Mathematical Modeling in Cultural Heritage: MACH2019 (Springer INdAM Series, 41) 3030580768, 9783030580766

Table of contents :
Preface
Contents
To Know Without Destroying?
1 The MADAR Project: Historical Heritage and Math Towards a Common Path
2 Potential and Weaknesses of the Porto Torres (Sardinia) Case Study
3 Construction Techniques, Features and Historical Phases of the Complex of Central Baths
4 To Know Without Destroying
References
Representative Volume Elements for the Analysis of Concrete Like Materials by Computational Homogenization
1 Introduction
2 Generation of the Geometry of the RVE
2.1 Spherical Aggregate Particles
2.1.1 Taking Process
2.1.2 Placing Process
2.2 Polyhedral Aggregate Particles
3 Micro–to–Macro Computational Homogenization
3.1 Periodic Boundary Conditions
4 Numerical Results
5 Conclusions
References
A New Nonlocal Temperature-Dependent Model for Adhesive Contact
1 Introduction
1.1 Rate-Independent Models
1.2 Rate-Dependent Models
1.3 Adhesive Contact with Nonlocal Effects
2 The Model and the PDE System
2.1 The State and Dissipative Variables
2.2 The Free Energy
2.3 The Dissipation Potential
2.4 The Balance Equations and the Constitutive Laws
2.5 The PDE System
2.6 Outlook to the Analysis
References
Chemomechanical Degradation of Monumental Stones: Preliminary Results
1 Introduction
2 A Mathematical Model of Marble Sulphation
2.1 A Simplified Chemical Reaction
2.2 The Hydrodynamical Model
3 The Variational Approach to Fracture
4 A Chemo-Mechanic Model
4.1 Mechanical Properties
4.2 Diffusivity
4.3 Mathematical Model
5 A Numerical Example
References
Modelling the Effects of Protective Treatments in Porous Materials
1 Introduction
2 Materials and Methods
3 Modelling of Capillary Imbibition
3.1 Boundary Conditions
4 Numerical Results and Comparison with Experimental Data
4.1 Numerical Approximation
4.2 Calibration of Parameters a, c
5 Conclusions and Perspectives
References
Mathematical Models for Infrared Analysis Appliedto Cultural Heritage
1 Introduction
2 Models
2.1 Optically Opaque Materials
2.2 Optically Semi-transparent Materials
3 Finite Element Implementation
4 Numerical Simulations and Comparison with Experimental Results
4.1 Optically Opaque Materials
4.2 Optically Semi-transparent Materials
References
Numerical Simulations of Marble Sulfation
1 Introduction
2 Mathematical Model
3 Numerical Approximation for Cartesian Domains
3.1 Accuracy and Efficiency
4 Numerical Approximation for Arbitrary Domains
4.1 Accuracy Test
4.2 Efficiency of Solvers
4.3 Domains Defined by Images
5 The Tickness of the Gypsum Crust as a Function of the Curvature
6 Conclusions
References
The Damage Induced by Atmospheric Pollution on Stone Surfaces: The Chemical Characterization of Black Crusts
1 Introduction, Black Crusts Formation Process
2 Analytical Methods Proposed for BCs Characterizations
3 Some Case Studies
4 Conclusion
References
Aging of Viscoelastic Materials: A Mathematical Model
1 Introduction
1.1 What Is Viscoelasticity?
2 Rheological Models for Viscoelasticity
3 Modeling the Aging
3.1 Derivation of the Constitutive Equation
3.2 The Motion Equation
3.3 The Kelvin–Voigt Limit
4 Mathematical Analysis of the Problem
References
A Quasi-Static Model for Craquelure Patterns
1 Introduction
2 Quasi-Static Evolutions by Alternate Minimization
2.1 Phase-Field Energy
2.2 Time-Discrete Evolution
2.3 Time-Continuous Evolution
3 A One-Dimensional Case Study
3.1 Continuous Displacement and Boundary Layer
3.2 Evolution of Cracks by Minimality
3.3 Second and Further Generations of Cracks by Minimality
4 Alternate Minimization in the One Dimensional Setting
5 Numerical Results for Uni-Axial Problems
5.1 Dyadic Structure for Short Bars
5.2 Periodic Patterns for Longer Bars
5.3 Irreversibility
6 Local Craquelure Patterns for a Real Life Specimen
References

Citation preview

Springer INdAM Series 41

Elena Bonetti Cecilia Cavaterra Roberto Natalini Margherita Solci Eds.

Mathematical Modeling in Cultural Heritage MACH2019

Springer INdAM Series Volume 41

Editor-in-Chief Giorgio Patrizio, Università di Firenze, Florence, Italy Series Editors Giovanni Alberti, Università di Pisa, Pisa, Italy Filippo Bracci, Università di Roma Tor Vergata, Rome, Italy Claudio Canuto, Politecnico di Torino, Turin, Italy Vincenzo Ferone, Università di Napoli Federico II, Naples, Italy Claudio Fontanari, Università di Trento, Trento, Italy Gioconda Moscariello, Università di Napoli “Federico II”, Naples, Italy Angela Pistoia, Sapienza Università di Roma, Rome, Italy Marco Sammartino, Universita di Palermo, Palermo, Italy

Springer INdAM Series This series will publish textbooks, multi-authors books, thesis and monographs in English language resulting from workshops, conferences, courses, schools, seminars, doctoral thesis, and research activities carried out at INDAM - Istituto Nazionale di Alta Matematica, http://www.altamatematica.it/en. The books in the series will discuss recent results and analyze new trends in mathematics and its applications. THE SERIES IS INDEXED IN SCOPUS

More information about this series at http://www.springer.com/series/10283

Elena Bonetti • Cecilia Cavaterra • Roberto Natalini • Margherita Solci Editors

Mathematical Modeling in Cultural Heritage MACH2019

Editors Elena Bonetti Dipartimento di Matematica Università degli Studi di Milano Milano, Italy

Cecilia Cavaterra Dipartimento di Matematica Università degli Studi di Milano Milano, Italy

Roberto Natalini Istituto per le Applicazioni del Calcolo Consiglio Nazionale delle Ricerche Roma, Italy

Margherita Solci Dipartimento di Architettura Universit`a di Sassari Alghero, Italy

ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-3-030-58076-6 ISBN 978-3-030-58077-3 (eBook) https://doi.org/10.1007/978-3-030-58077-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book collects some of the contributions presented in 2019 for the INdAM International Workshop “Mathematical modeling and Analysis of degradation and restoration in Cultural Heritage (MACH2019)”. The conservation of monuments, or more generally of cultural heritage, is a complex problem, which is deeply and intrinsically interdisciplinary. For instance, damage of monumental stones is a process in which the characteristics of the materials in combination with the environmental parameters are the main factors responsible for the decay of the artefacts. We need to take into account the effects of air pollution, which produce physical and chemical alteration processes due to the combined action of rain, wind, sunlight and freezing/unfreezing cycles. Besides, the pollutants form a deposit of particles and black encrustations. Another important aspect in the deterioration of the stone monuments is the mechanical degradation. So, to list some degradation phenomena occurring to artistic artefacts, we could mention corrosion and sulphation of different materials, damages and fractures, stress in thermomechanic systems and contact and adhesion problems. The traditional approach to all these problems has been driven, for many decades, by a strong focus on material and experimental sciences, in collaboration with art historians, architects and restorers. However, in the last few years, some research has been started to approach conservation of cultural heritage using mathematical models, numerical simulation and optimization. Models can describe the degradation processes, but they are also able to design more effective and optimized restoration strategies. Using simulation, it is possible to see the effects of different policies and support the authorities in charge to take more informed decisions. The main obstacle in using mathematical models is the lack of contacts and common background between different communities. On one hand, people traditionally working in cultural heritage usually did not receive a specific training in mathematics. On the other hand, in the mathematical community, it is difficult to understand the various problems arising in this area. For instance, even if a lot of research work has been done in modelling the behaviour of building subjected to strong mechanic stress, the scale of interest for cultural heritage is quite far v

vi

Preface

from the typical scales considered in civil engineering. Also, the usual standards considered in corrosion problems in industrial environments are not adequate to the care required to preserve artistic handcrafts. For all these reasons, we believe it was time to try to create a bridge between different groups of researchers, potentially interested in creating new tools to address the maintenance and the promotion of cultural heritage. After a series of quite useful, but occasional, experiences in the past few years, essentially provided by individual collaborations between researchers interested in specific topics, it was necessary to build up a more organized and organic community around some topics of general interest: modelling the evolution of damage under the pressure of external conditions, optimizing large-scale interventions to allow a sustainable fruition of monuments and improving the control of the existing protocols of restoration by an extended campaign of simulations. In this regard, the meeting at INdAM was quite successful, for the simultaneous presence of mathematicians, engineers, chemists and other researchers interested in these topics. The papers present in this book represent a balanced and worth reading selection of some of the works presented during the workshop. A first series of contributions presented in this book are aimed to use models to investigate mechanical properties of the materials. For instance, in the paper by Azzena and Busonera, a structural analysis tools is adapted to open a path to nondestructive investigation of historical structures, when archaeological research is not sufficient to provide the necessary framework. Another step in this direction is made in the paper by Bilotta et al., where Representative Volume Elements are introduced to devise computational homogenization of concrete structures in which coarse aggregates, mortar matrix and the interfacial transition zone are integrated to describe the macroscopic response of a structure under mechanical stress. In another direction, we find the contribution by Bonetti, Bonfanti and Rossi, which contains a survey on the mathematical modelling and analysis of adhesive contact and delamination. In the second part of this paper, the nonlocal adhesive approach by M. Frémond is followed to build up a new model for adhesive contact with thermal effects. Also, Negri considers the adhesion, in this case of a brittle layer on a stiff substrate. This yields a quasi-static limit model where craquelure patterns are governed in terms of an associated phase field. In the paper by Conti et al., we find a mathematical model to describe ageing effects occurring in viscoelastic materials, when a convolution kernel, which actually depends on ageing, takes into account for the delay effects due to the viscosity in the material. A second series of contributions describes the case when chemical processes became more relevant, sometimes the main source of degradation of artistic artefacts. Comite and Fermo report an in-depth study on the carbonaceous fraction in black crust samples coming from different Italian monumental stones. The identification of the main substances responsible for surface degradation phenomenon is crucial to define new conservative intervention strategies and also to provide a solid basis for new models of these phenomena. A partial result in this direction is contained in the paper by Bonetti, Cavaterra, et al., where a very complete model of concurrent chemical and mechanical degradation on natural stones, which also

Preface

vii

includes rugosity effects on the external surface, is presented. The main features of this approach are illustrated by specific numerical simulations, which show the synergic effects of chemical aggression and stress. Other numerical simulations for a similar model, considering only chemical effects, are the main concern in the work by Coco, Donatelli, et al. In this case, complex geometric features of the domain, possibly an artistic artefact, are solved on a Cartesian grid via a clever level-set formulation. So, it is possible to study, in a two-dimensional framework, the influence of the surface curvature on the growth of the external chemical crust. Models are used not only to describe the occurring processes inside of the materials but also to optimize the protective treatments, which try to slow down the natural damaging action of external factors. In their preliminary study, Bretti, De Filippo, et al. use experimental data obtained on limestones, before and after the application of a chemical protective, to calibrate a model of capillarity rise of water in stones. A final application of mathematics is to assess in a nondestructive way the internal state of materials. This is the aim of the paper by Caruso, Orazi, et al., where pulsed active infrared thermography is used for the analysis of cultural heritage artefacts like ancient bronze statuary and manuscripts. A mathematical model is implemented for an effective analysis of the thermographic signal. The modern science of conservation of cultural heritage has a very strong tradition in Italy, starting with the seminal contributions of Cesare Brandi, who, in his book Theory of Restoration, defines restoration as “the methodological moment in which the work of art is appreciated in its material form and in its historical and aesthetic duality, with a view to transmitting it to the future”. We are aware that this volume is just a first small step towards a more scientific and quantitative approach to conservation and restoration. Nevertheless, we hope that, thanks to a more intense and better organized dialogue with the experts in conservation science, it will be possible one day to add a “mathematical” dimension to the transmission of a work of art to the future. Milano, Italy Milano, Italy Roma, Italy Alghero, Italy July, 2020

Elena Bonetti Cecilia Cavaterra Roberto Natalini Margherita Solci

Contents

To Know Without Destroying? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Giovanni Azzena and Roberto Busonera Representative Volume Elements for the Analysis of Concrete Like Materials by Computational Homogenization .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Antonio Bilotta, Andrea Causin, Margherita Solci, and Emilio Turco A New Nonlocal Temperature-Dependent Model for Adhesive Contact .. . Elena Bonetti, Giovanna Bonfanti, and Riccarda Rossi Chemomechanical Degradation of Monumental Stones: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli, and Roberto Natalini Modelling the Effects of Protective Treatments in Porous Materials.. . . . . . Gabriella Bretti, Barbara De Filippo, Roberto Natalini, Sara Goidanich, Marco Roveri, and Lucia Toniolo Mathematical Models for Infrared Analysis Applied to Cultural Heritage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Giovanni Caruso, Noemi Orazi, Fulvio Mercuri, Stefano Paoloni, and Ugo Zammit

1

13 37

59

73

85

Numerical Simulations of Marble Sulfation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107 Armando Coco, Marco Donatelli, Matteo Semplice, and Stefano Serra Capizzano The Damage Induced by Atmospheric Pollution on Stone Surfaces: The Chemical Characterization of Black Crusts . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123 Valeria Comite and Paola Fermo

ix

x

Contents

Aging of Viscoelastic Materials: A Mathematical Model .. . . . . . . . . . . . . . . . . . . 135 Monica Conti, Valeria Danese, and Vittorino Pata A Quasi-Static Model for Craquelure Patterns .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 Matteo Negri

To Know Without Destroying? Giovanni Azzena and Roberto Busonera

Abstract This paper proposes some considerations that arise from an experiment conducted in the Archaeological Park of Porto Torres, where a predictive and noninvasive diagnostic tool based on a numerical code (TEAM) was tested. It is an innovative research topic that, in recent years, has seen mathematics contribute to artistic protection as well as in the hard field of the protection of historical buildings and archaeological structures. Starting from a point of view that arise from a long archaeological experience, it was proposed an application in a context in which the use of predictive models of damage might answer to a real need. Adapting the recent structural analysis tools to the complex needs and to the lack of archaeological research data could open a path towards new and different solutions, no longer intrinsically destructive, for the investigation of the historical structures. It could be a new point of view that starts from the construction of a common language between so different disciplines, towards a real communication and mutual contamination. Keywords Predictive models of damage · Archaeological protection · Turris Libisonis Archeological Park · Numerical simulation for ancient masonry

1 The MADAR Project: Historical Heritage and Math Towards a Common Path A protection of the archaeological field inspired exclusively by an idea of conservation can assume negative and self-defeating appearance if it is not pushed towards clear design guidelines. In this sense, the inspiration for a survey like the

G. Azzena () · R. Busonera Dipartimento di Architettura, Design e Urbanistica, Università di Sassari, Alghero, Italy e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_1

1

2

G. Azzena and R. Busonera

one developed in the MADAR project1 arises from a wide and obvious fork between the serious need for tools useful to protect historical buildings on one side, and the extraordinary potential and flexibility of design tools and structural analysis codes, which have been developed in the fields of numerical analysis and construction science. From this comparison, a question: is it possible that a tool which was born as an aid for the design of structure, could evolve into an available resource for historical buildings analysis? If the answer is yes, then which database should be provided in order to construct an adequate calculation code, which can process the information and be a safe and right device, able to read the history of a building and to predict the development of damage? An effective response should rely on a large amount of data incoming from historical sources, archives and past researches: basically, from a full and conscious knowledge of the monumental complex and of its functional and structural history. The case study of this investigation is a specific wall and its masonry. The masonry has peculiar features that make it different from more regular construction types, due to its inhomogeneous, anisotropic and highly asymmetric structure. Indeed, historical masonry analysis add difficulties, related to geometry and materials, to these complexities, due to the problems of developing a suitable investigation, that does not modify or damage the constructive complex (just for this reason, hard to be protected). Hence, the use of a computational and predictive code need to know even the geometric and material features, in order to rebuild them through indirect analyses. Therefore, the implementation of the code could rely, as much as possible, on a deep and extensive collection of direct and indirect preliminary data. In this context, even the strong need for classification, usually peculiar for archaeological research, can acquire an innovative and practical purpose, that is far from the enumerative risk which the keenness of the research is sometimes affected. However, because of the type of design reconstruction carried out by mathematical and information technology tools, it is possible that historical and material data could not be useful for each situation. The characteristics of the mathematical modeling marks a particular need of choice: no code would be able to process an unlimited amount of data: in the balance between the necessary richness of data base and the synthesis required by the management times, a very interesting space

1 This work arises from a collaboration between the authors and the regional office of Sardinia for the archaeological protection. We acknowledge the multidisciplinary research group that contributed to the development of the experimentation project about a numerical and structural modeling in the Archaeological Park of Porto Torres. In brief, Margherita Solci who was the scientific coordinator of the project, Emilio Turco for structural consultancy, Francesca Tedesco for the definition of the numerical code, James Rombi for computer processing, Alessandro Forci and Antonio Santonastaso for lithological analyses, Anna Camoglio e Giuseppe Padua for the relief of the monument. Specific thanks have to be addressed to Alessandra Urgu, who managed the architectural study. The research project was funded under the POR-FESR 2007–2013, and it has been coordinated by Giovanni Azzena for the archaeological field. A detailed description of the context and the technical results of the project is in [1].

To Know Without Destroying?

3

of reasoning, experimentation and theoretical investigation could be opened up. In brief, in order to understand when, how and why a building has been built [2, p. 25].2

2 Potential and Weaknesses of the Porto Torres (Sardinia) Case Study The case of Porto Torres shows many unusual aspects: at the end of a long process of study and analysis of the structures of the ancient city,3 it is now possible to know about 20% of the urban area, while the remaining 80% is still almost completely investigable. The process of studying the city life stages describes a decaying center perhaps as early as the fifth century A.C., when it has been moved to a position of less authority by the small medieval town, gathered around the church of San Gavino. The growth of the contemporary city brought together this settlement with the one located around the port area, heir of the very active Roman area. The join of the two villages has not affected the ancient city at all, and left the hill of the Roman city essentially free. New phases of development of the urban center also refer to events related to the realization of the SIR (Italian Resins Society), and the birth of the industrial area, that has had particular feedback (mostly negative) within the local history and both in urban planning and environment. Therefore, it is evident how the phases of urban evolution have affected the area of ancient Turris in many ways and how much the interference between the urban and the industrial pole development has been concentrated in the historical-environmental area already occupied by the ancient city (see Fig. 1). The archaeological area is now standing in territories where financial, political and administrative affairs are involved, and in which even the Ministerial Office for the Archaeological Protection had to make choices that were rarely aligned to the potential of urban development or to the possibilities of study, conservation and protection of the ancient remains. Nowaday, it could be possible to affirm that the complex of the Central Baths of Porto Torres (Fig. 2), as well as the Roman bridge and the Basilica of St. Gavino, is one of the most representative testimonies of the history of the city. Indeed, archaeological data would confirm its topographical centrality, as well as the symbolic role which is connected to the phases of life of the city and to the

2 Furthermore, with reference to the comparison between material reality and general sources, see [3]

Il muro, cioè la realtà materiale, si impone alla tradizione scritta. (The wall, that represents the material reality, imposes itself on written tradition). 3 A wide report on the studies concerning the Roman city can be found in [4–7].

4

G. Azzena and R. Busonera

Fig. 1 Topographical framework of Porto Torres. The archaeological area is recognizable with in the highlighted square, between the industrial sector and the contemporary city. Reproduced with permission from [8]

Fig. 2 View of the archaeological park of Porto Torres. Reproduced with permission from [8]

overlapping of different building and functions that marked the main events of the ancient Turris Libisonis, from its birth, up to the dissolution of its components. Then, it is clear why the Central Thermal Baths of Porto Torres could be an architectural element useful for the understanding of the ancient city and a fundamental database for the deepening of the researches: the re-reading and recomposition of the archive data has allowed the elaboration of new hypotheses

To Know Without Destroying?

5

Fig. 3 Thermal bath of the ancient Turris Libisonis (III century A.C.). Reproduced with permission from [8]

on the evolution of the city, and it suggested some methods for a potential reconstruction, or transformation, of the ancient complex. In addition to these main aspects, it should be added some considerations about the constructive and dimensional characteristics, its recent history and the investigation procedures that have characterized the knowledge process. This kind of study was dealt as a part of the MADAR project and it was useful in order to carry out a new support for the definition of a mathematical model, virtually able to define the response of oscillatory stress. It involved an architectural complex in the archaeological park of Porto Torres (Fig. 3), whose access was probably on the northern side of the building, bounded by a quadrangular arcade composed of a double row of brick pillars. Here, the rectangular room accessed from the arcade compartment reveal the area of frigidarium, where two symmetrical basins are still easily recognizable. On the other side, two small apodyteria have been obtained, whose walls have been taken into consideration for this study4 (Fig. 4). It was considered a straight-lined wall, with an east-west orientation and a various height between 0.45 m, measured at the west side, and 6.70 m at the east side. Therefore, close to the NE corner and between the apodyterium and the main entrance of the complex, is a communication portal, which is completely devoid of the brick curtains that should define its structural limits.5 Although only partially traceable, it is clear how much the structure has undergone numerous restorations (see [8, p. 96]), which have unfortunately accelerated

4 In this sense, the collaboration with the Regional Office of Sardinia for the archaeological protection was essential in the evaluation and selection of the structures to be tested. 5 Only in the basement perimeter, it is possible to identify some rows of bricks, both externally and internally [8, p. 94].

6

G. Azzena and R. Busonera

Fig. 4 Stratigraphic reading of the studied walls. Reproduced with permission from [10]

Fig. 5 Dimensional analysis of wall types. Reproduced with permission from [8]

the erosion process and the progressive damage, as well as compromising its ease of use. From the use of mines (see [7, p. 25]) to invasive interventions linked to excavation campaigns, which have affected archaeological deposits as well as bearing structures, all the problems to which the ancient building has survived are evident. The same solutions have also deleted a large amount of data relating to the structural characteristics of the building, that can hardly be recovered and that describe a general condition of mechanical and architectural weakness (Fig. 5). A large part of the building is dangerously lacking in the original decorative coatings and plaster, of which only a few small portions remain in structural bases

To Know Without Destroying?

7

Fig. 6 An example of sheet used for the census of mortars. Reproduced with permission from [8]

that are no longer protected by soil. Many other interventions of architectural consolidation have been carried out by the use of cement mortars (Figs. 6 and 7), that has been rejected by the ancient structures and have been not succeeded in repelling the harmful consequences of the sea and of the frequent acid rains (see [9, pp. 24–25]), whose infiltration inside the walls has caused the breaking of the surfaces with evident breakup of some structural parts.6

3 Construction Techniques, Features and Historical Phases of the Complex of Central Baths The very initial phase of the research, which is precisely connected to the “postdesign” nature of the model’s elaboration, involved the knowledge of techniques and materials used in the building process of the structure investigated. The contribution of archeology of architecture has been fundamental in order to find the right data for the creation of the mathematical model and to the full knowledge of the monument and the material problems arose through the stratification that characterizes it, that is an essential element for the creation of a suitable mathematical model. An important goal already achieved was the creation of an atlas for the classification and analysis of the wall types. From the stratigraphical investigations and the interpretation of direct and indirect sources, it was possible the full understanding of the constructive and destructive events. Still today, this aspect represents an essential knowledge for future planning interventions, that should be define from a depth awareness of the typology and characteristics of ancient materials, in order to arrive to a new, more modern and wiser solutions.

6 A condition that has basically accelerated the process of disintegration of materials, instead of slowing it down.

8

G. Azzena and R. Busonera

Fig. 7 Examples of mortars used in modern restoration. Reproduced with permission from [8]

Consequently, it was possible to understand the degree of alteration of the original configuration of the building, that was determined by the reuse of some part of it and events of collapse after the first phases of definitive abandonment. In addition to these phenomena, translations and rotations caused by deformations and breakup of masonry parts have been only partially arrested as a result of the conservation work carried out in the early twentieth century. Then, all the masonry techniques were detailed,7 while the stratigraphic analysis and the informations obtained from the sources, have helped to develop a diachronic reconstruction of the building stages, that have been linked to the natural erosion of the materials and to the incidence of other phenomena of degradation, caused by the long life of the building (see [10, pp. 143–152]). From this point, three macroperiods connected to as many moments and phases of use were identified.

7A

broad detail of this study is in [10, pp. 131–142].

To Know Without Destroying?

9

In the first period, it was recognized the moment of realization and the wall investigated by the experimentation (III–IV century A.C.).8 The thermal bath overlaps a portion of the oldest city, which was demolished to allow the installation of the new complex and which exploits the foundation systems defined by the remains of previous buildings.9 A special attention, which is still limited by contemporary interventions, arises from the analysis of the signs of the construction process,10 from which it was possible to propose phases and construction methods of the building. Moreover, the study of the materials has also made possible to recognize the use of local bricks, probably coming from ancient emporium, as well as the areas of stone production, not far from the ancient city: it could be a fundamental aspect, if read as a possibility for further and more detailed tests on the mechanical response of construction materials. Furthermore, the testimonies on plaster and coatings have allowed to rebuild a rich context in reference to the workers involved in the construction site. Then, the first moments of abandonment of the thermal complex, and a partial reuse as homes and craft industries, could be linked to a second phase.11 Between the fifth and sixth centuries, the signs of reuse are clear (see [12, pp. 314–315]), as well as the collapse of the vaults of some heated rooms and the breaking of the floors. It could be added even demolition operations and material erosion, caused by poor maintenance and the action of atmospheric agents.12 In a third phase the complex of thermal bath had to be already extremely modified. At the beginning of the twentieth century, an attempt to stop this degradation process was made, but almost all the interventions have often contributed to the erosive process, instead of arresting it. This last point of view should represent an important warning, linked to the need to understand the features of materials, in order to acquire useful data to predict the evolution of structural instability. The analysis involved only two structural elements, as a part of a larger complex that would require certainly more study. In spite of this, it was possible to achieve a clear knowledge, that was methodologically useful and potentially replicable for the study of any type of historical structure.

8 It was a period of great monumentalization throughout the territory of Sardinia, which was characterized by the construction of many thermal buildings. See [11, pp. 115–199]. 9 See the archaeological report written by G. Maetzke (Archaeological Superintendency Archive, binder called Porto Torres-Re Barbaro, n. 8–21). 10 The identified signs concern rectangular holes (21 cm per side). The distance between the rafters is about 100–110 cm, while that one between the support surfaces is about 120 cm ([10, p. 143]). 11 See again the archaeological report written by G. Maetzke (Archaeological Superintendency Archive, binder called Porto Torres-Re Barbaro, n. 8–21). 12 The action of counting was combined with that one of atmospheric agents which caused a complete detachment of the external curtains. Even for this reason, reconstructing the collapsing phases without a detailed report was rather hard ([10, pp. 146–147]).

10

G. Azzena and R. Busonera

4 To Know Without Destroying The investigations carried out let to integrate known data with new information relating to the dynamics and phases of the ancient building, while the research on construction techniques has been an important contribution to achieve the goals set by the MADAR project. These have been useful not only to understand the history of the monument, but to appreciate how human and natural interventions have affected the structure and have changed its essential characteristics over time. Due to the erosion process and restoration interventions, it has caused a progressive replacement of original elements with new ones, but these interventions have played a negative role in order to define a distorted view of the monument, without succeeding, however, in stopping the degradation process. As a consequence of a long process, ancient walls have not tolerated modern materials, and have caused a rejection from wall surfaces, made even worse by the closeness of the sea and the action of atmospheric agents. Even for this reason, all the collected data lead to think carefully about the coexistence of materials coming from different contexts, and suggest some guidelines to avoid future disorders. Starting from the data provided and appropriately selected, and from the subsequent survey phase, a specific finite element called TEAM was developed by the mathematicians involved in the project. It is a mathematical code which is able to describe a particular masonry element, and to provide its response arising from different types and intensities stresses, in order to identify the evolution of the damage and to make any future recovery intervention more precise. From this perspective, MADAR project could be a serious resource. The usual emergency conditions in ancient buildings protection make essential the possibilities offered by the proposed study, that could be a tool of knowledge able to guarantee a non-invasive and effective analysis, as well as being useful to prevent structural degradation, and let to plan more suitable actions. The study outlines an “exportable” model, that is born from a detailed knowledge of the phases and uses of any architectural artefact, and that let to suggest suitable solutions for material discontinuities and instability phenomena. Furthermore, the opportunity of creating a catalogue that shows the characteristics of the materials allows to manage data which are useful not only for mathematical modeling, but for the creation of a basic knowledge for any type of intervention. It is a kind of filing that does not use up the role of the mathematical tool, but it is able to increase its potential and possible applications. The main goal is to verify the structural conditions for a review about the response to different forces, as static and dynamic behavior of masonry buildings with different textures, arcs, vaults and domes that are close to collapse, thermomechanical analysis for a test of structural performance in according with the temperature conditions, erosion tracing. The final product is a functional tool that can be managed by those who directly deal with protection and which can be able to suggest a plan for targeted, non-

To Know Without Destroying?

11

invasive or expensive interventions and, as far as possible, compatible with the context. If the archaeological knowledge cannot ignore the destruction and alteration of the state of things, it could be possible turn to the use of mathematical modeling tools such as a new survey or a virtual excavation, which is able to allow the inevitable errors, without compromising the context, and to allow experimentations and simulations of the phenomena. In other word, a tool that let the physical intervention to be reduced to the essential, in order to keep the original shape safe. Acknowledgments The authors gratefully acknowledge financial support from Regione Sardegna (POR-FESR 2007-2013). Giovanni Azzena acknowledges the University of Sassari for funding his research in the frame of the Fondo di Ateneo per la ricerca 2019.

References 1. Cicalò, E., Solci, M. (eds.): Rinnovare la tutela. Modelli matematici e grafici per una ridefinizione delle prospettive. Gangemi, Roma (2016) 2. Giuliani, C.F.: L’edilizia nell’antichità. Carocci, Roma (2006) 3. Lugli, G.: L’esame critico del monumento negli studi di topografia romana. Historia 8, 387–409 (1933) 4. Azzena, G.: Turris Libisonis, la città romana. In: La Sardegna (Luoghi e tradizioni d’Italia), 368–380. Istituto Poligrafico e Zecca dello Stato, Editalia, Roma (1999) 5. Azzena, G.: Turrem pervenire. Ipotesi sui sistemi di accesso all’antica Turris Libisonis. In: Del Vais, C. (ed.) EPI OINOPA PONTON. Studi sul Mediterraneo antico in ricordo di Giovanni Tore, 659–668. S’Alvure, Oristano (2012) 6. Boninu, A.: L’impianto urbanistico di Turris Libisonis. In: Boninu, A., Le Glay, M., Mastino, A. (eds.) Turris Libisonis Colonia Iulia. Publications of the Department of History of the University of Sassari, Sassari (1984) 7. Boninu, A., Pandolfi, A.: Porto Torres. Colonia Iulia Turris Libisonis. Sassari (2012) 8. Azzena, G., Busonera, R., Urgu, A.: Matematica-Degrado-Archeologia. Archeologia dell’Architettura XXIII, 85–98 (2018) 9. Achenza, M., Sanna, U.: Il manuale tematico della terra cruda. Topografia del Genio Civile, Roma (2009) 10. Urgu, A.: Rappresentare per studiare la storia del costruito: il caso di due muri delle terme centrali di Porto Torres. In: Cicalò, E., Solci, M. (eds.) Rinnovare la tutela. Modelli matematici e grafici per una ridefinizione delle prospettive, pp. 125–156. Gangemi, Roma (2016) 11. Ghiotto, A.R.: L’architettura romana nelle città della Sardegna. Quasar, Roma (2004) 12. Maetzke, G.: Scavi e scoperte nel campo dell’Archeologia Cristiana negli ultimi dieci anni in Toscana e Sardegna. In: Atti del II Congresso di Archeologia Cristiana, Matera, 25–31 maggio 1969, 314–315. Roma (1971)

Representative Volume Elements for the Analysis of Concrete Like Materials by Computational Homogenization Antonio Bilotta, Andrea Causin, Margherita Solci, and Emilio Turco

Abstract The problem of devising an appropriate Representative Volume Element (RVE) for the analysis of concrete-like-materials is throughly discussed in the range of the elastic behavior. To this end, assuming concrete as a two-phases material (mortar and aggregates), the geometry of the RVE is automatically generated on the basis of spherical or polyhedral aggregates by proposing a new algorithm for the case of the polyhedral shapes. The associated apparent macro-response is evaluated by computational homogenization and its feasible use is pointed out. The subsequent numerical experimentation aims to highlight the influence of the relevant parameters such as the RVE’s size, aggregates shapes, constitutive moduli of the constituents and applied boundary conditions on the evaluation of the RVE’s macro-response. Keywords Computational homogenization · RVE for structural analysis · Models of concrete-like materials · Prediction of damage

1 Introduction Concrete is one of the most important building materials worldwide. From the designer’s point of view, it is usually considered as a homogeneous material and, under low load levels, the hypothesis of linear elastic behaviour is generally accepted [1]. On this basis, several phenomenological models have been proposed for structural design, as they may realistically predict the concrete behaviour also with respect to fracture processes. However, these models are not suitable for material optimization and they can neither describe nor allow to estimate the effects

A. Bilotta () DIMES, Università della Calabria, Rende, Italy e-mail: [email protected] A. Causin · M. Solci · E. Turco Dipartimento di Architettura, Design e Urbanistica, Università di Sassari, Alghero, Italy e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_2

13

14

A. Bilotta et al.

of the concrete composition on the macroscopic material behaviour (for instance, the propagation of microcracks occur on a smaller length scale than the structural length). Moreover, it often happens that the internal parameters of these material models cannot be directly measured in a physical experiment, see e.g. [2, 3]; as a result, their identification is generally difficult. The modelling of the heterogeneities of the concrete gives a deeper understanding of the physical processes determining the macroscopic behaviour and the research interest on this subject is very active [4]. In mesoscale simulations (see for example [5]) the numerical model explicitly represents the individual components of the heterogeneous internal structure of the concrete, such as the shape and the spatial distribution of the aggregates. In this way, specific material models can be assigned to each component and rather simple material formulations can be used for each material phase. Other important phenomena (see for example [6, 7]) can be also described. Among others: size effects on the nominal tensile strength, the stochastic scatter of the results in simulations with identical specimens but different aggregate configurations, the localization of damage and plasticity due to the heterogeneity of the material and also dissipation phenomena [8–10]. At the mesoscale level there are three different phases that can be identified [11]: coarse aggregates, mortar matrix and the interfacial transition zone (for brevity: ITZ) between aggregates and matrix. Each one of these components presents in itself a finer heterogeneous nature (the mortar matrix, for instance, mainly consists of hardened cement paste, air voids and all the fine aggregates which are not explicitly considered as an individual phase in the model), but in the present work this fact will be neglected. Furthermore, since the aim of the paper is to devise a Representative Volume Element (RVE) for a computational homogenization strategy, the ITZ has been neglected too and only the cement matrix and the coarse fraction of the aggregates are taken into consideration. Therefore, in this context a crucial point is an appropriate description of the internal material geometry. The model of the internal geometry of the concrete requires the description of the shape of the aggregate particles. These shapes have a significant influence on the stress distribution within the material and, consequently, on the cracks initiation and on the damage accumulation up to the macroscopic failure. Two main approaches to build a numerical mesoscale model can be distinguished: the first is based on image processing techniques [12, 13] while the second, followed also in the present work, is the artificial generation of the microstructure. Other works tackled with this kind of problem, in [14] the geometry of natural rounded aggregates has been simulated by using an appropriate morphological law, following any particle size distribution, with the aim to investigate the specific properties of the surface layer of a composite material such as concrete. In [15] the concrete aggregate particles are described on the basis of spherical harmonic functions and compared with 3D particle images acquired via X-ray tomography. In [11] the particles, the matrix material and the interfacial transition zone are considered as separate constituents and the particles are represented as ellipsoids, generated according to a prescribed grading curve and randomly placed into the specimen. Finally in [16] the threedimensional geometrical model for concrete is based on a Monte Carlo simulation

Concrete-Like RVEs for Computational Homogenization

15

method and the aggregate particles are spherical, generated from a certain size distribution and then placed into the RVE. In this paper the geometry of the RVE is generated on the basis of two algorithms. The first, similar to the one proposed in [16], is used for the creation of the spherical aggregate particles via a take-and-place approach: the particles are generated on the basis of a prescribed grading curve and then placed inside the RVE by checking their intersections with the RVE’s boundary. The second algorithm handles the generation of polyhedral aggregate particles and reverses the approach by following a place-and-take method. In particular, in a first step a Voronoi’s tessellation of the assigned RVE is performed and the polyhedral particles are randomly placed inside the volume; in the second step the particles are rescaled in order to match the volume occupied by the cement matrix and the required grading curve for the aggregates. In this way, the “placing” phase, usually particularly tedious in the case of polyhedral aggregate shapes, is completely avoided. The two algorithms quickly described above are the starting point for the study here presented with the aim to come up with feasible RVEs for the numerical analysis of concrete structures. The definition of an RVE capable to reproduce the real macroscopic properties of randomly heterogeneous materials is a problem which has been extensively considered in literature, see for example [16, 17]. However, very often, the approach used is statistical in nature leading to RVEs which must include a very large number of the composite micro-heterogeneities. In [18] a more pragmatic definition of the RVE is proposed by answering to the following question: which is the minimum structural component size that can be treated by macroscopically homogeneous constitutive representations. In this paper, in order to devise an adequate RVE for real structural analysis, the point of view proposed in [18] is adopted. It is important to stress the necessity of an effective evaluation of the concrete macro-response and, at the same time, the control of the computational costs. To this end, several concrete RVE’s generated on the basis of the algorithms described in Sect. 2 are discretized using the aligned approach in which the finite element boundaries are coincident with the interfaces of the material and therefore there are no material discontinuities within the elements. Within the framework of the first order computational homogenization [19], whose main points are recalled in Sect. 3, the direct computation of the boundary value problem defined at the micro-level is performed and the macro-properties of the concrete are determined. In the numerical experimentation presented in Sect. 4, several parameters affecting the mechanical response of concrete are investigated: the RVE’s size, the percentage of the aggregate particles, the shape of the inclusions, the difference between the constitutive parameters of the cement matrix and of the aggregate particles. Also, the influence of the applied boundary conditions (kinematic or periodic), have been considered due to their important role with respect to both the representativeness of the evaluated macro-properties and the computational costs implied for their evaluation. What can be concluded on the basis the work here presented is described in Sect. 5.

16

A. Bilotta et al.

2 Generation of the Geometry of the RVE Throughout this work, concrete is simulated as a two phases material consisting of a cement matrix and a set of aggregate particles. The particles are selected from an assigned size distribution and arranged in the concrete RVE without mutual intersections. Aggregates generally occupy a fraction varying from 60 to 80% of the concrete volume and can be divided into fine and coarse aggregates. Fine aggregates comprehend sand or small residuals produced by the crushing process of the stones; these particles pass through a 7 mm sieve. Coarse aggregates are particles with a diameter greater than 7 mm and they usually form from 25 to 40% of the concrete volume. In order to assure good workability and mechanical properties, mainly related to the minimization of the voids, the size distribution of the aggregates is usually chosen on the basis of grading curves prescribed by several Standard Specifications. Grading curves express the cumulative percentage passing through a series of sizes of sieve openings, one of these is the Fuller curve described by the simple equation P (d) = (d/dmax )n

(1)

where P (d) is the cumulative percentage passing a sieve with aperture diameter d, dmax is the maximum size of aggregate particles and n can vary from 0.45 to 0.7. Moreover, gravel aggregates usually have a rounded shape while crushed stone aggregates have an angular shape. In what follows, the geometry of the particles will be spherical or polyhedral. The generation of the geometry of the inclusions ist limited to the coarse fraction of the aggregate, assuming that the fine fraction is dispersed in the cement matrix. Aggregate percentages of the concrete volume greater than 40% will not be considered allowing the generation of RVE geometries and of the required finite element meshes by avoiding an excessive growth of the computational costs. In this respect the discretization of such complex microstructures can be afforded on the basis of aligned and not-aligned meshing approaches. Not-aligned meshing approaches, see for example [20], allow to assign the mesh to be used in the analysis and to control the required computational cost in advance, however the resulting resolution of the geometrical description is tied to the resolution of the finite element mesh used. On the contrary, the aligned meshing approaches, used for example in [11], have the advantage of explicitly representing the boundaries between aggregates and matrix, but they allow a weaker control on the sizes of the generated meshes. However, in order to better catch the differences between spherical and polyhedral shapes, the aligned meshing approach is here adopted.

Concrete-Like RVEs for Computational Homogenization

17

2.1 Spherical Aggregate Particles The generation of the RVE geometry in the case of spherical aggregate particles follows a scheme similar to the one proposed in [16] with a separation of the algorithm into two processes: taking process and placing process. The algorithm here adopted uses only a slightly different approach in the taking process used to define the desired percentage of the coarse aggregates Pc .

2.1.1 Taking Process The assigned diameter for the biggest sphere is used to initialize the generation of the sequence of the diameters, i.e. d1 = dmax . The diameters of the smaller spheres are generated in descending order until the desired volume of coarse aggregate Vc = Pc VRV E is reached by using the equation (see Eq. (1)) i (di+1 /dmax ) = 1 − n

j =1 Vj

Va

(2)

and by perturbing the obtained diameter di+1 through a random oscillation giving a variation of the obtained volume not greater than Vi − Vi+1 . In Eq. (2) Vj is the generic volume of the spheres previously generated and Va = Vc + Vf = Pa VRV E is the total volume of the aggregate particles which is fixed by assigning also a value for Pa in each simulation (Vf is the volume of the fine fraction of the aggregates). As already stated, the fine fraction of the aggregates is considered dispersed in the cement matrix.

2.1.2 Placing Process The subsequent placing process randomly arranges the family of spheres previously generated on the basis of the algorithm proposed in [16] in which the details of the strategy used for checking mutual intersections of particles are described.

2.2 Polyhedral Aggregate Particles In the case of polyhedral aggregate particles the geometry of the inclusions is generated by a Voronoi tessellation as described in the web resource [21] which also provides the open source code used in the present work. The tessellation of the RVE domain is obtained by assigning a set of randomly placed points to whom a cell of space is assigned in the way defined by the Voronoi’s algorithm. In this way the placing process is implicitly performed by the Voronoi tessellation, the taking

18

A. Bilotta et al.

process is then performed by rescaling each particle in order to satisfy Eq. (1). The whole process can be described as follows. • A first tessellation of the assigned RVE is generated by assigning the number of aggregate particles whose centroids are randomly placed in VRV E . • The obtained polyhedral aggregate particles are ordered in descending order with respect to their nominal sieve size d¯ (for each polyhedron, d¯ is the maximum distance between two vertices). • The polyhedron with the biggest d¯ is rescaled through a factor s1 in order to match the assigned dmax , i.e. s1 = dmax /d¯1 . The other polyhedra are subsequently rescaled on the basis of the Eq. (2) which furnishes for each polyhedron the new nominal sieve size and the related scaling factor si+1 = di+1 /d¯i+1 . • If the resulting aggregate volume Vc matches the desired value within a desired tolerance, the generation of the RVE geometry is completed; otherwise, the number of particles is increased or decreased and the generation process is repeated from the beginning. The algorithm is very simple because the placing process is completely avoided, i.e. there is no need to check for intersections between particles or between a particle and the RVE’s boundary. Moreover, only few attempts are needed to reach the desired Pc . However this kind of approach is limited to not so high values of aggregate percentage, i.e. not so high numbers of aggregate particles, otherwise the possibility to have also rescaling factors greater than one would require an explicit placing process. This is not a limitation for the present work in which only the coarse fraction of the aggregates is described. Of course, for the automatic generation of more generic microstructures many other types of algorithms are possibile; more details can be found in [22], where several ones are presented together with their ranges of applicability. Figure 1 compares the RVEs obtained by simulating spherical and polyhedral particles for two values of the aggregate percentage, namely 10 and 30%. In the captions the number of particles generated by the algorithms and the number of nodes of the discretization generated by the mesh generator here adopted, see [23], are also reported. As expected, the desired aggregate volume percentage is reached by the spherical particles with a smaller number of spheres and, as a consequence, a smaller size of the finite element mesh. The mesh is always constituted by 10–nodes tetrahedral elements and, for obvious reasons, the mesh of the cement matrix is not shown. In order to check if the desired distribution of the particle sizes is obtained, Fig. 2 shows the grading curves given by simulating spherical and polyhedral aggregates.

Concrete-Like RVEs for Computational Homogenization

19

(a)

(b)

(c)

(d)

Fig. 1 RVE geometries obtained with an RVE’s size = 45 mm. (a) Pc = 0.1, particles = 4, mesh nodes = 6528. (b) Pc = 0.3, particles = 36, mesh nodes = 45790. (c) Pc = 0.1, particles = 15, mesh nodes = 9340. (d) Pc = 0.3, particles = 140, mesh nodes = 146,447

20

A. Bilotta et al.

Fig. 2 Grading curves obtained with spherical (a) and polyhedral (b) aggregates on an RVE with size = 45. The curves are compared with two Fuller’s grading curves, Eq. (1), with an exponent equal to 0.45 and 0.7

3 Micro–to–Macro Computational Homogenization The concrete RVEs generated as described above are analyzed on the basis of the computational homogenization approach proposed in [19, 24]. At the micro-level the problem is formulated as a standard problem of a quasi-static continuum solid analysis. Equilibrium equations are defined on the RVE in absence of body forces, i.e. ∇·σ =0

(3)

together with the constitutive prescriptions σ = C : ε,

(4)

and the strain-displacement relationships ε=

 1 ∇u + ∇uT . 2

(5)

The displacement field u, the strain field ε and the stress field σ are evaluated numerically by a finite element solution on the basis of different types of prescribed conditions on the boundary of the RVE, i.e. static, kinematic or periodic. In particular kinematic and periodic boundary conditions are typically used in computational homogenization.

Concrete-Like RVEs for Computational Homogenization

21

The macro–stress  associated to the generic RVE is evaluated as follows, by averaging the stress field σ obtained by solving problem (3):  1 σ dV . (6) = VRVE VRVE Its tensorial components, see [19], can be evaluated by summing over the boundary nodes of the FEM discretization of the RVE by ij =

1



VRVE

p

p p

ti xj ,

i, j = 1 . . . 3 ,

(7)

where t p is the resulting external force and x p is the position of the boundary node p. As a kinematic counterpart of the macro–stress, the macro–strain is defined as  1 E= ε dV (8) VRVE VRVE and it is related to the macro–stress through the Hill–Mandel principle, see [5],  1 σ : ε dV . (9) :E= VRVE VRVE The constitutive relationships at the macro–level,  = CE, are defined through the macro–tensor C whose components are expressed (see [19]) as follows: Cij hk =



1 VRVE

p

p

pq q

xj Kih xk

i, j, h, k = 1 . . . 3 .

(10)

q

In this expression, the indexes p and q range over all the mesh nodes of the RVE boundary, x p and x q are the coordinates of the two nodes and the matrix K is the matrix obtained from the stiffness matrix of the RVE by condensing out all the internal degrees of freedom with respect to the degrees of freedom relative to the boundary nodes, Kpq is a sub-matrix collecting only the coefficients relative to nodes p and q.

3.1 Periodic Boundary Conditions In the case of periodic boundary conditions the following equations need to be imposed   u+ − u− = E x + − x − , t+ + t− = 0

(11) (12)

22

A. Bilotta et al.

where (•)+ and (•)− denotes quantities relative to a couple of opposite sides of the RVE and E is the  macro–strain applied to the RVE. For two opposite sides the quantity x + − x − is constant allowing to impose the conditions as schematically shown in Fig. 3 where the displacement parameters of four of the corner nodes of the mesh of the RVE can be used to compute the constant term of the right-hand side of Eq. (11). In particular, always by referring to the nodes of the mesh of the RVE, this constant term can be assumed to be equal to the quantity u2 − u1 for the couple of faces highlighted in Fig. 3a, being u1 and u2 the displacements of the two corner nodes used. Moreover the displacement parameters of the nodes located on the face containing corner node 1, see always Fig. 3a, are free while the displacement parameters relative the nodes located on the opposite face, i.e. the face

u+(u2 − u1 )

u+(u3 − u1 )

u

u u3

u1

u1

(b)

u

u3 +( 12

4 +u

u1

u 1)

3+

−2

u4

u

u1

3 +(

u4 − u1 )

u 12

u3

u1 u1

(c)

u3 +(

−u

u4

(u 4

+ u 12

u1

1)

3+

(u2

+u

4

−u

−2

u1 )

1)

−u

u14

u14 +(u3 − u1 )

u +(u4 − u1 )

(u2

1)

u14 +(u2 − u ) 1

(a)

u14 +(u2 + u3 − 2u1 )

u2

u2

u 12

3

u1

(d)

Fig. 3 Pairing of the degrees of freedom relative to the sides of the RVE: (a), (b) and (c). Pairing of the degrees of freedom relative to the edges of the RVE: (d)

Concrete-Like RVEs for Computational Homogenization

23

containing corner node 2, are dependent as stated by Eq. (11). The same reasoning is repeated for the couple of faces highlighted in Fig. 3b, c for which the couple of reference corner nodes 1,3 and 1,4 are used, respectively, to compute the constant term required in Eq. (11). Figure 3d depicts what happens for the nodes located on the edges of the RVE’s boundary. Considering for example the edge connecting corner nodes 1 and 3, the displacement parameters of the nodes located on this edge, denoted by the symbol u13 , are free and they are used to evaluate the displacement parameters on the other two edges parallel to it. For the edge located above the expression u13 + (u4 − u1 ) is obtained. The latter parameters are then used to evaluate the displacement parameters located on the other parallel edge obtaining the expression u13 + (u2 + u4 − 2u1 ). It is worth to be observed that Eq. (12) is imposed automatically by identifying the degrees of freedom in the way previously described. The imposition of the periodic boundary conditions has been implemented inside the 3D in-house code used for the present work but other works dealing with periodic boundary conditions are available in literature, see for example [25, 26]. Further Comments on Computational Costs The two level computational homogenization is, by its own nature, costly because it requires a FEM analysis at macro–level and further Ng FEM analyses for the micro–level, being Ng the total number of Gauss points used in the mesh of the macro–level. This is a fixed cost to be considered. But also the boundary conditions used to evaluate the homogenized response of the RVE play an important role with respect to the computations required, see Eq. (7) and (10). If we consider kinematic boundary conditions, the indexes p and q of Eq. (7) and (10) are relative to all the nodes of the boundary of the RVE. In the case of Eq. (7), the evaluation of each component of the stress tensor requires Nb multiplications, where Nb is the number of nodes relative to the boundary of the RVE. Then the computational cost required for the evaluation of the macro–stress tensor is equivalent to the cost of 6 scalar products performed on vectors with Nb components. Considering then Eq. (10), each component of the constitutive macro– tensor requires (Nb2 + Nb ) multiplications leading to a cost of 21 multiplications between a Nb × Nb matrix and a vector plus a scalar product. This is only a part of the overall cost required by Eq. (10) because the computation of the matrix K is required. The matrix K is obtained by condensing out all the free degrees of freedom with respect to all the constrained degrees of freedom which in the case of kinematic boundary conditions are relative to all the nodes of the RVE’s boundary, i.e. K = Kcc − KTac K−1 aa Kac

(13)

The previous expression is easily obtained by partitioning the stiffness matrix of the RVE with respect the active degrees of freedom, i.e. all the degrees of freedom relative to the internal nodes, and the constrained degrees of freedom, i.e. the degrees of freedom relative to the boundary nodes. Then the operations required are the

24

A. Bilotta et al.

assemblage of the sub-matrices Kcc and Kac , assemblages not usually performed in a finite element code in which only the matrix Kaa needs to be assembled, and the solution of Nc linear systems, where Nc is the number of constrained degrees of freedom equal to 3 × Nb . With very small differences, the above evaluations are also valid for static or mixed kinematic-static boundary conditions. Differently from the kinematic boundary conditions, in the case of periodic boundary conditions, the indexes p and q of Eqs. (7) and (10) vary only on the four corner nodes, i.e. Nb = 4 (these nodes are used as reference points and denoted with the numbers from 1 to 4 in Fig. 3). They also are the only constrained nodes of the mesh, while every other node, internal or on the boundary, is free. In this case Nc is exactly equal to 12 and the extra-costs related to the assemblage of Kcc and Kac and the solution of 12 linear systems is considerably easier than in the case of kinematic boundary conditions.

•

! Important Remark

In the case of simple linear elasticity the evaluation of the constitutive tensor can be performed avoiding Eq. (10) and the application of 6 simple strain fields can be used to evaluate 6 stress fields whose components give the required 36 constitutive coefficients.

4 Numerical Results In the following numerical experimentation several RVEs have been analyzed and the resulting macro–response has been compared by varying the following parameters: • • • •

the shape of the particles, spherical or polyhedral; the percentage of the coarse fraction of the aggregates, Pc ; the size of the RVE, L; the constitutive parameters of the cement matrix and of the aggregates, (Km , μm ) and (Ka , μa ) respectively; • the boundary conditions, kinematic or periodic. For each set of these parameters, five different instances of the RVE are randomly generated by following the procedures described in the Sect. 2. Each RVE has been discretized by using the open source code Gmsh [23] on the basis of 10–nodes tetrahedral finite elements. The finite element analysis has been performed by using the mixed tetrahedral element proposed in [27] and by applying boundary conditions deriving from an assigned uniform state of macro–strain, this is the typical down–

Concrete-Like RVEs for Computational Homogenization

25

scale step performed in a first order computational homogenization [19]. The finite element here used has been also tested in [28]. The macro–response of each RVE has been evaluated in terms of the constitutive macro–tensor, see Eq. (10), which is relative to a generic anisotropic material. In order to have a more synthetic view of this result and, at the same time, to highlight the dominant isotropic behavior which is expected for many of the RVEs considered, an eigenvalue analysis has been performed on the basis of the 6 × 6 matrix associated to this tensor if we write the constitutive relationships as ⎤ ⎡ C1111 11 ⎢ ⎥ ⎢C ⎢ 22 ⎥ ⎢ 2211 ⎢ ⎥ ⎢ ⎢33 ⎥ ⎢C3311 ⎢ ⎥=⎢ ⎢23 ⎥ ⎢C2311 ⎢ ⎥ ⎢ ⎣31 ⎦ ⎣C3111 12 C1211 ⎡

C1122 C2222 C3322 C2322 C3122 C1222

C1133 C2233 C3333 C2333 C3133 C1233

C1123 C2223 C3323 C2323 C3123 C1223

C1131 C2231 C3331 C2331 C3131 C1231

⎤⎡ ⎤ E11 C1112 ⎢ ⎥ C2212⎥ ⎥ ⎢ E22 ⎥ ⎥⎢ ⎥ C3312⎥ ⎢ E33 ⎥ ⎥⎢ ⎥. C2312⎥ ⎢2E23 ⎥ ⎥⎢ ⎥ C3112⎦ ⎣2E31 ⎦ C1212 2E12

(14)

The multiplicity of the obtained eigenvalues, γi for i = 1 . . . 6 sorted with respect to smallest magnitude, is compared with the multiplicity of the eigenvalues of an isotropic material whose eigenvalues are (μ, μ, μ, 2μ, 2μ, K), being μ and K the shear and the bulk modulus, respectively. A measure of the apparent multiplicities obtained is evaluated on the basis of the expressions m3 =

 3   γi − μγ  1− μγ

m2 =

i=1

 5   γi − 2μγ  1− 2μγ

(15)

i=4

where 1 γi 3 3

μγ =

(16)

i=1

is the apparent shear modulus of the macro–material. Values of m3 and m2 almost coincident with to 3 and 2, respectively, highlight isotropy in the apparent response of the material. In this case the apparent bulk modulus is simply given by Kγ = γ6

(17)

It is useful to recall at this point that when a behavior essentially isotropic is expected, the relationships tr σ /3 3K ∗ = and 2μ∗ = tr ε/3



σ   : σ   , ε   : ε  

(18)

26 Table 1 Bulk modulus and shear modulus assumed for the matrix phase and aggregate phase, GPa

A. Bilotta et al.

Set 1 Set 2

Km 10.42 6.44

μm 8.47 4.83

Ka 30.00 41.40

μa 22.50 31.04

Fig. 4 Analysis of RVEs with L = 90 mm, dmax = 30 mm and material set 1: variation of Kγ (a) and μγ (b) with respect to percentage of aggregate particles comparing spherical and polyhedral shapes

can be used to evaluate the apparent bulk and shear moduli. In Eq. (18) σ  and ε are the deviatoric parts of the micro–stress and the micro–strain fields, σ and ε,  1 calculated through the FEM analysis, while • = VRVE • dV . VRVE Table 1 reports the two sets of material parameters used for the matrix phase and particle phase of the RVEs. The set labeled with 1, adopted in [11], is relative to standard values for the cement matrix and particle aggregates. The second set was chosen in order to have a larger difference between the moduli of the two materials. Figures 4, 6, and 8 show the bulk modulus Kγ and the shear modulus μγ obtained by using the set 1 of materials and for the three different values of the RVE’s size here considered, i.e. 45, 60 and 90 mm. The maximum size of the included aggregate is 18 mm for L = 45 and 30 mm for the other two RVE’s sizes. Figures 5, 7, and 9 show the same results for the set 2 of materials. In the same figures the classical bounds by Reuss and Voigt [5] and by Hashin and Shtrikman [29] are reported. Figures 10 and 11 show the values of computed multiplicities for the material set 1 with respect to size of the RVE and the percentage of the aggregate particles, the same results are shown in Figs. 12 and 13 for the material set 2. From these figures an evident deviation from the isotropic behavior can be observed for the material set 2 for which the ratio of the constitutive properties of the constituents, mortar and aggregates, is higher.

Concrete-Like RVEs for Computational Homogenization

27

Fig. 5 Analysis of RVEs with L = 90 mm, dmax = 30 mm and material set 2: variation of Kγ (a) and μγ (b) with respect to percentage of aggregate particles comparing spherical and polyhedral shapes

Fig. 6 Analysis of RVEs with L = 60 mm, dmax = 30 mm and material set 1: variation of Kγ (a) and μγ (b) with respect to percentage of aggregate particles comparing spherical and polyhedral shapes

Finally, Figs. 14, 15, 16, and 17 compare the kinematic and periodic boundary conditions in the evaluation of the apparent properties of the RVE only for the RVE’s size equal to 45 mm. Similar results are obtained for the other RVE’s sizes and are not reported here.

28

A. Bilotta et al.

Fig. 7 Analysis of RVEs with L = 60 mm, dmax = 30 mm and material set 2: variation of Kγ (a) and μγ (b) with respect to percentage of aggregate particles comparing spherical and polyhedral shapes

Fig. 8 Analysis of RVEs with L = 45 mm, dmax = 18 mm and material set 1: variation of Kγ (a) and μγ (b) with respect to percentage of aggregate particles comparing spherical and polyhedral shapes

5 Conclusions Aiming to highlight the factors which mainly contribute to the determination of the apparent constitutive response of RVEs to be used in the contest of the first order computational homogenization, the problem of the choice of an effective RVE for the 3D analysis of concrete structures has been considered. In this regard the

Concrete-Like RVEs for Computational Homogenization

29

Fig. 9 Analysis of RVEs with L = 45 mm, dmax = 18 mm and material set 2: variation of Kγ (a) and μγ (b) with respect to percentage of aggregate particles comparing spherical and polyhedral shapes

Fig. 10 Analysis of RVEs with material set 1: variation of m3 comparing spherical (a) and polyhedral shapes (b)

evaluation of the macro-response relative to a generic Gauss point, or control point, in the FEM analysis of a concrete structure plays a crucial role, with respect both to the effective evaluation of the mechanical response and the computational burden to be sustained. To this end in the theoretical sections of the paper several aspects have been tackled with: (1) the automatic generation of the geometry of the RVE domain distinguishing between spherical and polyhedral shape of the included aggregates by proposing also an algorithm for the generation of the polyhedral shapes; (2) the

30

A. Bilotta et al.

Fig. 11 Analysis of RVEs with material set 1: variation of m2 comparing spherical (a) and polyhedral shapes (b)

Fig. 12 Analysis of RVEs with material set 2: variation of m3 comparing spherical (a) and polyhedral shapes (b)

evaluation of the macro-stress  and of the constitutive macro-tensor C highlighting the impact on the computational costs to be afforded in a typical 3D structural analysis; (3) the practical implementation of specific boundary conditions such as the periodic ones. On the basis of the numerical simulations here presented and with respect the aspects investigated, i.e. – the shape of the particles (spherical or polyhedral), – the percentage of the coarse fraction of the aggregates,

Concrete-Like RVEs for Computational Homogenization

31

Fig. 13 Analysis of RVEs with material set 2: variation of m2 comparing spherical (a) and polyhedral shapes (b)

Fig. 14 Comparison between kinematic and periodic boundary conditions, KBC and PBC respectively, in the analysis of RVEs with L = 45 mm, dmax = 18 mm and material set 1: variation of Kγ with respect to percentage of aggregate particles for spherical (a) and polyhedral shapes (b)

– the size of the RVE, – the constitutive parameters of the cement matrix and of the aggregates, – the boundary conditions applied to the RVE, the following conclusions can be drawn. • The shape of the inclusions affects the response of the RVE giving a stiffer response in the case of polyhedral aggregates, see Figs. 4, 5, 6, 7, 8, and 9. The

32

A. Bilotta et al.

Fig. 15 Comparison between kinematic and periodic boundary conditions, KBC and PBC respectively, in the analysis of RVEs with L = 45 mm, dmax = 18 mm and material set 1: variation of μγ with respect to percentage of aggregate particles for spherical (a) and polyhedral shapes (b)

Fig. 16 Comparison between kinematic and periodic boundary conditions, KBC and PBC respectively, in the analysis of RVEs with L = 45 mm, dmax = 18 mm and material set 2: variation of Kγ with respect to percentage of aggregate particles for spherical (a) and polyhedral shapes (b)

effect is more evident if we look at the apparent shear modulus μγ and it is more pronounced in the case of the material set 2 for which also the evaluation of the apparent bulk modulus Kγ highlights a such behaviour. Obviously for higher values of the percentage of the included aggregates the difference in the evaluation of the apparent moduli tends to increase.

Concrete-Like RVEs for Computational Homogenization

33

Fig. 17 Comparison between kinematic and periodic boundary conditions, KBC and PBC respectively, in the analysis of RVEs with L = 45 mm, dmax = 18 mm and material set 2: variation of μγ with respect to percentage of aggregate particles for spherical (a) and polyhedral shapes (b)

• The size of the RVE, if compared with the bigger size of the included aggregate, can be kept enough small allowing a certain reduction of the computational costs required by the analysis. In the numerical experimentation ratios L/dmax equal to 2, 2.5 and 3 have been considered obtaining essentially the same apparent response. This result confirms also what already observed in [18]. • Higher values of the ratios Ka /Km and μa /μm lead to an apparent response which cane be hardly assumed as isotropic. Certainly standard values of the material properties allow for a more confident use of the hypothesis of isotropy and then the adoption of the expressions (18) instead of the more computationally expensive Eq. (10) relative to all the coefficient of the constitutive matrix. • The experimentation on the boundary conditions here considered, kinematic or periodic, highlights small differences in the evaluation of the apparent shear modulus. The fundamental role played by the boundary condition is evident if we consider the drastic reduction of the computational costs relative to the Eq. (10) when applied in the case of periodic boundary conditions. Thinking about a successive development of the present study the analysis of concrete stuctures certainly requires the effects of the nonlinear behaviour of the material to be taken into account. In this regard fracture strength must obviously depend on the pre-critical cracking, which is not only dependent on the structure of the material, but also on the stress- and strain-gradients in the structure under consideration. Stress- and strain-gradients which are likely to be well described by using sharp polyhedral shapes for modeling the geometry of the aggregates.

34

A. Bilotta et al.

Acknowledgments Andrea Causin, Margherita Solci and Emilio Turco gratefully acknowledge the University of Sassari for funding their research in the frame of the Fondo di Ateneo per la ricerca 2019.

References 1. Neville, A.: Properties of Concrete. Pearson Prentice Hall, Upper Saddle River (1995) 2. Kezmane, A., Chiaia, B., Kumpyak, O., Maksimov, V., Placidi, L.: 3D modelling of reinforced concrete slab with yielding supports subject to impact load. Eur. J. Environ. Civil Eng. 21(7–8), 988–1025 (2017) 3. Chiaia, B., Kumpyak, O., Placidi, L., Maksimov, V.: Experimental analysis and modeling of two-way reinforced concrete slabs over different kinds of yielding supports under short-term dynamic loading. Eng. Struct. 96, 88–99 (2015) 4. Euro-c 2014: In: Bi´cani´c, N., Mang, H., Meschke, G., de Borst, R. (eds.) Computational Modelling of Concrete Structures. CRC Press, Boca Raton (2014) 5. Zohdi, T.I., Wriggers, P.: Introduction to Computational Micromechanics. Springer, Berlin, 2005 6. Bazant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton (1998) 7. Jirasek, M., Bazant, Z.P.: Inelastic Analysis of Structures. Wiley, Hoboken (2002) 8. Scerrato, D., Giorgio, I., Della Corte, A., Madeo, A., Dowling, N.E., Darve, F.: Towards the design of an enriched concrete with enhanced dissipation performances. Cement Concrete Res. 84, 48–61 (2016) 9. Giorgio, I., Scerrato, D.: Multi-scale concrete model with rate dependent internal friction. Eur. J. Environ. Civil Eng. 21(7–8), 821–839 (2017) 10. Scerrato, D., Giorgio, I., Della Corte, A., Madeo, A., Limam, A.: A micro-structural model for dissipation phenomena in the concrete. Int. J. Numer. Anal. Methods Geom. 39(18), 2037– 2052 (2015). nag.2394 11. Unger, J.F., Eckardt, S.: Multiscale modeling of concrete - from mesoscale to macroscale. Arch. Comput. Methods Eng. 18, 341–393 (2011) 12. Takano, N., Kimura, K., Zako, M., Kubo, F.: Multi-scale analysis and microscopic stress evaluation for ceramics considering the random microstructures. JSME Int. J. A Solid Mech. Mater. Eng. 46(4), 527–535 (2003) 13. Hollister, S.J., Kikuchi, N.: Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue. Biotechnol. Bioeng. 43(7), 586–596 (1994) 14. Wittman, F.H., Roelfstra, P.E., Sadouki, H.: Simulation and analysis of composite structures. Mater. Sci. Eng. 68, 239–248 (1985) 15. E.J. Garboczi, Three-dimensional mathematical analysis of particle shape using x-ray tomography and spherical harmonics: application to aggregates used in concrete. Cement Concrete Res. 32, 1621–1638 (2002) 16. Wriggers, P., Moftah, S.O.: Mesoscale models for concrete: Homogenisation and damage behaviour. Finite Elements Anal. Des. 42, 623–636 (2006) 17. Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679 (2003) 18. Drugan, W.J., Willis, J.R.: A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44(4), 497–524 (1996) 19. Kouznetsova, V.G., Brekelmans, W.A.M., Baaijens, F.P.T.: An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27, 37–48 (2001)

Concrete-Like RVEs for Computational Homogenization

35

20. Hain, M., Wriggers, P.: Numerical homogenization of hardened cement paste. Comput. Mech. 42(2), 197–212 (2008) 21. Rycroft, C.H.: Software library for three-dimensional computations of the Voronoi tessellation (2008). http://math.lbl.gov/voro++/ 22. Sonon, B., François, B., Massart, T.J.: A unified level set based methodology for fast generation of complex microstructural multi-phase {RVEs}. Comput. Methods Appl. Mech. Eng. 223– 224, 103–122 (2012) 23. Geuzaine, C., Remacle, J.F.: GMSH: A 3-d finite element mesh generator with built-in preand post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009) 24. Kouznetsova, V.G., Geers, M.G.D., Brekelmans, W.A.M.: Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput. Methods Appl. Mech. Engrg. 193, 5525–5550 (2004) 25. Tyrus, J.M., Gosz, M., DeSantiago, E.: A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models. Int. J. Solids Struct. 44(9), 2972–2989 (2007) 26. Xia, Z., Zhang, Y., Ellyin, F.: A unified periodical boundary conditions for representative volume elements of composites and applications. Int. J. Solids Struct. 40, 1907–1921 (2003) 27. Bilotta, A., Garcea, G., Leonetti, L.: A composite mixed finite element model for the elastoplastic analysis of 3D structural problems. Finite Elem. Anal. Des. 113(C), 43–53 (2016) 28. Tedesco, F., Bilotta, A., Turco, E.: Multiscale 3D mixed FEM analysis of historical masonry constructions. Eur. J. Environ. Civil Eng. 21(7–8), 772–797 (2017) 29. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963)

A New Nonlocal Temperature-Dependent Model for Adhesive Contact Elena Bonetti, Giovanna Bonfanti, and Riccarda Rossi

Abstract The aim of this note is twofold. First of all, we propose a very partial survey on the mathematical modeling and analysis of adhesive contact and delamination. Secondly, we advance a new model for adhesive contact with thermal effects that includes nonlocal adhesive forces and surface damage effects, as well as nonlocal heat flux contributions on the contact surface. In the derivation of the model, we follow the approach by M. Frémond applying it to nonlocal adhesive contact. Keywords Adhesive contact · Delamination · Rate-dependent processes · Nonlocal effects · Temperature

1 Introduction Adhesive contact and delamination have been intensively investigated in recent years both from the analytical and the mechanical viewpoint. First of all, a thorough understanding of these inelastic phenomena on surfaces plays an important role in the stability analysis for laminate structures, more and more used in industry. For instance, laminated materials enjoy remarkable energy absorption properties, and are therefore preferable to conventional metallic structures in designing energyabsorbing elements in vehicles, cf. e.g. [25]. Indeed, delamination is a progressive

E. Bonetti () Dipartimento di Matematica “F. Enriques”, Università di Milano, Milano, Italy e-mail: [email protected] G. Bonfanti Sezione di Matematica, DICATAM, Università di Brescia, Brescia, Italy e-mail: [email protected] R. Rossi DIMI, Università di Brescia, Brescia, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_3

37

38

E. Bonetti et al.

separation of bonded laminates, usually due to the degradation of the adhesive substance gluing them together. Since it is an inelastic process on a surface, from a mathematical standpoint adhesive contact and delamination can be modeled by resorting to a surface damage theory. From a broader perspective, models for joint volume and surface damage processes are and will be more and more relevant for the description of the degradation of monumental stones. Their deterioration is due to the harmful combination of environmental conditions and mechanical actions. In fact, physical and chemical mechanisms (such as, e.g., pollution) induce a progressive increase of the surface rugosity and the porosity of the external layers, with the formation of micro-cracks and fractures in crystal grains. A model describing this phenomenon has been first advanced and analyzed in the recent [13]. It would be interesting to gain further insight into this degradation process by mathematically modeling the damage of the external layers of monuments via a ‘surface damage approach’, in order to evaluate its influence on the behavior of the whole structure. In this note we are going to introduce a model for nonlocal adhesive contact with thermal effects that pertains to a class of models for adhesive contact and delamination originating from the approach by M. FRÉMOND, cf. [29] and the pioneering paper [28], also in the frame of the theory of generalized standard materials [30]. Typically, one considers two elastic bodies + , − ⊂ R3 (throughout this note, we shall confine the discussion to the 3D case, meaningful for the applications), possibly subject to viscosity and inertia, bonded along a prescribed contact surface C ; we set O := + ∪ − ∪ C . Neglecting thermal effects, the evolution, during a finite time interval (0, T ), of adhesive contact and delamination between + and − is described in terms of an internal variable χ : C × (0, T ) → [0, 1]. The parameter χ has in fact the meaning of a damage variable, as it describes the fraction of fully effective molecular links in the bonding. Namely, χ(x, t) =

 1 0

means that the bonding is

 fully intact completely broken

(1)

at the material point x ∈ C and the process time t ∈ (0, T ), with χ(x, t) ∈ (0, 1) for the intermediate states. The momentum balance for the displacement field u : (− ∪+ ) × (0, T ) → R3 is thus coupled with the flow rule for χ. The resulting PDE system can be derived via a generalized version of the principle of virtual power (cf. also Sect. 2 below); it has the abstract structure of a generalized gradient system ut t + μ∂V(ut ) + Du E(t, u, χ) 0

in U∗

a.e. in (0, T ),

(2a)

∂R(∂t χ) + Dχ E(t, u, χ) 0

in X∗

a.e. in (0, T ).

(2b)

Typically, for the momentum balance (2a) the ambient space U is H 1 (O\C ; R3 ) (or a subspace of the latter space, in order to account for Dirichlet conditions on the

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

39

displacement on a portion Dir of ∂O); the constant  ≥ 0 modulates the inertial term, while the dissipation potential V : U → [0, +∞), given by  V(ut ) := O\C

: (ut ) dx,

1 2 V (ut )

features the (positive definite, symmetric) viscosity tensor V, ( (ut ) denoting the classical strain rate tensor), and ∂V : U ⇒ U∗ is its subdifferential in the sense of convex analysis. The ambient space X for the flow rule (2b) can be (formally) taken as L2 (C ); the dissipation potential R : X → [0, +∞], with convex subdifferential ∂R : X ⇒ X∗ , typically consists of two contributions  R(χt ) :=

 R(χt ) dS + ν

C

I(−∞,0] (χt ) dS .

(3)

C

The indicator term I(−∞,0] enforces the constraint χt ≤ 0

a.e. in C × (0, T ),

(4)

translating the fact that the degradation of the adhesive bonds between + and − is irreversible; the dissipation density R : R → [0, +∞) is convex and will be discussed below. Finally, the driving energy functional E : (0, T ) × U × X → (−∞, +∞] for the adhesive contact/delamination model generally takes the form 

 E(t, u, χ) := O\C

1 2 E (u)

  I(−∞,0] ( u · n) dS

: (u) dx + λ C

+EC (χ) + Ecoup (u, χ) −

U∗

(5)

F(t), uU .

Namely, the elastic energy contribution, featuring the (positive definite, symmetric) elasticity tensor E, ( (u) denoting the linearized strain tensor), is added with a term that involves the normal component of the displacement jump [[u]] across C (i.e., [[u]] is the difference of the traces on C of u|+ and u|− ), as n denoted the outward unit normal vector to the boundary ∂. The indicator function I(−∞,0] ensures that along a solution to system (2) the non-interpenetration constraint between the bodies + and − , namely   u ·n≤ 0

a.e. in C × (0, T ),

(6)

holds. The “coupling” functional Ecoup is typically given by 

  J ( u , χ) dS

Ecoup (u, χ) := C

(7)

40

E. Bonetti et al.

with J : R3 × [0, 1] → [0, +∞] smooth, while the surface contribution EC may involve – a gradient, regularizing term; in that case, it is customary to assume that C is a “flat surface”, embedded in R2 (cf. also Sect. 2), on which the Hausdorff measure coincides with the Lebesgue one. Therefore, one writes dx in place of dS for integrals on C , and for the gradient terms the usage of Laplace–Beltrami operators can be avoided.  Hence, one can for instance choose a regularizing contribution of the form C 12 |∇χ|2 dx;  – the indicator term C I[0,1] (χ) dx enforcing the constraint χ ∈ [0, 1]

a.e. in C × (0, T );

(8)

– and, possibly, other smooth terms. Finally, the function F : (0, T ) → U∗ subsumes volume forces and applied tractions on the ‘Neumann part’ Neu of ∂O. The coefficients μ, ν, λ in (2a)–(2b), (3), and (5) are all non-negative and may thus switch off, or on, the viscosity contribution to the momentum balance, the unidirectionality constraint (4) for the evolution of χ, and the non-intepenetration condition (6), respectively. Similarly, inertial effects in (2a) are neglected if one takes  = 0. A major distinction has to be made between – rate-independent models, in which the dissipation density R is positively homogeneous of degree 1, namely R( χ) ˙ = R(χ) ˙

for all χ˙ ∈ R, ≥ 0

(9)

– and rate-dependent models, featuring a dissipation density with superlinear growth at infinity; typically, in adhesive contact models one considers a quadratic dissipation potential, i.e. (setting all physical constants equal to 1) R(χ˙ ) =

1 2 |χ| ˙ 2

for all χ˙ ∈ R.

(10)

Although the focus of this note is on a rate-dependent system for nonlocal, temperature-dependent adhesive contact, we will also very partially review the, intensively growing, literature on rate-independent models.

1.1 Rate-Independent Models Adhesive contact and delamination as rate-independent processes have been actively studied over the last 15 years since the pioneering paper [31], which addressed a quasistatic (i.e. without inertial effects) model where viscosity in u was

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

41

also neglected (viz., with  = μ = 0 in (2a)). In such a case, since the displacement variable is at equilibrium and the subdifferential operator ∂R : X ⇒ X∗ is 0(positively) homogeneous, system (2) is invariant upon (increasing) time rescalings, which reflects the modeling ansatz that the system possesses no internal time scale. In the 1-homogeneous case, though, system (2) is only formally written as a subdifferential inclusion holding pointwise in time. Indeed, the dissipation density R has linear growth at infinity, and thus one can in general expect only BV-time regularity for χ. Thus, χ may have jumps as a function of time, and its derivative χt need not be defined. In view of this, it is necessary to formulate (2) in a suitably weak way. In [31] the authors resorted to the commonest solvability concept for rate-independent processes, namely the notion of energetic solution (cf. [32]). It consists of an energy-dissipation balance (which, of course, only features the 1-homogeneous dissipation potential R, as viscosity and inertia in u are neglected) and of a (global) stability condition that can be indeed recast as a (global) minimization problem. The main result of [31] is the existence of solutions to the Cauchy problem for the energetic formulation. Without entering into further details, let us only mention that [31] addressed the case in which the ‘coupling energy’ from (7) is given by  Ecoup(u, χ) = C

κ 2 χ|

 2 u | dx,

(11)

with κ ≥ 0. This term penalizes displacement jumps in points with strictly positive χ but does not exclude them. Obviously, one expects that the blow-up of the coefficient κ will lead to a model with the brittle constraint   χ u =0

a.e. in C × (0, T ),

(12)

that allows for displacement jumps (i.e., [[u]] = 0) only at points where the bonding is completely broken (i.e., χ = 0), and otherwise imposes the transmission condition [[u]] = 0 on the displacements. Systems encompassing (12) are frequently referred to as brittle delamination models, as opposed to the adhesive contact models with the coupling energy from (11). The convergence of energetic solutions of the adhesive contact system examined in [31] to the brittle delamination system was rigorously proved in [42], relying on -convergence type arguments. In fact, the energetic formulation has an intrinsically variational character that allows for limiting procedures based on notions of variational convergence. This has been crucially exploited, for instance, for dimensional reduction analyses, cf. e.g. [33], dealing with the limit passage from bulk to surface damage as the thickness of an interface between two elastic bodies tends to zero, and [27], deriving models for both adhesive contact and brittle delamination in 2D plates as limits of delamination between 3D thin plates. More in general, the flexibility of the energetic concept allows for an easy coupling of delamination with other inelastic processes such as plasticity and phase transformations, as well as for

42

E. Bonetti et al.

enhancements to cohesive-type or mixed mode models. We refer to [41] (and the references therein) for a comprehensive survey of these aspects. The range of applicability of the energetic formulation can be broadened to encompass models coupling rate-independent evolution of the internal variable with rate-dependent evolution of the displacement, subject to viscosity and, possibly, inertia. This extended notion of energetic solution was exploited, for instance, in [40]. Relying on a further generalization of energetic solutions for rate-independent processes also subject to thermal effects (cf. [39]), in [37] a model for adhesive contact between two thermoviscoelastic bodies was analyzed; the extension to brittle delamination was carried out in [38]. The PDE systems analyzed in [37] and [38] couple (2) with the temperature equation in the bulk domain O\C ; in this case, the overall system has a more complex structure than that of a generalized gradient system and, accordingly, its energetics is more involved. A common feature of the coupled rate-independent/rate-dependent systems addressed in [37, 40] and [38] is that, due to the presence of inertial terms in the momentum balance, the unilateral non-interpenetration constraint (6) was not incorporated in the model  and, accordingly, the contribution C I(−∞,0] ([[u]] · n) dS was either neglected or replaced by a term only penalizing interpenetration, without excluding it.

1.2 Rate-Dependent Models Also rate-dependent delamination systems, featuring the quadratic dissipation potential (10), have been intensively studied over the last two decades. In [7] we first approached the study of FRÉMOND’s model for adhesive contact, neglecting inertia (i.e. setting  = 0 in (2a)), in favor of a more transparent formulation of the momentum balance that would account for the reaction forces associated with the constraints encompassed by the model. More precisely, we confined our analysis to the case of a single body  in adhesive contact with a rigid support, such that the contact surface C is a part of their common boundary. In this case, the jump of the displacement [[u]] coincides with its trace on C , hereafter simply denoted by u. We considered the driving energy functional 



E(t, u, χ) := 

1 2 E (u)





+ 

C

: (u) dx +

2 1 2 |∇χ| +I[0,1] (χ)−ws χ





 dx +



f(t) · u dx −

−

I(−∞,0] (u · n) dx C

C

κ 2χ

|u|2 dx

g(t) · u dx, Neu

in which the coupling contribution is given by (11), ws is a positive constant related to the cohesion on the adhesive, f is a volume force and g a traction applied on the Neumann part Neu of the boundary ∂ =  Dir ∪  Neu ∪  C . Imposing

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

43

zero Dirichlet boundary conditions on the Dirichlet boundary Dir we derived the following quasistatic PDE system for adhesive contact − div (V (ut )+E (u)) = f

in  × (0, T ),

u=0

on Dir × (0, T ),

(13a) (13b)

(V (ut )+E (u))n = g

on Neu × (0, T ), (13c)

(V (ut )+E (u))n + κχu + ∂I(−∞,0] (u · n)n 0 κ χt + ∂I(−∞,0] (χt ) − χ + ∂I[0,1] (χ) ws − |u|2 2 ∂ns χ = 0

on C × (0, T ),

(13d)

on C × (0, T ),

(13e)

on ∂C × (0, T ).

(13f)

Observe that (13d) generalizes the Signorini conditions from the basic unilateral contact theory, in that the term κχu represents the resistance to tension related to the action of microscopic bonds between the surfaces of the adhering solids. Let us stress that in system (13) all constraints on the variables u and χ are rendered by means of multivalued operators. This approach, though leading to some analytical difficulties, allows us to account for the internal reactions ‘activated’ by the constraints. The main result of [7] states the existence of solution to the Cauchy problem for (13), with the flow rule (13e) formulated as a subdifferential inclusion holding a.e. on C × (0, T ), and the momentum balance (13a) formulated with test functions from H 1 (; R3 ) having null trace on Dir . In particular, the weak formulation of (13a) with the boundary conditions (13b)–(13d) features a selection ξ from the (suitably realized) subdifferential ∂I(−∞,0] (u · n)n. This term has the meaning of a reaction force, activated ‘on the boundary’ of the non-intepenetration constraint, namely when u · n = 0. Dropping the inertial term in (13a) we were able to gain estimates for the term ξ ∈ ∂I(−∞,0] (u · n)n, and thus to encompass it in the momentum balance. Subsequently, in [45] (cf. also [43, 44]) an existence result was proved for a dynamic adhesive contact system with the non-interpenetration condition (6), in which the analytical difficulties attached to the coupling of inertia and a unilateral constraint were bypassed by resorting to a suitable weak solution concept introduced in [17]. The analysis in [7] has been extended in various directions. Indeed, the long-time behavior of system (13) has been addressed in [6], where the existence and properties of the associated ω-limit set have been investigated, while frictional effects have been included in the model studied in [9]. Furthermore, the coupling between adhesive contact and volume damage has been investigated in [16]. In the ratedependent context, the (asymptotic) relation between bulk and surface models has been addressed in [4, 5]. Therein, by means of the method of asymptotic expansions, models of ‘imperfect interfaces’ have been derived as limits of processes in a thin adhesive substrate between two bodies that undergoes a degradation process, as its thickness vanishes. The limiting models couple the evolution of the (bulk) displace-

44

E. Bonetti et al.

ment and temperature variables to that of a surface damage parameter, encompassing unilateral contact conditions along the interface. However, a rigorous dimensional reduction analysis from a model for bulk damage to a surface damage one, in the same spirit as [33], is still missing. Likewise, the limit passage from adhesive contact to brittle delamination (namely, the asymptotic analysis of the adhesive contact system (13) as the parameter κ blows up) is still open. In fact, for rate-dependent systems it is more challenging to perform this kind of analyses, for they would involve non-trivial limit passages in the momentum balance and in the flow rule. The model introduced in [7] has been extended in [8] to the temperaturedependent framework, considering heat generation effects in the phenomenon of adhesive contact, too. The thermal evolution of the system is assumed to be ruled by the heat exchange between the body and the adhesive substance through the contact surface. In particular, allowing for different temperatures in the bulk domain and on the contact surface, we assumed that temperature variables θ and θs are governed by two distinct entropy balance laws. The analysis carried out in [8] leads to an existence result for (the initial-value problem associated with) the following PDE system: ∂t (ln(θ )) − div(ut ) − θ = h  0 in (∂ \ C ) × (0, T ), ∂n θ = −k(χ)(θ − θs ) in C × (0, T ),

in  × (0, T ),

∂t (ln(θs )) − ∂t (λ(χ)) − θs = k(χ)(θ − θs )

in C × (0, T ),

(14c)

∂ns θs = 0

in ∂C × (0, T ),

(14d)

− div (E (u) + V (ut ) + θ I) = f

in  × (0, T ),

(14e)

u=0

in Dir × (0, T ),

(14f)

(E (u) + V (ut ) + θ I)n = g

in Neu × (0, T ),

(14g)

(E (u) + V (ut ) + θ I)n + κχu +∂I(−∞,0] (u · n)n 0

in C × (0, T ),

(14h)

χt − χ + ∂I[0,1] (χ) + γ  (χ) −λ (χ)(θs − θeq ) − κ2 |u|2

in C × (0, T ),

(14i)

∂ns χ = 0

in ∂C × (0, T ),

(14j)

(14a) (14b)

where λ, k, and γ are sufficiently smooth functions, whereas θeq > 0 is a phase transition temperature, κ a positive constant, and I the identity matrix. Let us comment that a peculiarity of the system is that, the evolution of the temperature variables θ and θs is governed by entropy, in place of internal energy, balance equations. While referring to, e.g., [14, 15] for a more accurate illustration of this approach, we may mention here that the entropy equations are recovered by rescaling the internal energy balance equations, neglecting some higher order

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

45

dissipative terms under the small perturbation assumption. As shown in [10, Sec. 2], this leads to a thermodynamically consistent model, where the strict positivity of θ and θs is enforced by the very form of the equations, cf. (14a) and (14c). On the other hand, the singular character of the terms ∂t (ln(θ )) and ∂t (ln(θs )) occurring in (14a) and (14c) does not allow but for poor time-regularity for θ and θs . Again, we note that the evolution of the temperatures is essentially governed by the heat exchange throughout the contact surface. More precisely, the entropy flux through C (namely the term k(χ)(θ − θs ) in (14b)) plays the role of a source of entropy in (14c). From the analytical point of view, this results in a nonlinear coupling between (14a)–(14b) and (14c) and gives rise to further technical difficulties. The thermodynamical modeling approach from [8] was also adopted in [10], where frictional contributions are further encompassed in a temperature-dependent model through a regularization of the classical Coulomb law, here generalized to the case of adhesive contact and assuming thermal dependence of the friction coefficient. The main result in [10] states the existence of solutions to a temperaturedependent system for adhesive contact with friction, in which all the constraints on the internal variables, as well as the unilateral contact conditions and the friction law, are rendered by means of subdifferential operators, in accordance with the approach developed in [7–9]. Concerning frictional contact problems, with or without adhesion, we recall an alternative approach which replaces the unilateral contact conditions (the Signorini contact conditions) rendered by (13d) with a normal compliance condition, allowing for the interpenetration of the surface asperities and thus for dispensing with the unilateral constraint on u · n. Analytically, the normal compliance law corresponds to a penalization of the subdifferential operator ∂I(−∞,0] in (13d). In this connection, we refer e.g. to [1], analyzing a dynamic model for frictional contact in thermoviscoelasticity with a power-law normal compliance condition, and the corresponding generalization of Coulomb’s law of friction. Contact with a deformable foundation is considered in [2] as well, where a dynamic contact problem for a thermoviscoelastic body, with frictional and wear effects on the contact surface, is investigated. A wide class of dynamic frictional contact problems in thermoelasticity and thermoviscoelasticity is also tackled in [24], with contact rendered by means of a normal compliance law. In the context of contact with a deformable foundation, we quote [35], where a dynamic frictional contact problem with adhesion is formulated by coupling a hyperbolic hemivariational inequality for the displacement and a first-order ODE for the adhesive field. By means of abstract results on variational-hemivariational inequalities, the existence and the regularity of a solution were proved (for these techniques and their applications in Contact Mechanics, see e.g. the monograph [34]). A different approach to frictional contact problems was developed in [23] and in [21] (cf. also the references in the monographs [22, 46]), where contact with a rigid foundation is modeled by the Signorini conditions in velocity form, that are indeed expressed in terms of ut rather than of u. In this context, existence results were obtained for contact problems including frictional and thermal effects.

46

E. Bonetti et al.

We also recall the contributions [19, 20, 36] on the analysis of systems coupling friction, adhesion and unilateral contact (modeled via the classical Signorini conditions), akin to FRÉMOND’s model (see [28, 29]). In [36], a consistent model describing unilateral contact, adhesion and friction was derived and the related quasistatic problem was written in terms of two variational inequalities and a firstorder ODE (indeed, local interactions in the adhesive substance were neglected); the main result therein provides the existence of solutions for an incremental formulation of the problem; some numerical schemes are also given. In [19, 20], based on [36], contact problems with adhesion and friction were considered in the quasistatic elastic case and in the dynamic viscoelastic case, respectively. Let us stress that, differently from our approach, in all of these contributions the existence of solutions is proved for formulations of the related PDE systems that involve variational inequalities, and not evolutionary differential inclusions, for the displacement. We conclude this short review on frictional contact problems with adhesion returning to the temperature-dependent case in order to illustrate our paper [11]. Differently from [10], therein the temperature equations for the bulk and the surface temperatures θ and θs are derived from the internal energy balance, without neglecting any higher order dissipative contribution. The presence of quadratic terms on their right-hand side that are only estimated in L1 ( × (0, T )) and in L1 (C × (0, T )), respectively, leads to considerable difficulties. By resorting to Boccardo-Gallouët [3] type estimates and under suitable growth conditions on the heat capacity and the heat conductivity of the system, the existence of a solution for the related initial-boundary value problem has been proved in [11].

1.3 Adhesive Contact with Nonlocal Effects In [12] we have further extended the analysis of the (isothermal) adhesive contact model by FRÉMOND, by assuming that also nonlocal forces act on the contact surface. More precisely, in addition to the interactions, on the contact surface, between damage (of the adhesive substance) at a point and damage in its neighborhood, in the model in [12] we have encompassed a nonlocal interaction among the adhesive substance, the body  and its rigid support. This results in an integral term further contributing to the resistance to tension in the generalized Signorini conditions (13d), and in a second integral term, coupled to it, in the flow rule for the adhesion parameter. The motivation for this enhancement of the model comes from experiments showing that elongation, i.e. a variation of the distance of two distinct points on the contact surface, may have damaging effects on the adhesive substance, cf. [26]. The aim of this note is to introduce a model further encompassing thermal effects in the nonlocal system studied in [12], still within the assumption that the temperatures in the body and in the adhesive substance are a priori different, as in the models from [8, 10, 11]. As we have seen, this ansatz leads to a different

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

47

approach to the modeling of the heat exchange between the body and the glue located on the contact surface. In particular, we are now going to assume a nonlocal interaction between the body and the adhesive as far as it concerns heat transfer, as well. Accordingly, the related PDE system (cf. (54) ahead) will contain integral terms in the boundary condition for the bulk absolute temperature on the contact surface, and in the equation for the surface temperature, too. We will derive it in the upcoming Sect. 2.

2 The Model and the PDE System In this section we present the modeling approach leading to the PDE system (54) ahead for an adhesive contact process in the presence of nonlocal thermomechanical effects. In the derivation of system (54), we shall refer to the theory introduced by FRÉMOND [29] and developed in [12, 26] for nonlocal adhesive contact models in the isothermal case. As in [12], we will confine the discussion to the reduced case in which only one body is considered in adhesive contact with a rigid support. We observe that this choice has the advantage of simplifying the exposition in comparison to the two-body case, without affecting the relevance of the model. During a time interval (0, T ), T > 0, we consider a thermoviscoelastic body located in a smooth and bounded domain  ⊂ R3 and lying on a rigid support on a part of its boundary, on which some adhesive substance is present. The contact surface C between the body and the support is part of the boundary of , given by ∂ =  Dir ∪  Neu ∪  C . Here Dir , Neu , and C are open subsets in the relative topology of ∂, each of them with a smooth boundary and disjoint one from each other. Without loss of generality, we suppose that C is a flat surface and identify it with a subset of R2 . That is why, all integrals on C will involve the Lebesgue measure, that coincides with the Hausdorff measure by the flatness requirement. We prescribe zero Dirichlet boundary conditions on Dir , while we assume that a traction is applied on Neu .

2.1 The State and Dissipative Variables The phenomenon of adhesive contact is modeled by state and dissipative variables, describing the thermomechanical equilibrium of the system and its evolution, respectively. In the bulk domain  the state variables are the absolute temperature θ and the symmetric linearized strain tensor (u) (θ, (u))

in  ,

(15)

48

E. Bonetti et al.

while the dissipative variables are in  .

(∇θ, (ut ))

(16)

On the contact surface C the state variables are (θs , χ, ∇χ, u)

on C ,

(17)

where θs is the absolute temperature of the adhesive substance, χ is the surface damage-type parameter from (1), its gradient ∇χ accounts for interactions of the degradation of the adhesive substance in a point, with the degradation in the neighborhood of that point and, with slight abuse of notation, we denote by u the trace of u on C . The surface dissipative variables are (∇θs , χt , θ − θs )

on C ,

(18)

where θ − θs represents the thermal gap on the contact surface (still denoting by θ the trace of the bulk absolute temperature on C ). Finally, on C we also consider state and dissipative variables describing the interaction between the body and the support along the contact surface: in order to distinguish between those describing local interactions and those rendering nonlocal interactions, we will now make explicit their dependence on the variables x ∈ C and (x, y) ∈ C ×C , respectively. In particular, the state variables attached to local interactions, defined pointwise in C , are (χ(x), u(x)),

x ∈ C ,

(19)

while the state variable describing nonlocal damaging effects is defined in C × C by g(x, y) := 2(x − y)u(x),

(x, y) ∈ C × C .

(20)

Analogously, we consider as dissipative variables the surface thermal gap defined pointwise in C , i.e. (θ (x) − θs (x)),

x ∈ C ,

(21)

as well as a dissipative variable defined in C × C by G(x, y) := 2(x − y)(θ (x) − θs (y)),

(x, y) ∈ C × C .

(22)

Let us point out that the terms g(x, y) := 2(x − y)u(x) and G(x, y) := 2(x − y)(θ (x) − θs(y)) render the nonlocal contributions to the degradation process of the adhesive. In particular, as analyzed in [26], g(x, y) takes into account the elongation

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

49

as a source of damage to the adhesive substance, while G(x, y) describes the effects due to the evolution of the thermal gap between two different points on the contact surface. These terms lead to integral contributions both in the normal reaction and in the flow rule for χ, and to a nonlocal heat flux contribution on the contact surface (see (34), (36), (47), and (51) below).

2.2 The Free Energy The free energy F of the system is given by the sum of three contributions: F = F(θ, (u), θs , χ, ∇χ, u, g) := F (θ, (u)) + FC (θs , χ, ∇χ) + Finter (χ, u, g),

(23)

with F , FC , and Finter the bulk, surface, and interaction free energies. We prescribe the bulk free energy as  F (θ, ε(u)) :=

 (θ (x), (u(x))) dx

with (24)



1  (θ, (u)) := θ − θ log(θ ) + θ tr( (u)) + (u) E (u), 2 (recall that E is the elasticity tensor). The surface free energy is given by  FC (θs , χ, ∇χ) :=

C

C (θs (x), χ(x), ∇χ(x)) dx

with

C (θs , χ, ∇χ) = θs − θs log(θs ) +I[0,1] (χ) + γ (χ) + 12 |∇χ|2 + λ(χ)(θs − θeq ),

(25)

where the indicator function I[0,1] of the interval [0, 1] imposes the physical constraint χ ∈ [0, 1], since I[0,1] (χ) = 0 if χ ∈ [0, 1] and I[0,1] (χ) = +∞ otherwise. The function λ is related to the latent heat and the constant θeq > 0 is a phase transition temperature. Moreover, the function γ , sufficiently smooth and possibly nonconvex, describes non-monotone dynamics for χ (it may model some cohesion in the material). Finally, we prescribe the interaction free energy as a sum between local and nonlocal contributions Finter (χ, u, g) := Flinter(χ, u) + Fnl inter (χ, g),

(26a)

50

E. Bonetti et al.

where the local contribution Flinter is given by  Flinter (χ, u)

:= C

l inter (χ(x), u(x)) dx

with

(26b)

l (χ, u) = I(−∞,0] (u · n) + κ2 χ|u|2 inter

and the nonlocal one consists of the integral functional on C × C  Fnl inter (χ, g) :=

C ×C

nl inter (χ(x), χ(y), g(x, y)) dx dy

nl inter (χ(x), χ(y), g(x, y)) =

with

2 1 2 − |x−y| g (x, y)H (χ(x), χ(y))e d 2 , 2

(26c)

where d a given constant and the interaction function H is assumed to be symmetric, i.e. H (x, y) = H (y, x).

2.3 The Dissipation Potential We follow the approach proposed by J.J. Moreau and prescribe the dissipated energy by means of a so-called pseudo-potential of dissipation, which is a convex, nonnegative functional, attaining its minimal value 0 at 0. More precisely, the dissipation potential is again given by the sum of a bulk, a surface, and an interaction part, i.e. P = P(∇θ, (ut ), ∇θs , χt , θ − θs , G) := P (∇θ, (ut )) + PC (∇θs , χt ) + Pinter (θ − θs , G) .

(27)

The volume part is given by  P (∇θ, (ut )) :=

 (∇θ (x), (ut (x))) dx

with



 (∇θ, (ut )) =

αb (θ ) 1 |∇θ |2 + (ut ) V (ut ), 2θ 2

(28)

(recall that V is the viscosity tensor), with αb the bulk heat conductivity coefficient. The surface contribution PC to the pseudo-potential of dissipation is  PC (∇θs , χt ) :=

C

C (∇θs (x), χt (x)) dx

C (∇θs , χt ) =

αs (θs ) 1 |∇θs |2 + |χt |2 , 2θs 2

with (29)

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

51

where αs denotes the surface heat conductivity coefficient. Finally, the interaction contribution Pinter also consists of a local and a nonlocal term, i.e. Pinter (θ − θs , G) := Plinter(θ − θs ) + Pnl inter (G) .

(30a)

The local contribution is given by  Plinter (θ − θs ) :=

C

linter (θ (x) − θs (x)) dx

linter (θ − θs ) =

with

k(χ) (θ − θs )2 , 2

(30b)

where the positive (and smooth) function k is a surface thermal diffusion coefficient. The nonlocal contribution is  Pnl (G) := nl with inter inter (G(x, y)) dx dy C ×C

nl inter (G(x, y)) =

2 1 − |x−y| |G(x, y)|2K(χ(x), χ(y))e d 2 . 2

(30c)

Here, the interaction function K is assumed to be symmetric, i.e. K(x, y) = K(y, x). For simplicity, from now on for the functions H and K in (26c) and (30c) we set H (χ(x), χ(y)) = K(χ(x), χ(y)) := χ(x)χ(y) . We note that in (26c) and (30c) the exponential term describes the attenuation of nonlocal interactions as the distance |x − y| between two points x and y on the contact surface increases.

2.4 The Balance Equations and the Constitutive Laws Now, we recover the equations of the system, written in the bulk domain and on the contact surface, by the general laws of Thermomechanics with the free energy (23) and the potential of dissipation (27). We derive the momentum balance equation for macroscopic movements (33) and the flow rule for the adhesive parameter (36) from the principle of virtual power, in which local and nonlocal microscopic forces responsible for the degradation of the adhesive substance are included. More precisely, for any virtual bulk velocity v with v = 0 on Dir and for any virtual microscopic velocity w on the contact surface, we

52

E. Bonetti et al.

define the power of the internal forces in  and C as follows 



Pint := −  +  +



 (v) d −

(Bw + H∇w) dx +



C ×C

C ×C

C

Rv dx C

2m(x, y)(x − y)v(x) dx dy

(31)

1 2 (Bnl (x, y)w(x) + Bnl (x, y)w(y)) dx dy.

Here,  is the Cauchy stress tensor, R the classical macroscopic reaction on the contact surface, B and H are local interior forces, responsible for the degradation of the adhesive bonds between the body and the support. The terms m(x, y) and i Bnl (x, y), i = 1, 2, are new scalar nonlocal contributions: they stand for internal microscopic nonlocal forces on the contact surface and describe the effects of the elongation as a source of damage. The power of the external forces is given by 



Pext :=

fv d + 

hv d,

(32)

Neu

where f is a bulk known external force, while h is a given traction on Neu . Note that here we have neglected any microscopic external force and any acceleration power. The principle of virtual power, holding for every virtual microscopic and macroscopic velocities and every subdomain in , leads to the quasistatic momentum balance − div  = f

in ,

(33)

supplemented by the following boundary conditions  n(x) = R(x) +

2(x − y)m(x, y) dy

in C ,

(34)

C

u=0

in Dir ,

n = h

in Neu .

(35)

Observe that in (34) the boundary condition for the stress tensor on the contact surface combines a local contribution involving the (pointwise) reaction R(x) and a nonlocal force (defined in terms of the new variable m(x, y)), related to the elongation. Again, the principle of virtual power leads to a micro-force balance on the contact surface given by   1  2 (y, x) dy in  , B(x) − div H(x) = C Bnl (x, y) + Bnl C H · ns = 0 on ∂C ,

(36)

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

53

where ns denotes the outward unit normal vector to ∂C . i Constitutive relations for , R, B, H, m, and Bnl , i = 1, 2, are given in terms of the free energies and the pseudo-potentials of dissipation. More precisely, the constitutive relation for the stress tensor  is  :=

∂ ∂ + = E (u) + θ I + V (ut ) ∂ (u) ∂ (ut )

(37)

with I the identity matrix, while the local reaction is l ∂inter = −κχu − ∂I(−∞,0] (u · n)n, ∂u

(38)

2 nl ∂inter − |x−y| (x, y) = −g(x, y)χ(x)χ(y)e d 2 . ∂g

(39)

R := − combined with m(x, y) := −

Concerning the microscopic forces B and H, we prescribe l ∂C ∂inter ∂C κ + + = ∂I[0,1] (χ) + γ  (χ) + λ (χ)(θs − θeq ) + |u|2 + χt ∂χ ∂χ ∂χt 2 (40) and let H be

B :=

H :=

∂C = ∇χ , ∂∇χ

(41)

1 and B 2 are (formally) defined as derivatives of  nl while the terms Bnl nl inter with respect to the values of the surface damage parameter in x and y ∈ C , respectively, as follows 1 (x, y) := − Bnl

2 nl ∂inter 1 − |x−y| = − g2 (x, y)χ(y)e d 2 , ∂χ(x) 2

(42)

2 (x, y) := − Bnl

2 nl ∂inter 1 − |x−y| = − g2 (x, y)χ(x)e d 2 . ∂χ(y) 2

(43)

The equations for the temperature variables are recovered from the first principle of thermodynamics, i.e. the internal energy balance written in the bulk domain and on the contact surface. The internal energy balance equation in  reads θ st + divq =

∂ (ut ) + h in , ∂ (ut )

(44)

54

E. Bonetti et al.

which h an external heat source. Here the entropy s is defined by the constitutive relation s := −

∂ = log(θ ) − divu , ∂θ

(45)

∂ = −αb (θ )∇θ ∂∇θ

(46)

and the heat flux q is given by q := −θ

(recall that αb is the heat conductivity coefficient in the bulk domain). The boundary conditions on q are prescribed on Dir ∪ Neu as q · n = 0, and on C as   l (q · n)(x) = θ (x) F (x) +

 2(x − y)M(x, y)dy ,

x ∈ C .

(47)

C

Here, the term  F (x) := F l (x) +

2(x − y)M(x, y)dy,

x ∈ C ,

(48)

C

split into local and nonlocal contributions, represents the total entropy flux through the contact surface. In particular, we prescribe F l (x) :=

∂linter (x) = k(χ(x))(θ (x) − θs (x)) ∂(θ − θs )

(49)

and M(x, y) :=

2 ∂nl − |x−y| inter (x, y) = G(x, y)χ(x)χ(y)e d 2 . ∂G

(50)

Finally, the internal energy balance on C is written as    θs (x)∂t ss (x) + divqs (x) = θs (x) F l (x) + C 2(y − x)M(y, x)dy ∂C (x)χt (x), x ∈ C + ∂χt

(51)

supplemented by no-flux boundary conditions qs · ns = 0 on ∂C . Here ss := −

∂C = log(θs ) − λ(χ) ∂θs

(52)

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

55

denotes the entropy on the contact surface and qs := −θs

∂C = −αs (θs )∇θs ∂∇θs

(53)

the heat flux (recall that αs is the heat conductivity coefficient on C ). We note that, like for the entropy flux through the contact surface (the term F in (48)), also for the entropy source involved on the right-hand side of (51), we have distinguished local and nonlocal contributions. With these choices, combining the previous constitutive relations with the balance laws, we derive the PDE system (54) below.

2.5 The PDE System By the previous constitutive relations and balance laws, we obtain the following boundary value problem θt − θ div(ut ) − div(αb (θ )∇θ ) = (ut ) V (ut ) + h, αb (θ )∇θ · n = 0,  αb (θ )∇θ · n = −θ k(χ)(θ −θs )  2  − |x−y| + 2(x − y)G(x, y)χ(x)χ(y)e d 2 dy ,

in  × (0, T ),

(54a)

in (Dir ∪Neu ) × (0, T ), (54b) in C × (0, T ),

(54c)

− div(E (u) + V (ut ) + θ I) = f,

in  × (0, T ),

(54d)

u = 0,

in Dir × (0, T ),

C

(54e) (E (u) + V (ut ) + θ I)n = h,

in Neu × (0, T ), (54f)

(E (u) + V (ut ) + θ I)n + κχu + ∂I(−∞,0] (u · n)n  2 − |x−y| 2(x − y)g(x, y)χ(x)χ(y)e d 2 dy 0, + C

∂t θs − θs λ (χ)χt − div(αs (θs )∇θs ) = |χt |2  + θs k(χ)(θ −θs )

in C × (0, T ),

(54g)

56

E. Bonetti et al.



2

+

2(y − x)G(y, x)χ(x)χ(y)e

− |x−y| 2 d

 dy ,

in C × (0, T ),

(54h)

C

αs (θs )∇θs · ns = 0,

in ∂C × (0, T ), (54i)

χt − χ + ∂I[0,1] (χ) + γ  (χ) κ + λ (χ)(θs − θeq ) − |u|2 2   2  1 − |x−y| g2 (x, y) + g2 (y, x) χ(y)e d 2 dy, in C × (0, T ), − 2 C ∂ns χ = 0,

(54j)

in ∂C × (0, T ), (54k)

where we have written explicitly the dependence of the unknowns (θ, θs , u, χ) on the variable x ∈ C only in the nonlocal terms involving integrals (with respect to the spatial variable y ∈ C ), cf. (54c), (54g), (54h), and (54j). Let us stress that, with respect to the ‘standard’ Frémond system for adhesive contact (see e.g. (14)), (54) encompasses integral terms in the flux boundary conditions (54c) for θ , in Eq. (54h), in the normal reaction (54g), and in the flow rule (54j) for χ. In particular, the thermal evolution of the system depends on a local contribution related to θ − θs evaluated at the same point x ∈ C , and on a nonlocal one described by the function G from (22) that involves the thermal gap between different points x and y on the contact surface. Analogously, the source of damage on the right-hand side of (54j) features local and nonlocal terms and, in particular, it may be different from zero even if u = 0, due to the integral contribution (in terms of the variable g from (20)) that renders the damaging effects of elongation.

2.6 Outlook to the Analysis The analysis of system (54) will be carried out in the forthcoming [18]. There, taking into account the L1 -character of the right-hand sides of the temperature equations (54a) and (54h), we will address the existence of a of weak solution for (54).

A New Nonlocal Temperature-Dependent Model for Adhesive Contact

57

References 1. Andrews, K.T., Kuttler, K.L., Shillor, M.: On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle. Eur. J. Appl. Math. 8, 417–436 (1997) 2. Andrews, K.T., Shillor, M., Wright, S., Klarbring, A.: A dynamic thermoviscoelastic contact problem with friction and wear. Int. J. Eng. Sci. 14, 1291–1309 (1997) 3. Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989) 4. Bonetti, E., Bonfanti, G., Lebon, F., Rizzoni, R.: A model of imperfect interface with damage. Meccanica 52, 1911–1922 (2017) 5. Bonetti, E., Bonfanti, G., Lebon, F.: Derivation of imperfect interface models coupling damage and temperature. Comput. Math. Appl. 77, 2906–2916 (2019) 6. Bonetti, E., Bonfanti, G., Rossi, R.: Well-posedness and long-time behaviour for a model of contact with adhesion. Indiana Univ. Math. J. 56, 2787–2819 (2007) 7. Bonetti, E., Bonfanti, G., Rossi, R.: Global existence for a contact problem with adhesion. Math. Methods Appl. Sci. 31, 1029–1064 (2008) 8. Bonetti, E., Bonfanti, G., Rossi, R.: Thermal effects in adhesive contact: modelling and analysis. Nonlinearity 22, 2697–2731 (2009) 9. Bonetti, E., Bonfanti, G., Rossi, R.: Analysis of a unilateral contact problem taking into account adhesion and friction. J. Differ. Equ. 253, 438–462 (2012) 10. Bonetti, E., Bonfanti, G., Rossi, R.: Analysis of a temperature-dependent model for adhesive contact with friction. Phys. D 285, 42–62 (2014) 11. Bonetti, E., Bonfanti, G., Rossi, R.: Modeling via internal energy balance and analysis of adhesive contact with friction in thermoviscoelasticity. Nonlinear Anal. Real World Appl. 22, 473–507 (2015) 12. Bonetti, E., Bonfanti, G., Rossi, R.: Global existence for a nonlocal model for adhesive contact. Appl. Anal. 97, 1315–1339 (2018) 13. Bonetti, E., Cavaterra, C., Freddi, F., Grasselli, M., Natalini, R.: A nonlinear model for marble sulphation including surface rugosity: theoretical and numerical results. Commun. Pure Appl. Anal. 18, 977–998 (2019) 14. Bonetti, E., Colli, P., Fabrizio, M., Gilardi, G.: Global solution to a singular integrodifferential system related to the entropy balance. Nonlinear Anal. 66, 1949–1979 (2007) 15. Bonetti, E., Colli, P., Fabrizio, M., Gilardi, G.: Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete Contin. Dyn. Syst. Ser. B 6, 1001–1026 (2006) 16. Bonetti, E., Frémond, M.: Analytical results on a model for damaging in domains and interfaces. ESAIM Control Optim. Calc. Var. 17, 955–974 (2011) 17. Bonetti, E., Rocca, E., Scala, R., Schimperna, G.: On the strongly damped wave equation with constraint. Commun. Partial Differ. Equ. 42, 1042–1064 (2017) 18. Bonfanti, G., Colturato, M., Rossi, R.: Existence results for a nonlocal temperature-dependent system modelling adhesive contact (2020). In preparation 19. Cocou, M., Rocca, R.: Existence results for unilateral quasistatic contact problems with friction and adhesion. M2AN Math. Model. Numer. Anal. 34, 981–1001 (2000) 20. Cocou, M., Schryve, M., Raous, M.: A dynamic unilateral contact problem with adhesion and friction in viscoelasticity. Z. Angew. Math. Phys. 61, 721–743 (2010) 21. Eck, C.: Existence of solutions to a thermo-viscoelastic contact problem with Coulomb friction. Math. Models Methods Appl. Sci. 12, 1491–1511 (2002) 22. Eck, C., Jarušek, J., Krbec, M.: Unilateral Contact Problems. Pure and Applied Mathematics vol. 270, Chapman & Hall/CRC, Boca Raton (2005) 23. Eck, C., Jaˇrusek, J.: The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat. Math. Meth. Appl. Sci. 22, 1221– 1234 (1999)

58

E. Bonetti et al.

24. Figueiredo, I., Trabucho, L.: A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity. Int. J. Eng. Sci. 1, 45–66 (1995) 25. Fleming, D.C., Vizzini, A.: The energy absorption of composite plates under off-axis loads. J. Compos. Mater. 30, 1977–1995 (1996) 26. Freddi, F., Frémond, M.: Damage in domains and interfaces: a coupled predictive theory. J. Mech. Mater. Struct. 1, 1205–1233 (2006) 27. L. Freddi, R. Paroni, Roubíˇcek, T., Zanini, C.: Quasistatic delamination models for Kirchhoff– Love plates. ZAMM Z. Angew. Math. Mech. 91, 845–865 (2011) 28. Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33, 1083–1103 (1996) 29. Frémond, M.: Non-Smooth Thermomechanics. Springer, Berlin (2002) 30. Halphen, B., Nguyen, Q.S.: Sur les matériaux standards généralisés. J. Mécanique 14, 39–63 (1975) 31. Koˇcvara, M., Mielke, A., Roubíˇcek, T.: A rate-independent approach to the delamination problem. Math. Mech. Solids 11, 423–447 (2006) 32. Mielke, A., Roubíˇcek, T.: Rate-Independent Systems. Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015) 33. Mielke, A., Roubíˇcek, T., Thomas, M.: From damage to delamination in nonlinearly elastic materials at small strains. J. Elast. 109, 235–273 (2012) 34. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013) 35. Migórski, S., Zeng, S.: Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model. Nonlinear Anal. Real World Appl. 43, 121–143 (2018) 36. Raous, M., Cangémi, L., Cocu, M.: A consistent model coupling adhesion, friction, and unilateral contact. Comput. Methods Appl. Mech. Eng. 177, 383–399 (1999) 37. Rossi, R., Roubíˇcek, T.: Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal. 74, 3159–3190 (2011) 38. Rossi, R., Thomas, M.: From an adhesive to a brittle delamination model in thermo-viscoelasticity. ESAIM Control Optim. Calc. Var. 21, 1–59 (2015) 39. Roubíˇcek, T.: Thermodynamics of rate-independent processes in viscous solids at small strains. SIAM J. Math. Anal. 42, 256–297 (2010) 40. Roubíˇcek, T.: Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal. 45, 101–126 (2013) 41. Roubíˇcek, T., Kružík, M., Mantiˇc, V., Panagiotopoulos, C.G., Vodiˇcka, R., Zeman, J.: Delamination and adhesive contacts, their mathematical modeling and numerical treatment. To appear as Chap.11 In: Mantiˇc, V. (ed.), Mathematical Methods and Models in Composites, 2nd edn. Imperial College Press, London 42. Roubíˇcek, T., Scardia, L., Zanini, C.: Quasistatic delamination problem. Contin. Mech. Thermodyn. 21, 223–235 (2009) 43. Scala, R.: Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish. ESAIM Control Optim. Calc. Var. 23, 593–625 (2017) 44. Scala, R.: A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint. Interfaces Free Bound. 19, 79–107 (2017) 45. Scala, R., Schimperna, G.: A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints. Eur. J. Appl. Math. 28, 91–122 (2017) 46. Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)

Chemomechanical Degradation of Monumental Stones: Preliminary Results Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli, and Roberto Natalini

Abstract The degradation of monumental stones resulting from the mutual interaction between mechanical actions and environment/pollution conditions is investigated here. In particular, the stone degradation is estimated as a function of the environmental conditions and the prediction of damaging phenomena, which can compromise permanently the fruition of monuments. This is done through a macroscopic phenomenological model which accounts for the main aspects of the problem: the chemical reaction and the mechanical behavior of stones. The sulphation reaction and the diffusion of the pollutant agents are described by suitable differential equations coupled with a variational formulation of fracture mechanics. The proposed model permits to evaluate how much aggressive atmospheric agents contribute to the decay of the mechanical properties of the stones as well as to establish the impact of the synergic chemical aggression and stress state. The latter is also influenced by the chemical reaction and by the evolving mechanical properties of the material. The main features of this approach are illustrated by specific numerical simulations. Keywords Calcium carbonate stone · Pollution · Damage · Chemo-mechanical model

E. Bonetti · C. Cavaterra Dipartimento di Matematica, Università degli Studi di Milano e IMATI-CNR, Milano, Italy e-mail: [email protected]; [email protected] F. Freddi () Dipartimento di Ingegneria e Architettura, Università degli Studi di Parma, Parma, Italy e-mail: [email protected] M. Grasselli Dipartimento di Matematica, Politecnico di Milano, Milano, Italy e-mail: [email protected] R. Natalini Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, Roma, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_4

59

60

E. Bonetti et al.

1 Introduction The decay of monumental stones is one of the main concern for people working in the field of conservation and restoration of cultural heritage. Degradation is a complex process in which the intrinsic properties of the material in combination with the environmental parameters are the main responsible factors. On the other hand, such factors are difficult to single out since they are the results of the interaction of various mechanisms. The stone degradation phenomenon generally starts with physical and chemical alteration processes due to the synergetic action of rain, wind, sunlight and freezing/thawing cycles [8, 21, 27]. Thus, the stone surface, which is initially smooth and clean, becomes progressively rough and the external layer more porous, with an increasing? loss of cohesion, formation of microcracks and fractures within and along the edge of the crystal grains. Moreover, the pollutants form deposits of particles and black encrustation, leaving secondary reaction products on stone surfaces. The consequence of these combined actions is a loss of cohesion with dwindling and scaling of stone material and with a general weakening of the surface mechanical strength. Actually, monumental materials are known to behave in a different way when in contact with a chemically reactive environment which is able to promote their decay [26]. The chemical attack may induce drastic changes at the microscopic scale, which can considerably modify the material elasticity and porosity, influencing their structural performance and modifying their external appearance [22]. In general, there is an associated and mutual mechanical action which is sympathetic to chemical transformations. While the progression of a chemical reaction can modify the strain and the material mechanical properties, also an applied state of stress and related deformation may alter the chemical affinity. This chemo-mechanical balance is classically associated with macroscopic indicators such as pressure, specific volume, and temperature. The effects of chemically aggressive environments on the mechanical properties of particular materials like ceramics, granite stones of different compositions or polymeric/ceramic composite materials have been studied for example in [24, 30]. The mechanisms of the chemical aggression of calcareous stones are illustrated in [6]. Among the reactions of the decay of calcium carbonate, sulphation is the most important phenomenon and produces a gypsum (Calcium sulphate, CaSO4 ) layer on the stone surface. Another well-known phenomenon is the chemical aggression on CaCO3 (Calcium Carbonate) due to the presence of SO2 (Sulphur Dioxide) in the surrounding polluted atmosphere. An initial model of the sulphation reaction [7] was essentially based on the Fick’s law and reaction-diffusion coupling, see also [3, 4] for a more general class of models including the effect of permeability. The simplest version of the model was numerically solved in one or two dimensions in [19], where a comparison with experimental data was also conducted. In [11] a new differential system is introduced, coupling bulk and surface evolution equations and including the effect of surface rugosity in the description

Chemomechanical Degradation of Monumental Stones: Preliminary Results

61

of marble sulphation of a monument. More precisely, it is assumed that the surface permeability coefficient depends on a phase parameter related to the local rugosity on the boundary of the monument. The evolution of this parameter is described by a time dependent differential equation on the boundary and possibly accounting for internal constraints on the phase variable through the presence of subdifferentials. The application of these models to configurations and shapes more complex than one or two-dimensional linear elements has not been yet applied or evaluated. Hence, understanding how accurately these models can predict actual degradation on complex geometries is a significant challenge. Other important aspect in the deterioration of the stone monuments is the mechanical degradation. The problem of mechanical degradation process and crack analysis should be considered within a mechanical framework: the cracks may be identified a posteriori as the regions where the elastic stiffness vanishes over localized bands. The literature dealing with the phenomenon of damage is very rich and covers different research fields and requires a non-trivial interplay between non-smooth mechanics, analysis of nonlinear partial differential equations, calculus of variations and computational mechanics. An effective way to describe diffuse damage process is given in terms of some phase variable evolution [2, 9, 10, 12, 15– 18]. Indeed, a fundamental advantage of gradient damage models is related to the existence of an intrinsic elastic limit stress compared to Griffith model and therefore the ability to retrieve crack nucleation. The link between these theories relies on the variational structure and the gradient damage models appear as an elliptic approximation of the variational fracture mechanics problems. Mathematical results based on Gamma-Convergence theory show that when the internal length of a large class of gradient damage models tends to zero, the global minima of the damage energy functional tend towards the global minima of the energy functional of Griffith brittle fracture [5, 12]. This works couples the two models, sulphation and variational fracture mechanics, to reproduce, starting from the environmental conditions and external actions, phenomena such as chemical transformation of the material, material de-cohesion, fracture initiation and evolution. The paper is organized as follows. In Sect. 2 the sulphation reaction is translated into a system of partial differential equations. The phase-field approach to fracture mechanics is presented in Sect. 3. The coupling chemo-mechanical model is described in Sect. 4 whereas a numerical example is presented in Sect. 5.

2 A Mathematical Model of Marble Sulphation Here we consider the mathematical model developed in [7] which macroscopically describes the two main aspects of sulphation process: the diffusion of the pollutant agent within the solid and the material transformation into gypsum. The model is derived by using some basic physical relations, as balance laws of the chemical reactions and Fick’s law, and by neglecting the permeability of the medium. All

62

E. Bonetti et al.

thermal effects are also neglected and the humidity within the air is sufficient to give rise to the reaction. Moreover, we assume that concentration variations of oxygen, water, and carbon dioxide do not alter the reaction.

2.1 A Simplified Chemical Reaction The path of reaction of SO2 with the calcite is assumed to be governed by a simplified one-step reaction 1 CaCO3 + SO2 + O2 + 2H20 → CaSO4 · 2H2O + CO2 . 2

(1)

In words, the reaction produces one molecule of calcium sulphate (gypsum) and carbon dioxide for one molecules of calcium carbonate and one of sulphur dioxide, adding oxygen and water.

2.2 The Hydrodynamical Model Let  be the domain occupied by the calcite specimen under consideration and denote by ρs the total concentration of SO2 , c the calcite density, and γ the gypsum density. Following [3] we suppose that ρs and c are governed by the balance laws 

∂t ρs + ∇ · (ρs Vs ) = − mkc ρs c , ∂t c = − mks ρs c ,

(2)

where Vs is the sulphur dioxide “fluid” velocity, k is the (constant) reaction rate, mc , ms are the molar masses of calcite and SO2 . Under the hypothesis of the total density conservation, it is possible to obtain the following relationship between s and c densities c+

mc mc γ = c0 + γ0 , mγ mγ

(3)

for some given initial densities c0 and γ0 , and setting mγ for the mass of a gypsum molecule. Let ϕ be the porosity of the calcite specimen, which is not assumed to be constant, because the transformation of calcite in gypsum changes the void volume (occupied by air and sulphur dioxide). Therefore, it is reasonable to consider it as a function of the amount of gypsum or, equivalently, as a function of the amount of calcite, that is ϕ = ϕ(c). As shown in [3], the specimen porosity during the reaction can be approximated by a linear combination of the porosity of the pure calcite specimen,

Chemomechanical Degradation of Monumental Stones: Preliminary Results

63

ϕ  the  porosity of the final sulphate product, ϕγ˜ , namely for c = 0 and γ˜ =  0 , and mγ mc c0 + γ0 , we suppose  c  . ϕ (c) = ϕγ˜ + ϕ0 − ϕγ˜ c0

(4)

We denote by s the porous concentration of SO2 , defined as the concentration taken with respect to volume of pores. The porous concentration is a function of the total concentration ρs = ϕ(c)s .

(5)

Simultaneously, the seepage velocity vs depends on the the fluid velocity Vs by the classical Dupuit–Forchheimer relation vs = ϕ(c)Vs .

(6)

Following [7] we assume that all the contribution induced by the pressure gradient to the seepage velocity, usually governed by Darcy’s law, are neglected. Therefore, we determine vs through the Fick’s law s vs = −D(c)Vs ,

(7)

where D(c) = dϕ(c)∇s, being d the scalar effective molecular diffusive coefficient. The following system of equations for the unknowns (s, c) is thus obtained 

∂t (ϕ (c) s) = − mkc ϕ (c) sc + d∇ · (ϕ (c) ∇s) , ∂t c = − mks ϕ (c) sc .

(8)

In the following, we assume that   the initial calcite density c0 is a positive constant. Then, setting a = c10 ϕ0 − ϕγ and b = ϕγ , we can rewrite the porosity function in the following form: ϕ (c) = ac + b.

(9)

Moreover, according to [7], we assume that ϕ0 > ϕγ , with a, b > 0 and 0 < ϕ (c) ≤ ac0 + b < 1. Model (8) gives the possibility of an accurate numerical approximation of the equations by finite elements or finite differences methods (see [7]). Other advantages of this formulation are: a better understanding of the involved physical processes and the possibility to adapt the model to more complex situations, where other damage factors are involved. One main factor is the possibility of determining the time asymptotic regime in one space dimension, which has been experimentally explored in [7].

64

E. Bonetti et al.

3 The Variational Approach to Fracture In this section we illustrate the formulation of the model for a damageable solid. The basic concepts of the phase-field approach to fracture are presented. Full details can be found in [13, 20]. Let  be a brittle damageable solid. For sake of conciseness, no external forces are assumed but only prescribed boundary displacements. The solid state is governed by two fields: the displacement u and the continuous scalar α :  → [0, 1] which represents the internal state of the material. Here α = 1 indicates that the material is intact while for α = 0 the cohesion is fully lost. This latter variable is mechanically equivalent to a damage variable. In the phase-field approach, the Griffith’s energy term of variational fracture formulation, which is proportional to the length of the crack surface  through the fracture toughness γ , is approximated as follows:   Gc meas() ∼ 

 1 2 2 ∇α wl + (1 − α)w dx , 2

(10)

where “meas” is the d − 1 Hausdorff measure [14], the parameter l ∈ R+ is a length scale that governs the localization band width of the phase-field regularizing the √ 3 2 sharp crack, · being the standard Euclidean norm, and w = 8l Gc is a material parameter defining the damage threshold. We consider the energy (10) but different choices are possible (see, for instance, [25]) although they do not change the present discussion and the related results. Recalling that (E(u)) = 12 CE(u)·E(u) is the quadratic strain energy density of the undamaged material where C is the fourth order elastic tensor and E(u) = ∇ s u is the symmetric part of the displacement gradient ∇u, the elastic energy term is modified and (almost) vanishes in fully damaged zones, so that we have    1 CE(u) · E(u) . (E(u)) ∼ α (E(u), α) = α 2 + kl 2

(11)

Alternative expression can be considered to describe specific failure processes [17]. Therefore, the regularized expression of the total energy functional is l (u, α) =

  

α + kl 2

    1 2 2 CE(u) · E(u) dx + wl ∇α + (1 − α)w dx , 2  2

 1

(12) where the coefficient kl = o(l) is introduced to ensure the coerciveness of the functional l .

Chemomechanical Degradation of Monumental Stones: Preliminary Results

65

The fracture problem consists of the minimization, under proper boundary conditions, of the functional (12), namely, min l (u, α) ,

(13)

(u,α)∈A¯

    A¯ = (u, α) ∈ W 1,2 ; Rd × W 1,2 (; [0, 1]) : u = u¯ on ∂u . Dirichlet boundary condition for the phase-field variable can also be prescribed on a boundary portion, depending on the mechanical problem, for instance to avoid boundary fractures that are energetically more convenient than internal cracks. If absent, natural Neumann boundary condition ∇α · n = 0 holds. In the phase-field approach, the irreversibility condition on the damage variable reads α˙ ≤ 0 ,

(14)

which does not allow healing processes in a broken material. The Euler–Lagrange equations are obtained by differentiating the functional (12) with respect to the displacement field and to the phase-field. This gives ⎧ ⎪ ⎨ div T = 0

in  ,

(15a)

α CE(u) · E(u) ⎪ ⎩ −1 − l 2 α + =0 w

in  ,

(15b)

with the Cauchy stress tensor T T=

∂ = (α 2 + kl )CE(u) . ∂E(u)

(16)

Finally, by standard arguments (see, e.g., [20]), when irreversibility (14) is taken into account and the evolution problem is considered, the phase-field Euler– Lagrange equation (15b) turns into the following Karush–Kuhn–Tucker conditions ⎧ α CE(u) · E(u) ⎪ ⎪ −1 − l 2 α + ≤ 0, ⎪ ⎪ w ⎪ ⎨ α˙ ≤ 0, ⎪   ⎪ ⎪ α CE(u) · E(u) ⎪ 2 ⎪ α˙ = 0. ⎩ −1 − l α + w

(17)

66

E. Bonetti et al.

4 A Chemo-Mechanic Model Here, the systems of Eqs. (8), (15a), and (17) are coupled together by considering the mutual interaction of the chemical and mechanical deteriorations from the phenomenological point of view. In particular, changes in the mechanical properties of the material and in the diffusivity coefficient will be considered. We restrict the analysis to the case of an isotropic material. Thus we have C = 2μI+λI⊗I, being μ and λ the Lamé coefficients, I the fourth-order identity tensor and I the second-order identity tensor.

4.1 Mechanical Properties At present no experimental characterization of the mechanical properties of the gypsum obtained as marble transformation is available to the authors, so we base our hypothesis on the data available in the literature and on phenomenological considerations. As reported in [28, 29], the gypsum presents weaker mechanical properties both in term of stiffness and strength if compared to marble. Moreover, the length scale in gypsum is much smaller than marble’s. For simplicity we assume a linear variation of the generic mechanical coefficient a(c) in function of the concentration of calcium carbonate c, that is a(c) = (amarble − agypsum )

c + agypsum c0

(18)

being amarble and agypsum the value for marble (c = c0 ) and for gypsum (c = 0), respectively. The linear transformation (18) is applied to the Young modulus E, the internal length scale l, and the damage threshold w. It should be also considered the fact that the structure of the gypsum crust presents a disordered structure full of defects resulting in mechanical properties that are much more similar to an aged material rather than a pristine one.

4.2 Diffusivity An important modeling choice to be made is how cracking (here described through a phase-field approach) can be coupled with diffusion. Among all the possible options, we adopt the assumption that the diffusivity coefficient d is a function of the phasefield, which corresponds to a one-way coupling where cracking affects the diffusive behavior but diffusion does not influence cracking. For instance, we can suppose     ϕ (c) d (α) = (a + bc) dc 1 − (1 − α)m + d0 (1 − α)m

(19)

Chemomechanical Degradation of Monumental Stones: Preliminary Results

67

where dc is the diffusivity of the undamaged material, d0 is the parameter controlling the diffusivity of the cracked material, taken as the intrinsic diffusivity of the diffusing species in the fluid phase, and m is a model parameter controlling the nonlinear trend of variation of the diffusivity.

4.3 Mathematical Model Recalling the chemical problem (8), the equilibrium equation (15a), the damage evolution system (17), and taking into account the coupling relations (18) and (19), we can formulate the following system of equations ⎧ ⎪ ∂t (ϕ (c) s) = − mkc ϕ (c) sc + d (α) ∇ · (ϕ (c) ∇s) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t c = − mks ϕ (c) sc , ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨div (α + kl )C (c) E(u) = 0 ,

α C (c) E(u) · E(u) ⎪ ≤ 0, −1 − γ (c) l 2 (c) α + ⎪ ⎪ w (c) ⎪ ⎪ ⎪ ⎪ ⎪ α˙ ≤ 0, ⎪ ⎪ ⎪   ⎪ ⎪ α C (c) E(u) · E(u) ⎪ 2 ⎪ ⎩ −1 − γ (c) l (c) α + α˙ = 0. w (c)

(20)

This system is analyzed in an uncoupled fashion. More precisely, at each time step the chemical problem is solved with a fully implicit finite element scheme for s and c, keeping α fixed. Subsequently, the mechanical equilibrium and damage evolution are obtained with an alternate minimisation algorithm which, in short, consists in solving a sequence of minimisation sub-problems for u at fixed α and for α at fixed u. The time is discretized in n constant time intervals. For the numerical simulations, a specific code has been written using FEniCS, an open-source finiteelement computing platform for solving partial differential equations, which leans on PETSc [1], a suite of data structures and routines for scalable (parallel) solutions, and TAO [23], a toolkit for advanced optimization problems which includes the GPCG algorithm.

5 A Numerical Example A two-dimensional rectangular specimen of marble  of sides W and H is considered. The setup is reported in Fig. 1. The considered domain is invested by a polluted air flow along the vertical boundaries whereas the other faces are isolated. The distribution of the pollutant SO2 is constant with a concentration equal to s0 .

68

E. Bonetti et al. ¯

Fig. 1 Rectangular prism in plane strain under uniaxial tension or compression (strain-driven test)

2

e2 e1

3

4

A

Table 1 Values of the adopted coefficients in the computation E (MPa) ν Marble 50,000 0.2 Gypsum 7000 0.2

w (MPa) l (mm) a b dc d0 m mc ms k 0.002 5 0.1 0.1 10 10000 100 64.06 100.09 1.e5 0.0001157 1

Moreover, the material is homogeneous so that the initial calcite concentration is c0 , whilst the concentration of SO2 within the solid is null. A constant vertical displacement is applied on the upper base 2 , while the lower base 1 is kept fixed and the vertical boundaries 3 and 4 are unconstrained and stress free. Moreover, the phase-field is fixed s = 1 on the bases in order to avoid rupture along the constrained boundaries. In summary, the boundary conditions for this case are ⎧ ⎪ u · e2 = 0, Tn · e1 = 0, α = 1 ⎪ ⎪ ⎨ u · e2 = u¯ e2 , Tn · e1 = 0, α = 1 ⎪ ⎪ ⎪ ⎩ Tn = 0, ∇α · n = 0, s = s0

on 1 , on 2 ,

(21)

on 3 and 4 ,

where, as in Fig. 1, e1 and e2 are the horizontal and vertical unit vectors respectively, n is the outward normal to the boundary, the applied displacement is u¯ = 0.01 mm. We consider the case W = 1 mm, H = 2 mm, with material constants reported in Table 1 and kl = 10−6 . The unitary time interval is divided into 1000 time steps i. The maps of α, c, and s are plotted for different time steps in Figs. 2, 3, 4, 5, 6, and 7. Up to time step 73 the undamaged solid is affected only by the sulphation process. Two small external layers of material present a vertical sulphation front that penetrates horizontally within the solids. These solid portions are affected by the decrement of the mechanical properties. In fact, despite the fixed deformation of the test, the material at time step 73 can no longer sustain the state of stress.

Chemomechanical Degradation of Monumental Stones: Preliminary Results

69

1

0

(a)

(b)

(c)

(d)

(e)

Fig. 2 Damage evolution for time steps (a–d) i ∈ {73, 100, 120, 130}, and (e) damage field color legend

(a)

(b)

(c)

(d)

Fig. 3 Damage evolution for time steps (a–d) i ∈ {200, 400, 600, 1000}

0

0

(a)

(b)

(c)

(d)

(e)

Fig. 4 Evolution of the concentration c for time steps (a–d) i ∈ {73, 100, 120, 130} and (e) c color legend

70

E. Bonetti et al.

(a)

(b)

(c)

(d)

Fig. 5 Evolution of the concentration c for time steps (a–d) i ∈ {200, 400, 600, 1000}

0

0

(a)

(b)

(c)

(d)

(e)

Fig. 6 Evolution of the concentration s for time steps (a–d) i ∈ {73, 100, 120, 130} and (e) s color legend

(a)

(b)

(c)

(d)

Fig. 7 Evolution of the concentration s for time steps (a–d) i ∈ {200, 400, 600, 1000}

Chemomechanical Degradation of Monumental Stones: Preliminary Results

71

Small fractures emanate horizontally from the external borders. As the sulphation process evolves, the cracks continue to propagates horizontally. At time step 100 and 120 the mutual interaction between chemical process and mechanical degradation is evident. In fact, fractures propagate in the zones affected by the chemical reaction and at the same time s penetrates in the damaged zones as clearly shown in Fig. 6b, c thus inducing the transformation of marble into gypsum near cracks, see Fig. 4b, c. At time step 130 a severe quasi-horizontal crack propagation interests two small fractures as illustrated in Fig. 2d. The rupture crosses the inner core of solid that is completely made of marble (c ≈ c0 ). Note that the diffusion zones of these fractures are larger than the transition width as evidenced by the fractures of the external layers. Observe that s0 deeply penetrates within the solid through these two cracks and consequently the sulphation interests the inner part of the solid. From time step 200 the evolution of the cracks is less severe and follows the evolution of the sulphation process. In fact, as clearly stated in Fig. 3b–d, the crack tips propagate stably in gypsum with a velocity similar to the evolution of the sulphation. The nonlinear interplay of the sulphation process Figs. 5b–d and 7b–d with fracture propagation is clear.

References 1. Abhyankar, S., Brown, J., Constantinescu, E.M., Ghosh, D., Smith, B.F., Zhang, H.: PETSc/TS: a modern scalable ODE/DAE solver library (2018). arXiv preprint arXiv:180601437 2. Alessi, R., Freddi, F.: Phase-field modelling of failure in hybrid laminates. Compos. Struct. 181, 9–25 (2017) 3. Alì, G., Furuholt, V., Natalini, R., Torcicollo, I.: A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I. Modeling and qualitative analysis. Transp. Porous Media 69(1), 109–122 (2007) 4. Alì, G., Furuholt, V., Natalini, R., Torcicollo, I.: A mathematical model of sulphite chemical aggression of limestones with high permeability. Part II: numerical approximation. Transp. Porous Media 69(2), 175–188 (2007) 5. Ambrosio, L., Tortorelli, V.M. Approximation of functionals depending on jumps by elliptic functionals via -convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990) 6. Amoroso, G., Fassina, V.: Stone decay and conservation: atmospheric pollution, cleaning, consolidation, and protection. Materials Science Monographs. Elsevier, Amsterdam (1983) 7. Aregba-Driollet, D., Diele, F., Natalini, R.: A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: numerical approximation and asymptotic analysis. SIAM J. Appl. Math. 64(5), 1636–1667 (2004) 8. Benavente, D., Cultrone, G.G.H.: The combined influence of mineralogical, hygric and thermal properties on the durability of porous building stones. Eur. J. Mineral. 20(4), 673–685 (2008) 9. Bonetti, E., Frémond, M.: Analytical results on a model for damaging in domains and interfaces. ESAIM Control Optim. Calc. Var. 17(4), 955–974 (2011) 10. Bonetti, E., Freddi, F., Segatti, A.: An existence result for a model of complete damage in elastic materials with reversible evolution. Contin. Mech. Thermodyn. 29(1), 31–50 (2017) 11. Bonetti, E., Cavaterra, C., Freddi, F., Grasselli, M., Natalini, R.: A nonlinear model for marble sulphation including surface rugosity: theoretical and numerical results. Commun. Pure Appl. Anal. 18(2), 977–998 (2019)

72

E. Bonetti et al.

12. Bourdin, B., Francfort, G.A., Marigo, J.J. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000) 13. Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1), 5–148 (2008) 14. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised Edition. Textbooks in Mathematics. CRC Press, Boca Raton (2015) 15. Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998) 16. Freddi, F., Frémond, M.: Damage in domains and interfaces: a coupled predictive theory. J. Mech. Mater. Struct. 1, 1205–1234 (2006) 17. Freddi, F., Royer-Carfagni, G.: Regularized variational theories of fracture: a unified approach. J. Mech. Phys. Solids 58(8), 1154–1174 (2010) 18. Freddi, F., Royer-Carfagni, G.: Phase-field slip-line theory of plasticity. J. Mech. Phys. Solids 94, 257–272 (2016) 19. Giavarini, C., Santarelli, M., Natalini, R., Freddi, F.: A non-linear model of sulphation of porous stones: numerical simulations and preliminary laboratory assessments. J. Cult. Herit. 9, 14–22 (2008) 20. Marigo, J.J., Maurini, C., Pham, K.: An overview of the modelling of fracture by gradient damage models. Meccanica 51(12), 3107–3128 (2016) 21. Marini, P., Bellopede, R.: Bowing of marble slabs: evolution and correlation with mechanical decay. Construct. Build Mater. 23(7), 2599–2605 (2009) 22. McCauley, R.: Corrosion of Ceramic and Composite Materials, 2nd edn. Corrosion Technology. CRC Press, Boca Raton (2004) 23. McInnes, L.C., Moré, J.J., Munson, T.: TAO Users Manual (2010) 24. Nara, Y., Kaneko, K.: Sub-critical crack growth in anisotropic rock. Int. J. Rock Mech. Mining Sci. 43(3), 437–453 (2006) 25. Pham, K., Amor, H., Marigo, J.J., Maurini, C.: Gradient damage models and their use to approximate brittle fracture. Int. J. Damage Mech. 20(4), 618–652 (2011). https://doi.org/10. 1177/1056789510386852 26. Shushakova, V., Fuller, E.R., Heidelbach, F., Mainprice, D., Siegesmund, S.: Marble decay induced by thermal strains: simulations and experiments. Environ. Earth Sci. 69(4), 1281–1297 (2013) 27. Smith, B., Gomez-Heras, M., McCabe, S.: Understanding the decay of stone-built cultural heritage. Progr. Phys. Geogr. Earth Environ. 32(4), 439–461 (2008) 28. Wong, L.N.Y., Einstein, H.H.: Crack coalescence in molded gypsum and carrara marble: part 1. macroscopic observations and interpretation. Rock Mech. Rock. Eng. 42(3), 475–511 (2009) 29. Wong, L.N.Y., Einstein, H.H.: Crack coalescence in molded gypsum and carrara marble: part 2—microscopic observations and interpretation. Rock Mech. Rock. Eng. 42(3), 513–545 (2009) 30. Yoshida, S., Matsuoka, J., Soga, N.: Sub-critical crack growth in sodium germanate glasses. J. Non-Cryst. Solids 316(1), 28–34 (2003)

Modelling the Effects of Protective Treatments in Porous Materials Gabriella Bretti, Barbara De Filippo, Roberto Natalini, Sara Goidanich, Marco Roveri, and Lucia Toniolo

Abstract The aim of this preliminary study is to understand and simulate the hydric behaviour of a porous material in the presence of protective treatments. In particular, here the limestone Lumaquela deAjarte is considered before and after the application of the silane-based product ANC. A recently developed mathematical model was applied in order to describe the capillary rise of water in stone specimens. The model was calibrated by using experimental data concerning the water absorption by capillarity in both treated and untreated stone specimens. With a suitable calibration of the main parameters of the model and of the boundary conditions, it was possible to reproduce the main features of the experimentally observed phenomenon. Keywords Mathematical modelling · Porous media · Stone protection · Predictive models

1 Introduction Natural stones used in historical buildings are open porous systems that are subject to different damaging processes because of their exposure to the environment. Most weathering processes affecting porous materials, such as salt-crystallization [1], freezing-thawing cycles [2], dissolution of soluble fractions [3], and swelling of clays [4], etc. are associated to liquid water penetration, either in the form of meteoric precipitation or groundwater moisture infiltration. In addition, the microstructural modifications induced by the above-mentioned processes may lead

G. Bretti () · B. D. Filippo · R. Natalini Istituto per le Applicazioni del Calcolo, CNR, Roma, Italy e-mail: [email protected]; [email protected]; [email protected] S. Goidanich · M. Roveri · L. Toniolo Dipartimento di Chimica, Materiali e Ingegneria Chimica, Politecnico di Milano, Milano, Italy e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_5

73

74

G. Bretti et al.

to an increase in water penetration, thus contributing to an ongoing decay process [5]. To reduce the effects of these deteriorating factors, strategies for stone protection have been recently developed pursuing a multi-functional approach, which is aimed at reducing water penetration on one side and the soiling and corrosion from air-borne pollutants on the other. Nowadays, one of the most important classes of protective treatments are silicon-based compounds, i.e., alkylsilanes or siloxanes, which penetrate into the pores of stone materials and hydrophobize them [6, 7] without causing relevant alterations of the deeper microstructure. Most of the protective performance relies indeed on the efficiency by which treatments modify the water/stone interface, adhering to the stone grains and adapting to their surface morphology, while the penetration depth was shown to have a looser correlation to the performance [8]. However, acquiring direct evidence of the modification of this interface is a difficult task. Capillary absorption measurements not only provide a fundamental indication for monitoring the progress of materials degradation but also critical parameters for evaluating the efficacy of protective treatments applied to stone materials with the aim of preventing or reducing water penetration [9]. The prediction of the effect of water-repellent treatments on the water absorption behaviour of stones is a highly challenging task. In particular, a considerable variety exists in the requirements set by the protection of stones depending on their characteristics, particularly mineralogical composition, porosity and pore size distribution. Therefore, the design of an effective treatment for a given lithotype should be tuned in order to meet its specific requirements. In this framework, the present paper describes a mathematical model using Darcy’s law to simulate the water uptake into a porous medium. The physical properties of the medium are described by two parameters representing the diffusion rate of water in the medium and the residual value of saturation that ensures the hydraulic continuity. These two parameters will be determined through the numerical calibration of the model—i.e. by comparing numerical results with the available experimental data on the limestone Lumaquela de Ajarte (Castile and León, Spain) before and after the application of a silane-based product. The chosen protective treatment, called ANC has been recently developed in the framework of the European project NanoCathedral [10, 11]. The paper is organized as follows: the second section will describe the materials considered and the experiments performed; in the third section the mathematical model is introduced and in the fourth section the numerical scheme applied to solve the system equations is described. Finally, the results obtained are discussed in section five.

2 Materials and Methods Ajarte stone (from Treviño, Castile and León, Spain) is a biomicrite with creamy greyish colour, composed of shell fragments in a carbonate matrix consisting of recrystallized fossils (>99% calcite). It is therefore composed mainly by CaCO3

Modelling the Effects of Protective Treatments in Porous Materials

75

Fig. 1 Specimen of Ajarte stone

and has an open porosity of 23.5% [10]. A picture showing an Ajarte stone sample is depicted in Fig. 1. The protective treatment ANC consists of silane monomers (40% w/w) in 2-propanol and contains a small fraction of T iO2 nanoparticles (0.12% w/w). The organosilica gel resulting from the hydrolysis and condensation of the silane units imparts water-repellent features to the product, while T iO2 nanoparticles add photocatalytic and self-cleaning properties [10]. The product was applied onto stone specimens by capillary absorption for 6 h, using a filter paper pad saturated with the treating material. Water absorption by capillarity in treated and untreated stone specimens was measured up to 96 h (30 specimens of 5 × 5 × 2 cm3 size) according to the EN 15801 standard [14]. Since Ajarte stone does not show any visible orientation of sedimentary planes, either by naked-eye or microscopy observation, no privileged direction was chosen upon preparing the specimens. Prior to the test, the specimens were immersed in deionized water for 1 h in order to remove any excess soluble salts, dried in oven at 65 ◦ C until constant weight and stored in a silica gel desiccator for another 24 h. Then, they were placed on water-soaked filter paper pads (Ahlstrom-Munktell laboratory filter paper, 1288 grade) and the water absorption was monitored gravimetrically. To limit the water consumption by evaporation, the test was conducted in closed vessels. The amount of water per unit area absorbed by the specimens at time t is defined 0) by W = (mi −m expressed in mg/cm2 , where mi − m0 is the amount of water S absorbed (expressed in mg) and S is the surface of the specimen in contact with the

76

G. Bretti et al.

Fig. 2 Laboratory experiment for the measurement of water absorption

soaked paper pad. W is graphically reported as Q(tk ), where tk is time expressed in s 1/2 . See Fig. 2 for the experimental setup.

3 Modelling of Capillary Imbibition Here a similar approach to the work presented by Bracciale et al. [12] was followed. In this case we consider the liquid water flow into an initially dry specimen of Ajarte stone. We aim at simulating the capillary absorption of the specimen with the mathematical model specified below. We denote the volume fraction occupied by the liquid and by the gas (composing the fluid) within the representative element of volume, by θl and θg , respectively. The following relation holds: n0 = θl + θg .

(1)

The water flow into a porous medium is given by the well-known Darcy’s law q = − k(s) μl (∇Pc (s) − ρl g), with Pc = Pc (θl /n) the capillary pressure, k the permeability of the porous matrix, μl the viscosity of the fluid and s = θl /n0 . Since stone specimens are small (2 cm height), gravity effects can be safely disregarded from Darcy’s law. There are many suggested experimental curves aiming to connect

Modelling the Effects of Protective Treatments in Porous Materials

77

the capillary pressure with the moisture content, but they are not completely trustable since a relation correlating capillary pressure with moisture content into the porous matrix still lacks. Therefore, as in our previous works, we introduce a function B, such that ∇B = −

k(·) ∇Pc (·), μl

(2)

expressed by:

B(s) =

 ⎧  2 ⎪ 2 1−s ⎪ c (31 − 2s) + (1 − a) , if s ∈ [a, 1], ⎨3 1−α 0, if s ∈ [0, a), ⎪ ⎪ ⎩ B(1) = 23 c(1 − a), if s > 1,

and B  (s) = max



  4Bmax (a − s)(s − 1), 0 . (1 − a)2

 Constants a and Bmax = c are physical properties of the porous material involved and will be determined later on. The quantity θl a · n, corresponding to s = a, is the residual value for saturation ensuring the hydraulic continuity, with B(s) a compactly supported function in [a, 1]. On the other hand c has the dimension of a diffusivity. The mathematical model describing the experiment consist of the water continuity equation, that in this setting, is given by

∂t θl = ∂zz B.

(3)

Equation (6) has to be coupled with reasonable initial and boundary conditions. For the imbibition experiment, we assume the conditions 

θl (z, 0) = 0, θl (0, t) = n0 ,

(4)

that is, the sample is initially dry (see Table 1 for the values of the parameters of the problem) while its bottom side is always saturated. To reproduce the loss of water at the upper boundary z = h1 due to evaporation, we derive θl (h1 , t) from the following relation θl = θ¯l . In the above condition, θ¯l is the moisture content of the ambient air (assumed constant), and can be considered as the value of humidity on the top side of the stone specimen. Note that to get θ¯l we used the formula for the saturated vapour density (SVD in [g/m3 ]) as a function of temperature T [◦ C]: SV D(T ) = 5.018 + 0.32321T + 8.1847 × 103T 2 + 3.1243 × 104 T 3 from which we can obtain the density of vapour in [g/cm3 ].

78

G. Bretti et al.

Table 1 Parameters of the problem Parameter n0 h1 h2 ρl

Description Open porosity of Ajarte stone Specimen’s height Specimen’s height immersed Density of water

Units – cm cm g/cm3

Value 0.235 2.0 0.2 1

Ref. [10]

[13]

The value of θ¯ is obtained from the formula [15]: SV D(T ) · n0 · RH, θ¯ = ρl

(5)

where RH is the relative humidity in the ambient air. In our simulation we suppose to have RH = 70%, since specimens are in a closed vessel. Actually, this value can be measured experimentally and we propose to do that in our future works.

3.1 Boundary Conditions Here we describe the initial and boundary conditions to complement the mathematical model. The immersed part of the specimen (for −h2 ≤ z ≤ 0) is pervious to water flow and we assume that it is initially saturated. In this way we can simply confine ourselves to mathematically describe the domain 0 ≤ z ≤ h1 . The water continuity equation is given by: ∂t θl = ∂zz B.

(6)

Here the porosity will remain constant and, thus, will not affect water flow. Equation (6) has to be coupled with reasonable initial and boundary conditions. For the experiment of imbibition, we assume the conditions 

θl (z, 0) = 0, θl (0, t) = n0 ,

(7)

that is, the sample is initially dry while its bottom side is always saturated. To reproduce the loss of water at the upper boundary z = h1 due to evaporation, we derive θl (h1 , t) from the condition: θl = θ¯l .

(8)

Modelling the Effects of Protective Treatments in Porous Materials

79

4 Numerical Results and Comparison with Experimental Data In this section we describe the numerical approximation of the model and results obtained by calibrating the parameters of the mathematical model.

4.1 Numerical Approximation In order to describe the numerical approximation to the Eq. (6), we mesh the interval [0, h] with a step z = Nh . Then we set λ=

t , zj = j z, j = 1, . . . , N, z

and we define wjk = w(zj , tk ) the approximation of the function w at the height zj and at the time tk . The simplest and consistent approximation of ∂z (r(z)∂z w) by means of Taylor expansions is the following first order approximation: j (r, w) :=

(rj + rj +1 )(wj +1 − wj ) − (rj −1 + rj )(wj − wj −1 ) . 2z2

(9)

From now on, we will omit for simplicity the subscript l of θ . Then, the discretization in explicit form of the Eq. (6) is: θjk+1 − θjk t

= j ((nk /n0 )2 , B k )

(10)

We remark that, differently from our previous work [12], in this experimental settings the evaporation from lateral sides in negligible, thus we did not include it in the model.

4.2 Calibration of Parameters a, c Now we describe the calibration procedure to determine a and c using the experimental data. We need to compute the total amount of water absorbed and lost by the specimen at time tk given by: 

h1

ρl θ (z, tk )dz, 0

(11)

80

G. Bretti et al.

thus we need to solve problem (6). We compute θ (z, tk ) numerically with the forward-central approximation scheme θjk+1 = θjk +

t (Ba,c (θjk+1 /n0 ) − 2Ba,c (θjk /n0 ) + Ba,c (θjk−1 /n0 )) z2

with the boundary condition at the top boundary (8), under the CFL condition t n0 n0 ≤ = , 2 2∂z Ba,c 2c z  h1 , {tk }k=1,...,Nmeas . At the with θjk = θ (zj , tk ), zj = j z, j = 0, . . . , N = z bottom boundary we use the Dirichlet imbibition condition θ0k+1 = nk+1 0 .

(12)

Then we compute the approximated values of the quantity of water in the specimen Qnum at time tk as follows. With the trapezoidal rule we compute the integral (11): k ⎛ Qnum =ρ k

z ⎝θ0k + 2 2

N−1 

⎞ θjk + θNk ⎠ ,

j =1

in order to compare the numerical quantity of water to experimental data Qk at time tk . The error to be minimized is then defined as E(a, c) =

1 Nmeas

N meas

(Qnum − Qk )2 k

k=1

Q2k

.

c applying the The calibration procedure has been carried out in MATLAB simulated annealing method. The parameters a and c in the function B describing the capillary pressure of the experiment can be found in Table 2. The other parameters are less determinant for the calibration of the model, since they take into account the water exchange on top of the specimen according to the relative humidity in the environment. c applying the simulated The fitting procedure has been carried out in MATLAB Table 2 Parameters obtained with a fitting procedure Parameter aNT cNT aT cT

Description Residual saturation of untreated Ajarte stone Diffusion rate of untreated Ajarte stone Residual saturation of treated Ajarte stone Diffusion rate of treated Ajarte stone

Units – cm2 /s – cm2 /s

Value 0.78 9.9541e–08 1.1e–01 8.6e–09

Modelling the Effects of Protective Treatments in Porous Materials

81

not treated case

550

model data

2

Quantity of water absorbed [mg/cm ]

500 450 400 350 300 250 200 150 100 50 0 0

100

200

300

time s

400

500

1/2

Fig. 3 Data fitting result for untreated Ajarte stone: comparison between data points (red circles) and fitting values (blue stars) obtained with the mathematical model for a = 0.78, c = 2.9505e−04

annealing method to the model to reproduce the experimental results obtained. The computational time for a single simulation with fixed parameters takes 900 s on an Intel(R) Core(TM) i7-3630 QM CPU 2.4 GHz. Table 2 lists the results obtained within an average error of about 1% for the untreated case and of 4% for the treated case. See Figs. 3 and 4 for a comparison with experimental data. As we can see, there is a good agreement between experimental and model values. We also underline that in the treated case we had at our disposal just the measurements on one specimen instead of an averaged value. In Fig. 5 we represent the volume fraction occupied by liquid for the untreated (on the left) and the treated case (on the right).

5 Conclusions and Perspectives In this paper we calibrated a mathematical model to describe the action of water absorption reduction in a porous stone treated with a protective product. The fitting procedure showed the effectiveness of our modelling approach, since we were able to simulate the amount of water absorbed by the specimen. In the presence of a protective treatment, the domain of the function B (the left endpoint is the residual saturation a) is larger, but the diffusion speed of the liquid is lower with respect to the untreated case. This represents a first promising result and in the future we will repeat the study by using different stone substrates in order to test and improve our mathematical model. In particular, we aim at developing a more accurate model

82

G. Bretti et al. treated case

30

Quantity of water absorbed [mg/cm2 ]

model data

25

20

15

10

5

0 0

100

200

300

time s

400

500

1/2

untreated case at time T=96 h

0.24

volume fraction occupied by liquid

volume fraction occupied by liquid

Fig. 4 Data fitting result for Ajarte stone treated with ANC: comparison between data points (red circles) and fitting values (blue stars) obtained with the mathematical model for a = 0.11, c = 8.6e − 09

0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 heigth of the specimen

treated specimen at time T=96 h

0.25 0.2 0.15 0.1 0.05 0

0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 height of the specimen

2

Fig. 5 The volume fraction θ occupied by liquid obtained with the mathematical model at the final time t = 96 h for the untreated (on the left) and the treated case (on the right)

where the capillary and the permeability functions are calibrated separately against experimental data. Our final goal is to end up with a sound simulation tool to forecast the effect of the protective treatments on different porous materials.

Modelling the Effects of Protective Treatments in Porous Materials

83

References 1. Doehne, E.: Salt weathering: a selective review. Geol. Soc. Lon. Spec. Publ. 205(1), 51–64 (2002). https://doi.org/10.1144/GSL.SP.2002.205.01.05 2. Scherer, G.W., Valenza, J.J.: Mechanisms of frost damage. In: Young, F., Skalny, J.P. (eds.) Materials Science of Concrete VII, pp. 209–246. Wiley, New York (2005) 3. Franzoni, E., Sassoni, E.: Correlation between microstructural characteristics and weight loss of natural stones exposed to simulated acid rain. Sci. Total Environ. 412, 278–285 (2011). https://doi.org/10.1016/j.scitotenv.2011.09.080 4. Ruedrich, J., Bartelsen, T., Dohrmann, R., Siegesmund, S.: Moisture expansion as a deterioration factor for sandstone used in buildings. Environ. Earth Sci. 63(7–8), 1545–1564 (2011). https://doi.org/10.1007/s12665-010-0767-0 5. Franzoni, E., Sassoni, E., Scherer, G.W., Naidu, S.: Artificial weathering of stone by heating. J. Cult. Herit. 14(3), e85–e93 (2013). https://doi.org/10.1016/j.culher.2012.11.026 6. Cnudde, V., Cnudde, J.P., Dupuis, C., Jacobs, P.J.S.: X-ray micro-CT used for the localization of water repellents and consolidants inside natural building stones. Mater. Charact. 53(2–4), 259–271 (2004). https://doi.org/10.1016/j.matchar.2004.08.011 7. Casadio, F., Toniolo, L.: Polymer treatments for stone conservation: methods for evaluating penetration depth. J. Am. Inst. Conserv. 43(1), 3–21 (2004). https://doi.org/10.2307/3179848 8. De Buergo Ballester, M.A., González, R.F.: Basic methodology for the assessment and selection of water-repellent treatments applied on carbonatic materials. Prog. Org. Coat. 43(4), 258–266 (2001). https://doi.org/10.1016/S0300-9440(01)00204-1 9. Charola, A.E.: Water repellents and other protective treatments: A critical review. In: Littmann, K., Charola, A.E. (eds.) Proceedings of Hydrophobe III - Third International Conference on Surface Technology with Water Repellent Agents, pp. 3–19. Aedificatio Verlag, Freiburg i. B. (2001) 10. Gherardi, F., Roveri, M., Goidanich, S., Toniolo, L.: Photocatalytic nanocomposites for the protection of European architectural heritage. Materials 11(1), 65 (2018) 11. Roveri, M., Gherardi, F., Brambilla, L., Castiglioni, C., Toniolo, L.: Stone/coating interaction and durability of Si-based photocatalytic nanocomposites applied to porous lithotypes. Materials 11(11), 2289 (2018). https://doi.org/10.3390/ma11112289 12. Bracciale, M.P., Bretti, G., Broggi, A., Ceseri, M., Marrocchi, A., Natalini, R., Russo, C.: Crystallization inhibitors: explaining experimental data through mathematical modelling. Appl. Math. Model. 48, 21–38 (2017). https://doi.org/10.1016/j.apm.2016.11.026 13. Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows, Porous Media, Society for Industrial and Applied Mathematics (2006). https://doi.org/10.1137/1. 9780898718942. http://epubs.siam.org/doi/abs/10.1137/1.9780898718942 14. EN15801: 2009. European Committee for Standardization: Conservation of Cultural Property, Test Methods, Determination of Water Absorption by Capillarity; European Committee for Standardization: Brussels, Belgium (2009) 15. HyperPhysics Website. http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, Georgia State University

Mathematical Models for Infrared Analysis Applied to Cultural Heritage Giovanni Caruso, Noemi Orazi, Fulvio Mercuri, Stefano Paoloni, and Ugo Zammit

Abstract Active pulsed infrared thermography is an effective technique consisting in moderately heating the specimen by means of the absorption of a visible light pulse and, then, in detecting the transient variation in the emitted infrared radiation by an infrared camera. Inhomogeneities and buried features eventually located into the specimen volume can be revealed by the recorded infrared images. Such a technique has been successfully applied to the analysis of cultural heritage artifacts like ancient bronzes and manuscripts. The former belong to the category of optically opaque materials, whereas the second to the one of optically semitransparent materials. For both the two considered categories, a mathematical model for the analysis of the thermographic signal is here presented, together with an implementation in Matlab environment using the finite element technique. The developed models are then used to analyse the experimental results and, hence, to obtain both qualitative and quantitative information about the investigated items. Keywords Pulsed thermography · Opaque materials · Semitransparent materials · Mathematical models · Non invasive investigation

1 Introduction Over the recent years, the interest for the conservation and preservation of has attracted growing attention. To this aim, several techniques for the non-destructive inspection of cultural heritage items have been developed and, among other, pulsed

G. Caruso () ISPC-CNR, Roma, Italy e-mail: [email protected] N. Orazi · F. Mercuri · S. Paoloni · U. Zammit Università di Roma Tor Vergata, Roma, Italy e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_6

85

86

G. Caruso et al.

thermography (PT) has been proven to be one of the most effective [1–5]. PT, which like other photothermal techniques allows the measurement of thermal parameters in different materials [6–8], has been successfully applied to the study of bronze statues [9–12], books [13] and paintings [14, 15]. PT relies on the locally resolved detection by means of an infrared (IR) camera of the transient change of the emitted IR radiation following the sample heating induced by the absorption of a visible light pulse. The presence of buried features can be detected thanks to the induced local variations in the amount of emitted radiation from the sample surface which are then displayed by the recorded IR images. As discussed later on, shallow features give rise to PT signal contrast showing at early delay time in comparison to that produced by the deeply buried ones. Therefore, the analysis of the PT signal time dependence can also allow to depth-resolve the position of the detected features. Such a possibility is not granted by other IR techniques which are widely employed in the field of cultural heritage evaluation such as reflectography. From the PT point of view, the samples can be divided into the following two main categories depending on the optical properties of their main constituent material. In optically opaque samples such as bronze statues the visible light absorption take place over a very thin layer under the sample surface. In addition, only the IR radiation emitted from the sample surface give rise to the PT signal since the one emitted from the sample volume gets absorbed by the sample itself. Under such circumstances, the PT signal is merely proportional to the temperature variation at the sample surface. When inspecting specimen made of opaque materials, the contrast in the thermographic images is therefore mainly originated from the presence of thermal properties inhomegeneities such as delaminations or voids into the sample volume. In fact, such inhomogeneities can locally affect the heat diffusion rate, thus giving rise to a different time dependence of the transient temperature variation at the sample surface above the buried inhomogeneity in comparison with the one observed in the other parts of the sample surface. Unlike the opaque ones, in optically semi-transparent samples such as manuscripts or documents both the visible light absorption and the IR emission occur through the volume of the investigated specimen. As shown in the following, the PT signal is no longer proportional to the temperature at the sample surface but, on the contrary, it is given by a weighted spatial average of the temperature profile into the sample. In this case, the contrast in the recorded IR images is mainly due to local differences in the optical properties, namely the visible absorption coefficient and the IR emissivity, of the buried features in comparison with the ones of the surrounding material. In the following, two theoretical models enabling the study of the PT signal obtained in opaque and semi-transparent samples, respectively are presented. The presented methods are then applied to analyse the PT data obtained in bronze statues and manuscripts. In the case of the bronze statues, the PT technique has been applied to investigate the additional parts which have been inserted into the main body of the statue after the main casting due to, for instance, to artistic reasons like the possibility to obtain features with different colour with respect to the main statue one. Such parts can be inserted by means of two different procedures, namely the metallurgical and the mechanical one. As shown later on, the analysis of the

Mathematical Models for Infrared Analysis Applied to Cultural Heritage

87

PT signal by means of the presented model allows to distinguish between the two different techniques which might have been adopted for the insertion, thus providing valuable information about the manufacture of the statue. As regards the model for the semi-transparent material, it has been applied to analyse the PT data obtained on a homogeneous paper layer containing graphical features buried into its volume. More specifically, the model has been used to evaluate the depth position of buried graphical features. In order to that, the optical and thermal parameters of the inspected material have been first evaluated by means of a calibration procedures which are also presented.

2 Models In this section mathematical models are developed for the analysis of the thermographic signal, taking into account the visible light absorption and consequent infrared emission in both optically opaque and semi-transparent materials.

2.1 Optically Opaque Materials In optically opaque materials the excitation light delivered by the flashes is totally absorbed in a very thin layer beneath the specimen illuminated surface. Moreover, the infrared radiation emitted by the sample volume is totally absorbed during its propagation inside the material. Accordingly, the infrared radiation detected by the camera is the one emitted by only the specimen surface and, as a consequence, the detected signal is just proportional to the surface temperature. Among others, active infrared thermography has proven to be very effective for the study of insertions in bronze statues [10, 16]. In this case, the sketch of a typical specimen is depicted in Fig. 1, representing a portion of a bronze statue of thickness H , whose domain is indicated with 2 , with an insertion (1 ) of thickness h. Let I indicate the interface between 1 and 2 , where n is the normal to I pointing outward from 1 . Finally, let If be the flash intensity and m the unit normal to the external contour of 2 pointing outwards. Fig. 1 Schematic representation of a portion of an optically opaque material containing an insertion. [Reprinted with permission from F. Mercuri et al. [16]. Copyright 2018, Elsevier]

88

G. Caruso et al.

The equations governing the heat generation and diffusion inside the specimen are: ∂T1 − k1 T1 = 0 in 1 ∂t ∂T2 − k2 T2 = 0 in 2 , ρ2 c2 ∂t

ρ1 c1

(2.1)

where t is the time, Ti is the temperature inside i and ρi , ci , ki are, respectively, the density, specific heat and thermal conductivity in i . Equations (2.1) are applied with the following boundary conditions − k2 ∇T2 · m = 0

on δl2 ,

−k2 ∇T2 · m = hc (Text − T2 )

on z = H ,

−ki ∇Ti · m = hc (Text − Ti ) − If δ(t) [−ki ∇Ti · n] = 0

on I ,

on z = 0 ,

i = 1, 2 ,

−k1 ∇T1 · n = C(T2 − T1 )

on I .

(2.2)

In (2.2) δl2 is the lateral surface of 2 , hc is the convection coefficient accounting for the heat losses, Text is the external temperature, C is the contact thermal conductance relevant to the interface I and [·] indicates the jump of · across I . A weak formulation of (2.1)–(2.2) can be easily obtained by multiplying each of (2.1) by, respectively, the test functions 1 and 2 , and integrating by parts, taking into account (2.2). It is thus obtained: 

     ∂T1 ∂T2 1 dv + 2 dv ρ1 c1 ρ2 c2 ∂t ∂t 1 2   + (k1 ∇T1 · ∇1 ) dv + (k2 ∇T2 · ∇2 ) dv 1

 +

δt2



(hc (Text − T2 ) − If ) d + 

2

δt1

(hc (Text − T1 ) − If ) d

(C[(T1 1 + T2 2 ) − (T1 1 + T2 2 )]) d .

+

(2.3)

I

As discussed later on, the weak formulation (2.3) is used for developing a finite element formulation and getting numerical solutions of the equations mentioned above.

Mathematical Models for Infrared Analysis Applied to Cultural Heritage

89

2.2 Optically Semi-transparent Materials In optically semitransparent materials the excitation light delivered by the flashes can penetrate inside the specimen, thus undergoing a volume absorption. Moreover the infrared radiation emitted by any layer inside the specimen, after undergoing a partial absorption during its propagation inside the specimen, can reach the detector. A typical specimen which can be investigated is depicted in Fig. 2, representing a paper sheet with a ink layer buried inside. Let  be the domain occupied by the paper sheet of thickness H , where n indicates the unit normal to its boundary pointing outward. By using a standard capacity concentration procedure the ink domain, being actually tridimensional, is here modeled by a surface  at a depth equal to d inside the paper sheet. As a matter of facts,  can be considered as a cut inside  which is thus a perforated domain. To this end, let  + and  − be, respectively, the upper and lower boundary of this cut in , both geometrically coincident with . Let us introduce a cartesian frame with its origin O located on the centre of the top specimen surface, where x indicates the coordinate in the plane parallel to the top and bottom specimen surfaces, whereas z is the vertical coordinate, increasing downwards. The light delivered by the flash can penetrate inside the specimen and undergoes a number of reflections at the top and bottom paper surfaces, depending on the reflection coefficient R. During its travel inside the specimen, the light beam undergoes a distributed absorption by the paper and a concentrated absorption by the ink layer each time it crosses the latter. Assuming a uniform light intensity If illuminating the specimen, the light intensity at a depth z inside the specimen is then given by I (x, z) = If (1 − R)e−αz + If (1 − R)Rγ 2 e−αH e−α(H −z) +If (1 − R)R 2 γ 2 e−2αH e−αz + . . . , I (x, z) = If (1 − R)γ e

−αz

+ If (1 − R)Rγ e

+If (1 − R)R 2 γ 3 e−2αH e−αz + . . . ,

0 0, defined as η (s) = t

 u(t) − u(t − s), u(t) − u(0),

s ≤ t, s > t.

With this position, calling μt (s) = −gt (s), and setting for simplicity 1 K = 1,  we obtain from Eq. (7), along with the boundary condition (9), the boundary value problem  ∞ ⎧ ⎨∂ u(t) − u(t) − μt (s)ηt (s)ds = 0, tt 0 ⎩ u(t)|∂ = 0.

t > 0,

(10)

Problem (10) is supplemented with the initial conditions  u(0) = u0 , ∂t u(0) = u1 ,

(11)

where the functions u0 , u1 :  → R are assigned data. Aiming to incorporate the boundary conditions, the correct functional setting requires to take the initial value of the displacement u0 in the Sobolev space H01 (),

144

M. Conti et al.

of functions which are square-summable along with their derivatives, and assume value zero on the boundary ∂ in the trace sense (see, e.g., [8]), whereas the initial velocity u1 is taken in the usual Hilbert space L2 () of square-summable functions on . Actually, in more generality, we will consider (10) for a wider class of kernels μt (s), of which −gt (s) is just a particular instance. More precisely, we require that the map (t, s) → μt (s) : R × R+ → R+ is sufficiently regular (say, of class C1 ) and satisfies a certain set of hypotheses (see [3]). Here, we just report the main assumptions: A1 For every fixed t ∈ R, the map s →  μt (s) is nonincreasing and summable, with total mass  ∞ κ(t) = μt (s)ds. 0

Besides, inft ∈R κ(t) > 0. A2 There exists δ > 0 such that μ˙ t (s) + μt (s) + δκ(t)μt (s) ≤ 0, for every t ∈ R and almost every s > 0. Here the dot and the prime denote the derivative with respect to t and s, respectively. A3 The function t → μ˙ t (s) satisfies the uniform integral estimate 1 sup 2 [κ(t)] t ∈R



∞

|μ˙ t (s)|ds < ∞.

0

Remark 2 Besides −gt (s), a further example of kernel complying with the conditions above is given by μt (s) =

1 e−s/ε(t ), [ε(t)]2

where ε ∈ C1 (R) is a positive nonincreasing function. Note that, if ε(t) → 0 as t → ∞, we are exactly in the situation where we recover the Kelvin–Voigt model. Then we can state the following well-posedness result. Theorem 1 In the hypotheses above, for every choice of initial data u0 ∈ H01 () and u1 ∈ L2 (), the initial-boundary value problem (10) and (11) admits a unique weak solution u ∈ C([0, ∞), H01())

with

∂t u ∈ C([0, ∞), L2 ()).

Aging of Viscoelastic Materials: A Mathematical Model

145

Hypotheses A1–A3 are actually overabundant for Theorem 1. For instance, the desired conclusion still holds if one replaces A2 with the much weaker assumption A2 There exists a function Q : R → [0, ∞), bounded on bounded intervals, such that μ˙ t (s) + μt (s) ≤ Q(t)μt (s), for every t ∈ R and almost every s > 0. If Q(t) ≡ 0, it is also true that the natural energy E(t) of the system, given by E(t) =

1 2

)





|∇u(t)|2 + 

|∂t u(t)|2 + 



∞

μt (s) 0

|∇ηt (s)|2 ds

* ,



is a decreasing function of time, meaning that the system is dissipative. Instead, if one invokes the stronger A2, the energy can be proved to decay exponentially fast. Namely, the following holds (see [3]). Theorem 2 In the hypotheses above, there exist constants M ≥ 1 and ω > 0, independent of the initial data, such that the energy E fulfills the estimate E(t) ≤ ME(0)e−ωt . It is interesting to observe that the classical counterpart of Theorem 2, for a kernel μ that does not depend on t, has been obtained only in relatively recent times (see [9–11]), within the assumption μ (s) + δμ(s) ≤ 0, which is exactly A2 when μt (s) is independent of t (and has unit total mass). Theorems 1 and 2, whose proofs can be found in [2, 3], are not mere extensions of their time-independent counterparts. The presence of the time-dependent kernel introduces essential difficulties, and the mathematical problem falls within the newly developed theory of dynamical systems on time-dependent spaces (see [1, 6]). In particular, several computations that are almost immediate in the classical framework require here new techniques and subtle approximation arguments.

References 1. Conti, M., Pata, V., Temam, R.: Attractors for processes on time-dependent spaces. Applications to wave equations. J. Differ. Equ. 255, 1254–1277 (2013) 2. Conti, M., Danese, V., Giorgi, C., Pata, V.: A model of viscoelasticity with time-dependent memory kernels. Am. J. Math. 140, 349–389 (2018) 3. Conti, M., Danese, V., Pata, V.: Viscoelasticity with time-dependent memory kernels, II: asymptotic behavior of solutions. Am. J. Math. 140, 1687–1729 (2018)

146

M. Conti et al.

4. Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297– 308 (1970) 5. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer, Berlin (1992) 6. Di Plinio, F., Duane, G.S., Temam, R.: Time dependent attractor for the oscillon equation. Discrete Contin. Dyn. Syst. 29, 141–167 (2011) 7. Drozdov, A.D., Kolmanovskii, V.B.: Stability in viscoelasticity. North-Holland, Amsterdam (1994) 8. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998) 9. Fabrizio, M., Lazzari, B.: On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rational Mech. Anal. 116, 139–152 (1991) 10. Liu, Z., Zheng, S.: On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 54, 21–31 (1996) 11. Muñoz Rivera, J.E.: Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52, 629–648 (1994) 12. Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical problems in viscoelasticity. Longman Scientific & Technical/John Wiley, New York (1987)

A Quasi-Static Model for Craquelure Patterns Matteo Negri

Abstract We consider the quasi-static evolution of a brittle layer on a stiff substrate; adhesion between layers is assumed to be elastic. Employing a phase-field approach we obtain the quasi-static evolution as the limit of time-discrete evolutions computed by an alternate minimization scheme. We study the limit evolution, providing a qualitative discussion of its behaviour and a rigorous characterization, in terms of parametrized balanced viscosity evolutions. Further, we study the transition layer of the phase-field, in a simplified setting, and show that it governs the spacing of cracks in the first stages of the evolution. Numerical results show a good consistency with the theoretical study and the local morphology of real life craquelure patterns. Keywords Phase-field fracture · Craquelure · Quasi-static evolution

1 Introduction Craquelure and crazing usually denote network of cracks which appear on thin superficial layers of materials (see Fig. 1). As a mathematical model for this kind of phenomena, we consider an elastic, brittle layer placed on a rigid adhesive substrate, which displaces the layer (and thus plays the role of a driving force). Having in mind the formation of patterns in a long time scale, we consider a quasi-static evolution driven by a time depending displacement of the substrate; in particular, we neglect diffusion of temperature and inertia. We employ a phase-field approach and we focus on a couple of complementary aspects: • characterization of quasi-static evolutions, obtained by time-discrete alternate minimization (staggered) schemes,

M. Negri () Dipartimento di Matematica, Università di Pavia, Pavia, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Bonetti et al. (eds.), Mathematical Modeling in Cultural Heritage, Springer INdAM Series 41, https://doi.org/10.1007/978-3-030-58077-3_10

147

148

M. Negri

Fig. 1 Networks of cracks with different morphologies. (a) Reproduced with permission. From The Texas Collection © Baylor University. (b) Photo from Pixabay

• pattern formation and crack spacing as the result of different generations of cracks. To our best knowledge, these aspects are not treated in the literature on this type of models; among the many, we mention [13] as the very first reference in this direction, [1] for multi-layer materials, [14] for the derivation from 3D models and [7] for a homogenization result. We assume that the reference configuration of the brittle film is an open bounded set  ⊂ R2 . The physical parameters are the Lamè coefficients λ > 0 and μ > 0, the fracture toughness Gc > 0 and the adhesion parameter β > 0 (which measures the elastic response of the adhesive between the brittle layer and the substrate). Finally, we consider an in-plane displacement g which gives, as a function of time, the displacement of the substrate (note that here g is a datum). For sake of simplicity we assume that g(t) is of the form t gˆ for some gˆ ∈ H 1 (, R2 ). The sharp crack energy associated to the system is, in some sense, a combination of Griffith energy [9] and Mumford–Shah functional [16]; in the (weak) setting of SBD 2 spaces it reads 

 F(t, u) =

1 2

1

W ((u)) dx + Gc H (Ju ) + β \Ju

|u − g(t)|2 dx

(1)



where W (depending on λ and μ) denotes the linear elastic energy density. In this framework the set of discontinuity points Ju represents the crack. Actually, in view of the numerical simulations, it is far more convenient to work with the phase-field approximation of F given by Bourdin et al. [5] and León-

A Quasi-Static Model for Craquelure Patterns

149

Baldelli et al. [13]  1 Fε (t, u, v) = 2 (v 2 + ηε )W ((u)) dx + 

 + Gc

1 2



ε−1 (v − 1)2 + ε|∇v|2 dx + β

 |u − g(t)|2 dx. 

(2) It well known [8] that the -limit of the phase-field energy Fε (t, ·, ·) is indeed the sharp crack energy F(t, ·). Now, let us introduce the time-discrete evolution which is used in the numerical simulations. Let tk = kτ for τ > 0. Known the configuration (uk−1 , vk−1 ) at time tk−1 the updated configuration (uk , vk ) at time tk is given by vk = limm→+∞ vk,m and uk = limm→+∞ uk,m , where uk,m and vk,m are the steps of the staggered scheme:  + , uk,m ∈ argmin F (tk , u, vk,m−1 ) , + vk,m ∈ argmin F (tk , uk,m , v ) : v ≤ vk−1 . The theoretical question we are interested is the characterization of the time continuous limit. A precise statement is contained in Sect. 2.3, assuming a stronger irreversibility condition, i.e. v ≤ vk,m−1 . Roughly speaking we will get an evolution t → (u(t), v(t)) which is possibly discontinuous and such that (u(t), v(t)) is a critical point of the energy, for every continuity point t. The technically difficult part, is instead the characterization of the behaviour in the discontinuity points, where the system makes an instantaneous (catastrophic) transition between (u(t − ), v(t − )) and (u(t + ), v(t + )) following some sort of staggered scheme. We finally provide some numerical results, which shows the pattern formation for a prototype problem and for the real life specimen of Fig. 1. It turns out that the pattern is formed by generations of cracks (which nucleate at different time) whose spacing follows a regular scheme: in the first generation, spacing of cracks is not predicted by minimizers (unless the domain is very short), it is instead dictated by the size of the boundary layer of the strain and of the phase-field function; further generations of cracks follow instead the dyadic behaviour of minimizers.

2 Quasi-Static Evolutions by Alternate Minimization 2.1 Phase-Field Energy First of all let us introduce more rigorously the phase-field setting: for ε > 0 and ηε > 0 we define the family of energies Fε : [0, T ]×H 1 (; R2 )×H 1 (, [0, 1]) →

150

M. Negri

[0, +∞) given by  Fε (t, u, v) =

1 2

(v 2 + ηε )W ((u)) dx + 

 + Gc

1 2

ε−1 (v − 1)2 + ε|∇v|2 dx + β



 |u − g(t)|2 dx. 

(3) Note that Fε (t, ·, ·) is separately quadratic, i.e. Fε (t, u, ·) and Fε (t, ·, v) are quadratic (positive) functionals; hence Fε (t, u, ·) and Fε (t, ·, v) are coercive and convex. On the other hand, Fε (t, ·, ·) is not (jointly) convex, however it is weakly lower semi-continuous in H 1 (; R2 ) × H 1 (, [0, 1]) (see e.g. [17, Lemma 2.1]). In particular, for each time t there exists a minimizer of the energy Fε (t, ·, ·). Now, let us turn to the -limit of Fε (t, ·, ·) as ε → 0+ . To this end, it ˜ ε : [0, T ] × L2 (; R2 ) × is convenient to define the extended functional F 2 L (, [0, 1]) → [0, +∞] as  Fε (t, u, v) ˜ Fε (t, u, v) = +∞

if u ∈ H 1 (; R2 ) and v ∈ H 1 (, [0, 1]), otherwise.

Then, by Chambolle [8] we known that the -limit of F (t, ·, ·), with respect to the topology of L2 (; R2 ) × L2 (), is the functional F˜ (t, ·, ·) : [0, T ] × L2 (; R2 ) × L2 (, [0, 1]) → [0, +∞] given by ˜ u, v) = F(t,

 F(t, u) +∞

if u ∈ SBD 2 () and v = 1 a.e. in , otherwise.

Let (uε , vε ) be a family of minimizers of Fε (t, ·, ·). Note that uε is bounded in L2 , then, by a fundamental result in the theory of -convergence, see e.g. [6, Theorem 3.3], we know that uε converge to a minimizer u of F(t, ·); more precisely, there exists a subsequence converging to u in the topology of L2 (; R2 ). On the contrary, not much is known on the convergence of the critical points of Fε (t, ·, ·) which will appear in the evolution.

2.2 Time-Discrete Evolution In this section we will describe the time-discrete evolution on which the numerical calculations are based. Here we can give only a brief description in the simplest possible setting, for complete proofs and generalizations the reader should make reference to [3, 11].

A Quasi-Static Model for Craquelure Patterns

151

For τ > 0 consider a time discretization tk = kτ . After setting the initial conditions u0 and v0 , at time t = 0, the configuration (uk , vk ), at time tk , is computed from (uk−1 , vk−1 ) by the following incremental scheme, known as alternate minimization algorithm. Let us introduce the further sequences uk,m and vk,m , with vk,0 = vk−1 and uk,0 = uk−1 . Then, for m ≥ 1 we define  + , uk,m ∈ argmin F (tk , u, vk,m−1 ) + , vk,m ∈ argmin F (tk , uk,m , v ) : v ≤ vk−1

(4)

Note that this scheme takes full advantage of the fact that F(tk , ·, ·) is separately quadratic, and indeed its numerical implementation is very convenient, even if in practice the algorithm may converge quite slowly. Then, (up to subsequences) we let vk = limm→+∞ vk,m and uk = limm→+∞ uk,m . Note that the distance between vk−1 and vk could be either small or large, in other terms, there is no a priori control on the speed (vk − vk−1 )/τ since the system is rate-independent. Moreover, note that (uk , vk ) is an equilibrium point for F (tk , ·, ·), unilateral w.r.t. v. Finally, it is important to comment on the constraint v ≤ vk−1 which models irreversibility. The convergence analysis under the latter constraint in the ”genuinely” rate-independent setting is still open. At the current stage a couple of alternative are feasible. The first consists in solving a discrete viscous parabolic evolution for the phase-field, with the contraint v ≤ vk−1 , followed by a vanishing viscosity procedure: by Almi et al. [4] and Almi et al. [2] this approach provides in the end a quasi-static evolution. The second option consists instead in replacing the constraint v ≤ vk−1 by the (stronger) constraint v ≤ vk,m−1 . In this way monotonicity is imposed at each alternate iteration. On the theoretical level, this assumption allows to give a full characterization of the evolution as the time step τ → 0, without passing through viscosity solutions. On the other hand, in some cases this assumption may be too strong in the numerical simulations; we will see an example in Sect. 5.3. Here we will follow the latter strategy.

2.3 Time-Continuous Evolution The time discrete scheme of the previous subsection gives for every τ > 0 a finite sequence (vk , uk ) in the points tk . We will denote by (uτ , vτ ) : [0, T ] → H 1 (; R2 ) × H 1 () an interpolation of (uk , vk ) in the points tk (here we will enter into the delicate technical issue about the choice of the interpolation). Our goal is to characterize the limit of (uτ , vτ ) as τ → 0. First, we give a “qualitative” description of the limit evolution, as a function of time. We have already remarked that in this setting there is no a priori control of the length of vτ (tk ) − vτ (tk−1 ) for tk = kτ . Indeed, fix t ∈ [0, T ] and let kτ s.t. t ∈ [kτ τ, (kτ + 1)τ ]. As τ → 0 it is clear that kτ τ → t and (kτ + 1)τ → t; on the contrary it may happen that vτ (kτ τ ) → v(t − ) and vτ ((kτ + 1)τ ) → v(t + ) where

152

M. Negri

v(t − ) = v(t + ). In other terms the limit evolution t → v(t) could be discontinuous in time. Note that this is observed in the numerical simulations. For this reason, in the limit as τ → 0 we expect an evolution t → (u(t), v(t)) of class BV in time. Now, let us briefly describe the behaviour in continuity and discontinuity points (for more details we refer to [3, 11]). If t ∈ [0, T ] is a continuity point for the evolution then (u(t), v(t)) is an equilibrium configuration for the system, i.e. ∂u F (t, u(t), v(t))[φ] = 0 forevery φ ∈ H 1 (, R2 ), ∂v F (t, u(t), v(t))[ξ ] = 0 forevery ξ ∈ H 1 ()withξ ≤ 0. Equilibrium of v is unilateral because, by irreversibility, only negative variations are allowed. Note that the configuration (u(t), v(t)) is not necessarily a (joint) global or a local minimizer of Fε (t, ·, ·), it is actually a separate minimizer, by the separate convexity of Fε (t, ·, ·). On the contrary, if t is a discontinuity point we expect an alternate “evolution”, connecting (u(t − ), v(t − )) and (u(t + ), v(t + )). More precisely, in the limit the path between (u(t − ), v(t − )) and (u(t + ), v(t + )) is made of infinitely many intermediate configurations vj  v(t − ) and uj → u(t − ) connected by a (reverse) alternate scheme: i.e.  + , uj ∈ argmin F (t , u, vj +1 ) , + vj ∈ argmin F (t , uj , v ) : v ≤ vj +1 . In particular the instantaneous transition between (u(t − ), v(t − )) and (u(t + ), v(t + )) is not simultaneous in u and v. Once again, this is confirmed by numerical results. In order to give a rigorous description of the limit evolution, it is necessary to introduce the derivatives of the energy Fε , which will provide the driving forces for the evolution. We follow in particular [3]. Clearly, we can take the partial derivatives of Fε ; moreover, we can define the following slopes |∂u F |H 1 (t, u, v) = sup{−∂u Fε (t, u, v)[φ] : φ ∈ H 1 (; R2 )}, |∂z− F |L2 (t, u, v) = sup{−∂v Fε (t, u, v)[ξ ] : ξ ∈ H 1 () ∩ L∞ () , ξ ≤ 0}. Note that the slope w.r.t. v is unilateral since, by irreversibility, we are interested only in negative variations. From the technical point of view it is fundamental that the slopes are weakly lower semicontinuous. Technically, we characterize the evolution in terms of a parametrized “balanced viscosity solution” [15]. First, in order to describe both the behaviour in continuity and discontinuity points we employ an arc-length parametrization s → (t (s), u(s), v(s)). In this setting, roughly speaking, discontinuity points (in time) correspond to intervals [s − , s + ] where t is constant, and thus t  = 0; vice-versa, continuity points (in time) correspond to

A Quasi-Static Model for Craquelure Patterns

153

points s where t  (s) > 0. Then, the evolution is characterized by the following set of conditions: (a) the map t : [0, S] → [0, T ] is non-decreasing and surjective, (b) (t, u, v) ∈ W 1,∞ ([0, S]; [0, T ] × H 1 (; R2 ) × L2 ()) with |t  (s)| + u (s)H 1 + v  (s)L2 ≤ 1 , (c) v is non-increasing and takes values in [0, 1], (d) if t  (s) > 0 then |∂u F |H 1 (t (s), u(s), v(s)) = 0

|∂z− F |L2 (t (s), u(s), v(s)) = 0,

and

(e) for every s ∈ [0, S] the following energy identity holds: 

s

∂t Fε (t (r), u(r), v(r)) t  (r) dr

F (t (s), u(s), v(s)) = F (0, u0 , v0 ) + 0



s



0 s

− −

0

|∂z− F |L2 (t (r), u(r), v(r)) v  (r)L2 dr |∂u F |H 1 (t (r), u(r), v(r)) u (s)H 1 dr.

For further details the reader may refer to [3].

3 A One-Dimensional Case Study As a preliminary study, it is interesting to focus on the behaviour and on the pattern generated by the minimizers of the sharp-crack energy, even if the evolution follows only partially this scheme. To this end it is useful to consider a simpler one-dimensional setting, where  = (−L, L) and g(t, x) = tx; the corresponding energy is then of the form  F (t, u) =

1 2

μ|u |2 dx + Gc #(Ju ) + β (−L,L)

 |u − g(t)|2 dx. (−L,L)

We will assume that the initial configuration (at time t = 0) is g(0) = 0.

(5)

154

M. Negri

3.1 Continuous Displacement and Boundary Layer First, let us consider a minimizer without cracks, i.e. with Ju = ∅. In this case, the configuration solves the Euler–Lagrange equation −μu + 2βu = 2βg(t) in (−L, L) with homogeneous Neumann boundary condition. The explicit solution is given by (see Fig. 2) u0 (t, x) = −taL sinh(λx) + tx,

for λ = (2β/μ)1/2 and aL =

1 . λ cosh(λL)

Then, a simple (but quite long) computation shows that  F (t, u0 (t)) = t 2 μ L −

1 λ

tanh(λL) = t 2 Fˆ (L).

In the sequel it will be important we understand the dependence of u0 on L. To this end, let us denote uL (x) = −aL sinh(λx) + x the continuous solution u0 in the interval (−L, L) for t = 1. Plotting the derivative uL (x) = −aL λ cosh(λx) + 1 = 1 −

cosh(λx) cosh(λL)

for different values of L (see Fig. 3) shows that uL changes in the boundary layers while it is almost constant in the “interior”. In order to provide some quantitative estimate of the size of the layer it is convenient to study how the derivative uL (x) scales with L. Thus, given a point x ∈ (0, L ) let us compute the (corresponding) point x ∈ (0, L) such that uL (x) = uL (x ). The latter identity reads cosh(λx) =

cosh(λL) cosh(λx ) . cosh(λL )

(6)

A plot of the behaviour of x as a function of L (see Fig. 4) shows that the distance between x and L is almost independent of L. Thus, we can expect that the length of

−L

Fig. 2 Plot of u0 = uL and u1

L

−L

L

A Quasi-Static Model for Craquelure Patterns

155 uL

uL

0 x

L

x

0

L

Fig. 3 Plot of uL for L = 6.5 and plot of uL for L = 12.5 x

0

L

L

Fig. 4 Plot of L and x (bold) from (6) as a function of L. Values are computed for L = 6.5 and x = 0.5. In this case we have L − x ∼ 5.5

the boundary layer remains almost constant with L. As we will see in the numerical simulations of Sect. 5, the “scale invariance” of the boundary layer explains the periodic spacing of the first generations of cracks better than the dyadic structure (explained in the next subsections), which occurs only for small values of the size L (more precisely, when L is comparable with the size of layer). In order to better explain the fact that L − x is almost constant, we can consider the case when x and L are large enough to approximate cosh with exp /2, then the above identity can be approximated by  exp(λx) ∼ exp(λL) exp(−λb)

for b = − ln

cosh(λx ) cosh(λL )

 > 0,

which gives x ∼ L − b. Roughly speaking, the transition layer is of constant size b.

3.2 Evolution of Cracks by Minimality The content of this section follows closely the study performed in [13]. In the case of a single crack let us assume (for the moment) that Ju = {0}, so that the interval (−L, L) splits into the subintervals (−L, 0) and (0, L); the minimizer, denoted by u1 (t), can then be computed using the previous Euler-Lagrange equation in the

156

M. Negri

subintervals (−L, 0) and (0, L) (see again Fig. 2). Then, the energy reads F (t, u1 (t)) = Gc + t 2 2Fˆ (L/2). Now we will compare the energy F (t, u0 (t)) of the continuous solution with the energy F (t, u1 (t)) of the discontinuous solution. To this end, it is interesting to consider in the (t, L) plane the set of “critical transition times”, satisfying t 2 (Fˆ (L) − 2Fˆ (L/2)) = Gc .

(7)

It is easy to see that the function 2 (L) = Fˆ (L) − 2Fˆ (L/2) =

μ λ

[2 tanh(λL/2) − tanh(λL)]

vanishes for L = 0 and is increasing for L > 0; hence it is positive and, given L, there exists a time  tL =

Gc Fˆ (L) − 2Fˆ (L/2)

1/2

 =

Gc 2 (L)

1/2 (8)

for which the two energies coincide. Further, t 2 (Fˆ (L) − 2Fˆ (L/2)) < Gc for t < tL while t 2 (Fˆ (L) − 2Fˆ (L/2)) > Gc for t > tL ; in other terms, F (t, u0 (t)) < F (t, u1 (t)) for t < tL ,

F (t, u0 (t)) > F (t, u1 (t))for t > tL ,

which justifies the name ”critical transition time” at length L. Before proceeding, let us check that for a single crack the least energy is always assumed when Ju = {0}. For L1 + L2 = L, we have to check the energy inequality    Gc + t 2μ L1 − λ1 tanh(λL1 ) + t 2μ L2 − λ1 tanh(λL2 ) > Gc + t 2μ L− λ2 tanh(λL/2) .

After some algebraic manipulations, the previous inequality boils down to 1 2 (tanh(λL1 )

+ tanh(λL2 )) < tanh(λ(L1 + L2 )/2),

which is true by the strict concavity of tanh. Now, let us consider the general case of m cracks (for m ≥ 1). By the previous symmetry argument it is not restrictive to assume that Ju = {−L + kL/(m + 1) : for 1 ≤ k ≤ m}. Arguing as above, the energy for m cracks reads F (t, um (t)) = mGc + t 2 (m + 1)Fˆ (L/(m + 1)).

A Quasi-Static Model for Craquelure Patterns

157

We want to compare F (t, u0 (t)) and F (t, um (t)). This time the set of critical points is defined by  t 2 Fˆ (L) − (m + 1)Fˆ (L/(m + 1)) = mGc , which gives the “critical transition time” 2 tm =

mGc . Fˆ (L) − (m + 1)Fˆ (L/(m + 1))

Let us show that the sequence tm is monotone increasing w.r.t. m, independently of L. To this end it is convenient to study the function s →

(s − 1) s tanh(λL/s) − tanh(λL)

2 (up to a positive multiplicative constant) when s = m + 1. which coincide with tm A simple calculation shows that this function is increasing and thus tm is monotone increasing as well. Note that t1 = tL . As a consequence, if the evolution is driven by energy minimization, in the first generation it appears a single crack in the center of the bar (−L, L). Indeed, for t ∈ [0, t1 ) we have F (t, u0 (t)) < F (t, u1 (t)) and thus u(t) = u0 (t). For t > t1 we have F (t1 , u0 (t)) > F (t1 , u1 (t)), moreover, being tm > t1 for every m > 1, we have F (t, um (t)) > F (t, u0 (t)), at least for t ∼ t1 . Hence, we expect that u(t) = u1 (t) for t ∈ (t1 , t2 ) where t2 denotes the time when the second generation of crack will occur.

3.3 Second and Further Generations of Cracks by Minimality When the first crack appears, at time t1 , the interval (−L, L) is splitted into the subintervals (−L, 0) e (0, L). By periodicity, this is equivalent to consider the behaviour of solutions in the interval (−L/2, L/2). Arguing exactly as in the previous section, just replacing L with L/2, provides the existence of the critical transition time tL/2 when the bar will split again. Note that tL/2 > tL because the function tL in (8) is decreasing w.r.t. L (remember that 2 is increasing with respect to L). Proceeding by induction, we find the times tL/2k (see Fig. 5) when the dyadic structure evolves; in the physical literature this is often called “halving” effect. To conclude this section, it is fair to remark that in this one dimensional example the transition between uL and uL/2 cannot occur following a continuous energy decreasing path, because of the activation threshold Gc which is payed as soon as the crack opens. In the numerical simulations, this topological problem is avoided by

158

M. Negri

tL/4 tL/2

tL

L

0 L/4 L/2

Fig. 5 Plot of tL = (Gc /2 (L))1/2

the phase-field regularization. Moreover, the uni-axial numerical results of Sect. 5 we show that the first generation of cracks does not follow this dyadic structure (at least for L large), rather the first crack pattern depends on the size of the boundary layer, described in the previous subsection.

4 Alternate Minimization in the One Dimensional Setting In this section we briefly study the evolution generated by the alternate minimization scheme in the one-dimensional setting of Sect. 3. For sake of simplicity we consider Dirichlet, instead of Neumann, boundary conditions. This example will be useful to understand the behaviour of the numerical results. We consider again the sharp crack energy F of (5) and the corresponding phase-field energy  Fε (t, u, v) = 12

(v 2 + ηε )μ|u |2 dx + (−L,L)



+ Gc 21

ε (−L,L)

−1

 2



(v − 1) + ε|v | dx + β 2

|u − g(t)|2 dx. (−L,L)

Here, we will further assume that u(t, ±L) = g(t, ±L) = ±tL. Fix tk = kτ assume that vk−1 = ck−1 is constant. We will show hereafter that the update vk is constant and that uk is constant, as well. Looking at this result in the time discrete scheme, it turns out that for Dirichlet boundary conditions there will be no nucleation of cracks if the initial phase-field v0 is homogeneous. Other interesting results on homogeneous states are contained in [18]. For sake of simplicity, we will solve the following alternate scheme without the irreversibility constraint, which would actually not change the qualitative result, 

+ , uk,m ∈ argmin F (tk , u, vk,m−1 ) : u(±L) = g(±tk L) + , vk,m ∈ argmin F (tk , uk,m , v ) .

A Quasi-Static Model for Craquelure Patterns

159

The Euler–Lagrange equation is of the form −ak−1 u + 2βu = 2βg(tk ) in (−L, L) with Dirichlet boundary conditions. The explicit solution is simply uk,m (x) = g(tk , x) = −tk x independently of ak−1 . Since uk is constant the Euler-Lagrange equation for v is of the form −v  + bv = c, whose solution vk,m is again a constant. In particular the staggered scheme finds a critical point after two iterations.

5 Numerical Results for Uni-Axial Problems In the following subsections are reported the numerical results obtained for a bar  = (−L, L) × (−H, H ) of variable length and fixed width 2H = 5. We have chosen fracture toughness Gc = 1.0, Young modulus E = 1.0, Poisson ratio ν = 0.15, adhesive constant β = 0.15. Moreover, we assume that the displacement of the substrate is of the form g(t, x) = (tx1 , 0). Since g is uni-axial we expect solutions in accordance with the theoretical arguments of Sect. 3. Since the datum g is monotone we neglected the irreversibility constraint in the alternate minimization scheme (4). With this choice crack patterns look sharper. A comparison between the solutions obtained with and without irreversibility constraint (see Sect. 5.3) show that the crack patterns behave in a similar way. Numerical results have been computed using FreeFem++ [10].

5.1 Dyadic Structure for Short Bars In this subsection we present and discuss the numerical results obtained with L = 6.5. The configuration of the phase field v shown in Fig. 6 are computed with the unconstrained version of scheme (4), i.e. without any constraint on v.

Fig. 6 Phase field v at time t = {3.0, 3.1, 3.7, 3.8} computed without irreversibility constraint; blue corresponds to sound material, i.e. v = 1, while yellow corresponds to cracks, i.e. v = 0

160

M. Negri

First, it is interesting to comment on the very first image. In this case there are no pre-existing fractures, hence nucleation is a fundamental ingredient in the evolution. Figure 6 shows that nucleation requires first a diffuse damage, with the phase-field v decreasing in a wide region, and then concentration, in a second stage (see also [12, Fig 4.8]). Note that, in the first image, the profile of v is closely related to the profile of uL (the derivative of the one dimensional solution uL ) presented in Sect. 3.1. Indeed, since the datum is uniaxial and Poisson ratio ν is small, we can expect that uL provides a good approximation of the strain energy density; then, in phase-field models the larger is the elastic energy density the smallest is the phase-field. This is why the profile of v is consistent with that uL (compare the first plot of Fig. 3 with the first image of Fig. 6). Then, the central crack nucleates and the domain splits. At that point, a similar evolution “restarts” in each subinterval, leading to the second generation of cracks. Note that in this case the “transition layer” for uL is, roughly speaking, of the same size of the half interval (0, L). For this reason the evolution generates a dyadic fracture pattern, clearly visible in the last image of Fig. 6. As we will see, the pattern of the first generation is not dyadic for larger L.

5.2 Periodic Patterns for Longer Bars In this subsection we present the numerical results obtained for L = 12.5. In the second picture, when the first generation of crack appears (at time t = 2.8) it is evident that the position of the first crack (from left to right) corresponds to the transition region of v, which, comparing with Fig. 3, corresponds to the transition layer of uL . Moreover, all the cracks, apart from the innermost, are equally spaced and their distance is approximately 5.5, which is the value expected from the estimates of the size of the transition layer (see Fig. 4). Note that, following the intermediate steps of the alternate minimization algorithm (at time t = 2.8), the outermost cracks nucleates firts, followed by the innermost. The reason behind the fact that cracks do not nucleate in the center of the bar is not fully clear, however, it should be found in the fact that the phase field v is almost constant in the inner part of the domain and thus its evolution does not promote concentration of strain and nucleation of cracks, as described in Sect. 4. The second generation of cracks follows instead the dyadic structure: this is due to the fact that the spacing of the cracks after the first generation is comparable with the size of the transition layer, as in the previous example. In conclusion, we could say that the first generation of cracks is determined by a characteristic length (the width of the transition layer) with cracks nucleating from the boundary to the interior, while the further generations evolve according to the dyadic scheme. Numerical results for longer bars confirm this behaviour. Finally we remark this type of evolution, triggered by the boundary layer, has been observed also in numerical simulations of brittle layered materials, see [1, Fig. 7], and soil drying (which share several features with our problem), see [19, Fig. 6].

A Quasi-Static Model for Craquelure Patterns

161

Fig. 7 Evolution for L = 12.5 at time t = 2.8, t = 3.1 and t = 4.3

5.3 Irreversibility In this section we briefly report the numerical results obtained with the scheme (4) under the constraint v ≤ vk−1 in the case L = 6.5. Clearly, the main difference is the fact that here the phase-field v cannot increase after the nucleation of cracks, because of the irreversibility constraint. From the results, it is evident that this evolution is very similar to the evolution of Fig. 6, at least as far as the structure of the crack pattern, the time and the nucleation sites. When the stronger constraint v ≤ vk,m−1 is applied, solutions are qualitatively similar, however the representation of the crack in the final solution is much wider; this is possibly due to the fact that the costraint prevents the solution to optimize the configuration of the crack after nucleation. In the case L = 12.5 the behaviour is similar to that of Fig. 7.

6 Local Craquelure Patterns for a Real Life Specimen In order to produce some more realistic craquelure patterns and crack morphologies, we ran several numerical experiment changing both the value of the adhesive parameter β and the datum g, of the form g(x) = Ax for A ∈ R2×2 . Even if the values are not provided by experimental measurements, it is clear from Fig. 9 that this quasi-static phase-field model captures quite well the local morphological features of the patterns in Fig. 1. Note that these plots do not show the entire evolution leading to the formation of the patterns but they show the final snapshot,

162

M. Negri

Fig. 8 Phase field v at time t = {3.0, 3.1, 3.7, 3.8} computed with the time irreversibility constraint v ≤ vk−1 (above) and with the step irreversibility constraint v ≤ vk,m−1 (below)

at a certain time T . In the evolution the cracks actually appear at different times, following a scheme reminiscent of the uni-axial case (Fig. 8). In the bi-axial setting it is more difficult to explain the behaviour of the evolution. However, some observations are due as far as crack junctions, which of course did not occur in the uniaxial setting. First, remember that for local minimizers of the (scalar) Mumford–Shah functional only Y-junctions (i.e. triple junctions) at 23 π angles occur, see [16, Theorem 2.1]. We may expect a similar behaviour in our linear elasticity context. However, in the first image only T-junctions occur, this is simply due to the fact that horizontal cracks nucleate in the first generation and vertical cracks in the second generation, therefore, once horizontal straight cracks are formed it is no longer possible to have 23 π angles. In this example, it would be natural to see orthogonal crossing of cracks; on the contrary cracks seem to “shift” passing from one horizontal stripe to the other. This feature could be due to remeshing and to the fact that horizontal cracks are not equally spaced, however, it is interesting to note that this feature occurs also in several points of the real life picture. In some cases (see the pictures on the right) the crack behaves instead as a local minimizer, splitting a T- or an X-junction into three (or more) Y-junctions,

A Quasi-Static Model for Craquelure Patterns

163

Fig. 9 Details of some numerical results obtained with different values of β and A compared with similar real life craquelure morphologies

with angles close to the optimal value 23 π, compare with [16, Figure 14]. In this way, small triangular regions are formed. Once again, it is worth to remark that this the case both in the numerical experiment and in the real life specimen (Fig. 9).

References 1. Alessi, R., Freddi, F.: Failure and complex crack patterns in hybrid laminates: a phase-field approach. Compos. B Eng. 179, 107256 (2019) 2. Almi, S.: Irreversibility and alternate minimization in phase field fracture: a viscosity approach. Z. Angew. Math. Phys. 71(4), 21 (2020) 3. Almi, S., Negri, M.: Analysis of staggered evolutions for nonlinear energies in phase field fracture. Arch. Ration. Mech. Anal. 236(1), 189–252 (2020) 4. Almi, S., Belz, S., Negri, M.: Convergence of discrete and continuous unilateral flows for ambrosio-tortorelli energies and application to mechanics. ESAIM Math. Model. Numer. Anal. 53(2), 659–699 (2019) 5. Bourdin, B., Francfort, G.A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000) 6. Braides, A.: Approximation of Free-Discontinuity Problems. Springer, Berlin (1998) 7. Braides, A., Causin, A., Solci, M.: A homogenization result for interacting elastic and brittle media. Proc. R. Soc. A 474, 20180118 (2019) 8. Chambolle, A.: A density result in two-dimensional linearized elasticity and applications. Arch. Ration. Mech. Anal. 167(3), 211–233 (2003) 9. Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. Roy. Soc. Lond. 18, 163–198 (1920) 10. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

164

M. Negri

11. Knees, D., Negri, M.: Convergence of alternate minimization schemes for phase field fracture and damage. Math. Models Methods Appl. Sci. 27(9), 1743–1794 (2017) 12. Kuhn, C.: Numerical and Analytical Investigation of a Phase Field Model for Fracture. Dr.Ing. Dissertation (2013) 13. León-Baldelli, A.A., Bourdin, B., Marigo, J.-J., Maurini, C.: Fracture and debonding of a thin film on a stiff substrate: analytical and numerical solutions of a one-dimensional variational model. Contin. Mech. Thermodynam. 25(2), 243–268 (2013) 14. León-Baldelli, A.A., Babadjian, J.-F., Bourdin, B., Henao, D., Maurini, C.: A variational model for fracture and debonding of thin films under in-plane loadings. J. Mech. Phys. Solids 70, 320–348 (2014) 15. Mielke, A., Rossi, R., Savaré, G.: BV solutions and viscosity approximations of rateindependent systems. ESAIM Control Optim. Calc. Var. 18(1), 36–80 (2012) 16. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989) 17. Negri, M.: A unilateral L2 -gradient flow and its quasi-static limit in phase-field fracture by alternate minimization. Adv. Calc. Var. 12(1), 1–29 (2019) 18. Pham, K., Marigo, J.-J.: Stability of homogeneous states with gradient damage models: size effects and shape effects in the three-dimensional setting. J. Elast. 110(1), 63–93 (2013) 19. Vo, T.D., Pouya, A., Hemmati, S., Tang, A.-M.: Numerical modelling of desiccation cracking of clayey soil using a cohesive fracture method. Comput. Geotech. 85, 15–27 (2017)