as on the physical properties of this material were reported by the Naval Ordinance Laboratory [1]. From Nitinol, these types of investigations were

Experimental characterization of shape memory alloys

extended to other types of alloys, aiming to understand deeply the pseudoelastic behavior and shape memory effect (SME) mechanisms [2e5]. Differential scanning calorimetry (DSC) testing is the most popular thermal analysis technique used to evaluate phase transformation temperatures and the latent heat related to them. It measures the difference in the amount of heat required to increase the temperature of a sample and reference as a function of temperature. Hence, it is also widely used to measure the initial and finish temperatures of the martensitic transformations. In Fig. 5.1 a typical DSC thermogram of an SMA is shown, with two transformation peaks. The power required to hold a constant heating rate for the alloy is represented on the y axis, whereas on the x axis, the temperature of the chamber is reported. The endothermic peak is associated with the transformation from twinned martensite to austenite and stands for additional heat to be supplied to maintain the constant heating rate. This transformation is identified by two temperatures: the austenite start temperature (As) and the austenite finish temperature (Af). The exothermic peak represents the transformation between austenite and martensite and is identified by starting and finish temperatures as well (Ms and Mf). All transformation temperatures are measured by means of the tangent method. Together with the martensite and austenite phases, an intermediate phase in the cooling curve can be detected, called R-phase (Fig. 5.2). In this case, the SMA goes through a multistage phase transformation with overlapping temperature ranges. This phenomenon makes it difficult to determine when the first transition actually ends or when the second one actually starts. Thus, Michael et al. [11] proposed a novel methodology with an adapting function allowing the deconvolution of multiple phase transformation peaks.

Qεz–f

•

Qεz

Heat flow

Baseline Mf

Tf

Ms As

Tf

Af Qεz–f

T

Figure 5.1 Schematic representation of a differential scanning calorimetry thermogram for a shape memory alloy material.

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A→R R→MR

Heat flow

144

R,M†→A 0

10

20

30 40 50 60 Temperature (°C)

70

80

90

Figure 5.2 Differential scanning calorimetry thermogram for a shape memory alloy element containing R-phase.

Transformation temperatures can be strongly affected by the mechanical history and mechanical energy stored by the sample, which could be responsible for precipitations, dislocations, and detwinning [6,7]. In these cases, the transformation peaks can be shifted in the thermogram (Fig. 5.3). Specifically, if the sample is strained and twinned martensite is formed, the temperatures characterizing the transformation from austenite to martensite are shifted to higher values, owing to the presence of twinned and detwinned martensite in the specimen. The shift of the DSC peak is more pronounced as the percentage of twinned martensite increases in the sample. This means that reverse phase transition temperatures increase with increasing strain. Mechanical deformation at low temperature induces stabilization of the martensite phase, demonstrated by the increase in the reverse phase transition temperatures. Martensite stabilization caused by cold deformation was observed by Liu and Favier [6]. Together with a variation in the transformation temperatures upon load cycling at room temperature, DSC experiments are also able to detect the influence that thermal cycles have on the transformation temperatures [12], in particular the effect of temperature on the martensite stabilization. In Fig. 5.4, the thermograms of NiTi alloys subjected to a thermal cycle under a constant load are reported [7]. During heating, the detwinned martensite initially obtained with the application of the load transforms into austenite and the wire recovers the initial deformation, working against the applied load. During cooling under constant load, austenite transforms back into detwinned martensite because of the temperature change. Upon load removal, therefore, the wire is in a detwinned martensite state and the DSC test measures the zero-stress phase transition temperatures of martensite formed under constant loading.

Heat flow

Experimental characterization of shape memory alloys

No strain 2% strain 4% strain 6% strain 8% strain 10% strain 40

60

50

70 80 Temperature (°C)

90

100

Figure 5.3 Differential scanning calorimetry thermogram of samples strained at room temperature at different strain levels [7].

Heat flow

Heating and cooling at constant strain constitute a complex loading history for the SMA wires. The two driving forces of the martensitic phase transition, stress and temperature, vary contemporarily during constrained recovery, which makes it difficult to

Room T Tmax = 90 °C Tmax = 110 °C Tmax = 130 °C 40

50

80 60 70 Temperature (°C)

90

100

Figure 5.4 Differential scanning calorimetry thermogram of samples cycled at a constant stress at different temperatures [7].

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separate their effects. On the other hand, an accurate interpretation of the constrained recovery behavior of SMAs in terms of phase transition temperatures is crucial in applications dealing with SMA hybrid composites [13,14]. Further results on the calorimetric characterization of constrained recovery SMAs can be found in the work by Li et al. [8], who performed DSC and recovery stress measurements on Nitinol wires that had been mechanically strained and cycled in constrained recovery to different maximum temperatures. It is possible to affirm that zero stress reverse phase transition temperatures for untrained SMAs are not constant, but depend significantly on the loading history of the material. In particular: - the reverse phase transition temperatures increase gradually with increasing strain; - the reverse phase transition temperatures increase gradually with increasing applied stress, and thermal cycling at a constant applied stress does not seem to affect the observed shift significantly [7,13].

5.4 Thermomechanical characterization To characterize the mechanical properties of an SMA and quantify the complex behavior of SMA materials, various loading paths are imposed while phenomena associated with the phase transformation are recorded, such as the SME and pseudoelasticity. This section focuses on applying and measuring three thermomechanical fields in particular: stress, strain, and temperature. A standard thermomechanical test of SMA requires histories of stress and temperature or strain and temperature to be prescribed while the evolution of the third quantity needs to be measured. To begin the discussion of thermomechanical SMA material characterization, it is appropriate to provide examples of actual material specimens designed for testing. In general, because SMAs are usually adopted in the form of wires able to provide high forces and displacements, the most common coupons for mechanical testing are wires equipped with special gripping systems. For the application of two-dimensional (2D) and 3D SMA materials, the dog bone configuration is used for standard mechanical tests [16].

5.4.1 Thermomechanical tests and parameters SMAs show complex behavior with strong nonlinear thermomechanical coupling that also depends on the history load. Thus, experimental characterization is complex and requires methods that are uncommon for conventional materials. Hence, SMAs require at least three types of material properties to describe three types of behaviors: the thermoelastic properties of austenite and martensite, the critical stress

Experimental characterization of shape memory alloys

and temperature states associated with the phase diagram, and the transformation strain evolution properties. The first parameters useful for structural applications are necessary to describe the material response when transformation or reorientation is not occurring. The second set of parameters helps evaluate the limits of the process of transformation between phases, depending on the current thermomechanical state (stress and temperature) and loading history of the material. Finally, the third set of parameters provides a relation between the current state of material transformation and the generation of transformation strain exhibited by the material. In the following discussion, experimental pseudoelastic behavior is analyzed in detail. It can be exploited by setting the temperature greater than Af, changing the stress, and evaluating the deformations (Fig. 5.5). In the first stage of loading, a linear response of the specimen is observed until a stress level (sMs) is attained at which the material starts to transform into martensite and a plateau is formed until a second level of stress (sMf). Then, the material has a linear response again with a different slope compared with the first one. When the specimen is unloaded, a similar response is observed: linear behavior, a plateau with the same strain length between stress levels sAs and sAf, and a linear response with the same slope as the initial loading.

800

Stress (MPa)

600

1

VMf

VMs

400

200

VAs T > Af

VAf

2

0 0

2

4 Strain (%)

6

8

Figure 5.5 Experimental example of constant temperature phenomenological transformation behavior in NiTi at a single temperature T higher than Af, C [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

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From these mechanical tests, it is important to consider carefully the differences characterizing the two different phases, such as the austenite at low stress/deformations the martensite at high stress/deformations. Together with an understanding of the elastic behavior of different phases, a second fundamental aspect to take into account is the strong relationship between stress and temperature. Each temperature determines a particular stress level initiating forward and reverse transformation during loading and unloading. Thus, one can figure out an experiment in which a fixed level of force is applied, but at different temperature, as described in DSC characterization. For this type of experiment, one has to remember the increase in stress level upon an increase in temperature. Fig. 5.6 shows an increase in the pseudoelastic plateau with the temperature, where T1 < T2 < T3. It is interesting to quantify the deformation associated with the temperature variation along the plateau region. In this case, it is useful to report the strain as a function of temperature during experiments at constant stress. Figs. 5.7 and 5.8 show the deformation versus temperature at zero stress and at varying stress levels. To characterize SMA materials fully, the required fundamental characteristics are the thermomechanical properties of austenite and martensite, the critical stress at a certain temperature, and the strain evolution as a function of temperature. The first properties can be evaluated by a stressestrain curve resulting from a loadingeunloading tensile test at T > Af, in which the Young’s modulus of austenite can be detected from the slope of the linear part of the curve at low strain; the Young’s 800

T 600 Stress (MPa)

148

400

200

T3 T2 T1

0 0

2

4

6

8

Strain (%)

Figure 5.6 Experimental example of constant temperature phenomenological transformation behavior in NiTi at multiple temperatures [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

Experimental characterization of shape memory alloys

1.6 0 = const AVs

Strain (%)

1.2

MVf

0.8

1 2

0.4 AVf

MVs 0.0 –30

0

30

60

90

Temperature (°C)

Figure 5.7 Experimental example of constant stress phenomenological transformation behavior in NiTi [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

modulus of martensite can be evaluated as the slope of the linear part of the curve during the unloading path, such as after the martensite is fully formed and sMf is overcome. From some applications, it is also necessary to evaluate the plastic yield and failure behavior of the material, as well as the thermal expansion coefficient. 1.6

300 MPa 200 MPa 120 MPa

Strain (%)

1.2

0.8

V 0.4

0.0 –30

0

30

60

90

Temperature (°C)

Figure 5.8 Experimental example of constant stress phenomenological transformation behavior in NiTi at multiple stresses [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10. 1007/978-0-387-47685-8_2).

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An extensive thermomechanical characterization can provide all of the necessary information to build the phase diagram of critical stresses and temperatures (Fig. 5.9), which involves determining the boundary limiting the transformation regions in a space where stress is a function of temperature [13,14]. This diagram is able to indicate where a specific transformation begins and ends and in which phase the material is at a specific temperature and at a given stress level. A set of parameters are needed to build the diagram fully. The basic information describing the material properties are the zero-stress transformation temperatures, such as the temperatures corresponding to the intersection between the boundary lines and the x-axis. These temperatures are Ms, Mf, As, and Af. The slopes of the boundary surfaces, known as stress influence coefficients, are another property necessary for the diagram. These slopes of transformation regions are CA identifying the transformation into austenite and CM identifying the transformation into martensite. The other parameters are the start and finish stresses limiting the phenomenon of the detwinning of martensite, such as ss and sf; these values can be temperature-dependent as well.

5.5 Complete experimental characterization of thermal and mechanical properties To achieve a complete characterization of the material according to the parameters and tests described earlier and to be useful for practical applications of the alloys, some experimental tests are essential [16]. These basic experiments also allow for an understanding of a specific material’s behavior after training or one induced by mechanical instability (Fig. 5.10).

Stress (V)

150

Detwinned martensite A to Md Vf Md to A

Mt to Md Vs Twinned martensite

Austenite CM

Mf

Ms

As

CA Af

Temperature (T)

Figure 5.9 Schematic representation of the phase diagram, with possible material properties defined [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

Experimental characterization of shape memory alloys

1 T1: Austenite

T

2

B

T3: Pseudoelastic

Stress

A

D

C T2: Martensite 3

Strain

Figure 5.10 Stressestrain curves of a shape memory alloy element at different temperatures.

The essential experiments mentioned earlier can be summarized in the following types: 1. Determination of transformation temperatures at zero stress. Zero-stress transformation temperatures are fundamental for the following testing process, because without this knowledge, it is impossible to configure further thermomechanical experiments. 2. Loadingeunloading tensile tests at T < Mf to determine the mechanical properties of the martensitic phase as well as of the limits of strain involved in the SME. This test is important for evaluating the elastic properties of the martensitic phase as well as for evaluating the recoverable strains under SME. 3. Loadingeunloading tensile tests at T > Af to determine the austenitic properties and pseudoelastic behavior. This test is important for evaluating the elastic properties of the austenitic phase, as well as for evaluating the pseudoelastic properties in the austenitic phase. 4. Evaluation of the properties of the SME at a nonzero stress level. This test provide information about the capability of the material to provide work output by generating and recovering transformations train under nonzero stresses. 5. Determination of cycling loading and the effects of fatigue cycling on the mechanical and recovery properties of the alloy. This test is useful for analyzing the stability of the material response under several cycles of loading and unloading. This type of cycling could generate the material training effect, which could modify the mechanical response of the material to further loading repetitions, so it is useful for applications where fatigue reliability is required.

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5.5.1 Test 1: differential scanning calorimetry As described in the previous sections, without knowledge of the stress-free transformation temperatures, it is difficult to know what phase is present in the alloy at any given stress and temperature state. To assess these temperatures, a DSC is used, as illustrated in the section on calorimetric analysis. In this test, a heatingecoolingeheating scan is carried out on the sample to cancel the mechanical history of the material resulting from production (wire drawing and quenching). The austenitic transformation temperatures measured during the first heating cycle are higher than those measured after the second cycle. Upon a third heating cycle, transition temperatures are stabilized and those values can be considered the actual transition temperatures of that material (Fig. 5.11).

5.5.2 Test 2: T < Mf Consider an NiTi specimen in the form of a wire, which is the most common shape for SMA materials used for actuation purposes. Before testing, heating to a temperature greater than Af followed by cooling below Mf is imposed on the wire to remove eventual detwinned martensite. Then, the wire is tested in a tensile machine and a temperature T < Mf is set during the experiment. This temperature is known from previous DSC measurements. 0.2

46.30 °C 0.1

Cooling

4.89 °C R-phase

Heat flow (W/g)

152

0.0

Transition to martensite

49.77 °C 14.09J/g

51.85 °C 11.36J/g

59.04 °C 11.48J/g

–0.1 Heating

Transition from martensite to austenite

–0.2 58.45 °C

64.39 °C 64.43 °C –0.3 –50

0

50

100 Temperature (°C)

150

200

250

Figure 5.11 Differential scanning calorimetry heatingecoolingeheating thermogram of an NiTi alloy containing R-phase.

Experimental characterization of shape memory alloys

During the tensile test, the wire is subjected to a loading path, in which a varying strain is applied to the sample and the corresponding stress is measured. The final information will be a stress versus strain curve, in which the elastic properties of the twinned martensite are determined. Moreover, with this test, it is possible to determine whether the material exhibits crucial shape memory behavior by verifying the capability of the material to exhibit a stress-free SME. Moreover, the yield and ultimate fracture properties can be assessed. Finally, to recover the strain, the wire is heated again at a temperature greater than Af (Fig. 5.12).

5.5.3 Test 3: T > Af The wire is tested in a tensile machine and a temperature T > Af is set during the experiment. This temperature is known from the previous DSC measurements. To examine pseudoelastic behavior, the wire is subjected to a loadingeunloading cycle [15]. In this way, the elastic modulus of the martensite and austenite can be evaluated. Furthermore, to evaluate the austenitic yield, the transformation from austenite to martensite has to be avoided and the wire has to be held at a temperature above Af. Finally, this test evaluates Md, the maximum temperature for the transition from austenite to martensite without plastic deformation (Fig. 5.13).

5.5.4 Test 4: fixed stress value Fundamental information for using SMA as an actuator consists of evaluating whether the material is able to produce displacement by applying a load. The SMA is analyzed in the austenitic phase at a temperature above Af and loaded at a constant stress that is kept while 700 T < Mf 600

Stress (MPa)

500 EA

400 300

EM

200 100 0 0

2

4

6

Strain (%)

Figure 5.12 Stressestrain curve of an NiTi sample at T < Mf [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

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600 500 Stress (MPa)

400 300 200 T > Af 100 0 0

1

2

3

4 5 Strain (%)

6

7

8

9

Figure 5.13 Stressestrain curve of an NiTi sample at T > Af.

the temperature is decreased, to induce the transformation to martensite. Then, the sample is heated again to attain the reverse transformation. In this way, it is possible to evaluate the resulting deformation that the material can provide. Fig. 5.14 reports an example of a constant stress mechanical test upon heating on an NiTi wire element.

5.5.5 Test 5: cycling To determine the effects of cyclic loading on the thermomechanical response of an SMA, the material is subjected to multiple transformation cycles at a constant temperature or constant stress. If the material has to be used as actuator, the sample is tested by applying several thermal cycles at a constant load (Figs. 5.15 and 5.16) and hence, performing training. 6 α

As

5 α

4 Strain (%)

154

Mf

1 3 2 2 α

α Ms

1

0 –30

–10

30 50 10 Temperature (°C)

Af

70

Figure 5.14 Constant stress test: strain is reported as a function of applied temperature [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

Experimental characterization of shape memory alloys

9 V const

8 7

Strain (%)

6 Cycle 80 5 4 Cycle 1

3 2 1 0 –20

–10

0

10

20

30

40

50

60

70

Temperature (°C)

Figure 5.15 Thermal cycling at constant stress [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

5.5.5.1 Effects of training Training can significantly affect DSC results because during extensive microscopic dislocations and local stresses are induced into the SMA with a subsequent change of local transformation temperatures. In this case, the DSC is unable to identify the global transition temperatures at zero stress and other techniques have to be considered. 6 200 MPa 5

150 MPa 100 MPa 50 MPa

Strain (%)

4

2.5 MPa

3

2

V

1

0 –50

–25

0

25 Temperature (°C)

50

75

100

Figure 5.16 Thermal cycling at different stress levels [16]. (From D. Hartl, D. Lagoudas, Thermomechanical Characterization of Shape Memory Alloy Materials, in: Shape Memory Alloys, Springer, Boston, MA, 2008. https://doi.org/10.1007/978-0-387-47685-8_2).

155

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Shape Memory Alloy Engineering

A useful tool that can be employed for this purpose consists of measuring the ER of the alloy while it is subjected to a thermomechanical test. A more detailed description of this technique will be reported in the next sections. However, after a set of training experiments, it is possible to make a phase diagram and establish a proper relation between applied stress and maximum transformation strain for the specific material. 5.5.5.2 Effects of mechanical and thermal rates on mechanical response SMA materials are characterized by strong coupled thermomechanical behavior that is also affected by the mechanical and thermal rate during testing. For example, during a pseudoelastic test under isothermal conditions, the transformation from austenite into martensite generates heat because it is an exothermic transition. This heat is dissipated if the loading is carried out slowly. On the other hand, if the loading is performed too quickly, the heat is not dissipated and the SMA material is subjected to an increase in temperature that adversely affects the experimental results. Thus, reasonable strain rates should be adopted in strain-driven tests. Furthermore, because in some cases the tests are performed under force control and stress-induced strains are evaluated, it is also important to highlight strain measurement systems. In this case, the deformations induced by the martensite reorientation originate in such positions and then spread along the length of the sample [9]. Thus, to measure macroscopic strain, it is preferable to use extensometers rather than strain gauges, with larger gauge lengths and the ability to obtain information over the entire length of a specimen [17e21]. Another significant issue is the heating or cooling rate in nonisothermal experiments that can be performed in different ways, depending on the geometry and size of the SMA material. For example, in the case of wires, it is carried out electrically or by using furnaces, whereas cooling is done with chilled water [17] or a laboratory coolant [18]. To avoid undesirable local heating and transition phenomena that can affect the test results, in this kind of experiment, it is important to adopt slow heating or cooling rates and ensure uniform heating or cooling. The cyclic stressestrain behavior [22] of an SMS is a requirement for designing a superelastic component such as bolts, nuts, and washers. In a superelastic regime, the behavior depends on the composition of the alloy. As example, the copper SMA (CuAl-Ni-Mn-Ti) has a bilinear evolution during loading that is almost linear during unloading, whereas most SMAs are characterized by well-defined plateau stress in both the loading and unloading paths.

Experimental characterization of shape memory alloys

5.6 Electrical resistance measurements The effects of heat treatment and thermal cycling on the various SMA phases under stress-free conditions were widely studied by Uchil et al. [23,24], who found that Rphase can form in a specified range of heat treatment temperatures depending on the alloy’s composition. To detect the transformation temperatures, ER measurements can be useful and a good probe for identifying both temperature- and stress-induced transformations involving SMA crystallographic phases. Uchil et al. [23] investigated 40% cold-worked Nitinol phase transformations as a function of heat treatment by measuring ER variations with temperature. They observed that resistivity increases at martensite / austenite transformation and decreases at austenite / martensite transformation. Moreover, if the R-phase is present, resistivity approaches a maximum value, and during the cooling step, when resistivity increases, it is possible to evaluate the R-phase start and finish temperatures. Antonucci et al. [10] implemented an experimental setup to measure the electrical properties of NiTi alloys with R-phase upon variations in temperature (Fig. 5.17). They found that as a function of temperature, ER during cooling shows a rapid and large increase corresponding to transformation from austenite to R-phase (Fig. 5.18). Moreover, the degree of transformation of different NiTi phases during the whole thermal cycle (heating and cooling) was evaluated from both DSC and resistivity data, with good agreement between them. The comparison of calorimetric and electrical measurements pointed out the importance of temperature dependence of NiTi ER for identifying particular transformations that otherwise could not be identified by calorimetric techniques. ER measurement has proved to be a good probe for identifying R-phase and its start and finish transition temperatures. Moreover, the presence of a mixed phase composed of martensite and R-phase was detected from ER measurements, but not DSC analysis, in which only a large peak transition in the cooling part of the heat-flow curve is present. ER measurements clearly revealed the presence of R-phase during both heating and cooling; R-phase resistance was higher than austenite and martensite ones.

Figure 5.17 Electrical resistivity measurements experimental setup [10].

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Shape Memory Alloy Engineering

95 Ms 90 Resistivity (P: cm)

158

Rf

85 Cooling

80

Rs

Mf

Af

75 Heating 70 –20 –10

As

0 10 20 30 40 50 60 70 80 90 100 1100 Temperature (°C)

Figure 5.18 Electrical resistivity versus temperature diagram of NiTi alloy containing R-phase [10].

DSC and ER measurements are still valid techniques for detecting the transformation temperatures and amount of martensite associated with SME [25e27]. However, some authors [28] proved limitations on the size and geometry of test samples and have proposed a method, based on electromechanical impedance measurement, that can provide more accurate results than those of DSC and ER measurements.

5.7 Morphology characterization techniques SMA phase transformation can be investigated by techniques as well, which are able to provide information about the morphology of transforming phases owing to applied external forces, mainly thermal and mechanical. A big problem related to the mechanics of SMAs is the variety of deformations that can be activated in a thermomechanical transformation, whereas an SMA element is subjected to a thermomechanical load, because these depend on current stressestraintemperature conditions and the thermomechanical history. Although properties related to these transformations are relatively well-known, mechanics of transformations occurring in the polycrystalline environment are less understood. From normal macroscopic measurements, it is difficult to separate the elastic and inelastic contributions (from the respective phases) from the overall deformation. A useful tool proved to be neutron diffraction, which, combined with micromechanics modeling of SMA, gives information about the deformation and transformation processes in NiTi polycrystals.

Experimental characterization of shape memory alloys

Although SMAs generally operate under conditions of multiaxial stress in applications, most in situ investigations based on neutron diffraction on SMAs have been limited to homogeneous stress states owing to the uniaxial load. Currently, there are investigations to determine the thermoelastic deformation mechanisms under conditions of heterogeneous stress during heating and application of uniaxial or torsional load. Neutron diffraction has the advantage of using neutrons that can penetrate the bulk material; hence, experimental information averaged on the whole gauge volume of the specimen subjected to thermomechanical stresses can be obtained [29]. Experimental information that can be obtained by using this technique involves the evolution of elastic lattice strains and phase fraction in oriented grains and phases of transforming polycrystalline SMAs [30,31]. The principle of the method is depicted in Fig. 5.19. The specimen is loaded mechanically and thermally inside the neutron diffractometer. Integral intensities and positions of reflections of austenite and martensite phases are measured at selected states of stress, temperature, and strain, and their evolution with these external forces can be monitored [32e35]. Neutron powder diffraction data contain large number of relatively strong reflections in all three phases: R-phase, B190 , and B2. Hence, they allow a more precise evaluation of the structures, lattice parameters, distortions, or phase compositions. Moreover, use of high-resolution instruments proved essential for investigating B2 / R / B190 phase transformations during cooling or loading, because individual reflections of these three phases overlap and R-phase reflections further split owing to continuously changing rhombohedral distortion with temperature [36].

N eu

Austenite

Axial

PS D

Tensile stress

tro

400

n am

300

Martensite

be

Axial

200 100

2dhkl sin(4) = no

0 0.00

0.1

0.02 0.03

2.0

0.1

d-spacing [A] Temperature

Figure 5.19 Principle of in situ neutron diffraction measurements on polycrystalline shape memory alloys.

159

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Shape Memory Alloy Engineering

Sittner et al. [36] showed that in situ neutron diffraction experiments during thermomechanical tests carried out using the techniques of time-of-flight and high-resolution strain are interesting novel tools providing information about: 1. identification of deformation mechanisms; 2. load partitioning among grains and phases; and 3. martensite variant selection by the applied stress. This technique has enabled the evolution of the phase state and of martensite domain variants upon stress-induced martensite transformation of Co-Ni-Ga single crystals to be determined with an emphasis on elastically stretched martensite. In the case of Co49Ni21Ga30 [37] single crystals, the SMA showed a recovery of elastic strains under compression up to 11% in the crystallographic direction [001]. This recoverable strain includes both martensitic phase transition and elasticity of martensite. The absence of a suitable slip system seems to promote this excellent behavior. Thus, in situ neutron diffraction enables an analysis of stress-induced phase transformation and martensite elasticity in [001]-oriented Co49Ni21Ga30 [38]. L
opez-García et al. used this analysis to investigate the origin of these effects by studying atomic order and magnetic arrangement in the austenite phase of Ni50Mn34In16 and Ni45Co5Mn37In13 in two different atomic order states induced by thermal treatments [39,40]. Many authors have extensively studied the mechanisms of pseudoelastic behavior and the related phase transformations. The modeling of behavior in complex thermal and multiaxial loading conditions remains challenging because the associated phase transformations are unbalanced phenomena, because of the hysteretic nature of the transitions, which require adequate modeling. For example, the Ni-Ti alloy is also well-known to develop a possible R-phase intermediate between austenite and martensite, associated with a specific free transformation tension. Phase transformation under applied stress induces localization instability so that the spatial distribution of the phases is highly heterogeneous. Therefore, it is interesting to determine the presence of the different phases locally according to the experienced local voltage. This information is accessible from combined x-ray diffraction (XRD) and a digital image correlation (DIC) if the XRD is executed in situ and if the analyzed points can be repositioned on the DIC deformation map. A particular difficulty of XRD analysis is that because of the selection of the martensitic variants, the crystallographic plot is critical to the quantification of phases. To circumvent this difficulty, a correct orthogonal decomposition technique is proposed, also known as principal component analysis, to identify the main modes in which the spectra are decomposed. These modes are further interpreted as combinations of austenite, martensite, and R-phase, to obtain a solid determination of crystallography. This tool is used to map the different phases during a uniaxial loading test performed on an SMA Ni-Ti strip. The combined DIC and XRD observations confirm the coexistence of austenite, R-phase, and martensite with the transformation plateau [41,42].

Experimental characterization of shape memory alloys

Other advanced techniques that examine the SMA microstructure include the European Synchrotron Radiation Facility (ESRF) and transmission electron microscopy (TEM). ESRF is a third-generation source of synchrotron x-rays that characterizes a material microstructure in 3D without destroying it. This source generates high-energy x-rays at 8e150 keV, which can penetrate volumes of aluminum at several centimeters and steel at several millimeters for the highest energies. The high flux and low divergence of synchrotron radiation can be exploited by reducing the size of the beam dimensions down to the submicrometer range for imaging with high spatial resolution. These characteristics of synchrotron sources and detection equipment result in extremely fast measurements and in situ dynamic studies in polycrystalline grains and subgrains during deformation [40,42]. The microstructural evolution of deformed martensite in SMAs was also studied by TEM to clarify the mechanisms of deformation. The technology uses an accelerated electron beam, which traverses a thin specimen to reveal the structure and morphology. In an TEM, an electron gun fires an electron beam. The gun accelerates electrons at extremely high speeds using electromagnetic spirals and voltages up to several million volts. The electron beam is focused on a thin and small beam by an optical capacitor, which has a high aperture diaphragm that eliminates high-angle electrons. Reaching their highest brightness, the electrons zoom through the ultrathin specimen and the parts of the beam are transmitted according to the sample’s transparency to the electrons. The lens focuses on the part of the beam that is emitted from the sample in an image. Another component of TEM is the vacuum system, which is essential to ensure that electrons do not collide with the gas atoms. A low vacuum is first achieved by using a rotor pump or diaphragm pump, which allows low enough pressure for a diffusion pump to operate; this then reaches the vacuum level required for the operation. The image produced by the TEM, called a micrograph, is seen through the projection on a screen that is phosphorescent. Once radiated by the electron beam, this screen emits photons. A camera positioned below the screen can be used to capture the image, or digital locking can be achieved with a charge-coupled unit camera. The microstructural evolution of deformed martensite in the Zr50Cu50 SMA was studied by TEM and the mechanism of deformation was finally explained. Before deformation, intervariants in the Zr50Cu50 alloy showed a (021) type I twin relationship, and the dominant substructures inside the martensite variant were (001) compound twins. Distinct deformation mechanisms were observed at different deformation stages. Detwinning of the (001) compound twins was the dominant mechanism during the primary stage. As the compressive strain increased, the detwinning area expanded but did not result in complete detwinning. At higher strain levels, stress induced the formation of many (021) and (201) nanoscale deformation twins, originating from 1/10 partial dislocations; the mobility of martensite during heating was considered the dominant deformation mechanism [43].

161

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Shape Memory Alloy Engineering

5.8 Conclusion SMAs exhibit some unique properties; for instance, they can recover macroscopic shapes previously imposed at a characteristic temperature or by particular loading cycles. Transformation behavior can be studied using different complementary methods such as DSC, ER, and thermomechanical tests that are useful and good techniques for identifying both temperature- and stress-induced transformations. Moreover, techniques such in situ neutron diffraction and radiation synchrotron provide information about the morphology of transforming phases owing to applied thermal and mechanical external forces.

Bibliography [1] W. Buehler, R. Wiley, The Properties of TiNi and Associated Phases, Tech. Rep., U.S. Naval Ordnance Laboratory, 1961. [2] K. Otsuka, C.M. Wayman (Eds.), Shape Memory Materials, Cambridge University Press, Cambridge, 1999. [3] J. Perkins, Shape Memory Effects in Alloys, Plenum Press, New York, 1975. [4] H. Funakubo (Ed.), Shape Memory Alloys, Gordon and Breach Science Publishers, 1987. [5] X. Ren, K. Otsuka, Universal symmetry property of point defects in crystals, Phys. Rev. Lett. 85 (5) (2000) 1016e1019. [6] Y. Liu, D. Favier, Stabilisation of martensite due to shear deformation via variant reorientation in polycrystalline NiTi, Acta Mater. 48 (2000) 3489e3499. [7] G. Faiella, V. Antonucci, F. Daghia, M. Giordano, F. Daghia, E. Viola, Effect of the loading history on shape memory alloy transformation temperatures, Adv. Sci. Technol. 59 (2008) 57e62. ISSN: 16620356. [8] Y. Li, L.S. Cui, X.B. Xu, D.Z. Yang, Constrained phase-transformation of a TiNi shape-memory alloy, Metall. Mater. Trans. 34A (2003) 219e223. [9] P. Feng, Q. Sun, Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force, J. Mech. Phys. Solid. 54 (8) (2006) 1568e1603. [10] V. Antonucci, G. Faiella, M. Giordano, F. Mennella, L. Nicolais, Electrical resistivity study and characterization during NiTi phase transformations, Thermochim. Acta 462 (2007) 64e69. [11] A. Michael, Y.N. Zhou, M. Yavuz, M.I. Khan, Deconvolution of overlapping peaks from differential scanning calorimetry analysis for multi-phase NiTi alloys, Thermochim. Acta 665 (2018) 53e59. [12] E. Acar, M. Çalıskan, H.E. Karaca, Differential scanning calorimetry response of aged NiTiHfPd shape memory alloys, Appl. Phys. A 125 (2019) 239, https://doi.org/10.1007/s00339-019-2543-7. [13] F. Daghia, Active Reinforced Composites with Embedded Shape Memory Alloys, PhD thesis, 2008. [14] F. Daghia, G. Faiella, V. Antonucci, Thermomechanical modelling and experimental testing of a shape memory alloy hybrid composite plate, Adv. Sci. Technol. 59 (2008) 41e46. ISSN: 1662-0356. [15] Emiliavaca, C.J. de Ara
ujo, C.R. Souto, A. Ries, Characterization of shape memory alloy microsprings for application in morphing wings, Smart Mater. Struct. 28 (2019), 13pp. [16] D.C. Lagoudas, Shape Memory Alloys Modeling and Engineering Applications, Springer, 2008. [17] J. Shaw, S. Kyriakides, Thermomechanical aspects of NiTi, J. Mech. Phys. Solid. 43 (8) (1995) 1243e1281. [18] D.A. Miller, D.C. Lagoudas, Thermo-mechanical characterization of NiTiCu and NiTi SMA actuators: influence of plastic strains, Smart Mater. Struct. 9 (5) (2000) 640e652. [19] Y. Liu, Y. Li, K. Ramesh, Rate dependence of deformation mechanisms in a shape memory alloy, Philos. Mag. A 82 (12) (2002) 2461e2473. [20] P. Popov, K. Ravi-Chandar, D. Lagoudas, Dynamic loading of polycrystalline shape memory alloy rods, Mech. Mater. 35 (7) (2003) 689e716.

Experimental characterization of shape memory alloys

[21] J. Nemat-Nasser, W. Choi, G. Guo, J. Isaacs, Very high strain-rate response of a NiTi shape-memory alloy, Mech. Mater. 37 (2e3) (2005) 287e298. [22] G.D. Travassos, L.F.A. Rodrigues, C.J. de Ara
ujo, Fabrication and thermomechanical characterization of a new CueAleNieMneTi shape memory alloy bolt, J. Braz. Soc. Mech. Sci. Eng. 39 (2017) 1269e1275. [23] J. Uchil, K.P. Mohanachandra, K. Gesh Kumara, K.K. Mahesh, Mater. Sci. Eng. A 251 (1998) 58e63. [24] J. Uchil, K. Ganesh Kumara, K.K. Mahesh, Mater. Sci. Eng. A 332 (1e2) (2002) 25. [25] B. Cao, T. Iwamoto, A new method to measure volume resistivity during tension for strain rate sensitivity in deformation and transformation behavior of Fe-28Mn-6Si-5Cr shape memory alloy, Int. J. Mech. Sci. 146e147 (2018) 445e454. [26] B. Cao, Takeshi Iwamoto an experimental investigation on rate dependency of thermomechanical and Stress-induced martensitic transformation behavior in Fe-28Mn-6Si-5Cr shape memory alloy under compression, Int. J. Impact Eng. 132 (2019) 103284. [27] B. Lynch, X.-X. Jiang, A. Ellery, F. Nitzsche, Characterization, modeling, and control of Ni-Ti shape memory alloy based on electrical resistance feedback, J. Intell. Mater. Syst. Struct. 27 (18) (2016) 2489e2507. [28] M.F. Cunha, J.M.B. Sobrinho, C.R. Souto, A.J.V. dos Santos, A.C. de Castro, A. Ries, L.D. Nathan, Sarmento, Transformation temperatures of shape memory alloy based on electromechanical impedance technique, Measurement 145 (2019) 55e62. [29] S.L. Raghunathan, M.A. Azeem, D. Collins, D. Dye, In situ observation of individual variant transformations in polycrystalline NiTi, Scripta Mater. 59 (2008) 1059e1062. [30] R. Vayadyanathan, M.A.M. Bourke, D.C. Dunand, Phase fraction, texture and strain evolution in superelastic NiTi and NiTieTiC composites investigated by neutron diffraction, Acta Mater. 47 (1999) 3353e3366. [31] P. Sittner, P. Luk
as, D. Neov, M.R. Daymond, V. Nov
ak, G.M. Swallowe, Stress induced martensitic transformation in CuAlZnMn polycrystals investigated by two in situ neutron diffraction techniques, Mater Sci. Eng. A A324 (2002) 225e234. [32] C. Leinenbach, A. Arabi-Hashemi, W.J. Lee, A. Lis, M. Sadegh-Ahmadi, S. Van Petegem, T. Panzner, H. Van Swygenhoven, Characterization of the deformation and phase transformation behavior of VC-free and VC-containing FeMnSi-based shape memory alloys by in situ neutron diffraction, Mater. Sci. Eng. A 703 (2017) 314e323. [33] A.P. Stebner, H.M. Paranjape, B. Clausen, L.C. Brinson, A.R. Pelton, In situ neutron diffraction studies of large monotonic deformations of superelastic Nitinol, Shap. Mem. Superelasticity 1 (2015) 252e267. [34] O. Benafan, D.J. Gaydosh, R.D. Noebe, S. Qiu, R. Vaidyanathan, In situ neutron diffraction study of NiTie21Pt high-temperature shape memory alloys, Shap. Mem. Superelasticity 2 (2016) 337e346. [35] H. Yang, D. Yu, Y. Chen, J. Mu, Y.D. Wang, K. An, In-situ TOF neutron diffraction studies of cyclic softening in superelasticity of a NiFeGaCo shape memory alloy, Mater. Sci. Eng. A 680 (2017) 324e328. [36] P. Sittner, P. Luk
as, V. Nov
ak, M.R. Daymond, G.M. Swallow, In situ neutron diffraction studies of martensitic transformations in NiTi polycrystals under tension and compression stress, Mater. Sci. Eng. A 378 (2004) 97e104. [37] Reul, C. Lauhoff, P. Krooß, M.J. Gutmann, P.M. Kadletz, Y.I. Chumlyakov, T. Niendorf, W.W. Schmahl, In situ neutron diffraction analyzing stress-induced phase transformation and martensite elasticity in [001]-Oriented Co49Ni21Ga30 shape memory alloy single crystals, Shap. Mem. Superelasticity 4 (2018) 61e69. [38] J. L
opez-García, V. S
anchez-Alarcos, V. Recarte, J.I. P
erez-Landaz
abal, O. Fabelo, E. Cesari, J.A. Rodríguez-Velamaz
an, Routes for enhanced magnetism in Ni-Mn-In metamagnetic shape memory alloys, Scripta Mater. 167 (2019) 21e25. [39] Z.-B. Li, B. Yang, Y.-D. Zhang, C. Esling, X. Zhao, L. Zuo, Crystallographic insights into diamondshaped 7M martensite in NieMneGa ferromagnetic shape memory alloys, IUCrJ Mater. Comput. (2019). ISSN 2052-2525.

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[40] S. Kabra, J. Kelleher, W. Kockelmann, M. Gutmann, A. Tremsin, Energy-dispersive neutron imaging and diffraction of magnetically driven twins in a Ni2MnGa single crystal magnetic shape memory alloy, J. Phys. Conf. (2016) 746. [41] M. Dubois, M.-H. Mathon, V. Klosek, A. Gilles, A. Lodini, Neutron diffraction study of a CuAlBe shape memory alloy, Solid State Phenom. 172e174 (2011) 178e183 (XRD). [42] M. Ko€k, S.B. Durgun, E.O.€ zen, Thermal analysis, crystal structure and magnetic properties of Crdoped NieMneSn high-temperature magnetic shape memory alloys, J. Therm. Anal. Calorim. 136 (2019) 1147e1152. [43] W. Gao, X. Yi, B. Sun, X. Meng, W. Cai, L. Zhao, Microstructural evolution of martensite during deformation in Zr50Cu50 shape memory alloy, Acta Mater. 132 (2017) 405e415.

Further reading [1] M. Schmidt, J. Ullrich, A. Wieczorek, J. Frenzel, A. Schutze, G. Eggeler, S. Seelecke, Thermal stabilization of NiTiCuV shape memory alloys: observations during elastocaloric training, Shap. Mem. Superelasticity 1 (2015) 132e141.

CHAPTER 6

Manufacturing of shape memory alloys Mohammad Elahinia1, Mohammadreza Nematollahi1, Keyvan Safaei Baghbaderani1, Adelaide Nespoli2, Francesco Stortiero3 1

The University of Toledo, Toledo, OH, United States; 2CNR ICMATE Unit of Lecco, Lecco, Italy; 3GFM S.P.A., Mapello, Bergamo, Italy

6.1 Introduction The manufacture of shape memory alloys (SMA) includes many different working processes, and each step is crucial for the functional properties of the final device. Starting from the process of melting, each preparation method affects the mechanical and thermal characteristics of the SMA product. We principally consider the traditional working preparation and thermomechanical optimization of the material, but also we focus on some innovative processes that have opened interesting perspectives into the realization of new applied solutions and devices in many industrial fields. The chapter particularly focuses on NiTi alloys and NiTi-derived ternary systems, which are the most important SMA metallic materials for which a relevant number of manufacturing processes have been developed to fulfill the applicative outlooks of these smart alloys.

6.2 Melting process of shape memory alloys Many studies have been carried out to investigate the best melting process for NiTi alloys, which enables the satisfactory control of chemical composition and the content of impurities. Little change in the chemical composition of even only 0.1 (atm%), drastically affects the transition temperatures of NiTi alloys and their possibility of being used in the area of application. Moreover, the homogeneity of the final ingot is an important and challenging result owing to the sensitivity of shape memory effect (SME) properties of the alloy stoichiometry. Therefore, accuracy in preparing NiTi alloys is fundamental to avoid products with functional characteristics that are far from the final desired use. From the point of view of contamination control, the principal impurities related to the raw materials and melting process are C and O. C is highly soluble in nickel and has high affinity with titanium, and O forms carbides and/or oxides with Ti (principally TiC and Ti4Ni2Ox) [1]. The high content of TiC impurities has important consequences for the transition temperatures and temperature transition range as well as fatigue properties [1]. Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00006-6

© 2021 Elsevier Ltd. All rights reserved.

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The most common industrial melting process [2,3] of NiTi is vacuum induction melting (VIM), which ensures good homogeneity thanks to electromagnetic stirring. However, the main drawback of this method is carbon contamination derived from the use of a graphite crucible. During VIM, NiTi melts dissolve carbon and TiC particles form during solidification. Different possibilities for minimizing C picked up from the crucible have been investigated [4], such as study of the charging sequence or multiple use of the same crucible, in which a coating of alloy is created and reduces the release of C. The other important melting method is vacuum arc remelting (VAR), in which the starting elements are positioned into a consumable electrode. The melting process is carried out in a water-cooled copper crucible; in this case, there is no contamination from the crucible, but the ingots have to be melt several times to ensure good homogeneity of the final alloy. The need for high-quality material imposed on the biomedical applications field by ASTM F-2063-05 gives precise limits for C, O, and N content. Consequently, this quality of material has been pursued and different kinds of melting sequence methods [2] have been investigated to achieve good control of the chemistry, homogeneity and segregated microstructures, and the inclusion content of the final NiTi ingots. The most commonly used complete melting process is VIM/VAR, in which the VIM process is followed by VAR or multiple VAR. Remelting in a copper crucible maintains low carbon levels, and good control of vacuum and raw materials results in NiTi alloy that is in agreement with ASTM standards. The need for the low content of impurities described in ASTM F-2063-05 became crucial for non-biomedical application devices as well, because the development of NiTi innovative solutions demands precise control of quality of the starting alloys to prepare low-dimensional products and systems with high fatigue resistance properties. The quality of NiTi alloys has highly increased with respect to commercial material available not long ago. Moreover, in the published literature, results obtained from melting processes such as electron beam (EB) or plasma arc melting (PAM) are described [5e7]. Figure 6.1 depicts the plasma torch in the PAM process.

Figure 6.1 Plasma arc melting process. (From CNR ICMATE Unit of Lecco).

Manufacturing of shape memory alloys

EB and PAM have good results concerning the purity of the final alloy, but the cost of the process and the lack of complete homogeneity of the product make these processes useless for industrial production. However, these processes are advantageous for preparing other SMA such as ferromagnetic SMA, for which there are applications based only on single-crystal or thin-film samples. In addition, EB and PAM are used to produce pure alloys that are subsequently employed in other processes such as sputtering deposition of thin film or melt spinning, powder metallurgy, and metal injection molding (MIM) [8e11].

6.3 Traditional working process of shape memory alloy materials The most important starting point to consider for all manufacturing processes of NiTi devices is that the NiTi is an intermetallic system; therefore, the working protocol of this alloy is a complex combination of thermomechanical steps in which the maximum cold work sustained by the material is about 40% and each thermal treatment must be followed by fast cooling (water quenching or other) to fix the microstructural properties and avoid secondary phase segregation such as Ti2Ni-TiNi3. Figure 6.2 depicts typical steps of the working process of an NiTi product. According to this scheme, here we consider only the working processes; subsequently, we will further examine shape memory thermal treatment and possible aging thermal procedures. Starting from the melting process, a first hot working step is necessary to reduce the size of the start ingot to set the homogenization of the microstructure and reduce the solidification grain texture adjustment (grain size adjustment). This process is usually carried out from about 800 to 950
C, normally using protective metallic sheets (Cu or steel) to reduce surface oxidation. Moreover, in this first working step, the billet is prepared to produce bars, composites, or tubes, using a suitable core, as Figure 6.3 shows. Then, this first passage is followed by a series of cold working processes, with intermediate annealing thermal treatments between 500 and 800
C. Each NiTi product follows an ad hoc preparation protocol, which makes it possible to assess a particular microstructure with a controlled mixture of appropriate grain size, precipitates, and dislocation content suitable to give the final NiTi a controlled range of functional properties. NiTi does not sustain cold work more than 40e45%. These preparation phases of the working process are long and articulated because it is possible to change and combine different protocol steps. Moreover, the smaller the size is of the final product, the longer the work will be, which affects the final price of the NiTi component [12]. This special alloy can be worked by traditional rolling and drawing procedures down to a very low size (20e30 mm). In the case of a low NiTi size, the development of an ad hoc working process protocol is necessary, starting from melting with good control of the

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Figure 6.2 Rough guide diagram of working process of NiTi product. CW, cold work; VAR, vacuum arc remelting; VIM, vacuum induction melting.

Figure 6.3 Examples of hot extrusion products.

Manufacturing of shape memory alloys

content of impurities, which can represents a common reason for breaking, and then the use of diamond dies, adequate payoff and pay-on spooling systems, and special load cells to build equipment with high flexibility to optimize NiTi drawing procedures [13]. Thermal treatment suitable for working can be carried out in air or a controlled inert atmosphere, but not under reducing conditions using H2 or Ar/H2, because the H content causes embrittlement of the alloy. The fulfillment of sheets with a low dimension has also been a focus of interest, but the traditional working process is limited because it is not possible to reduce large surfaces, and commercially available products have a maximum width of about 100 mm. The need for more complex applications implies a geometry that cannot be obtained from bars, wires, or sheets. Therefore, the possibility of machining NiTi by traditional processes has been investigated [14e16]. In this context, it is important to establish some considerations: (1) Characteristic transformation temperatures (TTs) of NiTi alloy that is used need to be evaluated with respect to the machining temperature. There are some important consequences: movement of the sample during machining owing to pseudoelasticity or strain recovery caused by possible overheating, which has a direct effect on precision and the tolerances of the final size. (2) Work hardening and the intrinsic NiTi intermetallic microstructure can show an irregular and unexpected change in hardness and in the workability of the material. This causes breakage in the sample, but also the machine tool. Therefore, it is necessary to calibrate the speed of work and the choice of suitable accessories. If these considerations are assessed, it is possible to obtain good results by machining NiTi samples: for example, excellent tolerance and finish can be achieved in turning operations using suitable carbide tooling and a controlled working speed. Milling and drilling performed with similar caution are also employed at the microscale [17]. General abrasive and electrochemical methods can also be used to prepare the sample and define the final surface requirements [18]. However, the cutting process developed over the past years has created a lot of opportunities to achieve NiTi products in a wide range of geometries and sizes. Electrodischarge machining is a method that offers a major opportunity to control the tolerance and precise size of the final sample in NiTi [19], even if the working time and costs are not negligible. The surface layer altered by the process can be remove by chemical and electrochemical etching to give good control of the bulk properties of the obtained sample. Moreover, there is a low size limit owing to the dimension of the wire used in the electrodischarge process.

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Laser cutting is one of the most important cutting processes in the metallics field. With technological improvements related to focalization of the laser beam, this technique has become greatly interesting for the preparation of NiTi devices, particularly in the use of laser beams in fiberglass. This technique allow excellent size control and tolerance, good performance at the micrometric scale, and a reduced thermal altered zone that in SMA has an important effect on microstructural conditions and functional properties [20]. Moreover, interesting progress has been achieved in the removal of dross and melted material and in the effect of surface preparation before laser cutting. Laser technology may be one of the most promising working processes of NiTi. In addition, an attractive perspective is represented by water jet cutting with abrasive particles [21]. With this process, it is possible to obtain good control of size and tolerance with no consequences due to overheating of the sample, which is completely avoided. However, the force developed by the impact of the water jet against the material causes unwanted deformations in the sample. Therefore, this method is a good solution only for bulk specimen preparation. Finally, to give a possibly complete overview of the principal machining process, it necessary consider the soldering of NiTi alloy. The principal problem of NiTi soldering is that part of the material melts, which changes the microstructure of the sample locally, both in the SME and in the pseudoelasticity. In some cases, it also alters the chemical composition. Soldering methods such as laser, EB, or Tungsten Inert Gas (TIG) welding, which minimize the melted zone, are preferred [22]. Figure 6.4 shows an example of the thermomechanical performance of an NiTi sample after laser welding.

600

Type 2 Material T=50°C

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0

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ε

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Figure 6.4 Thermal and mechanical characterization of NiTi sample after Nd-YAG laser welding (From CNR ICMATE Unit of Lecco).

Manufacturing of shape memory alloys

In actuator applications, in which the shape of the wire is usually preferred as the NiTi active element, it is better to consider mechanical joining by crimps; in this way, there is no thermal alteration of the microstructure and functional properties are kept constant for 106 cycling tests. When NiTi has to join to another element, there are two important consequences: (1) The interface between the two materials is a fragile connection because it does not reach strains in the same way as the rest of the material. Use of a material with proper interference fit is a possible way to reduce this problem. (2) The formation of intermetallic compounds within the heat affected zone results in brittle behavior, and in many cases this solution is acceptable only for applications subjected to low forces and stresses. For example, soldering with steel is troublesome because of the formation of intermetallic TiFe and TiFe2. Other welding process such as diffusion bonding and friction welding give only feeble results.

6.4 Technologies for preparing shape memory alloy products To overcome problems in preparing NiTi devices and open new possibilities in tuning microscopic material conditions, some innovative and nonconventional working procedures have been studied and developed. One method for obtaining NiTi ribbons and skipping the long working process by rolling or drawing is melt spinning, which consists of rapid solidification from a melted alloy [23,24] (Fig. 6.5). In this way, the goal is to obtain ribbons ready to be used directly, without the complicated thermomechanical process of assessing the peculiar properties of these alloys. The principal characteristic of these ribbons is a smaller grain size; in some cases, it also introduces a region of amorphous material and a texture due to the rapid cooling process. NiTi binary systems and NiTiebased ternary alloys have been investigated and results are promising, even though mechanical and shape memory properties are lower than those of the bulk material [25e27]. In many cases, adding a rolling process or thermal treatment can improve performance, and the deeper investigation into the process parameter gave a better size and control of microscopic features [27]. A large new field of investigation concerns the preparation of NiTi alloys by powder metallurgy. These efforts have the objective of fabricating near-net shape components using different kinds of methods such as sintering under pressure, self-propagating high-temperature synthesis (SHS), and MIM [8,28e31]. With these processes, the porosity obtained ranges from 40% to 70% and the pore size is 60e500 mm. Porous NiTi products are principally used for biomedical applications owing to their potential as implant materials, which promote osteosynthesis, but they are also interesting

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Ejecon gas

Crucible Heang system Melt

Nozzle Free melt jet

Molten alloy puddle

Solidified ribbon

Rotang substrate Wheel

Figure 6.5 Melt spinning process scheme.

for the study of innovative microstructures to bulk devices with a controlled density. Some commercial products are available, particularly in the orthopedics field: for example, an NiTi porous implant for spinal fusion produced by SHS process by Actipore, Biortex, in Canada. In many cases, principal problems are the oxygen content, control of the chemical final composition, the formation of secondary phases, and the microstructural homogeneity of the final material [32]. Generally, the mechanical response is limited and affected by the strong possibility of inducing cracks in the porous region, which damage the sample and limit strain in pseudoelastic or shape recovery performance [33,34]. Another important field of research and implementation in new technologies is the production of NiTi thin films. Usually, the films are less than 10 mm thick and are deposited on silicon, glass, or polymeric substrates by sputter deposition. Commercial devices consisting of NiTi film on silicon substrate are then fabricated by photolithographic techniques [35]. Control of the composition of thin film is the crucial experimental aspect in this preparation method using either prealloyed or elemental targets. Also, optimization of the microstructure is an important step because films in the as-deposited state are generally amorphous and require subsequent heat treatment to recrystallize [36]. The field of application of thin films is a scientific research activity; applications can be found in miniature pneumatic valves, spacers in flat panel displays, and other microelectromechanical systems.

Manufacturing of shape memory alloys

A final mention in this overview is devoted to an innovative process, the preparation of SMA foams, which can develop new interesting strategies associating particular thermal conductivities and functional properties [37]. The process manufactures open-cell metal foams by liquid infiltration in a leachable bed of silica-gel particles. In this way, it is possible to obtain a uniform microstructure in the ligaments and a regular and well-reproducible open-cell morphology. Moreover, the investigation of mechanical properties showed that recovered monoaxial compressive strains up to 3% are reached.

6.5 Thermomechanical process to optimize the functional properties of shape memory alloys All thermic and mechanical processes involved in the final shape setting of an SMA material are key to developing an NiTi device with suitable functional properties. This forming protocol gives the material its final shape and operating parameters such as recovered strain and stress and activation temperatures in both SME and pseudoelastic applications. Thermal and mechanical optimization of the functional performance of an SMA sample produces a studied and quasiplanned combination of effects on the grain size, on working defects such as dislocations, and on the amount and kind of precipitate. All of these factors influence the microstructural conditions that result in the functional performance. An assessment of a suitable preparation protocol is necessary and is specific for each kind of device to realize precise and repeatable action in the final application. The industrial field considers these protocols a fundamental resource of know-how and expertise in the SMA sector. Therefore, the development of a thermomechanical process to optimize a new device is not divulged or published; in some cases, specially developed protocols have been patented. Consequently, references in the literature are rare, and only experimental experience can provide useful knowledge about these particular phases of preparation of an SMA device. For these reasons, we will resume and show general criteria that provide operating conditions to obtain a defined influence on the microscopic level in the alloy. Within this synthetic frame, it is possible to choose the parameter control of the method of preparation suitable for fixed objectives to be realized in the application. According to the scheme presented in Otsuka and Ren [38], thermal treatment can be applied to NiTi cast samples or a material obtained by hot deformation. In this case, the thermal process acts on the grain size and on the formation of precipitates, giving the sample the required final shape, adjustment of transition temperatures, and/or mechanical hardening. According to the desired function of the device, these thermal treatments can be combined or changed with time and temperature to modulate the properties in the case of pseudoelastic application, an actuator, or high-fatigue resistance devices.

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Some general considerations are that: - Usually, the temperatures involved can be 300 and 600
C and the time can be from some minutes to 1 h. - Temperature higher than 600
C refines the microstructure under fully annealed conditions, which corresponds to a regular and homogeneous grain structure without dislocations and precipitates. This is a microscopic condition for a quick and precise shape recovery effect that is suitable for a free recovery condition, for example, but with poor mechanical properties. - Intermediate temperatures (300 to 580
C) and longer times influence all precipitates’ presence and their transition temperatures. For example, long thermal treatment has to be considered when shape forming requires more of one thermal process or when it is necessary to start from a particular chemical NiTi composition and achieve different functional temperatures. In these cases, the effect on mechanical and fatigue properties has to be considered and studied. - Experimental tests must follow a thermal process to give the necessary control to functional performance and consequently to adjust and optimize the protocol under development. Differential scanning calorimetry (DSC) is the principal method of characterization that can indirectly explain and represent the effect of thermal treatment on an alloy’s microstructure. It is possible to see a change in the calorimetric response of coldworked NiTi after different thermal treatments in Figs. 6.6, 6.7. Each curve corresponds to a specific microscopic condition induced in the material. If the temperature and time are increased, in addition to a change in the microscopic condition, it is possible to evaluate the effect of precipitates, which after suitable thermal treatment can result in a material response different from the one observed after full annealing. These figures show the extraordinary wealth of modulation in the microstructure of NiTi and also the sensibility and difficulty of preparing a device with SMA, because it is necessary to have deep knowledge of the material and excellent control of the process parameters (PPs) to establish a reproducible preparation protocol of devices. In shape recovery applications, the working conditions creates many possibilities, and then a lot of protocols can be developed to satisfy the recovery temperature, stress or strain, and cyclic resistance conditions. In working requests for pseudoelastic applications, it is possible to investigate these effects of change in microstructure on the mechanical properties, and therefore to study how to program different thermal shape assessments according to the operating conditions. Figure 6.8 shows the stressestrain curve for a NiTi wire after different times and temperatures of annealing.

Manufacturing of shape memory alloys

Figure 6.6 Comparison of differential scanning calorimetry curves registered for Ni49Ti51 alloy from cold-worked condition to annealed states (From CNR ICMATE Unit of Lecco).

The mechanical performances are different, and for pseudoelastic applications, it is better to start from a cold-worked condition and to focus only on a change in the grain size and the distribution of dislocations. The precipitates may negatively affect the plateau and residual strain [39]. Finally, a particular thermal process can be carried out to stabilize the actuator functional parameter or fatigue resistance [40]. In this case as well, the choices are numerous and at first are needed to facilitate grain orientation. It is possible to conduct thermal treatment under stress to induce a texture and optimize the mechanical response in a particular direction, or to conduct thermal or mechanical cycling to assess and fix the temperature and the strain recovered. In some cases, thermal treatment at low temperatures can contribute to stabilizing the transformation parameter or increasing the fatigue properties. Finally, innovative studies are presented using the laser technologies to make local and microcontrolled thermal processes for development in the microactuation field. Different methods are involved in the final surface preparation after shape setting. The oxide layer can be removed mechanically, chemically, and electrochemically, each of which has different effects on the roughness and polishing. Abrasive processes are principally involved in eliminating the first oxide layer; polishing can be obtained by tumbling or chemically or electrochemically using suitable mixtures of HF, HNO3, and H2O for chemical etching and perchloric acid and

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Figure 6.7 Comparison of differential scanning calorimetry curve registered for Ni51Ti49 alloy from cold-worked condition to annealed states; (a): effect of the temperature of the annealing route, (b): effect of both temperature and time of the annealing route (From CNR ICMATE Unit of Lecco).

Manufacturing of shape memory alloys

Figure 6.8 Stressestrain curves registered for an Ni50.8Ti wire at room temperature after thermal treatment at 300 to e550
C for 15, 30, and 60 min (From CNR ICMATE Unit of Lecco).

2-butoxyethanol for electrochemical polishing. Some attention must be paid to controlling the effects of these methods; in particular, tumbling and electrochemical polishing have some influence on fatigue properties for mechanical hardening the surface and introducing H during the process. Although the thermomechanical constitutive behavior of SMA has been studied extensively, there is a lack of knowledge about their failure mechanisms, specifically crack growth and fracture behavior. As discussed by Haghgouyan et al. [41], this is mainly because applying conventional fracture mechanics theories to SMA is not straightforward. As discussed by Haghgouyan et al. [42], the fracture behavior of SMA is complex owing to the presence of phase transformation, the reorientation of martensitic variants,

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the possibility of transformation-induced plasticity, and inherent thermomechanical coupling. Because the thermomechanical behavior of SMA is extremely sensitive to their microstructure, slight changes in processing variables introduces new challenges to interpreting the data and further complicates an assessment of their fracture properties.

6.6 Additive manufacturing There is a vast literature on the applications of SMA [43e45]. Increasing interest in developing new devices and new solutions based on SMA drives researchers to investigate the possibility of using additive manufacturing (AM) techniques in preparing NiTi alloy samples. A good result would be the capacity to obtain samples using a quick and precise process such as the three-dimensional (3D) printing used with polymer materials. Unfortunately, the situation for the NiTi is not so simple. The necessarily high melting temperatures involve experimental equipment that is more complicated from a different point of view: the materials that are used, the source of energy for melting, and precise control of the atmosphere. Moreover, it is well known that control of the final microstructure is a crucial aspect that affects the functional properties of the final product. The process explained earlier showed that an NiTi material obtained from powder by MIM or SHS can be considered a good product only for a small field of application and that the functional properties are lower than those of the bulk material. There has been scientific interest in multiple methods of AM, including selective laser melting (SLM), direct energy deposition (DED), and EB melting (EBM), which offer new possibilities of melting and preparing near-net shape products that give more flexibility in making more complex shapes (Fig. 6.9). The following discussion mentions AM methods briefly and then presents advances in AM of NiTi alloys (Fig. 6.9).

Figure 6.9 The ability of additive manufacturing to make NiTi complex shapes [46].

Manufacturing of shape memory alloys

6.6.1 Additive manufacturing techniques SLM is a powder bedebased technique of AM that is a powerful tool for the free-form fabrication of metals. A schematic of the SLM process is shown in Fig. 6.10. The process starts by layering a 3D computer-assisted drawing file of the part into a number of layers based on the layer thickness, which is a process parameter that can be defined by the user. A roller or scraper coats the substrate plate with a defined layer thickness of the powder. Then, selected areas in each layer are melted and fused together with a high-power laser. To avoid picking up impurities, the chamber is purged by an inert gas (i.e., nitrogen, argon). Laser power (P), scanning speed (V), hatch spacing (h), and layer thickness (t) are the most effective PPs that change the properties of the final part. The combination P of these parameters is defined as the energy density (E ¼ J/mm3), which shows the Vht amount of the energy received by a unit volume of the material. For a dense and defectfree part, PPs should be optimized for each material. In addition to having a dense part, the structural and functional thermomechanical properties of the fabricated parts are important. Other parameters include, but are not limited to, the scanning strategy, powder particle size distribution, oxygen level of the chamber, chamber and bed temperature, and laser beam diameter, which influence the properties of the printed parts. SLM is capable of producing complex shape parts with high density and good size accuracy for a vast range of alloys. However, the limited size of the printed parts restricts the use of this technology in some applications [47]. Direct energy deposition, also known as laser engineered net shaping, laser cladding, direct light fabrication, and shape deposition manufacturing, is a flow-based AM

Figure 6.10 Demonstration of selective laser melting process [48].

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technique in which the material is directly deposited onto a build plate or an existing part. Unlike having a powder bed in SLM, in DED powder is sprayed inside the laser-induced molten pool through nozzles that surround the laser, as depicted in Fig. 6.11 After the melt pool solidifies, the bead is attached to the substrate and then the base table or laser head moves to the next position to build layers on top of each other. In DED machines, typically both the laser and nozzles are mounted coaxially on a four- or five-axis arm, which gives the ability to move in different directions and in any angle. Relatively fast builds, good process for repairing parts, and the ability to handle different powders to create functionally graded materials are advantages of DED. However, the poor surface finish is a drawback, because most parts fabricated through DED need significant postprocessing. Important parameters involved in DED include, among others, laser power, laser spot size, scanning speed, and powder feed rate, all of which affect the properties of the final products and need to be optimized. EBM is similar to SLM, except the heat source is from an EB. In EBM, fabrication takes place in a vacuum chamber and the powder bed is heated to several hundred degrees Celsius, which frees the part from residual stresses.

6.6.2 Selective laser melting of shape memory alloys To date, the most commonly used material for AM of SMA has been NiTi; more recently, it is NiTiHf, and the most investigated method is SLM. The following discussion summarizes work on SLM NiTi. As mentioned, four main parameters affect the quality and properties of parts: laser power (P), laser scanning speed (v), hatch distance (h), and layer thickness (t). Most of the literature has been on optimizing and realizing the effects of these PPs to achieve defect-free parts and then to tailor the thermomechanical properties of NiTi-based alloys.

Figure 6.11 Schematic of direct energy deposition process [49].

Manufacturing of shape memory alloys

6.6.3 Printability Figure 6.12 shows some defects, such as cracking and delamination, which can happen during rapid solidification and thermal stresses during AM of NiTi and NiTiHf alloys [50,51]. It has been shown that the scanning speed and laser power were the most important factor controlling the formation of defects, and that by optimizing these two, defectfree dense AM SMA were achieved. Figure 6.12(b) presents a printability map of AM NiTi, showing the lesser importance of hatch spacing and layer thickness. Furthermore, extremely rapid superheating and undercooling during SLM generates an ultrafine microstructure and high residual stresses inside the as-built material that may cause a warping effect, (Fig. 6.13). Fabricating defect-free parts leads to the successful fabrication of lattice structures, which is important for biomedical applications.

6.6.4 Transformation temperatures Transformation temperatures (TTs) as a key feature of NiTi-based alloys are highly affected by the microstructure [52]. Hence, SLM processes that affect the microstructure have a strong influence on TTs [45e50]. Composition and precipitation are two more important factors discussed subsequently. As-fabricated SLM parts have higher TTs with respect to powders TTs [51,53,56,57]. Energy density as an effective parameter in the SLM technique has a significant role in TTs. Ni evaporation caused by the density of laser energy was likely the reason for such rise [48e50]. The nickel content in Ni-rich NiTi has a dominant role, so that TTs increase 20
C when the Ni content is decreased by 0.1 atm%. Furthermore, as Fig. 6.14 shows, the higher the Ni atomic percent is, the more the energy density affects TTs. For example, TTs for Ni50.7Ti rose remarkably by increasing the energy density, whereas this shift was not significant for Ni50.2Ti (Fig. 6.14). However, several works show differences between parts fabricated with the same energy density but different PPs [53,55,56,58]. In this regard, Mathew Speirs et al. [58] studied different PPs (laser power, scanning speed, and hatch space) to build SLM parts with the same energy density. It was reported that an increase in scanning speed and power resulted in a drop in TTs, whereas the energy density kept constant. However, it was shown that higher power and scanning speed intensifies Ni evaporation based on the chemical composition analysis, which is supposed to affect TTs in the opposite trend. As discussed previously in several works, the trend cannot be explained by changes in composition. Because of the thermal history of exposure to the SLM part, several Ti-rich or Ni-rich precipitates can be formed. In the SLM process, the heat-affected zone of the adjacent laser passes or of the following layers have an aging effect on the alloys, so that thermal loading can result in the creation of precipitates based on the thermal history. Ni4Ti3 and Ni3Ti2 are two metastable Ni-rich precipitates of Ni-rich NiTi composition

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Figure 6.12 (a) and (c) Defective and nondefective selective laser melted NiTi and NiTiHf coupons, respectively. (b) Plot of linear energy density (P/v) versus hatch spacing, showing the scatter of conditions of parts based on the process parameters [50,51].

Shape Memory Alloy Engineering

(a)

Manufacturing of shape memory alloys

Figure 6.13 Example of warping effect due to internal stresses (From CNR ICMATE unit of Lecco).

350

Ni 49.7Ti

325 Ni 50.2Ti 300 Ni50.7Ti

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400 200 600 800 Energy density ω v / (J/mm3)

Austenite start temperature (K)

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that are formed in the aging process between 200 and 700
C [59,60]. Ji Ma et al. computationally studied the effect of the thermal history caused by different hatch spacings on the TTs of Ni50.9Ti49.1 [59]. Based on the thermal modeling, they showed the probability of Ni4Ti3 nanoprecipitates forming, which can explain the change in TTs; however, the formation of Ni4Ti3 precipitates during SLM has not yet been reported. The formation of these precipitates not only changed the composition but also acted as barriers in the matrix of the material to prevent phase transformation. The density and distribution of the Ni4Ti3 precipitate over the interior grains or boundaries highly affected the transformation behavior [38,61]. Owing to the active behavior of Ti in reacting to other components, a gain in impurities (oxygen, nitrogen, and carbon) is another possible factor affecting TTs [62]. As a result, in addition to the material composition, Ni-rich or Ti-rich precipitates and secondary phases formed during the SLM process have a significant role in the transformation behavior of the near-equiatomic NiTi alloy. These precipitates have significant effects on TTs by facilitating martensite formation caused by induced incoherent stress, or changing the composition through the depletion of Ni or Ti.

350

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Figure 6.14 Effect of energy density on the transformation temperature for three different NiTi compositions [56].

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Figure 6.15 Optical microscopy image of NiTi samples fabricated by selective laser melting with different hatch spacings of (a) 80 mm, (b) 120 mm, and (c) 180 mm. By changing the H, the microstructures have been changed [63].

6.6.5 Thermomechanical behavior of selective laser melting fabricated parts Besides TTs, the thermomechanical responses of selective laser melted parts are highly affected by PPs owing to differences in the resultant microstructures. Parts with a similar energy density showed different superelasticity behavior at Afþ 10
C [55]; moreover, changing only one parameter such as hatch spacing can change the microstructure and thus the mechanical behavior [63], as illustrated in Fig. 6.15. Basically, by changing the hatch spacing from 80to 180 mm, the texture and mechanical properties of the asfabricated parts changed significantly. Hence, not only is it possible to fabricate dense and functional parts, tailoring the properties by changing PPs is a capability of processing SMAs with AM techniques. Another important factor that determines the thermomechanical behavior of NiTi is texture. The [001] direction is a preferred growth orientation for cubic metals such as B2 NiTi during solidification. Moreover, the superelastic, fatigue, and creep behavior of NiTi alloys is enhanced in the [001] direction [64e66]. By optimizing parameters, a strong texture of [001] was achieved by changing the hatch spacing to 80 mm. Figure 6.16 shows three inverse pole figure maps of samples fabricated with different PPs and the correspondent cyclic behavior of them at Afþ 10
C. The sample with the stronger texture had superior properties, and 5.20% of stabilized recoverable strain under compression was achieved. In addition to good pseudoelastic recovered strains, significant shape memory responses could be achieved. As an example, Fig. 6.17(a) reports the results of a strain recovery test at different stresses of an as-built NiTi specimen fabricated by SLM (E ¼ 74 J/mm3). Tests were conducted in the three-point bending configuration at a heatingecooling rate of 2
C/min. This specimen was obtained starting from an NiTi micrometric powder with 50.2 atm% of Ni (Af ¼ 28.5
C); the final part presents shape memory behavior at room temperature with an Af of 54.5
C (Fig. 6.17b). An increasing number of applications with high operational temperature have resulted in the demand for high-temperature SMA (HTSMA) [43]. Adding a third element such as Au, Pt, Hf, or Zr to NiTi binary alloy causes a TT shift of the alloys

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Figure 6.16 Inverse pole figure maps and texture plot of selective laser melted parts fabricated with powder at 250 W, a scanning speed of 1250 mm/s, and hatch spacings of (a) 80 mm, (b) 120 mm, (c) 180 mm and their correspondent cyclic behavior at Afþ 10
C [63]. SLM, selective laser melting.

Manufacturing of shape memory alloys

Compressive Stress (MPa)

800



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Figure 6.17 (a) Strain recovery test of an as-built NiTi specimen produced through the selective laser melting process (E ¼ 74 J/mm3). (b) Differential scanning calorimetry of the starting powder and final job (From CNR ICMATE unit of Lecco).

[67,68]. Because of the low price of Hf compared with Pt and Au, as well as it's good thermomechanical stability Hf addition has become of interest to many researchers [69,70]. Despite the high demand for HTSMA, little work has been reported on AM of NiTiHf. Elahinia et al. [54] first succeeded in fabricating NiTiHf parts through SLM. A slightly Ti-rich Ni49.8Ti30.2Hf20 cast ingot was homogenized at 1050
C for 72 h and then hot-extruded (at 900
C) to improve the thermomechanical properties of the as-cast ingot. Electrode induction-melting gas atomization by TLS Technik GmbH (Bitterfield, Germany) was employed to produce an NiTiHf powder. There was a drop in TTs from the as-extruded to the selective laser melted part, which was explained by oxygen pickup that occurred during the SLM process. Oxygen pickup resulted in the formation of Ti4Ni2Ox or TiO as secondary phases that depleted Ti from the matrix and caused the TTs to shift. A transformation strain of up to 1.5% was reported under isobaric tests. In other work, Nematollahi et al. [51] conducted a comprehensive optimization study on the SLM of Ni-rich NiTiHf. Thirty different sets of PPs were selected to fabricate the NiTiHf coupans. Based on the PPs and energy density, different types of defects, including spherical and irregular pores, delamination, and cracks, were shown (Fig. 6.12). Also, as shown in Fig. 6.12, the more energy density existed, the more size deviation up to 30% was achieved. The wide range in TTs between 100 and 400
C that was reported shows that, similar to NiTi, TTs of NiTiHf are significantly affected by PPs. A chemistry analysis of the as-fabricated powder proved the large amount of Ni evaporation during the SLM process for the parts fabricated with a high energy density. However, energy density is not the only parameter affecting TTs; it was shown that laser power influenced TTs for two parts whereas the energy density was the same.

Manufacturing of shape memory alloys

6.6.6 Heat treatment of selective laser melted NiTi Heat treatment as a powerful postprocessing technique can be used to enhance the thermomechanical behavior of selective laser melted NiTi parts. The duration of heat treatment and temperature are the two main factors that influence the final properties of heat-treated samples [71,72]. Metastable Ni4Ti3 particles are a common secondphase of Ni-rich NiTi created during aging treatment and affect the thermomechanical properties of the alloy. Figure 6.18 shows the effect of the aging temperature and time on the TT of NiTi parts fabricated through SLM. An increase in TTs was shown from the ingot to the as-fabricated parts that resulted from Ni evaporation. At 350 to 500
C, the two stages of DSC peaks showed the presence of R-phase formed during aging. However, at 550 and 600
C, no sign of R-phase was detected. Also, two-stage transformations were formed after 30 minutes of aging at 450
C. Both temperature and duration changed the morphology of the Ni4Ti3 precipitates formed, resulting in multistage TT. Ni4Ti3 precipitates mostly formed along the grain boundary, but ae the oxide regions as

Figure 6.18 The effect of aging on transformation temperatures of Ni-rich NiTi: (a) effect of aging temperature; (b) effect of aging time [72].

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well (Ti4Ni2O). Raising the aging time duration resulted in growth in the size and number of precipitates; hence, the Ni4Ti3 precipitates only present in boundary and also covered the whole grain. When precipitates grew, the intermetallic particles spacing became larger, resulting in the deterioration of precipitate impacts on martensite transformation. In addition, the mechanical response of AM parts can be improved by heat treatment. As shown in Fig. 6.19, solution annealing (S) at 950
C for 5.5 h followed by aging at 350
C significantly improved the recovery strain of the NiTi SLM part compared with the as-received one. However, aging at 450
C caused a large residual strain that was not favorable for stable actuation. Therefore, selecting a proper heat treatment process is important for achieving good thermomechanical behavior. There are other aspects of AM and SMA, such as modeling [73,74], fracture, fatigue and failure [75e77], and polishing effects [78]. Interested readers are encouraged to refer to the cited articles.

6.7 Ecocompatibility of shape memory alloys We consider the problem of ecocompatibility for NiTi alloys, which are the only type of SMA alloy that has begun to have widespread commercial use. The ecological impact of NiTi is not still considered in the literature, probably because the volume of industrial applications of this material is not currently large. However, we can discuss some comments: • The raw material is sufficiently available in the world and the costs of Ni and Ti are actually justifiable from an economic point of view for different kinds of industrial application. • The energy cost for production is comparable to that of other metallic alloys. The process production is currently at a good level of refinement; quasiusual melting processes are used.

Figure 6.19 Influence of two heat treatment process on the superelasticity of selective laser melted Ni-rich NiTi [71]. SLM, selective laser melting.

Manufacturing of shape memory alloys

•

The preparation of product is longer and more complicated than for other metallic alloys; therefore, the great expense in this case is a point that must be considered • To use an NiTi product such as an actuator, in many cases it is necessary to increase the temperature in the material by the Joule effect. Then, the efficiency of this system is not high and the ecocompatibility requirements are well-satisfied only at the miniscale and microscale • The possibility of recycling NiTi alloys is low because remelting the alloys increases the content of impurities, lowering the alloys’ quality, particularly for biomedical applications. Moreover, it is difficult to maintain control of the chemical composition in the remelting process, which cannot occur only between two or more alloys of the same nominal composition. A different point of view is to discuss the possibility of increasing the ecocompatibility of the application for which SMA alloys are used. For example, in the building industry, many devices based on SMA can reduce energy dispersion in air-conditioning and heating systems, or they can increase the time when sunlight is employed. Moreover, some solutions for lighting systems, external and at the home, are studied to optimize the use of light with the precise aim of better energy use. Some examples will be presented in Chapter 8.

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[55] S. Saedi, N.S. Moghaddam, A. Amerinatanzi, M. Elahinia, H.E. Karaca, On the effects of selective laser melting process parameters on microstructure and thermomechanical response of Ni-rich NiTi, Acta Mater. 144 (2018) 552e560. [56] C. Haberland, M. Elahinia, J.M. Walker, H. Meier, J. Frenzel, On the development of high quality NiTi shape memory and pseudoelastic parts by additive manufacturing, Smart Mater. Struct. 23 (10) (2014) 104002. [57] J. Sam, B. Franco, J. Ma, I. Karaman, A. Elwany, J.H. Mabe, Tensile actuation response of additively manufactured nickel-titanium shape memory alloys, Scripta Mater. 146 (2018) 164e168. [58] M. Speirs, et al., On the transformation behavior of NiTi shape-memory alloy produced by SLM, Shape Mem. Superelasticity 2 (4) (2016) 310e316. [59] J. Ma, et al., Spatial control of functional response in 4D-printed active metallic structures, Sci. Rep. 7 (2017) 46707. [60] J. Khalil-Allafi, A. Dlouhy, G. Eggeler, Ni4Ti3-precipitation during aging of NiTi shape memory alloys and its influence on martensitic phase transformations, Acta Mater. 50 (17) (2002) 4255e4274. [61] X. Wang, S. Kustov, K. Li, D. Schryvers, B. Verlinden, J. Van Humbeeck, Effect of Nanoprecipitates on the Transformation Behavior and Functional Properties of a Tie50.8 at.% Ni Alloy with MicronSized Grains, Acta. Materialia. 82 (2015) 224e233. [62] P. Olier, F. Barcelo, J.L. Bechade, J.C. Brachet, E. Lefevre, G. Guenin, Effects of impurities content (oxygen, carbon, nitrogen) on microstructure and phase transformation temperatures of near equiatomic TiNi shape memory alloys, J. Phys. IV 7 (C5) (1997) C5eC143. [63] N.S. Moghaddam, et al., Achieving superelasticity in additively manufactured NiTi in compression without post-process heat treatment, Sci. Rep. 9 (1) (2019) 41. [64] Y. Liu, Z.L. Xie, J. Van Humbeeck, L. Delaey, Effect of texture orientation on the martensite deformation of NiTi shape memory alloy sheet, Acta Mater. 47 (2) (1999) 645e660. [65] S. Paula, et al., Study of the textural evolution in Ti-rich NiTi using synchrotron radiation, Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 246 (1) (2006) 206e210. [66] K. Gall, H. Sehitoglu, R. Anderson, I. Karaman, Y.I. Chumlyakov, I.V. Kireeva, On the mechanical behavior of single crystal NiTi shape memory alloys and related polycrystalline phenomenon, Mater. Sci. Eng. A 317 (1e2) (2001) 85e92. [67] J. Ma, I. Karaman, R.D. Noebe, High temperature shape memory alloys, Int. Mater. Rev. 55 (5) (2010) 257e315. [68] H.E. Karaca, E. Acar, H. Tobe, S.M. Saghaian, NiTiHf-based shape memory alloys, Mater. Sci. Technol. 30 (13) (2014) 1530e1544. [69] D.R. Angst, P.E. Thoma, M.Y. Kao, The effect of Hafnium content on the transformation temperatures of Ni49Ti51-xHfx. Shape memory alloys, J. Phys. IV 5 (C8) (1995) C8eC747. [70] G.P. Toker, et al., Shape memory behavior of NiTiHf alloys fabricated by selective laser melting, Scripta Mater. 178 (2020) 361e365. [71] S. Saedi, A.S. Turabi, M.T. Andani, C. Haberland, H. Karaca, M. Elahinia, The influence of heat treatment on the thermomechanical response of Ni-rich NiTi alloys manufactured by selective laser melting, J. Alloys Compd. 677 (2016) 204e210. [72] S. Saedi, A.S. Turabi, M.T. Andani, N.S. Moghaddam, M. Elahinia, H.E. Karaca, Texture, aging, and superelasticity of selective laser melting fabricated Ni-rich NiTi alloys, Mater. Sci. Eng. A 686 (2017) 1e10. [73] Mehrpouya, Mehrshad, et al., A prediction model for finding the optimal laser parameters in additive manufacturing of NiTi shape memory alloy, Int. J. Adv. Manuf. Technol. (2019) 1e9. [74] Taheri Andani, Mohsen, Modeling, simulation, additive manufacturing, and experimental evaluation of solid and porous NiTi. Diss, University of Toledo, 2015. [75] M. Speirs, B. Van Hooreweder, J. Van Humbeeck, J.P. Kruth, Fatigue behaviour of NiTi shape memory alloy scaffolds produced by SLM, a unit cell design comparison, J. Mech. Behav. Biomed. Mater. 70 (2017) 53e59. [76] P. Bayati, A. Jahadakbar, M. Barati, M. Nematollahi, L. Saint-Sulpice, M. Haghshenas, S.A. Chirani, M.J. Mahtabi, M. Elahinia, Toward low and high cycle fatigue behavior of SLM-fabricated NiTi: considering the effect of build orientation and employing a self-heating approach, Int. J. Mech. Sci. (2020) 105878.

Manufacturing of shape memory alloys

[77] A. Bagheri, M.J. Mahtabi, N. Shamsaei, Fatigue behavior and cyclic deformation of additive manufactured NiTi, J. Mater. Process. Technol. 252 (2018) 440e453. [78] C.A. Biffi, P. Bassani, M. Nematollahi, N. Shayesteh Moghaddam, A. Amerinatanzi, M.J.,. Mahtabi, A. Tuissi, Effect of ultrasonic nanocrystal surface modification on the microstructure and martensitic transformation of selective laser melted nitinol, Materials 12 (19) (2019) 3068.

Further reading [1] S.E. Saghaian, A. Amerinatanzi, N.S. Moghaddam, A. Majumdar, M. Nematollahi, S. Saedi, H.E. Karaca, Mechanical and shape memory properties of triply periodic minimal surface (TPMS) NiTi structures fabricated by selective laser melting, Boil. Eng. Med. 3 (2018) 1e7. [2] S. Saedi, A.S. Turabi, M.T. Andani, C. Haberland, M. Elahinia, H. Karaca, Thermomechanical characterization of Ni-rich NiTi fabricated by selective laser melting, Smart Mater. Struct. 25 (3) (2016), 035005. [3] M. Mohajeri, et al., Nickel Titanium Alloy Failure Analysis under Thermal Cycling and Mechanical Loading: A Preliminary Study, 2018 arXiv preprint arXiv:1803.01110. [4] B. Haghgouyan, et al., Crack Growth Behavior in Niti Shape Memory Alloys under Mode-I Isothermal Loading: Effect of Stress State, 2018 arXiv preprint arXiv:1812.02362.

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CHAPTER 7

Fatigue and fracture Ferdinando Auricchio1, Carmine Maletta2, 3, Giulia Scalet1, Emanuele Sgambitterra4 1

Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universit
a degli Studi di Pavia, Pavia, Italy; 2European Organization for Nuclear Research (CERN), Geneva, Switzerland; 3Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy; 4DIMEG, University of Calabria, Rende, Italy

7.1 Introduction Since their discovery [1], shape memory alloys (SMAs) have found many advanced applications in medical and engineering devices [2e5]. However, in most of their current and potential future applications, SMAs are subjected to fatigue thermomechanical loadings involving cyclic phase transformations. These cause material damage at both the structural and functional level; that is, they trigger crack formation and propagation mechanisms and deteriorate recovery capabilities. A deep knowledge of the evolution of both structural and functional properties of SMAs, under thermomechanical fatigue loadings, represents an important need for designing effective and reliable SMA-based smart components in many high-demanding applications. SMAs exhibit unusual fracture and fatigue responses compared with common engineering metals, owing to their stress- and/or thermally induced microstructural evolutions. As a consequence, well-known theoretical models and standard testing procedures to analyze crack formation and propagation mechanisms, under static or fatigue loads, cannot be directly applied to SMAs. Further complexity comes from the large strain inhomogeneity owing to localized phase transformations occurring even under nominally uniform loading conditions. In addition, marked functional fatigue damage usually occurs during repeated actuations through either thermal and/or mechanical loads, and special methods must be developed for designing SMA components with long-term performance and durability. This chapter gives an overview of methods and models for measuring and predicting both the structural and functional fatigue response of SMAs, together with significant results taken from the scientific and technical literature. In particular, functional fatigue is briefly reviewed in Section 7.3 with a special focus on pseudoelastic SMAs, structural fatigue is examined in Section 7.4 with a highlight on the effects of localized phase transformation phenomena, and finally, crack formation and propagation mechanisms, in terms of stable fatigue crack growth and fracture toughness, are reviewed in Section 7.5.

Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00007-8

© 2021 Elsevier Ltd. All rights reserved.

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7.2 List of symbols a Macroscopic internal variable tensor z Martensite volume fraction bs Mesoscopic shear stress b s h Mesoscopic hydrostatic stress sa Stress amplitude sm Mean stress su Ultimate tensile stress W Dissipated energy Wrec Recovered energy W t Total energy in the stabilized cycle WeD Tensile elastic energy in the stabilized cycle Tmax Maximum temperature in the stabilized cycle W first Dissipated energy in the first cycle s0f Fatigue strength coefficient ε0f Fatigue ductility coefficient s0f Shear fatigue strength coefficient Nf Number of cycles to failure Neq Equivalent fully reversed fatigue life smax Maximum stress sba Bending stress amplitude sa Shear stress amplitude due to torsion sbf Bending fatigue limit sf Torsion fatigue limit sa von Mises equivalent stress amplitude sn;max Maximum normal stress smax Maximum shear stress amplitude a Pmax Maximum hydrostatic pressure min Rs [ ssmax Stress ratio εa Strain amplitude εm Mean strain εmax Maximum strain εa Total equivalent strain amplitude ε Total equivalent strain εtra Equivalent transformation strain amplitude εtr Equivalent transformation strain εel Equivalent elastic strain gmax Maximum shear strain amplitude a εna Normal strain amplitude εn;max Maximum normal strain amplitude a Dεp Plastic strain per cycle εea Elastic strain amplitude εia Inelastic strain amplitude min Rε [ εεmax Strain ratio Af Austenite finish temperature Ms Austenite starting temperature Md Austenite desist temperature T Temperature εres Residual strain εrec Recovered strain

Fatigue and fracture

sAM Martensite start stress s EA Elastic modulus of the austenite EM Elastic modulus of the martensite E Effective elastic modulus E Young’s modulus y Poisson’s ratio

7.3 Functional fatigue A critical issue limiting the application of SMAs in several sectors is the functional fatigue damage occurring during repeated actuations through either thermal and/or mechanical cycles. Although these phenomena were observed in early works by Melton et al. [6] and Miyazaki et al. [7], the term “functional fatigue” was proposed only in 2004 by Eggeler et al. [8]. It indicates functional damage occurring during repeated transformation cycles (i.e., the evolution of strain recovery capabilities and thermal or mechanical hysteresis). After these early works, several studies were carried out to better understand these phenomena [9e15], as well as to analyze the role of localized phase transformation on functional damage at the component scale. In the following subsections, the main features related to the functional fatigue damage of SMAs are reviewed, with a special focus on pseudoelastic SMAs.

7.3.1 Shakedown effect Shakedown is a necessary condition for the safety assessment of structures subjected to cyclic loading. In particular, shakedown analysis classifies the stressestrain response of a pseudoelastic SMA structure under cyclic loading (below plastic yield) in the following categories [16e18]: (1) elastic shakedown, if SMA response is purely elastic (neither martensite reorientation nor phase transformation); (2) alternating phase transformation without martensite reorientation, if the cyclic evolution of martensite volume fraction is stabilized and martensite reorientation strain is not evolving; (3) alternating martensite reorientation without phase transformation, if the cyclic evolution of martensite reorientation strain is stabilized and martensite volume fraction is constant; (4) alternating phase transformation and martensite reorientation, if the cyclic evolution of both martensite volume fraction and martensite reorientation strain is stabilized; and (5) transformation ratcheting, if a cyclic accumulation of strain (including peak and residual strain) is taking place. As an example, responses (1) and (2) are represented in Fig. 7.1(a) and (b), respectively. Depending on such a classification, shakedown behavior is associated with high cycle fatigue (HCF) and low cycle fatigue (LCF) regimes in the fatigue diagram that are respectively defined by a number of cycles to failure, Nf , lower than 103 and between 104 and 106 . Shakedown theorems for SMAs and nonsmooth mechanics have been proposed [16,19e22].

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(b)

M

Stress

(a) Stress

198

A+M

A+M

A

Strain

Strain

Figure 7.1 Two examples of stressestrain responses of a pseudoelastic SMA structure subjected to cyclic loading: (a) elastic shakedown (neither martensite reorientation nor phase transformation) in three different regions of the pseudoelastic path; (b) alternating phase transformation without martensite reorientation. A, Austenite volume fractions; M, martensite volume fraction.

7.3.2 Cyclic stressestrain response The functional fatigue in pseudoelastic SMAs can be defined in terms of the following parameters, as schematically shown in Fig. 7.2: • recovered strain, εrec : recoverable upon unloading by the pseudoelastic effect; • residual strain, εres : ratcheting-like irrecoverable strain after mechanical unloading; • recovered energy, Wrec : energy released by the material after mechanical unloading (i.e., the area below the unloading branch of the hysteresis loop);

Figure 7.2 Schematic representation of evolution of stressestrain response of a pseudoelastic shape memory alloy from the first cycle to the stabilized cycle.

Fatigue and fracture

•

dissipated energy, W : energy dissipated by the material during the loadinge unloading stage (i.e., the area between the loading and unloading paths of the hysteresis loop); • martensite start stress, sAM s : represents the stress required to start the transformation from austenite to martensite (i.e., the intersection between the initial elastic path and the tangent to the stress plateau); • effective elastic modulus, E: represents the elastic modulus of the material in the initial stage of the mechanical response (i.e., the slope of the initial elastic response in the stressestrain diagram for each fatigue cycle). All of these parameters are important for the fatigue characterization of SMAs because they represent a quantitative measure of the cyclic evolution of the pseudoelastic properties of the alloy. Their variation can have a big influence, especially for applications in which the stroke of components or the damping capability represents a design parameter [23]. If a cyclic load is applied to SMAs, the typical evolution of the residual and recovered strain that one can obtain is reported in Fig. 7.3. In particular, figures represent their trend with fatigue cycling for different values of the maximum applied strain, εmax , ranging in the pseudoelastic regime (the one typically employed for damping applications). Fig. 7.3 shows that a rapid increase in εres , and a consequence decrease of εrec , occurs during the first few cycles, and the higher the maximum applied strain, the higher the rate of the evolution of such parameters. This phenomenon occurs in both stress- and strain-controlled experiments, but only if martensitic phase transformation is involved during the fatigue history [7]. However, the generation of residual strain can be attributed to two different reasons: (1) plasticity due to dislocation motions in the material [24,25], and (2) the formation of retained martensite that is unable to transform back in the parent austenitic phase [26]. It is still possible to distinguish between the two mechanisms by performing a thermal flash test [27], which consists of heating the material beyond the martensite desist temperature, Md, for 30 s after cycling, and measuring the evolution of the residual strain. The recovery strain after heating is related to retained martensite, whereas the unrecovered one is attributed to dislocation motions. As reported in Fig. 7.2, repeated mechanical loading causes not only a modification of the strain recovery capability but also a change in the shape of the whole stressestrain loop. In particular, after stabilization, mechanical hysteresis gets narrower, generating a marked reduction of the intrinsic capability in dissipating energy. In addition, if one compares the first stressestrain loop with the one obtained after the stabilization of the material, it is also possible to observe a reduction in the stresses required for the direct and reverse transformation and an increase in the mechanical slope during the transformation.

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Figure 7.3 Evolution of deformations with fatigue cycling as a function of the maximum applied strain: (a) evolution of the residual strains, (b) evolution of the recovered strains.

Figure 7.4 Evolution of the energy with fatigue cycling as a function of the maximum applied strain: (a) evolution of the dissipated energy, (b) evolution of the recovered energy. Fatigue and fracture

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Figure 7.5 Evolution of the martensite start stress with fatigue cycling as a function of the maximum applied strain. (Reprinted from C. Maletta, E. Sgambitterra, F. Furgiuele, R. Casati, A. Tuissi, Fatigue properties of a pseudoelastic NiTi alloy: Strain ratcheting and hysteresis under cyclic tensile loading, International Journal of Fatigue, 66, 78-85, Copyright (2014), with permission from Elsevier [14].)

Fig. 7.4 reports the evolution of recovery energy Wrec (Fig. 7.4(a)) and dissipated energy W (Fig. 7.4(b)), whereas direct stress transformation sAM is reported in Fig. 7.5 as a s function of the fatigue cycles and for different values of maximum applied strain, εmax . It can be observed that sAM s , Wrec , and W decrease with cycling and, as is true for the deformations, their evolution rate depends on the strain amplitude until complete stabilization. The reason for this behavior is attributed to the accumulation and saturation of residual strain. With an increase in the number of cycles, the slip deformations generate an increase in the dislocation density, which obstructs the generation of further martensite in a manner similar to strain hardening in plasticity. As a consequence, the slope of the transformation path in the stressestrain curve during loading increases. In addition, because the value of the maximum and minimum stressestrain is typically fixed, the size of the hysteresis loop necessarily decreases. When the residual strain accumulation stops, the hysteresis loop stabilizes in a way comparable to plastic shakedown [17] until the specimen fails. Furthermore, accumulated martensite generates an internal stress field close to the variant bands that, assisting the applied stress, causes the decrease in stress required for inducing further transformation by cycling. Finally, an evolution of the effective elastic modulus, E, during the mechanical cycles can be observed (Fig. 7.6). In particular, starting from the initial value of the austenitic microstructure, EA ¼ 68 GPa, it decreases in the first few cycles, with more evidence

Fatigue and fracture

Figure 7.6 Evolution of effective elastic modulus with fatigue cycling as a function of the maximum applied strain. (Reprinted from C. Maletta, E. Sgambitterra, F. Furgiuele, R. Casati, A. Tuissi, Fatigue properties of a pseudoelastic NiTi alloy: Strain ratcheting and hysteresis under cyclic tensile loading, International Journal of Fatigue, 66, 78-85, Copyright (2014), with permission from Elsevier [14].)

when a big strain is applied. This behavior is justified by the generation of a heterogeneous microstructure during the stabilization process, such as the formation of stabilized martensite, detwinning, grain reorientation, and slips deformations [7,28]. This heterogeneous microstructure is characterized by an effective elastic modulus that lies between the one of the austenite (EA) and martensite (EM), as illustrated in Fig. 7.6. One can observe that if a high strain is applied, εmax ¼ 4:5%, the effective modulus achieves stabilization around 45 GPa, which is close to the elastic modulus of the martensite. In this loading condition, the amount of martensite that is generated and that accumulates cycle by cycle is high; therefore, the mechanical response of the material is practically dictated by the big volume fraction of residual martensite. It was pointed out earlier that the functional evolution of SMAs is faster during the first cycles, and then it tends to saturate. Stabilization typically occurs after 100e140 cycles independently on the applied strain amplitude [14]. This is an unusual behavior compared with common metals; in the latter, stabilization of the mechanical stressestrain response, due to continuous plastic deformations, is related to the cycles to failure, and consequently, to the strain amplitude [29]. However, stabilization of SMAs is strain independent only if the load frequency is not high, because the evolution of SMA is strongly rate dependent [15,30]. If a pseudoelastic SMA is loaded using different strain rates, the response that one can obtain is reported in Fig. 7.7.

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Figure 7.7 Strain rate-dependent cyclic stressestrain curves evaluated at different fatigue cycles. (Reprinted from O. Ammar, N. Haddar, L. Dieng, Experimental investigation of the pseudoelastic behaviour of NiTiwires under strain- and stress-controlled cyclic tensile loadings, Intermetallics, 81, 52-61, Copyright (2017), with permission from Elsevier [15].)

In particular, figures demonstrate that independently of the loading rate, the mechanical response shows the typical functional evolution of a SMA; however, there is some change according to the loading rate. First, with a higher strain rate, stabilization of the material is quicker; furthermore, the stress required to start the transformation from austenite to martensitic increases with the strain rate moving from 530 MPa for the lowest applied strain rate f ¼ 3:3$104 s1 to 631 MPa when the strain rate is increased up to 1:6$102 s1. This behavior has to be attributed to the thermomechanical coupling effect involving SMAs. This latter is generally related to the thermoelastic austenite to martensite phase transformation, according to which the forward phase transformation is exothermic whereas the reverse one is endothermic. Two heat sources are responsible for the temperature variation of the material during stress-induced transformation: (1) the intrinsic dissipation and (2) the latent heat. When the SMA is subjected to cyclic loading, the accumulation of the mechanical dissipation heats the material and the absorption and release of the latent heat lead to temperature variations that influence the mechanical response of the material, as reported in Fig. 7.6. When the temperature increases, austenite becomes more stable and a higher stress is required to allow the direct phase transformation to start; when the temperature decreases, however, martensite becomes more stable and the stress must then be decreased further to allow the reverse phase transformation [31]. Consequently, the dissipated energy changes with the applied loading rate. Under cyclic loading, the temperature of the material oscillates cyclically as a function of the loading rate. Higher loading rates lead to a weaker heat exchange with the surroundings, which generates more heat accumulation inside the material

Fatigue and fracture

(i.e., an increase in its local temperature). Thus, rate dependence seems to result from strong thermomechanical coupling, pseudoelasticity degeneration, and temperature dependence [32]. All of these mechanisms exist not only in uniaxial cases, but also when multiaxial loadings are applied to SMA components. Knowledge of the behavior of SMAs under such operating conditions is greatly interesting; as reported in Robertson et al. [33], multiaxial stresses are often encountered in the service process of implanted stents at human joints. The functional evolution of SMAs in multiaxial loading exhibits a behavior similar to the uniaxial case [34e38]. It means that all parameters previously described show a similar trend. However, when working with nonproportional multiaxial loading, compared with the uniaxial and proportional multiaxial one, transformation ratcheting is much higher [39]. The reason can be attributed to two different effects: i. During nonproportional multiaxial cyclic loading, the directions of principal stresses change continuously in each cycle. This mechanism in polycrystalline SMAs enables martensitic transformation in multiple grains with different crystallographic orientations leading to more progressive transformation ratcheting in the first initial cycles compared with the uniaxial case. In this latter, in fact, the direction of principal stress is always the same, and transformation ratcheting occurs in only a few grains favorably orientated in the loading direction. ii. During multiaxial cyclic loading, cyclic transformation is activated in more martensite variants compared with the uniaxial cases, which results in more remarkable transformation-induced plasticity that occurs not only at the interfaces between austenite and martensite phases but also at the interfaces between different martensite variants [40]. Considering this particular behavior of SMAs and their change in mechanical response with cycling, it could be interesting to find a way to get a more stable response under operating conditions. A key parameter that considerably influences the fatigue response of SMAs is the final treatment of the material. Several different methods are proposed in the literature to realize a more stable SMA using different treatment techniques [41]. The combination of cold work and aging treatments, instead of annealing and aging treatments, is the one recommended to achieve an optimized state of microstructure that guarantees a more stable SMA under cyclic loading.

7.4 Structural fatigue The structural fatigue of SMAs refers to degradation of the mechanical properties of the alloy under cyclic loading, which may lead to failure owing to the nucleation and propagation of defects.

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Different experimental approaches (either stress- or strain-controlled) can be employed to study the structural fatigue of SMAs. Early experimental studies focused on bending and rotating-bending fatigue failures of wires. Later, because of the wide interest in NiTi alloys in the biomedical field, several studies focused on tube, dog bone, or diamond-shaped samples, similar to those used to manufacture stents or stent subcomponents. The selection of the most appropriate experimental setup is generally based on parameters such as the type of loading to which the component is subjected in reality, as well as material mechanical behavior. Most work on the fatigue of NiTi SMAs have involved strain-controlled setups and thus adopted a strain-life approach. Many biomedical applications (e.g., stents) are subjected to cyclic deformations, and strain-controlled testing procedures are more appropriate for their fatigue analysis. However, NiTi alloys are also often used for force-carrying applications (e.g., reinforcing bars in concrete bridges, endodontic files), in which stress-controlled loading induces an increase in deformation in the first tens or hundreds of cycles, and SMA fatigue response is studied by employing a stress-life approach. Based on experimental observations, a variety of life-prediction models have been also proposed for SMAs. Most are formulated in a one-dimensional framework in terms of uniaxial strain or stress quantities; some three-dimensional models in terms of equivalent strain or stress quantities or energetic quantities have been also formulated to examine the multiaxial state of deformation, which is common in SMA applications. In the case of energetic quantities, as the dissipated energy, an energy-based approach is adopted. In the following, a review of the experimental and modeling contributions based on the strain-life, stress-life, and energy-based approaches related to the fatigue crack initiation stage will be presented and discussed for austenitic and martensitic NiTi alloys (i.e., alloys tested at a temperature, T , respectively, above Af and below Ms ). Contributions to fatigue crack propagation and growth will be discussed in the next section. For a comprehensive review, the reader is referred elsewhere [8,33,42e45].

7.4.1 Strain-life approaches Strain-life studies first investigated the fatigue behavior of martensitic and austenitic specimens tested at different temperatures under constant strain amplitude, εa , revealing fatigue lives of martensitic NiTi specimens higher than those of austenitic NiTi specimens [6,46e48]. Melton and Mercier [6] studied the fatigue behavior of Ni55Ti45 specimens (Ms ¼ 30  C) at room temperature (w22  C) under fully reversed conditions and proposed a Manson-Coffin law to model LCF results: nεp ¼ aðNf Þb

(7.1)

Fatigue and fracture

where nεp is the plastic strain per cycle, Nf is the number of cycles to failure, and a and b are fatigue parameters. Miyazaki et al. [46] performed rotary bending tests on martensitic and austenitic Ni50Ti50 specimens at different testing temperatures. A reverse relationship between the testing temperature and fatigue life was found for T > Af and εa > 0:7%. Moreover, a convergence of strain-life curves at about 105 cycles was found, implying the independence of the strain-life curves to temperature in the longlife regime. Tobushi et al. [48] performed rotating-bending tests on Ni55.3Ti44.7 wires (Af ¼ 50  C) at temperatures of 30, 60, and 80  C and adopted a power law to describe the obtained data for εa > 0:8% (Nf < 104 ): εa ¼ aðNf Þb

(7.2)

where a and b are fatigue parameters. To consider the dependence of the fatigue life on the testing temperature, T , Tobushi et al. [49] and Matsui et al. [50] defined a as: a ¼ as $10aðT Ms Þ

(7.3)

where as and a are material parameters. To examine martensite transformations in austenitic SMAs, which are not considered by the law reported in Eqn. (7.3), Maletta et al. [51] proposed the following modified Manson-Coffin model: εa ¼ εea þ εia ¼ Cð2Nf Þc þ Dð2Nf Þd

(7.4)

where εea and εia are the elastic and inelastic strain amplitudes, respectively, whereas C, D, c, and d are material parameters. The Coffin-Manson model was applied to stress-induced martensitic NiTi alloys and may not be applicable to fully austenitic or thermally martensitic ones. Kollerov et al. [52] applied rotating-bending tests on NiTi alloys with Ni contents of 55.7%, 55.8%, and 54.7% with strain amplitudes between 0.75% and 4.8%. Except for one alloy, all specimens were austenitic at the testing temperature of w21  C. Kollerov et al. [52] proposed a modified Coffin-Manson model in terms of the maximum recoverable strain, which can be completely recovered after unloading for austenitc Nitinol, or by heating to a temperature above Af , for thermally martensitic Nitinol. In the modified equation, the strain exponent, b, depends on the stress plateau and modulus of elasticity, whereas the strain coefficient, a, is a function of the maximum recoverable strain, stress plateau, and modulus of elasticity. A good correlation was found for 2% < εa < 5%. Pelton et al. [53] applied rotating-bending tests on Ni50.8Ti49.2 specimens with Af ¼ 2  C tested at 25, 23, and 60 C. The obtained fatigue data of austenitic NiTi alloys were divided into four regimes in log-log εa  Nf plots (Fig. 7.8):

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Figure 7.8 Strain-life fatigue data related to Ni50.8Ti49.2 specimens with Af ¼ 2  C tested at 25, 23, and 60  C. (Reprinted from A.R. Pelton, J. Fino-Decker, L. Vien, C. Bonsignore, P. Saffari, M. Launey, M.R. Mitchell, Rotary-bending fatigue characteristics of medical-grade Nitinol wire, Journal of the Mechanical Behavior of Biomedical Materials, 27, 19-32, Copyright (2013), with permission from Elsevier [53].)

1. very low-cycle regime (Nf  103 ), related to very high strain amplitudes. The material is fully martensitic and exhibits both elastic and plastic behaviors. The fatigue life decreases almost linearly with increasing strain amplitude. 2. low-cycle regime (Nf w103 ), related to high strain amplitudes. The material is in the mixed austenite/stress-induced martensite phase state. The fatigue life is almost independent of the strain amplitude, owing to the dominating role of stress, rather than strain, in the stress plateau region. 3. midcycle regime (103  Nf  105 ), related to medium strain amplitudes. The fatigue behavior is related to the transition between the linear elastic behavior in austenite phase to stress-induced martensite phase transformation. The fatigue life constantly decreases as the strain amplitude increases. 4. high cycle regime (Nf  105 ), related to low strain amplitudes. The material behaves elastically. The fatigue life increases slightly as the strain amplitude decreases. A log-log linear relationship (Eqn. 7.2) was used in first, third, and fourth regions with b coefficients that decrease with an increase in temperature. The b values were 0:3 0:38 in the first and third regions, indicating that the rate of fatigue degradation was almost the same in these regions, whereas they were between 0.03 and 0.07 in the fourth region, indicating faster degradation of the material with increasing strain. Effects of tensile mean strain on NiTi fatigue life were shown to be beneficial or detrimental, depending on the applied strain level. The so-called constant-life diagram is commonly used to compare results for different combinations of mean strains, εm , and strain amplitudes, εa ; at a certain run-out life (e.g., 106 or 107 cycles), as performed, for example, by Tolomeo et al. [54] and Pelton et al. [55] on austenitic SMAs at 37 C (Af ¼ 28 and 30  C, respectively). According to Tolomeo et al. [54], the alternating strain increases with an increase in mean strain up to a certain mean strain amplitude

Fatigue and fracture

(e.g., 3%); beyond 3% mean strain, the tolerable alternating strain decreases as the mean strain increases without a failure of the specimen. Pelton et al. [55] confirmed similar effects and included compressive mean strains calculated from finite element analyses of stent-like components. Their findings indicate that the alternating strain decreases as the mean strain increases up to about 1e1.5%. Above such a mean strain, the alternating strain tends to increase with an increase in the mean strain level, owing to the formation of stress-induced martensitic phase. These findings were also confirmed by Morgan et al. [56]. Tabanli et al. [57] observed a shorter fatigue life for austentic NiTi specimens on the stress plateau region as the mean strain increased. They considered the sharp edges between different phases to be a possible cause for the shorter fatigue life in the presence of mean strains. Results by Pelton et al. [58], related to austenitic diamond-shaped and microedog bone specimens, are reported in Fig. 7.9. For mean strains between 1.5% and 3%, fatigue life increases from 0.4% to 0.6% strain amplitude; a dotted line at 0.6% strain amplitude between 3% and 7% mean strain indicates that there are insufficient data for a complete analysis. Above 7% mean strain, constant-life data exhibit a negative slope. According to Pelton et al. [58], the enhanced fatigue life between 1.5% and 7% mean strains results from microstructural effects of stress-induced martensite, and the mixed stress-induced martensite/austenite alloy is able to accommodate a greater amount of strain amplitude, leading to longer fatigue-lives than either austenite or martensite. The effects of metallurgical phases on the fatigue behavior of the austenitic NiTi alloys in the presence of mean strains were studied by Pelton [41]. Another consequence of stress-induced phase transformation in austenitic NiTi alloys is that the modulus of elasticity is phase dependent [33]. Such dependence significantly

Figure 7.9 Constant life diagram from diamond stent sub-component and microdogbone specimens fatigue testing. Conditions that survived the 107 cycle testing are shown as open triangles/squares. (Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Journal of Materials Engineering and Performance Nitinol Fatigue: A Review of Microstructures and Mechanisms, A. R. Pelton, 20:613-617, Copyright (2011) [41].)

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influences the material cyclic response in the presence of mean strains. The summarized results on mean stress effect clearly indicate that Goodman or Soderberg models, which are classically used for metals, cannot be directly applied to SMAs. Accordingly, Mathabi and Shamsaei [59] evaluated the applicability of various mean stressestrain correction models for a fatigue life analysis of austenitic NiTi alloys under strain-controlled fatigue tests with Rε ¼ e1, 0, and 0.5. They analyzed the data by relating the strain amplitude, εa , to an equivalent fully reversed fatigue life, Neq , using a Coffin-Manson-type equation as: εa ¼

s0f E

ð2Neq Þb þ ε0f ð2Neq Þc

(7.5)

where s0f is the fatigue strength coefficient, ε0f is the fatigue ductility coefficient, E is the Young’s modulus, and b and c are material parameters. Neq was calculated based on the following mean stress correction models: • the Goodman (G) equivalent fully reversed life, Neq;G , written as:  Neq;G ¼ Nf

sm 1 su

1=b (7.6)

where sm is mean stress, su is the ultimate tensile stress of the material, and b is the fatigue strength exponent; • the SmitheWatsoneTopper (SWT) equivalent fully-reversed fatigue life, Neq;SWT , written as:  Neq;SWT ¼ Nf

1  Rs 2

1=2b (7.7)

where Rs is the stress ratio and b is the fatigue strength exponent; • the Walker (W) equivalent fully reversed fatigue life, Neq;W , written as:  Neq;W ¼ Nf

1  Rs 2

ð1gÞ=b (7.8)

where g is a fitting constant, which is adjustable based on the sensitivity of the material to mean stress; and • The Kwofie (K) equivalent fully reversed fatigue life, Neq;K , written as:

Fatigue and fracture

 Neq;K ¼ Nf e

 a ssmu

=b

(7.9)

where a accounts for the sensitivity of the fatigue life to mean stress; Also, Mathabi and Shamsaei [59] analyzed Morrow’s strain-based mean stress correction model: εa ¼

s0f  sm E

ð2Nf Þb þ ε0f ð2Nf Þc

(7.10)

and the SWT damage parameter: εa smax ¼

 0 2 sf E

ð2Nf Þ2b þ s0f ε0f ð2Nf Þbþc

(7.11)

where smax is the maximum stress. Mathabi and Shamsaei [59] plotted the strain amplitude against equivalent fully reversed fatigue lives based on the described models (Fig. 7.10). As shown, none of the employed mean stress correction models in strain-life analysis provided satisfactory results for austenitic NiTi. Studies on the torsional fatigue of Nitinol alloys are still limited in the current literature (e.g., [60,61]). Unlike rotating-bending and uniaxial fatigue, the torsional fatigue behavior of both martensitic and austenitic Nitinol is similar to that of standard metals, whereas the effect of mean shear strain is not necessarily detrimental. However, more comprehensive experimental and analytical works are needed to evaluate the effect of stress-induced martensite under different shear load ratios. Runciman et al. [61] studied Ni50.8Ti49.2 with Af ¼ 16 and 22  C at testing temperatures of 25 and 37  C, respectively. Tests were continued to failure or run out (107 cycles) under Rε ¼ e1, 0, 0.2, and 0.6. Shear strain-life fatigue data were described by using a Coffin-Manson model in terms of equivalent quantities, as: εa ¼ g þ aðNf Þb

(7.12)

where g, a, and b are material parameters and εa is the total equivalent strain amplitude. The total equivalent strain, ε, is defined as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 (7.13) ε ¼ ðεt Þ2 þ ðεs Þ2 3 where subscripts t and s denote, respectively, tensile and shear stress quantities. Such a formulation did not work well for very high mean strains (Rε w0:88  0:99). Accordingly, an alternative approach for 1  Rε  0:99 was proposed, as:

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Figure 7.10 Strain amplitude versus equivalent fully-reversed fatigue lives for different strain-based mean stress correction models: Goodman, SWT, Walker, Kwofie, and SWT parameter. (Reprinted from Mohammad J. Mahtabi, N. Shamsaei, A modified energy-based approach for fatigue life prediction of superelastic NiTi in presence of tensile mean strain and stress, International Journal of Mechanical Sciences, 117, 321-333, Copyright (2016), with permission from Elsevier [59].)

εtra ¼ aðNf Þb

(7.14)

where εtra is the equivalent transformation strain amplitude. The equivalent transformation strain εtr is defined in terms of the total equivalent strain, ε, and the equivalent elastic strain, εel :

Fatigue and fracture

εtr ¼ ε  εel with:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 4 2 ε ¼ εelt þ εels 3 el

(7.15)

(7.16)

Such an approach provided good predictions for torsion, tension/tension, and bending over the entire range of Rε s. To consider the effects of the Poisson’s ratio, y, the equivalent von Mises (VM) strain was used [42]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1 þ yÞ2 ðεt Þ2 þ 3ðεs Þ2 (7.17) ε¼ 1þy However, the model was not validated on multiaxial loading with both axial and shear strains applied simultaneously. Few studies address the multiaxial fatigue behavior of NiTi alloys. The VM equivalent strain in uniaxial fatigue data is a common approach to multiaxial predictions. However, in general, such a method does not work well for nonproportional loading and typically underestimates fatigue damage. Berti et al. [62] investigated the fatigue behavior of Nitinol peripheral stents through different fatigue criteria (strainequivalent or critical plane-based approaches) under proportional and nonproportional loads. In particular, Berti et al. [62] considered: • the VM fatigue index, calculated from the alternate values of the principal strains, is: VM ¼ •

1 pffiffiffi 2ð1 þ nÞ 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDε1  Dε2 Þ2 þ ðDε2  Dε3 Þ2 þ ðDε3  Dε1 Þ2

(7.18)

the Fatemi-Socie (FS) fatigue index, a function of the maximum shear strain amplin;max (both acting on the plane tude, gmax a , and the maximum value of normal stress, s of the maximum shear strain amplitude):   sn;max FS ¼ gmax 1 þ k a sy

(7.19)

where sy is the material monotonic yield strength and k is a material parameter. • The Brown-Miller (BM) fatigue index, defined as an equivalent shear strain amplitude, given by a combination of the maximum shear strain amplitude, gmax a , and the normal strain amplitude, εna , both occurring in a cycle on the plane experiencing the maximum shear strain amplitude:

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BM ¼ gmax þ Sεna a

(7.20)

where S is a material parameter. • The SWT fatigue index, defined in terms of the maximum normal stress, sn;max , and the maximum normal strain amplitude, εan;max , both occurring on the plane of maximum normal strain: SWT ¼ sn;max εan;max

(7.21)

The variables used in these models, as the equivalent strain amplitude, are taken from the stable cyclic stressestrain hysteresis loop. The constant life results by Berti et al. [62] showed that the VM criterion is always the most conservative, whereas the critical planebased criteria (i.e., FS, BM) are less conservative and vary depending on the applied loading. The basic variables used in these models are the strain amplitude and equivalent strain amplitude, which are taken from the stable cyclic stressestrain hysteresis loops of NiTi SMAs. More comprehensive studies regarding the multiaxial fatigue of NiTi alloys that consider the effects of multiaxial deformation paths as well as phase transformation are needed to formulate more accurate multiaxial fatigue predictive models. The effect of prestrain on the fatigue life of Nitinol was analyzed in Senthilnathan et al. [63] for different testing cases, namely rotary bending, tensionetension, and diamond specimens. In rotary bending fatigue, wire samples were pre-trained in tension from 0 to 10 with 2% increments before they were subjected to fully reversed loading at different strain amplitudes. The fatigue life was shown to increase with amount of prestrain, especially when the material starts deforming plastically. In tensionetension fatigue, wire samples were prestrained in tension (4%, 6%, 9%, 10%, and 11%), released to 2% mean strain, and then tested at different strain amplitudes. Although the resulting strain amplitudes were smaller compared with rotary bending fatigue test, the effect of prestrain on fatigue life improvement was noteworthy. In diamond specimen testing, the samples were prestrained in both tension and compression to 9% before they were subjected to cycling at 3.5% mean strain. In the case of prestraining leading to compressive residual stresses at locations exposed to tension during cycling, the fatigue life improved, whereas in the case of prestraining leading to tensile residual stresses at locations exposed to tension during cycling, fractures appeared. Senthilnathan et al. [63] observed that, compared with standard metals, austenitic NiTi alloys create unusually large residual stresses after inhomogeneous deformations that arise from macroscopic inhomogeneities (bending or torsion), voids, hard inclusions, and crystallographic alignment of grains. Moreover, they observed that stress concentration sites can be made less or more effective depending on the deformation history.

7.4.2 Stress-life approaches Stress-life representations of the fatigue data are less often used for Nitinol alloys.

Fatigue and fracture

In general, austenitic NiTi alloys exhibit greater fatigue resistance in the stress-life approach, whereas martensitic NiTi alloys typically have greater lives in the strain-life approach. The greater fatigue lives observed in stress-life analyses can be attributed to the greater stress carrying capacity of austenitic NiTi alloys compared with martensitic ones [6,64,65]. Work by Melton and Mercier [6] includes a stress-life approach to study the 107 fatigue limit of specimens with different nominal compositions and martensite starting temperatures, Ms . Force-controlled tests were performed under fully reversed conditions at room temperature (w22  C). Results reveal that the fatigue limit at room temperature decreases with an increase in Ms . Miyazaki et al. [64] studied Ni50.8Ti49.2 specimens with Af ¼ 27  C at different testing temperatures. These data indicate that for the stress-life fatigue approach and T > Af , the fatigue life increases as the testing temperature increases. Stress-life fatigue data by Kim and Miyazaki [46] on Ni50.9Ti49.1 with Af ¼ 60 and 37  C were analyzed by Pelton et al. [65], who reported a temperature-dependent fatigue limit that increased with increasing testing temperature toward Af and a drastic reduction in fatigue life with increasing stress amplitude. Stress-life data for austenitic NiTi wires were described by a linear relationship in the semilog plot. Moumni et al. [66,67] also reported that the stress-life fatigue results for superelastic Ni48.7Ti51.3 dog bone shape can be expressed in a linear form in the semilog plot. Moreover, the fatigue lives under fully reversed conditions were shown to be greater than those lives under pulsating loading (Rs ¼ 0). Another study examining the effects of nonzero mean stress at room temperature (w22  C) on the fatigue life of NiTi tubes with Af between 3 and 6  C was performed by Tabanli et al. [68]. Uniaxial tensile experiments were performed in the strain-controlled condition at 20 Hz frequency in a noncontrolled environment. Data from Tabanli et al. [68] do not indicate a specific relation between mean stresses and fatigue life for austenitic Nitinol alloys. The results, which were different from those obtained in other studies [66], may be attributed to the greater testing frequency in the experiment by Tabanli et al, which may induce latent heat in the specimen and affect material properties. Nayan et al. [12] also investigated the uniaxial tensile fatigue behavior of Ni50.7Ti49.1 flat dog bone shapes (As ¼ 1  C, Af ¼ 23  C) at room temperature (w22  C) and Rs ¼ 0:1. Stress-life data indicate two linear sections that have a common point at a stress level close to the critical stress required for inducing martensitic transformation. Kang et al. [13] studied the influence of mean stresses on the fatigue behavior of martensitic NiTi alloys. Similar to other metals, their observations show that compressive mean stress improves fatigue resistance, whereas tensile mean stresses seem to be detrimental. Mathabi and Shamsaei [59] analyzed some stress-based mean stress correction models to calculate the equivalent fully reversed stress amplitude, sa , employed in the power-law equation: sa ¼ s0f ð2Nf Þb

(7.22)

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•

In particular, Mathabi and Shamsaei [59] considered: the Goodman (G) equivalent fatigue strength, sa;G : sa;G ¼

•

(7.23)

the SWT equivalent fully reversed stress, sa;SWT : sa;SWT ¼

•

sa sm 1 su

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sa smax

(7.24)

the Walker (W) equivalent fully reversed stress, sa;W : sa;W ¼ ðsa Þg ðsmax Þ1g

(7.25)

where g is a fitting constant, adjustable based on material sensitivity to mean stress; and • the Kwofie (K) equivalent fully reversed stress, sa;K :  sa;K ¼ sa e

 a ssmu

(7.26)

where a accounts for the sensitivity of the fatigue life to mean stress. None of the considered mean stress correction models provided satisfactory results. Despite all of these contributions, there are few data in the literature to generalize the effects of the stress ratio on the stress-life fatigue behavior of Nitinol alloys. Predki et al. [69] studied the effects of specimen type on the torsional fatigue behavior of austenitic Nitinol alloys. Those authors observed greater fatigue lives for solid specimens, which is agrees with the observations reported for the effects of a cross-sectional shape on the torsional fatigue of steels. Jensen [70] studied the multiaxial fatigue of martensitic NiTi alloys (Ni49.9Ti50.1) under proportional tension-torsion tests. The tests were conducted at 23 C to specimen failure or run out (105 cycles) and five fully reversed axial stress amplitudes (740, 560, 500, 250, and 125 MPa) were combined with a fully reversed shear stress amplitude of 250 MPa. Jensen [70] suggested the following relation between the VM equivalent stress amplitude, sa , and the fatigue life: sa ¼  85:766 ln Nf þ 1859:8

(7.27)

Jensen [70] also reported that the fatigue lives for the combinations of 250 and 125 MPa axial stress and 250 MPa shear stress were similar to those obtained under pure torsion loading. Therefore, no significant effect from axial loading may result in the presence of considerable torsional loads. Moreover, they reported that the fatigue

Fatigue and fracture

lives for the combinations in which tension was more dominant were lower compared with other loading conditions. Mahtabi and Shamsaei [71] analyzed the torsional and in-phase multiaxial fatigue data generated by Jensen [70] through the following multiaxial stress-based and critical plane fatigue models: • a classical uniaxial stress-based model: sa ¼ s0f ð2Nf Þb

(7.28)

The equivalent stress amplitude, sa , was first defined as the VM equivalent stress: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 (7.29) sa ¼ pffiffiffi s2xa þ s2ya þ s2za þ 3 s2xya þ s2xza þ s2yza 2 where sxa , sya , and sza are alternating normal stresses and sxya , sxza , and syza are shear stress amplitudes. Then, sa was defined using the VM equivalent stress and the hydrostatic stress, according to the Sines model:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 seq ¼ pffiffiffi s2xa þ s2ya þ s2za þ 3 s2xya þ s2xza þ s2yza þ mðsmx þ smy þ smz Þ 2 (7.30) where smx , smy , and smz are mean normal stresses and m is the coefficient of the mean stress influence; • the Gough model: 

sba sbf

2

 2 sa þ ¼ 1 sf

(7.31)

where sba is bending stress amplitude, sa is the shear stress amplitude due to torsion, sbf is the bending fatigue limit, and sf is the torsion fatigue limit; • the Findley stress-based critical plane model: 

sa þ ksn;max

 max

¼ s0f ð2Nf Þb

(7.32)

where k is a material constant that indicates the sensitivity of the material to normal stress, and s0f and b are the shear fatigue strength coefficient and exponent, respectively; • the McDiarmid stress-based critical plane model: smax þ a

sA;B sn;max ¼ s0f ð2Nf Þb 2su

(7.33)

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Shape Memory Alloy Engineering

where smax is the maximum shear stress amplitude, sA;B is the fatigue strength for case A a (cracking is along the surface, such as in torsion) or case B (cracking is into the surface, such as cracking under bending loads); and • the Fatemi-Socie stressestrain-based critical plane model:  smax a

sn;max 1þk sy



¼ s0f ð2Nf Þb

(7.34)

where k is a material constant. No fatigue failure is expected without the presence of alternating shear loads. Mahtabi and Shamsaei [71] found that the predictions for torsional and biaxial loadings might reach a 20e100 times error band, implying that classical fatigue models are inappropriate for NiTi SMAs. Significantly nonconservative fatigue life predictions were found for the VM model. Because the fatigue data are based on tests conducted in fully reversed condition, Mahtabi and Shamsaei [71] obtained the same results for the VM and Sines models. The Gough model was not used because it requires bending and torsion fatigue limits, which were unavailable. Findley and McDiarmid models have only stress terms in the fatigue parameter and are more appropriate for an HCF regime. They provided considerably nonconservative life predictions. The Fatemi-Socie model has both stress and strain terms and may be used for HCF and LCF regimes. Among all of the models, the Fatemi-Socie-type model provided a better prediction. Song et al. [37,72] focused on the nonproportional multiaxial fatigue failure of NiTi microtubes. They considered various stress levels and nonproportional multiaxial paths (square, hourglass type, butterfly type, rhombic, and octagonal). The fatigue lives under multiaxial loading were shown to be much shorter than those under uniaxial loading and to depend significantly on the loading paths; the square path was more damaging than the others. In general, more comprehensive experimental and analytical investigations involving in-phase and out-of-phase load paths are needed to better understand the multiaxial fatigue behavior of NiTi SMAs and formulate a reliable multiaxial fatigue model. Auricchio et al. [17] investigated the cyclic response of SMAs under macroscopic elastic shakedown and proposed a criterion for HCF. The criterion is multiaxial and is based on a multiscale analysis of phase transformation between austenite and martensite the framework of standard generalized materials. It affirms that a structure subjected to cyclic loading has an infinite lifetime expressed as an elastic shakedown state at both the macroscopic and mesoscopic scale, if: maxfbs ðx; tÞ þ aðaðxÞÞb s h ðx; tÞg  bðaðxÞÞ t>t0

(7.35)

Fatigue and fracture

for all points x of the structure. If condition (7.35) is not respected in a point, fatigue crack will initiate and the structure will have a finite lifetime. Here, bs is the mesoscopic shear stress and b s h is the mesoscopic hydrostatic stress. The material parameters requested to calibrate the Dang Van line (i.e., a and b) should be material dependent (through internal variable a) and independent on the testing conditions (Fig. 7.11). The criterion was verified on uniaxial experimental data by Pelton [41] by using the Souza-Auricchio model and was applied to stents [73]. Later, Moumni et al. [18] proposed a shakedown-based HCF criterion for SMAs, obeying the Zaki-Moumni model, defined as: GðxÞ þ aPmax  1

(7.36)

where Pmax represents the maximum value of the hydrostatic stress and a is a material constant. The fatigue factor, G, is defined as: 8     > 1 w * r *  1 w * r *  > > >  þ  þ    if 0  z  1 < 2 b 2 b c c (7.37) GðxÞ ¼ > * > v > > if z ¼ 0 : b0 where b and c are two material constants corresponding to fatigue limits with respect to phase transformation and martensite orientation, which define the maximum size of the safe domain of the material. The safe domain consists of a finite hypercylinder parallel to the direction of martensite orientation. When z ¼ 0, the martensitic phase is absent and the elastic domain reduces to a hypersphere.

Figure 7.11 (Left) Illustration of the Dang Van (DV) criterion in the bs eb s h plane. The stress space is split in the infinite and finite lifetime by the DV line (red: light gray in printed version). (Right) Set of DV lines in the bs eb s h plane for a fixed number of cycles, N. Each line is defined by a fixed value of the martensite volume fraction 0 < z < 1. (Reprinted from F. Auricchio, A. Constantinescu, C. Menna, G. Scalet, A shakedown analysis of high cycle fatigue of shape memory alloys, International Journal of Fatigue, 87, 112-123 , Copyright (2016), with permission from Elsevier [17].)

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In work by Mostofizadeh et al. [74] a self-heating method is based on temperature measurements has been proposed as a faster alternative to classical fatigue tests. One-dimensional thermomechanical constitutive equations were implemented into the self-heating method using a probabilistic two-scale model. The approach predicts the S/N curve of raw and heterogeneous specimens for any failure probability in a shorter time compared with classical methods.

7.4.3 Energy-based approaches Some fatigue failure models have been established by taking the dissipation energy at the stabilized cycle, W , as the basic variable. Moumni et al. [66,67] proposed an empirical energy-based fatigue failure model to predict the fatigue life of austenitic NiTi alloys: W ¼ aðNf Þb

(7.38)

where a and b are material parameters. The criterion was validated for tensile/ compressive experiments with different Rs , but not for other loading cases. Moreover, this energy-based damage parameter does not consider the effects of tensile mean stress separately and assigns the same damage parameter for two tests with different maximum stresses as long as the hysteretic areas are the same. Moreover, the proposed model does not assign fatigue damage for the cyclic tests in the linear elastic regime, whereas experimental observations revealed fatigue failure in this case. Later, Morin et al. [75] extended such a criterion to account for the influence of the maximum hydrostatic pressure, Pmax : W þ gPmax ¼ aðNf Þb

(7.39)

where g is a material parameter. Kan et al. [76] modified Eqn. (7.38) by replacing the power-law equation with a logarithmic one:   W Nf ¼ a ln (7.40) b Mahtabi and Shamsaei [59] defined the total energy density, W t , as the sum of dissipated energy density, W , and tensile elastic energy density, Weþ : W t ¼ W þ Weþ

(7.41)

The proposed energy approach provides reasonable correlations among the fatigue data for various strain ratios. The correlation of fatigue data from various strain ratios based on the first cycle energy parameter was also analyzed and was almost as good as the one based on the stable response.

Fatigue and fracture

Song et al. [72] considered the dissipation energies in the stabilized cycle, W , and in the first one, W first :   W a ln b ! (7.42) Nf ¼ W first l ln h where l is the nonproportionality factor of the multiaxial loading path, h and b are the material parameters, respectively, and a is a reference energy parameter introduced to ensure the nondimensionality of the natural log function. For an isothermal condition, the area of hysteresis depends only on the mechanical loads; however, when the effect of frequency is considered, the area of hysteresis could be greatly influenced by thermomechanical coupling and hysteresis dissipation-based criteria may be not applicable. A strain energy-based criterion that uses continuum damage mechanics has been proposed by Zhang et al. [77]. A thermomechanical life-prediction model that depends on the maximum temperature in the stabilized cycle, Tmax , has been proposed by Zhang et al. [78]: Nf ¼ aðTmax Þb

(7.43)

7.4.4 Global versus local damage mechanisms As demonstrated in experimental work [79e82], phase transformation phenomena in SMAs are triggered and developed locally, mainly owing to inhomogeneities at the grain scale, especially in commercial polycrystalline alloys. This leads to large strain inhomogeneity even under nominally identical loading conditions, such as in uniaxial samples. Localized transformations occur by L€ uders- like bands, resulting in marked differences between nominal and local strain values. Unfortunately, these local effects cannot be easily measured or numerically simulated, but they cause an accumulation of local damage that leads to the formation and propagation of fatigue cracks. Unfortunately, in most studies in the literature the functional and structural fatigue properties of SMAs are correlated to global and/or nominal strain; that is, they are obtained from measurements carried out at a typical macroscopic engineering scale. This results in an unusual fatigue response of SMAs in terms of strain-life curves, which can be attributed to local transformation phenomena. In particular, a Z-shaped trend can be observed in pseudolastic SMAs [7,53,82,83]. This is made of different fatigue regimes: (1) an LCF zone in the fully transformed martensitic region (M), (2) an HCF zone in the predominantly elastic austenite phase (A), and (3) an unusual transition zone between LCF and HCF with almost constant fatigue life, corresponding mostly to the stressinduced transition plateau (A-M). This unexpected transition zone is particularly evident

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only when testing the material in a wide strain range (i.e., from elastic austenite [HCF] to fully transformed martensite [LCF]). To analyze the evolution of local transformations in SMAs, as well as their effects at the macroscopic scale, special local measurement methods were used, such as digital image correlation (DIC), infrared thermography (IR) [84e88] and synchrotron x-ray diffraction (XRD) [81,89]. Furthermore, nanoindentation was also found to be sensitive to local phase transition phenomena and material damage [90,91]. These experimental works actually defined a new research trend for a deeper understanding of effective key mechanisms for both functional and structural fatigue damage in SMAs. Fig. 7.12 shows the functional evolution of the material in terms of residual and recovery strain (εres , εrec ) at both the global and local scales, as obtained from DIC measurements in the uniaxial fatigue testing of a pseudoelastic SMA. Fig. 7.12(a) reports the cyclic stressestrain curve at the first loading cycle for a maximum deformation, εmax ¼ 5%. The strain maps obtained from DIC in different points of the stressestrain curve are also illustrated. Marked local effects were observed in both loading and unloading transformation plateaus, owing to both direct (B2eB190 ) and reverse (B190 eB2) stress-induced transformations. L€ uders-like transformation bands were observed, causing a marked difference between the global and local strain response. Furthermore, the effects of localized transformation on cyclic ratcheting were analyzed, as shown in Fig. 7.12(b). The accumulation of residual strain during cycling was related to irreversibility at the local scale. The latter can be regarded as sites for the accumulation of fatigue damage. Fig. 7.13 shows the evolution of strain recovery capabilities as a function of maximum strain (εmax ) after material stabilization at the global (Fig. 7.13a) and local (Fig. 7.13b) scales. In particular, the trends of residual strain (εres ¼ εmin ), strain amplitude (εa ¼ εrec =2), and mean strain (εm ¼ εmin þ εa ) are illustrated. In the following discussion, symbols (ε) and (ε) denote global and local measurements, respectively. Marked differences at the two scales were observed. Monotonic increasing trends were obtained at the global scale, even when εres increased rapidly from the early stage of stress-induced martensitic transformation (εmax  2%) and εrec tended to stabilize at the end of the transformation plateau (εmax y6%), with maximum values around 4%. The scenario at the local scale is much different, because εrec shows an almost flat trend just beyond the initiation of stress-induced transformation (εmax  2%). In particular, stress-induced transformation saturates locally in the transformation bands even at the beginning of the global transformation plateau (i.e., εmax is close to the end of the stress-induced transformation plateau [w6%], starting from εmax z2%), as also illustrated in Fig. 7.12(a). Fig. 7.14 reports the effects of global and local strain on strain-life curves (εa  Nf ) in the whole LCF to HCF regime. In particular, in Fig. 7.14(a), fatigue life data are plotted as a function of the global initial strain amplitude (εa0 ¼ εmax =2) (i.e., the value at the first loading cycle). This is a standard representation of fatigue data obtained from rotating

Figure 7.12 Global versus local strain distribution obtained by digital image correlation (DIC): (a) strain evolution during the first loading cycle for εtot ¼ 5%, and (b) the evolution of residual strain εres as a function of the number of cycles. Fatigue and fracture

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Figure 7.13 Evolution of mean strain, strain amplitude, and residual strain versus global maximum strain (εmax ): (a) global strain values (εm ; εa ; εres ), and (b) local strain values (εm ; εa ; εres ) by digital image correlation.

Figure 7.14 Low to high cycle strain-life curves (εa  Nf ) computed by global and local strain values: (a) global initial strain amplitude (εa0 ¼ εmax =2); (b) local stabilized strain amplitude (εa ). Fatigue and fracture

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bending tests, in which the evolution of the effective strain amplitude cannot be directly measured. Initial deformations are imposed geometrically from the curvature of the wire; strain ratcheting effects occurring during the test cannot be considered. Four different regions are observed: region I (Nf < 3*103 ) defining the LCF behavior of fully transformed martensite (M), region II (Nf z3*103 ) corresponding to the transformation plateau (A-M), region III (3*103 < Nf < 105 ) representing LCF-HCF behavior in the predominantly elastic regime of austenite (A), and region IV (Nf > 105 ) for the lowest value of strain amplitude [82]. In region II, corresponding to a global maximum strain range εmax ¼ 2  6%, fatigue life seems to be mainly unaffected by the strain amplitude [7,53,83]. This strain range is highlighted by the dashed area in the figure; it represents the region where local maximum deformation (εmax ) is saturated in the transformation bands, as illustrated in Fig. 7.14(b). Furthermore, data in regions I and III are fitted by power law relations (εa ¼ AðNf Þa ). The values of coefficient A and exponent a are shown in the figure. These latter are similar to the results reported in Pelton [53] for a similar alloy composition and testing temperature. The scenario completely changes when plotting fatigue life data as a function of the local stabilized strain amplitude, εa (Fig. 7.14(b)), measured in the transformation bands. In particular, region II completely disappears, because local strain amplitudes in such a region are almost constant (εa z2%) (Fig. 7.13b), and the corresponding points move to region I. The global maximum strain region, εmax ¼ 2  6%, corresponds to a small variation of εa (i.e., the dashed area in Fig. 7.14(a) becomes narrow). This result is consistent with the physics of the problem, because just two distinct finite fatigue regions are observed, linked to the two crystallographic phases of the material. Region I represents the LCF behavior of martensite, and region III defines the LCF-HCF behavior of austenite. Furthermore, fatigue life in region I seems to be mainly unaffected by the strain amplitude (a w 0) as a consequence of the almost stable local pseudoelastic response of the alloy for εmax > 2% (Fig. 7.13).

7.5 Crack formation and propagation mechanisms Stress- and/or thermally induced transformation phenomena in SMAs have an important role on crack formation and propagation mechanisms under both fatigue and static loadings. In particular, nearecrack tip phase transformations significantly affect stressestrain fields; consequently, they have a marked influence on both stable fatigue crack propagation and fracture toughness. In the following subsections, nearecrack tip transformations mechanisms are described together with special methods (both analytical and experimental) for capturing localized phase transformation phenomena. Subsequently, these methods are applied to analyze the fatigue crack propagation and fracture toughness of commercial NiTi alloys.

Fatigue and fracture

7.5.1 Nearecrack tip transformations Stress-induced phase transformations in SMAs cause a complex crack tip stress distribution compared with common engineering metals, as schematically depicted in Fig. 7.15. This results from large transformation strain as well as the different elastic properties of the two crystallographic phases (B2 and B19’). Three different regions are observed near the crack tip: (1) a fully transformed martensitic zone at the very crack tip (r < rM) with a martensite volume fraction of xM ¼ 1; (2) a transformation region (rM < r < rA) with 0 < xM < 1; and (3) an austenitic untransformed region (r > rA), where xM ¼ 0. As a consequence, common fracture mechanics theories and standard procedures based on linear elastic fracture mechanics (LEFM) or elastic plastic fracture mechanics (EPFM), cannot be directly applied to SMAs. Within this context, research was carried out with the aim of capturing the effects of stress- and/or thermally induced phase transformations on crack formation and propagation mechanisms [92e94]. It was always found that nearecrack tip microstructural transitions have a significant role in the evolution of both static and fatigue cracks. To understand these features better, special and/or ad hoc investigation techniques were applied to analyze local nearecrack tip transformations directly, such as XRD [84,95e97], IR [88,89], and DIC [85e87,98]. Furthermore, special analytical methods were developed within the framework modified LEFM and EPFM theories [99e103], such as the one described in the following section.

Figure 7.15 Crack tip stress distribution and transformation region in shape memory alloys (SMAs). LEFM, linear elastic fracture mechanics.

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7.5.1.1 Analytical model

An ad hoc analytical approach was developed [101] based on modified LEFM relations and the assumption of small-scale transformation. The method enables a prediction of the extent of crack tip transformations, stress distribution, and the resulting stress intensity factor (SIF). In the case of mode I loading, the martensitic and austenitic radii (rM and rA ) are given by: !2 2 ð1  n  bnÞð1  bÞKIe rM ¼ (7.45) p ð1  bÞEA εL þ ðb þ 1Þð1  2nÞsAM þ ð1  bÞsAM s f 2ð1  bÞKIe2   sAM psAM þ sAM s s f pffiffiffiffiffiffiffiffiffiffiffiffiffi   AM  sAM r þ 4ð1  n  bnÞK 2 EA εL þ a1 M Ie 2rM =p M sf s   þ rM þ AM þ sAM ð1  bÞa1 M þ ðb þ 1Þð1  2nÞ ss f

rA ¼

(7.46)

where aM ¼ EM =EA is the Young’s modulus ratio, n is the Poisson’s ratio, b ¼ 0 for plane stress and b ¼ 2n for plane strain, and KIe is the effective stress intensity factor (i.e., it is related to the effective crack length, ae, similar to Irwin’s correction of the LEFM): ae ¼ a þ Dr   ð1  bÞ2 KIe 2 Dr ¼ rA  2p sAM s

(7.47) (7.48)

The crack tip stress distribution in both austenitic and martensitic regions (sA and sM ) can be obtained from equilibrium and compatibility equations:

sMi ðrÞ ¼ gi

KIe sAi ðrÞ ¼ gi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðr  DrÞ pffiffiffiffiffiffiffi AM  sAM 2ð1  n  bnÞKIe = 2pr  EA εL þ a1 M sf s ð1  bÞa1 M þ ðb þ 1Þð1  2nÞ

(7.49)

(7.50)

where gi ¼ 1 for i ¼ 1,2 and gi ¼ b for i ¼ 3. The mode I austenitic SIF, namely KIA , can be directly obtained from stress distribution (Eqn. 7.49) by considering the distance from the effective crack tip (er ¼ r  Dr), according to Irwin’s assumption: pffiffiffiffiffiffiffi KIA ¼ lim 2per sA ¼ KIe (7.51) er/0

Fatigue and fracture

The mode I martensitic SIF, namely KIM , can be obtained from Eqn. (7.50): pffiffiffiffiffiffiffi KIM ¼ lim 2pr sM ¼ r/0

2ð1  n  bnÞ KIe ð1  bÞa1 M þ ðb þ 1Þð1  2nÞ

(7.52)

Knowledge of the extent of the transformation region in terms of both rM and rA is required to calculate KIA and KIM by an iterative approach, similarly to Irwin’s correction for elastic-plastic materials. 7.5.1.2 Digital image correlation method

DIC method enables the direct measurement of nearecrack tip displacement and strain fields. In addition, the effective stress intensity factor can be estimated by the numerical fitting of displacement data [86]. In particular, the analytical solution of the nearecrack tip displacement field fug ¼ fux uy gT (Fig. 7.16) for an isotropic material under mode I loading is given by: " # j11 j12 j13 1 0 (7.53) fug ¼ ½jfUg ¼ fKI TABx By gT j23 0 1 j21 j22 where KI is the mode I stress intensity factor, T is the T-stress parameter, A is the rigid body rotation term, and Bx and By are the rigid body motions along x and y

Figure 7.16 A single-edge crack subjected to a remote stress, together with the corresponding neare crack tip horizontal and vertical displacements. (Reprinted from E. Sgambitterra, C. Maletta, P. Magaro, D. Renzo, F. Furgiuele1, H. Sehitoglu, Effects of Temperature on Fatigue Crack Propagation in Pseudoelastic NiTi Shape Memory Alloys, Shape Memory and Superelasticity, 5, 278-291, Copyright (2019), with permission from Springer [86].)

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axis. Functions jij can be expressed in terms of the polar coordinate centered at the crack tip: 8 rffiffiffiffiffiffiffiffiffiffiffi  

  > r  > > q 1 2 q > j ¼ 1$ > $cos ðk  1Þ þ sin > 11 > m 2p 2 2 2 > > > > ffiffiffiffiffiffiffiffiffiffiffi r    

>   > > 1 r q 1 2 q > > $ ðk þ 1Þ  cos j ¼ $sin > 21 > m 2p 2 2 2 > > > < vr $sinq j22 ¼  (7.54) 2mð1 þ vÞ > > > > > r > > $cosq j12 ¼ > > > 2mð1 þ vÞ > > > > > j13 ¼ r$sin q > > > > > > : j23 ¼ r$cos q where m ¼ E=½2ð1 þvÞ is the shear modulus of elasticity, k ¼ ð3 vÞ=ð1 þvÞ for plane stress, and k ¼ ð3 4vÞ for plane strain. The T-stress parameter is included owing to the size of the investigation windows, which is larger than the K-dominant zone. If Eqn. (7.53) is applied to the m measurement points of the DIC, a system of 2m linear equations is obtained: *  * u ¼ j fUg (7.55)  * where j is a 2m  5 matrix obtained by computing the matrix ½j of Eqn. (7.53) in the m points and the vector u* contains the corresponding 2m displacement components. Equation (7.55) represents an overdetermined system of linear equations (i.e., with five unknowns and 2m equations), and the least-square method can be used to obtain   an estimate of the unknown parameters fUg by the pseudoinverse of the matrix j* :  T  1  * T * j u (7.56) fUg ¼ j* j* Equation (7.56) can be solved only if the coordinates of the physical crack tip ðx0 ; y0 Þ are known. This can be obtained from high-resolution images, and numerical fitting can be carried out. However, the physical crack length usually does not provide the best fit of the displacements field because of large crack tip nonlinearities that cause a virtual increase in the crack length, namely, the effective crack length, ae, as discussed in the previous section. Better fitting can be obtained by considering the effective location of the crack tip ðxe ; ye Þ, by moving the origin of the coordinate system of Eqn. (7.54) (Fig. 7.16). This can be done by trial-and-error approaches or by a nonlinear regression method as described in Sgambitterra et al. [86].

Fatigue and fracture

7.5.2 Fatigue crack propagation Fig. 7.17 reports the results of fatigue crack propagation experiments in a pseudoelastic SMA at three different testing temperatures below the martensite desist temperature (T ¼ 25, 45, and 65  C), as obtained from eccentrically loaded single-edge crack specimens, according to the standard ASTM E647. In particular, the figure reports the evolution of the crack length during fatigue propagation experiments at the three testing temperatures, starting from an initial crack length to width ratio (a/W) around 0.2. The propagation rate decreases with an increase in temperature, resulting in an improved fatigue response. This is attributed to the effects of the temperature on the crack tip transformation mechanisms, as discussed in the previous section. Pseudoelastic SMAs always exhibit the same martensitic microstructure at the very crack tip below the martensite desist temperature (T < Md); therefore, differences in crack propagation mechanisms are attributed to the stressestrain fields. This result is in agreement with fracture toughness experiments, as discussed in the previous section, in which the SMA exhibits an increase in fracture toughness with an increase in the testing temperature below the martensite desist (T < Md). Unfortunately, standard methods based on linear elastic and/or elastic plastic theories are incapable of capturing the effects of temperature; therefore, they fail in the analysis of both static and fatigue cracks. To this aim, fatigue crack propagation data were analyzed by the DIC regression method and the modified LEFM model, as described in previous sections.

Figure 7.17 Evolution of the normalized crack length, a/W, as a function of the number of cycles for the three testing temperatures (25, 45, and 65  C). (Reprinted from E. Sgambitterra, C. Maletta, P. Magaro, D. Renzo, F. Furgiuele1, H. Sehitoglu, Effects of Temperature on Fatigue Crack Propagation in Pseudoelastic NiTi Shape Memory Alloys, Shape Memory and Superelasticity, 5, 278-291, Copyright (2019), with permission from Springer [86].)

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Fig. 7.18 reports the evolution of the stress intensity range (DKI ) as a function of the normalized crack length (a/W) as obtained from the regression method (Fig. 7.18(a)), analytical model, and LEFM (Fig. 7.18(b)) at the three testing temperatures. The effective SIF range is mainly unaffected by the testing temperature for small cracks (a/W < 0.4), whereas the temperature effects become significant when the extent of the crack increases. In particular, the lower the temperature is, the higher the SIF range is, owing to the increased nearecrack tip transformation zone and effective crack length. These effects are well-captured by the analytical model, as shown in Fig. 7.18(b). The regressed data and analytical model always provide similar results, whereas the LEFM fails by a large amount, especially at a low temperature (25  C) and large crack lengths (a/ W > 0.5). Differences tend to vanish when the testing temperature increases, as shown in Fig. 7.18(b), because the material approaches the linear elastic solution as a result of the Clausius-Clapeyron relation (i.e., the marked increase in transformation stress). Fig. 7.19 shows the crack propagations curves (da=dN vs. DKI ) for the three testing temperatures, as obtained from the LEFM method (Fig. 7.19(a)), DIC regression method (Fig. 7.19(b)), and modified LEFM analytical model (Fig. 7.19(c)). In  addition, propagation data within the steady state regime were fitted to the Paris law (da dN ¼ C DKIm ). The values of exponent m and the threshold stress intensity range (DKth ) are also reported in the figures and summarized in Fig. 7.19(d). The three methods provide similar results on the SIF threshold. In addition, the latter can be considered a temperature-independent parameter with an average value around 4.6 MPa m1/2. This is the expected result because, owing to stress-induced crack tip transformations, nonlinear effects in the near-threshold region become negligible. Furthermore, the obtained value is within the wide dispersion band obtained from previous experiments. Values from 1.9 to 5.4 MPa m1/2 are reported in Robertson et al. [33] for pseudolastic NiTi. This large range is attributed to several material and testing factors affecting crack tip transformation mechanisms, including the material composition, crystalline texture, transformation temperature, testing temperature, loading frequency, loading ratio, testing environment, and specimen geometry. In addition, most of these parameters are not independent because of the marked thermomechanical coupling in SMAs. Results on the exponent m show a different trend with respect to DKth (i.e., with significant differences among the three prediction methods). Regressed data are temperature independent, with an average value around 2.5. Contrary, marked temperature effects are observed from LEFM predictions: that is, m decreases when the testing temperature increases from about 3.2 at 25 to 2.7  C and 2.6 at 45 and 65  C, respectively. The analytical model provides results similar to those of the regression method at 45 and 65  C, whereas a slightly higher value, around 2.8, was captured at 25  C. This is attributed to the marked nonlinearities occurring at 25  C: that is, the assumptions of small-scale transformation [104] are not fully satisfied.

Figure 7.18 Mode I stress intensity range, DKI , as a function of normalized crack length a/W: (a) data obtained from digital image correlation (DIC) regression method; (b) data obtained from analytical model and linear elastic fracture mechanics (LEFM). (Reprinted from E. Sgambitterra, C. Maletta, P. Magaro, D. Renzo, F. Furgiuele1, H. Sehitoglu, Effects of Temperature on Fatigue Crack Propagation in Pseudoelastic NiTi Shape Memory Alloys, Shape Memory and Superelasticity, 5, 278-291, Copyright (2019), with permission from Springer [86].) Fatigue and fracture

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Figure 7.19 Crack propagation curves Da=DN versus DKI obtained at the three testing temperatures (25, 45, and 65  C): (a) linear elastic fracture mechanics (LEFM) results; (b) regressed results; (c) analytical results; (d) exponent m and of the threshold stress intensity range (DKth ) versus testing temperature. (Reprinted from E. Sgambitterra, C. Maletta, P. Magaro, D. Renzo, F. Furgiuele1, H. Sehitoglu, Effects of Temperature on Fatigue Crack Propagation in Pseudoelastic NiTi Shape Memory Alloys, Shape Memory and Superelasticity, 5, 278-291, Copyright (2019), with permission from Springer [86].)

Fatigue and fracture

7.5.3 Fracture toughness As demonstrated in several literature studies, crack tip transformations have a significant role in the fracture properties of SMAs, as also observed for stable fatigue crack propagation (see the previous section). Thermally and/or mechanically induced transformation has a marked effect on the nearecrack tip stress/strain field; consequently, it changes both stable and unstable crack growth mechanisms. As a consequence, special investigation methods and theoretical models should be applied to study the fracture properties of SMAs, the methods illustrated in Section 7.5.1. Fig. 7.20 reports results from monotonic fracture experiments [87] of a commercial * ) as a function of the testing tempseudoelastic NiTi SMA, in terms of critical SIF (KIC perature within the pseudoelastic regime (Af < T < Md ). The superscript asterisk indicates that calculations do not satisfy constraints given by standard methods (ASTM E399) for fracture toughness measurements of metallic alloys. In addition, the DIC method and analytical model, both described in Section 7.5.1, were used to calculate the critical SIF. * is Their values, KIDIC C , KIA C and KIM C , are shown in the figure. An increase of KIC 1/2  observed with an increase in temperature, from about 30 MPa m at T ¼ 25 C to about 37.5 MPa m1/2 at T ¼ 65  C. For the sake of comparison, Fig. 7.20 also shows results obtained in previous investigations [87,89].

Figure 7.20 Critical stress intensity factor (KIC) as a function of the testing temperature: comparison of results from linear elastic fracture mechanics (LEFM), digital image correlation (DIC), and a modified LEFM analytical model.

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* with temperature seems to disagree with the literature assumption The trend of KIC that stress-induced crack tip transformations in SMAs represent a toughening effect. Crack tip transformations increasingly lessen when with an increase in temperature owing to the Clausius-Clapeyron relation, and they completely disappear at the martensite desist temperature (T > Md). This is in accordance with systematic experimental results reported in Gollerthan et al. [89] and is shown in Fig. 7.20. In Gollerthan et al. [89], a marked increase in the critical stress intensity factor is observed at T > Md with respect to pseudoelastic (austenite) and pseudoplastic (martensite) alloys. On the contrary, the critical values of the austenitic SIF calculated according to the analytical model, KIA C , seems to be almost temperature independent. The same consideration applies to the martensitic SIF, KIA M , because the ratio KIA =KIM is a material constant (Eqn. 7.52). In addition, the figure shows that critical values of the effective SIF estimated by DIC, KIDIC C , are close to the austenitic SIF and are temperature independent. This demonstrates that the analytical model is able to capture the effects of complex thermomechanical loading conditions in SMAs (i.e., in terms of applied stress and temperature). In particular, the increase in critical SIF, based on LEFM, with an increase in the testing temperature cannot be attributed to a change in the material properties at the crack tip, but temperature has a significant role in the effective crack tip stressestrain distribution. Material properties at the very crack tip are unaffected by the temperature (i.e., cracks always grow in the high-stress detwinned martensitic phase). Based on the analytical approach, a temperature-independent critical value of SIF can be defined, as shown in Fig. 7.20, and this can be considered a material property. The analytical model can be used to define critical conditions for fast fracture in SMAs (i.e., by comparing temperature-dependent SIF with its temperature-independent critical value).

7.6 Conclusions SMA-based components are usually subjected to fatigue loading conditions resulting from repeated thermal actuations or stress-induced transformation cycles. The number of cycles could range from an LCF (106), depending on the needs of the application. Under such loading conditions, SMAs exhibit structural damage (that is, the formation and propagation of fatigue cracks), but also cyclic degradation of their functional properties (namely, functional fatigue). Therefore, designing an SMA component against fatigue is a complex task, because long-term performance is affected by damage at both the structural and functional levels. The task becomes even more complex because damage mechanisms are significantly affected by the unique stress- and or thermally induced phase transition phenomena in SMAs.

Fatigue and fracture

Main features related to fatigue and fracture properties of SMAs are that: - The cyclic stressestrain response exhibits a marked evolution resulting in degradation of the strain recovery and damping capabilities. However, a stable functional response is always observed after the first transformation cycles (around 100); - Classical fatigue theories and testing standards cannot be directly applied to analyze fatigue and fracture properties of SMAs owing to their unique thermomechanical response and crack tip transformation mechanisms; - A variety of experimental and modeling contributions can be found in the current literature regarding the structural fatigue of austenitic and martensitic SMAs. The choice of the approach (strain-, stress-, or energy-based) depends on the loading condition in the application of interest. Most reviewed approaches focus on cyclic uniaxial and bending loading conditions (including mean stressestrain effects), whereas torsional and nonproportional multiaxial fatigue requires further deepening. Moreover, studies are limited on the effect of prestresseprestrain, variable strainestress amplitude, variable temperature, or variable loading rate on SMA fatigue behavior. - Fatigue damage are significantly affected by local deformation mechanisms occurring near phase transformation sites. These deformations are significantly different with respect to global/nominal strain and cause the fatigue response to be significantly different with respect to common engineering metals; - Nearecrack tip stress distribution is significantly affected by stress- and/or thermally induced transformation phenomena. These have a marked effect on both fatigue crack propagation and fracture toughness of SMAs. Because of the complex thermomechanical response of the alloy, special procedures and experimental approaches are required to estimate the fracture properties of SMAs. Based on these features, despite the amount of research that has been performed, more investigations are required for a deep understanding of the fatigue and fracture behavior of SMAs. This is also needed to fill the lack of standard methods required to characterize their fatigue resistance and fracture properties. Available standards in the SMA field are related only to their thermomechanical characterization [105,106].

Bibliography [1] W.J. Buehler, J.V. Gilfrich, R.C. Wiley, Effect of low-temperature phase changes on the mechanical properties of alloys near composition TiNi, J. Appl. Phys. (1963), https://doi.org/10.1063/ 1.1729603. [2] T.W. Duerig, The use of superelasticity in modern medicine, MRS Bull. (2002), https://doi.org/ 10.1557/mrs2002.44. [3] J.M. San Juan, M.L. N o, C.A. Schuh, Superelasticity and shape memory in micro- and nanometerscale pillars, Adv. Mater. (2008), https://doi.org/10.1002/adma.200701527.

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Modelling Editor:

Sonia Marfia

Dipartimento di Ingegneria, Universita di Roma Tre, Rome, Italy

List of chapters 8. 9. 10. 11.

1D SMA models SMA constitutive modeling and analysis of plates and composite laminates Advanced constitutive modeling SMAs in commercial codes

CHAPTER 8

1D SMA models Sonia Marfia1, Andrea Vigliotti2 1

Dipartimento di Ingegneria, Universit
a di Roma Tre, Rome, Italy; 2Innovative Materials Laboratory, Centro Italiano Ricerche Aerospaziali, Capua, Italy

8.1 Introduction Several one-dimensional (1D) and 3D phenomenological models [1] able to reproduce shape memory alloy (SMA) constitutive behavior have been proposed. An overview of the phenomenological models can be found in Lagoudas [1], Auricchio et al. [2], Khandelwal and Buravalla [3], and Paiva and Savi [4]. In this chapter, attention is focused on 1D SMA models. 1D models, which are characterized by a simpler formulation with respect to 3D ones, are able to account accurately for all significant features of the thermomechanical response of the SMA, such as pseudoelastic and shape memory effects, the reorientation of martensite variants, the different behaviors in tension and compression, and the different elastic properties of the SMA phases, introducing a reduced number of material parameters with a clear physical meaning. Moreover, most structural elements in SMA applications can be designed using 1D constitutive relations, including the bending and axial response of beams. In the literature, many 1D models have been proposed; some are briefly described in the following discussion. Tanaka [5] proposed a 1D model introducing an internal variable, representing the fraction of martensite, considering exponential hardening laws for the evolution of the internal variable during phase transitions and assuming the material properties to be constant during phase transformations. The model did not consider single-variant martensiteemultivariant martensite phase transformation. Achenbach [6] presented a model considering three phases (austenite, tensile martensite, and compressive martensite) and introducing Helmholtz free energy for individual phases. They adopted the method of potential energy wells to determine probabilistically when a phase transformation occurs. Liang and Rogers [7] proposed a model based on the assumption of the martensite fraction as an internal variable whose evolution was governed by phase kinetic laws given by a cosine-based function to obtain a better fit to the experimental data. They considered the material properties to be constant during the phase transitions, as in Tanaka et al. [5]. Abeyaratne and Knowles [8] proposed a model considering as an internal variable the strain present in each phase, to study the initiation and propagation of phase transformation in the context of a 1D bar.

Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00008-X

© 2021 Elsevier Ltd. All rights reserved.

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Brinson [9] proposed a model based on the introduction of two internal variables to model both multivariant and single-variant martensite. The evolution of the internal variables was governed by a cosine hardening law and variable elastic stiffness during the phase transformations. In particular, Voigt-type nonconstant material functions were adopted to describe the constitutive behavior. Ivshin and Pence [10,11] modeled the transformation between austenite and a singlevariant martensite considering the phase fraction as a state function. In particular, they considered hysteretic and nonhysteretic phase transformations and complete transformations as well as loadingeunloading paths leading to inner loops. Bekker and Brinson [12,13] presented a model formulated in terms of possible thermomechanical paths in stress-temperature space. A cosine hardening function was adopted for the phase transformation. Auricchio and Lubliner [14] proposed a 1D model in the framework of general plasticity, modeling pseudoelastic and shape memory effects. They introduced two internal variables representing the single-variant martensite fraction and the multivariant martensite fraction whose evolution was governed by the stress and temperature. Auricchio and Sacco [15,16] presented a pseudoelastic 1D constitutive model based on the stress-temperature phase diagram and a beam finite element. The model was based on the introduction of one internal variable corresponding to the single-variant volume fraction. The kinetic laws governing the internal volume evolution were assumed to be a function of the stress and temperature [14]. Different elastic moduli for the austenite and martensite were considered and different homogenization techniques were studied. Auricchio and Sacco [17] presented an SMA model based on Auricchio and Sacco [15,16], able to describe pseudoelastic and shape memory effects. Two internal variables corresponding to the volume fractions of single-variant martensite and multivariant martensite were introduced as in Auricchio and Lubliner [14]. The kinetic rules governing the evolution of the internal variables were assumed to be a function of the strain and temperature. The elastic moduli of SMA were evaluated considering different elastic properties for the austenite and martensite and adopting a Reuss homogenization. Chenchiah and Sivakumar [18] proposed a thermomechanical model to predict the shape memory and pseudoelastic behaviors considering two SMA variants. The model was based on two internal variables: the phase fractions of the variants of martensite, whose evolution was described geometrically in phase space as a function of the stress, strain, and temperature loading history. It was shown in this work that for any specified constant ratio of the variants of martensite, this model reduced to a one-variant thermomechanical model similar to the IvshinePence model [10]. Govindjee and Kasper [19] proposed a model introducing the martensitic variants in tension and compression and adopting constant stiffness and thermal expansion coefficients to describe the pseudoelastic and shape memory effects. They introduced a memory parameter in terms of the extreme value of the variant fraction achieved in the current active transformation zone. To account for the behavior of the material at high values of stress, they adopted a linear isotropic hardening plastic model assuming the evolution of the plastic variables to be independent of any phase transformations.

1D SMA models

Rajagopal and Srinivas [20] considered the transformation from austenite to one preferred variant of martensite considering two distinct reference configurations (one was the austenitic (parent) phase and the other was the martensitic (product) phase), and modeling pseudoelastic and shape memory effects. The constitutive equations were developed by introducing a Helmholtz potential and dissipation mechanisms. In particular, nondissipative and dissipative phase transformations were considered. Raniecki et al. [21] and Rejzner et al. [22] presented a pseudoelastic model to study the bending response of a beam with a symmetric cross-section. The explicit analytical equations for the momentecurvature hysteresis loop were derived by studying the movement of the beam planes separating the regions occupied by a single phase (austenite or martensite) and their mixture (austenite and martensite) in the course of phase transitions. Marfia et al. [23] proposed a simple SMA model based on Auricchio and Sacco [17] employed in a new layer-wise finite element to study SMA as a reinforcement of beams. Ikeda et al. [24] proposed a simple but accurate macroscopic constitutive model of SMAs for unidirectional loading, which had a physical background and was derived from a grain-based microscopic model. This model was based on microscopic aspects and included the memory effect of deformation history. Stressestrain relationships were simulated for some representative strain cycles considering the inner loops. Pavia et al. [25,26] proposed a model based on the introduction of four phases: tensile and compressive martensite, austenite, and multivariant martensite. The free energy of the SMA was introduced as a combination of the free energy of the different phases considering the SMA as a mixture, as presented by Fremond [27] Tensile-compressive asymmetry and internal subloops were also taken into account. Chang et al. [28] presented a 1D strain-gradient continuum model of an SMA wire element including possible unstable mechanical behavior and thermomechanical coupling. The phenomenological constitutive model, based on two internal variables, was derived in the context of irreversible thermodynamics from a free energy function. Buravalla and Khandelwal [29] proposed a 1D constitutive model of SMAs considering nonconstant material functions and martensite fraction as the internal variable. In this work, they discuss the importance of introducing suitable memory parameters, defined as the distance of a point of the load path inside the transformation zone from the finish boundary, to capture the extent of transformation under the considered loading path. Auricchio et al. [30] proposed a phenomenological 1D model, which considered tensionecompression asymmetries as well as elastic properties depending on the phase transformation level, combined with a good description of the pseudoelastic and shape memory behaviors.

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Evangelista et al. [31] proposed some modifications to the original 3D model proposed by Souza [32] and modified by Auricchio and Petrini [33] and Evangelista et al. [34], to derive its consistent 1D formulation and clarify the mechanical meaning of the material parameters governing the constitutive model. The transformation strain was assumed to be the internal variable. A robust time integration algorithm was developed in the framework of the finite element method and a new beam finite element was proposed. Nallatambi et al. [35] presented a thermodynamically consistent three-phase (austenite, plus-martensite, and minus-martensite) model in a uniaxial framework. The driving force equations were obtained as a direct consequence of mechanical dissipation inequality coming from the second law of thermodynamics. Rizzoni et al. [36] presented a theoretical and experimental investigation of the bending recovery performances for a commercial NiTi shape memory alloy strip. To model the strip bending response, they adopted a 1D phenomenological constitutive equation for the shape memory material [14,15], based on the introduction of (twinned and detwinned) martensite and austenite volume fractions as internal variables. Under the assumption of uniform bending, they calculated a quasiclosed form solution for the stress and martensite fraction distributions in a shape memory beam during bending and subsequent shape recovery. Marfia and Rizzoni [37,38] proposed a thermodynamically consistent model based on two martensite variants that are assumed to be internal variables. The evolutive laws of the internal variables are assumed to be a function of the stress and temperature. Analytical and numerical procedures were developed to study the axial and bending behavior of SMA elements, considering the different responses in tension and compression and the different elastic properties of the SMA phases. Barrera et al. modified the SouzaeAuricchio model to capture the onset of functional fatigue in the material [39]. The model accounted for the growth of macroscopic strain owing to the accumulation of microscopic plastic strain. A novel micromechanical constitutive model was proposed by Chao et al. capable of reproducing the cyclic deformation of polycrystalline NiTi in a variety of thermomechanical cyclic conditions [40]. A micromechanical constitutive model based on the plasticity of a single NiTi crystal capable of reproducing cyclic deformation behavior was proposed [41]. The model for the evolution of martensite and austenite is based on thermodynamics, and the Morie Tanaka homogenization method has been used to model the different solid phases. Another work proposed a phenomenological model for the behavior of porous SMAs [42], which was able to reproduce the response of SMA under proportional and nonproportional loading at a reduced cost compared with more complex micromechanical models.

1D SMA models

Malagisi et al. [43] proposed two constitutive models able to reproduce the mechanical behavior of SMA considering the simultaneous presence of normal and shear stresses, and their coupling without developing a full 3D model. Karamooz-Ravari et al. modified and validated an existing small strainebased constitutive model to capture the cyclic response of selective laser meltingemanufactured NiTi alloys [44]. The model used microplane theory to define the active strain tensors, whereas plastic strain was updated by means of the associated flow rule and isotropic hardening, with all model parameters obtained from differential scanning calorimetric measurements. Ashrafi proposed a phenomenological model capable of reproducing the evolution of plastic strain in cyclic loading [45]. In addition, a research article developed a 3D model capable of delivering a complete description of the behavior of severely prestretched SMAs, including one-way and two-way phenomenology, plasticity, the dependence of mechanical properties on temperature, the evolution of transition temperatures in the presence of applied stresses, and the consequent thermal hysteresis [46]. In this chapter, the SMA cyclic response, which represents one of the most interesting features of the material, is studied. From experimental evidence [47e55], the SMA response progressively changes during cycles. In particular, mechanical cycling shifts the pseudoelastic hysteresis loop downward, reducing its height and width and increasing the amount of permanent deformations. The cyclic changes are more significant during the early stages of cyclic loading, and as the number of cycles increases, the material response approaches a saturation limit beyond which the pseudoelastic response does not change significantly. Thus, after so-called material training (i.e., after a certain number of stressetemperature loadingeunloading cycles), the material response reaches a limit and stable path. From a phenomenological point of view, the training effects on the stressestrain curve result in two aspects: • a progressive decrease in the initial and final stress thresholds of the phase transformations from austenite to martensite, and vice versa; • a progressive increase in residual permanent deformation after unloading. Both of these phenomena have a micromechanical interpretation that is still under investigation. In particular, a micromechanical interpretation discussed in the literature is based on the conjecture that residual strains and stresses resulting from the development of dislocations induced by plastic phenomena occur in the material during the cycles; the presence of residual stresses causes the turnout of a residual permanent martensite phase, which accumulates during training. On the basis of this interpretation, progressive change in the SMA response under cyclic loading results from the oriented residual stresses occurring during the arrangement of dislocations. The nucleation and growth of preferential martensite variants are favored by these residual stresses. In other words, because the generation of dislocations is strictly linked to the development of plasticity, it can be

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deduced that the training effects are connected to stresses induced in the material microstructure due to the residual plastic strains. The presence of stresses in the microstructure does not allow complete martensiteeaustenite transformation, leading to a progressive increase in residual permanent oriented martensite variants that occur at the end of a loadingeunloading cycle because of the arrangement of dislocations. The presence of permanent deformation due to plastic strains and the residual martensite variants occurring during material training enables reversible spontaneous shape change to occur during cooling and heating processes without applying an external load, which is known as the two-way memory effect (TWME). This effect can be exploited to realize new potential applications as reversible fasteners, temperaturesensitive actuators, retrievable medical implants, toys, and novelty items. SMA models that take into account the training effect and the TWME have been presented in the literature. Among others, Tanaka et al. [56] extended a 1D model, also developed by Tanaka [5], to study the cyclic uniaxial response of SMAs subjected to thermal and/or mechanical loads. The model was based on three internal variables: local residual stress and strain and the residual volume fraction of martensite accumulated during cycles. Lexcellent and Bourbon [57] proposed a 1D pseudoelastic model to analyze the tensile cyclic response of SMA, introducing a new internal state parameter representing the instantaneous residual martensite volume fraction. They assumed that a portion of the martensitic volume fraction was not recovered after each cycle. Thus, residual strain was attributed only to residual martensite. This behavior could reach saturation with the number of cycles. Abeyaratne and Sang-Joo Kim [58] developed a 1D model based on an energetic approach, assuming that the critical value of the driving forces required for nucleating and propagating the phase transformations was affected by the defect density, which depended on the number of loading cycles. Bo and Lagoudas [59e62] described a 1D model, based on a framework provided earlier by Boyd and Lagoudas [63], which accounted for the training effects introducing irreversible plastic strains. This was the first model that allowed plastic strain and transformation strain to develop simultaneously during the forward and reverse transformations. The model was able to take into account the evolution of the plastic strains and the TWME and to describe the minor hysteresis loops. Lexcellent et al. [57,63] developed a model in the framework of thermodynamics introducing a training term in the expression of the free energy of self-accommodated martensite. Auricchio and Sacco [64] proposed a thermomechanical coupled model based on the constitutive model proposed by Auricchio and Lubliner [14], able to describe the material differences properly in tension and compression. The transient thermomechanical problem was solved considering the thermal evolution in the framework of a beam model. Auricchio et al. [65] proposed a 1D SMA model to study the cyclic response of the material based on the SMA constitutive model proposed by Auricchio and Sacco [17]. They considered the variation in stress thresholds of the austeniteemartensite phase transitions and the residual martensite

1D SMA models

fraction during the cycles until a stable value was reached. Sun and Rajapakse [66] modified the Brinson constitutive model [9] to examine the stabilized isothermal response of an SMA element subjected to cyclic tensile stress or strain loading. They studied the influence of the loading frequency, temperature, static strain offset, and strain amplitude on pseudoelastic hysteresis loops and energy dissipation, developing dynamic analyses of a single degree of freedom system. Azadi-Borujeni et al. [67] proposed a 1D pseudoelastic constitutive model to consider the cyclic effects. The effect of temperature on cyclic behavior was also considered. The constitutive model was adopted within a finite element framework to study the dynamic pseudoelastic response of an NiTi wire subjected to cyclic loading. The models presented in this chapter assume the presence of two different variants of martensite: thermal or twinned or multivariant martensite, and stress or detwinned or single-variant martensite. Thermal martensite forms when the alloy is cooled, starting from the austenite state, below the initial martensite formation temperature of the alloy, Ms , under stress-free conditions. This variant is characterized at the microscopic level by the presence of multiple domains in which the alloy has the same crystallographic configuration that differs only in the orientation. The presence of such twin domains gives this variant the name of twinned martensite and is responsible for the pseudoelastic effect. The stress martensite forms when a given stress is applied to thermal martensite. Upon the application of stress, the martensite domains reorient themselves along the direction of the stress. In this process, the alloy yields a large strain that can be recovered in the austenite transformation. Because the twin domains disappear upon the application of stress, the stress martensite is also called detwinned martensite. In this chapter, we present two approaches to the modeling of 1D behaviors of SMAs: • In Section 8.3, a simplified approach [68] is described based on the phenomenological models presented by Lyang and Rogers [7,69] and Brinson [9]. This approach assumes that a direct algebraic relation exists between the martensite fractions and the thermomechanical state variables. Thus, it is not necessary to integrate the state equation in time to determine the status of the alloy, but it is assumed that the alloy is always in thermodynamic equilibrium. This model is valid in quasistatic conditions when it is possible to assume that the stress and strain rates are negligible in relative terms with respect to the rate of the austeniteemartensite transformation. • In Section 8.4 a model is presented that is able to reproduce the pseudoelastic and shape memory behavior of SMA and to consider many of the special features of SMA behavior, such as the different responses in tension and in compression, the different elastic properties of austenite and martensite, the reorientation process. Furthermore, the model is able to simulate cyclic behavior, including training and TWME [65]. This approach is based on the time integration of the transformation kinetic equations that are a function of the strain and temperature.

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8.2 List of symbols s Stress Mf final thermal martensite transformation temperature ε Strain As Initial austenite transformation temperature (stress free) E Young’s modulus Af Final austenite transformation temperature (stress free) x Total martensite volume fraction ss Initial detwinned martensite transformation stress xT Thermal martensite (twinned) volume fraction sf Final detwinned martensite transformation stress xd Stress martensite (detwinned) fraction or single variant martensite fraction CAS Clausius-Clapeyron constants for the austenite to stress martensite transformation xA Austenite volume fraction CSA Clausius-Clapeyron constants for stress martensite to austenite transformation xR Residual martensite volume fraction u Reorientation parameter u Transformation tensor k Training parameter a Thermal expansion coefficient xrev Reversible martensite fraction T Temperature sSS Reorientation stress threshold EM Young’s modulus of martensite g Reorientation material parameter EA Young’s modulus of austenite AS SA SA bAS Material constants measuring material ability to be trained s , bf , bs , bf ES Young’s modulus of single-variant martensite gAS , gSA Accumulated martensite fraction during phase transformations Ms Initial thermal martensite transformation temperature pAS , pSA , aAS , aSA Material parameters HAS , H SA Phase transformation activation factors GAS , GSA Variables function of strain and temperature governing the evolution of single-variant volume fraction during transformation phases z Absolute value of strain AS Strain thresholds of austeniteesingle variant martensite phase transformation SAS f , Ss AS Variables influencing strain thresholds of austeniteesingle-variant martensite RAS f , Rs phase transformation SA SSA f , Ss Strain thresholds of single-variant martensiteeaustenite phase transformation SA RSA f , Rs Variables influencing strain thresholds of single-variant martensiteeaustenite phase transformation

1D SMA models

8.3 Simple nonkinetic models In this section, we will describe the implementation of the model developed by Liang and Rogers [7], which accounts for the formation of a single variant of martensite, the detwinned martensite. According to this model, detwinning of martensite and the domain reorientation does not involve mechanical work. Subsequently, we describe the extension to this model introduced by Brinson [9,70], which considers the formation of both variants of martensite and the transformation of the twinned martensite into detwinned martensite, which can occur only at the expense of some mechanical work applied to the alloy. Finally, numerical test cases are presented that implement the multiple martensite variant models.

8.3.1 Single martensite variant model The model developed by Liang and Rogers [7] accounts for the formation of a single martensite variant, detwinned martensite, even when no stress is applied. It follows that it is not strictly adherent to experimental observations; nevertheless, it represents a useful simplification that can be employed as a starting point to illustrate the finite element implementation of shape memory materials. According to Lyang and Rogers, only three state variables are necessary to describe the composition of the alloy: the martensite fraction x, where x ¼ 1 for the alloy in a full martensite state and x ¼ 0 for austenite; s ¼ the stress; and T ¼ the temperature. The equilibrium equation for a continuum can be derived following the same reasoning outlined in Liang and Brinson. Considering the dependency of the Young’s modulus on x, we obtain s  s0 ¼ EðxÞε  Eðx0 Þε0 þ uðxÞx  uðx0 Þx0 þ aðT  T0 Þ

(8.1)

where EðxÞ is the Young’s modulus, uðxÞ is the transformation tensor, and a is the thermal expansion coefficient. The expression for UðxÞ was given by those authors as: uðxÞ ¼ EðxÞεL

(8.2)

where εL is the maximum transformation strain. As summarized by Brinson and Huang [71], two options are available to model the change in the elastic modulus of the alloy as a function of the martensite fraction: 1. According to Sato and Tanaka [72] and Liang and Rogers [69], it can be assumed that the austenite fraction and martensite fraction are working in parallel along the wire. In this case, the elastic modulus of the mixture will be similar to the scheme of two springs connected in parallel (the Voigt scheme): EðxÞ ¼ xEM þ ð1  xÞEA

(8.3)

where EM is the elastic modulus of the martensite and EA is the elastic modulus of the austenite.

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2. Alternatively, according to Ivshin and Pence [10,11] and Auricchio Fugazza and Des Roches [73,74], the different fractions should be considered to be working in series (the Reuss scheme). In this case, the dependence of the elastic modulus by the martensite fraction can be expressed as: EðxÞ ¼

EA EM EM þ xðEA  EM Þ

(8.4)

In this chapter, we assume that martensite and austenite act in parallel. This choice does not affect the implementation of the material model, and it is possible to switch to the series arrangement at any time. We now need to introduce a thermomechanical relation to model the transformation of martensite in austenite, and vice versa. To this end ,we partition the s  T plane into five zones, separated boundaries, as illustrated in Fig. 8.1. The boundaries are straight lines that intercept on the temperature axis: the four critical temperatures of phase transformation at zero stress (As and Af , the start and finish transformation temperatures for austenite; and Ms and Mf , the start and finish transformation temperatures for martensite). The slopes of boundary lines CAS and CSA are material constants that need to be determined by experimental measures; likewise, the critical Temperatures. According to Liang and Rogers, the transformation is driven by both stress and temperature changes and can occur only in specific zones on the s  T plane, specifically: 1. austenite to martensite transformation, i.e., x_ > 0 is possible only in region 2 and only if the status is approaching the boundary line passing through Mf ; 2. martensite to austenite transformation, i.e., x_ < 0 is only possible in zone 4 and only if the status is approaching the boundary line passing through Af . A good approximation to experimental observation was found by Lyang and Rogers based on a cosine law. For zone 2, the result is:
  1  x0 p s 1 þ x0 x¼ cos (8.5) T  Mf  AS þ Ms  Mf 2 2 C

Figure 8.1 Transformation zones for the single-variant martensite model.

1D SMA models

where x0 is the martensite fraction when entering the transformation zone. The transformation described by Eqn. (8.5) relative to zone 2 is valid only for martensite production (i.e., for x_  0). This constraint provides us with the following necessary condition regarding the increase in T and s to allow martensite production: dT


ds CSA

(8.8)

To calculate the evolution of the martensite fraction along the load path and assess whether variations in x occur, it is convenient to write the transformation and equilibrium equations in a rate form. These rate equations can then be integrated at each load increment, verifying whether the conditions for s, T, ds, and dT allow alloy transformation. To this aim, we can write the differential increment of Eqn. (8.1) as: vE vu dx þ Edε þ x dx þ Udx þ adT vx vx   vE vu ds ¼ ε þ x þ u dx þ Edε þ adT vx vx

ds ¼ ε

(8.9) (8.10)

where the differential increase in the martensite fraction can be written as: dx ¼

vx vx ds þ dT vs vT

(8.11)

The partial derivative terms in Eqn. (8.11) can be obtained from Eqns. (8.5) and (8.7), depending on the position of the status point. In this approach, the kinetics of the martensiteeaustenite transformation is neglected because we link the martensite fraction directly to the thermomechanical state through Eqns. (8.5) and (8.7). Because we solve mechanical equilibrium in a displacement-based finite element approach, we need to express the stress as a function of the strain and temperature. To this end, we substitute

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Eqn. (8.11) into Eqn. (8.10) and obtain the following expression for the differential increase in stress as a function of the differential of the strain and temperature: 
  vE vu vx þu Edε þ Q þ ε þ x dT vx vx vT   ds ¼ (8.12) vE vu vx þu 1 ε þx vx vx vs Equations (8.11) and (8.12) are a system of differential equations that can be integrated numerically, given the initial conditions (s0 , ε0 , x0 , T0 ), to find the element stress for any given strain and temperature, at each increase in the solution. To progress through NewtoneRaphson iteration, it is necessary to determine the local slope of the stressestrain curve, which we can obtain from Eqn. (8.12) as:  vs E    ¼ (8.13) vE vu vx  vε T 1 ε þx þu vx vx vsT

8.3.2 Multiple martensite variant model In this section we describe a finite element implementation of the multivariant model developed by Brinson [9]. This model expands the Liang and Rogers model accounting for the formation of two martensite variants: 1. Twinned martensite, which forms while cooling below Ms under stress free conditions. In twinned martensite, twin domains exist at a grain size whose crystallographic characteristics differ only for the local orientation. We call this variant thermal martensite; and 2. Detwinned martensite, which forms while cooling below Ms if a compressive or tensile stress above a certain threshold is applied to the material. Under this condition, all domains orient themselves according to the direction of the applied loads and the material exhibits typical large, plastic-like strain. The reorientation process is called detwinning because it implies the disappearance of all twin variants; we call this variant stress martensite. We separate the total martensite fraction into the sum of the detwinned martensite, or stress martensite xd , and twinned martensite, or thermal martensite xT : x ¼ xT þ xd

(8.14)

1D SMA models

Figure 8.2 Transformation zones for the multiple-variant model.

In the case of multivariant martensite, along with the critical temperatures, two additional material parameters are necessary to describe the behavior of the material: critical AS stresses sAS s and sf , indicating, respectively, the minimum stress necessary to form stress martensite, and the stress value above which no thermal martensite can exist. Similar to what was done in the previous section, the s  T plane can be divided into different regions, each characterized by different kinetics rules, as shown in Fig. 8.2. The figure illustrates both the positive and negative stress half-planes because the behavior can generally be different in compression and tension. Nevertheless, in this section, we will assume that the alloy behaves in the same way in both compression and tension: that is, ss þ ¼ ss  ¼ ss , and sf þ ¼ sf  ¼ sf ; and CAS;þ ¼ CAS; and CSA;þ ¼ C SA; . Similar to the austeniteemartensite transformation, here we assume that the kinetics of martensite detwinning upon the application of stress is significantly faster than the rate of change of the mechanical variables; thus, the alloy is always under quasistatic equilibrium conditions. To evaluate the production of xd and xT in correctly zone 7, where they are simultaneously admissible, we consider the total fraction, x, and the stress martensite fraction, xS , to be independent variables, whereas thermal martensite can be obtained by difference from (Eqn. 8.14) at any time. This approach has the advantage of dealing with two species, both of which are nonnegative rate in zone 7, while the thermal martensite can be produced at the expense of austenite during cooling at constant

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or decreasing stress, or destroyed by transformation into stress martensite, for an increase in isothermal stress. The transformation kinetics in each zone can be described as follows: • Zone 6. When the stress is below ss , only the formation of thermal martensite (x_ > 0, x_ T > 0 x_ d ¼ 0) is possible, and only for a decrease in temperature (dT < 0); • Zone 7. Stress martensite formation is possible (x_ d  0) only if stress is increasing (ds > 0); thermal martensite formation is also possible (x_ T  0) only if the temperature is decreasing (dT < 0). Thermal martensite fraction can also decrease when it is converted into stress martensite. The total martensite fraction cannot decrease in this zone (x_  0); • Zone 8. The material is completely martensite, existing in both the stress and thermal variants. Only the production of stress martensite is possible (x_ d  0) for an increase in stress (ds > 0), which occurs at the expense of thermal martensite (x_ T  0). The total martensite fraction is constant (x ¼ 0); and • Zone 9. The material is completely martensite, existing in the stress and temperature variants; no further transformation is possible. The stress level is below the minimum critical stress (x_ d ¼ 0, x_ T ¼ 0, x ¼ 1). Implementation of the constitutive law is similar to the single martensite variant case. As shown by Brinson [9], the equilibrium equation for the multivariant martensite SMA model is given by: s  s0 ¼ EðxÞε  Eðx0 Þε0 þ uðxÞxd  uðx0 ÞxS0 þ QðT  T0 Þ

(8.15)

Likewise, for the single-variant martensite model, it is possible to write the rate form of the equilibrium equation by differentiating Eqn. (8.15), obtaining:   vE vu ds ¼ Edε þ ε þ xd dx þ Udxd þ adT vx vx   vE vx vu vx vx þ xd þ u d þ a dT Edε þ ε vx vT vx vT vT ds ¼ (8.16) vE vx vx vu vx 1ε  u d  xd vx vs vx vs vs where the differentials of the total martensite fraction and the stress martensite fraction in the first part of Eqn. (8.16) are given by: 8 vx vx > > dT þ ds < dx ¼ vT ds > > : dxd ¼ vxd dT þ vxd ds vT ds

1D SMA models

Similar to the single-variant martensite model, it is possible to integrate Eqn. (8.16) with respect to strain and temperature to calculate stress as: Zε

s  s0 ¼

E dε vE vx vxd vu vx  u  x 1  ε d ε0 vx vs vx vs vs vE vx vu vx vx ZT ε þ xd þu dþa vx vT vx vT vT dT þ vE vx vxd vu vx 1  ε  u  x d T0 vx vs vx vs vs

Finally, the Jacobian of the stress with respect to the strain is given by:  ds E ¼  vE vx vx vu vx dε T  u d  xd 1ε vx vs vx vs vs

(8.17)

(8.18)

8.3.3 Numerical test cases In the next sections, we discuss a number of test cases implementing the material model presented here. The physical parameters of the alloys used for the simulation are: EA ¼ 67:0 $103 MPa

Mf ¼ 2:85  C CAS ¼ 8:0 Mpa= C εmax ¼ 0:067

EM ¼ 26:3 $103 MPa Ms ¼ 5:9  C a ¼ 0:55 Mpa= C

As ¼ 34:5  C 

Af ¼ 49:0 C

CSA ¼ 8:0 Mpa= C

ss ¼ 100 Mpa sf ¼ 170 Mpa

8.3.3.1 Material model tests We consider here an SMA wire subject to different thermomechanical loads. Fig. 8.3(a) shows the results of the simulation of tensile loading followed by unloading at different temperatures below Ms . The initial conditions were obtained assuming that the alloy was cooled from a full austenite state, and T > Ms under a stress-free condition at the test temperature. Thus, we consider different x and xT values for each test. During loading, the alloy exhibits different moduli of elasticity depending on the test temperature because of the difference in the initial martensite fractions. During unloading, the alloy is always in a full martensite state and the slope of the s ε is the same for all temperatures. Fig. 8.3(b) reports the time histories of the material status variables for the test executed at 0  C. During the initial cooling, when the temperature entered zone 6, at time 100, the formation of only thermal martensite starts. At step 250 the mechanical load starts and no further change in the martensite fraction occurs until the stress reaches ss , at about step 350. At this point, the stress martensite forms and the thermal martensite fraction is reduced until full conversion into stress martensite for s ¼ sf . No further change occurs

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Figure 8.3 Stressestrain diagram for loading and unloading of shape memory alloy at different temperatures below Ms (a). Time history of temperature and martensite fraction (b, upper) and of stress and martensite fraction (b, lower) for test at 0  C.

during unloading, which is not shown in the figure, in which the test temperature is always below As . Fig. 8.4 shows the result of a numerical test executed on the same alloy at different temperatures above Mf . The entire superelastic behavior shows up only for temperatures above Af , whereas for lower temperatures, only a fraction or no martensite is converted into austenite, depending on the test temperature. Because the test temperature is above Mf , only stress martensite forms. Results for these tests are in accordance with similar tests performed by Brinson [9]. Fig. 8.5 presents the results of a numerical test of cyclic loading unloading on the SMA alloy for a test temperature equal to 10  C, between Ms and As . First a tensile force has been applied to the alloy in complete austenite condition, x ¼ 0, until a tension of about 150 Nmm2 was reached. This tension is above the lower boundary for martensite

Figure 8.4 Stressestrain diagram for loading and unloading of an shape memory alloy wire at different temperatures above Ms (a); time history of stress and martensite fractions (b).

1D SMA models

Figure 8.5 Cyclic loadingeunloading of shape memory alloy around ss, stressestrain diagram (a); time step evolution of martensite fraction and stress (b).

formation at the test temperature, s1 , marked with the dotted line. Subsequently, a number of unloadingeloading cycles with an amplitude of 20 Nmm2 have been applied. In Fig. 8.5(a), during the unloading legs of the cycles, the alloy followed a straight line because no variation of the martensite fraction was possible. In the loading leg, martensite production is triggered as soon as tension exceeds s1 . At every repeated cycle, the slope of the unloading line changes, reflecting the change in the elastic modulus due to the changes in the alloy’s composition. Fig. 8.5(b) shows the time history of stress and martensite fraction. The transformations in the martensite fractions start when the stress crosses the s1 threshold with a positive slope. Another interesting observation about this test is that cyclic loads are capable of producing relevant transformation strain, even when the stress levels are well below the upper transformation boundary, every time the alloy enters the transformation region, the transformation restarts, and a portion of austenite transforms into martensite. Fig. 8.5 shows the results of a numerical test of recovery under constant force. The initial test temperature was 10  C. Starting from austenite state, a force has been applied to produce complete transformation into martensite. Subsequently, a number of thermal cycles have been applied to oscillate the alloy between austenite and martensite. In Fig. 8.6(b) upon heating, the alloy is seen to recover fully from the transformation strain. 8.3.3.2 Actuation of a truss structure In this section, we describe results of the simulation of a truss structure actuated via an SMA wire. The structure has been modeled with rod elements made of a linear elastic material; only one SMA wire is considered (Fig. 8.7(a)). In the first case, we consider the SMA initially in a full stress martensite state (xd ¼ 1, xT ¼ 0) with no preload applied. This state corresponds to point A in Figs. 8.7(b) and 8.8. During heating, from point AeB, the stress martensite transforms into austenite and a fraction of the

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Shape Memory Alloy Engineering

Figure 8.6 Stressestrain diagram for a tensile load application followed by a warmingecooling cycle, stressestrain diagram (a); evolution of the stress-temperature plane (b, upper) and time-step evolution of deformation and martensite fraction (b, lower).

Figure 8.7 Shape memory alloyeactivated truss, no initial preload. Initial and deformed state (a); evolution of the alloy on the T-s plane (b).

transformation strain is recovered. In the following leg, the SMA wire is cooled to its initial temperature, legs BeC and CeD. During cooling, the alloy enters zone 2 at point C, where the production of stress martensite starts, and stress decreases. When the temperature goes below Ms , the stress level is also below sf . For this reason, no further production of stress martensite is possible and thermal martensite starts forming, as shown in Fig. 8.8(b). In the design of SMA actuated structures, the formation of thermal martensite implies a reduction in the mechanical work that the alloy is capable of producing, because the transformation of thermal martensite into austenite generates no strain recovery.

1D SMA models

Figure 8.8 Shape memory alloyeactivated truss, no initial preload. Tip vertical displacement (a); timestep evolution of martensite fractions (b).

In the next case, an initial stress above sf has been applied to the alloy. As shown in Figs. 8.9(b) and 8.10 at the end of cooling phase, leg BeCdA, the stress in the alloy remains above the upper threshold for stress martensite formation and no thermal martensite arises. The behavior of the alloy is stable and repeatable since the first cycle, and all the SMA employed produces mechanical work. This example shows that accurate finite element simulations of the SMA wires coupled with the structure are necessary to be able to predict and design SMA-based actuators properly. During actuation, the thermomechanical history to which the wire is subject can be complex because of the nonlinear nature of the wire itself and the hosting structure. Results of this test are in accordance with a similar test executed by Brinson et al. [75].

Figure 8.9 Shape memory alloyeactivated truss, initial preload (a); T-s plane, no initial preload (b).

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Figure 8.10 Shape memory alloyeactivated truss, initial preload, tip vertical displacement (a); timestep evolution of martensite fractions (b).

8.4 Advanced models with training effect In this section, the 1D SMA model proposed by Auricchio et al. [65] is presented. To get a simple and effective formulation to model the pseudoelastic behavior as well as the shape memory effect, the analysis is restricted to the case in which the temperature is greater than Ms . Thus, only austeniteesingle-variant martensite A/S and singlevariant austenite phase S/A transformations are considered [23]. The austenite and the single variant martensite volume fractions are denoted as xA and xd , respectively. Because it is xA þ xd ¼ 1, it results in: xA ¼ 1  xd

(8.19)

In the following formulation, the single-variant martensite volume fraction is chosen to be the independent internal variable governing the phase transformations. Furthermore, the proposed model takes into account the different behaviors in the tension and compression of the SMA. The presented 1D uniaxial SMA model is able to describe the pseudoelastic and shape memory effects but also the SMA cyclic response. The micromechanical interpretation that considers the progressive increases in permanent deformation during cyclic loading owing to an increase in the irreversible residual martensite fraction is adopted [65]. Thus, the martensite fraction is assumed to be the sum: xd ¼ xrev þ xR

(8.20)

where xrev represent the reversible martensite fraction and xR is the residual irreversible martensite fraction.

1D SMA models

8.4.1 Strain decomposition and elastic relation The additive decomposition of total strain ε results in: ε ¼ εe þ xd u  xR ðu  kÞ þ aðT  T0 Þ

(8.21)

where u is the reorientation parameter and k is a training parameter whose value is set in the next subsection. The stressestrain law is introduced as: s ¼ EðxS Þ½ε  xd u þ xR ðu  kÞ  aðT  T0 Þ

(8.22)

where E is the elastic modulus, given by: Eðxd Þ ¼

EA ES ES þ xd ðEA  ES Þ

(8.23)

with EA and ES as the Young’s moduli for the austenite and single-variant martensite, respectively. Equation (8.23) is obtained from the homogenization theory adopting the Reuss scheme [15,17].

8.4.2 Kinetic rules To describe the kinetics rules that characterized the thermomechanical response of SMAs, the following material parameters are introduced: • εmax is the recoverable strain representing a measure of the maximum deformation obtainable aligning all single-variant martensite in one direction; it is set as  εmax ¼ εþ max in tension and εmax ¼ εmax in compression; AS SA • C and C are the Clausius-Clapeyron constants for the phase transformations A/S and S/A, respectively; they are set as CAS ¼ CASþ and CSA ¼ CSAþ in tension and CAS ¼ CAS and CSA ¼ CSA in compression, as represented in Fig. 8.2; • ss and sf are the starting and final stress for the A/S phase transformation at temþ þ  perature T ¼ TAM s ; they are set as ss ¼ ss and sf ¼ sf in tension and ss ¼ ss  and sf ¼ sf in compression, as represented in Fig. 8.2. In particular, the increase in the single-variant martensite is set as: AS SA x_ d ¼ x_ d þ x_ d

(8.24)

where x_ d and x_ d represent the single-variant martensite volume fraction rates occurring during the austeniteesingle-variant martensite and single-variant martensiteeaustenite AS SA phase transformations, respectively. x_ d and x_ d cannot be simultaneously different from zero. AS

SA

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Shape Memory Alloy Engineering

The phase transformation A/S, inducing the single-variant martensite production of AS x_ d , is governed by the evolution equation: _ AS

G _xAS ¼ pAS 1  xAS HAS AS

aAS d d Sf  GAS

(8.25)

where: GAS ¼ z  SAS f ¼

CAS T E

sf  CAS MS sS  CAS MS AS þ RAS S ¼ þ RAS f S S ES E

(8.26) (8.27)

with

if

if

s0

s > > < RAS s > > > : AS Rf 8 z ¼ > > > < RAS s > > > : AS Rf

ε ¼ xd u  xR ðu  kÞ ¼ u  xR ðu  kÞ ε

(8.28)

¼ xd u þ xR ðu  kÞ ¼ u þ xR ðu  kÞ

The phase transformation S/A, inducing the single-variant martensite production of _xSA , is governed by the evolution equation: d _ SA

SA G SA  x x_ d ¼  pSA xSA H R SA

aSA d Sf  GSA

(8.29)

where: GSA ¼ z  SSA f ¼

CSA T E

CSA Af CSA Af þ RSA þ RSA SSA ¼ f s s EA E

(8.30) (8.31)

1D SMA models

with:

if

if

8z ¼ > > > < RAS s0 s > > > : AS Rf 8z ¼ > > > < RAS s > > : AS Rf

ε ¼ xd u  xR ðu  kÞ ¼ xR k ε

(8.32)

¼ xd u þ xR ðu  kÞ ¼ xR k

The quantities pAS , pSA , aAS , and aSA are material parameters that can be evaluated performing a uniaxial tensile test. Furthermore, HAS and HSA are activation factors, defined as: ( AS 8 _ >0 > < 1 when G AS AS H ¼ (8.33)  SAS SAS > s G f : 0 otherwise ( SA 8 _ < 1 when G SA AS H ¼ (8.34)  SSA SSA > f G S : 0 otherwise The austeniteemartensite and martensiteeaustenite phase transformations cannot occur at the same time; thus, only one of two evolutive Eqns. (8.25) and (8.29), is not trivial, which has to be solved during the phase transformation. During the loading histories, the single-variant martensite undergoes a reorientation process when a transition occurs from tensile to compressive stress, or vice versa. To predict these effects, a simple model is proposed. Reorientation parameter u is introduced, whose variation is governed by the evolutive Eqn. (8.17):   g½εmax sgnðsÞ  u absðsÞ  sSS when u_ ¼ absðsÞ > sSS (8.35) 0 otherwise where g is a material parameter measuring the reorientation process rate and sSS is a limit stress that activates the reorientation process. sSS can assume different values in tension, sSS ¼ sSSþ , and in compression, sSS ¼ sSS .  Training parameter k is set as k ¼ εþ max in tension and k ¼ εmax in compression. It is assumed that the initial and final stress thresholds of the phase transformations and the residual martensite fraction depend on the loading history, as discussed in the following section, to simulate the training effects properly on the material.

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8.4.3 Modeling of training effects The characteristic behavior of SMA under cyclic loading is illustrated in Fig. 8.11, where an experimental test on an NiTi wire of circular section, produced by OMRCO (Glendora, CA), is described. The experimental stressestrain curve, reported in Fig. 8.11, represents the typical pseudoelastic mechanical response under cyclic loading of NiTi SMA. This and several other tests on the same material were performed in the Laboratory of Biological Structure Mechanics (Labs) at the Politecnico of Milano (Italy). In Fig. 8.11, only the first and 50th of 100 performed cycles are represented. The other cycles are not reported so as to obtain a clearer representation of the training effects. The whole cyclic test demonstrated that by the 50th cycle, the material response has already reached a limit and stable path. The initial and final stress thresholds of austeniteemartensite and martensiteeaustenite phase transformation decreased during cyclic loading of different quantities. Thus, each value of stress is bounded between two values. In particular, for AS are limited between the the phase transformation A/S, the values of sAS s and sf following values: - for no-trained material (i.e., virgin) ss ¼ sAS s1

sf ¼ sAS f1

(8.36)

ss ¼ sAS s2

sf ¼ sAS f2

(8.37)

- for fully trained material

Figure 8.11 Stressestrain response of an OMRCO wire of a circular section of NiTi shape memory alloy under cyclic loading: experimental results.

1D SMA models

AS AS AS with sAS s1  ss2 and sf 1  sf 2 . In the phase transformation S/A, the initial and final stress thresholds decrease as a consequence of the increase in temperatures As and Af between two limit values, denoted as: - for no-trained material (i.e., virgin)

As ¼ As1

Af ¼ Af 1

(8.38)

As ¼ As2

Af ¼ Af 2

(8.39)

- for fully trained material

with As1  As2 and Af 1  Af 2 . Thus, in the proposed model, it is assumed that ss , sf , As , and Af evolve between the two limiting values, reported in Eqns. (8.36e8.39) after an exponential law in terms of accumulative measures of the occurrence of phase transformation. In particular, it is set as: i

AS

h AS AS SA ss ¼ sAS (8.40)  s  s g 1  exp  b s1 s1 s2 s i

AS

h AS AS SA (8.41) sf ¼ sAS  s  s g 1  exp  b f1 f1 f2 f h

i AS As ¼ As1 þ ðAs2  As1 Þ 1  exp  bSA g (8.42) s h

i AS g (8.43) Af ¼ Af 1 þ ðAf 2  Af 1 Þ 1  exp  bSA f where:

   AS  g_ AS ¼ x_ S 

   SA  g_ SA ¼ x_ S 

(8.44)

AS SA SA and bAS s , bf , bs , and bf are material constants [65] measuring the material ability to be trained. They can be measured experimentally during cyclic uniaxial tensile tests, such as the ones illustrated in Section 8.4.6. Evolution of stress values ss and sf occurs during the S/A phase transformation, whereas the evolution of temperature values As and Af occurs during the S/A phase transformation. Equation (8.40) is plotted in Fig. 8.12 for different values of bAS s . Similar curves can be derived for sf , As , and Af . Analogous considerations are made for residual martensite fraction xR . From the experimental stressestrain curve reported in Fig. 8.11, parameter xR is bounded between two values; for no-trained material (i.e., virgin), it is xR ¼ 0; for fully trained material, it is xR ¼ xL .

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Figure 8.12 Evolution of ss as a function of gSA for different values of bsAs.

Thus, it is assumed that residual martensite fraction xR evolves between the two limiting values, xR ¼ 0 and xR ¼ xL , following an exponential law in terms of an accumulative measure of the occurrence of phase transformation. In particular, it is set as: 

xR ¼ xL 1  exp bR gAS (8.45) where gAS is given by the first of Formulas (8.44) and bR is a material constant measuring the material ability to be trained. The evolution of the residual martensite fraction xR occurs during the A/S phase transformation. Equation (8.45) is plotted in Fig. 8.13 for different values of bR .

8.4.4 Time-discrete model The evolutive equations governing SMA phase transformations of the models described in the previous sections are integrated developing a step-by-step time algorithm. In particular, once the solution at time tn is known, a backward-Euler implicit integration procedure is adopted to evaluate the solution at time tnþ1 ¼ tn þ Dt. In the following discussion, subscript “n” indicates a quantity evaluated at time tn , whereas no subscript indicates a quantity evaluated at current time tnþ1 . The Delta symbol indicates variable increases at time step Dt. The discretization form of the evolutive Eqn. (8.44) is:

  AS AS SA SA SA SA gAS ¼ gAS þ x  x ¼ g þ x  x g (8.46) n d d;n n d d;n

1D SMA models

Figure 8.13 Evolution of xR as a function of gAS for different values of bR.

In Eqn. (8.46), it is assumed that conditions (8.33) and (8.34) for phase transformation activation are satisfied. The time-discrete form of evolutionary Eqns. (8.25) and (8.29) are written in a residual form is:

aAS

AS AS AS R1 ¼ lAS  G  pAS ð1  xd Þ GAS  GAS ¼ 0 (8.47) S d f ;n n H

aSA

SA SA SA R2 ¼ lSA  S  pSA ðxd  xR Þ GSA  GSA ¼ 0 G d f ;n n H

(8.48)

where the martensitic fraction variation is defined as: Ztnþ1 lAS d

¼ tn

AS x_ d dt

Ztnþ1 lSA d

¼

SA x_ d dt

(8.49)

tn

such that SA xd ¼ xd;n þ lAS d þ ld

(8.50)

Only one of two residual Eqns. (8.47) and (8.48), is not trivial, which has to be solved during the phase transformation, because, as has been pointed out, when HAS ¼ 1, HSA ¼ 1; and when HSA ¼ 1, HAS ¼ 0. The two residual equations, (8.47) and (8.48), are nonlinear, so they are linearized by a NewtoneRaphson technique. An iterative procedure is developed to evaluate lAS d and k lSA at each time step. In the following discussion, apex denotes quantities at the k-th d

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iteration, and apex kþ1 , those at the current iteration. The delta symbol d indicates the quantity variation between two consecutive iterations (i.e., dld ¼ ldkþ1  lkSd ). By linearization, Eqns. (8.47) and (8.48) become:   dR 1 0 ¼ R1kþ1 ¼ Rk1 þ AS  dlAS (8.51) dld  d k  dR2  0 ¼ R2kþ1 ¼ Rk2 þ SA  dlSA (8.52) dld  d k

For

HAS

¼ 1, substituting formulas (8.23) and (8.50) in Eqn. (8.47) results in:

 AS

 AS;k a AS;k AS  p  l (8.53) Rk1 ¼ lAS;k L 1  x LAS;k d;n 1 2 d d

 ! AS;k AS AS aAS

l a C TðE  E Þ dR1  A S AS;k 1þ d  ¼ L1  dlAS EA ES LAS;k d 1 k

 1 AS TðE  E Þ 1  xd;n  lAS;k C A S d A þ pAS @LAS;k þ 2 EA ES 0

where LAS;k ¼ SAS f ;n  z þ 1 ¼ z LAS;k 2

h

i  CAS T ES þ xd;n þ lAS;k  E Þ ðE A S d

EA ES h

i  CAS T ES þ xd;n þ lAS;k  E Þ ðE A S d EA ES

 GAS n

(8.54)

(8.55)

(8.56)

For HSA ¼ 1, substituting formula (8.23) and (8.50) into Eqn. (8.52) results in:

aSA

 Rk2 ¼ lSA;k  pSA xd;n þ lSA;k  xR LSA;k (8.57) LSA;k 1 2 d d  !

 SA SA SA lSA;k dR2  SA;k a d a C TðEA  ES Þ 1  ¼ L1  dlSA EA ES LSA;k d k 1  1  x xd;n  lSA;k CSA TðEA  ES Þ R d A þ pSA @LSA;k þ 2 EA ES 0

(8.58)

1D SMA models

where LSA;k ¼ z  SAS f ;n  1 ¼ z LSA;k 2

h

i  CSA T ES þ xd;n þ lSA;k  E Þ ðE A S d

EA ES h

i  CSA T ES þ xd;n þ lSA;k  E Þ ðE A S d EA ES

 GSA n

The single-variant martensite variation at each iteration results in:  !1  !1   dR dR   1 2 k SA dlAS R dl ¼  Rk2  d ¼  1 d AS  SA  dld dld k

(8.59)

(8.60)

(8.61)

k

SA so, variables lAS d and ld can be updated as:

þ dlAS ldAS;kþ1 ¼ lAS;k d d

lSA;kþ1 ¼ lSA;k þ dlSA d d d

(8.62)

Then, the iterative procedure goes on until a convergence test is satisfied (i.e., when the values of the residuals computed by Eqns. (8.53) and (8.57) are less than a prefixed tolerance). Setting exponentials aAS ¼ 1, aSA ¼ 1, and coefficients pAS ¼ 1, pSA ¼ 1, residual Eqns. (8.47) and (8.48) become linear. In this case, the evolution of xd can be evaluated with an explicit formula without using the NewtoneRaphson technique illustrated earlier. The result is:


CAS

AS T  Gn 1  xd;n z  ES þ xd;n ðEA  ES Þ EA ES AS ld ¼ (8.63)



CAS AS AS T Sf ;n  Gn þ 1  xd;n ðEA  ES Þ EA ES 


CSA

SA  xd;n  xR z  ES þ xd;n ðEA  ES Þ T  Gn EA ES (8.64) ¼ lSA d



CSA SA SA T Sf ;n  Gn  xd;n  xR ðEA  ES Þ EA ES Equations (8.63) and (8.64), setting xR ¼ 0 and z ¼ jεj, give the evolution equaSA tions of lAS d and of ld according to the SMA model presented by Marfia et al. [23]. The time integration of evolutive Eqn. (8.35) of the reorientation parameter u when it occurs (i.e., when jsj > sSS ) gives: (

Dtgðεmax  uÞ s  sSS when s > sSS u ¼ un þ (8.65)

Dtgðεmax þ uÞ s þ sSS when s < sSS

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Then, parameter u is evaluated solving the second-order equation obtained by substituting the expression of s given by Formula (8.22) into Eqn. (8.65) [17, 23]. The training equations do not contain rate terms; hence, they are evaluated at time tnþ1 :

i AS

h AS AS SA  s  s g 1  exp  b ss ¼ sAS (8.66) s1 s1 s2 s

i AS

h AS AS SA  s  s g 1  exp  b (8.67) sf ¼ sAS f1 f1 f2 f h

i AS g (8.68) As ¼ As1 þ ðAs2  As1 Þ 1  exp  bSA s h

i AS g (8.69) Af ¼ Af 1 þ ðAf 2  Af 1 Þ 1  exp  bSA f 

(8.70) xR ¼ xL 1  exp bR gAS Within a generic time step [tn , tnþ1 ], the time-discrete model is solved adopting a specific return-map algorithm that should consider a proper evaluation of the activation factors in a time-discrete setting.

8.4.5 Algorithmic tangent modulus In the following discussion, the construction of the tangent modulus consistent with the time-discrete models presented in the previous subsection is discussed [76]. The differentiation of constitutive Eqn. (8.22) gives consistent tangent modulus Et : ds ¼ Et dε

(8.71)

with Et ¼ E* A½ε  xd u þ xR ðu  kÞ  aðT  T0 Þ þ E½1  Aðu  Cðu  kÞÞ  Bðxd  xR Þ

(8.72)

where E* ¼

vE vxd

vu B ¼ vε

A ¼

vxd vlAS vlSA ¼ d þ d vε vε vε



vx C ¼ R ¼ HAS xL bR exp bR gAS vxd

(8.73)

In particular, it is: E* ¼

vE EA  ES ¼ E2 vxd EA ES

(8.74)

1D SMA models

It results: if HAS ¼ 1:



AS

AS sgnðεÞ L1  LAS pAS 1  xd;n  lAS 2 a d A¼ AS aAS þ1

AS * AS

AS C AS AS aAS AS 1  x L1 þ pAS LAS E T L þ L d;n  ld 1 L2  p 1 2 E2 (8.75)

with LAS 1 LAS 2



CAS T ES þ xd;n þ lAS d ðEA  ES Þ ¼ zþ EA ES 

CAS T ES þ xd;n þ lAS d ðEA  ES Þ ¼ z  GAS n EA ES SAS f

(8.76) (8.77)

if HSA ¼ 1:



SA

SA SA sgnðεÞ pSA xd;n þ lSA d  xR L1  L2 a A¼ SA aSA þ1

CSA * SA

SA SA SA aSA SA x L1 þ pSA LSA E T L þ L d;n þ ld  xR 1 L2  p 1 2 E2 (8.78) with

 SA SA T E þ x ðE C þ l  E Þ S A S d;n d LSA  SSA 1 ¼ z f ;n EA ES 

CSA T ES þ xd;n þ lSA d ðEA  ES Þ ¼ z   GSA LSA 2 n EA ES

(8.79) (8.80)

if s > sSS : B¼

E* A½ε  xd u þ xR ðu  kÞ  aðT  T0 Þ þ Eð1 þ ðC  AÞbÞ ½1 þ Dtgðs  sSS Þ=Dtgðεmax  uÞ þ Eðxd  xR Þ

(8.81)

if s  sSS : B¼

E* A½ε  xd u þ xR ðu  kÞ  aðT  T0 Þ þ Eð1 þ ðC  AÞuÞ ½1  Dtgðs  sSS Þ=Dtgðεmax þ uÞ þ Eðxd  xR Þ

(8.82)

8.4.6 Numerical results The beam finite elements developed in Marfia et al. [23] and implemented in the numerical code FEAP [77] are adopted to perform interesting applications to investigate the cyclic behavior of the SMA material.

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8.4.6.1 Shape memory alloy cyclic behavior Some numerical applications are developed to assess the ability of the presented model to simulate typical SMA pseudoelastic strainestress responses under cyclic loading conditions. Experimental data relative to an NiTi alloy are considered because NiTi SMAs probably the most frequently used in commercial applications. In particular, the experimental data presented in Ref. [78] are considered. The material is a commercial pseudoelastic NiTi straight wire with a circular cross-section 2.01 mm in diameter. From an inspection of the experimental data, parameters characterizing the tensile behavior of the material for the proposed SMA model are: EA ¼ 70000 MPa Ms ¼ 10



C

ES ¼ 10000 MPa Mf ¼ 5



C

εþ max ¼ 0:08 As1 ¼ 30

 

sSS;þ ¼ 30 MPa

C

Af 1 ¼ 31



C

Af 2 ¼ 31







As2 ¼ 30

¼ 600 MPa sAS;þ s1

sAS;þ ¼ 380 MPa s2

sAS;þ ¼ 700 MPa f1

CAS;þ ¼ 6 MPa= C CSA;þ ¼ 8 MPa= C a ¼ 0:00002

C

C C

sAS;þ ¼ 700 MPa f2 xL ¼ 0:012

aAS ¼ 1:1

pAS ¼ 1:1

aSA ¼ 1:0

pSA ¼ 0:97

¼ bAS ¼ 0:1 bAS s f

bSA ¼ bSA ¼ 0:1 s f

bR ¼ 0:5

 (8.83)

The data related to compressive behavior are not reported because only tensile behavior is investigated in the following discussion. In particular, two cyclic loading histories are considered. Fig. 8.14(a) reports the experimental stressestrain response of the material subjected to loading cycles characterized by the same maximum elongation equal to 7.0%. As it can be seen in Fig. 8.14(a), during each cycle, incomplete phase transformations A/S and S/A occur, and during training, there is a significant decrease in initial stress threshold ss and a relevant increase in residual martensite fraction xR . Fig. 8.14(b) represents the pseudoelastic response obtained by the presented model. The numerical response reported in Fig. 8.14(b) is in good accordance with the experimental behavior reported in Fig. 8.14(a). Stress value ss and residual martensite fraction xR vary more significantly in the first cycle; then, after 20 cycles, the values of ss and xR tend to stabilize. In Fig. 8.15(a), the evolution of stress value ss is plotted in terms of the accumulated strain during the S/A phase transformation, defined as gSA ¼ gSA εmax ; and in Fig. 8.15(b), residual martensite fraction xR versus the accumulated strain during the A/S phase transformation, set as gAS ¼ gAS εmax , is represented. The experimental

1D SMA models

(a)

(b)

Figure 8.14 Pseudoelastic response of an NiTi alloy subjected to equal loading cycles with 7% maximum elongation: experimental results (a); numerical results (b).

data and numerical results are represented with diamonds and with a continuous line, respectively. The experimental and numerical data are in good accordance; thus, the choice of model parameters is satisfactory. In particular, the model is able to simulate the evolution of stress value ss and of residual martensite fraction xR during cyclic loading. Furthermore, both the decrease in stress threshold value ss and the increase in residual martensite fraction xR tend to stabilize after the first 20 cycles. Fig. 8.16(a) presents the experimental stressestrain response of the material subjected to different loading cycles; the cycles are characterized by an increasing value of the maximum elongation, which varies from 2% to 8%. In Fig. 8.16(b), the mechanical pseudoelastic response obtained by the proposed model is represented. The numerical results (Fig. 8.16b) are in good accordance with the experimental results (Fig. 8.16a).

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(a)

(b)

Figure 8.15 Evolution of ss as a function of the accumulated strain gSA (a); evolution of xR as a function of the accumulated strain gAS (b).

Fig. 8.17(a) plots the stress value ss in terms of accumulated strain gSA , and in Fig. 8.17(b), residual martensite fraction xR versus accumulated strain gAS is represented. The variation of ss and of xR occurring during the S/A and A/S phase transformations, respectively, are influenced by two opposite effects. Variations in parameters ss and xR tend to decrease as these parameters are modeled adopting exponential functions (8.40) and (8.45). On the other hand, they tend to increase as a consequence of the increase in the maximum deformation of the loading cycles.

1D SMA models

(a)

(b)

Figure 8.16 Pseudoelastic response of an NiTi alloy subjected to loading cycles with increasing maximum elongation from 2% to 8%: experimental results (a); numerical results (b).

Also, in this case, the experimental and numerical data are in good accordance; thus, the choice of model parameters appears satisfactory. Then, experimental data related to the OMRCO wire of the circular section of NiTi SMA, represented in Fig. 8.11, are considered. The experimental stressestrain curve represents the pseudoelastic mechanical response of the wire subjected to loading cycles characterized by the same maximum elongation equal to 6%. The parameters of the material are set on the basis of the experimental data. In the following discussion, the tensile behavior of the wire is investigated. Thus, only parameters characterizing the tensile response are set as:

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(a)

(b)

Figure 8.17 Evolution of ss as a function of accumulated strain gSA (a); evolution of xR as a function of accumulated strain gAS (b).

EA ¼ 33000 MPa

ES ¼ 18300 MPa

εþ max ¼ 0:05

sSS;þ ¼ 30 MPa

Ms ¼ 4  C

Mf ¼ 3  C

As1 ¼ 8  C

Af 1 ¼ 11  C

As2 ¼ 10 ¼ 200 MPa sAS;þ s1

sAS;þ ¼ 90 MPa s2



C

sAS;þ ¼ 200 MPa f1

Af 2 ¼ 14  C sAS;þ ¼ 140 MPa f2

CAS;þ ¼ 12 MPa= C CSA;þ ¼ 15 MPa= C a ¼ 0:00002  C

xL ¼ 0:0125

aAS ¼ 1:1

pAS ¼ 1:0

aSA ¼ 1:0

pSA ¼ 0:97

¼ bAS ¼ 0:1 bAS s f

bSA ¼ bSA ¼ 0:1 s f

bR ¼ 0:5 (8.84)

1D SMA models

Figure 8.18 Stressestrain response of an OMRCO wire under cyclic loading: comparison of experimental and numerical results.

In Fig. 8.18, the stressestrain response of the OMRCO wire is represented; in particular, the first and the 50th cycles are plotted. The experimental and numerical results are in good accordance. The presented model shows the variation in the initial and final stress thresholds of the phase transformations and the increase in permanent deformation. 8.4.6.2 Two-way memory effects A cantilever beam is analyzed, characterized by an elastic core and two SMA layers, one on the top and the other on the bottom, represented in Fig. 8.19.

SMA layer

0.1 mm

Elastic core

0.8 mm

SMA layer

0.1 mm

10 mm Figure 8.19 Cantilever beam: geometry and boundary conditions. SMA, shape memory alloy.

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200

E=30000 MPa

Bending Moment [MPa]

284

E=20000 MPa E=10000 MPa

150

100

50

0 0.00

0.06

0.03

0.09

0.12

x 100 0.15

Transversal displacement [mm]

Figure 8.20 Bending moment versus transversal displacement for three different values of Young’s modulus of elastic core.

It is assumed that the SMA has the same mechanical behavior in tension and compression, so only the data characterizing the tensile behavior of material are reported: EA ¼ 33000 MPa

ES ¼ 18300 MPa

εþ max ¼ 0:08

sSS;þ ¼ 30 MPa

Ms ¼ 10  C

Mf ¼ 5  C

As1 ¼ 28  C

Af 1 ¼ 32  C

As2 ¼ 28  C

Af 2 ¼ 32  C

sAS;þ ¼ 80 MPa f1

sAS;þ ¼ 80 MPa f2

¼ 40 MPa sAS;þ s1

sAS;þ ¼ 40 MPa s2

CAS;þ ¼ 6 MPa= C CSA;þ ¼ 8 MPa= C a ¼ 0:00002  C

xL ¼ 0:0125

aAS ¼ 1:0

pAS ¼ 1:0

aSA ¼ 1:0

pSA ¼ 1:0

¼ bAS ¼ 0:2 bAS s f

bSA ¼ bSA ¼ 0:2 s f

bR ¼ 0:5 (8.85)

Three analyses are performed using different values for the Young’s modulus of the elastic core: • first case Ec ¼ 10000 MPa • second case Ec ¼ 20000 MPa • third case Ec ¼ 30000 MPa The beam length is equal to 10.0 mm, the rectangular cross-section is 1.0 mm  1.0 mm, and each SMA layer has a thickness equal to 0.1 mm, so the elastic core thickness is 0.8 mm.

1D SMA models

The layer-wise beam finite element, developed in Marfia et al. [23], is adopted to perform the numerical analyses. To obtain a fully trained shape memory material, the beam is subjected to an initial history of 10 cycles of bending loadingeunloading, with a maximum bending moment equal to 200 Nmm at a constant temperature T ¼ 60  C. The whole training takes t ¼ 20 s. In Fig. 8.20, the loadingeunloading bending moment is plotted versus the transversal displacement of the free end of the beam for the three examined cases. After 10 cycles, the mechanical response of the beam is stable in all three cases as the SMA material is completely trained. Then, the beam is subjected to two cycles of temperature changes from 11 to 100  C. In Fig. 8.21, transversal displacement of the beam free end is plotted versus time for the three different values of the elastic core Young’s modulus. At the end of the bending training (i.e., at time t ¼ 20 s), the beam remains in a deformed configuration because of permanent strain accumulated during the cycles. After that time (i.e., for t > 20 s), two thermal cycles are performed. Thus, the beam swings between two configurations: the beam is able to recover a greater part of the permanent strain during cooling, whereas it goes back to the deformed shape during heating. Moreover, the influence of the Young’s modulus of the elastic core on the described two-way shape memory effect is small.

x 100 0.15

E=10000 MPa

w [mm]

0.12

0.09

E=20000 MPa

0.06

E=30000 MPa

0.03

0.00 0

3

6

9

12

15

18

21

24

t [s] Figure 8.21 Transversal displacement versus time for three different values of Young’s modulus of elastic core.

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8.5 Conclusions In this chapter, two approaches for the modeling of SMA alloys have been described. One approach adopts a phenomenological description of the alloy, which assumes that the alloy is always in thermodynamic equilibrium and does not require time integration of the transformation kinetic equations. The implemented model is able to simulate the behavior of an SMA wire including the evolution of both thermal and stress martensite. The procedure presented here is based on the derivation of an evolutionary form of the constitutive and equilibrium equations obtained by calculating the different production terms of the martensite variants in the different zones of the stress temperature plane, and integrating these terms along the thermomechanical load path to which the alloy is subject. The other approach considers the transformations from austenite to single-variant martensite and from single-variant martensite to austenite, considering the influence of the temperature. The martensite volume fraction is assumed to be an internal variable whose variation is ruled by kinetic laws governed by the strain and temperature. The martensite reorientation is also taken into consideration. Different elastic properties are considered for the austenite and martensite and the elastic moduli of the SMA are evaluated following a Reuss scheme. Furthermore, the different behaviors in tension and compression of the material are modeled. Moreover, the model is able to describe the cyclic response of the SMA. In particular, the micromechanical interpretation that assumes that the training effects are due to the residual permanent martensite fraction is adopted. Hence, the material training under cyclic loading is modeled adopting exponential evolutions of the austeniteemartensite transformation stress thresholds and of the residual permanent deformation expressed in terms of the residual permanent martensite fraction. The time integration of the evolutive equations is performed adopting a backward Euler scheme, and the finite time step is solved through a modified return-map algorithm. The procedure presented in this chapter follows the approach of displacement-based structural finite element programs, and it has been outlined how to evaluate internal stress and the tangent stiffness matrix necessary for the iterative solution with the full Newtone Rapson method. The procedure is suitable for integration into custom-made finite element codes or in codes that allow user-provided subroutines. A number of examples of numerical model have been presented to verify the compliancy of the outcomes compared with experimental results. Results are also presented for the simulation of SMA wires actuating an elastic truss, which shows the importance of an accurate simulation when designing active structures based on an SMA actuator. The capacity of a model to reproduce experimental data concerning the cyclic response and the two-way shape memory effect has also been assessed. The ability of the models to describe the material response for a complex loading history (partial loadingeunloading pattern) has been numerically tested.

1D SMA models

References [1] D.C. Lagoudas, Shape Memory Alloys: Modeling and Engineering Applications, Springer, New York, 2008 s.l. [2] F. Auricchio, et al., Shape Memory Alloys, Advances in Modelling and Applications, CIMNE, Barcelona, 2001 s.l. [3] A. Khandelwal, V. Buravalla, Models for shape memory alloy behavior: an overview of modeling approaches, Int. J. Struct. Changes Solids (2009) 111e148. [4] A. Paiva, M.A. Savi, An overview of constitutive models for shape memory alloys, Math. Probl Eng. (2006). [5] K. Tanaka, S. Kobayashi, Y. Sato, Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys, Int. J. Plast. (1986) 59e72. [6] M. Achenbach, A model for an alloy with shape memory, Int. J. Plast. (1989) 371e395. [7] C. Liang, C.A. Rogers, One-dimensional thermomechanical constitutive relations for shape memory materials, J. Intell. Mater. Syst. Struct. 1 (1990) 207e234. [8] R. Abeyaratne, J.K. Knowles, A continuum model of a thermoelastic solid capable of undergoing phase-transitions, J. Mech. Phys. Solid. (1993) 541e571. [9] L.C. Brinson, One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable, J. Intell. Mater. Syst. Struct. (1993) 229e242. [10] Y. Ivshin, T.J. Pence, A constitutive model for hysteretic phase transition behavior, Int. J. Eng. Sci. (1994) 681e704. [11] Y. Ivshin, T.J. Pence, A thermomechanical model for a one variant shape memory material, J. Intell. Mater. Syst. Struct. (1994) 207e234. [12] A. Bekker, L.C. Brinson, Temperature-induced phase transformation in a shape memory alloy: phase diagram based kinetics approach, J. Mech. Phys. Solid. (1997) 949e988. [13] A. Bekker, L.C. Brinson, Phase diagram based description of the hysteresis behavior of shape memory alloys, Acta Mater. (1998) 3649e3665. [14] F. Auricchio, J. Lubliner, A uniaxial model for shape-memory alloys, Int. J. Solid Struct. (1997) 3601e3618. [15] F. Auricchio, E. Sacco, A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite, Int. J. Non Lin. Mech. (1997) 1101e1114. [16] F. Auricchio, E. Sacco, A superelastic shape-memory-alloy beam model, J. Intell. Mater. Syst. Struct. (1997) 489e501. [17] F. Auricchio, E. Sacco, A temperature-dependent beam for shape-memory alloys: constitutive modelling, finite-element implementation and numerical simulations, Comput. Methods Appl. Mech. Eng. (1999) 171e190. [18] I.V. Chenchiah, S.M. Sivakumar, A two variant thermomechanical model for shape memory alloys, Mech. Res. Commun. (1999) 301e307. [19] S. Govindjee, E.P. Kasper, Computational aspects of one-dimensional shape memory alloy modeling with phase diagrams, Comput. Methods Appl. Mech. Eng. (1999) 309e326. [20] K.R. Rajagopal, A.R. Srinivasa, On the thermomechanics of shape memory wires, Z. Angew. Math. Phys. (1999) 459e496. [21] B. Raniecki, J. Rejzner, C. Lexcellent, Anatomization of hysteresis loops in pure bending of ideal pseudoelastic SMA beams, Int. J. Mech. Sci. (2001) 1339e1368. [22] J. Rejzner, C. Lexcellent, B. Raniecki, Pseudoelastic behaviour of shape memory alloy beams under pure bending: experiments and modelling, Int. J. Mech. Sci. (2002) 665e686. [23] S. Marfia, E. Sacco, J.N. Reddy, Superelastic and shape memory effects in laminated shape-memoryalloy beams, AIAA J. (2003) 100e109. [24] T. Ikeda, et al., Constitutive model of shape memory alloys for unidirectional loading considering inner hysteresis loops, Smart Mater. Struct. (2004) 916e925. [25] A. Paiva, L. Pacheco, et al., A constitutive model for shape memory alloys considering tensilecompressive asymmetry and plasticity, Int. J. Solid Struct. (2005) 3439e3457.

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[26] M.A. Savi, A. Paiva, Describing internal subloops due to incomplete phase transformations in shape memory alloys, Arch. Appl. Mech. (2005) 637e647. [27] M. Fremond, Mechanique des milieux continus: materiaux
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1D SMA models

[49] B. Strnadel, et al., Effect of mechanical cycling on the pseudoelasticity characteristics of Ti-Ni and TiNi-Cu alloys, Mater. Sci. Eng.A-Struct. Mater. Prop. Microstruct. Process. (1995) 187e196. [50] H. Tobushi, et al., Thermomechanical properties due to martensitic and R-phase transformations of TiNi shape memory alloy subjected to cyclic loadings, Smart Mater. Struct. (1996) 788e795. [51] Y. Liu, et al., Strain dependence of pseudoelastic hysteresis of NiTi, Metall. Mater. Trans.: Phys. Metall. Mater. Sci. (1999) 1275e1282. [52] M.A. Iadicola, J.A. Shaw, An experimental setup for measuring unstable thermo-mechanical behavior of shape memory alloy wire, J. Intell. Mater. Syst. Struct. (2002) 157e166. [53] M.A. Iadicola, J.A. Shaw, The effect of uniaxial cyclic deformation on the evolution of phase transformation fronts in pseudoelastic NiTi wire, J. Intell. Mater. Syst. Struct. (2002) 143e155. [54] S. Nemat-Nasser, W.G. Guo, Superelastic and cyclic response of NiTiSMA at various strain rates and temperatures, Mech. Mater. (2006) 463e474. [55] J. Ma, et al., Superelastic cycling and room temperature recovery of Ti74Nb26 shape memory alloy, Acta Mater. (2010) 2216e2224. [56] K. Tanaka, et al., Phenomenological analysis on subloops and cyclic behavior in shape-memory alloys under mechanical and or thermal loads, Mech. Mater. (1995) 281e292. [57] C. Lexcellent, G. Bourbon, Thermodynamical model of cyclic behaviour of Ti-Ni and Cu-Zn-Al shape memory alloys under isothermal undulated tensile tests, Mech. Mater. (1996) 59e73. [58] R. Abeyaratne, S.J. Kim, Cyclic effects in shape-memory alloys: a one-dimensional continuum model, Int. J. Solid Struct. (1997) 3273e3289. [59] Z.H. Bo, D.C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part I: theoretical derivations, Int. J. Eng. Sci. (1999) 1089e1140. [60] D.C. Lagoudas, Z. Bo, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part II: material characterization and experimental results for a stable transformation cycle, Int. J. Eng. Sci. (1999) 1141e1173. [61] Z.H. Bo, D.C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part III: evolution of plastic strains and two-way shape memory effect, Int. J. Eng. Sci. (1999) 1175e1203. [62] Z.H. Bo, D.C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part IV: modeling of minor hysteresis loops, Int. J. Eng. Sci. (1999) 1205e1249. [63] G. Boyd, D.C. Lagoudas, A thermodynamical constitutive model for shape memory materials - PartI. The monolithic shape memory alloy, Int. J. Plast. (1996) 805e842. [64] C. Lexcellent, et al., The two way shape memory effect of shape memory alloys: an experimental study and a phenomenological model, Int. J. Plast. (2000) 1155e1168. [65] F. Auricchio, E. Sacco, Thermo-mechanical modelling of a superelastic shape-memory wire under cyclic stretching-bending loadings, Int. J. Solid Struct. (2001) 6123e6145. [66] F. Auricchio, S. Marfia, E. Sacco, Modelling of SMA materials: training and two way memory effects, Comput. Struct. (2003) 2301e2317. [67] S. Sun, R.K.N.D. Rajapakse, Simulation of pseudoelastic behaviour of SMA under cyclic loading, Comput. Mater. Sci. (2003) 663e674. [68] B. Azadi-Borujeni, R.K.N.D. Rajapakse, D.M. Maijer, Modelling of the cyclic behaviour of shape memory alloys during localized unstable mechanical response, Smart Mater. Struct. 18 (2009). [69] A. Vigliotti, Finite element implementation of a multivariant shape memory alloy model, J. Intell. Mater. Syst. Struct. (2010) 685e699. [70] C. Liang, C.A. Rogers, One-dimensional thermomechanical constitutive relations for shape memory materials, J. Intell. Mater. Syst. Struct. (1990) 207e234. [71] L.C. Brinson, R. Lammering, Finite-element analysis of the behavior of shape-memory alloys and their applications, Int. J. Solid Struct. (1993) 3261e3280. [72] L.C. Brinson, M.S. Huang, Simplifications and comparisons of shape memory alloy constitutive models, J. Intell. Mater. Syst. Struct. (1996) 108e114. [73] Y. Sato, K. Tanaka, Estimation of energy-dissipation in alloys due to stress-induced martensitictransformation, Res. Mech. (1988) 381e393.

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[74] F. Auricchio, D. Fugazza, R. DesRoches, A 1D rate-dependent viscous constitutive model for superelastic shape-memory alloys: formulation and comparison with experimental data, Smart Mater. Struct. (2007) S39eS50. [75] F. Auricchio, D. Fugazza, R. Desroches, Rate-dependent thermo-mechanical modelling of superelastic shape-memory alloys for seismic applications, J. Intell. Mater. Syst. Struct. (2008) 47e61. [76] L.C. Brinson, et al., Analysis of controlled beam deflections using SMA wires, J. Intell. Mater. Syst. Struct. (1997) 12e25. [77] J.C. Simo, T.J.R. Hughes, Computational Inelasticity, Springer-Verlag, New York, 1998 s.l. [78] R.L. Taylor, FEAP - A Finite Element Analysis Program, Berkeley, Department of Civil & Environmental Engineering, University of California, Vol. 8.2. [79] C. M (Ed.), Proceeding of the Final Workshop of ISTECH Project: Shape Memory Alloy Devices for Seismic Protection of Cultural Heritage Structures, 2000. Ispra, Italy.

CHAPTER 9

SMA constitutive modeling and analysis of plates and composite laminates Elio Sacco1, Edoardo Artioli2

Dipartimento di Strutture per l’Ingegneria e l’Architettura (DiSt), Universit
a di Napoli “Federico II”, Naples, Italy; Dipartimento di Ingegneria Civile e Ingegneria Informatica (DICII), Universit
a degli Studi di RomaeTor Vergata, Rome, Italy 1 2

9.1 Introduction The study of polycrystalline shape memory alloys (SMA) has been a scientific research topic of utmost importance over the past half century, attracting engineers, physicists, mathematicians, and material scientists from industry and academia. Mathematical modeling of the special thermomechanical response of SMA represents an important issue for designing new applications and performing virtual testing of SMA devices. In fact, the literature devoted to the subject of modeling pseudoelastic (PE) behavior, the shape memory effect (SME), and the two-way effect (TWE) has reached considerable dimensions. Several approaches have been proposed in the literature for modeling SMA behavior; one of the most attractive approaches to the constitutive modeling of SMA is based on continuum thermodynamics with internal variables. This type of model has attracted a great deal of attention by researchers and engineers over the years and represents the typical constitutive solver core in most engineering finite element (FE) modeling simulations owing to numerical robustness and efficiency, even if it can describe only the macroscopic material response. Such models, based on the stressetemperature phase diagrams for SMA derived from experimental data, are generally accurate. Continuum thermodynamics is a consolidated and well-established framework for developing constitutive models consistently with the fundamental principles of the thermodynamics [1e6]. The main feature of this approach is to introduce appropriate energy densities (Gibbs free energy or Helmholtz free energy) depending on internal variables, describing the physics of the material internal structure from a macroscopic point of view, and on observable control variables. The constitutive equations are derived writing the state equations, which define entities conjugate to the control variables and to the internal variables, and the evolution equations of the internal variables. Among others, one of the first applications of such an approach to SMA in the realm of small deformation is the work by Tanaka and Nagaki [7] in the so-called assumed phase Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00009-1

© 2021 Elsevier Ltd. All rights reserved.

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transformation kinetics framework. In fact, the martensitic volume fraction is employed as an internal variable to describe the extent of phase transition, expressed as a given exponential function of current stress and temperature. Subsequent models owing to this idea, and still limited to the small deformation case, can be found in Tanaka and Iwasaki [8]. Fremond [9] developed a model for three-dimensional (3D) media able to reproduce both PE and SME with the aid of three internal variables that obey internal constraints related to the coexistence of the different phases. At the beginning of the 1990s, innovative engineering SMA applications (e.g., medical, mechanical, aerospace applications [10]) promoted the development of efficient models and computational tools. Significant contributions in this promising research field were offered by the work of Raniecki et al. [11] in the context of the PE behavior of polycrystalline SMA and by Liang and Rogers [12], who discussed a constitutive model based on the combination of both micromechanics and macromechanics and martensite volumetric fraction to describe the phase transformation process. The work by Graesser and Cozzarelli [13] modeled the stressestrain response associated with martensitic twinning hysteresis and austeniteemartensite/martensiteeaustenite pseudoelasticity, expressing the growth of inelastic strain in a rate-type formulation similar to viscoplastic laws. The article by Ivshin and Pence [14] presented a model for rate-independent hysteresis and referred to temperature-induced phase transitions such as those that occur in shape memory metals. The dependence on the hysteresis envelope for complete transitions is examined to permit fitting with experimental data. Boyd and Lagoudas [15] proposed a revisit of the Tanaka constitutive model [7] in the 3D setting. Raniecki and Lexcellent [16], among others, derived a relation for the martensitic volume fraction defined in terms of the norm of inelastic strain occurring during the phase transformation (transitional strain). The 3D character of the phase transformation was thus taken into account by the model adopting a strain-like tensor internal variable. In 1995, two extensive experimental studies were published that considerably affected SMA modeling research activity: work by Shaw and Kyriakides [17] presented several aspects of the thermomechanical behavior of NiTi wires, whereas work by Sittner et al. [18] presented an extensive multiaxial test campaign on CueAleZneMn alloys. Boyd and Lagoudas [19] proposed a constitutive model able to reproduce pseudoelasticity and the SME due to martensitic transformation and reorientation, using a free energy function and a dissipation potential. Following the work by Raniecki and Lexcellent [16], Leclercq and Lexcellent [20] presented a macroscopic description for simulating the global thermomechanical behavior of SMA; two internal variables were considered: the volume fraction of self-accommodating (pure thermal effect) and oriented (stress-induced) product phases.

SMA constitutive modeling and analysis of plates and composite laminates

Lubliner and Auricchio [21] applied the generalized plasticity theory to SMA and proposed a constitutive model accounting for multiaxial loading, based on DruckerPrager flow potential; the model was capable of reproducing PE behavior and applied to a simplified representation of the behavior of SMA, with numerical examples. Auricchio and Taylor [22] proposed a large strain FE approach for the analysis of medical devices employing pseudoelasticity resorting to the constitutive model proposed in Lubliner and Auricchio [21]. Manach and Favier [23] proposed a phase transformation function depending on J2 and J3, the second and third invariants of the deviatoric stress tensor, able to account for the tension-compression asymmetry. Raniecki and Lexcellent [24] tuned the J2eJ3 dependency determining thermostatic properties and phenomenological constants for an NiTi alloy using experimental data. The pioneering work by Souza et al. [25] appeared in 1998 and proposed a model capable of describing 3D solids undergoing stress-induced phase transformations typical of polycrystalline shape memory materials. It had been conceived within the framework of generalized standard materials theory and was formulated with the formalism of convex analysis and optimization. Micromechanical aspects, not usually included in phenomenological models, were considered by the multivariant model of Huang and Brinson [26] based on thermodynamics and micromechanics for SMA single crystals. This model is derived using the concept of a thermodynamic driving force, and it is shown to exhibit appropriate responses for uniaxial results on single crystals. In particular, both pseudoelasticity and the SME are captured. Experimental tests drawn by Liu et al. [27] presented results showing remarkable differences in stressestrain curves of polycrystalline martensitic NiTi SMA under tension and compression; under tension, a flat stress plateau occurs, whereas under compression, the material exhibits strain hardening and no flat stress plateau is observed. Cyclic deformation with tensionecompression also shows the asymmetric stressestrain loop. Lim and McDowell [28] extended this investigation to more complex multiaxial loading conditions such as proportional/nonproportional axial-torsion loading. Gall and Sehitoglu [29] proposed a thorough investigation of the role of texture in tensione compression asymmetry in polycrystalline NiTi. Their evidence quantitatively demonstrated that texture measurements coupled with a micromechanical model can accurately predict tensionecompression asymmetry in NiTi SMA. Within this approach, predicted critical transformation stress levels and transformation stressestrain slopes under both tensile and compressive loading are consistent with experimental results. Following these observations, among others, Qidwai and Lagoudas [30] introduced an extended phase transformation function that depends on the three invariants: I1 (the first invariant of the stress tensor), J2, and J3 (second and third invariants of the stress deviator tensor). The same authors [31] detailed a comprehensive

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study on the numerical implementation of the SMA constitutive response using return mapping algorithms for the constitutive model based on previous work by Boyd and Lagoudas [19]. A return map integration algorithm for large strain models based on the deformation gradient multiplicative decomposition into elastic and transformation parts was proposed by Auricchio [32] in a thermodynamically consistent manner. Here, the internal variables are the martensite volume fraction accounting for only a single variant and the deformation gradient part associated with phase transition; the control variables are the Kirchhoff stress and the temperature. Helm and Haupt [33] proposed a model able to represent material behavior multiaxiality, one-way SME, two-way SME, and PE and pseudoplastic behavior as well as the transition range between pseudoelasticity and pseudoplasticity. To distinguish among different deformation mechanisms, case distinctions are introduced into the evolution equations. A successful implementation of the so-called SouzaeAuricchio model is developed in Migliavacca et al. [34] with a focus on coronary stent simulation. Bouvet et al. [35] presented a constitutive model for simulating PE behavior under general multiaxial loading paths, considering the tensionecompression asymmetry effect and the thermomechanical coupling of PE behavior. Lagoudas and Entchev [36] presented a rate-independent SMA model entailing a modification of Gibbs free energy, which accounts for transformation and plastic strains, as well as for shape and size variations in the hysteresis loop with repeated transformation cycling. The model adopted numerical integration via return mapping. Auricchio and Petrini [37,38] proposed a model with evolution associated with a Prager-Lodeetype transformation function, accounting for a different response in tension and compression. The article contains profound insights regarding a more rigorous comprehension of material state branch detection and presented numerous boundary value problems of SMA biomechanical devices under working conditions. M€ uller and Bruhns [39] proposed a finite-strain model describing the PE response, based on a self-consistent Eulerian theory of finite deformations using the logarithmic rate. The mass fraction of martensite is introduced as the internal state variable. The constitutive equation is integrated by corotational integration to preserve objectivity and implemented into an FE code. Auricchio et al. [40] proposed a model capable of including permanent inelastic effects combined with PE and shape memory behaviors, along with extensive numerical tests on robustness and accuracy for loading histories at the material point level. The model by Popov and Lagoudas [41], based on a modified phase transformation diagram, takes into account both the direct conversion of austenite into detwinned martensite and the detwinning of self-accommodated martensite. The model is suitable for performing numerical simulations on SMA materials undergoing complex thermomechanical loading paths in stressetemperature space. The model is based on thermodynamic potentials and uses three internal variables to predict phase transformation and detwinning of martensite.

SMA constitutive modeling and analysis of plates and composite laminates

Panico and Brinson [42] proposed a model that explains the effect of multiaxial stress states and nonproportional loading histories. The model was able to account for the evolution of both twinned and detwinned martensite. Moreover, this model considers the reorientation of the product phase according to the loading direction. Within the framework of generalized standard materials with internal constraints, Zaki and Moumni [43] proposed the martensite volume fraction and martensite orientation strain tensor as internal variables to explain self-accommodation, orientation, and reorientation of martensite, as well as pseudoelasticity and one-way SME. The model by Ziolkowski [44] extends the approach of Raniecki and Lexcellent [16] to explain finite deformation, still considering multiplicative decomposition of total deformation gradient into elastic and phase transformation parts and the ensuing additive decomposition of the Eulerian strain rate tensor. The Helm model [45], also based on finite deformation, considers the thermomechanical coupling phenomena. Pan et al. [46] presented a rateindependent, finite deformation crystal mechanics constitutive model for martensitic reorientation and detwinning, for the relative FE implementation; the article focuses on the experimental response of an initially textured and martensitic polycrystalline TieNi rod under a variety of uniaxial and multiaxial stress states. Vieille et al. [47] proposed a set of FE tests to validate a 3D numerical model of PE behavior of SMA at finite strain by comparison with experimental results. They considered tensile tests on CuAlBe perforated strips and bulging tests on CuAlBe sheets, as well as image correlation and infrared thermography analysis to pinpoint the thermomechanical couplings of the material. Moumni et al. [48] simulated the PE response of SMA as well as one-way SME. In addition, they compared the results with experimental data reported in Sittner et al. [18]. Stein and Sagar [49] endeavor a micromechanical approach to describe the behavior of SMA. The model considers n phase variants that are represented by n energy functions and addresses the problem of finding the minimum of the sum of no-convex energy functionals. Extending the small strain model by Helm and Haupt [33], Reese and Christ [50] suggested a finite strain phenomenological model and coded it into an FE solver to simulate NiTi stents. Evangelista et al. [51] proposed some modifications to the SouzaeAuricchio model [25,37], presenting a consistent thermodynamical formulation and clarifying the mechanical meaning of the material parameters governing the constitutive model. Numerical applications and comparison with experimental data available in the literature are carried out. Then, Evangelista et al. [52] developed a finite strain constitutive model based on multiplicative decomposition of the deformation gradient into elastic and inelastic parts, with evolution for the right CauchyeGreen strain connected to transformation phenomena associated with a limit function of the von Mises type. The model is time-integrated, resorting to an exponential map and solved with the Newton Raphson method. More recently, Arghavani et al. [53] extended the small strain model proposed by Panico and Brinson [42] to the finite deformation regime, which basically results in an extension of model of Evangelista et al. [52],

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accounting for a careful detection of the evolution branch through an innovative nucleation-completion paradigm that avoids using a regularized transformation strain norm and enables the identification of the end of the reverse transformation in the discrete procedure. The proposed model tests both the exponential and logarithmic mapping integration procedures for robustness and numerical efficiency. A further improved version of this finite strain constitutive model is in Arghavani et al. [54], who work reformulated the governing equations at a time-continuous level in terms of the stretch associated with transformation, leading to a numerical integration procedure that adopts only symmetric tensors; as a result, a gain in numerical efficiency is obtained. Much attention has been paid to the cyclic response of SMA to evaluate the capability of devices during the service life. Qui et al. [55] presented a consistent thermodynamic SMA model able to reproduce anisotropic martensitic transformation, introducing a suitable J2 eJ3 -type phase transformation surface within a correction tensor. A residual martensitic volume fraction is considered to capture the accumulation of strain with the increasing number of cycles. Consistent modeling of the presence of permanent strain occurring during cyclic loadings has been proposed in Ashrafi [56]. Inelastic strain resulting from plastic effects have been introduced in a 3D SMA model from Scalet et al. [57] to reproduce the TWE. A further interesting problem for long-life SMA devices is damage and fracture caused by the material. In this framework, Bahrami et al. [58] developed a PE constitutive model based on the approach of Boyd and Lagoudas able to account for plastic deformation and fracture behavior using the GursoneTvergaardeNeedleman model, investigating the ability of the proposed model to predict the initiation of fractures. Petrini and Bertini [59] presented a phenomenological model able to describe the accumulation of inelastic strains owing to fatigue and plasticity, introducing two limit functions for defining the phase transformation domain and plastic region. Increasing use in commercially valuable applications has motivated strong interest in the analysis of plate and shell structural components. The commercial availability of a wider class of product shapes such as fibers, wires, and strips has made engineering applications exploiting PE, SME, and TWE a top engineering research and development branch. Common applications refer to plates, shells, and panels often composed of elastic laminae coupled with SMA fiber-reinforced laminae or coupled with hybrid composite materials containing SMA-embedded wires or strips. In fact, new composites have been developed in the last decades embedding SMA inclusions. Potential applications of embedded SMA include the control of the external shape, of the stiffness, of the damage state, of the vibration, of the buckling and damping properties of the composites. In particular, the SMA reinforcements have the following benefits: (l) increase the buckling load significantly and improve the post-buckling behavior; (2) reduce deflections and stresses in plates subjected to low-velocity impact; (3) change the natural frequencies of structures.

SMA constitutive modeling and analysis of plates and composite laminates

Layered composite SMA plates are obtained as stacking sequence of thin layers, some of which contain SMA wires. The SMA wires or fibers can be suitably oriented in the laminate to obtain the desired performance along certain directions of the plate. Several macromechanical studies of the behavior of structural elements integrated with composite SMA layers have been developed. In particular, the thermal buckling behavior of laminated composite elements with embedded SMA wires has been investigated developing analytical approaches, numerical tools, and experimental studies to evaluate the increase in the critical buckling load and the reduction in thermal buckling deformation using the activation force of embedded SMA wires. Paine and Rogers [60] embedded small amounts of SMA fibers into brittle composite materials to improve the impact properties of the composite. Impact tests were performed on hybrid composite materials; they showed that the presence of SMA fibers is able to avoid perforation of the composite under impact. Birman et al. [61] presented a study of composite plates subjected to low-velocity impact. The plates are reinforced with prestrained SMA fibers; with an increase in temperature, the martensite phase in the SMA is transformed into austenite, reducing deflections and stresses in the plate when subjected to a low-velocity impact. The problem of optimizing SMA fiber orientation is treated. Birman [62] investigated the effectiveness of composite and SMA stiffeners on the stability of composite cylindrical shells and rectangular plates subjected to a compressive load. He demonstrated that composite stiffeners are more efficient in cylindrical shells, whereas SMA stiffeners may be preferable in plates or in long shallow shells. Dano [63] developed numerical and experimental studies on composite unsymmetric laminates reinforced with SMA, investigating multiple configurations and stability issues. Thompson and Loughlan [64] illustrated the manufacturing methodology of hybrid SMAecarboneepoxy plates and performed thermomechanical finite-element analysis to predict the temperature profile within the laminates and demonstrate that the presence of SMA fibers in a laminate significantly improves postbuckling behavior. Wei et al. [65] illustrated the principles and basic design concepts of using SMA composite thin films and laminates, with an emphasis on manufacturing issues. Chen and Levy [66] investigated the effect of temperature on the frequency, loss factor, and control of a flexible beam with a constrained viscoelastic layer and SMA layer; they showed the important role the temperature in the SMA had on the behavior of such structures. Lee et al. [67] studied the effect of embedded SMA wires on the characteristics of thermal buckling. The authors performed numerical analyses of laminated composite shells with embedded SMA wires. Lee and Choi [68] developed analytical studies concerning the thermal buckling and postbuckling behavior of a composite beam with embedded SMA wires, and remarked on the ability of SMA actuators to improve the buckling and postbuckling response. An experimental investigation on the buckling control of a laminated composite beam by means of embedded SMA wires was presented by Sup Choi et al. [69], who showed the effectiveness of

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using embedded active SMA actuators to enhance the buckling response of composite beams. A numerical study, based on FE analyses, of the influence of the presence of SMA fibers in a composite laminate on natural frequencies and thermal buckling was developed by Ostachowicz et al. [70]. Thompson and Loughlan [71] conducted an extensive experimental and numerical program, also outlining the manufacturing methodology of hybrid SMAecarbon/epoxy plates. Numerical investigations regarded thermal as well as mechanical FE analysis with the aim of predicting the temperature profile and the buckling load. A discussion of the process and mechanism of the SME of SMA embedded in smart hybrid composites has been presented by Yang [72]. The buckling and postbuckling behavior of laminated composite shells reinforced with SMA wires has been numerically investigated by Lee and Lee [73], who performed FE analyses using a commercial code. The activation of eccentric SMA wires is able to increase the value of the critical buckling load. Psarras et al. [74] and Parthenios et al. [75], in a series of two papers, evaluated residual thermal stresses of composites incorporating SMA wires; in particular, they adopted the technique of laser Raman spectroscopy, quantifying the generated compressive loads during SMA activation. They conducted the tests at different activation temperatures, observing that for higher wire volume fractions, the high compressive recovery stresses generated during electrical resistive heating of the wires can lead to failure of the composite coupons. Sun et al. [76] studied the response of a polymericematrix composite plate with embedded SMA wires subjected to uniform lateral pressure. The analysis, which was conducted adopting the one-dimensional SMA Brinson model implemented in an FE code, investigated the influence of boundary conditions, ply orientations, prestrains of SMA wires, and the effect of thermoviscoelasticity of the polymeric matrix on the response of the smart composite plate. Dano and Hyer [77] presented a model and designed suitable experiments to investigate, within the context of structural morphing, the possibility of controlling the snap-through of unsymmetric composite laminates containing SMA wires. The change in configuration and the snap-through of the laminate is controlled by heating the SMA wires. The authors also noted the satisfactory repeatability of the experimental results and the ability of the model to predict the experimental measurements. Sensitivity FE analyses have been performed by Zak et al. [78] to evaluate the influence of various geometrical parameters, material properties, and boundary conditions on the dynamic performance of multilayered composite plates with embedded SMA wires. The research demonstrates that the dynamics of SMA composites can be successfully controlled by the optimal selection of geometrical parameters and material properties. Tsoi et al. [79] developed experimental tests to address the effectiveness of SMA wires embedded into composite laminates against damage caused by a low-velocity impact. The authors remarked that the presence of SMA wires improves the overall response of the laminate, mainly when it is subjected

SMA constitutive modeling and analysis of plates and composite laminates

to higher-energy impacts. The influence of the materials and processing conditions on the actuation properties of adaptive hybrid SMA composites was tested by Choi and Salvia [80], who considered four sets of asymmetrical composite systems based on a glass epoxy laminate. Zhang et al. [81] fabricated epoxy resin composites filled with NiTi alloy short fibers and particles and performed experimental tests to derive their mechanical properties. Moreover, a laminated plate model with SMA fillers was proposed to predict mechanical behavior; experimental and numerical results were compared and are showed the satisfactory ability of the model to simulate the composite’s response. The doctoral thesis written by Daghia [82] presented a simple model for simulating the behavior of composite laminates with embedded SMA wires. Experimental tests were also performed considering different possible choices of materials and manufacturing processes, and demonstrated the satisfactory accuracy of the theoretical model in predicting the experimental behavior. Using an FE code, Kuo et al. [83] investigated the buckling problem of a rectangular composite laminate containing SMA fibers. They verified that concentrating the SMA fibers in the center of the rectangular laminate, the buckling load can be significantly increased. Ganilova and Cartmell [84] developed a composite laminate with a periodic arrangement of SMA wires able to perform active control of the vibrations of the plate. The active property tuning (APT) method and active strain energy tuning (ASET) method are considered. The authors noted that the ASET generally gives a higher performance than the APT. Shiau et al. [85] performed FE analyses to investigate the free vibration behavior of buckled cross-ply and angle-ply laminates reinforced with SMA fibers. The effect of prestrain and the volume fraction of the SMA fibers is evaluated, showing that postbuckling deflection and the natural frequencies of the laminate can be significantly modified, activating SMA fibers. Cho and Rhee [86] developed nonlinear FE analyses of hybrid laminate composite shells reinforced with SMA wires subjected to structural and thermal loading. Numerical results are compared with experimental ones. The authors discussed the influence of the volume fraction of SMA, temperature and the ply angles of the wires on the overall behavior of the laminate. The nonlinear free vibration of thermally postbuckled laminated composite spherical shells reinforced with SMA wires has been investigated in Panda and Singh [87], who performed FE analysis for different geometries and lamination schemes. The postbuckling behavior of SMA hybrid composite-laminated beams subjected to uniform heating is investigated in Asadi et al. [88], who developed an (approximated) analytical solution of the nonlinear problem. The influence of the layup, SMA volume fraction, SMA prestrain, boundary condition, and thickness of the layer in which SMA fibers are embedded has been presented. The layer-wise theory has been adopted to study rectangular SMA composite laminates subjected to impact loadings in Shariyat and Niknami [89]. The postbuckling behaviors of doubly curved three-phase composite shells in a hygrothermal environment have been investigated in Karimiasla et al. [90]

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by employing a multiple-scale perturbation method. The third-order shear deformation theory within the von-Karman nonlinear shell theory has been considered. Different SMA volume fractions, rises in temperature, laminations, and curvature ratios have been investigated. Micromechanical studies have also been developed to derive the overall behavior of composite SMA, examining the microstructure and material properties of the constituents. In fact, micromechanical and homogenization analyses enable the overall nonlinear behavior of the composite SMA to be determined, giving fundamental information that can be used to optimize the material design related to a specific application. Among others, Boyd et al. [91] studied the interaction between the embedded SMA and the matrix to design composites. Taya [92] proposed an analytical model based on Eshelby’s formulation for predicting the response of smart composites, including SMA fiber composites. A micromechanical approach for deriving the overall behavior of SMA composites was proposed by Cherkaui et al. [93], who considered an SMA inclusion embedded in a ductile matrix, adopting the self-consistent homogenization technique. The authors remarked on the importance of the micromechanical approach, which can be used to tailor smart composites. Kawai [94] studied the effects of the inelastic deformation of the matrix on the overall behavior of a unidirectional TiNi SMA fiber composite and on the local PE response of embedded SMA fibers under isothermal loading and unloading conditions. The average behavior of the SMA composite is evaluated with the micromechanical method of cells. Lu and Weng [95] proposed a two-level micromechanical analysis to study the influence of the shape and volume concentration of SMA inclusion on the overall behavior of SMA composites. The first level corresponds to the smaller SMA level at which phase transformations occur. The second level is the larger scale, consisting of SMA inclusions in a polymeric matrix. Briggs and Ponte Casta~ neda [96] studied the effective behavior of active composites obtained by embedding aligned SMA fibers in a linear elastic matrix, using a homogenization technique based on variational estimates. A micromechanical investigation into predicting the yield stress of SMA fiberemetal composite has been conducted by Lee and Taya [97], adopting Eshelby’s equivalent inclusion method with MorieTanaka’s mean field theory to compute residual stresses and strains in the matrix and fibers. Predicted yield stresses were compared with experimental evidence. Gilat and Aboudi [98] developed a microemacro approach to study the response of plates made of composite SMA subjected to thermal loading. They adopted the Lagoudas SMA model within the method of cells to derive the overall constitutive behavior of the composite. Marfia and Sacco [99] presented a micromechanical study of composites with embedded SMA fibers; they developed analytical and numerical homogenization procedures to determine the overall response of the composite. Marfia [100] proposed a microemacro approach to analyze the behavior of structural elements reinforced with

SMA constitutive modeling and analysis of plates and composite laminates

composite SMA, based on the implementation of suitable nonlinear analytical and numerical homogenization techniques able to derive the overall constitutive behavior of the composite at the Gauss point level of a 3D FE. Marfia and Sacco [101] developed a full microemacro (i.e., multiscale) approach to studying the response of smart laminates, obtained as a stacking sequence of fiberreinforced composite laminae and composite SMA layers. The analysis is performed implementing a mixed interpolation of tensorial component laminate FE, based on the first-order shear deformation laminate theory. Numerical analyses concerning the response of different laminated plates have been presented. A micromechanical analysis has been performed in Tang and Felicelli [102] to reproduce the effective timedependent PE responses of composites consisting of a thermos-viscoelastic polymer matrix with SMA reinforcement. The chapter is organized as follows: After introducing the list of symbols, the SMA model is presented in the framework of finite and small deformation; then, a higherorder SMA plate model in a finite strain context is presented, and a laminate plate model is discussed considering the von Karman theory. Finally, numerical applications are presented for possible simple SMA devices.

9.2 List of symbols Fe Deformation gradient elastic part Ft Deformation gradient transformation part F Deformation gradient U Reference body domain T Absolute temperature E GreeneLagrange strain tensor Ee GreeneLagrange strain tensor elastic part Et GreeneLagrange strain tensor transformation part I Second-order identity tensor Je Helmoltz elastic free energy density Jt Helmoltz transformation free energy identity J Helmoltz free energy density I1 First invariant of elastic GreeneLagrange strain tensor I2 Second invariant of elastic GreeneLagrange strain tensor l; m Lame constants b SMA material parameter relating critical stress to temperature h SMA material parameter describing uniaxial stress-transformation strain curve slope εL SMA material parameter indicating maximum uniaxial transformation strain S Second PiolaeKirchhoff stress tensor Se Second PiolaeKirchhoff elastic stress tensor h Entropy density c M Mandel stress tensor g Thermodynamic reaction conjugate to transformation strain saturation constraint f ðTÞ Transformation function in deviatoric stress space

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R Radius of limit domain z_ Transformation rate parameter Y Relative stress tensor H Unit relative stress tensor εe Elastic part of Green strain tensor εt Transformation part of Green strain tensor ℂ Isotropic elastic fourth-order tensor bt Tensile SMA material parameter bc Compressive SMA material parameter τ Back-stress tensor fτ ðτÞ Limit function in deviatoric stress space J2 Second invariant of stress deviator J3 Third invariant of stress deviator st Tensile stress parameter for limit function sc Compressive stress parameter for limit function ðX1 ; X2 Þ In-plane plate reference Cartesian coordinates Z Transverse reference Cartesian coordinate M Plate midplane U Plate domain vðX1 ; X2 ; ZÞ Plate displacement field ðu; 4; jÞ Plate displacement field components wrt transverse coordinate H gradient  0Plate1displacement  H ; H ; H2 Plate displacement gradient components wrt transverse coordinate Pðv; dvÞ Total potential energy for plate MðkÞ k-th order stress moment for plate ne Number of elastic layers for SMA laminate ns Number of SMA composite layers for SMA laminate ε Strain tensor for SMA laminate εM In-plane strain tensor for SMA laminate εo Membrane part of in-plane strain tensor for SMA laminate εc Bending part of in-plane strain tensor for SMA laminate εg Out-of-plane strain tensor for SMA laminate s Stress tensor for SMA laminate p Inelastic strain tensor for SMA laminate sM In-plane part of stress tensor for SMA laminate pM In-plane part of inelastic stress tensor for SMA laminate sg Out-of-plane part of stress tensor for SMA laminate pg Out-of-plane part of inelastic stress tensor for SMA laminate M ℂ In-plane constitutive tensor g C Out-of-plane constitutive tensors ðO; x1 ; x2 ; x3 Þ Reference coordinate frame at microscale level for SMA laminate ℂM Matrix material elasticity tensor aM Thermal expansion coefficient ℂW SMA inclusion elasticity tensor p SMA inclusion inelastic strain d k SMA inclusion prestrain εt Representative volume element average transformation strain ε Representative volume element average strain

SMA constitutive modeling and analysis of plates and composite laminates

s Representative volume element average stress V Representative volume element measure ℂ Overall elasticity tensor ðN; M; QÞ Stress resultants for SMA laminate ðA; B; DÞ Membrane, coupling, and bending constitutive tensors for stress resultants of SMA laminate G Shear constitutive tensor for stress resultant of SMA laminate

9.3 Three-dimensional phenomenological constitutive model for shape memory alloys 9.3.1 Finite strain constitutive model This section presents the phenomenological thermomechanical finite strain SMA constitutive model proposed by Evangelista et al. [52]. The model is deduced introducing a free energy function depending on internal variables able to describe the state of the phase transformation and represent the history dependence of SMA behavior. The model is based on the assumption of the local multiplicative split of the deformation gradient into an elastic part and a phase transformation part, denoted as Fe and Ft , respectively [103]. Accordingly: FðX; tÞ ¼ Fe ðX; tÞFt ðX; tÞ

(9.1)

at a typical material point position X˛U. In particular, the proposed model assumes temperature T and GreeneLagrange strain tensor E as the control or observable variables and the transformation GreeneLagrange strain tensor as the internal variable, Et. Here, 1 Et ¼ ðCt  IÞ 2

(9.2)

in which I is the second-order identity tensor and Ct is the transformation right CauchyeGreen tensor, defined as: Ct ¼ FTt Ft

(9.3)

Tensor function Et describes the strain associated with the phase transformation occurring in the SMA and accounts for the history of the material. The elastic GreeneLagrange tensor is introduced as: 1 Ee ¼ ðCe  IÞ 2

(9.4)

in which Ce ¼ FTe Fe is the elastic right CauchyeGreen tensor, which turns out to be: 1 Ce ¼ FT t CFt

(9.5)

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Once the state variables have been defined, the existence of a thermodynamic potential from which the state laws can be derived is postulated. Thereby, a free energy function J for the SMA material is considered, assumed to be isotropic with respect to tensors E and Et, and introduced through the convex potential as: JðE; Et ; T Þ ¼ Je ðEe Þ þ Jt ðEt ; T Þ

(9.6)

Elastic strain energy Je due to the isotropy assumption is written in terms of invariants I1 ¼ trEe and I2 ¼ 12 trE2e I12 of tensor Ee : Je ¼

 1 ðl þ 2mÞI12 þ 4mI2 2

(9.7)

in which l and m are the Lam
e constants. The elastic strain energy is zero when the material is not elastically deformed (i.e., when Ee ¼ 0, in which 0 is the secondorder null tensor). Energy connected to the phase transformation Jt is an extension of the finite strain of the model proposed by Souza et al. [25], and is given by: 1 Jt ðEt ; T Þ ¼ bhT  Mf ikEt k þ hkEt k2 þ IεL ðEt Þ 2

(9.8)

in which b is a material parameter related to the dependence of critical stress on the temperature, h$i is a Macaulay bracket that denotes the positive part of the argument, Mf is the finishing temperature of the austenite-martensite phase transformation, T is the temperature, k,k represents the Euclidean norm of the argument, and h is a material parameter defining the slope of the linear stressetransformation strain relation in the uniaxial case. The symbol IεL ðEt Þ denotes the indicator function introduced to satisfy the constraint on the transformation GreeneLagrange strain tensor norm:  0 if kEt k  εL (9.9) IεL ðEt Þ ¼ þN if kEt k > εL in which εL is a strain-like material parameter linked to the maximum transformation strain reached at the end of the phase transformation during a uniaxial strain test. GreeneLagrange transformation strain tensor Et describes the strain associated with phase transformation, in particular with the conversion from austenite or twinned martensite to single-variant martensite. The norm of Et should be bounded between zero for the case of a material without single-variant martensite and εL for the case in which the material is fully transformed in single-variant martensite. The specific form (Eqn. 9.8) adopted for energy Jt can describe PE behavior as well as the SME.

SMA constitutive modeling and analysis of plates and composite laminates

The state laws of the model are derived by differentiating free energy function J with respect to the observable and internal variables: S ¼ F1 t

vJe T T F ¼ F1 t Se Ft vEe t

(9.10)

vJ vT

(9.11)

h¼ 

in which h is the entropy. T is introduced as the thermodynamic force: c T  b T ¼ F1 t M Ft

(9.12)

and is the state variable that drives the phase transformation processes associated with transformation GreeneLagrange tensor Et. The tensors c M and b in Eqn. (6.12) are the symmetric Mandel stress and the back stress tensors given by: vJe c M ¼ ð2Ee  IÞ vEe

b ¼

vJt vEt

(9.13)

The proposed formulation satisfies the second principle of thermodynamics in the reference configuration, and hence, the ClausiuseDuhem inequality [1,2]. Stresses S and b depend only on E and Et. After some calculations, the expression of the second PiolaeKirchhoff stress tensor becomes: S ¼ lI1 ð2Et þ IÞ1 þ mð2Et þ IÞ1 ð2Et þ IÞð2Et þ IÞ1

(9.14)

Similarly, the back stress is given as: b ¼ ðbhT  Mf i þ hkEt k þ gÞ

Et kEt k

with g belonging to the subdifferential of the indicator function IεL ðEt Þ: 8 if kEt k < εL > < 0 vIεL ðEt Þ ¼ Rþ if kEt k ¼ εL > : B if kEt k > εL

(9.15)

(9.16)

Note that parameter g automatically controls the end of the austenite-martensite phase transformation. A limit function f describing the onset of inelastic deformations is introduced as:    f ðTÞ ¼ dev Ft TFTt   R (9.17)

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Shape Memory Alloy Engineering

in which R is the radius of the elastic domain, stress tensor Ft TFTt represents the push forward of tensor T to the current configuration, and devð,Þ ¼ ,  1 3 ðtr ,ÞI is the deviatoric part of the resulting spatial tensor operator. Using the associative normality rule, the evolution of the transformation strain tensor can be expressed as: =

306

vf E_ t ¼ z_ vT

(9.18)

which directly gives the evolution of the internal variable in the reference configuration. The derivative of inelastic potential f with respect to variable T defines the direction of the transformation strain rate, whereas z_ is a transformation multiplier rate. The model is completed by the classical KuhneTucker conditions: z_  0

f 0

z_ f ¼ 0

(9.19)

The fulfillment of these conditions enables an evaluation of transition multiplier z and, as a consequence, the evolution of the internal variable. The evolutive equation can be represented in terms of control variable E and internal variable Et tensors that are defined in the reference or undeformed configuration. After some calculations, the results are: devðYÞ _ Et Þ½2Et þ I with H ¼ E_ t ¼ zHðE; kdevðYÞk

(9.20)

in which Y ¼ 2ðES Et aÞ þ S  b is the relative stress tensor [52]. The material parameters of the proposed model are characterized by a clear physical meaning and are determined by simple experimental tests, as described in Evangelista et al. [51]. The model is able to reproduce both phase transformation austenite / single-variant martensite and single-variant martensite / austenite. Moreover, the macroscopic SMA model is based on the use of the evolution of transformation strain Et, which is zero when the material is in the austenite or twinned martensite phase; on the contrary, it is different from zero when single-variant martensite is present. Thus, the model does not distinguish between austenite and twinned martensite. Evolutive law (9.20) allows the variation of kEt k from 0 to εL inducing austenite / single-variant martensite transformation, but also from εL to 0 when the reverse transformation austenite / single-variant martensite occurs. A discrete numerical scheme is adopted to perform time integration of the material model. In particular, an exponential integration procedure [104,105], based on the algorithm proposed by Reese and Christ [50], is developed.

SMA constitutive modeling and analysis of plates and composite laminates

9.3.2 Small strain constitutive model The strain quantities defined in the previous section are nonlinear expressions in terms of body motion; they lead to nonlinear governing equations. These governing equations can be linearized to enable a reformulation of the SMA constitutive model proposed within the framework of finite strain in the small deformation regime, linearizing the strain and stress measures preserving the nonlinear material response. In this section, the phenomenological thermomechanical SMA model, proposed and discussed in Souza et al. [25], Auricchio and Petrini [37], and Evangelista et al. [51], is presented. The model is deduced from the finite strain model presented in the previous section [106]. Within the framework of infinitesimal deformation, GreeneLagrange strain tensor E is substituted by infinitesimal strain tensor ε, transformation strain Et, and infinitesimal transformation strain εt , so that elastic strain Ee is substituted by εe ¼ ε  εt . Convex potential J, given by Eqn. (9.6), is written as function of ε, εt , and temperature T . The elastic strain energy (Eqn. 9.7) takes the form: 1 Je ¼ ℂðε  εt Þ: ðε  εt Þ 2

(9.21)

in which ℂ is the isotropic elastic fourth-order tensor. The symbol “:” denotes the double tensor contraction. The energy connected to the phase transformation (Eqn. 9.8) is written in the form: 1 Jt ðεt ; T Þ ¼ bðuÞhT  Mf ikεt k þ hkεt k2 þ IεL ðεt Þ 2

(9.22)

bðuÞ ¼ hðuÞbt þ ð1  hðuÞÞbc

(9.23)

where in which u is the trace of the strain tensor ε, the function hðuÞ ¼ 0 if u  0 and hðuÞ ¼ 1 if u > 0, and parameters bt and bc are the tensile and compressive parameters governing the SMA response. The chosen form for the transformation energy (Eqn. 9.22) enables the reproduction of tensionecompression asymmetry in phase transformation shown by experimental evidence, especially for NiTi and CuZnAl alloys. The Cauchy stress and relative or transformation stress tensors assume the expressions: s¼

τ¼ 

vJ ¼ ℂðε  εt Þ vε

vJ vkεt k ¼ s  ½bhT  Mf i þ hkεt k þ vIεL ðεt Þ vεt vεt

(9.24) (9.25)

307

308

Shape Memory Alloy Engineering

SA MS

Vt,f

Et

AM Mf

MA

Ms A s

Af

Vc Vc,f

Et

AS

Vt

Ec Ec

Figure 9.1 Phase transition zones in tension and compression.

Eqs. (9.24) and (9.25) state that stress s and transformation stress τ are quantities thermodynamically conjugate to strain-like variables ε and εt , respectively. Yield function fτ , which is able to catch the asymmetric behavior of SMA during tensionecompression tests, is introduced: pffiffiffiffiffiffi J3 fτ ðτÞ ¼ 2J2 þ m  R (9.26) J2 in which J2 and J3 are the second and the third invariants of τdev , which is the deviatoric part of relative stress tensor τ. In Eqn. (9.26), parameter R is the radius of the elastic domain in the deviatoric space whereas m is a material parameter, with m  0:46 to guarantee yield surface convexity. Parameters R and m are defined as: rffiffiffiffiffi rffiffiffi 2 sc st 27 sc  st R¼2 m ¼ (9.27) 3 st þ sc 2 st þ sc in which st and sc are the uniaxial critical transformation stresses in tension and compression, respectively, as schematically illustrated in Fig. 9.1. Evolution of transformation strain is governed by the associative normality rule: vfτ ðτÞ ε_ t ¼ z_ vτ

(9.28)

in which z_ is the transformation multiplier rate. From the analysis of the flow rule from Eqn. (9.28), considering Eqn. (9.26), it can be pointed out that transformation strain ε_ t is deviatoric; thus, the condition of

SMA constitutive modeling and analysis of plates and composite laminates

incompressibility during inelastic flow is recovered. This is not a generally accepted assumption; an interesting discussion on this point can be found in an article by Qidwai and Lagoudas [30], who noted that small transformation strain volume change can be found experimentally. On the other hand, it is not simple to evaluate and satisfactory to describe this small volume change occurring during the transformation phase, and thus, with the aim of recovering a simpler model, it is often considered negligible [25,107,108]. The model is completed introducing classical complementarity KuhneTucker conditions: z_  0

fτ  0 z_ fτ  0

(9.29)

that reduce the rate problem to a constrained optimization problem. The full 3D, characterized by 3D SMA constitutive and evolutive laws, can be simplified in a 3D model governed by 1D evolutionary equations. This assumption arises from the observation that in many cases, SMA devices are subjected to a load condition that is prevalent along a particular direction. This is the case of SMA wires embedded in layered composite plates, for which the highest value of transformation strain occurs just along those directions. If the 3D model is specialized to the case of uniaxial phase transition in the SMA material, it can be assumed that the phase transformation tensor takes the form: 2 3 1 0 0 6 7 1 6 7 0 7 60  ε t ¼ l6 (9.30) 7 2 6 7 4 5 1 0 0  2 in which only one scalar variable, l, is used to describe phase transition occurring in the material. Because of the particular relationship between the components of the transformation strain tensor, 1 εt22 ¼ εt33 ¼  εt11 2

εt12 ¼ εt13 ¼ εt23 ¼ 0

the components of the transformation strain rate tensor satisfy the constraints:  ε_ t11 þ 2_εt22 ¼ 0 ε_ t11 þ 2_εt33 ¼ 0

(9.31)

(9.32)

309

310

Shape Memory Alloy Engineering

Considering evolutionary Eqn. (9.28) for internal variable εt , relations (9.32) are satisfied when: s22 ¼ s33 ¼ s12 ¼ s13 ¼ s23 ¼ 0

(9.33)

into the derivative of activation function fτ ; thus, the phase transition results ruled by the value of stress component s11 ¼ sl . Therefore, derivatives of the yield function with respect to thermodynamic force are: pffiffiffi pffiffiffi pffiffiffi vfτ 6 3sl  9 3hl þ 2mz bhT  Mf i 2 ¼  9z vτ11 sgnðlÞz (9.34) vfτ vfτ 1 vfτ ¼ ¼  2 vτ11 vτ22 vτ33 with: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi

2 pffiffiffi pffiffiffi pffiffiffi h z¼ 2 sl  3 ðbhT  Mf iÞsgnðlÞ þ3lh  2sl þbhT  Mf i 6 sgnðlÞ þ 3 l 2 (9.35) in which sgnðhÞ ¼ 1 if h < 0 and sgnðhÞ ¼ 1 if h > 0. The components of thermodynamic force τ become:

τ11 ¼ sl  e b τ22 ¼ τ33 ¼ with

1e b τ12 ¼ τ13 ¼ τ23 ¼ 0 2

#rffiffiffi rffiffiffi 3 2 e sgnðlÞ b ¼ bhT  Mf i þ h jlj þ g 2 3 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi and g ¼ 0 if 0  3=2jlj < εL , g  0 if 3=2jlj ¼ εL . Components of the deviatoric part of thermodynamic force tensor τ are:

(9.36)

"

(9.37)

2 1 dev 1 1 e dev dev dev dev e dev τdev 11 ¼ sl  b τ22 ¼ τ33 ¼  τ11 ¼  sl þ b τ12 ¼ τ 13 ¼ τ23 ¼ 0 3

2

3

2

(9.38) so that the yield function depends only on the first component of the deviatoric thermodynamic force tensor as: rffiffiffi 3 dev m dev fτ ðτÞ ¼ (9.39)

τ þ τ11  R 2 11 3

SMA constitutive modeling and analysis of plates and composite laminates

Finally, activation of the phase transformation (i.e., the evolution of the inelastic strain (Eqns. 9.28 and 9.29)) is ruled by the associative law: vfτ l_ ¼ z_ vτ11

_τ0 fτ  0 zf

z_  0

(9.40)

9.4 Plate and laminate models for shape memory alloy applications An initially flat plate is defined by its reference domain, U:    t t U ¼ ðX1 ; X2 ; ZÞ ˛ R3 = Z ˛  ; ; ðX1 ; X2 Þ ˛ M 3 R2 2 2

(9.41)

where t is the constant thickness of the plate, Z is the transverse coordinate orthogonal to the undeformed midplane M, identified by Z ¼ 0, and ðX1 ; X2 Þ are midplane (or simply in-plane) reference coordinates.

9.4.1 Finite deformation plate model The following displacement field is assumed for the plate: vðX1 ; X2 ; ZÞ ¼ uðX1 ; X2 Þ þ Z4ðX1 ; X2 Þ þ Z 2 jðX1 ; X2 Þ

(9.42)

T

in which u ¼ fu1 ; u2 ; u3 g represents the displacement vector of midplane points, whereas 4 [ f41 ; 42 ; 43 gT and j [ fj1 ; j2 ; j3 gT are functions of the in-plane coordinates. Thus, the three components of the displacement field are assumed to be parabolic through the thickness of the plate; this model represents an extension of the kinematical model proposed by Arciniega and Reddy [109]. Introducing gradient operator VX ¼ fvX1 vX2 vZ g, the displacement gradient is evaluated as: H ¼ v5VX ¼ H0 þ ZH1 þ Z 2 H2

(9.43)

with H0 ¼ u5Vs þ 45p3 H1 ¼ 45Vs þ 2j5p3 H ¼ j5V 2

(9.44)

s

in which Vs ¼ fvX1 ; vX2 ; 0gT is the in-plane gradient operator, p3 ¼ f0; 0; 1gT is the unit vector orthogonal to the midplane, and the symbol 5 denotes the dyadic product. The deformation gradient is written in the form: F ¼ x5Vx ¼ ðX þ vÞ5Vx ¼ F0 þ ZF1 þ Z 2 F2

(9.45)

311

312

Shape Memory Alloy Engineering

in which x ¼ X þ v is the actual position of point X and F0 ¼ I þ H0 , F1 ¼ H1 , and F2 ¼ H2 . Accordingly, the right CauchyeGreen strain tensor takes the form: h   T  T i T C ¼ FT F ¼ F0 F0 þ Z F0 F1 þ F1 F0 þ h   T  T i T Z 2 F0 F2 þ F1 F1 þ F2 F0 þ (9.46) h  i h  i   T T T Z 3 F1 F2 þ F2 F1 þ Z 4 F2 F2 whereas the GreeneLagrange strain tensor is: 1 E ¼ ðC  IÞ 2

(9.47)

Variation in the GreeneLagrange strain tensor assumes the form: dE ¼

2 X 2 X

Z ðiþjÞ dEij

(9.48)

i¼0 j¼0

with dEij ¼

1 h i T j  j T i i F dH þ dH F 2

(9.49)

The weak form of static equilibrium, formulated in the reference configuration, is recovered applying the virtual displacement principle [110]. It states that the body is in equilibrium if and only if the virtual work of the external forces is equal to the virtual work of the internal forces for any virtual compatible displacement: Z Z Z Pðv; dvÞ ¼ S: dEdV  b$dvdV  p$dvdV ¼ 0 cdv compatible (9.50) U

U

vt U

in which S is the second PiolaeKirchhoff stress tensor, b is the body force vector per unit volume, and p is the surface traction vector acting on vt U, which is a part of whole boundary vU of the body. The symbol “,” denotes the scalar product. Substituting Eqn. (9.48), into the expression for the internal virtual work (9.50), we obtain: 1 0 Z Z Z X 2 X 2 2 X 2 X S: dEdV ¼ S: @ Z ðiþjÞ dEij AdV ¼ MðiþjÞ : dEij dA U

U

i¼0 j¼0

M i¼0 j¼0

(9.51)

SMA constitutive modeling and analysis of plates and composite laminates

where the stress resultant tensors are defined as: Z t=2 ðkÞ Z k SdV with k ¼ 0; .; 4 M ¼ t=2

(9.52)

in which Mð0Þ represents the normal and shear forces, Mð1Þ denotes the bending and torsional moments, and Mð2Þ , Mð3Þ , and Mð4Þ represent higher-order moments. This formulation is suitably approximated by a displacement-based FE model that is adopted for the simulation and analysis of SMA devices in Section 9.5.1. Details on this procedure can be found in Artioli et al. [110].

9.4.2 Small deformation laminate model A composite laminated plate domain, U, given by Eqn. (9.41), is considered. The laminate is made up of n layers identified by through-the-thickness coordinates Zk and Zkþ1 , in which Z1 ¼ t=2 and Znþ1 ¼ t=2. Two different types of layers are considered: orthotropic linear elastic laminae and SMA composite laminae, characterized by the special thermomechanical response described in the previous section. The typical SMA layer of the laminate contains fibers oriented in a random direction in the X1  X2 plane. Finally, the laminate is constituted of ne elastic layers and ns SMA composite layers. To analyze the structural behavior of the laminate, the first-order shear deformation theory (FSDT) is considered [110]. It is based on the following displacement representation form: v1 ðX1 ; X2 ; ZÞ ¼ u1 ðX1 ; X2 Þ þ Z41 ðX1 ; X2 Þ v2 ðX1 ; X2 ; ZÞ ¼ u2 ðX1 ; X2 Þ þ Z42 ðX1 ; X2 Þ

(9.53)

v3 ðX1 ; X2 ; ZÞ ¼ u3 ðX1 ; X2 Þ obtained by simplifying Eqn. (9.42), in which u1 and u2 are the in-plane displacements, u3 is the transverse deflection, and 41 and 42 are rotations of the typical fiber orthogonal to the midplane. The constitutive behavior of layers containing SMA fibers is deduced developing a suitable homogenization procedure; in such a way, the layer with SMA fibers is considered in the laminate equation as a homogeneous lamina. As a consequence, two different scales are introduced: the scale of the structure (i.e., of the laminate) and the scale of the SMA inclusions (i.e., the fibers). Thus, an analysis of the laminate response is derived implementing a multiscale procedure. The strain and stress quantities at the macroscale (the scale of the laminate) are denoted in the following discussion with an overbar to distinguished them from those quantities at the microscale (the scale of the SMA inclusions).

313

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Shape Memory Alloy Engineering

Thus, strain ε at the typical point of the laminate is split in in-plane εM ¼ εo þ Zεc and out-of-plane εg contributions, setting: 2 3 1 2 1 u þ þ u þ u u u u 1;1 3;1 1;2 2;1 3;1 3;2 7 6 2 2 7 6 εo ¼ 6 7 5 41  1 2 u2;2 þ u3;2 u1;2 þ u2;1 þ u3;1 u3;2 2 2 (9.54) 3 2  1 ( ) 41;1 41;2 þ 42;1 7 6 u3;1 þ 41 2 1 7 6 εc ¼ 6 7 εg ¼ 5 41 2 u3;2 þ 4  2 42;2 41;2 þ 42;1 2 in which εo is the in-plane strain, εc is the strain resulting from to the curvature, and εg is the contribution caused by out-of-plane shear strain. In-plane strain accounts for nonlinear terms according to the von Karman theory [111], which considers small strain and moderate rotations effects. Analogously, stress tensor s and inelastic strain tensor p at the typical point of the laminate are split in the in-plane, sM and pM , and out-of-plane, sg and pg , contributions. M g Introducing the in-plane and out-of-plane constitutive tensors, ℂ and C , characterized by dimensions 2  2  2  2 and 2  2, respectively, the stressestrain law for the i-th inelastic layer is written as:  MðiÞ  M gðiÞ sM ¼ ℂ ε  pM sg ¼ C ðεg  pg Þ (9.55)

9.4.2.1 Shape memory alloy composite material Once the mechanical response of the components are assigned, the overall nonlinear behavior of the composite is derived by performing micromechanical analysis and using a suitable homogenization procedure. To this end, a representative volume element (RVE) of a composite material is considered characterized by a volume, V, and with M and W as the matrix and the inclusion, with volume V M and V W , respectively. A reference coordinate system ðO; x1 ; x2 ; x3 Þ at the microscale level is introduced with the x1 -axis oriented along the fiber direction, as illustrated in Fig. 9.2. The mechanical response of the matrix material M is assumed to be governed by a linear elastic constitutive law. In particular, the stressestrain relationship results in:   s ¼ ℂM ε  aM ðT  T0 ÞI (9.56)

SMA constitutive modeling and analysis of plates and composite laminates

[1 M

[3

k W :

[2

Figure 9.2 Shape memory alloy (SMA) composite obtained by embedding SMA fibers into a matrix.

where ℂM is the elasticity matrix and the term aM ðT T0 ÞI accounts for the inelastic strain induced by a possible thermal deformation in the matrix, in which aM is the expansion coefficient, T0 is the reference temperature, and T is the actual temperature. The stressestrain relationship of the inclusion W is assumed to be of the form: s ¼ ℂW ðε  pÞ

(9.57)

where ℂW is the elasticity matrix of the SMA inclusion and p is the inelastic strain given by: p ¼ aW ðT  T0 ÞI þ dk þ εt

(9.58)

that is, obtained as the sum of thermal deformation aW ðT T0 ÞI, of the possible prestrain d k of the SMA wire in the direction of fiber axis k and of transformation strain εt . In preparing a composite SMA, the prestrain is obtained by prescribing an extension to the SMA wires before curing the composite. Once the composite is cured, the extensional load is removed from the wires, inducing a self-equilibrated stress state in the composite SMA, characterized by a tensile stress in the SMA and a compressive stress in the matrix. To reproduce this stress state in the model, inelastic strain d k, with d < 0 (i.e., a contraction), is enforced on the SMA wires embedded in the composite. 9.4.2.2 Homogenization for the nonlinear composite Once the stressestrain laws of the RVE constituents are defined according to Eqns. (9.56)e(9.58), the homogenization problem is formulated prescribing the average strain ε in the RVE, the thermal expansion strain in the matrix and in the inclusion, and the prestrain in the inclusion. Thus, the problem consists of determining strain field εðxÞ, transformation strain field εt ðxÞ, stress field sðxÞ, and average stress s in the RVE, such that Z Z 1 1 s¼ sðxÞdV ε ¼ εðxÞdV (9.59) V V V

V

315

316

Shape Memory Alloy Engineering

related by the equation: s ¼ ℂðε  pÞ

(9.60)

in which ℂ is the overall elasticity matrix, satisfying the condition s33 ¼ 0, and p the overall inelastic strain [99e101]. The problem can be solved under the assumption of periodicity conditions [112,113] adopting transformation field analysis to account for the nonlinearities [114,115]. ℂ is initially evaluated in the material axes coordinate reference system ðx1 ; x2 ; x3 Þ; then, it is rotated in the ðX1 ; X2 ; ZÞ coordinate system. Introducing the stress resultant as: Z t=2 Z t=2 Z t=2 o c o c N¼ ðs þ Z s ÞdZ M ¼ Zðs þ Z s ÞdZ Q ¼ sg dZ t=2

t=2

t=2

(9.61) the laminate constitutive law is written in the form: ns Z Zjþ1 X  MðiÞ  M o c N ¼ Aε þ Bε þ ℂ ε  pM dZ Zj

j¼1

M ¼ Bεo þ Dεc þ g

Q ¼ Gε þ

ns Z X j¼1

ns Z X

Zℂ

MðiÞ  M

 ε  pM dZ

(6.62)

Zj

j¼1 Zjþ1

Zjþ1

C

gðiÞ

ðεg  pg ÞdZ

Zj

where: A¼

ne Z X j¼1

Zjþ1 Zj

ℂ

MðiÞ

dZ

B ¼

ne Z X j¼1

Zjþ1

Zℂ

Zj

MðiÞ

dZ

C ¼

ne Z X j¼1

Zjþ1

Z2ℂ

MðiÞ

dZ

Zj

A, B, and D are the 2  2  2  2 membrane, coupling, and bending constitutive tensors obtained considering only the elastic layers of the laminate, whereas ne Z Zjþ1 X gðiÞ C dZ G¼ j¼1

Zj

is the 2  2 shear constitutive tensor. A detailed description of the FSDT model and of the related governing kinematical parameters and equations can be found in Reddy [111].

SMA constitutive modeling and analysis of plates and composite laminates

A nonlinear version of the MITC4 (four-node mixed interpolation of tensorial components) plate FE, proposed by Bathe and Dvorkin [116] for homogeneous plates and extended to the laminate problems by Alfano et al. [117], is developed for the analysis of SMA composite laminate elements [101]. The FE model mentioned earlier will be subsequently adopted to simulate SMA laminate buckling and bending problems.

9.5 Numerical results 9.5.1 Finite deformation analysis of shape memory alloy plates In this section, some numerical simulations are illustrated in an attempt to analyze the capabilities of the presented FE scheme in predicting the thermomechanical structural behavior of plate and shell structures. The analysis has two main targets: first, the numerical calibration of the adopted constitutive model in representing effective 3D material behavior; and second, an assessment of the efficiency and reliability of the approach in describing the structural behavior of 3D structural devices in the large displacement regime. Computations are performed by implementing the finite deformation plate theory presented in Section 9.4.1 in conjunction with the SMA constitutive model of Section 9.3.1, in the research-oriented FE code FEAP [118,119]. Numerical simulations are carried out referring to SMA material properties reported in Table 9.1 [52,104]. 9.5.1.1 Thermomechanical ring actuator device A semicircular, SMA, thick, round arch, clamped along one edge and free along the opposite, is analyzed here. The geometry is given in Fig. 9.3. The arch internal and external radii are Ri ¼ 25 mm and Re ¼ 35 mm, respectively; a unit width is considered. The arch is subjected to a uniformly distributed cyclic load applied on the free edge, whose result is horizontal shear force F oriented toward the arch center. For the purpose of the current analysis, a uniform 2  20-element mesh (Fig. 9.5) of 5  5 node (biquadratic) elements is used to discretize the structure, and three different isothermal cyclic loadingeunloading histories are considered at increasing room temperature. With reference to Fig. 9.1, the analysis is carried out in stress control for three different values of temperature: T > Af , Ms < T < As , and T < Mf , kept constant during the numerical simulation. Moreover, for the case T < As , the SME Table 9.1 Shape memory alloy material properties adopted for numerical simulations.

E ¼ 53,000 MPa Mf ¼ 223 K b ¼ 2.1 MPa K1

h ¼ 1000 MPa Ms ¼ 239 K

n ¼ 0.36 As ¼ 248 K

εL ¼ 0.04 Af ¼ 260 K

317

Shape Memory Alloy Engineering

Figure 9.3 Thermomechanical ring actuator device: geometry, boundary, and loading conditions.

is exploited, properly applying a variation in temperature after mechanical loading has been set to zero. In Fig. 9.4, the value of applied force F is plotted versus the horizontal displacement component of point A on the free edge, u1A. In the first case, the arch is subjected to an increasing value of the force until a maximum equal to 45 N is reached, maintaining the temperature constant at 285 K; then, the load is removed, allowing the element to recover its initial shape. Fig. 9.5 shows the undeformed and deformed configurations of the arch structure. Therefore, if the material is loaded at a temperature above Af, 45 40 35 30 25

F [N]

318

20 15 10 5 T = 223 K 0 -5

0

T = 240 K T = 285 K 0.5

1

1.5

2

2.5

u1A [mm]

Figure 9.4 Thermomechanical ring actuator device: force versus point A horizontal displacement plot at increasing temperatures.

SMA constitutive modeling and analysis of plates and composite laminates

Figure 9.5 Thermomechanical ring actuator device: undeformed initial configuration and maximum deflection configuration at t ¼ 1.0 and T ¼ 285.

macroscopic nonlinear large deformations occur and are totally recovered during unloading, because the product phase (i.e., the single-variant martensite) is not stable at this value of temperature. The typical hysteretic loop in terms of stress and strain (pseudoelasticity), is clearly observed. In the second case (i.e., Ms < T < As), the arch is subjected to the same type of loadingeunloading cycle with a maximum value of the force equal to 30 N, maintaining the temperature constant at 240 K, whereas in the third case (i.e., T < Mf ), the maximum value of force F is 20 N and the temperature is fixed at 223 K. At the removal of force F (t ¼ 1) in the second and third cases, the arch has not recovered its undeformed shape. The initial shape is obtained by heating the structure until the temperature reaches the maximum value of T ¼ 300 K. Finally, the arch is cooled to the initial temperature to recover the initial conditions. In this case, the ability of the model to reproduce the SME is tested. The simple test shows the ability of the method to reproduce the PE and the SMEs for an SMA device of technical relevance. Archshaped, thermally controlled actuators are being prototyped to assess their ability to control the activation of airplane wing flap morphing [119]. In this advanced research context, it is mandatory to perform an accurate FE simulation of the SMA device to assess its performance as an actuator and optimize its design. Therefore, the current numerical procedure can be a powerful numerical tool for such an analysis owing to its reliability in an accurate estimation of the arch state of stress and strain under mixed-control thermomechanical loading conditions in the large displacement setting.

319

320

Shape Memory Alloy Engineering

9.5.1.2 Thermomechanical spring actuator Attention is now focused on a single coil of a typical SMA actuator that exploits the SME: the example chosen is widely used in engineering applications, such as in the automotive and electromechanical fields. The problem is similar to the one originally proposed in Auricchio and Petrini [37] and Evangelista et al. [52], in which a helical spring made of SMA material clamped along one end is first elongated by applying a force per unit of area Q applied at the free cross-section end and subsequently subjected to a cyclic thermal load. The system under consideration is represented in Fig. 9.6; it is a circular, initially flat, single coil with internal and external radii Ri ¼ 3.0 mm and Re ¼ 4.0 mm, respectively; the reference cross-section is rectangular and measures 1.0  0.5 mm2. The structure represents a single coil of an SMA thermomechanical spring subjected to a typical loading history exploiting the SME of the material. The assumed mixed stressetemperature loading history is summarized in Table 9.2; as usual, changes in temperature and applied traction are assumed to be piecewise linear functions of t. The initial boundary value problem is solved with a 20 7  7-node element mesh; the loading history is discretized in 100 equal time steps over the first stack and with 2  200 equal time steps over the following two stacks. The plan and perspective views of the coil in undeformed configuration (t ¼ 0) and in maximum elongation configuration (t ¼ 1) are represented in Fig. 9.7. In Figs. 9.8 and 9.9, respectively,

Figure 9.6 Thermomechanical coil: geometry, boundary, and loading conditions.

Table 9.2 Thermomechanical coil loading history for applied traction and temperature.

t [e] Q [N/mm2] T [K]

0 0 285

1 8.25 285

2 8.25 800

3 8.25 285

SMA constitutive modeling and analysis of plates and composite laminates

Figure 9.7 Thermomechanical coil: plan view and perspective view of adopted discretization in undeformed configuration and at maximum elongation configuration at time t ¼ 1 and T ¼ 285 K. 0

1

u3A [mm]

2

3

4

5

6

7 0

0.5

1

1.5

2

2.5

3

t [s]

Figure 9.8 Thermomechanical coil: loaded end axial displacement versus time.

the plots of loaded edge transverse displacement u3A versus time and of applied load versus displacement are reported. This case confirms the ability of the proposed method to describe the thermomechanical behavior of complex 3D structural devices undergoing a 3D state of stress efficiently, because it is clear that the spring is subjected to membrane bending and shear stresses globally.

321

Shape Memory Alloy Engineering

9

8 7

6

2

q [N/mm ]

322

5

4 3 2 1 0 0

1

2

3

4

u

[mm]

3A

5

6

7

Figure 9.9 Thermomechanical coil: applied load versus displacement.

9.5.2 Small deformation analysis of shape memory alloy laminates In this section, some structural examples concerning the buckling analysis of a plate in cylindrical bending and the flexural response of symmetric and unsymmetric laminates are studied applying the plate FE model for SMA laminates of Section 9.4.2. The SMA material properties are set on the basis of experimental data presented by Sittner et al. [18]. In all applications, the following material properties are considered for the NieTi alloy fibers: T0 ¼ 245 K

ðÞ

E W ¼ 53000 MPa

nW ¼ 0:36

εL

¼ 0:04

h ¼ 1000 MPa

bt ¼ 2:1 MPaK1

bc ¼ 1:8 MPaK1 sc ¼ 72 MPa st ¼ 56 MPa Mf ¼ 223 K

where E W and nW are the Young’s modulus and Poisson ratio, respectively. On the basis of this set of material parameters, the following SMA properties can be derived: As ¼ 230 K Af ¼ 250 K in which As and Af are the starting and finishing temperatures of the martensite-austenite phase transformation in the case of stress-free material. The thermal expansion coefficient is taken to be aW ¼ 0:000001 K1 .

SMA constitutive modeling and analysis of plates and composite laminates

A composite characterized by a low-stiffness polymeric matrix is considered. In particular, the material properties of the elastic matrix and of the SMA are: EM ¼ 3600 MPanM ¼ 0:305aM ¼ 0:0  C1

9.5.2.1 Buckling analysis The problem of a laminate buckling, simply supported along two opposite edges and subjected to axial distributed forces N, as schematically illustrated in Fig. 9.10, is analyzed in the case of cylindrical bending. Two different laminations are considered: the first is unsymmetric and is characterized by an elastic isotropic layer (core) and one layer made of composite SMA (CSMA) at the bottom, so that the stacking sequence of the layer is core/CSMA; the second one is a symmetric laminate, with the stacking sequence CSMA/core/CSMA. The material properties of the elastic layers are: Ec ¼ 5000 MPa nc ¼ 0:2 The geometrical properties of the plate are: L ¼ 300 mm b ¼ 20 mm hs ¼ 1 mm hc ¼ 3 mm in which L is the span of the plate in cylindrical bending, b is the width, and hs and hc are the thicknesses of the composite SMA layer and of the elastic core, respectively. Computations are performed adopting the self-consistent homogenization technique. The plate geometry, loading condition, and considered lamination schemes are reported in Fig. 9.10. The following loading history is prescribed thus: • an initial inelastic prestrain d ¼ h 0:04, in which 0  h  1, is assigned to the SMA fibers and is kept constant during the whole analysis; the prestrain d induces

unsymmetric lamination

initial imperfection C

y

z

300 mm

N

Elastic material

1 mm 20 mm

x symmetric lamination

SMA composite

3 mm

1 mm

SMA composite

3 mm

Elastic material

1 mm

SMA composite 20 mm

Figure 9.10 Geometry and loading scheme for buckling analysis of a plate in cylindrical bending. SMA, shape memory alloy.

323

324

Shape Memory Alloy Engineering

a self-equilibrated stress state in the composite SMA, which leads to the austenitemartensite phase transformation in the SMA wire; • an initial imperfection in terms of transversal displacement, wCi , of the midpoint of plate C, is imposed, applying a small transversal load; • an increasing compressive axial displacement, u1, at point B is prescribed; • when the axial displacement reaches an assigned value, the composite SMA layer placed at the bottom of the laminate is heated until temperature T ¼ Tmax, inducing the martensite-austenite phase transformation in the SMA fibers; • keeping temperature T ¼ Tmax constant, compressive axial displacement u1 at point B is still increased. The loading history is summarized in Table 9.3, in which the variation in the prescribed quantities is assumed to be linear in each time interval. Regarding the prescription of the imperfection, Table 9.3 shows that the amount of imperfection is constant from time t ¼ 2 to 5 s; indeed, during that time interval, the transversal displacement of point C is not constant, because of the buckling effect, but transversal load inducing at time t ¼ 2 s at displacement ui3;C is taken to be constant. 9.5.2.1.1 Unsymmetrical laminate

The case of unsymmetric lamination is investigated, performing different analyses. Initially, no-prestrain and a constant temperature are considered (i.e., h ¼ 0 and T ¼ T0 ¼ Tmax), whereas u11;B ¼ 2.5 mm. The analysis is performed until time t ¼ 3 s. In Fig. 9.11, axial force N is plotted versus transversal displacement wC of point C. When no initial imperfection is prescribed (i.e., wCi ¼ 0), the plate buckles, developing positive deflections (i.e., toward the side of the SMA composite layer) because of unsymmetric lamination. To obtain buckling in the opposite side, initial imperfection ui3;C ¼ 0.8 mm is prescribed. As expected, the critical load is the same when the plate buckles in the two directions; moreover, it is close to the value computed using the classical Euler formula, which gives about 74 N. Then, several analyses are performed to investigate the effects of the phase transformations of the SMA fibers on the buckling behavior of the plate; the phase transformations can be governed by the values of the prestrain and the maximum temperature. Table 9.3 Buckling analysis of a laminate: analysis inputs at various time steps.

Time [s] Prestrain Imperfection [mm] Axial displacement [mm] Temperature [K]

0 0 0 0 T0

1 d 0 0 T0

2 d ui3;C 0 T0

3 d ui3;C u11;B T0

4 d ui3;C u21;B Tmax

5 d ui3;C u31;B Tmax

SMA constitutive modeling and analysis of plates and composite laminates

80 initial imperfection 70 no initial imperfection

60

N [N]

50

40

30

20

10

0 –8

–6

–4

–2

0

2

4

6

8

10

u3C [mm]

Figure 9.11 Buckling analysis of unsymmetric laminate: axial force versus transversal displacement of point C with or without assuming an initial imperfection.

In Figs. 9.12 and 9.13, axial force N versus transversal displacement wC of point C and axial force versus axial displacement u1;B of point B are represented for the cases reported in Table 9.4. The results of case 1 were reported in Fig. 9.11, but the analysis results reported in Fig. 9.12 are carried out for greater values of axial displacement. After reaching a maximum axial load, a softening phase follows as austenite-martensite transformation occurs, so that a tangent stiffness of the plate is obtained, characterized by lower values of the elastic parameters. When a prestrain, h ¼ 0.5, is prescribed and an analysis is performed, keeping the temperature constant (i.e., case 2), a lower critical load is recovered with respect to case 1. In fact, as a partial phase transformation occurs during the prestrain in case 2, the elements of the tangent stiffness of the plate have lower values than the ones characterizing the full austenite or full martensite situations; as a consequence, a decrease in the critical load is induced. This reduction in tangent stiffness, and thus in the critical load, does not appear in case 3, for which the phase transformation is completed during the prestrain, so that the tangent stiffness of the plate results is equal to the initial one. For this reason, the critical load for cases 1 and 3 are almost the same, but case 3 does not have a softening branch.

325

326

Shape Memory Alloy Engineering

Figure 9.12 Buckling analysis of the unsymmetric laminate: axial force versus transversal displacement for different values of the prestrain and maximum temperature.

Figure 9.13 Buckling analysis of the unsymmetric laminate, axial force versus axial displacement for different values of the prestrain and maximum temperature.

SMA constitutive modeling and analysis of plates and composite laminates

Table 9.4 Loading history for the buckling problem. Case

h

ui3;C [mm]

Tmax [K]

u11;B [mm]

u21;B [mm]

u31;B [mm]

1 2 3 4 5 6

0 0.5 1 0.75 1 1

0 0.7 0.5 0.5 0.5 0.5

223 223 223 400 400 445

3.5 3.5 3.5 0.025 0.025 0.025

1.625 1.625 1.625

3.5 3.5 3.5

To enhance the buckling behavior of the laminate, a prestrain inducing the austenitemartensite phase transformation and a variation in temperature able to activate reversephase transformation can be exploited. Cases 4, 5, and 6 demonstrate the effectiveness of using SMA fibers to control the buckling behavior. Comparing the results obtained for cases 4 and 5, which are characterized by the same temperature history, the maximum critical load is obtained in case 4 for a lower value of the initial fiber prestrain. To increase the critical load reached in case 5 for h ¼ 1, a higher value of temperature variation is prescribed in case 6. To understand the complex mechanical behavior of the smart structure better, the ratio fm ¼ kεt k=εL , representing the equivalent martensite volume fraction, is introduced. In Figs. 9.14 and 9.15, the equivalent martensite volume fraction fm versus time and the normal stress in the fiber. sSMA , versus temperature are represented for cases 4e6.

Figure 9.14 Buckling analysis of the unsymmetric laminate: equivalent martensite volume fraction versus time for different values of the prestrain and maximum temperature.

327

Shape Memory Alloy Engineering

600

case 6: ω =1.0, Tmax= 445 K case 5: ω =1.0, Tmax= 400 K

500

400

σ SMA [MPa]

328

300

200 case 4: ω =0.75, Tmax= 400 K 100

0

250

300

350

400

450

T [K]

Figure 9.15 Buckling analysis of the unsymmetric laminate: normal stress in the shape memory alloy fibers versus temperature for different values of the prestrain and maximum temperature.

Both fm and sSMA are computed at the Gauss point closest to the midspan of the laminate. During the prestrain, the austenite-martensite phase transformation occurs in the SMA fibers, reaching an equivalent volume fraction: fm ¼ 0:8 in case 4 and fm ¼ 1 in cases 5 and 6. When the fiber is heated (i.e., 3 s  t  4 s), a partial martensite-austenite phase transformation occurs in the fibers, inducing an increase in the critical load in all three cases; in particular, the highest reduction in the equivalent martensite volume fraction is obtained for case 4 during heating, inducing a maximum increase in critical load. In Fig. 9.15, the slope of the curve for SMA fiber stress versus temperature during the martensite-austenite phase transition appears the same in all three cases. As mentioned, in case 4, the martensite-austenite phase transformation occurs more significantly than in the other cases; in fact, normal stress in the SMA fiber is lower when heating begins and the phase transition starts when the temperature reaches about 285 K. On the contrary, for cases 5 and 6, because normal stress in the SMA fibers is higher, the phase transition starts only for a temperature about 375 K. In Figs. 9.12 and 9.13 the mechanical response in case 4e6 is characterized by a softening branch after the peak occurrence at a constant temperature equal to T ¼ Tmax. The axial load tends toward the critical load of cases 1 and 3. In fact, if the analyses are carried out increasing axial displacement at a constant temperature, T ¼ Tmax, the equivalent fraction of martensite tends to 1, so the axial load tends to the critical one computed in cases 1 and 3.

SMA constitutive modeling and analysis of plates and composite laminates

9.5.2.1.2 Symmetric laminate

Next, the case of symmetric lamination is studied. In addition, for symmetric lamination, several analyses are performed to investigate the effects of prestrain and temperature variation on the buckling behavior of the plate. The loading history illustrated in Table 9.5 is prescribed. In particular, it is assumed that: • the initial prestrain is assigned to the SMA fibers of the CSMA laminae inducing austenite-martensite phase transformation; • when axial displacement reaches an assigned value, only the CSMA layer placed at the bottom of the laminate is heated until temperature T ¼ Tmax, inducing a martensiteaustenite phase transformation in the SMA fibers. In Fig. 9.16, axial force N versus transversal displacement u3,C of point C is represented for the cases reported in Table 9.5. Table 9.5 Loading histories for symmetric laminate. Case

u

ui3;C [mm]

Tmax [K]

u11;B [mm]

u21;B [mm]

u31;B [mm]

1 2 3

0 0.75 1

0 0.097 0.097

223 320 350

1.4 0.05 0.05

0.65 0.65

1.4 1.4

450

400

350

80% prestrain - Tmax= 350 K

300

N[N]

250

200

80% prestrain - Tmax= 320 K

150 no prestrain

100

50

0 0

2

4

6

8

10

u3C [mm]

Figure 9.16 Buckling analysis of the symmetric laminate: axial force versus transversal displacement for different values of the prestrain and maximum temperature.

329

330

Shape Memory Alloy Engineering

Results are obtained qualitatively similar to those computed performing analyses for unsymmetric lamination. For this reason, no further computations are developed. • In case 1 the critical load is close to the value computed using the classical Euler formula, which gives about 190 N. After the peak value of axial load N, the mechanical response of the laminate is characterized by a softening branch due to the phase transformation occurring in the SMA fibers. • In cases 2 and 3, an increase in the critical load is obtained by inducing the austenitemartensite phase transformation during the fiber prestrain and reverse phase transition by heating the SMA fibers of the SMA composite layer placed at the bottom. • The increase in the critical load is more evident for case 3 than for case 2; in fact, case 3 is characterized by a higher value of maximum temperature than case 2, which induces higher values of the equivalent martensite volume fraction, and hence, greater martensite-austenite phase transition in the fibers. • In cases 2 and 3, the mechanical response after the peak is characterized by a softening branch, with the axial load tending to the critical load computed in case 1. 9.5.2.2 Analysis of a composite laminate An application of a layered composite SMA plate is presented to investigate the SME. In particular, a square cantilever plate is considered that is clamped along one edge and free on the other three edges, subjected to different loading and thermal conditions. The aim of the application is to demonstrate the possibility of significantly governing transversal displacement of the cantilever plate performing temperature cycles in the composite SMA layers. In all of the following computations, the load is applied at a constant temperature and is kept constant during temperature changes in the composite SMA layers. In the following numerical applications, a laminate is considered that is characterized by an elastic isotropic layer (core) and two layers made of composite SMA, one on the top and one on the bottom. The geometrical properties of the plate are: L ¼ 300 mm hs ¼ 5 mm hc ¼ 10 mm

(9.63)

No prestrain in the fibers is considered. Computations are performed adopting the self-consistent homogenization technique. 9.5.2.2.1 Laminate subjected to applied moment

First, a laminate characterized by a sequence 0/core/0 is analyzed. The plate is initially subjected to an increasing value of the bending moment, as represented in Fig. 9.17, until a maximum value is reached; then, a temperature cycle is prescribed according to the loading and temperature history reported in Table 9.6.

SMA constitutive modeling and analysis of plates and composite laminates

C m

A B

Figure 9.17 Geometry and loading scheme of a square laminate subjected to applied moment. Table 9.6 Clamped laminate subjected to bending moment:, summary of analysis for inputs at various time steps.

t [s] m [Nmm/mm] T [K]

0 0 223

1 1333 223

2 1333 400

3 1333 223

In Fig. 9.18, the transversal displacement of the middle point, u3,A, and of the corner, u3,B, of the free edge is plotted versus time. In the same figure, the results are resulted for the full 3D SMA model and the 3D model with 1D evolution (3D-1D). From the figure, it can be verified that the 3D model leads to greater values of the displacements with 18

16

14

3

u [mm]

12

10

8

6 u3A 3D model

4

u3A 3D-1D model u3B 3D model

2

u3A 3D-1D model 0 0

0.5

1

1.5

2

2.5

3

t [s]

Figure 9.18 Laminate subjected to applied moment: transversal displacement of free edge of cantilever plate versus time.

331

Shape Memory Alloy Engineering

respect to the 3D-1D model. The maximum transversal displacement is reached at time t ¼ 1 s, when the maximum value of the bending moment is achieved at temperature T ¼ 223 K and the austenite-martensite phase transformation has occurred in the SMA fibers. From t ¼ 1 to 2 s, the temperature is increased and the martensiteaustenite phase transformation occurs in the SMA fibers; as a consequence, the transversal displacement is reduced. From time t ¼ 2 to 3 s, the temperature decreases to the initial value and the plate goes back to the deformed shape reached at t ¼ 1 s. The good agreement between results obtained for the full 3D and the 3D-1D models can be remarked. In Fig. 9.19, the bending moment is plotted versus the transversal displacement of the free edge of the plate for the 3D solution. The first branch is nonlinear because of the nonlinear behavior of the SMA wires owing to the phase transformation. Then, changing the temperature, the beam swings between two configurations: the beam is able to recover part of the transversal displacement during heating, and it returns to the deformed shape during cooling. The SMA actuation is significant as transversal displacement during the heating process is recovered, at more than 30% of the maximum value.

1400

1200

1000

m [Nmm/mm]

332

800

600

400

200

u3A u3B

0 0

2

4

6

8

10

12

14

16

u3 [mm]

Figure 9.19 Laminate subjected to applied moment: bending moment versus transversal displacement of free edge of cantilever plate.

SMA constitutive modeling and analysis of plates and composite laminates

q

C

A B Figure 9.20 Geometry and loading scheme of a square laminate subjected to transversal load on the free edges. Table 9.7 Clamped laminate subjected to transversal load on the free edges:, summary of analysis inputs at various time steps.

t [s] q [N/mm] T [K]

0 0 223

1 18 223

2 18 400

3 18 223

9.5.2.2.2 Laminate subjected to transversal load on the free edges

Then, the plate is subjected to distributed transversal load on the three free edges of the laminate, as illustrated in Fig. 9.20, according to the loading and thermal history reported in Table 9.7. Two analyses are performed considering the following lamination sequences 0/core/ 0, and 90/core/90. Results are obtained adopting only the 3D SMA model. In Fig. 9.21 the transversal displacement of point A versus time is plotted for the cases 0/core/0 and 90/core/90 with dash-dot and solid line, respectively. It can be pointed out that in the case 0/core/0 the fibers more significantly influence the behavior of the laminate than in the case 90/core/90 because during the increase of the transversal loading the austenite-martensite phase transformation occurs almost completely along the fibers oriented in the direction 0 degrees. As a consequence, in the case 0/core/0, during the thermal cycles, the transversal displacement is significantly reduced when the temperature is increased, inducing the martensite-austenite phase transformation of the SMA fibers. The plate goes back to the deformed shape reached at t ¼ 1 s when the temperature is decreased at the initial value. In the case 90/core/90, during loading, the austenitemartensite phase transformation occurs in a very limited part of the fibers so only a reduced part of the transversal displacement is recovered during heating. In Fig. 9.22 the distributed load is plotted versus the transversal displacement for the two different analyses. It can be noted that in the case 0/core/0 the nonlinear mechanical response of the composite is strongly influenced by the phase transformations occurring in the SMA fibers while the case 90/core/90 is characterized by an almost linear response.

333

Shape Memory Alloy Engineering

120

100

60

u

3A

[mm]

80

40

20 90°/core/90° 0°/core/0° 0 0

0.5

1

1.5

2

2.5

3

t [s]

Figure 9.21 Laminate subjected to transversal load on the free edges: transversal displacement wA of point A versus time. 20 18 16 14 12

q [N/mm]

334

10 8 6 4 2

90°/core/90° 0°/core/0°

0 0

20

40

60

80

100

u3A [mm]

Figure 9.22 Laminate subjected to transversal load on the free edges: distributed load q versus transversal displacement wA of point A.

SMA constitutive modeling and analysis of plates and composite laminates

9.5.2.2.3 Laminate subjected to torsional load

Finally, the plate is subjected to a torsional load, as illustrated in Fig. 9.23, according to the loading and thermal history reported in Table 9.8. Several analyses are performed considering different orientations of the fibers: 0/core/0, 90/core/90, 45/core/45, and 45/core/45. In Fig. 9.24(a) and (b) and in Fig. 25(a) and (b), transversal displacement of the points A, B and C versus time and the distributed load versus the transversal displacement of the points A, B and C are represented, respectively, for the cases 0/core/0 with dash-dot line, 90/core/90 with solid line, 45/core/45 with dot line and 45/core/45 with dash line. It can be pointed out that the displacement of the point A is represented only for the case 45/core/45 as for the other analyses it results equal to zero. Fig. 9.24(b) shows that crossply laminates, i.e. 90/core/90 and 0/core/0 laminations, are characterized by higher values of the initial stiffness with respect to the other laminations. In the case 45/core/45, the mechanical response of the composite SMA is more significantly influenced by the SME in the SMA fibers than in the other cases. In fact, in Fig. 9.24(a) and (b) and Fig. 9.25(a) and (b), in 0/core/0, 90/core/90, and 45/core/45, transversal displacements of points B and C are partially recovered during heating because of the martensite-austenite phase transformation occurring in the SMA fibers, but for 45/core/45, this recovery is more significant than in the other cases. For the analysis of 45/core/45, displacements of points A and C recover during heating, inducing an increase in the displacement of point B owing to stiffness in the y-direction of the plate. The displacement of point B is not influenced by phase transformations occurring in the fibers because they are inclined at an angle of 45 degrees versus point C in both laminae.

-q C A

q B

Figure 9.23 Geometry and loading scheme of a square laminate subjected to torsional load.

Table 9.8 Clamped laminate subjected to torsional load, summary of analysis for inputs at various time steps.

t [s] q [N/mm] T [K]

0 0 223

1 18 223

2 18 400

3 18 223

335

Shape Memory Alloy Engineering

(a)

50 45 40 35

[mm]

30

u

3B

25 20 15 10

90°/core/90°

5

0°/core/0° -45°/core/45° 45°/core/45°

0 0

0.5

1

1.5

2

2.5

3

t [s]

(b)

10

0

-10 -45°/core/45° u3C 45°/core/45° u3A

-20

45°/core/45° u3C 90°/core/90° u3C

3

u [mm]

336

-30

0°/core/0° u3C

-40

-50

-60

0

0.5

1

1.5

2

2.5

3

t [s]

Figure 9.24 Laminate subjected to torsional load; (a) transversal displacement w of point B versus time, (b) transversal displacement w of point A and C versus time.

SMA constitutive modeling and analysis of plates and composite laminates

(a)

60

50

q [N/mm]

40

30

20

45°/core/45°

10

90°/core/90° –45°/core/45° 0°/core/0°

0 0

10

20

30

40

50

u3B [mm]

(b) 60

50

q [N/mm]

40

30

20 45°/core/45° u3A 90°/core/90° u3C –45°/core/45° u3C

10

45°/core/45° u3C 0°/core/0° u3C 0 –60

–50

–40

–30

–20

–10

0

10

u3 [mm]

Figure 9.25 Laminate subjected to torsional load: (a) q versus transversal displacement of point B; (b) q versus transversal displacement of points A and C.

337

338

Shape Memory Alloy Engineering

Computations are performed also considering the 3D-1D simplified model for laminates 0/core/0 and 45/core/45. Because no significant differences have been noted with respect to the results obtained adopting the full 3D model, the plots are not included in the figures to leave them more legible. Transversal displacement of point B versus time is represented, respectively, for cases 0/core/0 and 45/core/45. In 45/core/45, the mechanical response of the SMA composite is more significantly influenced by the SME in the SMA fibers than for the other cases. In fact, during heating, transversal displacement of point B is partially recovered because of the martensite-austenite phase transformation occurring in the SMA fibers, but for 45/core/45, this recovery is more significant than for the other cases. The use of the simplified SMA model does not induce significant differences in the solution with respect to those recovered adopting the full 3D SMA model.

9.6 Conclusions Constitutive models for SMA and effective numerical procedures for analyzing plate and laminate composite materials embedding SMA components are an active research field in engineering and material science. The target of this research is the development of efficient and reliable numerical tools for the analysis and design of innovative complex mechanical and biomedical devices. This chapter focused on some developments in that research branch. In particular, 3D phenomenological constitutive models for SMA able to reproduce PE and SME in both the small strain and finite strain frameworks were presented, together with innovative plate/shell FE models for SMA homogeneous and fiber-reinforced SMA composite laminates, adopting a microemacro approach and homogenization methods. A large set of numerical tests confirming the robustness and computational efficiency of the proposed methodology has been presented. Selected tests are also provided to show the capability of analyzing engineering devices undergoing a complex thermomechanical loading history, with special emphasis on the buckling phenomenon of laminated plate elements made of SMA fibers embedded into an elastic matrix. The numerical results show that it is possible to enhance the buckling behavior increasing the critical load by applying a prestrain in the SMA fibers and by heating the SMA fibers. Furthermore, some numerical applications are developed to show the possibility of controlling the transversal deformation of laminate plates subjected to bending or torsional loading prescribing thermal cycles in the SMA fibers. The results point to the importance of fiber orientation and the beneficial effects of composite SMA on the mechanical response of laminate plates in many innovative applications. Research on SMA is an active field with many modeling problems under investigation. Current research directions are the development of innovative constitutive models,

SMA constitutive modeling and analysis of plates and composite laminates

the devising of new state variable sets able to reproduce internal material behavior thorough agreement with experimental tests, and the development of efficient nonlinear numerical solvers of the material state update problem at the integration point level.

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SMA constitutive modeling and analysis of plates and composite laminates

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CHAPTER 10

Advanced constitutive modeling Giulia Scalet, Ferdinando Auricchio Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universit
a degli Studi di Pavia, Pavia, Italy

10.1 Introduction Constitutive modeling of shape memory alloys (SMAs) is an active field of research continuously supporting the design process in several areas of application from aerospace and automotive to medicine and robotics. On the one hand, knowledge of the mechanical and functional response of SMA-based devices through computer-based simulations can provide helpful information to avoid unexpected failures as well as optimize performance. On the other hand, such knowledge is essential to reduce the number of timeconsuming and costly experimental tests usually required by design-build-testebased methods. Over the years, several constitutive modeling approaches have been proposed to describe the mechanical and functional response of SMAs. They are generally classified as (1) microscopic thermodynamics models, (2) micromechanics-based macroscopic models, and (3) macroscopic phenomenological models. The reader is referred to other contributions [1e6] for a comprehensive review. To give an acceptable description of SMA behavior, constitutive models need to describe at least pseudoelasticity (PE) and the shape memory effect (SME), which include the evolution, orientation, and reorientation of martensitic variants. In this chapter, PE and SME are defined using the term “primary effects.” However, such constitutive models may be limited in predicting SMA behavior under complex conditions such as cyclic thermomechanical loading, severe prestrain or stress causing plastic yielding, multiaxial nonproportional loading, or thermal cycling under low or zero stress. Under such conditions, the complex evolution of various material phases may cause the occurrence of other effects in SMA behavior in addition to the primary ones, such as phase-dependent elastic properties, stress- and temperature-dependent phase transformations, texture-induced anisotropy, different kinetics between forward and reverse transformation, two-way SME, transformation-induced plasticity, yield and plastic deformation, tensionecompression asymmetry, minor hysteresis loops, return point memory effects, and thermomechanical coupling effects [7]. This chapter defines all of these additional effects using the term “secondary effects.” It is clear from this discussion that advanced constitutive models, which include the description of both primary and secondary effects, should be proposed to provide an

Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00010-8

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accurate and comprehensive prediction of SMA behavior. Although primary effects are well-known and widely described by current models, secondary effects require further efforts from the modeling community, because they are less described for several reasons. In particular, the additional description of secondary effects leads to an increase in the number of nonlinear equations and involved parameters in formulating models. Because of the complexity in solving the resulting equations as well as in using a set of physical parameters to be easily calibrated, secondary effects are rarely considered jointly and simultaneously. Motivated by such a framework, this chapter aims to review and discuss modeling contributions to the description of secondary effects, available from the literature. In particular, attention will be paid to macroscopic models derived based on continuum thermodynamics and the introduction of internal variables that are able to describe the global response of SMAs, and thus which are suitable for analyzing engineering applications based on polycrystalline SMAs. The chapter is organized as follows. First, the framework of continuum thermodynamics with internal variables within small strain will be briefly presented. Then, several secondary effects will be described, together with the related modeling contributions. For each effect, numerical examples from the literature will be discussed. Finally, conclusions and a summary will be provided.

10.2 List of symbols ε Total strain tensor T Absolute temperature a Set of internal variables j Free-energy function je Elastic free-energy function jch Chemical free-energy function jint Interaction or mixing free-energy function jc Free-energy function due to variables’ constraints h Entropy s Stress tensor X Thermodynamic force tensor F Yield function q Heat flux vector xA Austenite volume fraction xM Total martensite volume fraction xTM Twinned martensite volume fraction xDM Detwinned martensite volume fraction xDM; t Tension-induced detwinned martensite volume fraction xDM; c Compression-induced detwinned martensite volume fraction xMr Total martensite volume fraction induced by reverse transformation xMf Total martensite volume fraction induced by forward transformation xR R-phase volume fraction

Advanced constitutive modeling

εe Elastic strain tensor εth Thermal strain tensor εtr Transformation strain tensor εF Forward transformation tensor εR Reverse transformation tensor εre Reorientation tensor εpt Parent phase transformation tensor εt Transformation strain tensor generated during twinning εd Transformation strain tensor generated during detwinning dtr Transformation strain direction tensor εtwin Inelastic strain tensor due to the accommodation of twins between martensite variants εt Mean transformation strain tensor εtwin Mean twins accommodation strain tensor εL Positive parameter defining maximum transformation strain obtainable through alignment of martensite variants E eq Effective Young’s modulus EA Austenite Young’s modulus EM Total martensite Young’s modulus Geq Effective shear modulus K eq Effective bulk modulus p Geometric parameter of the Mori-Tanaka homogenization scheme s von Mises equivalent stress εtrip Transformation-induced plastic strain tensor xM,rev Reversible martensite volume fraction xM,irr Residual irreversible martensite volume fraction εp Plastic strain tensor Nf Number of cycles to failure W d Dissipated energy associated with the stabilized hysteresis cycle d Damage variable

10.3 Three-dimensional macroscopic modeling with internal variables Within the framework of continuum thermodynamics with internal variables under small strain, the state of an SMA material medium at a given point and time instant can be defined by a set of observable and internal variables [8]. Observable variables are the total strain, ε, and the absolute temperature, T . Internal variables, grouped in set a, describe the physics of an SMA material internal structure from a macroscopic point of view and are chosen depending on the type of effect that needs to be described. Set a will be specified for each discussed effect in the following subsections. Once both observable and internal variables are chosen, a free-energy function, j (typically the Helmholtz free-energy), is defined as the sum of different contributions: j ¼ jðε; T ; aÞ ¼ jel þ jch þ jint þ jc

(10.1)

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where jel is the elastic energy, jch is the chemical energy related to entropic changes associated with phase transformation, jint is the interaction (or mixing) energy derived from micromechanical or metallurgical considerations, and jc represents any energy term related to the presence of constraints on internal variables. Based on the defined free-energy function, a standard thermodynamical procedure [8,9] is applied to derive the necessary constitutive equations, here expressed in the general form: h ¼ hðε; T ; aÞ

(10.2)

s ¼ sðε; T ; aÞ

(10.3)

X ¼ Xðε; T ; aÞ

(10.4)

where h, s, and X are, respectively, the entropy, stress, and thermodynamic force associated with the internal variables. If thermal conduction is included in the model formulation, a constitutive equation (the Fourier law) relating the heat flux, q, and the temperature gradient, VT , is also derived in the form: q ¼ qðε; T ; a; VT Þ

(10.5)

The derived equations must satisfy the second law of thermodynamics, usually expressed by the Clausius-Duhem inequality [8]. Then, model formulation is completed by the definition of the dissipation mechanism [9]. The most common approach introduces a yield function, F, and a set of evolution equations for the internal variables in the form:
 a_ ¼ f ε; T ; a; ε_ ; T_ (10.6) that are completed by classical Kuhn-Tucker conditions. In the following sections, we will review the contributions belonging to the modeling class described earlier and describing several secondary effects typical of SMA behavior. In particular, we will focus on martensitic phase transformation, phase-dependent elastic properties, smooth thermomechanical response, stress-dependent transformation strain magnitude, asymmetric forwardereverse transformation and tensionecompression response, anisotropy, plasticity, minor loops, damage and fatigue, as well as thermomechanical coupling. As will be shown, the description of these effects will be included following different approaches based, for example, on the introduction of appropriate internal variables or on the modification of the yield function.

10.3.1 Martensitic phase transformation SMA primary and secondary effects are the consequence of martensitic phase transformations between the parent phase, called austenite, and the different variants of the product phase, called martensite [10]. The latter can appear in two states: (1) the twinned

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(or self-accommodated) martensite, and (2) the detwinned (or oriented) martensite. On the one hand, twinned martensite is formed by cooling under no applied stress, because martensite variants accommodate themselves through twinning, with no observable macroscopic inelastic strains. On the other hand, detwinned martensite is produced under an applied stress, because martensite variants reorient (or detwin) into a single variant, with observable macroscopic inelastic strains. Therefore, detwinned martensite can be formed either from phase transformation of austenite under an applied stress or from reorientation of twinned martensite. As an example, the schematic one-dimensional stress-temperature phase diagram reported in Fig. 10.1 shows different transformation regions among austenite (A), twinned martensite (TM), and detwinned martensite (DM) material phases and some possible loading paths (dashed lines). To describe martensitic phase transformations, constitutive model generally introduce a set of internal variables. The most common approach consists of using both scalar and tensorial variables to treat the growth and orientation or reorientation of variants as two different physical processes. Therefore, adopting standard literature terminology [10], the use of both scalar and tensorial variables distinguishes between phase transformation and martensite reorientation, as described in the following discussion. 10.3.1.1 Phase transformation To describe each possible phase transformation experienced by the SMA and the related phenomena (e.g., phase-dependent elastic properties, as detailed next), constitutive models generally consider polycrystalline SMA as a mixture of two or more phases. Accordingly, set a is assumed to be composed of two or more scalar internal variables and the free-energy function, j, reported in Eq. (10.1), is generally formulated as a

Figure 10.1 Schematic one-dimensional stress-temperature phase diagram [11,12].

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weighted sum of the elastic and chemical energies of the single phases, plus the remaining energy terms. A standard choice in basic constitutive models [13e18] is to adopt two scalar internal variables, xA and xM , to represent the volume fraction of austenite and total martensite in an SMA volume element, respectively, such that: 0  xA  1

(10.7)

0  xM  1

(10.8)

xA þ xM ¼ 1

(10.9)

Model formulation is then restricted to just one independent scalar variable, xM , by letting xA ¼ 1  xM and 0  xM  1. Although most models use such a scalar variable to describe the percentage of the SMA material transformed to martensite, they are unable to distinguish between the martensite fraction associated with macroscopic transformation strain (i.e., detwinned martensite) and the martensite fraction not associated with macroscopic transformation strain (i.e., twinned martensite). To include such a distinction, several authors [11,12,19e22] proposed an additive split of the total martensite volume fraction, xM , into twinned and detwinned variants, respectively, xTM and xDM , subjected to the following constraints: xM ¼ xTM þ xDM

(10.10)

0  xM  1

(10.11)

0  xTM  1

(10.12)

0  xDM  1

(10.13)

To include tensionecompression asymmetry (see Subsection 10.3.6), some models [23e26] further split the volume fraction of detwinned martensite, xDM , into tensioninduced and compression-induced parts, respectively, xDM ; t and xDM ; c . Chatziathanasiou et al. [27] proposed decoupling reverse and forward transformations by distinguishing between the martensite volume fraction induced by reverse and forward transformation, respectively, xMr and xMf , such that: xM ¼ xMf  xMr

(10.14)

0  xM  1

(10.15)

In addition to martensitic transformations, NiTi-based SMAs may transform from austenite into monoclinic martensite either directly or via the rhombohedral R-phase. The presence of the R-phase affects the mechanical behavior of SMAs. However, only few modeling efforts have been made to include a description of the R-phase,

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and they were mostly restricted to the PE response [28e30]. Sedlak et al. [31] introduced two scalar internal variables, xM and xR , to represent the volume fraction of total martensite and R-phase in an SMA volume element, respectively, such that: 0  xM  1

(10.16)

0  xR  1

(10.17)

0  xR þ xM  1

(10.18)

Accordingly, the volume fraction of austenite is given by xA ¼ 1  xM  xR . In this way, Sedlak et al. [31] were able to capture the change in the elastic behavior and entropy during the austenite-R-phase transformation, while neglecting the transformation strain associated with the R-phase (usually reaching, at most, one-tenth the martensitic transformation strain [31]). Model formulations with different material phases as scalar internal variables decouple the different phase transformation mechanisms and thus achieve more freedom for a completely independent, flexible description of the behavior of SMA polycrystals. Such decoupling becomes important for the analysis and design of some SMA-based devices, for which one needs to know and understand the interactions between physical phenomena during nonproportional multiaxial mechanical and thermal loading. As an example, Frost et al. [32] performed an experimental and a numerical investigation on helical spring actuators. The latter were first loaded by different forces of 100, 180, 250, 400, 600, and 800 mN and then exposed to thermal cycling under constant applied force. Finite element simulations performed by Frost et al. [32] were based on the constitutive model by Sedlak et al. [31]. Figure. 10.2 shows the role of the R-phase for a case with a low applied force (i.e., 250 mN): the transformation from austenite to R-phase is described by a step-like change between 0 and 20  C in the hysteresis loop. Finally, the inclusion of two or more phases leads to a more complex mathematical formulation, because it increases the number of nonlinear and highly coupled equations as well as of equalityeinequality constraints (Eqs. 10.7e10.18), and thus requires ad hoc numerical techniques for robust and efficient model implementation into software.

10.3.1.2 Martensite reorientation Despite the importance in modeling volume fractions for describing phase transformation in SMAs, a set a composed only of scalar variables, as proposed, for example, by Fremond [33], is generally inadequate to describe reorientation of martensite in a macroscopic setting. SMA polycrystals exhibit not only phase transformation but also reorientation of martensitic variants when experiencing stress-induced martensitic transformation under multiaxial nonproportional loading. Therefore, macroscopic inelastic strain is not only the product of phase transformation; it is also affected by reorientation.

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Figure 10.2 Experimental and numerical stroke-temperature curves related to NiTi helical spring actuators. (Based on experimental and numerical data taken from the work by Frost et al. [32].)

A common modeling approach is to use one or more tensorial variables to account for the reorientation of martensitic variants, in addition to scalar internal variables that describe phase transformation. Accordingly, an additive split of the total strain in an elastic, thermal, and transformation contribution may then be adopted: ε ¼ εe þ εth þ εtr

(10.19)

where the tensor, εtr, is introduced to describe reversible martensitic phase transformations combined with martensite reorientation. Tensor εtr can be assumed to be traceless to describe volume-preserving phase transformation processes. Along this line, some authors [18,34,35] used the transformation strain itself, εtr, and the total martensite fraction, xM , as internal variables. The evolution of εtr was linked to both stress and martensite fraction through a plasticity-like flow rule. To distinguish further between forward and reverse phase transformation and reorientation, Chatziathanasiou et al. [27] split εtr into three contributions: εtr ¼ εF þ εR þ εre

(10.20)

The evolutions of the forward and reverse, and reorientation transformation tensors (respectively, εF , εR , and εre Þ were linked to both stress and martensite fractions (Eqs. 10.14 and 10.15) through plasticity-like flow rules. Panico and Brinson [10] partitioned the transformation strain, εtr, into two contributors to distinguish between parent phase transformation as well as reorientation [10]:

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εtr ¼ εpt þ εre

(10.21)

and adopted the volume fraction of detwinned and twinned martensite as scalar internal variables (Eqs. 10.10e10.13). Alternatively, Popov and Lagoudas [11] distinguished between stress-induced transformation strain (associated with A / DM transformation), εt , and the transformation strain generated during detwinning (associated with TM / DM transformation), εd , by adopting the following decomposition of transformation strain, εtr : εtr ¼ εt þ εd

(10.22)

The evolution of εt and εd was linked to the twinned and detwinned martensite fractions (Eqs. 10.10e10.13) and to the stress tensor, s, through a plasticity-like flow rule. Alternatively, some authors split the transformation strain, εtr, as: εtr ¼ xDM dtr

(10.23)

εtr ¼ xM dtr

(10.24)

or:

Here, dtr is a tensorial variable that describes the direction of the transformation strain owing to martensite detwinning and reorientation [12] or the mean transformation strain of martensite [16,17,31,36]. Chemisky et al. [37] defined the transformation strain, εtr , as the sum of an inelastic strain due to martensitic transformation, εt , and an inelastic strain due to the accommodation of twins between martensite variants, εtwin , as: εtr ¼ εt þ εtwin ¼ xDM εt þ xTM εtwin

(10.25)

where εt and εtwin denote, respectively, mean transformation strain and mean twins accommodation strain over the martensitic volume. Differently from these models, which separate the description of pure phase transformation from that of a pure reorientation mechanism, the model originally proposed by Souza et al. [38] and then generalized by Auricchio and Petrini [39] adopts only one tensorial internal variable, the transformation strain itself, εtr , to describe martensitic phase transformation. The adopted variable is then subjected to the following constraint: kεtr k  εL

(10.26)

in which εL is a positive parameter defining the maximum transformation strain attainable through alignment of the martensite variants and k$k is the Euclidean norm. Accordingly, the model originally proposed by Souza et al. [38] and then generalized by Auricchio and Petrini [39] distinguishes only between a generic parent phase (not associated with macroscopic strain) and a generic product phase (associated with macroscopic strain). Moreover, the internal variable tensorial character considers reorientation in an approximated form [39]. Owing to its simplicity, the model may

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lead to a constrained approach if additional secondary effects need to be included in the formulation, because scalar and directional features are strongly linked. Figure 10.3 compares the performances of different modeling approaches under nonproportional loading. In particular, combined tension-torsion tests at two axial prestresses of 194 and 379 MPa and at two constant temperatures of 40 and 50  C are considered. Numerical curves are taken from the results reported in the works by Auricchio et al. [12], Sedlak et al. [31], Auricchio et al. [40], and Rio et al. [41] and are compared with the experimental curves obtained in the framework of the SMA round-robin modeling initiative within the S3T EUROCORES program [7]. At the axial prestress of 379 MPa, the deformation mechanism is based on martensite reorientation, whereas at 194 MPa, the behavior is superelastic [7]. The models by Auricchio et al. [12] and Sedlak et al. [31] distinguish between phase transformation and martensite reorientation

Figure 10.3 Experimental and numerical torqueeangular displacement and axial straineangular displacement curves at two constant temperatures of 40 and 50  C and two constant axial prestresses of 194 and 379 MPa. (Experimental and numerical data are taken from the works of Sittner et al. [7], k et al. [31], and Rio et al. [41].) Auricchio et al. [12,40], Sedla

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Figure 10.3 Cont’d.

and thus capture such deformation mechanisms better than the models by Auricchio et al. [40] and Rio et al. [41].

10.3.2 Phase-dependent elastic properties To examine the variations in stiffness among austenitic, martensitic, and R-phases, constitutive models generally introduce an equivalent elastic modulus by using standard homogenization schemes, such as the Reuss, Voigt, and Mori-Tanaka modeling schemes. The Reuss model considers the SMA material to be a composite in which the different phases are disposed in series. Then, the effective Young’s modulus, E eq , of the SMA is given by the general definition:

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Np X 1 xi ¼ Eeq Ei i¼1

(10.27)

in which Np is the total number of phases and E i is the Young’s modulus of the i-th phase. Similarly, in a three-dimensional framework, the effective shear, G eq , of the SMA is expressed in the form: Np X 1 xi ¼ G eq Gi i¼1

(10.28)

in which G i is the shear modulus of the i-th phase. The effective bulk modulus, K eq , is generally assumed to be equal for all phases [12,31]. The Reuss model has been adopted in several models [12,14,16,18,22,25,31,42e47]. The Voigt model considers the SMA material to be a composite in which the different phases are disposed in parallel. Then, the effective Young’s modulus, E eq , is given by the general definition: E ¼ eq

Np X

xi Ei

(10.29)

i¼1

The Voigt model has been adopted in some models [19,24,48e50]. Both Voigt and Reuss modeling schemes neglect grain interactions that are accounted for by the Mori-Tanaka model. The latter considers the SMA material to be a matrix of one phase containing Eshelby inclusions of the other phase. Accordingly, considering austenite (A) and martensite (M) phases, the effective Young’s modulus, Eeq , is given by [14]: " # M M MA M A EM AM þ xM E ð1  x Þ þ x E ð1  x ÞE Eeq ¼ (10.30) þ 2 ð1  xM ÞE M þ xM EMA E A ð1  xM ÞEM E AM þ xM E A where: E MA ¼

EM E A þ pðE M  EA Þ

(10.31)

E AM ¼

EA þ pðE A  E M Þ

(10.32)

EM

in which p is a parameter depending on the geometry of the inclusion, whereas E A and E M are the Young’s modulus of austenite and martensite, respectively. Auricchio and Sacco [14] compared the Reuss and Mori-Tanaka schemes and found that the Reuss model was more accurate compared with experimental data. In particular,

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the Mori-Tanaka model was found to be accurate for xM < 0:2 (corresponding to the case of martensite inclusions in an austenite matrix) and for xM > 0:8 (corresponding to the case of austenite inclusions in a martensite matrix) [14]. Gong et al. [51] found good agreement between the elastic moduli computed with an extended MoriTanaka model and experimental data for a porous CuAlMn SMA containing oriented oblate spheroid pores of different specimen sizes, porosities (25e70%), and pore sizes, manufactured by the sintering-evaporation process. All of the described homogenization schemes lose accuracy in the case of highly anisotropic materials, for which statistical approaches should be adopted [5].

10.3.3 Smooth thermomechanical response Martensitic phase transformation may start and finish in a smooth and gradual manner in a stressestrain and strainetemperature diagram, because of the different crystallographic orientations of the grains, the heterogeneity of internal stresses generated by the processing and transformation histories, and/or the presence of nontransforming precipitates [18]. To capture such a smooth transition, models generally adopt polynomial [52], exponential [20,53], and trigonometric [13,19] laws for the evolution of the martensite volume fraction or for the definition of the transformation hardening function. Lagoudas et al. [54] proposed a model unifying all of these laws to describe the transformation hardening function and compared their performance. The results obtained by Lagoudas et al. [54] are summarized in Fig. 10.4. The exponential hardening law is plotted for two different calibrations of material constants [54]. Figure 10.4(a) shows that the stressestrain curves of all models are similar and give the same amount of total hysteresis, except for the curve obtained with the exponential law labeled “transf. temp. given.” Although the stressestrain curves are similar, significant differences are evident when the values of

Figure 10.4 (a) Uniaxial stressestrain behavior for different hardening laws; (b) heat energy versus temperature during A / M transformation. (From Lagoudas et al. [54], Taylor & Francis, reprinted by permission of the publisher (Taylor & Francis Ltd, http://www.tandfonline.com).)

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the specific heat due to phase transformation are compared, as shown in Fig. 10.4(b). In particular, the cosine hardening law gives the best results compared with experimental data. Lagoudas et al. [18] proposed a power evolution law in terms of the martensite volume fraction for the hardening energy, allowing for continuous first derivatives between the elastic and transformation regimes in the stressestrainetemperature response. Auricchio et al. [12] examined such a smooth transition by adopting a power law dependence of the transformation criterion radius on the detwinned martensite volume fraction. In general, the polynomial, exponential, and trigonometric laws do not always describe experimental results effectively, whereas a good match has been found for power laws [12,18].

10.3.4 Stress-dependent transformation strain magnitude The inelastic strain produced during martensitic transformation is maximized if the martensitic transformation occurs at sufficiently high applied stress (i.e., full transformation), such as in PE paths or during thermal cycling under a high constant force [18]. In such cases, the maximum number of favored variants is produced. However, if the martensitic transformation occurs at a lower applied stress, the generated martensite is not fully oriented and the inelastic strain produced during phase transformation is not maximized. From these considerations, it is clear that the magnitude of the recoverable inelastic strain depends on the applied stress. Generally, models [10e12,19] consider two distinct volume fractions of martensite (i.e., twinned and detwinned martensites) to describe a stress-dependent magnitude in thermally induced paths under applied stress values varying between the critical uniaxial start and finish stresses required for detwinning of twinned martensite. Some authors [18,37,43,55] adopted a unique martensite volume fraction and an additional internal variable describing the average transformation strain in the martensitic phase. In this approach, the average transformation strain magnitude varies linearly [37,55] or nonlinearly [18,43] with the stress. Alternatively, Auricchio et al. [56] proposed the dependence of the transformation criterion radius on temperature for the model originally proposed by Souza et al. [38]. The dependence of the transformation strain magnitude on the applied stress is particularly important for developing actuators. Figure 10.5 clearly shows the evidence of such an effect on NiTi spring actuators subjected to thermal cycling under different applied force levels. In particular, Fig. 10.5 shows a comparison performed by Auricchio et al. [57] among three different approaches: (1) the model by Souza et al. [38] (labelled as SA in the figure), which does consider the dependence of the transformation strain magnitude on the applied stress; (2) its extension [56] (labelled as Mod. SA in the figure), based on a yield surface radius depending on temperature; and (3) the model by Auricchio et al. [12] (labelled as AB in the figure), which considers two volume fractions of martensite. The comparison of experimental and numerical curves, demonstrates the

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Figure 10.5 Comparison between numerical and experimental spring relative elongation-temperature curves for thermal-cycling tests on 1.5 mm wire springs at constant loads of 3, 5, and 8 N. (Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Journal of Materials Engineering and Performance, Auricchio et al. [57].)

inability of the model originally proposed by Souza et al. [38] to catch SMA behavior for the three low-force levels.

10.3.5 Asymmetric forwardereverse transformation Asymmetry between forward and reverse phase transformation has clearly an impact on the transformation hysteresis area. As an example, the width of the hysteresis loop may decrease by increasing the applied temperature in PE paths or by increasing the applied stress level in thermal cycling tests under a constant applied load. When considering the schematic phase diagram reported in Fig. 10.6(a), such an asymmetry can be described by two different slopes of the forward and reverse transformation regions. The diagram shows how the asymmetry affects the width of the hysteresis loop in PE paths at two different temperatures (blue [dark gray in printed version] vertical curves) and in thermal cycling tests under two different constant applied loads (red [light gray in printed version] horizontal curves). Asymmetric forwardereverse transformation has rarely been addressed (e.g., in models based on a transformation criterion whose critical value is different for forward and reverse transformation [12,17,27]); for the reported approach, the critical value can be stress- and/or temperature-dependent. Figure 10.6 illustrates the change in the hysteresis area in thermal cycling tests under a constant load and in isothermal loading paths. Experimental data [7,58] are compared with numerical predictions obtained by Lagoudas et al. [18] and Auricchio et al. [12], whose models consider such an asymmetry, and by Souza et al. [38], whose model does not describe this effect. As observed, the models by Lagoudas et al. [18] and Auricchio et al. [12] correctly predict the change in experimental width, whereas the model by Souza et al. [38] predicts the same hysteresis loop width in PE tests at different temperatures (Fig. 10.6c and d).

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Figure 10.6 (a) Schematic one-dimensional phase diagram with different slopes for forward and reverse transformation regions. Comparison of experimental data and numerical simulations for (b) thermal cycling tests at two different applied stresses and (c, d) isothermal loading paths at two different temperatures. ((b) Adapted from Lagoudas et al. [18], Copyright (2012), with permission from Elsevier. (c, d) Adapted from Auricchio et al. [12], Copyright (2014), with permission from Elsevier.)

10.3.6 Asymmetric tensionecompression behavior SMA behavior may display tensileecompressive asymmetry, which manifests through different critical stresses to trigger martensitic transformations, different slopes of the transformation regions in the phase diagram, and different transformation strain magnitudes. To describe asymmetric tensionecompression behavior, constitutive models generally rely on introducing a specific form for the yield function, F, associated with the transformation surface. For isotropic SMA materials, yield surfaces are commonly defined in terms of invariants of the deviatoric stress or of the thermodynamic forces associated with internal variables (e.g., martensite fraction or transformation strain).

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Among these criteria, the von Mises one is expressed in terms of the second invariant, J2 , and is one of the most widely adopted. However, such a criterion does not account for the asymmetric behavior of SMAs, and other yielding surfaces have been proposed. In particular, considering that martensitic transformations are volume invariants, such yield forms generally do not depend on the first invariant, I1 . Accordingly, to describe tensionecompression asymmetry in isotropic SMAs, yielding surfaces depending on the second and third invariants, respectively, J2 and J3 , have been proposed [15,39,59e61]. As an example, Patoor et al. [59] adopted the Prager criterion in the form: ! J3 F ¼ J2 1 þ b 3  R 2 (10.33) 2 J2 where b is a material parameter and R is the elastic domain radius. The Prager Eq. (10.33) describes a convex surface for a specific range of b; if b ¼ 0, the von Mises criterion is recovered. Zaki [62] adopted the mathematical framework by Raniecki and Mr oz [63], who proposed a transformation criterion in the form: 3n

F ¼ ð J2 Þ 2  cð J3 Þn  s3n c

(10.34)

where n, c, and sc are material parameters. Specifically, for every odd n and for every c different from zero, the yield function (10.34) is sign-sensitive. The model by Zaki [62] is thus able to describe asymmetric orientation stress thresholds, asymmetric maximum orientation strains, as well as asymmetric strain hardening. Bouvet et al. [64] proposed a criterion in the form: F ¼ sgðys Þ  s0

(10.35)

Here, s is the von Mises equivalent stress, ys is the third invariant of the stress, and the function gðys Þ is expressed as: # " cos1 ð1  að1  ys ÞÞ (10.36) gðys Þ ¼ cos 3 where a is a material parameter determined from yield stresses in pure tension and compression. Expression yields a convex criterion for a ˛½0; 1. For a ¼ 0, the von Mises criterion is obtained; for a ¼ 1, the criterion has the maximum tensioncompression asymmetry and a triangular shape in the stress deviator plane. An equivalent transformation strain, depending on the third invariant and coherent with the equivalent stress, was then defined by Bouvet et al. [64]. The criterion (10.35) has been adopted and generalized in several contributions [65,66].

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Auricchio et al. [46] included tensionecompression asymmetry in the onedimensional formulation of the model by Souza et al. [38], by using a modified norm, j$ji , in the form: jaji ¼ ð1 þ ci Þjaj  ci a

(10.37)

where ci is a given constant such that ci > 1=2 to ensure norm positivity. For ci ¼ 0 the standard (symmetric) norm is recovered; for ci > 0 the slope for a < 0 is greater than that for a > 0. The norm introduced in Eq. (10.37) can be extended to a three-dimensional setting (e.g., the Prager-Lode J2 /J3 norm). To include pressure dependence, Qidwai and Lagoudas [67] proposed two transformation functions based on the first invariant, I1 . The first is a J2  I1 criterion in the form: 2  pffiffiffiffiffiffi (10.38) F ¼ b 3J2 þ gI1 þ p  Y 2 where b and g are material parameters, Y is the elastic domain radius, and p is the thermodynamic force associated with the martensite volume fraction. The second criterion is a J2  J3  I1 criterion in the form: !2 #2 " pffiffiffiffiffiffi J3 þ uI1 þ p  Y 2 (10.39) F ¼ h 3J2 1 þ y 3=2 ð3J2 Þ where h and y are parameters related to the asymmetry, u is a parameters that captures the volumetric effect, and 2 is a constant. For y ¼ 0 (or 2 ¼ 0Þ and u ¼ 0, the form of the J2 function is recovered; for y ¼ 0 (or 2 ¼ 0), the form of J2  I1 function is recovered. Alternative approaches [23e26] split the volume fraction of detwinned martensite, xMD , as the sum of tension-induced and compression-induced parts, respectively, xMD; t and xMD; c , to include tensionecompression asymmetry (see Section 10.3.1).

10.3.7 Anisotropy In the case of textured SMAs (e.g., CueZneAl SMAs), the assumption of isotropic material behavior in the previous subsection no longer holds [65]. Macroscopic models taking SMA anisotropy into account for are rare in the literature. Generally, they consider an anisotropic yield surface [65,68]. Taillard et al. [65] modified the equivalent stress and equivalent transformation strain discussed in the previous subsection. In particular, the anisotropic equivalent stress, seq;ani , was developed from the isotropic equivalent stress, seq ¼ sgðys Þ (Eq. 10.35) by e , obtained by an affine transformation: substituting stress s with a dilated stress, s e ¼ D: s s

(10.40)

where D contains constant material parameters. Owing to the linearity of the tensor transformation, the anisotropic yield surface, F, is convex. Similarly, the equivalent transformation strain was proposed by considering a generalization of the equivalent transformation.

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Following the approach by Taillard et al. [65], Sedlak et al. [31] introduced an equivalent transformation strain: hεtr i  1

(10.41)

by adopting a specific form for function h$i , depending on the second and third invariants of the transformation strain obtained by an affine transformation. Sadjadpour and Bhattacharya [69] studied transversely isotropic as well as tensione compression asymmetric SMAs; their work was later generalized by Kelly et al. [70]. Chatziathanasiou et al. [61] generalized the approach by Patoor et al. [59], based on the Prager criterion (Eq. 16.33), as:
* !1=n qffiffiffiffiffiffiffiffiffiffiffiffi J s 3 F ¼ J2 ðs* Þ 1 þ b 3  ks (10.42) 2 J2 ðs* Þ where n is a positive real number and s* is a second-order tensor defined as: s* ¼ Rs : s  xs

(10.43)

Here, Rs is a fourth-order dimensionless tensor containing constants and xs is a secondorder tensor with dimensions of stress. A nonassociated evolution rule that captures incompressibility and strain anisotropy was then adopted.

10.3.8 Plasticity As discussed and detailed in the following discussion, the plastic behavior of SMAs can be described by two different mechanisms: (1) transformation-induced plasticity (TRIP), and (2) plastic yielding. Both mechanisms are dissipative and mostly driven by shear deformation as detwinning; however, detwinning occurs by a lattice change without alteration to neighboring atoms, whereas plasticity includes such an alteration. Unlike strain due to phase transformation and detwinning, plastic strain resulting from the latter mechanism cannot be recovered. Both the mechanisms are described and discussed in the following discussion. Special attention will be then given to two-way SME, which is a phenomenon associated with plasticity in SMAs. 10.3.8.1 Transformation-induced plasticity TRIP accumulates during cyclic thermomechanical transformation and is classically explained by the GreenwoodeJohnson and Magee mechanisms [71]. The first mechanism is based on a volume change between austenitic and martensitic phases, in which plastic flow is induced in the weaker phase according to internal stresses; the second mechanism is based on the activation of specific variants depending on the local stress, resulting in an orientation of the transformation strain. Therefore, TRIP does not require

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exceeding the yield strength of either phase, but only for local stresses at the austenitee martensite interface to become sufficiently high to cause localized slip dislocations. During cyclic loading, plastic deformation pins specific martensite variants and prevents their reverse transformation to austenite, reducing the amount of recoverable inelastic strain. In general, TRIP increases with cycling and saturates on a stable value after a certain number of cycles. Degradation effects are also associated with the phenomenon (e.g., manifesting in a reduction in stresses triggering martensitic transformation in PE loops). To include TRIP description, one-dimensional [44,72e74] and three-dimensional [21,43,66,71,75e84] models have been proposed. The most common approach [71,76,82,83] is to consider an additive split of the total strain as: ε ¼ εe þ εth þ εtr þ εtrip

(10.44)

where εtrip is used to represent the accumulated plastic strain induced by phase transformation (i.e., the TRIP strain). In addition, an internal variable to describe the internal stress tensor generated during the training process may be used [43,76,82]. In general, TRIP strain is commonly expressed in terms of parameters and variables related to the number of loading cycles. In a one-dimensional framework, Auricchio et al. [44] split the martensite fraction, xM , into a reversible and a residual irreversible fraction, respectively, xM ; rev and xM ; irr , and adopted an internal variable, b, describing the change of martensite reorientation. Accordingly, the total strain was decomposed as in Eq. (10.44), with εtr ¼ xM b and εtrip ¼ xM ; irr ðb kÞ, and with k a training parameter. Alternatively, some models [66,77,79,80,84] introduce an internal variable, εtrip , giving a measure of the part of εtr that cannot be recovered when unloading to a zero-stress state. The additive split remains the same as in Eq. (10.19) and the evolution of εtrip is strictly linked to εtr . Lexcellent et al. [21] modified the free-energy function by adding an experimental expression depending on the maximum stress during training. This captures the response at the end of the training process but it does not describe the cycling loading itself. The work by Karamooz-Ravari et al. [73] applied a one-dimensional microplane model to additive manufactured SMAs. 10.3.8.2 Plastic yielding Plastic yielding arises from an applied stress or a temperature variation when the yield surface of either phase is reached. To describe plastic yielding, one-dimensional [23,24] and three-dimensional [49,50,71,83,85e89] models have been proposed. These models generally introduce an additional tensorial variable, namely the plastic strain, εp , such that: ε ¼ εe þ εth þ εtr þ εp

(10.45)

Advanced constitutive modeling

The evolution of εp is usually accounted for by means of a generalized plasticity theory and associative flow rules. Moreover, its formation may take place either simultaneously with martensitic transformation or when martensitic transformation is completed. Most models focus on plastic yield in the austenite state [49,50,86,88]; a few works also focus on the martensitic state [23,24,85,87,89]. In addition to εp , work by Hartl and Lagoudas [85] uses the plastic hardening energies to measure the increase in the free-energy due to isotropic hardening as plastic yield progresses in austenitic and martensitic regions, as well as the plastic back stress to couple transformationeplastic yield in the formulation. Moreover, Hartl and Lagoudas [85] defined the bounds on the martensitic volume fraction, xM , that are influenced by the increase in plastic deformation: the lower bound on xM is defined as the fraction of retained martensite or martensite that cannot be converted back into austenite (usually taken to be 0), whereas the upper bound is the maximum fraction of martensite that can be generated from austenite (usually taken to be 1). The irrecoverable martensitic volume fraction, xM ; irr , is introduced to represent the retained martensite and is postulated to be a function of the effective plastic strain. In particular: xM  xM ; irr xM  1

if x_ < 0 if x_ > 0

(10.46) (10.47)

The works by Petrini et al. [88] and Oliveira et al. [71] further couple the yield plastic strain with TRIP and martensitic transformation. 10.3.8.3 Two-way shape memory effect The two-way SME describes the ability of the SMA to change shape reversibly during subsequent thermal cycles between transformation temperatures under zero mechanical load. Such an effect is particularly exploited to create fasteners, actuators, and couplers. The two-way SME is not a natural feature of SMAs, but it can be induced in the material by means of a specific thermomechanical cyclic treatment (the so-called training process) or a one-time martensite deformation. The training process subjects the SMA to cyclic loading (causing transformation from austenite to preferentially oriented martensite or from deformed martensite to austenite); it is the most widely applied method to induce the two-way SME. The strain, which is not recovered at the end of each loading cycle, accumulates during training until saturation, and the two-way SME is the result of dislocation arrays during the training process. Accordingly, a trained SMA is able to switch configurations between stable austenite and oriented martensite by heating or cooling with no applied load. The role of the TRIP phenomenon and the importance of its modeling are clear in this context. In this regard, several works [43,44,76,82] studied the two-way SME experimentally and theoretically. As an example, the finite strain model by Xu et al. [82] is used to simulate an SMA initially loaded under a constant tensile load of 200 MPa and then subjected to 100 thermal cycles with temperatures varying from 310 to 440 K. Once the 100

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Figure 10.7 Thermal cycling under a 200-MPa constant load and stress-free two-way shape memory effect (SME) response for a NiTi shape memory alloy. Numerical [82] and experimental [90] results for (a) selected cycles, (b) cycle 1, (c) cycle 10, (d) cycle 20, (e) cycle 40, (f) cycle 70, (g) cycle 100, and (h) two-way SME cycle. (Reprinted from Xu et al. [82], Copyright (2020), with permission from Elsevier.)

Advanced constitutive modeling

thermal cycles are completed, an additional thermal cycle under stress-free conditions is performed to examine and model the two-way SME of the material. Figure 10.7 compares the strainetemperature curves obtained experimentally by Atli et al. [90] and numerically by Xu et al. [82]. A large amount of TRIP strain (around 11%) is accumulated during the 100 thermal cycles and causes the onset of the two-way SME. Work by Auricchio et al. [44] modeled the two-way SME in a cantilever beam characterized by an elastic core and two SMA layers, positioned on the top and bottom of the beam. The beam is subjected to an initial 10 cycles of bending loadingeunloading at high temperature. At the end of the bending training, the beam remains in a deformed configuration because of the accumulated permanent strain. Then, the beam is subjected to two thermal cycles, recovering part of the permanent strain during cooling. If a stress is applied to a trained SMA during thermal cycling, the resulting behavior is called assisted two-way SME. The applied stress favors the nucleation of oriented martensite variants and cooling results in a macroscopic inelastic strain that is recoverable by heating above the finish austenitic temperature. One-time martensite deformation treatment deforms the SMA in the martensitic phase, unloading the material, and then in undergoing thermal cycling. Such a method generally causes two-way SME strains of limited magnitude; however, it has been shown by Liu et al. [91] that the two-way SME strains can be increased by imposing severe martensitic deformations (e.g., causing plastic yielding). Differently from the two-way SME induced by a thermomechanical cyclic treatment, behavior induced by the one-time martensite deformation treatment has been less

Figure 10.8 Results of finite element analysis of the two-way SME in a C-shaped fastener performed by Scalet et al. [89]. Horizontal displacementetemperature curve for a fastener extremity. (Reprinted from Scalet et al. [89], Copyright (2019), with permission from Elsevier.)

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studied. To the best of our knowledge, only work by Scalet et al. [89] models the twoway SME in severely prestrained NiTi SMAs. The response of these alloys in terms of the transformation temperatures, one-way SME, and two-way SME is affected by the (yielding) plastic strain occurring during severe martensitic prestrain and subsequent thermal cycles without an applied load. Figure 10.8 shows the result of the finite element analysis of a C-shaped fastener subjected first to a severe martensitic predeformation and then, upon unloading, to a heatingecooling cycle. The onset of plastic strain in the martensitic state is responsible for the observed macroscopic (two-way SME) strain during subsequent cooling (denoted as dtw in the figure). The magnitude of the generated two-way strain depends on the applied martensitic predeformation. Further modeling efforts need to be addressed to model the two-way SME, given the complexity of the phenomena taking place (e.g., finite deformations, plasticity, phase transformation, and reorientation).

10.3.9 Minor loops SMA behavior under thermomechanical cycling is hysteretic, and major or minor transformation cycles may be observed. A major (or complete) transformation cycle is achieved if the SMA undergoes through the entire phase transformation process, whereas a minor (or partial) transformation cycle is created inside the major cycle if the SMA is subjected to a load that is reversed before the phase transformation is completed. The size and shape of the partial loop may change depending on the applied history, and three different scenarios may be observed [92]: (1) partial loop with single reversal point, (2) closed loop, and (3) partial loop with multiple reversal points. Figure 10.9 shows the three scenarios for a thermal cycling test under a constant applied load. In the figure, the major transformation cycle is defined by an order number equal to zero, whereas subsequent minor cycles are defined by an ascending order

Strain

368

Scenario 1

1

P1

0 0

Scenario 3

Scenario 2

1

P1 0

0

2

1 P3

P1

0 P2

P = 1st rev. point 1

3

0 P4

2 4 P2

P = 2nd rev. point 2 P = 3rd rev. point 3 P = 4th rev. point 4

Temperature

Figure 10.9 Partial transformation loop scenarios in the case of a thermal cycling test under constant applied load: partial loop with a single reversal (rev.) point (scenario 1), closed loop (scenario 2), and partial loop with multiple reversal points (scenario 3). See the work by Karakalas et al. [92] for further detail.

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number. In the first scenario, the SMA first cools down and forward transformation starts; then, at reversal point P1, cooling stops and the SMA is heated until reverse transformation is completed. Accordingly, a first-order branch inside the major transformation cycle is observed during heating. In the second scenario, as before, the SMA is first cooled down; then, at reversal point P1, the cooling stops and the SMA is heated until reversal point P2. At this new point, the SMA is cooled again. Accordingly, a first-order branch inside the major transformation cycle is observed during heating, whereas a second-order branch is formed during the second cooling. As the second cooling goes on, the secondorder branch evolves toward P1 and a closed loop is formed. Such an SMA feature is known as reversal memory, because the material is able to remember the previous reversal points. In the third and last scenario, the process is similar to the previous one, but different reversal points are reached in this case. Accordingly, multiple branches of increasing order are observed. In the most general case, the SMA remembers all previous reversal points. As cooling proceeds, the fourth-order branch evolves, first toward P3, and then toward P1, thus merging with the major cooling branch. Such a feature in the case of higher-order partial transformation branches is known as wiping-out memory. Similar considerations hold for PE loading cases. Most constitutive models focus on predicting the major transformation cycle; limited attention has been paid to predicting partial loops. In general, when approaching the modeling of minor loops, the following considerations should be considered: (1) computational cost increases in case of higher-order partial transformation branches, because all the reversal points experienced by the material should be memorized; and (2) a high number of experimental data may be required to increase model accuracy. Available modeling approaches can be classified as (1) algebraic expressions, (2) differential equations and Duhem-Madelung models [42,92], (3) Preisach models, (4) purely phenomenological models [20,24,72,93], and (5) state equations based on principles of thermodynamics and/or plasticity [37,55,94]. For details, refer to the work by Karakalas [92]. Models belonging to category 1 are mostly based on phenomenological observations and are used for actuator control applications. They do not provide information regarding the evolution of quantities characterizing SMA behavior, such as the transformation strain. Models of category 2 are advantageous for implementing hysteresis into SMA dynamics and analyzing system stability, for example [92]. The martensitic volume fraction for minor hysteresis loops is related to the martensitic volume fraction for the major hysteresis loop. Models of category 3 are the most employed operator-based hysteresis models for ferromagnetic materials. The material is assumed to be composed of a large number of hysteretic elements, and the global response is obtained by the single element’s responses. Such models are able to describe partial transformation loops as well as the wiping-out

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memory features; however, their accuracy depends on the selection of weighting function values and the number of experiments needed for calibration (generally under various thermomechanical loading conditions). Models of category 4 are suited to describe the effects of multiaxial thermomechanical loading paths and cyclic loading in SMAs, because of the introduction of internal state variables in their formulation. However, such models may give poor predictions of the minor loop hysteretic response. In this framework, work by Bo and Lagoudas [93] proposes a phenomenological connection between categories 3 and 4 and uses the methodologies developed in 3 to account for minor hysteresis loops in thermomechanical models of group 1. Chemisky et al. [37] generalized the approach by Peultier et al. [55], which defines the critical threshold for minor loops proportional to the martensitic volume fraction, to consider a critical threshold depending on the martensitic volume fraction and a minimal thermodynamic threshold. Paiva et al. [24] used an indicator function to establish constraints related to minor loops. A three-dimensional phenomenological model able to account for the return-point memory effect during isothermal conditions was presented by Bouvet et al. [94] and Saint-Sulpice et al. [66]. Work by Karakalas [92] generalizes the model by Lagoudas et al. [18] by introducing the concept of hysteresis scaling [42] into the hardening function evaluation as well as an enhanced approximation to account for the return-point memory and memory wipe-out in single, closed partial transformation cycles. The results are compared with several experimental data, giving accurate results in a reasonable amount of time. Work by Ikeda et al. [45] proposes a one-dimensional shift and skip model to account for inner loops. Mehrabi et al. [95] proposed a phenomenological model based on microplane theory to capture the effects of loading history on the transformation start conditions. Nascimento et al. [96] proposed a hysteresis model based on the limiting loop proximity approach. Constitutive modeling of the partial transformation cycles is essential in case of SMAs undergoing a high number of cycles without reaching full transformation (e.g., in solidstate actuators or shape-adaptive structures) [97].

10.3.10 Damage and fatigue The prediction of the fatigue life of an SMA component remains an open problem in the current literature, owing to the complex behavior of the alloy under cyclic loading. Fatigue in SMAs can be divided into structural fatigue (i.e., fatigue due to high cyclic loads) and functional fatigue (i.e., fatigue due to repeated martensitic transformation leading to a deterioration in functional properties). In general, the prediction of structural fatigue determines the SMA component lifetime and is associated with the development of microcracks during transformation. On the contrary, the prediction of functional fatigue computes changes in a given SMA

Advanced constitutive modeling

component over its lifetime and relies on the simulation of irrecoverable strains upon cycling (i.e., TRIP strains). Moreover, it is possible to subdivide functional fatigue into (1) pseudoelastic fatigue due to stress-induced phase transformation and (2) actuation fatigue due to thermally induced phase transformation. A detailed discussion of both fatigue responses is beyond the scope of this chapter; the reader can refer to Chapter "Fatigue and fracture" of this book. Much research has been done to model pseudoelastic fatigue, in part because of the large employment of SMAs in the biomedical field, although less research has been dedicated to actuation fatigue, in part owing to the complexity and time needed to conduct tests involving thermally-induced phase transformation cycles. Regarding multiaxial pseudoelastic fatigue, it is worth recalling the contribution by Moumni et al. [98], in which a relation between the amount of dissipated energy associated with the stabilized hysteresis cycle, W d , and the number of cycles to failure, Nf , was established to predict low-cycle failure: W d ¼ aNfb

(10.48)

where a and b are material parameters. The validity of such a criterion has been proven by Moumni et al. [98] in the case of uniaxial loading. A good match between numerical and experimental data is found and it is shown that the dissipated energy decreases with an increasing number of cycles. Regarding actuation fatigue, scientific contributions generally aim to predict the lifetime of an SMA actuator by deriving a relationship among various quantities related to phase transformation (e.g., applied stress, irrecoverable strain, or actuation work) and the number of cycles to failure. Few works model the microstructural phenomena leading to failure during the actuation fatigue lifetime. Specifically, works by Chemisky et al. [99] and Philips et al. [100] proposed a three-dimensional deterministic and isotropic model based on continuum damage mechanics, to couple damage evolution, phase transformation, and TRIP strain. In particular, they introduced an additional scalar variable, d, describing isotropic damage accumulation induced during phase transformation. Chemisky et al. [99] further distinguished between the damage induced during forward and reverse transformation. The main difference between the two contributions lies in the adopted relationship between the number of cycles, Nf , and the damage variable, d. From experimental evidence [100], the evolution of internal damage in an SMA actuator seems to consist of three regimes: the nucleation phase that evolves logarithmically, the growth phase that is linear, and the coalescence phase that grows in an exponential manner (Fig. 10.10). Accordingly, various combinations of linear, logarithmic, and exponential functions were studied by Philips et al. [100] to determine the best function to describe the evolution of internal damage. Figure 10.10 shows the results of the linear, logarithmic plus linear, exponential plus linear, and logarithmic plus exponential curve

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Figure 10.10 Internal damage versus percentage of actuation fatigue lifetime diagram. Comparison of experimental data and internal damage evolution model. Results obtained by Philips et al. [100]. (Reprinted from Phillips et al. [100], Copyright (2019), with permission from Elsevier.)

fit models provided by Philips et al. [100]. Results indicate that the linear assumption used by Chemisky et al. [99] is valid for approximately 75% of the actuation fatigue lifetime, but it is unable to capture the nucleation of internal damage from the beginning of life to 2% actuation fatigue lifetime or exponential growth at the end of the fatigue life. The exponential plus linear model has a similar weakness at the beginning of life. On the other hand, the logarithmic plus linear model is capable of predicting rapid nucleation at the beginning of the actuation fatigue lifetime but does not properly capture the growth and coalescence phases. The logarithmic plus exponential model best captures nucleation, steady growth, and exponential coalescence; thus, it is adopted by Philips et al. [100] as:

Advanced constitutive modeling

  N c4 N þ c3 e Nf d ¼ c1 log c2 Nf

(10.49)

where c1 , c2 , c3 , and c4 are material parameters. The damage variable can be considered anisotropic (i.e., the damage is assumed to adopt preferred orientations), leading to the use of a tensorial damage variable. However, no models are available that consider anisotropic damage. Furthermore, no contributions based on probabilistic approaches are currently available.

10.3.11 Thermomechanical coupling The heat exchanged during phase transformation may result in self-heating or selfcooling of the SMA, depending on the transformation direction. Forward transformation is accompanied by a release of latent heat and is therefore exothermic, whereas reverse transformation is endothermic. Self-heating or self-cooling has a direct consequence on the phase transformation process: on the one hand, the generation of heat during forward transformation stabilizes the austenite phase by increasing its temperature, and thus by increasing the stress required to proceed with the transformation; on the other hand, the adsorption of latent heat during reverse transformation stabilizes the martensite phase by decreasing the stress. Such behavior is further influenced by other factors, such as the dissipation due to phase transformation and inelastic strains and by conditions affecting the exchange of heat with the environment, including the loading rate. Modeling thermomechanical coupling is necessary in case of sufficiently high loading rates, for which the temperature in the SMA during phase transformation is no longer constant. In such a case, temperature variation affects the size of the hysteresis loop, which is connected to the dissipated energy, and thus, to fatigue life. Thermomechanical coupling is generally accounted for by considering latent heat and intrinsic dissipation due to phase transformation as the heat source in one-dimensional [101e104] and three-dimensional [27,47,105,106] models. Peigney and Seguin [105] proposed an analysis of a variational formulation of nonisothermal evolution problems in generalized standard materials. In particular, the authors focused on the timeincremental step and augmented the time-discrete equations by means of additional terms to let the corresponding nonlinear system be symmetric. These additional terms turned out to be consistent in the case of continuous temperature evolution. The incremental step was then solved by minimization. Auricchio et al. [107] proposed a variational frame for the model by Souza et al. [38], making sense at both the continuous and discrete levels simultaneously. Finally, these models disregard localized dissipation that accompanies the formation of L€ uders-like phase transformation fronts. The nucleation and propagation of these fronts at different loading rates and thermal environmental conditions were considered by Iadicola and Shaw [108], for example.

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10.4 Conclusions Constitutive models for SMAs are required to predict a great variety of stressestraine temperature responses depending on the applied loading conditions, owing to the complex material behavior. This chapter has reviewed current macroscopic constitutive models with internal variables able to describe primary and secondary effects in SMA behavior under small strains. In particular, the focus has been on phase transformation, martensite reorientation, phase-dependent elastic properties, smooth thermomechanical response, stress-dependent transformation strain magnitude, asymmetric forwarde reverse transformation and tensionecompression response, anisotropy, plasticity, minor loops, and damage and fatigue, as well as thermomechanical coupling. For each effect, modeling approaches have been recalled and discussed by presenting significant results and further developments. From the reviewed literature, it can be observed that models describing several secondary effects simultaneously are limited. Moreover, high levels of accuracy in the description of different physical mechanisms have been reached; in particular, models that employ a large number of material parameters show generally the best performances. However, the high number of parameters (not always associated with physical quantities) may require costly calibrations. Moreover, the inclusion of several effects in model formulations necessitates a suitable numerical algorithms to treat nonsmooth functions and/or constraints on internal variables and to guarantee numerical convergence. Therefore, a good balance between accuracy and efficiency always has to be taken into account. Finally, attention has been paid to NiTi-based polycrystalline SMAs, which are the most widely used alloys. However, a wide number of contributions are available to describe other types of SMAs, such as high-temperature, iron-based, copper-based, magnetic, Fe-based, and porous SMAs. Moreover, some contributions are available for finite strain (not reviewed here), even if further work is required owing to the lack of contributions regarding the two-way SME.

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[30] C.W. Chan, S.H.J. Chan, H.C. Man, P. Ji, 1-D constitutive model for evolution of stress-induced R-phase and localized L€ uders-like stress-induced martensitic transformation of super-elastic NiTi wires, Int. J. Plast. (2012) 85e105, 32-33. [31] P. Sedlak, M. Frost, B. Benesova, T. Ben Zineb, P. Sittner, Thermomechanical model for NiTibased shape memory alloys including R-phase and material anisotropy under multi-axial loadings, Int. J. Plast. 39 (2012) 132e151. [32] M. Frost, P. Sedlak, L. Kaderavek, L. Heller, P. Sittner, Modeling of mechanical response of NiTi shape memory alloy subjected to combined thermal and non-proportional mechanical loading: a case study on helical spring actuator, J. Intell. Mater. Syst. Struct. 27 (14) (2016) 1927e1938. [33] M. Fremond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. [34] D. Helm, P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics, J. Intell. Mater. Syst. Struct. 40 (2003) 827e849. [35] V. Panoskaltsis, S. Bahuguna, D. Soldatos, On the thermomechanical modeling of shape memory alloys, Int. J. Non Lin. Mech. 39 (2004) 709e722. [36] X. Gu, W. Zaki, C. Morin, Z. Moumni, W. Zhang, Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases, Int. J. Solid Struct. 54 (2015) 82e99. [37] Y. Chemisky, A. Duval, E. Patoor, T. Ben Zineb, Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation, Mech. Mater. 43 (7) (2011) 361e376. [38] A. Souza, E. Mamiya, N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations, Eur. J. Mech. A Solid. 17 (1998) 789e806. [39] F. Auricchio, L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: solution algorithm and boundary value problems, Int. J. Numer. Methods Eng. 61 (2004) 807e836. [40] F. Auricchio, S. Morganti, A. Reali, SMA numerical modeling versus experimental results, in: European Symposium on Martensitic Transformations (ESOMAT), 2009, pp. 1e6, 08004. [41] G. Rio, D. Favier, Y. Liu, Elastohysteresis Model Implemented in the Finite Element Software HEREZHþþ, European Symposium on Martensitic Transformations, ESOMAT, 2009, pp. 1e6, 08005. [42] Y. Ivshin, T.J. Pence, A thermomechanical model for a one variant shape memory material, J. Intell. Mater. Syst. Struct. 5 (1994) 455e473. [43] Z. Bo, D. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part III: evolution of plastic strains and two-way shape memory effect, Int. J. Eng. Sci. 37 (9) (1999) 1175e1203. [44] F. Auricchio, S. Marfia, E. Sacco, Modeling of SMA materials: training and two way memory effects, Comput. Struct. 81 (24) (2003) 2301e2317. [45] T. Ikeda, F.A. Nae, H. Naito, Y. Matsuzaki, Constitutive model of shape memory alloys for unidirectional loading considering inner hysteresis loops, Smart Mater. Struct. 13 (2004) 916. [46] F. Auricchio, A. Reali, U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Comput. Methods Appl. Mech. Eng. 198 (2009) 1631e1637. [47] C. Morin, Z. Moumni, W. Zaki, A constitutive model for shape memory alloys accounting for thermomechanical coupling, Int. J. Plast. 27 (2011) 748e767. [48] K. Tanaka, R. Iwasaki, A phenomenological theory of transformation superplasticity, Eng. Fract. Mech. 4 (21) (1985) 709e720. [49] W. Yan, C. Wang, X. Zhang, Y. Mai, Theoretical modeling of the effect of plasticity on reverse transformation in superelastic shape memory alloys, Mater. Sci. Eng. 354 (2003) 146e157. [50] B. Zhou, A macroscopic constitutive model of shape memory alloy considering plasticity, Mech. Mater. 48 (2012) 71e81. [51] S. Gong, Z. Li, Y. Zhao, An extended Mori-Tanaka model for the elastic moduli of porous materials of finite size, Acta Mater. 59 (2011) 6820e6830.

Advanced constitutive modeling

[52] J.G. Boyd, D.C. Lagoudas, A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy, Int. J. Plast. 12 (6) (1996) 805e842. [53] K. Tanaka, S. Kobayashi, Y. Sato, Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys, Int. J. Plast. 2 (1986) 59e72. [54] D.C. Lagoudas, Z. Bo, M.A. Qidwai, A unified thermodynamic constitutive model for SMA and finite element analysis of active metal matrix composites, Mech. Compos. Mater. Struct. 3 (2) (1996) 153e179. [55] B. Peultier, T. Ben Zineb, E. Patoor, Macroscopic constitutive law for SMA: application to structure analysis by FEM, Mech. Mater. 38 (2006) 510e524. [56] F. Auricchio, A. Coda, A. Reali, M. Urbano, SMA numerical modeling versus experimental results: parameter identification and model prediction capabilities, J. Mater. Eng. Perform. 18 (2009) 649e654. [57] F. Auricchio, G. Scalet, M. Urbano, A numerical/experimental study of nitinol actuator springs, J. Mater. Eng. Perform. 23 (7) (2014) 2420e2428. [58] X. Wu, G. Sun, J. Wu, The nonlinear relationship between transformation strain and applied stress for nitinol, Mater. Lett. 57 (2003) 1334e1338. [59] E. Patoor, M. El Amrani, A. Eberhardt, M. Berveiller, Determination of the origin for the dissymmetry observed between tensile and compression tests on shape memory alloys, J. Phys. IV 2 (1995) 495e500. [60] Y. Gillet, E. Patoor, M. Berveiller, Calculation of pseudoelastic elements using a non-symmetrical thermomechanical transformation criterion and associated rule, J. Intell. Mater. Syst. Struct. 9 (1998) 366e378. [61] D. Chatziathanasiou, Y. Chemisky, F. Meraghni, G. Chatzigeorgiou, E. Patoor, Phase transformation of anisotropic shape memory alloys: theory and validation in superelasticity, Shape Mem. Superelasticity 1 (2015) 359e374. [62] W. Zaki, An approach to modeling tensile-compressive asymmetry for martensitic shape memory alloys, Smart Mater. Struct. 19 (2) (2010) 025009. [63] B. Raniecki, Z. Mr oz, Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect, Acta Mech. 195 (2008) 81e102. [64] C. Bouvet, S. Calloch, C. Lexcellent, Mechanical behavior of a Cu-Al-Be shape memory alloy under multiaxial proportional and nonproportional loadings, J. Eng. Mater. Technol. 124 (2002) 112e124. [65] K. Taillard, S. Arbab Chirani, S. Calloch, C. Lexcellent, Equivalent transformation strain and its relation with martensite volume fraction for isotropic and anisotropic shape memory alloys, Mech. Mater. 40 (2008) 151e170. [66] L. Saint-Sulpice, S. Arbab Chirani, S. Calloch, A 3D super-elastic model for shape memory alloys taking into account progressive strain under cyclic loadings, Mech. Mater. 41 (1) (2009) 12e26. [67] M.A. Qidwai, D.C. Lagoudas, On thermomechanics and transformation surfaces of polycrystalline NiTi shape memory alloy material, Int. J. Plast. 16 (2000) 1309e1343. [68] R. Mehrabi, M.T. Andani, M. Elahinia, M. Kadkhodaei, Anisotropic behavior of superelastic NiTi shape memory alloys; an experimental investigation and constitutive modeling, Mech. Mater. 77 (2014) 110e124. [69] A. Sadjadpour, K. Bhattacharya, A micromechanics-inspired constitutive model for shape-memory alloys, Smart Mater. Struct. 16 (2007) 1751e1765. [70] A. Kelly, A.P. Stebner, K. Bhattacharya, A micromechanics-inspired constitutive model for shapememory alloys that accounts for initiation and saturation of phase transformation, J. Mech. Phys. Solid. 97 (2016) 197e224. [71] S. Oliveira, V.M. Dornelas, M.A. Savi, P. Pacheco, A. Paiva, A phenomenological description of shape memory alloy transformation induced plasticity, Meccanica 53 (2018) 2503e2523. [72] K. Tanaka, F. Nishimura, T. Hayashi, H. Tobushi, C. Lexcellent, Phenomenological analysis on subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads, Mech. Mater. 19 (4) (1995) 281e292.

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[73] M.R. Karamooz-Ravari, M.T. Andani, M. Kadkhodaei, S. Saedi, H. Karaca, M. Elahinia, Modeling the cyclic shape memory and superelasticity of selective laser melting fabricated NiTi, Int. J. Mech. Sci. 138e139 (2018) 54e61. [74] C. Li, Z. Zhou, Y. Zhu, A uniaxial constitutive model for NiTi shape memory alloy bars considering the effect of residual strain, J. Intell. Mater. Syst. Struct. 30 (8) (2019) 1163e1177. [75] D. Lagoudas, P. Entchev, Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. Part I: constitutive model for fully dense SMAs, Mech. Mater. 36 (2004) 865e892. [76] W. Zaki, Z. Moumni, A 3D model of the cyclic thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solid. 55 (11) (2007) 2427e2454. [77] F. Auricchio, A. Reali, U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. J. Plast. 23 (2007) 207e226. [78] W. Zaki, S. Zamfir, Z. Moumni, An extension of the ZM model for shape memory alloys accounting for plastic deformation, Mech. Mater. 42 (2010) 266e274. [79] N. Barrera, P. Biscari, M. Urbano, Macroscopic modeling of functional fatigue in shape memory alloys, Eur. J. Mech. A Solid. 45 (2014) 101e109. [80] M. Peigney, G. Scalet, F. Auricchio, A time integration algorithm for a 3D constitutive model for SMAs including permanent inelasticity and degradation effects, Int. J. Numer. Methods Eng. 115 (9) (2018) 1053e1082. [81] P. Sittner, P. Sedlak, H. Seiner, P. Sedmak, J. Pilch, R. Delville, L. Heller, L. Kaderavek, On the coupling between martensitic transformation and plasticity in NiTi: experiments and continuumbased modelling, Prog. Mater. Sci. 98 (2018) 249e298. [82] L. Xu, A. Solomou, T. Baxevanis, D. Lagoudas, Finite strain constitutive modeling for shape memory alloys considering transformation-induced plasticity and two-way shape memory effect, Int. J. Solid Struct. (2020) 1e18. https://doi.org/10.1016/j.ijsolstr.2020.03.009. [83] L. Heller, P. Sittner, P. Sedlak, H. Seiner, O. Ty, L. Kaderaveka, P. Sedmak, M. Vronka, Beyond the strain recoverability of martensitic transformation in NiTi, Int. J. Plast. 116 (2019) 232e264. [84] M. Ashrafi, Constitutive modeling of shape memory alloys under cyclic loading considering permanent strain effects, Mech. Mater. 129 (2019) 148e158. [85] D. Hartl, D.C. Lagoudas, Constitutive modeling and structural analysis considering simultaneous phase transformation and plastic yield in shape memory alloys, Smart Mater. Struct. 18 (2009) 104017. [86] D. Jiang, C. Landis, A constitutive model for isothermal pseudoelasticity coupled with plasticity, Shape Mem. Superelasticity 2 (2016) 360e370. [87] X. Jiang, B. Li, Finite element analysis of a superelastic shape memory alloy considering the effect of plasticity, J. Theor. Appl. Mech. 55 (4) (2017) 1355e1368. [88] L. Petrini, A. Bertini, F. Berti, G. Pennati, F. Migliavacca, The role of inelastic deformations in the mechanical response of endovascular shape memory alloy devices, Proc. IME H J. Eng. Med. 231 (5) (2017) 391e404. [89] G. Scalet, F. Niccoli, C. Garion, P. Chiggiato, C. Maletta, F. Auricchio, A three-dimensional phenomenological model for shape memory alloys including two-way shape memory effect and plasticity, Mech. Mater. 136 (2019) 103085. [90] K. Atli, I. Karaman, R. Noebe, G. Bigelow, D. Gaydosh, Work production using the two-way shape memory effect in NiTi and a Ni-rich NiTiHf high-temperature shape memory alloy, Smart Mater. Struct. 24 (12) (2015) 125023. [91] Y. Liu, Y. Liu, J. Van Humbeeck, Two-way shape memory effect developed by martensite deformation in NiTi, Acta Mater. 47 (1) (1998) 199e209. [92] A. Karakalas, T. Machairas, A. Solomou, D. Saravanos, Modeling of partial transformation cycles of SMAs with a modified hardening function, Smart Mater. Struct. 28 (2019) 035014. [93] Z. Bo, D.C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part IV: modeling of minor hysteresis loops, Int. J. Eng. Sci. 37 (1999) 1205e1249.

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[94] C. Bouvet, S. Calloch, C. Lexcellent, A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loadings, Eur. J. Mech. A Solid. 23 (2004) 37e61. [95] R. Mehrabi, M. Shirani, M. Kadkhodaei, M. Elahinia, Constitutive modeling of cyclic behavior in shape memory alloys, Int. J. Mech. Sci. 103 (2015) 181e188. [96] M. Nascimento, C. De Ara ujo, L. de Almeida, J. da Rocha Neto, A. Lima, A mathematical model for the strain-temperature hysteresis of shape memory alloy actuators, Mater. Des. 30 (2009) 551e556. [97] A.A. Karakalas, T.T. Machairas, D.A. Saravanos, Effect of shape memory alloys partial transformation on the response of morphing structures encompassing shape memory alloy wire actuators, J. Intell. Mater. Syst. Struct. 1e17 (2019). [98] Z. Moumni, W. Zaki, H. Maitournam, Cyclic behavior and energy approach to the fatigue of shape memory alloys, J. Mech. Mater. Struct. 4 (2009) 395e411. [99] Y. Chemisky, D.J. Hartl, F. Meraghni, Three-dimensional constitutive model for structural and functional fatigue of shape memory alloy actuators, Int. J. Fatig. 112 (2018) 263e278. [100] F.R. Phillips, R.W. Wheeler, A.B. Geltmacher, D.C. Lagoudas, Evolution of internal damage during actuation fatigue in shape memory alloys, Int. J. Fatig. 124 (2019) 315e327. [101] S. Zhu, Y. Zhang, A thermomechanical constitutive model for superelastic SMA wire with strainrate dependence, Smart Mater. Struct. 16 (2007) 1696e1707. [102] F. Auricchio, D. Fugazza, R. Desroches, Rate-dependent thermo-mechanical modelling of superelastic shape-memory alloys for seismic applications, J. Intell. Mater. Syst. Struct. 19 (2008) 47e61. [103] M.T. Andani, M. Elahinia, A rate dependent tensionetorsion constitutive model for superelastic nitinol under nonproportional loading; a departure from von Mises equivalency, Smart Mater. Struct. 23 (2014) 015012. [104] S. Hashemi, H. Ahmadian, S. Mohammadi, An extended thermo-mechanically coupled algorithm for simulation of superelasticity and shape memory effect in shape memory alloys, Front. Struct. Civ. Eng. 9 (2015) 466e477. [105] M. Peigney, J. Seguin, An incremental variational approach to coupled thermo-mechanical problems in inelastic solids. Application to shape-memory alloys, Int. J. Solid Struct. 50 (2013) 4043e4054. [106] H. Yu, M.L. Young, Three-dimensional modeling for deformation of austenitic NiTi shape memory alloys under high strain rate, Smart Mater. Struct. 27 (2018) 015031. [107] F. Auricchio, E. Boatti, A. Reali, U. Stefanelli, Gradient structures for the thermomechanics of shape-memory materials, Comput. Methods Appl. Mech. Eng. 299 (2016) 440e469. [108] M.A. Iadicola, J.A. Shaw, Rate and thermal sensitivities of unstable transformation behavior in a shape memory alloy, Int. J. Plast. 20 (2004) 577e605.

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CHAPTER 11

SMAs in commercial codes Silvestro Barbarino1, Luca Esposito2 1

Data Analytics, Joby Aviation Inc., Santa Cruz, CA, United States; 2Department of Engineering, University of Campania Luigi Vanvitelli, Aversa (CE), Italy

11.1 Introduction Shape memory alloys (SMAs) have been selected as active elements for developing of several innovative actuators [1e4] because of their compactness and better ratio of power to weight in certain cases compared with traditional ones. The next chapters will discuss some applications. Superelasticity (or pseudoelasticity) and shape memory effect (SME) are the features mainly responsible for these benefits, but also their shortcomings. Challenges associated with simulating highly nonlinear, coupled (thermal, mechanical, and electrical) problems early in the design stage are not negligible and have heavily affected the interest of many industrial companies and limited the adoption of SMAs in commercial applications. Every industrial product, from toys to space rockets, undergoes extensive design, prototyping, and testing before starting production and eventually becoming commercially available. This procedure ensures that all possible design, quality, and cost issues are discovered and solved before the end user gets the final product. The early design stage is becoming increasingly crucial to ensure the realization of a successful product, and the extensive use of finite element (FE) modeling addresses most of these issues, including keeping costs low. Integration of the highly nonlinear, time-dependent, thermomechanical behavior of SMAs has been successful in FE solvers such as COMSOL Multiphysics, ANSYS, Abaqus, and MSC Nastran. This software is usually oriented toward multidisciplinary FE simulations (e.g., a combination of structural, thermal, electric, aerodynamic, and electromagnetic phenomena) and exposes these capabilities to the designer, facilitating the implementation of customized solutions. ANSYS is the only software able to implement both superelastic and SMA behavior. However, even if SMA is not readily available in Nastran as a material, a procedure will be presented in the next sections to integrate SMA modeling within the Nastran solver by means of MSC Marc, stand-alone software for advanced nonlinear computation. Its implementation for the SME is based on Saeedvafa and Asaro’s thermomechanical model.21 Studies can be found in the literature using MSC Marc.22.

Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00011-X

© 2021 Elsevier Ltd. All rights reserved.

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11.2 Superelastic shape memory alloys within SIMULIA abaqus solver Starting from version 6.5, Abaqus provided a built-in user material model to model the behavior of superelastic alloys such as Nitinol accurately. This model, for both Abaqus/ Standard and Abaqus/Explicit, has been well-tested and performs robustly for all applicable elements. The model is based on an additive strain decomposition in which total strain is taken as the sum of elastic strain and transformation strain. Because transformation strains are large compared with typical elastic strains in a metal, the material exhibits typical superelastic behavior. The different elastic constants for austenite and martensite phases are accounted for, and user control of volumetric transformation strains is allowed. Temperature effects are included as well. The material formulation does not incorporate plasticity or the SMEs of these alloys. The model is uniaxial and is based on the work of Auricchio and Taylor [6] and Auricchio et al. [7]. Different behaviors under tension and compression can be specified. The model also allows for user control of volumetric transformation strain in cases for which the user requires different behaviors under tension and compression. If this quantity is not specified, it is assumed to be zero and nonassociated Drucker-Prager type formulation is used. Supported elements include three-dimensional (3D) solids, plane strain, axisymmetric, plane stress, 3D shells, 3D membranes, and 3D beams. The material model can be used with analysis procedures that support mechanical behavior. The following procedures are commonly used in typical applications involving superelastic alloys: *STATIC, *COUPLED TEMPERATURE-DISPLACEMENT, and *DYNAMIC, EXPLICIT. The Nitinol VUMAT contains no nonthread-safe statements such as data, save, and common. Thus, it is safe to use this VUMAT in parallel execution for both thread-based parallelization and message passing interface (mpi)-based parallelization.

11.3 Integration of shape memory alloys within COMSOL Multiphysics solver COMSOL Multiphysics is a versatile solver that combines FE analysis with multiple physics interfaces. Multiple studies can be found in the literature that implement the superelastic effect for SMAs in COMSOL. For example, Thiebaud et al. [4] used a phenomenological model based on the previously developed Raniecki model [8] in COMSOL Multiphysics software to simulate superelastic behavior and the dynamic response of SMAs. The structural mechanics module of the software is coupled with its partial differential equation (PDE) module to solve the weak form of the phase transformation problem. Also, the stiffness and damping properties of the material under offset harmonic strain loading are evaluated using an equivalent complex Young’s modulus approach.

SMAs in commercial codes

Tabesh [9] adopted Tanaka’s model [10] instead to prepare a 3D framework to evaluate the functionality of components made of SMAs. To this end, three application modules (i.e., the structural mechanics, PDE, and heat transfer modules) are tied together to solve the problem. The structural mechanics module solves equations of elasticity with dependent variables of displacement components in Cartesian coordinates. The kinetics of martensite transformation is modeled through the PDE module with a detwinned martensite fraction and twinned martensite fraction in positive and negative axial directions as dependent variables. From this module, transformation strains are obtained and integrated into solid mechanics equations. Also, the heat transfer module is responsible for solving the heat equation and providing the solution as temperature to the SMA PDE module. The approach used by Barbarino et al. [5] dealt with the SME and used COMSOL capability to interface with MATLABÒ for the purpose. In particular, COMSOL Multiphysics can export (and import again) the structural FE model in MATLABÒ workspace as a structured variable (a structural parametric analysis and its geometry must first be created). This FEM structure, as it is called in COMSOL, can then be accessed in MATLABÒ and modified. Functions can be added, each representative of the stresse strain curve of the SMA at a given temperature previously estimated using the SMA model of choice. Once these functions are loaded back in COMSOL, subdomain expressions can be defined that interpolate these functions in the wanted temperature range to be simulated. These expressions, one for each component of the stresses vector, are necessary because of the nonlinear nature of the problem and the NewtoneRaphson iterative method used to achieve convergence (which may require intermediate steps at temperatures not provided as a function, in which no stress-strain curve is defined). Finally, the module that manages the material model (the Equation System subdomain) is modified to consider the new expressions for calculating the stress, based on the subdomain expressions previously described. The analysis is executed on the basis of the temperature parameter, which will vary within the desired range. At each iteration (chosen temperature), the solver will select the appropriate stressestrain curve based on the temperature (or a 2D interpolation of the two closest curves) and determine the effective stress and strain within the SMA elements for that temperature. The direct simulation method UMFPACK is most effective for simulating any structure implementing SMA elements.

11.4 Integration of shape memory alloys within ANSYS solver The ANSYS suite spans the entire range of physics, offering a wide portfolio of engineering solutions. Regardless of the type of simulation, each model can be represented by means of both the classical graphics user interface (GUI) and the ANSYS Parametric Design Language (APDL), a powerful scripting language that provides many conveniences such as parameterization, macros, conditioning and looping, and complex math operations.

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An SMA is a metallic alloy that remembers its original shape. Upon loading and unloading cycles, SMA can undergo large deformation without showing residual strains, the pseudoelasticity effect (PE), also often called superelasticity. Moreover, SMA can recover its original shape through thermal cycles, the SME. Such distinct material behavior results from to the material microstructure in which two different crystallographic structures exist, one characterized by austenite (A) and another by martensite (M). Austenite is the crystallographically more-ordered phase, and martensite is the crystallographically less-ordered phase. The key characteristic of SMA is the occurrence of a martensitic phase transformation. Both effects, pseudoelasticity and shape memory, are available in the ANSYS suite by modeling the material options (accessed defining table via the TB, SMA command). The material option for superelasticity is based on Auricchio et al. [23], in which the material undergoes large deformation without showing permanent deformation under isothermal conditions. The material option for the SME is based on the 3D thermomechanical model for stress-induced solid phase transformations [7,24,25].

11.4.1 Constitutive model for superelasticity From a macroscopic point of view, phase-transformation mechanisms involved in superelastic behavior are described in Fig. 11.1, where sAS and sAS are the starting s f SA and final stress values for forward phase transformation and ss and sSA f are the starting and final stress values for reverse phase transformation, respectively. Moreover, εL is the maximum residual strain and a is a parameter measuring the difference between material responses in tension and compression. To model the superelastic behavior of SMAs, the data table is initialized by using the table TB, SMA command’s SUPE option. Then, when defining the elastic behavior in the austenite state via the MP command, the superelastic SMA option is described by six constants that define stressestrain behavior in loading and unloading for the uniaxial stress-state. For each data set, the temperature is defined by the TBTEMP command. Then, constants C1 through C6 are set via the TBDATA command, as shown in Table 11.1. The following APDL commands are used to define the elastic properties of the austenite phase:

SMAs in commercial codes

V

VAS ∫ VAS s VSA s VSA ∫ HL

H

Figure 11.1 Idealized stressestrain diagram of superelastic behavior. Table 11.1 Superelastic options constants. Constant

Meaning

Property

C1 C2 C3 C4 C5 C6

sAS s sAS f sSA s sSA f εL

Starting stress value for forward phase transformation Final stress value for forward phase transformation Starting stress value for reverse phase transformation Final stress value for reverse phase transformation Maximum residual strain Parameter measuring difference between material responses in tension and compression

a

11.4.2 Constitutive model for shape memory effect The SME is based on a 3D thermomechanical model for stress-induced solid phase transformations, presented in Langbein and Welp [1], Banks [2], and Bokaie [3]. Within the framework of classical irreversible thermodynamics, the model is able to reproduce all of the primary features relative to shape memory materials in a 3D stress state. Fig. 11.2 shows the admissible paths for elastic behavior and phase transformations. The austenite phase is associated with horizontal region abcd (the Parent Phase in Fig. 11.2). Mixtures of phases are related to surface cdef. The martensite phase is

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eT Product Phase e

H

h

s f g

d

a W

b c

R X

M

W + H' M h tr

S–X

Mixture of Phases

Parent Phase

Figure 11.2 Admissible paths for elastic behavior and phase transformations.

represented by horizontal region efgh (the Product Phase in Fig. 11.2). Point c corresponds to the nucleation of the martensite phase. Phase transformations take place only along line cf. Saturated phase transformations are represented by paths along line fg. Horizontal region efgh contains elastic processes except, of course, those on line fg. A backward Euler integration scheme is used to solve the stress update and the consistent tangent stiffness matrix required by the FE solution to obtain a robust nonlinear solution. Because the material tangent stiffness matrix is generally unsymmetric, the unsymmetric NewtoneRaphson option, set by the NROPT, UNSYM command, is suggested to avoid convergence problems. To model the SME behavior of SMAs, the data table is initialized using the TB, SMA command’s MEFF option. Once the elastic behavior in the austenite state is defined via the MP command, the SME option is described by seven constants that define the stresse strain behavior of the material in loading and unloading cycles for the uniaxial stress-state and thermal loading. For each data set, the temperature is set by the TBTEMP command; then, constants C1 through C7 are defined via the TBDATA command.

11.4.3 Supported elements A wide library of elements is able to support SMA material models with the superplasticity option, such as plane, solid, and solid-shell elements for which 3D stress states are applicable (including 3D solid elements, solid-shell elements, 2D plane strain, axisymmetric elements, and solid pipe elements). Moreover, the memory effect option is available with beam, shell, plane, solid, and solid-shell elements (including 3D solid elements, solid-shell elements, 2D plane stress and strain, axisymmetric elements, and solid pipe elements).

SMAs in commercial codes

11.4.4 Results The solution results are available in terms of stress, elastic strains, transformation strains, εtr , expressed as plastic strain EPPL, the ratio of the equivalent transformation strain to maximum transformation strain expressed as part of nonlinear solution record (NL) and processed as component EPEQ of NL, and elastic strain energy density as part of the strain energy density record SEND. 11.4.4.1 Validation example: superelastic effect The structure proposed by Auricchio et al. [-] has been modeled and solved to show the feasibility of the suite in modeling the superelastic behavior of SMA. The obtained results have been compared with those presented by the authors. In detail, a square block of length, height, and width L has been constrained in the X direction on the left face, constrained in the Y direction rear on the bottom face, and constrained in the Z direction on the rear face (3D case only). It has been uniaxially loaded with tensile stress s1 and unloaded on the top face. The stressestrain response for an NiTi alloy has been determined by using by using 2D four-node, 2D eight-node, and 3D eight-node structural solid elements. Fig. 11.3 shows the geometry and boundary conditions of the considered structure. The adopted geometry, loads, materials and SMA properties are listed in Table 11.2. A 2D axisymmetric analysis has been performed first using a single four-node PLANE182 element and then using a single eight-node PLANE183 element. 3D analysis has been then achieved using SOLID185 elements. The simulated stressestrain responses have been compared with the linear model in work by Auricchio et al. [7] and listed in Table 11.3. The obtained results are in agreement with those presented by Auricchio et al. [7] with an average error of about 1%.

D

C

σ1

L

A

B

L

Figure 11.3 Geometry and boundary conditions of the considered structure.

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Table 11.2 Adopted geometry, loads, materials, and shape memory alloy properties. Load

Geometry

Material properties

Shape memory alloy

s1 ¼ 600 MPa

L ¼ 10 mm

E ¼ 60,000 MPa n ¼ 0:3

sAS s ¼ 520 MPa sAS f ¼ 600 MPa sSA s ¼ 300 MPa sSA f ¼ 200 MPa εL ¼ 0.07 a¼0

Table 11.3 Comparison of results. Target

Mechanical ANSYS Parametric Design Language

Ratio

520.00 0.010 600.00 0.080 300.00 0.074 200.00 0.003

522.013 0.010 599.992 0.08 300.016 0.075 197.500 0.003

1.004 1.046 1.000 1.000 1.000 1.013 0.988 1.029

520.00 0.010 600.00 0.080 300.00 0.074 200.00 0.003

522.012 0.010 600.044 0.08 299.993 0.075 197.587 0.003

1.004 1.046 1.000 1.000 1.000 1.013 0.988 1.029

520.00 0.010 600.00 0.080 300.00 0.074 200.00 0.003

522.007 0.010 599.996 0.08 300.0086 0.075 197.689 0.003

1.004 1.046 1.000 1.000 1.000 1.013 0.988 1.029

Results using PLANE182

SIG-SAS EPTO-SAS SIG-FAS EPTO-FAS SIG-SSA EPTO-SSA SIG-FSA EPTO-FSA Results using PLANE183

SIG-SAS EPTO-SAS SIG-FAS EPTO-FAS SIG-SSA EPTO-SSA SIG-FSA EPTO-FSA Results using SOLID185

SIG-SAS EPTO-SAS SIG-FAS EPTO-FAS SIG-SSA EPTO-SSA SIG-FSA EPTO-FSA

SMAs in commercial codes

11.4.5 Validation example: shape memory effect The structure proposed by Souza et al. [24] has been modeled and solved to show the feasibility of the suite for modeling the shape memory behavior of SMA. In detail, a block is composed of SMA material. The block, which has an initial temperature of 253.15 K, has been loaded to 70 MPA to obtain a detwinned martensitic structure, and then unloaded until it was stress-free to obtain a martensitic structure in which residual strain remains. The temperature has been then increased to 259.15 K to recover residual strain and regain the austenitic structure. The final stress and strain have been obtained and compared with the reference solution. Fig. 11.4 shows the geometry and boundary conditions of the considered structure. The adopted geometry, loads, materials, and SMA properties are listed in Table 11.4. The problem has been solved using 3D SOLID185 elements. The shape memory material has been defined using the TB, SMA command. The block has been subjected to 70 MPa uniaxial tension; then, loading has been removed. Finally, the temperature has been increased to recover residual strain. The final stress and strain state have been acquired using the *GET command in the postprocessing section. The simulated stresse strain results are compared with those reported in Souza et al. [24] and are listed in Table 11.5, showing perfect agreement. The results in terms of pressure-displacements are plotted in Fig. 11.5.

11.5 Integration of shape memory alloys within MSC Nastran solver MSC Nastran software [11] does not include a built-in SMA material model. Its programming interface, a direct matrix abstraction program (DMAP), provides the ability to modify Nastran’s prewritten solution sequences or write customized solution

Figure 11.4 Geometry and boundary conditions of the considered structure.

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Table 11.4 Adopted geometry, loads, materials and shape memory alloy properties.

Load

1 Apply pressure in X direction to 70 MPa 2 Remove pressure in X direction 3 Increase temperature to T0 ¼ 253.15 K

Geometry

Material properties austenite phase

Material properties martensite phase

Parameters for phase transformation

W¼5m

E ¼ 70,000 MPa

E ¼ 70,000 MPa

h ¼ 500 MPa

v ¼ 0.33

R ¼ 45 MPa b ¼ 7.5 MPa/K T0 ¼ 253.15 K m¼0

Table 11.5 Results comparison.

S_X EPEL_X EPPL_X

Target

Mechanical ANSYS Parametric Design Language

Ratio

0.0 0.0 0.0

0.0 0.0 0.0

e e e

Figure 11.5 Obtained results in terms of pressure-displacements.

SMAs in commercial codes

sequences to solve specialized problems. DMAP delivers a high-level, highly flexible, powerful programming language that allows users to expand MSC Nastran capabilities by writing their own applications and installing their own custom modules. The downside is that MSC Nastran DMAP has its own programming language (mostly based on Fortran) and grammatical rules, which may represent an obstacle for many end users who are unwilling to invest time in learning a new framework. The procedure presented in this chapter has the advantage of requiring only basic knowledge in Nastran operation and the generation of a text input file. Also, most of the programming and iterative logic is commanded through MathWorks Matlab.30 MATLABÒ is a programming environment for algorithm development, data analysis, visualization, and numerical computation widely used in the scientific community. For the motivations previously discussed, it is difficult to find in the literature a procedure to study and integrate SMA elements with a generic structure simulated in Nastran. The modeling and simulation of SMAs start from the selection of the right model able to predict the phenomenon of interest properly. As mentioned, the focus of this chapter is on SMA for use as actuators: therefore, SME is to be considered. Because no such model is implemented in MSC Nastran FE code, an external routine in MATLABÒ simulates the behavior of the alloy integrated within the surrounding structure. Several constitutive models have been presented in previous chapters and are available in the literature. These models range from 1D (axial to torsional) to 3D and from quasistatic to dynamic (at a low frequency) and are able to capture and predict SMA behavior with different degrees of complexity and accuracy. However, phenomenological models are usually preferred for integration within FE simulations because they use engineering constants or experimentally easily measured quantities to describe the thermomechanical behavior of SMAs. Therefore, they can easily be understood and integrated by the designer within the FE model. The integration procedure presented here assumes Liang and Rogers’ 1D model [12] to predict the SMA behavior, as in the applications presented in the following discussion. However, the general and modular approach adopted here allows for any SMA model to be implemented, as long as the MATLABÒ routine is prepared with the proper inputs and outputs toward the Nastran code. Fig. 11.6 illustrates the iterative scheme at the base of this approach. MSC Nastran FE simulations are set up for a nonlinear thermal static analysis. Although the nonlinear behavior of the SMA elements is not directly implemented in Nastran, SMA actuation induces large displacements in the structure that must be accounted for. Each SMA element is modeled with traditional beam elements that are assigned a negative expansion coefficient and loaded with a unit thermal gradient. Each SMA contracts as phase transition occurs from martensite to austenite, and strain is recovered. SMA activation is simulated by increasingly negative values of the associated

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SMA effecve recoverable strain and acvaon temperature F.E. Model (Thermal)

SMA Model (Liang & Rogers)

SMA applied force (internal stress) vs. strain

Figure 11.6 Simulation scheme adopted for shape memory alloy (SMA) integration within the finite element (F.E.) approach: iterative procedure.

thermal coefficient. To guarantee complete activation of the SMA (complete recovery of the imposed contraction), a large Young’s modulus is considered for those elements (a magnitude larger than other materials involved in the simulation). This last step is necessary because the FE code solves for the elastic equilibrium between the SMA element and the remaining structure, and ensures the expected effects will occur on the structure. Therefore, by artificially increasing the Young’s modulus of the FE elements representative of the SMA actuator, the designer ensures that the Nastran solver applies the complete amount of the associated thermal coefficient (and not a reduced amount because of the elastic equilibrium). The MATLABÒ routine will then implement the SMA model to determine the effective stressestrain behavior of the element representative of the SMA actuator and update its status in the FE model. Several nonlinear analyses are performed on this FE model for increasing values of the thermal coefficient of each SMA element, to relate the forces to be provided by the SMA actuator (or internal stresses) with the displacements transmitted to the structure (in terms of SMA recovered strains). The approach here described uses a Matlab routine to generate several FE models, based on a common input files in which the SMA beam elements can be clearly identified, and then runs them in batch using the Nastran solver. Output files generated by the FE code can then be read and the data imported in MATLABÒ. Therefore, a curve associated with the structural stiffness of the entire FE model may be obtained for each embedded SMA element. This curve (dash-dot curved line in Fig. 11.7) represents the stiffness as seen by each SMA actuator in different positions of the structure. An SMA actuator will experience this stress versus strain resistive behavior from the surrounding structure. As shown in Fig. 11.7 (dotted red [light gray in printed version] curved line), for growing absolute thermal coefficients (shape recovery, and therefore reducing strains), the structure deforms accordingly and the required actuation force increases. Then, each stiffness curve is compared with the SMA stress versus strain behavior for increasing temperatures values, implemented in a MATLABÒ routine according to the adopted SMA model (Liang and Rogers [12] in this example). The SMA element will exhibit growing recoverable strains only until the complete transformation into austenite

SMAs in commercial codes

Figure 11.7 Simulation scheme adopted for shape memory alloy integration within the finite element approach: graphical representation.

is attained and equilibrium is reached with the surrounding structure. This point is represented by the intersection of the structural stiffness curve (dotted red [light gray in printed version] curved line) with the SMA constitutive material law curve at high temperature (dotted curve). For complete actuation, the intersection must happen in the linear region of austenite phase (Fig. 11.7). This point is the working condition of the selected SMA actuator. Each SMA element embedded in the structure can be subjected to different temperature and stresse strain levels, and have a different working condition. However, among all possible SMA curves at different temperatures, the MATLABÒ routine has to find the best combination between complete actuation and the minimum required activation temperature. This is an efficiency requirement that translates into the minimum necessary thermal energy to be supplied to each SMA actuator to achieve its maximum effective recoverable strain. This best condition corresponds to the intersection point taking place exactly at the austenite finish stress level (upper end of the linear curve in austenite phase during unloading), as shown in Fig. 11.7.

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Finally, the thermal coefficient of each SMA element in the FE simulation is updated to meet the solution found by this procedure. At this point, the iteration is concluded and the MATLABÒ routine can generate and run MSC Nastran to solve the final FE simulation. From a graphical point of view, increasing the temperature makes the SMA material curve move up in the stressestrain plane and the working point shifts left, over the lower boundary of the SMA material curve (austenite transformation). By cooling the alloy, the process is reversed under the action of the structural recalling force, and the SMA transforms again into martensite. In this case, the working point runs over the higher part of material curve (martensite transformation). In the loading or recovery phase, the transformation may be then said to be temperature or stress driven, respectively. To understand the proposed approach fully, it is beneficial to recall some concepts. The use of SMA elements as actuators is based on the SME. The SME can take place if (1) the SMA has been prestressed and a recoverable strain is present; (2) a temperature gradient is applied to induce the martensite to austenite phase change; and (3) to achieve cyclic actuation, the prestress described in (1) must be applied again to the SMA elements upon cooling. Therefore, the main FE simulation starts with SMA elements already prestrained (those that possess the strain that will be recovered during actuation). If in this initial condition the SMA is in a stress-free state, no full cyclic actuation is possible. Indeed, upon cooling, each SMA element will be only partially prestrained, depending on the structural stiffness and regardless of the prestrain available at the first cycle. This is the typical case that occurs when the FE structural model with embedded SMA elements is designed starting from its initial stress-free configuration. For the process to be fully cyclic, the initial condition point should be moved at a prestressed level featuring a complete martensite state (martensite finish stress level), shown as a second intersection point between the structural stiffness (dash-dot curved line) and the SMA constitutive law at an environmental temperature (solid curve) in Fig. 11.7. This ensures that a complete martensiteeaustenite transformation is accessible both ways upon heating and cooling. This also shows that a fully cyclic actuation necessarily depends on temperature, in particular because of the martensite finish level dependence on the environmental temperature. In a fully cyclic SMA simulation, the effective recoverable strain of each SMA actuator is given by the difference in strain levels between the prestrain state and the complete actuation state. From the FE simulation perspective, the real challenge is to design the desired structural model with SMA actuators that have the not only desired initial prestrain but also the required initial prestress to enable fully cyclic actuation. Therefore, the structure with embedded SMA elements must be derived in a new stress-free configuration (the one to use as input for the FE solver), which, once the embedded SMA elements are preloaded at the martensite finish stress level, is able to

SMAs in commercial codes

achieve a new equilibrium condition. This new equilibrium condition corresponds to the real initial configuration from which SMA activation produces further deformation. For the designer to estimate the stress levels correctly within the SMA elements and the entire structure, this is the only viable solution. The addition of artificial prestress to the SMA elements only in the FE model does not consider the overall increase in stress in the entire surrounding structure, producing an underestimation of the stress levels of non-SMA components. The estimate of the structural geometry needed for this new initial, stress-free configuration can be challenging in itself, depending on the structural stiffness and complexity, and can require a prior iterative design. Then, the SMA elements themselves can be used in the initial FE simulation to produce the desired level of prestress and the new equilibrium configuration. This can be done using the method just discussed, based on an initial thermal coefficient for each SMA element. Therefore, the presence of this artificial predeformation in the overall FE structure will show as an artificial, additional prestrain in the SMA actuators. Referring to Fig. 11.7, the effective recoverable strain is approximately 3% (the difference in strain levels between the initial and final operating conditions), but an additional 2% is initially needed to prestress the SMA element. Overall, by implementing this procedure, it is possible to estimate the mechanical behavior of any 1D SMA element working as an actuator, complete activation temperature, and subsequent stressestrain levels within the structure owing to actuation.

11.6 Applications Three sample applications are presented in this section. The first two applications describe the use of SMAs as actuators of larger structures. In some cases, the integration of SMAs as both actuation and structural load-bearing elements is also presented. The examples discussed here all use the 1D model of Liang and Rogers’ [12] to predict the thermomechanical behavior of SMAs. Strictly speaking, Liang and Rogers’ model cannot predict the SME because of the lack of separate modeling of the twinned (temperature-induced) and detwinned (stress-induced) martensite phases. Such a capability is typical of Brinson’s 1D model [13], which is otherwise similar to Liang and Rogers’ model. However, in these applications, the SMA is already initially prestressed and the focus is on its high-temperature behavior, to estimate how much of its recoverable strain can be used for actuation. Liang and Rogers’ model is sufficient in those cases and proved accurate enough against experimental tests. Another important aspect is related to the model’s assumptions, which must be consistent with the working modality of the actuators: 1D behavior of the material (strain recovery only along the axial direction) and no bending actions (usually this is ensured by hinge connections between the SMA element and the surrounding structure). The 1D

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SMA elements to be integrated within the structure are modeled as beam elements. Applications will be shown using the MSC Nastran solver and the ad hoc procedure previously described. The third application concerns the modeling of an SMA spring using ANSYS software. Also in this case, the spring can be used as an actuator. Attanasi and Auricchio [26] proposed a seismic isolation device based on superelastic material components manufactured using SMA, consisting of a flat sliding bearing and a superelastic material lateral restraining system manufactured using SMA helical spring components. Here, the shape memory behavior of the spring has been analyzed under prescribed displacements.

11.6.1 Aeronautical stiffened panels This application is a good example of validating the proposed approach for a simple structure without initial prestress. It was presented by Pecora et al. [14] and Barbarino et al. [15]. The objective of this work was to evaluate the actuation performance of aeronautical skin panels by integrating SMA wires and ribbons, assuming a fixed maximum recoverable deformation of 3.3% in length for each element. Four different configurations were introduced, based on Al 7075T6 aluminum panels (500  300  1.0 mm in size) and were constrained on one shorter side while being free on the others. Also, two configurations underwent genetic optimization. Performance was evaluated in terms of vertical displacement and rotation of the free edge of the panel, and the complete activation temperature of the SMA actuators. Details are available elsewhere and will not be discussed here. Instead, a single configuration will be discussed in detail (defined as Architecture 1 in the publications), with particular a focus on the simulation approach. This solution, as depicted in Fig. 11.8, is composed of three couples of SMA ribbons connected to stiffeners running in the transverse direction. The SMA ribbons are independent: six ribbons are displaced in three couples to improve the uniformity of actuation of the panel. The left edge in Fig. 11.8 is constrained. By properly selecting the SMAs to be activated, this solution is capable of multiple stable configurations. In the publications, all of the symmetrical combinations (with ribbons within same couple always in the same state) of SMA activation have been considered. For simplicity, only one case will be discussed here, with the middle couple (C2 in Fig. 11.8) of SMA ribbons activated and the others passive. The procedure can be then repeated for each couple of SMAs and possible combinations. Because of the symmetrical structure, SMA ribbons within the same couple are subject to the same stress levels upon activation, and their behavior can be investigated as a single actuator. Instead, different SMA couples are subject to different conditions. Fig. 11.9 shows details of the geometry and mesh of the FE structural model to be solved. In detail, the panel and stringers are modeled as surfaces and assigned shell elements; the SMAs are instead modeled as curves and assigned beam elements.

SMAs in commercial codes

Figure 11.8 Graphical representation of Architecture 1 configuration.

Figure 11.9 Finite element model for Architecture 1 configuration. SMA, shape memory alloy.

Also, SMA elements are connected through hinges to the stringers (a rotational depth of field is allowed in the direction parallel to the stringers) and the left edge has a fixed boundary condition. Each couple of SMA ribbons is assigned a specific material property. For the purpose of this example, only the central couple of elements will be activated; therefore, it is associated with a different material. Indeed, only the material associated with this active couple of SMAs will have a variable thermal coefficient, which will be instead zero for the others (Fig. 11.10). The basic FE model for the proposed structure embedding SMA actuators is therefore ready, and the iterative procedure previously described can be applied. The simulation is performed as a nonlinear thermal analysis with an applied unitary thermal load to simulate SMA element activation (only C2). Several runs are executed to increase absolute values

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Figure 11.10 MSC Nastran input file for Architecture 1 configuration: shape memory alloy modeling details.

of the thermal coefficient, up to the maximum recoverable strain of 3.3% in length. Next, analyzing the output files generated by Nastran solver, the transmitted load deformation of the actuator curve can be obtained, which is equivalent to the structural rigidity of the architecture. The MATLABÒ routine can then estimate the minimum complete activation temperature for the SMA ribbons composing couple C2 and the effective recoverable strain, as illustrated in Fig. 11.11. Finally, the FE model can be updated with the effective recoverable strain applied in terms of the thermal coefficient to the SMA ribbons of couple C2 and executed by the Nastran solver. Actual performance can then be estimated for this particular case, as shown in Fig. 11.12. The same procedure can be applied for the other SMA couples and all possible activation combinations. Further details and validation against experimental tests will not be reported here but are available in the full publications [14,15].

11.6.2 Airfoil variable camber trailing edge An application for a morphing wing trailing edge, initially presented by Barbarino et al., is discussed here [16]. Morphing capability was introduced to replace a conventional flap device. A compliant rib structure was designed based on SMA actuators able to sustain external aerodynamic loads and simultaneously allow controlled wing shape modification.

SMAs in commercial codes

Figure 11.11 Shape memory alloy (SMA) behavior estimated by Liang and Rogers model implemented in MATLABÒ.

In particular, the rib is composed of different plates interconnected through titanium spring plates. These are in charge of providing enough rigidity to the structure, ensuring the required preload for SMA recovery upon cooling and allowing for large relative rotations (X springs exhibit nonlinear behavior because of the variation in position of the rotation pivot during deflection). The active elements are SMA wires located under the springs connecting two consecutive rib plates. Upon activation, their contraction generates a momentum action around the springs, with consequent induced rotations of the rib parts and camber variations. A single hinge is represented in Fig. 11.13. An optimization study has been performed for each of the elastic hinges by considering parameters such as the springs’ properties, their angular position’ and their location on the rib. The SMA actuator mechanical properties and geometry have been taken into account as well. Also, the effect of external aerodynamic loads has been considered. More details can be found in the full publication. This work expands the previous example by considering the need for initial prestress, critical to achieve cyclic actuation. Therefore, it discusses a case in which an initial unloaded geometrical configuration was estimated that, once prestressed, reached the actual neutral condition. Further deformation is then possible up to the fully actuated state. Actuation performance is calculated in terms of trailing edge-induced displacement as a difference between the latter two states. The introduction of an initial prestress in the structure, and therefore in the embedded SMA actuators, poses some challenges. Prestress required for cyclic use strictly depends on the selected SMA actuator and desired initial prestrain. For a complete austenitemartensite phase transformation, which maximizes the SMA actuation capabilities, prestrain must be set at least at the martensite finish value; otherwise, partial loops are

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400 Shape Memory Alloy Engineering

Figure 11.12 Final FE simulation of proposed configuration with SMA ribbons recovering the effective strain (shading is representative of total displacement). (a) Architecture 1 side view. (b) Architecture 1 isometric view.

SMAs in commercial codes

Figure 11.13 Single elastic hinge scheme adopted for parametric study. SMA, shape memory alloy.

possible. Even when assuming a specific SMA, the prestress value corresponding to the martensite finish stress level is temperature dependent. The lowest working temperature, usually the environmental one, can be chosen for the purpose. Once the necessary prestress level is known, the required preload that the surrounding structure has to apply to the SMA element (to achieve such a prestress) depends on the chosen cross-section of the actuator itself. In turn, this choice depends on how stiff the overall structure is that the SMA has to actuate, and ultimately on how much actuation strain (capability) is needed. Therefore, the choice of a suitable SMA element to actuate a certain portion of the structure should be established in advance. Choice may be restricted by the commercial availability of a specific SMA type, and in this case the surrounding structure must be designed to be able to work with the available actuator. On the other hand, as proposed in the article, a single representative actuation unit can be parametrically studied for simplicity. The investigation of the overall structure with multiple SMA actuators is still needed (because the stressestrain behavior may be different for each and external loads may be present), but the designer can get a better understanding of the dependencies involved and iteratively achieve the final working design. The final challenge associated with the need to prestress the SMA elements to allow for fully cyclic actuation is related to the FE simulation of the entire structure. This involves determining a new stress-free configuration to be used as input for the FE model, characterized by an initial deformed geometry. The initial state must be determined so that in a prestressed but not activated condition, the required recovery stress field is present within the elastic elements and SMAs. The SMA elements can be used by introducing an artificial initial thermal coefficient to simulate prestress and achieve this new equilibrium state. Then, the effective recoverable strain to accomplish complete activation is summed as an additional quantity to the thermal coefficient associated with the SMA element.

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The designer must be careful in scaling the effective recoverable strain to be assigned to the selected SMA element in terms of an additional thermal coefficient. In the initial, stress-free state, the FE geometry of the structure corresponds to what would look like for a built prototype able to be achieve the neutral configuration, and then the actuated one. However, this is not true for the SMA elements. The FE model of the stress-free configuration will need longer SMA elements to be able to interconnect several parts of the structure before any stress is applied. As previously explained, an artificial thermal coefficient allows the equilibrium state to be reached. Only at this point (equilibrium state) will the SMA length correspond to a real SMA being prestrained. Therefore, the effective recoverable strain calculated by the constitutive law implemented in the MATLABÒ routine will assume this as the initial, prestrained length. Instead, the FE model in MSC Nastran will assume as initial length of the SMA the one in the stress-free configuration, and the thermal coefficient is calculated according to this length. A suitable scaling rule must be considered when exchanging between the two lengths. For the proposed example, Fig. 11.14 illustrates these three different configurations.

Figure 11.14 Rib main states: (a) neither active nor preloaded shape memory alloy, (b) neutral position, (c) rib completely actuated.

SMAs in commercial codes

In this application, the stress-free configuration can easily be determined by rotating the rigid rib plates around each elastic hinge a given amount in the opposite direction of the expected actuation. In other structural designs, the initial geometry to be used as input for the FE solver may be more difficult to estimate. Moreover, although the stress-free and neutral conditions are unique, the actuation configuration presented here assumes activation of all embedded SMA elements. Different intermediate states between neutral and totally extended configurations are possible, according to the SMA elements considered active. Following the same procedure previously presented in this chapter, each SMA element is responsible for the actuation of a different portion of the compliant rib and undergoes a different stressestrain behavior. Therefore, the iterative procedure to estimate the effective recoverable strain and minimum complete activation temperature must be repeated for each embedded SMA. Finally, it must be verified that the surrounding structure is able to deform from the stress-free state to equilibrium and finally to the actuated state without an excessive increase in internal material stress levels or failure. Also, fatigue life may need to be investigated because of the large strain induced in the structure by the SMAs.The approach presented in this book is able to integrate and predict the behavior of SMA elements undergoing SME embedded within a generic structure simulated in MSC Nastran FE code. The iterative procedure and general considerations discussed here have been used also for other works. In particular, several aeronautical applications [17e22] focused on the design of a variable camber trailing edge or morphing flap based on SMA actuation.

11.6.3 Shape memory alloy spring In the following example, a helical spring subject to prescribed displacement has been considered and analyzed by means of ANSYS software. The adopted geometrical parameters, loads, and material properties are listed in Table 11.6. A standard hexahedral element with eight nodes (3 degrees of freedom for each node and linear shape functions) has been used. To model the behavior of SMAs, the data table is initialized by using the table TB, SMA command’s MEFF option. Then, when the elastic behavior is defined in the austenite state via the MP command, the superelastic SMA option is described by six constants that define the stressestrain behavior in loading and unloading for the uniaxial stress-state. For each data set, the temperature is defined by the TBTEMP command. Then, constants C1 through C6 are set via the APDL TBDATA command:

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Table 11.6 Adopted geometrical parameters, loads, and material properties. Geometry

Radius [mm] Pitch [mml Wire diameter Number of coils

7 7 0.75 5 Load

Time Prescribed displacement Uz [mm] Body uniform temperature T [K]

0 0 253.15

1 6 253.15

2 0 253.15

3 0 259.15

Material properties

Young’s modulus Poisson coefficient Austenite modulus Hardening Reference Elastic limit Maximum Martensite modulus

[MPa] [MPa] [MPa] [K] [MPa] [MPa]

70,000 0.33 70,000 500 253.15 45 0.03 70,000

!SMA: material EY=70E3 !Young Modulus [Mpa] ni=.33 !Poisson coefficient MP,EX,1,70E3 !Austenite Modulus [MPa] MP,PRXY,1,0.33 !Poisson’s Coefficient C1=500 !Hardening Parameter [MPa] C2=253.15 !Reference Temperature [K] C3=45 !Elastic Limit [MPA] C4=7.5 ![MPa] C5=0.03 !Max Transformation Strain C6=70E3 !Martensite Modulus [MPA] C7=0 !Symmetrical Behavior, M = 0 TB,SMA,1,,7,MEFF TBDATA,1,C1,C2,C3,C4,C5,C6,C7

In Fig. 11.15, the geometry (left) and the mesh (center) are shown, with a detail of the section of the wire (right) to demonstrate the effectiveness of the adopted discretization. The results obtained in terms of force versus applied displacements are shown in Fig. 11.16, in which point A corresponds to the change in phase of the material, point B (time ¼ 1 in Table 11.6) to unloading, and point C (time ¼ 2 in Table 11.6) to application of the recovery temperature (time ¼ 3 in Table 11.6).

SMAs in commercial codes

Figure 11.15 Geometry (left), mesh (center), and a detail of the section of the wire (right).

Figure 11.16 Force versus applied displacements.

Moreover, the results in terms of deformed shape (left) and plastic strain in the wire (center), and in the section of the wire (right) are shown in Fig. 11.17 with reference to point A (upper left), B (upper right), C (lower left) and the origin of the reference system (lower right), respectively. As expected, ANSYS software is able to reproduce all of the features relative to shape memory materials. In particular, the obtained results highlight the change in phase of the

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Figure 11.17 Deformed shape (left) and plastic strain in the wire (center), and in the section of the wire (right) with reference to point a (upper left), b (upper right), c (lower left) and the origin of the reference system d (lower right), respectively.

material from point A in Fig. 11.16, corresponding to a reduction in the material stiffness. Moreover, at the end of unloading, a plastic strain remains (point C). Finally, recovery of the spring shape is obtained owing to the increase in the applied temperature, with a related plastic strain two orders lower.

11.7 Conclusions Integration of SMAs within commercial codes was presented in this chapter. Procedures for COMSOL Multiphysics, ANSYS, and MSC Nastran were discussed. Abaqus and MSC Marc were also briefly considered. Integration with MSC Nastran was suggested and discussed in detail. Some applications were then illustrated to clarify the proposed approach. An example of the implementation of SMA was presented in an ANSYS environment.

Bibliography [1] S. Langbein, E.G. Welp, One-module actuators based on partial activation of shape memory components, J. Mater. Eng. Perform. 18 (5-6) (2009) 711e716. [2] R.M. Banks, Energy Conversion Systems, US Patent US03913326, 1975. [3] Bokaie M. “Latch-Release Pin Puller with Shape-Memory-Alloy Actuator,” website: http://www. nasatech.com/Briefs/Feb98/LEW16511.html, printed April 17, 2003.

SMAs in commercial codes

[4] F. Thiebaud, M. Collet, E. Foltete, C. Lexcellent, Implementation of a multi-axial pseudoelastic model in FEMLAB to predict dynamical behaviour of shape memory alloys, in: Proceedings of the COMSOL Conference; 2005. Paris, France, 2005. [5] S. Barbarino, S. Ameduri, L. Lecce, A. Concilio, Wing shape control through an SMA-based device, J. Intell. Mater. Syst. Struct. 20 (3) (2009), https://doi.org/10.1177/1045389X08093825, 283e296. [6] F. Auricchio, R.L. Taylor, Shape-memory alloys: modelling and numerical simulations of the finitestrain superelastic behavior, Comput. Methods Appl. Mech. Eng. 143 (1997), 175e194. [7] F. Auricchio, R.L. Taylor, J. Lubliner, Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior, Comput. Methods Appl. Mech. Eng. 146 (1997), 281e312. [8] B. Raniecki, C. Lexcellent, K. Tanaka, Thermodynamic models of pseudoelastic behaviour of shape memory alloys, Arch. Mech. 44 (1992) 261. [9] M. Tabesh, Finite Element Analysis of Shape Memory Alloy Biomedical Devices (Master’s thesis), University of Toledo, May 2010. [10] K. Tanaka, A thermomechanical sketch of shape memory effect: one-dimensional tensile behaviour, Res. Mech. 18 (3) (1986), 251e263. [11] MSC Software Corporation. Website: http://www.mscsoftware.com. [12] C. Liang, C.A. Rogers, One-dimensional thermo-mechanical constitutive relations for shape memory material, J. Intell. Mater. Syst. Struct. 1 (2) (1990), 207e234. [13] L.C. Brinson, One-dimensional constitutive behaviour of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable, J. Intell. Mater. Syst. Struct. 4 (2) (1993), 229e242. [14] R. Pecora, L. Lecce, M. Riccio, A. Concilio, Numerical optimization of wing skin panels actuation based-on SMA. XIX AIDAA national congress, September 17e20, in: Forlì (FC), Italy, Conference proceedings on CD, article N.232, p.10, 2007. [15] S. Barbarino, L. Lecce, E. Calvi, S. Ameduri, Numerical and experimental investigation on S.M.A. Actuated wing skin panels. XIX AIDAA National Congress, September 17e20, in: Forlì (FC), Italy, Conference proceedings on CD, article N.231, p.10, 2007. [16] S. Barbarino, R. Pecora, L. Lecce, A. Concilio, S. Ameduri, E. Calvi, A novel SMA-based concept for airfoil structural morphing, J. Mater. Eng. Perform. 18 (5) (2009) 696e705, https://doi.org/10.1007/ s11665-009-9356-3. [17] S. Barbarino, S. Ameduri and R. Pecora, Wing camber control architectures based on SMA: numerical investigation, In: International conference on smart materials and nanotechnology in engineering (SMN2007), July 1e4, 2007, Harbin (China). Proceedings of SPIE, vol. 6423; 2007. p. 8, https:// doi.org/10.1117/12.779397. 64231E-1. [18] S. Barbarino, S. Ameduri, R. Pecora, L. Lecce, A. Concilio, Airfoil morphing architecture based on shape memory alloys, in: Conference on smart materials, adaptive structures & intelligent systems (SMASIS08); October 28e30, 2008. p. 9. Ellicott City (Maryland, USA), Conference proceedings on CD, 2008 paperSMASIS2008-480. [19] S. Barbarino, S. Ameduri, R. Pecora, L. Lecce, A. Concilio, Design of an actuation architecture based on SMA technology for wing shape control, in: Actuator 2008 conference; June 9e11, 2008. pp. 961e964. Bremen (Germany), Conference proceedings and on CD, poster P144, 2008. [20] S. Barbarino, W.G. Dettmer, M.I. Friswell, Morphing trailing edges with shape memory alloy rods, in: Proceedings of 21th international conference on adaptive structures and technologies (ICAST 2010); October 4e6, 2010. State College (PA), 2010. [21] S. Barbarino, R. Pecora, L. Lecce, A. Concilio, S. Ameduri, L. De Rosa, Airfoil structural morphing based on S.M.A. actuator series: numerical and experimental studies, J. Intell. Mater. Syst. Struct. 22 (10) (2011), https://doi.org/10.1177/1045389X11416032, 987e1004. [22] R. Pecora, S. Barbarino, A. Concilio, L. Lecce, S. Russo, Design and functional test of a morphing highlift device for a regional aircraft, J. Intell. Mater. Syst. Struct. 22 (10) (2011) 1005e1023, https://doi.org/10.1177/1045389X11414083. [23] F.A. Auricchio, Robust integration-algorithm for a finite-strain shape-memory-alloy, Int. J. Plast. 17 (2001) 971e990.

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[24] A.C. Souza, E.N. Mamiya, N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations, Eur. J. Mech. Solid. 17 (1998) 789e806. [25] F. Auricchio, F. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Int. J. Numer. Methods Eng. 55 (2005) 1255e1284. [26] G. Attanasi, F. Auricchio, Innovative superelastic isolation device, J. Earthq. Eng. 15 (S1) (2011) 72e89.

Further reading [1] L.F. Campanile, T. Melz, R. Keimer, W. Wadehn, Actuator for Producing Controlled Surface Structure Deformation Has Flexible Kinetic Arrangement with Stiffened Supporting Surfaces on Upper, Lower Sides Normal to Preferred Direction, EU Patent DE10155119A1, 2003. [2] Q. Li, S. Seelecke, Thermo-mechanically coupled analysis of shape memory actuators, in: Proceedings of the COMSOL Users Conference; 2007. Boston, MA, 2007. [3] S. Shrivastava, Simulation for thermomechanical behavior of shape memory alloy (SMA) using COMSOL multiphysics, in: Proceedings of the COMSOL Users Conference; 2006. Bangalore, India, 2006. [4] M. Collet, Modeling implementation of smart materials such as shape memory alloys and electro-active metamaterials, in: Proceedings of the COMSOL Conference; 2008. Hannover, Germany, 2008. [5] S. Yang, S. Seelecke, Modeling and analysis of SMA-based adaptive structures, in: Proceedings of the COMSOL Conference; 2008. Boston, MA, 2008. [6] Barrett, P.R., Fridline, D. User Implemented Nitinol Material Model in ANSYS.” Website: http:// www.caeai.com/all-downloads.php. [7] Barrett, P.R., Cunningham, P. Super elastic alloy eyeglass frame design using the ANSYS workbench environment. Website: http://www.caeai.com/all-downloads.php. [8] P. Terriault, F. Viens, V. Brailovski, Non-isothermal finite element modeling of a shape memory alloy actuator using ANSYS, Comput. Mater. Sci. 36 (4) (2006), 397e410. [9] M. Sreekumar, T. Nagarajan, M. Singaperumal, Modelling and simulation of a novel shape memory alloy actuated compliant parallel manipulator, J. Mech. Eng. Sci. 222 (6) (2008), 1049e1059. [10] Han, L.H., Lu, TJ. 3D finite element simulation for shape memory alloys. In: Yang, W. (Ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials. ISBN 978-1-402-04130-6, pp. 227e236. [11] F. Richter, O. Kastner, G. Eggeler, Implementation of the MVullereAchenbacheSeelecke model for shape memory alloys in ABAQUS, J. Mater. Eng. Perform. 18 (5e6) (2009), 626e630. [12] Y. Chemisky, A. Duval, B. Piotrowski, T. Ben-Zineb, E. Patoor, Numerical tool based on finite element method for SMA structures design, in: Proceedings of ASME 2008 conference on smart materials, adaptative structures & intelligent systems (SMASIS), October 28e30, Turf Valley Resort, Ellicott City, MD, paper SMASIS08-485, 2008. [13] X.-Y. Gong, A.R. Pelton, T.W. Duerig, N. Rebelo, K. Perry, Finite element analysis and experimental evaluation of superelastic nitinol stent, in: A.R. Pelton, T.W. Duerig (Eds.), Proceedings of the international conference on shape memory and superelastic technologies (SMST-03), May 5e8, Pacific Grove, California, CA, 2001, 443e451. [14] N. Rebelo, N. Walker, H. Foadian, Simulation of implantable nitinol stents, in: Abaqus Users’ conference proceedings, vol. 143; 2001. 421e434, 2001. [15] F. Auricchio, M. Conti, S. Marconi, A. Reali, J.L. Tolenaar, S. Trimarchi, Patient-specific aortic endografting simulation: from diagnosis to prediction, Comput. Biol. Med. 43 (2013) 386-94. [16] M. Saeedvafa, R.J. Asaro, LA-UR-95e482, Los Alamos report, 1995. [17] Choudhry, S., Yoon, J.W. A general thermo-mechanical shape memory alloy model: formulation and applications. Materials processing and design: modeling, simulation and applicationsdNUMIFORM 2004d Proceedings of the 8th international conference on numerical methods in industrial forming processes. AIP Conference Proceedings, 712, pp. 1589e1594, Columbus (OH). https://doi.org/ 10.1063/1.1766756.

SMAs in commercial codes

[18] G. Lin, ANSYS User Material Subroutine USERMAT Standard Package, ANSYS Inc., Canonsburg (PA), 1999. [19] MathWorks Inc. Website: http://www.mathworks.com. [20] F. Auricchio, D. Fugazza, R. DesRoches, Numerical and experimental evaluation of the damping properties of shape-memory alloys, J. Eng. Mater. Technol. 128 (3) (2006) 312e319.

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SECTION 4

Actuators Editor:

Eugenio Dragoni University of Modena and Reggio Emilia, Reggio Emilia, Italy

List of chapters 12. 13. 14.

Design and development of advanced SMA actuators Design and industrial manufacturing of shape memory alloy components Design of SMA-based structural actuators

CHAPTER 12

Design and development of advanced SMA actuators Eugenio Dragoni, Andrea Spaggiari University of Modena and Reggio Emilia, Reggio Emilia, Italy

12.1 Introduction Shape memory alloys (SMAs) are widely used for miniactuation and microactuation owing to the outstanding power densities they can develop compared with traditional drives (Fig. 12.1(a)) [1]. In contrast, conventional actuators outperform shape memory technologies in the range of medium- and high-output strokes and forces. When loaded under tension in the form of straight wires, SMA actuators can generate considerable force, but the useful stroke is a tiny fraction (3e5%) of their length [2] (top left corner in Fig. 12.1(b)). When wound as helical springs, SMA wires contribute to much bigger strokes but at the expense of the output force [3e5] (bottom right corner in Fig. 12.1(b)). This chapter aims to overcome these limitations by proposing SMA actuators based on either the use of smart springs (conventionally shaped springs made of SMA) or the application of smart designs (conventional SMA wires used unconventionally). In both cases, the devices are developed for maximum force capacity in a compact package, so as to move the working point toward the top right corner of the plot shown in Fig. 12.1(b). To be helpful to the everyday engineering designer, the proposed approach to developing high-performance actuators is based on simple, yet test-proven assumptions concerning the material behavior. As shown in Fig. 12.2, the true complex stressestrain response of the martensite (low-temperature) and austenite (high-temperature) states of the SMA (Fig. 12.2(a)) is replaced by straight lines (Fig. 12.2(b)). In the simplified model, the austenite line emanates from the origin with slope Ea (austenitic modulus) whereas the martensite line starts from a small offset stress, s0, with slope Em (martensitic modulus). Typical values for these material parameters are Ea ¼ 20e40 GPa, s0 ¼ 25e35 MPa, and Em ¼ 0.9e1.1 GPa. Because the offset stress s0 is small, further simplification of the diagrams consists of the austenite and martensite lines radiating from the origin with the material behavior completely defined by only the moduli Ea and Em. Although extremely simple, these assumptions are test-proven and allow realistic and quantitative models of the actuators to be achieved, which are extremely useful for developing and optimizing devices. All the actuator designs described in this chapter are based on either of these linearized material models, irrespective of the particular method used to Shape Memory Alloy Engineering, Second Edition ISBN 978-0-12-819264-1, https://doi.org/10.1016/B978-0-12-819264-1.00012-1

© 2021 Elsevier Ltd. All rights reserved.

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Figure 12.1 (a) Comparison of shape memory alloy (SMA) actuators with alternative technologies in terms of specific power and overall mass. (b) Typical behavior of specific force versus specific stroke for SMA actuators.

Figure 12.2 (a) True stressestrain behavior of a shape memory alloy wire (hot state) and (b) elementary approximation with basic material parameters (cold state).

change the temperature of the alloy (electrical current dissipation, environmental heating, or radiation) [6]. To accomplish a meaningful design, it is also important to consider the fatigue response of SMAs under both strain-based and stress-based cycles [7e9]. The effect of stress, strain, heating rates, and loading conditions heavily influences the material properties [10e12], and therefore the most suitable condition should be tailored for each application considered. This chapter is organized into two main sections (12.3 and 12.4). The first main section (12.3) summarizes the working principle of an SMA actuator on switching from low to high temperature and emphasizes the role of the backup element needed to restore the shape of the actuator upon cooling. The comparison of the efficiency of the classical

Design and development of advanced SMA actuators

backup element will foster the merits of an alternative backup concept, explained later, based on the principle of elastic compensation. The second main section (12.4) describes in detail the concept, design, simulation, fabrication, and experimental characterization of advanced, high-performance SMA actuators. The family of actuators presented is divided into two subgroups. The first capitalizes on the smart use of SMA springs with classical geometry and the second capitalizes on the smart design of nonconventional actuators. In both cases, the aim is to obtain engineering devices with high output in terms of both force and stroke.

12.2 List of symbols a, c Characteristic dimensions A Austenite transformation temperature, area b Width C Spring index d, D Diameters E Young’s modulus f Frequency, coefficient of friction, deflection F Force G Shear modulus h Height, convection coefficient H Height I Electrical current k Spring rate K Spring rate, empirical coefficient l, L Length m Mass M Martensite transformation temperature n Number of coils N Number of waves p Pressure, prestretch P Load, power Q Tangential force R Radius, electrical resistance s Dimensionless parameter S Stroke t Time, thickness T Temperature, torque u Displacement V Volume x, Dx Position, incremental position (stroke) X Generic spring property, enthalpy z Stress ratio Z Number of turns a, b Angle

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g d ε l m n x s s f

Slenderness ratio Spring deflection Strain Slenderness ratio Surface/mass ratio Poisson’s ratio Density, void ratio Normal stress Shear stress Winding angle

12.3 Classical backup systems Although two-way SMAs exist that can be toggled between two different shapes by acting only on the temperature, high-performance actuators invariably rely on oneway shape memory material. For a one-way SMA to be used as an actuator, the active shape memory element must be coupled with a contrasting backup element (dashed line in Fig. 12.2), which serves to restore the deformation of the active element upon cooling (Point A in Fig. 12.2(b)). Without a backup element, the active SMA element would remain permanently in the memorized shape achieved at the end of the first heating step (point B in Fig. 12.2(b)). The backup force can be exerted on the active shape memory elements in three classical ways (Fig. 12.3): by means of a constant force (Fig. 12.3(a)), using an elastic spring (Fig. 12.3(b)), and through a second shape memory spring (Fig. 12.3(c)). The backup constant force can be supplied by a weight (as in Fig. 12.3(a)), by a constant-pressure pneumatic actuator, or by a constant-force spring [13e17]. In the case of two antagonistic SMA springs (Fig. 12.3(c)), the primary and backup springs are activated alternatively so that the resisting force exerted by the passive cold spring is reduced to a minimum. Fig. 12.4 compares the efficiency of the three backup systems in terms of stroke for a given output force, F, assumed to be constant for both directions of motion of the

Figure 12.3 Classical backup systems for shape memory alloy (SMA)-based actuators: (a) constant force, (b) elastic spring, and (c) secondary SMA spring.

Design and development of advanced SMA actuators

(a)

(b)

Backup

(c)

(d)

Figure 12.4 Stroke performance for given output force of SMA actuators with various backup systems: elastic spring (a), constant force (b), secondary SMA spring (c), elastic compensation (d).

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actuator. For simplicity, the material model for the SMA is reduced to straight lines emanating from the origin for both austenite and martensite (s0 ¼ 0; see Introduction). The standard arrangement with backup provided by a linear elastic spring (Fig. 12.4(a)) is taken as the reference and the characteristic line is reproduced in all diagrams for the constant force (Fig. 12.4(b)) and the secondary SMA (Fig. 12.4(c)). All diagrams show that the effect of the external force (offset lines below and above the backup line) is to reduce the stroke compared with the ideal situation with no external force (points A and B in Fig. 12.2(b)). Fig. 12.4 shows that for a given primary SMA element and external force, the highest stroke S is achieved by an SMA backup spring (Fig. 12.4(c)), followed by the constant-force backup (Fig. 12.4(b)) and elastic backup spring (Fig. 12.4(a)). Dragoni and Scir
e [18] developed a theory for a novel backup system based on the concept of elastic compensation (Fig. 12.4(d)), which can be applied to augment any of the classical backup systems in Fig. 12.4(aec). The compensation unit works like a spring with negative stiffness and allows greater strokes to be achieved with respect to all classical arrangements. The working principle behind this unique concept will be described in detail in Paragraph 12.4.4 together with application examples.

12.4 Advanced actuators This section illustrates the concepts, design, modeling, manufacture, and testing of highperformance SMA actuators developed by adopting the approaches and assumptions introduced in the former sections. The concepts of the actuators are inspired by two principles: (1) the enhancement of classical springs made smart using SMAs instead of passive metals; and (2) the application of a smart design to capitalize on intrinsic virtues of SMA wires made efficient by using them unconventionally way. The first three subsections that follow belong to the first family; the last five cases belong to the second family.

12.4.1 Wave springs This section describes the design procedure of wave springs made from SMAs as a means to enhance the mechanical, thermal, and electrical performances of SMA actuators. Fabricated by winding an undulated strip in multiple, closely packed coils (Fig. 12.5(a)) [19e21], wave springs represent a valid alternative to more traditional springs (e.g., helical or Belleville) (analyzed in Maletta et al. [22] considering SMA materials) to provide high stiffness under substantial loads, especially when the axial dimensional constraints are challenging. An analytical model for the multiphysics behavior of SMA wave springs is described in Spaggiari and Dragoni [23]. A comparison of helical and wave SMA springs is provided considering the two springs to share the same properties: material, mean diameter, outer diameter, maximum applied force, maximum allowable deflection, and maximum stress. The wave spring considered in

Design and development of advanced SMA actuators

Figure 12.5 (a) Multiturn wave spring and (b) side view with primary design variables.

the analysis is a crest-to-crest wave spring with multiple turns and no-shim ends, as described in Fig. 12.5(b), together with the main design variables (mean diameter D, radial width b, and thickness t of the cross-section; numbers of waves per turn, N; and number of turns, Z). To understand the mechanical behavior of a wave spring, it is convenient to consider the bending behavior of the turns, estimate the stresses, and compute the deflection of the spring, as in Spaggiari and Dragoni [23]:
p 3PD smax ¼ tan (12.1) 2bt 2 N 2N f ¼Z

PD3 D  b $Ka Ebt 3 N 4 D þ b

(12.2)

where the empirical coefficient Ka is estimated according to the procedure described in Spaggiari and Dragoni [23] and equals: Ka ¼

3arctgð12  2NÞ þ 15 5

(12.3)

The electrical resistance of the wave spring depends on the electrical behavior of the contact points between the crests of the waves. Two ideal cases may be considered, with the contacts behaving as open circuits (infinite electrical resistance, Roc) or as a short circuit (zero electrical resistance, Rsc): pDZ bt pD Z $ ¼r 2Nbt 2N

ROC ¼ r RSC

(12.4) (12.5)

The thermal properties of a wave spring could be modeled by considering only convective cooling; thus, the cooling time, twc, of a body made of SMA with volume Vw, density x, and external area Aw is given by:

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 twc ¼

xVw Aw



  cAM Ms  T r ln h Mf  Tr

(12.6)

where cAM is a fictitious transformation-specific heat that includes the enthalpic contribution, h is the coefficient of convection, and Tr is the ambient temperature. The fictitious specific heat, cAM, in Eqn. (12.6) is defined as: cAM ¼

cA þ cM XAM þ 2 Ms  Mf

(12.7)

where cA and cM are the austenitic- and the martensitic-specific heats, respectively, and XAM is the transformation enthalpy from austenite to martensite. From Eqns. (12.4) and (12.5), Eqn. (12.6) becomes:   bt cAM Ms  Tr twc ¼ (12.8) x ln 2ðb þ tÞ h Mf  T r It is possible to compare the properties of the wave spring with a traditional helical spring made of the same SMA material. The mechanical properties of the helical spring (section diameter d, mean coil diameter D, and number of active coils n) are defined as in Shigley et al. [24], using the Bergstr€asser curvature correction factor. The first constraint to be enforced is the equivalence of the maximum stresses. The von Mises criterion is chosen to equalize the maximum stresses, bending for the wave spring and torsion for the helical spring. The stress condition can be cast as: pffiffiffi Nt 2 4C  3 p 3
p  tan (12.9) ¼ 4C þ 2 16 2N d2 The condition of equal deflection is enforced by: 8nPD3 PD3 D  b ¼ Z $Ka Gd4 Ebt 3 N 4 D þ b and then it is possible to write an equivalence in terms of turns: pffiffiffi
p Z t 5pN 3 ð1 þ nÞ 3 4C  3 C þ 1 ¼ $ $tan n d 3arctgð12  2NÞ þ 15 4C þ 2 C  1 2N

(12.10)

(12.11)

Eqns. (12.9) and (12.11) ensure the mechanical equivalence requested and represent the key tools for comparing the two springs. Once the values for N and C are decided, the dimensionless thickness t/d is calculated. Next, with the known values for N, C, and t/d, Eqn. (12.11) gives the ratio Z/n. Because the spring index normally varies between 5 and 12, the effect of this variable on the results is weak and can be disregarded by considering a fix spring index C ¼ 8. Based on this assumption, Fig. 12.6(a) reports the dimensionless ratio Z/n and the inverse of ratio t/d as functions of N. It is possible to normalize

Design and development of advanced SMA actuators

Figure 12.6 (a) Dimensionless geometrical properties and (b) generic nondimensional properties of wave springs contrasted with classical helical springs.

the properties of the wave spring on those of the helical spring, so that the generic propwave erties X is always defined as X * ¼ XXhelical , as shown in Fig. 12.6(b). Compared with the traditional helical design, the wave geometry has two crucial advantages: (1) the higher electric resistance (implying simpler supply electronics), and (2) the lower cooling time (leading to a higher working frequency). Although generally longer and heavier than the helical counterpart, the wave spring has no match when it is called to generate medium to low forces in tight axial spaces. The structured design procedure proposed in Spaggiari and Dragoni [23] allows the designer to identify through a step-by-step procedure the optimal wave spring that satisfies multiphysics design requirements and constraints.

12.4.2 Hollow helical springs This section describes how to enhance the mechanical, thermal, and electric performance of actuators by providing analysis and design equations and experimental validation for helical SMA springs with a hollow round section. Emptied of inefficient material from its center, the hollow section features a lower mass, lower cooling time, and lower heating energy than its solid counterpart for a given strength, stiffness, and stroke [25]. A comparison with a traditional solid section spring is carried out by considering the geometries in Fig. 12.7(a,b). In addition to the standard variables used in the analytical development of helical springs, two useful auxiliary variables needed are the spring index, Ch , and the void ratio, x, defined as in Fig. 12.7(c): Ch ¼

Dh dho

(12.12)

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(a)

(b)

(c)

Figure 12.7 Helical spring with solid (a) and hollow (b) section. Auxiliary design variables (c).

x¼

dhi dho

(12.13)

The void ratio, x, measures the hollowness of the section of the spring and ranges from 0 (solid spring) to 1 (zero-thickness hollow spring). The analytical development carried out in Ref. [25] leads to the deflection (Eqn. 12.14) and the spring rate (Eqn. 12.15): C 3 Nh 8F  $ h G dho 1  x4   G dho 1  x4 Kh ¼ $ 8 Ch3 Nh

dh ¼

(12.14)

(12.15)

Following the same procedure as in Paragraph 12.4.1 the electrical, thermal, and mechanical properties of the hollow spring are retrieved and reported in Spinella and Dragoni [25]. Because of the particular geometry of the hollow spring, some considerations are needed regarding the three instability modes of the hollow spring. The first mode can produce instability if the deflection becomes too large. The second mode results from buckling of the wall under excessive shear stresses, whereas the third mode could occur during manufacturing, when the straight tubing blank is wound into coils and the wall kinks. The critical displacement for the first mode is reported in Gobbi and Mastinu [26]. With reference to the second mode of instability (shear buckling of the wall), the critical shear stress producing local buckling is:   4Gð1 þ nÞ 1  x 3=2 scr ¼ $ (12.16) 3ð1  n2 Þ3=4 1 þ x Finally, considering the third mode of instability, the critical forming stress, scr , is:

Design and development of advanced SMA actuators

pffiffiffi 4 2E 1  x scr ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi$ 9 1  n2 1 þ x2

(12.17)

A comparison of hollow versus solid spring depends on the void ratio and is done by using dimensionless groups, defined as X * ¼ XXhollow and applied to the ratio m (¼ S/m) of solid surface S to mass; the slenderness ratio, l, the fundamental frequency, f, the electrical resistance, R, the cooling time, t, and the number of turns, N. Only the body length, L, critical stress, smax, spring index, C, spring stiffness, K, and deflection, d, stay the same for both springs. Fig. 12.8 shows these relationships and plots their graphical trends against the void ratio. The dashed line represents an instability limit for typical SMA values (x < 0.81). To validate the model experimentally, two SMA compression springs were built: one solid (Fig. 12.9(a)) and one hollow (Fig. 12.9(b)) [27]. The mechanical equivalence was computed on the basis of the void ratio, x ¼ 0.77, selected for the hollow spring, exploiting the relationships in Fig. 12.8. Hollow and solid springs were coiled at room temperature by winding the blanks on a grooved mandrel and applying a small axial pull. After winding, the extremities of the springs were constrained to the mandrel and then the assembly was heated to 425  C for 30 min and then cooled in water at 25  C for shapesetting. Fig. 12.9(c) shows the overall characteristics, nominal and measured, of the springs. The performances of the hollow and the solid springs are compared from the standpoints of their mechanical and mass properties, cooling times, and electrical properties. The properties were retrieved by a loadingeunloading cycle at three temperatures (40,

Figure 12.8 Effect of void ratio on dimensionless properties of the hollow spring.

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Figure 12.9 (a) Solid spring, (b) hollow spring, and (c) nominal versus measured values for both parts.

Figure 12.10 (a) Forceetime curves during heatingecooling of hollow and solid springs under 10 mm deflection. (b) Comparison of the main spring properties.

60, and 95  C) and by applying Joule heating on the constrained spring with an applied deflection of 10 mm (Fig. 12.10(a)). Fig. 12.10(b) reports the main comparative results. The mass of the hollow spring is 30% lower compared with the solid spring, in good agreement with the prediction given by Fig. 12.8. The spring rate of the hollow spring is only 6% lower than that of its solid counterpart owing to manufacture imperfections. The ratio between the two values (K*) is about 0.94, well in line with the theoretical figure of 1. The ratio between the spring rate and the mass of the active coils, m ¼ K/ ma, shows that the hollow spring has a better specific stiffness (þ33%), which leads to a more efficient structure. The cooling time of the hollow spring, thc ¼ 42 s, is nearly four times less than the cooling time of the solid competitor, tsc ¼ 163 s. This is the greatest advantage of the hollow spring: that it improves the critical weak point of SMAs (i.e., the long actuation cycle time). Fig. 12.8 shows that the theoretical ratio between the cooling times of hollow and solid spring is twice the measured one, which means that the deactivation time is faster than expected. A possible explanation involves the

Design and development of advanced SMA actuators

transformation temperature change, as discussed in Spinella et al. [27]. The hollow spring shows better performances than the solid spring, especially in terms of reduced mass (37% lighter) and cooling time (four times lower). Because of the faster response upon cooling, the hollow springs represents an interesting design option for SMA actuators with a high stroke and high operating frequency [28]. The technological feasibility of thin-walled SMA springs, although more complex than solid-section springs, is demonstrated and corroborates the theoretical predictions.

12.4.3 Negator spring This section describes the merits of shape memory Negator springs as powering elements for solid-state actuators. A Negator device (Fig. 12.11(a)) is a strip of flat spring material, which has been given a curvature so that in its relaxed condition the spring is in the form of a tightly wound spiral [29]. The unique characteristic of Negator springs is the nearly constant force needed to unwind the strip for large, theoretically infinite deflections, thus overcoming a major limitation of SMAs. Furthermore, the large area to volume ratio improves the cooling time compared with wire-based solutions. First, following the procedure detailed in Spaggiari and Dragoni [14], an analytical model of the Negator spring is developed for a general SMA material. Then, test results on a prototype SMA spring are presented and discussed in relation to the theoretical model. Modeled as a beam in pure bending according to the schematic in Fig. 12.11(b), the classical steel spring can be analyzed as in Votta and Lansdale [29] to give the constant load, P, needed to unwind the system: P¼

Ebt 3 24R2

(12.18)

Figure 12.11 (a) The Negator spring, (b) main design variables, and (c) material properties considered in the experimental tests and the approximate model (solid line).

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When considering an SMA, the definition of the elastic modulus is not trivial. Whereas the austenitic behavior could be approximated as linear (Ea), the martensitic behavior is nonlinear. By considering the properties retrieved in Spaggiari et al. [16], a smooth approximation is provided by the exponential law based on two moduli (Ema and Emb) and a strain value (εg ¼ 0.25%), with the shape and the values shown in Fig. 12.11(c).
 s ¼ Ema $εg $ 1  eε=εg þ Emb $ε (12.19) By combining the procedure adopted in Feodosyev [30] for bending an elastoplastic Euler-Bernoulli beam, the moment of the martensitic Negator spring can be obtained as explained in Spaggiari and Dragoni [14], giving an equivalent modulus hEmi (Eqn. 12.20), which, entered into Eqn. (12.18), leads to the martensitic force Pm:   2  t R R 2 2R$εg hEm i ¼ Emb þ 12$ $Ema $εg $e 2εg $ þ 1 t t ! (12.20)  2 R R 2 þ 3$ Ema $εg 1  8 $ $ εg t t The available actuation force is obtained easily using Eqn. (12.21), whereas the stroke is virtually infinite: Pm ¼

bt 3 ðEa  hEm iÞ 24R2

(12.21)

A finite element (FE) analysis of the SMA Negator spring was used to validate the analytical model. The system is modeled as a strip coiled on a rigid cylindrical surface, using the Abaqus thick shell elements, which allow the frictionless self-contact needed. The material model in the FE simulation exactly reproduced the smooth law of Fig. 12.11(c) and was implemented according to a hyperelastic law following the Marlow model (uniaxial test data), as in Scir
e Mammano and Dragoni [31]. The hot, activated material was considered fully austenitic with an elastic linear behavior and the cold deactivated material was described as fully martensitic with the previously described hyperelastic behavior (Fig. 12.11(c)). Stressestrain curves intermediate between these limit states, which would have required a more complex approach [32,33], were disregarded. The results of the FE analyses are reported in Fig. 12.12 for the spring in the martensitic state, with the contours of the equivalent stresses shown in Fig. 12.12(a) and the displacements in the direction of the force illustrated in Fig. 12.12(b,d). The austenitic behavior, modeled as a pure linear material, is not reported here for the sake of brevity. The theoretical forces calculated from Eqn. (12.21), using either Ea or hEm i, underestimate the FE prediction by less than 10% [14].

Design and development of advanced SMA actuators

Figure 12.12 (a) Equivalent von Mises stress in the Negator spring, and (b) applied displacements.

A Negator spring made of SMA was manufactured and experimentally tested to demonstrate the feasibility of this actuator. An Ni49Ti51 (atm%) SMA alloy processed at CNR IENI (National Research Council - Institute for ENergetics and Interphases, Lecco Unit) with transformation thresholds above room temperature was produced by vacuum induction melting in a carbon crucible. After hot rolling, some rods were straight cold rolled, with intermediate annealing, down to 0.2-mm strips. The sample strips exhibited some undulations owing to the rolling process needed to obtain the desired thickness. The strip width was 31 mm and the length was 390 mm. After winding on a spindle with radius of 15 mm, the martensite finish temperature, Mf, of around 11  C and an austenite finish temperature, Af, of 73  C were detected with differential scanning calorimetry. The Negator spring was tested on a uniaxial tensile machine equipped with a climatic chamber (Fig. 12.13(a)), and the forceedisplacement curves

Figure 12.13 (a) Characterization of the spring in the climatic chamber, and (b) comparison between experimental curves (solid lines) and analytical model (dashed lines) at different temperatures.

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illustrated in Fig. 12.13(b) were obtained for both martensite state (lower curve) and austenite state (upper curve). The theoretical forces (shown in Fig. 12.13(b) with dashed lines) closely match experimental measurements for the austenite (Fth_A ¼ 4.08 N versus Fexp_A ¼ 4.19 N) and provide a fair agreement for the martensite (Fth_M ¼ 1.53N versus Fexp_M ¼ 1.75 N). Therefore, the analytical equations disclosed can be used for design purposes. The mechanical characteristic of this actuator exhibits two great advantages compared with typical SMA actuators. The Negator spring stroke is long (a feature that limits many competitor applications) and the output force is constant, the best condition for an actuator. Moreover, the high area to volume ratio grants a better cooling time compared with other configurations. In addition, only a little portion of the material needs to be heated, which leads to faster cooling of the device. Because the thickness and width are two independent parameters, it is possible to control the force and cooling time as independent variables, which is impossible with the other solutions presented in the sections on wave springs and hollow helical springs. The main drawbacks of SMA Negator springs are the complex manufacturing procedure, the cost of the material itself, and the need for a backup element. This last disadvantage could be overcome considering the rolamite architecture, as discussed in Spaggiari and Dragoni [34]. The results confirm the applicability of this kind of geometry to the SMA actuators, and the analytical model is confirmed to be a powerful design tool to dimension and predict the spring behavior in both the martensitic and austenitic range.

12.4.4 Elastic compensation 12.4.4.1 Compensation concept The force delivered by shape memory actuators backed up by classical systems, as in Fig. 12.4(a-b), varies linearly with the displacement and the maximum and minimum values taking place at the ends of the stroke (points A and B, respectively, in Fig. 12.2). By contrast, the external load that needs to be overcome is usually constant over the stroke. A classical SMA actuator is thus designed so that the minimum output force is higher than the external load [25]. This implies a reduction in the useful stroke with respect to the limit value dictated by the maximum deformation that the shape memory elements can withstand. The concept of elastic compensation introduces a balancing device that draws energy from the SMA element in positions where the internal force is high, and then returns this energy in positions where the force is low [35e37]. As in the field of manipulators [38] and constant-force supports [39], the compensation system capitalizes on bistable mechanisms coupled with an elastic spring element. This combination features negative elastic characteristics (the opposite of a traditional elastic spring), which means that the generated force decreases as the deformation of the mechanism increases (Fig. 12.4(d)). The same

Design and development of advanced SMA actuators

Figure 12.14 Working principle of the compensation system: (a) actuator with two antagonistic shape memory alloy (SMA) wires compensated by a transverse elastic spring; (b) forceedisplacement diagrams of the parts for different actuation conditions.

principle has been successfully applied in the field of electroactive polymer actuators by introducing compliant mechanisms [40]. To understand better how a compensation system operates, consider the simple actuator in Fig. 12.14, which consists of two antagonistic SMA wires. The two wires, with free length l0, are fixed to the frame at A and B and are linked at point O. The connection in O is made at a low temperature after applying an overall prestretch, p, to the wires, so that the heating (activation) of each wire leads to shortening of that wire and lengthening of the other one (while kept cool). The compensation system in Fig. 12.14(a) consists of a preloaded conventional spring, which pushes transversely to the wires on the horizontally guided point O. Let us focus on the variation of horizontal force F (positive if directed to the left in Fig. 12.14(a)), which all the elements exert on point O, when its position x changes. We will consider three operating conditions: wire 1 activated and wire 2 deactivated (first column in the graphs in Fig. 12.14(b)); wire 1 deactivated and wire 2 activated (second column in Fig. 12.14(b)); and both wires deactivated (third column of graphs in Fig. 12.14(b)). The trend of the net force produced by the two SMA wires (first row of the graphs in Fig. 12.14(b)) is linear in all three cases: always positive when wire 1 is activated, always negative when wire 2 is activated, and alternately negative and positive when no wire is activated. The distance between positions C and D (Fig. 12.14(b)), in which the force vanishes when one or the other wire is activated, defines the maximum theoretical stroke, Dx, of the uncompensated actuator. This stroke can be achieved only when the external force is zero. For an increasing external force, the stroke decreases. For

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the precise value of the external force, F0, shown in Fig. 12.14(b), the useful stroke of the uncompensated actuator falls to zero, because even the slightest movement of O from x ¼ 0 would cause the internal force to fall below F0. The force applied on point O by the compensation system alone (second row in Fig. 12.14(b)) has the same downward antisymmetric trend in the three operating cases and features zero force at x ¼ 0. For small displacements, the trend is approximately linear with positive forces for negative x positions, and negative forces for positive x. In other words, the compensating spring tends to push the output port O away from its otherwise stable position, x ¼ 0. By superimposing the actions of the wires and the compensator, we get the total force exerted on point O by the entire actuator (third row in Fig. 12.14(b)). By choosing the appropriate (negative) slope of the characteristics of the compensator, the useful force of the actuator can be made constant, with magnitude F0, for both directions of motion (first and second columns in the third row in Fig. 12.14(b)). The compensated actuator can thus displace across the entire theoretical stroke Dx an external force (F0) that the uncompensated actuator would not even be able to move. If both wires of the compensated actuator are deactivated (third column in Fig. 12.14(b)), the characteristics of the actuator assume the bistable trend described by the segment ef, with equilibrium positions at points C or D. These end positions must be ensured by hard stops so as to prevent overstraining of the SMA elements. The bistable system resulting from this architecture has three additional advantages: the existence of a certain equilibrium position even when power is shut off, the enforcement of precise binary positioning, and the possibility of easy control strategies. The advantages of the compensation system shown in Fig. 12.14 for the case of two SMA actuator elements are also true for SMA actuators with a single SMA element, although the overall improvement in terms of force and stroke is less pronounced (Fig. 12.4). In this case, a single mechanical hard stop is required, and the behavior without power supply becomes monostable. The theory behind the elastic compensation concept is detailed in Scir
e Mammano and Dragoni [37]. The remainder of this section will illustrate the implementation of this idea in two SMA actuators by means of a classical mechanism (rocker arm) and compliant members (buckled beams). Numerical examples and proof-of-concept prototypes are presented to illustrate the advantages of both compensated actuators over their classical uncompensated counterparts. 12.4.4.2 Rocker arm actuator The compensation mechanism shown in Fig. 12.15(a) is made of a rocker arm R hinged in G to frame T. The elastic compensation spring Sc (with free length L 0 Trad and spring rate kTrad ) connects end F of the shortest side of the rocker arm to point E of the frame.

Design and development of advanced SMA actuators

Figure 12.15 Compensation mechanism based on oscillating rocker arm: (a) concept; (b) forcee displacement relationships. SMA, shape memory alloy.

Point O at the extremity of the longest side of the rocker arm represents the output port of the actuator and is used to connect the primary elastic elements of the actuator (SMA1 and SMA2), which are also fixed at points P and Q to the frame. In the case of a singleSMA actuator, the active element (SMA1) is placed at the bottom of the device, element SMA2 disappears, and the contrasting function is provided only by the conventional compensation spring, Sc. Disregarding the SMA springs, the mechanism in Fig. 12.15 has a position of unstable equilibrium corresponding to the configuration where axis EF of the compensation spring Sc passes through the hinge point G of the rocker arm. In this position, the force exerted by the compensator on port O is null. In the case of an actuator with a single active element, compensation spring Sc is always placed to the right of hinge G to exert a contrasting force on the SMA element, necessary to deform it in the disabled state. For an actuator with two contrasting SMA elements, spring Sc is located in a position of unstable equilibrium (line EF passing through G) when point O is at the center of the stroke (S). In this way, the compensation mechanism helps active element SMA1 for the lower half of the stroke and element SMA2 for the upper half of the stroke. Line OA in Fig. 12.15(b) represents the characteristics of the austenitic (hot) SMA1 element. Segment BC represents the martensitic (cold) response of the antagonistic SMA2 element. Line DE is the characteristics of the compensation spring, with point D corresponding to the center point of the total stroke S of the actuator. Line EG represents an ideally constant external load of amplitude FON1. The situation beyond point D is obtained by extrapolating linearly all characteristic lines shown in Fig. 12.15(b). Line

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AE corresponds to the characteristics of the SMA1 element and of the compensation spring combined. At any position x, the difference between lines AE and BC gives the net output force of the actuator (Fnet ON1) when element SMA1 is activated. When element SMA1 is disabled and SMA2 is enabled, the chart becomes similar to Fig. 12.15(b), with all lines mirrored with respect to the center line AD. The optimal performance of the actuator in Fig. 12.15(b) is achieved when lines AE and BC become parallel to each other, so that the net output force of the actuator (Fnet ON1) equals the external load (FON1) at any position x. According to Mammano and Dragoni [41], the net output forces generated by the actuator by activating either SMA1 or SMA 2 are given by the expressions (symbols as in Fig. 12.15(b)): Fnet Fnet

ON1 ON2

¼ F0comp  F0m2  kmb2 S þ ðkcomp þ ka1 þ kmb2 Þx

(12.22)

¼ F0comp þ F0m1  ka2 S þ ðkcomp þ ka2 þ kmb1 Þx

(12.23)

Eqns. (12.22) and (12.23) show that for kcomp ¼ (ka2 þ kmb1) ¼ (ka1 þ kmb2), output forces Fnet ON1 and Fnet ON2 are independent of displacement x. This condition is automatically achieved when using identical SMA elements (ka1 ¼ ka2 and kmb1 ¼ kmb2). Fig. 12.16 shows the test bench with the compensated actuator in the foreground and the uncompensated actuator in the background. The compensated actuator is equipped with hard stops, placed symmetrically with respect to the center position of output port O. Both rocker arms are made of aluminum alloy and are supported by ball bearings at

Figure 12.16 (a) Front view (b) and back view of the experimental rig featuring the compensated actuator (bottom) and uncompensated actuator (top).

Design and development of advanced SMA actuators

hinges G to minimize friction. Likewise, the compensation spring is connected in E to the frame and in F to the rocker arm through ball bearings. For each actuator, the external load force is provided by a magnetic hysteresis rotary brake located at the bottom of the main plate, together with the electronics (Fig. 12.16(b)). The brakes are connected to output ports O of the two actuators by means of nylon cables, guided by antifriction pulleys and wound on a drum fitted to the input shaft of each brake. By acting on the potentiometers shown in Fig. 12.16(a), both actuators were tested for several levels of resisting force. The angular position of the rocker arms was measured during operation by contactless rotary position sensors fitted to rotation axes G of both actuators (Fig. 12.16(b)). Heating of the SMA springs was achieved by the Joule effect with electrical current supplied by two independent electronic boards. Fig. 12.17 compares the theoretical (solid lines) and experimental (open circles) output forces generated by the uncompensated (Fig. 12.17(a)) and compensated (Fig. 12.17(b)) actuators for three operating conditions: SMA1 ON þ SMA2 OFF, SMA1 OFF þ SMA2 ON, and SMA1 OFF þ SMA2 OFF. In both charts, position x of the output port ranges from zero to the nominal design stroke (S ¼ 100 mm) of the compensated actuator. A direct comparison of the net strokes and stroke times are presented in Fig. 12.18(a,b) for the compensated (solid circles) and uncompensated (open circles) actuators. Besides showing a good correlation between theory and experiment, Fig. 12.17 confirms that for a given output force (the height of the shaded rectangle), the net stroke (the width of the shaded rectangle) is much greater for the compensated actuator than for the compensated actuator. This is better seen in Fig. 12.18(a), which shows that the ratio

Figure 12.17 Comparison of theoretical and experimental output forces generated by (a) uncompensated and (b) compensated actuators for three operating conditions. SMA, shape memory alloy.

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Figure 12.18 Comparison of compensated and uncompensated actuator performance in terms of (a) net stroke and (b) stroke time.

between the strokes is 2.5 times at zero load and 22 times under the maximum force of 1.5 N. Furthermore, the net stroke of the compensated actuator is independent of the applied force, which simplifies both the design and the selection of the actuator for a given application. Another remarkable advantage of the compensated actuator lies in the quicker time needed to complete a single stroke (Fig. 12.18(b)). Presumably, because of the lesser difference between internal and external forces on approaching the end equilibrium positions, the uncompensated actuator keeps moving, although slowly, for a longer time before coming to rest. Owing to the considerable volume occupied by the device, which would be a major concern in many practical applications, this study is to be regarded only as a first step to demonstrating the feasibility of the compensation technology. Based on the same general theory, a more compact architecture is presented in the following section. 12.4.4.3 Buckled beams actuator Mechanisms with rigid members have the disadvantage of large dimensions and jeopardize the intrinsic simplicity of the solid-state SMA principle. To remove this limitation, this section proposes a modular two-SMA actuator compensated by compliant mechanisms [42] in the form of buckled beams [18]. The specific architecture addressed was inspired by a compensating device first proposed by Plante et al. [43] in the field of dielectric-elastomer actuators and by Jee et al. [44] in the category of SMA actuators. The basic module of the actuator is composed of four identical flat beams with hinged ends, compressed axially beyond their buckling limit (Fig. 12.19). The beams are arranged in parallel pairs, with each pair symmetrically placed across the common axis of two antagonistic sets of helical SMA springs. The outer hinges of the beams and the outer ends of the SMA springs are fixed to the frame, whereas the inner hinges of the beams and

Design and development of advanced SMA actuators

Figure 12.19 Characteristic positions of the actuator compensated by buckled beams with two shape memory alloy (SMA) elements: (a) center position, (b) leftmost position; (c) rightmost position.

the common inner point of the springs are fixed to a slider, which moves within the frame as a drawer. The beams serve the double function of guiding the slider linearly and providing the elastic compensating force with negative slope (Fig. 12.14(a)). Moving from the center position in Fig. 12.19(a) (u ¼ 0), slider C covers the entire stroke S for which it is designed. Two characteristic configurations are shown in Fig. 12.19(b) (leftmost position, u ¼ umin) and in Fig. 12.19(c) (rightmost position, u ¼ umax ¼ umin þ S). The latter two configurations are achieved depending on which of the two SMA springs is activated. In the leftmost position (Fig. 12.19(b)), the spring SMA1 is activated and its traction force is supplemented with the horizontal force generated by the compensating beams. This situation is reversed in the rightmost position (Fig. 12.19(c)) when spring SMA2 is activated. Following the design process detailed in Scir
e Mammano and Dragoni [18], the concept in Fig. 12.19 was embodied by the layout and prototype shown in Fig. 12.20(a,b). The slider C in Fig. 12.20(a) translates horizontally, guided as a drawer

Figure 12.20 (a) Layout and (b) closeup of prototype beam-compensated actuator.

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within frame T. The four compensating springs, mounted transversely to the slider, do not lie on the same plane, but are offset through the thickness of the slider. The letters at the ends of the beams in Fig. 12.20(a) identify the corresponding beams in Fig. 12.19. The particular arrangement of the beams is aimed at minimizing the width of the slider (hence of the actuator) and eliminating the torque about the longitudinal axis of the slider. The hinges at the ends of the beams are obtained by direct contact of the thin beams against V sockets cut into the frame and the slider. To maximize the force generated by the actuator, four pairs of antagonistic SMA springs (such as ABFG in Fig. 12.20(a)) are used in the design. Two pairs of springs (visible in Fig. 12.20(a)) are placed on the top face of the slider and the other two pairs (not visible in Fig. 12.20(a), but partially visible in Fig. 12.20(b)) are placed on the bottom face to provide symmetry and eliminate tilting moments on the slider. Finally, the attachment points (FeB and HeE in Fig. 12.20(a)) of the SMA springs to the slider are split for each pair of antagonistic springs so as to minimize the longitudinal length of the actuator. The overall dimensions of the actuator are 78  60  30 mm3 in the closed configuration and 104  60  30 mm3 in the fully extended configuration for a net stroke of about 26 mm. Testing of the actuator was performed under displacement-control and constantforce conditions. The displacement-control tests capitalized on a horizontal electromechanical machine, allowing the actuator weight to be easily supported by means of a vertically adjustable polytetrafluoroethylene (PTFE) plate. During displacementcontrol, the force generated by the actuator was measured for each position of the slider (C in Fig. 12.20(a)) over the entire extension of instroke and outstroke. For the constantforce tests, the frame of the actuator was fixed to a horizontal plane and an adjustable weight was attached to the slider through a wire wound on a pulley. The overall stroke, stroke time, and response time were measured for each loading weight. The reaction time was defined as the time between the start of the electrical supply and the beginning of the movement of the slider. The results obtained for both test types are shown in Fig. 12.21(a) (displacement-control) and Fig. 12.21(b) (constant-force). The experimental curve in Fig. 12.21(a) demonstrates that the compensated actuator generates constant forces in both directions of motions over the entire stroke. The agreement with the theoretical forces (1.5 N, dashed lines) is excellent during outstroke (extension curve) and satisfactory for the instroke (contraction curve), with the experimental force somewhat higher (þ7%) than the predictions. Fig. 12.21(b) shows that the actuator consistently achieves the maximum output stroke of 26 mm up to the maximum design force of about 1.5 N. Although beyond this force threshold the measured stroke goes down, the behavior of the compensated actuator is much better than classical actuators for which the net stroke decreases steadily for increasing loads. A larger dependence on the applied load is observed in Fig. 12.21(b) for both response time (open circles) and stroke time (solid circles). In particular, the stroke time increases

Design and development of advanced SMA actuators

Figure 12.21 (a) Theoretical and experimental forceedisplacement curves for the actuator. (b) Measured performance of the prototype in terms of stroke, stroke time, and response time for varying applied force.

abruptly for loads just above the nominal design force (1.5 N). For applied loads below the nominal capacity, Fig. 12.21(b) shows that the response time represents an important fraction of the total stroke time (about 90% for loads up to 0.9 N and over 80% for loads between 0.9 and 1.5 N). This behavior suggests that both times can be significantly reduced by adopting a smarter supply strategy than the basic constant-current scheme used in this work. Optimal supply strategies based on pulse-width modulation techniques or the like are widely employed and are extensively described in the literature [45,46].

12.4.5 Wire-on-drum This section presents the analytical model of a linear/rotary solid-state actuator formed by a shape memory wire wound over a cylindrical drum [31]. The analytical model for the drum actuator (sketch in Fig. 12.22) investigates the behavior of the actuator backed up by either a constant force or a constant couple (F0 and C0 in Fig. 12.22(a,b)). The analytical model assumes bilinear stressestrain behavior of the wire in the martensitic state and a linear elastic response in the austenitic state (Fig. 12.22(c)). Based on simple equilibrium conditions, the model calculates the stress and strain distributions in the wire when subjected to a constant external backup force and undergoing frictional sliding forces at the contact with the drum. According to the calculation reported in Scir
e Mammano and Dragoni [31] it is possible to consider the forces acting on an infinitesimal wire portion wound on the drum as in the configuration in Fig. 12.22(a). Fig. 12.23 shows the free-body diagram of an infinitesimal wire element of angular amplitude da, lying on the drum of radius R0. This gives a radius of curvature of the wire axis R ¼ R0 þ d/2 (where d is the diameter of the wire), which will be used in the following discussion as the effective radius of

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(c) σ (a)

(b) A

O

A C0

A

Austenite

Ea O

Martensite σg

F0

B Ema

O

C

Emb ε

εg

Figure 12.22 Concept of the wire-on-drum actuator in two configurations: (a) for linear motion; (b) for rotary motion; and (c) simplified double linear model used.

Figure 12.23 Schematic used for the analytical model of the wire-on-drum arrangement.

the drum. The wire is assumed to be in the martensitic state and is subjected to the forces generated by the externally applied backup force F0. The elemental wire element in Fig. 12.23 undergoes the following forces: axial forces T and T þ dT, applied at the ends of the element; radial contact force N; and tangential friction force Q. If p is the contact force per unit length of the wire and f is the coefficient of sliding friction, forces N and Q are given by N ¼ pRda and Q ¼ f N ¼ f pRda. The mathematics involving the material model described in Fig. 12.22(c) leads to simpler results if the Young’s moduli Ea, Ema, and Emb are grouped in the dimensionless parameters s1 and sm as: s1 ¼ Ea =Ema

and

sm ¼ Emb =Ema

(12.24)

Angle ag, defined as the portion of wire strained beyond the elastic threshold of the martensite (point B in Fig. 12.22(c)), is found to be ag ¼ ln(sg/s0)/f as described in Scir
e Mammano and Dragoni [31], where s0 ¼ F0/A is the stress induced in the wire by the backup force at a ¼ 0.

Design and development of advanced SMA actuators

The martensitic axial strain at position a, εm, and the austenitic recovery strain, εar, with the condition that the angle is lower than ag, are given by:

   sg f a s0 ef a sm  1 (12.25) 1 1 e εm ðaÞ ¼ ha  ag i sm Ema s0 sg s0 f a fa εar ¼ e s1  e sm þ s1 ðsm  1Þ (12.26) Ea sm s0 The effective stroke of the actuator, Dx, is calculated assuming that the wire is heated above Af:  sg s 1 Rs0 s1  f alim1 f alim1 1e (12.27) Dx ¼ þ 1e þ f ðsm  1Þalim1 fEa sm s0 s m where alim is computed following the procedure reported in Scir
e Mammano and Dragoni [31]. The theory was validated against FE results and laboratory tests on a prototype. In the FE simulation, the circular drum was modeled with a single, fully constrained rigid curve. The wire was described by a chain of 1884 shear flexible beam elements (each 0.5 mm long) with the axis lying on the drum profile. The SMA was described hypothetically by means of a temperature-dependent hyperelastic material, corresponding to Marlow’s constitutive model, following the shape of the curve in the martensitic region below 20  C and the austenitic curve above 95  C (Fig. 12.22(c)). Fig. 12.24(a,b), corresponding to friction coefficient f ¼ 0.1, shows strong agreement between the analytical and FE stressestrain results for both wire temperatures (ON and OFF states) over the entire winding span (n ¼ 3 turns / aA ¼ 18.85 rad). A similar degree of agreement was obtained for many other frictional coefficients. The overall picture

Figure 12.24 Comparison of analytical results and finite element (FE) results for (a) wire stresses and (b) wire strains versus winding angle. Effect of frictional coefficient f and number of turns n on theoretical stroke (c).

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of stroke (normalized over the drum diameter) as a function of the frictional coefficient is shown in Fig. 12.24(c). We can see that the system presents a stroke equal to a free SMA wire when the friction coefficient is zero. Furthermore, Fig. 12.24(c) shows that to have a distinct advantage from this configuration, it is crucial to lower the friction coefficient as close to zero as possible. Results from the friction experiments between the SMA wire and the drum (Fig. 12.25(a)) show that SMAs on PTFE have a friction coefficient around 0.0575 (both in martensite and austenite states), as detailed in Scir
e Mammano and Dragoni [31]. The comparison of predicted and measured strokes for the proof-of-concept actuator in Fig. 12.25(b) (drum diameter 2R0 ¼ 25 mm; coefficient of friction f ¼ 0.056; and martensite prestrain ¼ 3%) shows good agreement, with an experimental stroke of around 1.9 mm and a predicted one around 1.5 mm. Although limited to a single coefficient of friction (f ¼ 0.056), the test results confirm the order of magnitude of the analytical forecasts. For that particular coefficient of friction, the maximum stroke is already achieved for a single turn of wire, and it is pointless to increase the number of turns to increase the net stroke. The prototype shown in Fig. 12.25(c) lowers the coefficient of friction by replacing the sliding contact of the wire against the drum with peripheral pulleys mounted on ball bearings placed along a helical path around the drum. The prototype in Fig. 12.25(c) has four turns of eight pulleys freely rotating on ball bearings. The pulleys guide the wire over an equivalent radius of 50 mm. Under a prestrain of 2.8%, the actuator generates a stroke

Figure 12.25 (a) Experimental rig for polytetrafluoroethylene shape memory alloy (SMA) friction measurements for stroke measurements for the proof-of-concept linear actuator. (b) Prototype of wire-onpulleys SMA actuator.

Design and development of advanced SMA actuators

of 10 mm (20% of the virtual drum diameter). This figure compares well with the stroke produced by a sliding version of the actuator (with an equal drum diameter, equal number of turns, and equal prestrain) for a coefficient of friction f ¼ 0.013. The model developed and validated in this section shows that for a friction coefficient of 0.05 (typical of metal-on-PTFE contact), a drum 100 mm in diameter generates a net stroke of 7.0 mm when working against a constant weight that induces a maximum strain of 3% in the wire. This is a big improvement on the 2.8-mm stroke that a 100-mm straight wire would produce under the same load. The model also indicates that a friction coefficient as low as 0.01 would increase the stroke to 34 mm for the same drum. Rolling contact alternatives as in Fig. 12.25(c) should be tested and validated as feasible engineering solutions to achieve such low friction values.

12.4.6 Compliant bow This section presents a bow-like compliant actuator (Fig. 12.26(a)) aimed at improving the specific performance of shape memory wires in terms of force-displacement output, as described in Mammano and Dragoni [47]. Conceptually, the actuator is formed by two straight elastic beams mutually hinged at the ends with a prestretched SMA wire between them (Fig. 12.26(b)). Heating the alloy shortens the wire, which in turn makes the beams buckle outward in a symmetric double-arched configuration. Transverse displacement of the beams amplifies the contraction of the wire while producing the output force. This compliant design presents an incremental ratio between the bow deflection and wire contraction, which decreases for increasing wire contractions, generating a nearly constant output force despite the rapid decrease in wire tension upon shortening. A high specific stroke to size ratio, which is difficult to obtain with simple SMA, can be obtained by stacking multiple actuators, as shown in Fig. 12.26(c). The remainder of this section compares the predictions of an analytical model with FE results and measurements on a proofof-concept prototype. (a)

(b)

B

E, I F

SMA Wire Elastic beam

SMA Wire

C

D

A

f

B FF

(c)

F

C

D A

xa Martensite

xa Austenite

Martensite

Austenite

Figure 12.26 (a) Single-bow, (b) two-bow, and (c) multiple stacked bow actuators shown in slightly deformed martensitic configuration (left) and highly deformed austenitic (right) configurations. SMA, shape memory alloy.

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The analytical model is based on the buckled beam theory and assumes that the SMA wire contraction coincides with the vertical displacement of an axially loaded beam. For a beam of general length l, kinematic ratio s, between horizontal displacement xa and vertical stroke f (Fig. 12.26(b)), can be computed from the elastica theory given in Timoshenko and Gere [48]. Using the approximation explained in Scir
e Mammano and Dragoni [18], in which it is acceptable if the f/L ratio of SMA is below 5% (maximum practical SMA stretch), the expression for s reads: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi vxa 5ð2  f =lÞ2 f 5 s¼ ; for /0 sz ¼ (12.28) vf 4pf =lð4  f =lÞ l 4pf =l By considering the linear model for SMA shown in Fig. 12.22(c) and Eqn. (12.24), and following the procedure in Mammano and Dragoni [47], the force in both the activated austenitic state, FON, and the deactivated martensitic state, FOFF, normalized over the critical instability load, Pcr, is: 

   FON 2p xa s1 Ema Aw p
xa 2 1 þ εmax $ ¼ εmax 1  2 (12.29) 5 l 5 l Pcr Pcr εmax

    FOFF 2p xa Ema Aw p
xa 2 1 þ εmax 2 ¼ εmax sg ð1  sm Þ þ sm 1  $ 5 l 5 l Pcr Pcr εmax (12.30) Results from Eqns. (12.29) and (12.30) show that when external loads are applied to the actuator (FON T s 0, FOFF T s 0), the net stroke, S, is necessarily lower than the maximum displacement achieved under zero external load (Fig. 12.27(a)). Based on the thorough design procedure described in Mammano and Dragoni [47], the

Figure 12.27 (a) Plot of Eqn. (12.29) (upper curve) and Eqn. (12.30) (lower curve) as a function of the relative stroke. (b) Comparison of analytical models (solid lines) and FE model (dots).

Design and development of advanced SMA actuators

bow-like actuator can be designed and dimensioned according the required force-stroke output behavior. The design procedure follows three main steps: (1) the cross-sections of wire and beams are first determined to provide the design forces; (2) the optimal normalized stroke of the actuator, defined as the stroke over the beam length, is calculated; (3) the lengths of wire and beams are scaled to satisfy the actual design stroke. The bow-like actuator was modeled by FEs using the Abaqus 6.10 package. The model was two-dimensional with the SMA wire discretized using beam elements as in the Wire-on-Drum section. Fig. 12.27(b) shows the results of FE analyses in terms of forces measured at the center node of the movable beam (the node where the incremental displacement was applied). Results for the enabled actuator are shown with empty squares, and for the disabled actuator, with empty circles. The net stroke of the actuator predicted by the FE simulations was SEF ¼ 24.2 mm, close the analytical prediction of 25 mm. Moreover, numerical simulations show that the actual stroke of the actuator is strongly affected by the ratio jFOFF T/FON Tj. The smaller this ratio, the greater the stroke; the best performance was achieved when FOFF T ¼ 0. The theoretical model was also validated by tests on the prototype, designed as in Fig. 12.28(a). The results are shown in Fig. 12.28(b); the experimental points (open circles) fit the analytical curves (solid lines) nicely. The 2.05 N force in the ON state is 9% lower than the predicted maximum value of 2.25 N. The displacement in ON state is 7.2 mm, only 7% less than the predicted displacement of 7.8 mm. The prototype amplification factor (i.e., the amplification of the SMA wire contraction) is about 3.6.

Figure 12.28 (a) Functional prototype of a bow-like actuator in ON/OFF state. (b) Comparison of analytical predictions with experimental data.

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The wire-activated, bow-like SMA actuator presented in Mammano and Dragoni [47] can generate reasonable forces and strokes with a simple compliant architecture. The flexural elastic beams serve the multiple purpose of acting as a mechanical frame, being a backup force for the SMA wire, and amplifying the wire contraction. The analytical model developed, confirmed by FE simulations, shows that the wire contraction can be amplified by factors up to 6, depending on the ratio between the output forces requested by the actuator for outstroke and instroke. The theoretical model is confirmed by the experimental testing, with closer agreement for the force-displacement profiles in the enabled state than in the disabled state.

12.4.7 Overrunning clutches This section presents the conceptual design, modeling, prototyping, and testing of an innovative rotary motor featuring SMA wires and overrunning clutches (OCs). The device (Fig. 12.29(a)) is composed of an SMA wire wound on a low-friction cylindrical drum according to the concept presented in the Wire-on-Drum section. The drum is mounted onto a shaft through the OC2 and is contrasted by a backup spring [49,50]. The SMA wire contracts upon heating, causing rotation of the drum, which is transferred to the shaft owing to the engaged OC2. During the backup-recoiling phase, the drum rotates backward without moving the shaft, owing to the frame-supported OC1. A model for the quasistatic simulation of the motor is provided and the experimental characterization of a prototype device features three active drums, a rotary sensor, and an angular brake to apply the external load. The aim of this embodiment is to provide a mechanical system able to exploit the features of SMA wires (high specific force and controllability) without their main disadvantages of low stroke and intermittent movement. The particular combination of the SMA wire-on-drum provides a longer stroke if it designed following the method presented in Scir
e Mammano and Dragoni [31]; the OC prevents

Figure 12.29 (a) Conceptual scheme of the shape memory alloy (SMA) wire on drum plus overrunning clutches (OCs). (b) Modular architecture exploited to provide continuous rotary motion.

Design and development of advanced SMA actuators

the system from rotating back when the SMA is not engaged, mimicking rotary inchworm behavior. The basic concept can be improved by parallelizing the unit modules to provide either smoother continuous movement or an increase in the output torque. The schematic of the parallelization scheme is provided in Fig. 12.29(b). Once the prestrained wire has been tightly wound on the drum, the unidirectional clockwise rotation of the shaft is obtained by repeating two-step cycles of alternated heating and cooling of the wire. During heating, the wire contracts and effectively rotates the drum and shaft clockwise. During cooling, the backup spring rotates the drum counterclockwise and the clutch prevents the shaft from moving back while elongating the SMA wire. At the end of cooling step, the motor is ready to perform another cycle. Serial activation of the modules in Fig. 12.29(b) results in smoother rotation of the output shaft with respect to a single-stage motor. Similarly, parallel activation of the modules increases the output torque while conserving the stepwise rotation of the unit stage. If the number of modules is large, it is also possible to combine serial and parallel activation of subsets of modules and partially overlap the cycle time of the modules within each subset to achieve more regular rotation. The design procedure exploits the method described in the Wire-on-Drum section and the material model depicted in Fig. 12.22(c). To estimate the wire stretch in martensite and austenite states, the stress acting in the wire due to the torques must be computed. The stress in the austenitic wire, s1, is: s1 ¼

Tb þ TL þ T1 R$Aw

(12.31)

where R is the winding radius of the wire, Aw is the cross-sectional area of the wire, T1 is the friction torque of OC1, TL is the externally applied torque, and Tb ¼ Fb Rb is the restoring torque generated by the backup spring (Fb is the constant backup force and Rb is the arm of Fb with respect to the axis of rotation of the drum). Upon deactivation, the stress s0 in the martensitic wire is: s0 ¼

Tb  T2 R$Aw

(12.32)

where T2 is the friction torque of OC2. Both stresses s1 and s0 are measured in the free wire length between the frame and the drum in Fig. 12.29(a). Once these two stresses are defined, the procedure of the Wire-on-Drum section and the flowchart of Scir
e Mammano and Dragoni [31] can be applied to express the net rotation angle of the drum, Da, which corresponds to the linear stroke expressed by Eqn. (12.27). The two angular values are given by Da1 and Da2:   s1 s0    1 f alim1 f alim1 Da1 ¼ s1 1  e þ 1e þ zf ðsm  1Þalim1 (12.33) fEa sm

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Shape Memory Alloy Engineering

    s1 s0 1 f alim2 f alim2 s1 1 e þs1 s0 1 e  Da2 ¼ ðsm  1Þ½1þzðln z 1Þh0  ag i fEa sm (12.34) In Eqns. (12.33) and (12.34), parameters alim1 and alim2 represent the limit alternative winding angles over which the strain in the wire is recovered, as reported in Scir
e Mammano and Dragoni [49]. The three-dimensional (3D) computer-aided design (CAD) model in Fig. 12.30(a) includes the sensor and brake used for experimental characterization. The frame was manufactured by rapid prototyping using a 3D printer based on FDM (fused deposition modeling) technology. OC, overrunning clutch; SMA, shape memory alloy. Testing of the prototype in Fig. 12.30(b) measured the stroke under five different resisting torques (Fig. 12.31(a)) and the speed of the system under an input current of 800 mA for the same torque levels (Fig. 12.31(b)). Fig. 12.31(a) also compares test measurements with the theoretical stroke provided by Eqns. (12.33) or (12.34). Endurance tests documented in Scir
e Mammano and Dragoni [49] were found to be in agreement with fatigue tests performed on straight wires [8,10,11]. Fig. 12.31(a) shows that although the restoring force of the backup spring is not precisely constant as assumed by Scir
e Mammano and Dragoni [31], the analytical predictions fit the experimental data closely (error below 2.5%). Fig. 12.31(b) shows that regardless of the applied torque, the mean velocity of the motor initially increases and then decreases with the supply time. For a short supply time, the wire does not have enough time to cool down and recover the strain before the next supply cycle starts again. For longer supply times, the modules remain idle after the cooldown. The curves in Fig. 12.31(b) indicate that the peak angular velocity decreases with the applied torque while the supply time needed to achieve the peaks increases with the torque, as expected.

Figure 12.30 (a) Three-dimensional computer-aided design model of rotary motor. (b) Physical prototype used for testing.

Design and development of advanced SMA actuators

Figure 12.31 (a) Three-dimensional model of the prototype. (b) Physical realization used for the experimental tests.

Figure 12.32 Optimized miniactuator (net dimensions of 60  20  35 mm).

Fig. 12.32 displays an optimized miniaturized version of the motor (with no brake and sensor) leading to a net envelope of 60  20  35 mm. This actuator has a specific stroke of 0.0086 deg/mm3 and a specific output torque of 0.48  103 Nmm/mm3, which places this solution in the high end of all actuators known so far. In particular, the specific output work per cycle of 0.3  105 J/mm3 is about three times the highest value reported in Nespoli et al. [1].

12.4.8 Pushepull rubber block This section describes the concept, develops the theory and shows an experimental validation for a pushepull actuator made by winding a thin shape memory wire on a solid

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Figure 12.33 (a) Pushepull rubbereshape memory alloy (SMA) actuator in the relaxed position, (b) after wire heating, and (c) in the recovery position.

rubber cylinder. The incompressibility of the rubber converts the thermomechanical contraction of the heated wire in twice as much axial strain available in the rubber core. The intrinsic elastic backup provided by the core allows pushepull action of the device. Based on the assumption of both material and geometric linearity, a simple theoretical model is developed, which culminates in a simple closed-form equation for the output stroke of the actuator. Fig. 12.33 shows the SMA wire in the three key stages of actuator construction and operation. A finite length of prestretched SMA wire is wound tightly on a cylindrical rubber core (Fig. 12.33(a)) and its ends are fixed to the rubber. Upon heating, the contraction of the wire is possible only if the rubber core is squeezed radially (Fig. 12.33(b)) and elongates axially. Upon deactivation of the SMA wire, the rubber core partially recovers its undeformed shape (Fig. 12.33(c)). The SMA wire can be modeled as in Scir
e Mammano and Dragoni [31,37]. Fig. 12.34(a) shows the freebody diagram of the rubber core under the action of the radial pressure of the SMA wire, pSMA, and of the external axial force, Fext, or its equivalent external pressure, pext. The external actions in Fig. 12.34(a) produce a generalized plane state of hydrostatic stress in the rubber core. Considering the generalized Hooke’s law, and letting εT (