Challenges in Mechanics of Time Dependent Materials, Volume 2: Proceedings of the 2020 Annual Conference on Experimental and Applied Mechanics [2] 3030595412, 9783030595418

Table of contents :
Preface
Contents
Chapter 1: Characterization of the Viscoelastic Response of Closed-Cell Foam Materials
1.1 Introduction
1.2 Experimental Methods
1.3 Results and Discussion
1.4 Conclusion
References
Chapter 2: Time-Dependent Yielding of Polymer Thin Films Under Creep
2.1 Introduction
2.2 Free Volume Nonlinear Viscoelastic Model
2.3 Creep Tests
2.4 Results and Discussion
2.5 Conclusion
References
Chapter 3: Non-Newtonian Fluid-Like Behavior of Poly(Ethylene Glycol) Diacrylate Hydrogels Under Transient Dynamic Shear
3.1 Introduction
3.2 Materials and Methods
3.3 Analytical Procedure and Results
3.4 Conclusion
References
Chapter 4: The Interfacial Shear Strength of Carbon Nanotube Sheet Modified Carbon Fiber Composites
4.1 Introduction
4.1.1 Surface Wetting
4.2 Background
4.2.1 Scalable Nanofabrication of CNT Sheet Wrapped Carbon Fiber Composite
4.3 Analysis
4.3.1 Fiber Push-Out Experiment and FEM Simulations
4.3.2 Fiber Push-in Nanoindentation
4.4 Conclusion
References
Chapter 5: Analytical Assessment of Creep Behavior of European Species in Outdoor Conditions
5.1 Context and Problematic
5.2 Analytical Modeling of Notched Beam Deflection in 4-Points Bending Test
5.2.1 Modeling of the Compliance of Notched Beam
5.2.2 Taking into Account of the Moisture Content (w) Effect
5.3 Results and Discussions
5.4 Conclusions
References
Chapter 6: Room Temperature Stress Relaxation of a Quenched and Tempered Steel
6.1 Introduction
6.2 Background
6.3 Analysis
6.4 Conclusion
References
Chapter 7: Improved Load Duration in Split Hopkinson (Kolsky) Bar Technique Using a Serpentine Type Striker Bar
7.1 Introduction
7.2 Striker Bar Design
7.3 Dynamic Compression Experiments
7.4 Conclusions and Future Work
References
Chapter 8: Experimental Shear Property Characterization of Agarose Hydrogel and Polydimethylsiloxane (PDMS)
8.1 Introduction
8.2 Materials and Methods
8.3 Results and Discussion
8.4 Conclusion
References
Chapter 9: Viscoelastoplastic Oxidative Multimode Damage Model for Fibrous Composite Materials at Extreme Temperatures
9.1 Kinematic/Kinetic Definitions
9.2 Thermodynamics/Constitutive Development
9.3 Oxidation Model Description
9.4 Conclusion
References
Chapter 10: Effect of Thermal and Mechanical Damage on Phase Separation, Crosslink Density, and Polydispersity of Polyurea Variants
10.1 Introduction and Background
10.2 Polyurea Formulations and Sythesis
10.3 ATR-FTIR Characterization
10.4 Crosslink Density
10.5 Gel Permeation Chromatography (GPC)
10.6 Conclusion
References
Chapter 11: Time-Resolved Characterization of Taylor Impact Testing
11.1 Introduction
11.2 Experimental
11.3 Results
11.4 Conclusion
References
Chapter 12: A Novel Method of Validating Polymer Relaxation Using Hopkinson Bar and Quasi-Static Loading
12.1 Introduction
12.2 Experimental Technique
12.3 Material Preparation
12.4 Modelling
12.5 Results
12.6 Conclusion
References
Chapter 13: Visco-Elasto-Plastic Characterization of PVC Foams
13.1 Introduction
13.2 Tested Material
13.3 Material Modeling
13.4 Results
13.5 Conclusions
References
Chapter 14: Virtual Dynamic Mechanical Analysis
14.1 Introduction
14.2 Simulation Method
14.3 Test Sample Preparation
14.4 Results and Discussion
14.5 Conclusions
Chapter 15: Direct Extraction of Mode I and Mode II Traction–Separation Relationships of Polymer Modified Bitumen Using Rigid Cantilever Beam Experiments Combined with StereoDIC
15.1 Introduction
15.2 Materials and Methods
15.2.1 Sample Preparation
15.3 Results and Discussion
15.4 Conclusions
References
Chapter 16: Glass Fiber Composites (GFCs) in Infrastructure: Developing New Measurement Methods to Meet the Challenge of 100 Year Service-Life-Prediction
16.1 Introduction
16.2 Progress Toward a Realistic Estimate of F-M IFSS (The NIST Approach)
16.2.1 Using Automated Data Acquisition to Acquire Tier 1 Data
16.2.2 Probing the Effect of Fiber–Fiber Interactions on the Fiber Fragmentation Process
16.3 Future Directions
References

Citation preview

Conference Proceedings of the Society for Experimental Mechanics Series

Meredith Silberstein Alireza Amirkhizi  Editors

Challenges in Mechanics of Time Dependent Materials, Volume 2 Proceedings of the 2020 Annual Conference on Experimental and Applied Mechanics

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society's two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society's Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research. More information about this series at http://www.springer.com/series/8922

Meredith Silberstein  •  Alireza Amirkhizi Editors

Challenges in Mechanics of Time Dependent Materials, Volume 2 Proceedings of the 2020 Annual Conference on Experimental and Applied Mechanics

Editors Meredith Silberstein Department of Mechanical and Aerospace Engineering Cornell University Ithaca, NY, USA

Alireza Amirkhizi UML North Campus, Dandeneau Hall 219 University of Massachusetts Lowell Lowell, MA, USA

ISSN 2191-5644     ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-030-59541-8    ISBN 978-3-030-59542-5 (eBook) https://doi.org/10.1007/978-3-030-59542-5 © The Society for Experimental Mechanics, Inc 2021 This work is subject to copyright. All rights are reserved 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

Challenges in Mechanics of Time-Dependent Materials represents one of seven volumes of technical papers presented at the 2020 SEM Annual Conference and Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics held in Orlando, FL, September 14–17, 2020. The complete Proceedings also includes volumes on Dynamic Behavior of Materials; Fracture, Fatigue, Failure and Damage Evolution; Advancement of Optical Methods and Digital Image Correlation in Experimental Mechanics; Mechanics of Biological Systems and Materials, Micro- and Nanomechanics and Research Applications; Mechanics of Composite, Hybrid and Multifunctional Materials; and Thermomechanics and Infrared Imaging, Inverse Problem Methodologies and Mechanics of Additive and Advanced Manufactured Materials. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics, the Mechanics of Time-Dependent Materials, Fracture, Fatigue, Failure, and Damage Evolution being some of these areas. The Time-Dependent Materials track was organized to address constitutive, time (or rate)-dependent constitutive and fracture/failure behavior of a broad range of materials systems, including prominent research in progress in both experimental and applied mechanics. Papers concentrating on both modeling and experimental aspects of Time-Dependent Materials are included. The track organizers thank the presenters, authors, and session chairs for their participation and contribution to these tracks. The support and assistance from the SEM staff is also greatly appreciated. Ithaca, NY, USA Lowell, MA, USA 

Meredith Silberstein Alireza Amirkhizi

v

Contents

1 Characterization of the Viscoelastic Response of Closed-Cell Foam Materials����������������������������������������������������� 1 Jialiang Tao, Xiuqi Li, Alexander K. Landauer, David Henann, and Christian Franck 2 Time-Dependent Yielding of Polymer Thin Films Under Creep ����������������������������������������������������������������������������� 5 Veli Bugra Ozdemir and Kawai Kwok 3 Non-Newtonian Fluid-Like Behavior of Poly(Ethylene Glycol) Diacrylate Hydrogels Under Transient Dynamic Shear �������������������������������������������������������������������������������������������������������������������������������������������������������������17 K. Upadhyay, K. Luo, G. Subhash, and D. E. Spearot 4 The Interfacial Shear Strength of Carbon Nanotube Sheet Modified Carbon Fiber Composites �����������������������25 Xuemin Wang, Tingge Xu, Monica Jung de Andrade, Ihika Rampalli, Dongyang Cao, Mohammad Haque, Samit Roy, Ray H. Baughman, and Hongbing Lu 5 Analytical Assessment of Creep Behavior of European Species in Outdoor Conditions���������������������������������������33 Claude Feldman Pambou Nziengui, Rostand Moutou Pitti, Joseph Gril, and Éric Fournely 6 Room Temperature Stress Relaxation of a Quenched and Tempered Steel �����������������������������������������������������������39 Karl-Heinz Lang 7 Improved Load Duration in Split Hopkinson (Kolsky) Bar Technique Using a Serpentine Type Striker Bar�����������������������������������������������������������������������������������������������������������������������������������������������������������45 Richard Leonard III, Luliang Zhang, Luke Luskin, Josh Loukus, Haitham El Kadiri, Hongjoo Rhee, and Wilburn Whittington 8 Experimental Shear Property Characterization of Agarose Hydrogel and Polydimethylsiloxane (PDMS) �������51 D. W. Millar, M. M. Mennu, K. Upadhyay, C. Morley, and P. G. Ifju 9 Viscoelastoplastic Oxidative Multimode Damage Model for Fibrous Composite Materials at Extreme Temperatures�����������������������������������������������������������������������������������������������������������������������������������������������������������������55 Richard B. Hall and Robert A. Brockman 10 Effect of Thermal and Mechanical Damage on Phase Separation, Crosslink Density, and Polydispersity of Polyurea Variants���������������������������������������������������������������������������������������������������������������������59 Vahidreza Alizadeh and Alireza V. Amirkhizi 11 Time-Resolved Characterization of Taylor Impact Testing �������������������������������������������������������������������������������������63 Phillip Jannotti, Nicholas Lorenzo, and Chris Meredith 12 A Novel Method of Validating Polymer Relaxation Using Hopkinson Bar and Quasi-Static Loading ���������������69 T. R. Commins and C. R. Siviour 13 Visco-Elasto-Plastic Characterization of PVC Foams�����������������������������������������������������������������������������������������������75 Marco Sasso, Fabrizio Sarasini, Edoardo Mancini, Attilio Lattanzi, Jacopo Tirillò, Claudia Sergi, and Emanuele Farotti

vii

viii

Contents

14 Virtual Dynamic Mechanical Analysis�����������������������������������������������������������������������������������������������������������������������83 J. C. Moller, N. Hagerty, T. Nguyen-Beck, S. Hawkins, A. Maffe, R. J. Berry, and D. Nepal 15 Direct Extraction of Mode I and Mode II Traction–Separation Relationships of Polymer Modified Bitumen Using Rigid Cantilever Beam Experiments Combined with StereoDIC���������������������������������89 Sreehari Rajan, Troy Myers, and Michael A. Sutton 16 Glass Fiber Composites (GFCs) in Infrastructure: Developing New Measurement Methods to Meet the Challenge of 100 Year Service-Life-Prediction �������������������������������������������������������������������������������������95 G. A. Holmes and J. W. Gilman

Chapter 1

Characterization of the Viscoelastic Response of Closed-Cell Foam Materials Jialiang Tao, Xiuqi Li, Alexander K. Landauer, David Henann, and Christian Franck

Abstract  Foam-like materials are ubiquitously employed in impact protection applications due to their relatively low density and high dissipation capacity under increased strain-rate loading. In our previous work, we have investigated the coupling between the volumetric and distortional responses of open-cell elastomeric foams under the application of large deformations and under quasi-static rates and proposed a hyperelastic model for the equilibrium response. Here, we present data collected from a set of experiments under quasi-static and moderate strain rates for a closed-cell elastomeric foam, which allows the model to be extended to higher loading rates. Two different load frames were constructed to test materials under different loading rates. One is applicable for strain rates between 10−4 and 10−1 1/s and the other to achieve maximum strain rates up to 102 1/s. The data is used to develop a finite viscoelasticity model for closed-cell elastomeric foams. Keywords  Closed-cell foam · Rate sensitivity · Material testing · Finite deformation · Hyperelastic model

1.1  Introduction Elastic foams are materials made up of two different phases: a solid matrix phase made from a viscoelastic polymeric elastomer and an air-filled fluid phase. Due to their porous structure, elastomeric foam materials are highly compliant and able to return to their original shape after undergoing large deformations involving significant volumetric change. Furthermore, due to the viscoelasticity of the elastomeric matrix, which is caused by coupling of both two phases, many elastomeric foams display a highly dissipative and time-dependent mechanical response, which led to their widespread application for impact protection. Considering the ubiquitous application of elastomeric foams in many impact-­related consumer goods, developing a predictive constitutive model is of value—in particular in applications that seek to maximize protection performance while minimizing material use. A common approach for modeling the large-deformation, viscoelastic mechanical response of soft materials is based on a Zener-like approach, which invokes a number of non-­equilibrium mechanisms each modeled using a multiplicative decomposition of the deformation gradient [1, 2]. For example, models of this type have been successfully applied to chloroprene and nitrile rubbers [1] and polyurea [3] over a range of strain rates. However, applications of this modeling approach to highly compressible elastomeric foam materials have not been undertaken. Our previous work [4] considered the quasi-static response of elastomeric foams and proposed a hyperelastic model for the equilibrium response, and current work seeks to extend both experimental characterization and constitutive modeling to a wide strain-rate range in which viscoelastic effects are dominant and more terms are introduced into model. In this work, we focus on collecting experimental data on the load/unload stress– strain response in simple tension and compression for a highly compressible elastomeric foam material over a wide range of strain rates in order to inform model development. (For brevity, modeling details are not discussed in this abstract.)

J. Tao (*) · C. Franck Department of Mechanical Engineering, College of Engineering, University of Wisconsin – Madison, Madison, WI, USA e-mail: [email protected] X. Li · A. K. Landauer · D. Henann School of Engineering, Brown University, Providence, RI, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Silberstein, A. Amirkhizi (eds.), Challenges in Mechanics of Time Dependent Materials, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-59542-5_1

1

2

J. Tao et al.

1.2  Experimental Methods Ethylene-vinyl acetate (EVA) closed-cell foam sheets were produced with the same specifications and from a single batch. All the specimens were cut from the center region of the sheets to avoid potential edge effects in the fabricated sheets and potential inhomogeneity between different parts of a sheet. Compression specimens were cut with an approximate 1.1 cm by 1.1 cm cross section and 1.3 cm thickness. Tension specimens were fabricated following the ASTM D3574 standard. Speckle patterns on the specimens for digital image correlation (DIC) technique were applied an airbrush. Two different load frames were used to test specimens under different strain rates. For quasi-static rates (10−3–10−1 strain/s), tests were implemented by a load frame with a slow motor and a high-resolution camera. For moderate strain rates (10−1–102 strain/s), tests were carried out via a different load frame, which has a heavy-duty motor and high-speed camera. Images of specimen deformation were recorded during experiment process and resultant displacement fields were computed based on recorded images via a custom digital image correlation algorithm (qDIC) [5]. Global strain, including axial strain and lateral strain, was measured based on plane fitting and used to generate stress–strain data as shown by the solid lines in Fig. 1.1. For the sake of validating the material model, three types of inhomogeneous tests were implemented over a range of specified loading rates and compared with the model predictions. The validating tests consist of tension of a specimen with circular holes cut out, spherical indentation, and shear tests with or without compression. Inhomogeneous tension tests are conducted through tension specimens following the ASTM D3574 standard, but with some holes cut out along sheet thickness direction. Spherical indentation tests were conducted on specimens with a 5 cm × 5 cm cross section and a 1.5 cm thickness using a spherical indenter with a 4  mm diameter. Shear tests with or without pre-compression were implemented, and the same equivalent shear strain amplitude was achieved in both cases. Sinusoidal loading profiles were used in all three types of validation tests using a range of different loading frequencies and using the appropriate testing machine for each loading rate.

1.3  Results and Discussion

strain rate 1.0×10 -3 s -1 Experiment Model

0

-500

-1000 -0.6

-0.4

-0.2

0

0.2

0.4

1000

strain rate 1.0×10 -3 s -1 Lateral engineering strain

Lateral engineering strain

0.1

0

-0.1

-0.2 -0.6

-0.4

-0.2

0

0.2

Axial engineering strain

0.4

Experiment Model

500 0 -500 -1000 -0.6

Axial engineering strain

b

strain rate 1.0×10 -2 s -1

0.1

-0.4

-0.2

0

0.2

0.4

500

0

strain rate 1.0×10 -1 s -1 Experiment Model

-500

-1000 -0.6

-0.4

-0.2

0

0.2

Axial engineering strain

Axial engineering strain

strain rate 1.0×10 -2 s -1

strain rate 1.0×10 -1 s -1

0

-0.1

-0.2 -0.6

Axial engineering stress [kPa]

500

Lateral engineering strain

Axial engineering stress [kPa]

a

Axial engineering stress [kPa]

Representative graphs of the resultant axial stress–strain response and axial-lateral strain response are plotted in Fig. 1.1 with different loading–unloading rates in both tension and compression. The axial stress–strain curves show some similar features across quasi-static and moderate rates, such as typical non-linearity in the tension regime and linear-­plateau-­stiffening response in the compression regime. With rising rates, the foam material shows more significant hysteresis, which is

-0.4

-0.2

0

0.2

Axial engineering strain

0.4

0.4

0.05 0 -0.05 -0.1 -0.15 -0.6

-0.4

-0.2

0

0.2

0.4

Axial engineering strain

Fig. 1.1  Representative axial stress–strain (a) and axial-lateral strain data (b) for axial engineering strain rates of 10−3, 10−2, and 10−1. For different rates, axial stress–strain curves illustrate typical linear-­plateau-­stiffening response in compression region. Higher strain rate increases stiffness in both compression and tension loading and does not influence hysteresis significantly. The axial-lateral strain data indicate similar response under different strain rates

1  Characterization of the Viscoelastic Response of Closed-Cell Foam Materials

3

consistent with other soft materials. The calibrated model fits are shown as dashed lines in Fig. 1.1 as well, indicating that the model can capture these features. Three validation tests (not shown here) show minor difference between experimental data and model prediction, which demonstrates that the model is capable of accurately capturing the experimental response.

1.4  Conclusion For a rate-sensitive closed-cell foam material, we achieved experiments across five decades of elevated strain rates. Two separate load frames were constructed and applied for two loading rate regimes to carry out reliable and stable loading– unloading profiles. The similarity in non-linear response and increased hysteresis for the foam material is shown in different strain rates. The collected data was used to develop a material model, and additional data from experiments in loading situations involving inhomogeneous deformation was used for the purpose of model validation. Acknowledgements  Material samples as well as financial support were provided by Reebok and are gratefully acknowledged. The authors also acknowledge funding from the Army Research Office under grant W911NF-16-1-0084 and an NSF Graduate. Research Fellowship and DGE 1058262 (AKL). Part of this research was conducted using computational resources at the Center for Computation and Visualization at Brown University.

References 1. Bergström, J., Boyce, M.C.: Constitutive modeling of the large strain time-dependent behavior of elastomers. J.  Mech. Phys. Solids. 46, 931–954 (1998) 2. Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1998) 3. Shim, J., Mohr, D.: Rate dependent finite strain constitutive model of polyurea. Int. J. Plast. 27, 868–886 (2011) 4. Landauer, A.K., Li, X., Henann, D.L., Franck, C.: Experimental characterization and hyperelastic constitutive modeling of elastomeric foam. J. Mech. Phys. Solids. 133, 103701 (2019) 5. Landauer, A.K., Patel, M., Henann, D.L., Franck, C.: A q-factor-­based digital image correlation algorithm (qDIC) for resolving finite deformations with degenerate speckle patterns. Exp. Mech. 58, 815–830 (2018)

Chapter 2 Time-Dependent Yielding of Polymer Thin Films Under Creep Veli Bugra Ozdemir and Kawai Kwok

Abstract  This paper focuses on developing a time-dependent yield criterion for a polymer thin film, namely StratoFilm 420, under long-term creep conditions. StratoFilm 420 is a linear low-density polyethylene film used in constructing super-pressure balloons in the Ultra-Long Duration Balloon (ULDB) program of NASA. Yielding of StratoFilm 420 under long-term loading has been found to cause balloon failure and limit the operational duration. Knowledge of time-dependent yielding behavior of StratoFilm 420 under different stresses and temperatures is essential to design reliable balloon structures. In this paper, a previously developed nonlinear viscoelastic model for the film and its model parameter calibration are first described. The experimental procedure for characterizing the onset of permanent deformation is presented. The yield limit is determined using the strain recovery method using creep tests with different durations and measuring the corresponding residual strains. The variation of yield strains is presented with respect to stress levels and temperature. A free energy-based criterion combined with nonlinear free volume model is proposed for predicting yielding of StratoFilm 420. Keywords  Creep yielding · Polymer thin films · Free volume model · Super-pressure balloon film · Time-to-failure

2.1  Introduction Polymer thin films are widely used in large lightweight structures including deployable solar sails, inflatable solar concentrators, and balloon structures. One notable example of polymer thin film applications is the super-pressure balloon developed by the NASA Balloon Program Office for science missions in the stratosphere [1]. These super-­pressure balloons maintain positive gage pressure for long durations of time, providing a low-cost platform for atmospheric observation and experiments. Super-pressure balloons are subjected to pressurization cycles and thermal loads due to the diurnal temperature change, and the balloon envelope is subjected to constantly varying thermomechanical stresses. Current super-pressure balloon designs employ linear low-­density polyethylene based StratoFilm 420 for its ductility compared to polyester and polyimide [2]. The time-­dependent mechanical response of StratoFilm 420 is critical to the long-term reliability of balloons. In particular, an accurate time-dependent yielding criterion is vital for avoiding permanent deformation of the balloon envelope. Yielding of polymers has been studied extensively. Previous work has mainly focused on the effect of temperature and strain rate effects on yielding [3], whereas some studies were focused on the effect of hydrostatic stress state on the yield stress behavior [4–6]. The first challenge in understanding yielding in polymer is the experimental determination of yield point from mechanical stress–strain data. A number of methods have been proposed in the past. The offset methods have been used [7–9] with offset strain values ranging from 0.3% to 2%. Other methods define the yield point as the load-drop point, or the point at the knee of the stress-strain curve when there is no apparent load-drop point [10, 11], or as the at the knee of the stress–strain curve constructed from backward extrapolation. Finally, the strain recovery method in which specimens are loaded up to different points and the unrecovered inelastic deformation is measured after some recovery time has also been employed to locate the yield point [3, 12, 13]. Previous studies have also investigated the effect of strain rate and hydrostatic pressure on yielding [10]. Several approaches for developing yielding criteria have been employed in the literature. One approach is to extend the theory of metal plasticity to polymers, such as pressure modified Tresca and pressure modified von Mises criteria. Quinson V. B. Ozdemir (*) · K. Kwok Department of Mechanical and Aerospace Engineering, College of Engineering, University of Central Florida, Orlando, FL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Silberstein, A. Amirkhizi (eds.), Challenges in Mechanics of Time Dependent Materials, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-59542-5_2

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V. B. Ozdemir and K. Kwok

et al. [13] have experimentally shown that there is a relation between deformation mode and yielding criterion to be used. Another approach is based on molecular considerations of yielding such as the Eyring theory [14]. Numerous molecular mechanisms underlying macroscopic yielding are proposed with the aim to relate the macroscopic applied stress with microscopic yielding mechanisms via micromechanical models [15]. Yielding of viscoelastic materials is not only affected by stress, strain rate, and temperature but also by the duration of the applied stress. It is possible to yield a viscoelastic material if stresses that are lower than yield stress are maintained for long durations of time, such as in creep conditions. Naghdi and Murch [16] developed a viscoelastic failure theory for the timedependent failure of polymers. Their work is later followed and modified by Crochet [17]. Reiner and Weissenberg developed an energetic criterion using the deviatoric free energy. This paper presents initial experimental results on the onset of yielding StratoFilm 420 under creep at different stress levels and temperatures, as well as an application of the Reiner–Weissenberg for yield prediction. This paper is organized as follows. A brief background on the nonlinear viscoelastic model developed for the super-pressure balloon film by Li et al. [18] is first given. The experimental procedure on determination of yield points is then described. Finally, the free energy concept of Reiner and Weissenberg is applied and investigated for describing the experimentally determined yield points.

2.2  Free Volume Nonlinear Viscoelastic Model Polymers display time and temperature dependent mechanical properties. Linear viscoelastic models are commonly used to capture the time and temperature effects in the material response. For one-dimensional case, equations of linear viscoelasticity are given in Eqs. (2.1) and (2.2). t′

 ( t ) = ∫D ( t ′ − s )



0

t′

dσ dT ds + ∫α ds ds ds 0

(2.1)

where D is the creep compliance and α is the coefficient of thermal expansion. Eq. (2.1) is referred to as Boltzmann superposition integral and relates stress history and thermal effects to strain. t′ is reduced time at the reference temperature T0. Time and reduced time are related to each other by t

t′ = ∫



0

ds a (T ( s ) )

(2.2)

where a(T) is the shift factor at a given temperature. In linear viscoelasticity, shift factors are commonly represented in the form of Arrhenius type equation for temperatures below the glass transition temperature, and in the form of Williams– Landel–Ferry [19] equation for temperatures above Tg. Arrhenius and WLF equations assume the shift factors as a function of temperature only. Over the years, due to the limitations of classical approaches on predicting moderate to high strain levels, different approaches have been developed to account for the effects of deformation on material response. Knauss and Emri [20] developed a nonlinear free volume model to account for deformation effects on the shift factors. In the free volume approach, the shift factors are not only a function of temperature but also deformation [21]. At temperature T, the free volume in the material is represented as f = f0 + α v ( T − T0 ) + δ vθ



(2.3)

where f0 is the free volume at a reference temperature T0, αv is the volumetric coefficient of thermal expansion, and θ is the mechanical dilatation, i.e.

θ = ii



(2.4)

Shift factors are then calculated from log a =

Bd  1 1   −  2.303  f f0 

(2.5)

2  Time-Dependent Yielding of Polymer Thin Films Under Creep

7

where Bd is the material constant. The free volume model takes mechanical deformations into account. If the material undergoes positive mechanical dilatation, the free volume in the material increases and polymer chains can more easily move with respect to each other, therefore material shows an accelerated response. The nonlinear free volume model used in this study was revised previously to include the effects of shear deformations and the shift factors are given as Bd  α v ( T − T0 ) + δ vθ  Bs log a = −  − 2.303 fd  fd + α v ( T − T0 ) + δ vθ  2.303 fs

  eff   fs +  eff

  

(2.6)

where ϵeff is the effective deviatoric mechanical strain [18]. The compliance matrix is given by Eq. (2.7), noting that although the polymer thin film is under plane stress conditions, thickness direction (Di3) terms are also needed to calculate deformation effects.  D11 ( t ) D12 ( t ) D13 ( t ) 0    D12 ( t ) D22 ( t ) D23 ( t ) 0   D (t ) =  D13 ( t ) D23 ( t ) D33 ( t ) 0    0 0 D66 ( t )   0



(2.7)

Compliance terms are represented using Prony series as



t n −  D ( t ) = D0 + ∑D j  1 − e aτ i  j =1 

   

(2.8)

where D0 is the instantaneous compliance, Dj are the Prony coefficients, and τi are the corresponding retardation times. Assuming the material is thermorheologically simple, the same shift factor applies to all retardation times. In the free volume model, shift factors are a function of deformations as well as temperature. The free volume model assumes that as the material undergoes deformation, available free volume in the polymer increases, causing easier motion of the polymer chains. Nonlinear free volume model was applied to StratoFilm 420 by Li et al. [18]. In their study, the material was characterized by different constant strain rate tests at various temperatures. The material properties were calibrated up to yielding.

2.3  Creep Tests The material used in this study is StratoFilm 420 (SF420) obtained from the NASA Balloon Program Office. SF420 is a linear low-density polyethylene (LLDPE) based material and produced by co-extrusion of three layers [2]. The outer layers include UV inhibitors to negate the effect of radiation during the operation of the balloon. The film has a total thickness of 38 μm. The middle layer contributes 60% to the overall thickness and each of the outer layers contributes 20%. SF420 has a glass transition temperature of −95 °C [22], therefore in operational conditions for the super-­pressure balloon, SF420 is in the rubbery state. SF420 displays mechanical anisotropy, being slightly stiffer in the transverse direction (TD) compared to the machine direction (MD). Previously it has been shown that material displays higher strength in constant strain rate u­ niaxial tests in TD compared to MD [3]. For this study, tests are carried out on specimens aligned in the machine direction in order to be conservative. The experimental setup for creep tests is shown in Fig. 2.1. The tests were carried out using a MTS Model 43 tensile testing system equipped with a thermcraft environmental chamber. Three dimensional digital image correlation (DIC) was used to obtain the strain field. A custom made light panel was used to illuminate the speckled specimens with diffused backlighting. A 250 N load cell was used in all experiments. The analog output from load cell is calibrated and synchronized with the imaging of the DIC. Two additional K-type thermocouples are used to monitor temperature continuously. Specimens of dimensions 25.4 mm by 254 mm are cut by a precision cutter in both machine and transverse directions from SF420 sheet. The cut specimens were lightly speckled with black paint to achieve random speckle patterns for the digital image correlation. To be able to attach specimens to mechanical grips without introducing damage, specimens were taped to hard plastic grips which were attached to the mechanical grips. Once the specimen was attached to the tensile testing machine, it was put in a taut position for image focusing. The distance between the camera system and the specimen was adjusted and cameras were re-focused on the specimen with a high confidence interval. The exposure rate of the cameras was re-adjusted to not overexpose images due to the light panel. The imaging rate of DIC was set to 0.1 Hz.

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Fig. 2.1  Test setup for testing of StratoFilm 420: 1—stereo system, 2—External thermocouples, 3—Light panel, 4—DAQ module for thermocouples, 5—Mechanical wedge action grips, 6—Environmental chamber

After setting up the DIC, crosshead reading was zeroed and then the specimen was put to slack configuration by lowering the crosshead. Putting the specimen in slack configuration ensures the specimen was not stressed due to cooling before the experiment. The desired temperature level was input in MTS, and the temperature was monitored with the environmental chamber’s thermocouples and two external K-type thermocouples. After the temperature in the environmental chamber reaches the pre-set value, 40 min was allowed to have a uniform temperature field in the chamber, also allowing the thermal contraction of the extension rods. Crosshead was re-adjusted such that the specimen was slightly tensed to remove creases but not exceeding the value of 0.1 N. The new gage length was calculated from the new crosshead value and original gage length, and crosshead velocity was calculated to yield the nominal strain rate of 0.001% s−1. Specimens were stretched with the nominal strain rate to the predetermined load level corresponding to the desired engineering stress and load is kept constant for determined creep duration. At the end of the test, specimens were unloaded by a load rate of 0.5 N/s and at zero load, crosshead was moved to the original position immediately. After specimens were removed from the mechanical grips, a recovery time of 24 h at the room temperature was allowed before measuring the residual deformations. Residual deformations are measured by an electronic caliper with a resolution of 0.01 mm. The residual strain was calculated as the ratio of residual deformation and original gage length. Recovery time of 24 h is selected based on the rule of thumb that recovery time is more than the test times [12]. While processing the images, an area of interest is chosen to cover most of the specimen between the grips. Strains are analyzed using Vic-3D software and post-processed to achieve Lagrange strains. A sample strain field and speckle pattern can be seen in Fig. 2.2. Variation in the strain field is approximately 10% of the nominal value due to noise associated with the test setup. The observed strain field is averaged over the area of interest for future analyses.

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Fig. 2.2  Sample longitudinal (top) and transverse (bottom) Green–Lagrange strain fields during test for specimen oriented in MD. Some experimental noise can be observed at the boundaries of the area of interest

Creep tests are carried out with a constant strain rate of 0.001% s−1 until the specimen reaches the predetermined stress level of 5 MPa. Creep tests for model calibration have a creep duration of 30 min at 5 MPa. For the given temperatures and the stress level, no yielding is observed. Creep calibration tests are performed in both machine and transverse directions. To determine the deformation effect on the free volume, the compliances in thickness direction need to be determined. However, measuring the out of plane compliances (D13 and D23) experimentally is difficult due to small thickness of the specimen. Therefore, Kwok and Pellegrino [22] used the in-plane response of the material from various tests to predict out of plane compliance terms with the free volume model. In their paper, prediction of out of plane compliances was postulated as an optimization problem to minimize the differences between model prediction and experimental data. The same approach is adopted in this paper to determine D13 and D23 with the creep tests carried out at 5 MPa at four different temperatures (−30 °C, −10 °C, 10 °C, 22 °C). Determination of D13 and D23 is crucial since the material response is predicted by free volume modified shift factor via Eq. (2.6). Effects of out of plane compliances directly influence the material model. If the nonlinear material model is to be used to determine time-to-failure subsequently, the creep response must be accurately predicted. For determination of D13 and D23 under creep, the other compliance values and material parameters obtained previously are kept unmodified. To determine the compliance parameters, the experimental results from the −30 °C, −10 °C, 10 °C, and 22 °C tests were compared with the model prediction in an optimization problem. Optimization was achieved via an unconstrained optimization solver with the interior-point algorithm. Considering the operational conditions for the super-pressure balloon, higher weights for errors are applied at the colder temperature tests. The objective function for optimization for D13 and D23 can be given as follows:   model ( t k ) −  experiment ( t k )  minimize f = ∑wi ∑     experiment ( t k ) Ti tk   2



(2.9)

where inner sum is for the differene in single temperature test evaluated at discrete time steps and outer sum is over the range of temperatures with wi being the weight factors. The model prediction compared to experimental data is shown in Fig. 2.3. The yield point is defined as the onset of non-recoverable deformation since no clear load-drop observed in the experiments with 0.001% s−1 strain rate. The same was observed by Bosi and Pellegrino [3]. The commonly used approach [8, 10] of defining the onset of yielding for polymers as load-­drop point cannot be used for SF420, and the strain recovery method was employed. In the strain recovery approach, it is essential to differentiate between recoverable viscoelastic strains and non-recoverable plastic strain. Determination of yield points for a given stress and temperature under creep condition must require performing creep tests with different creep durations and measurement of corresponding residual strains. Obtaining yield points at each temperature–stress pair requires minimum of three tests with different creep durations. Testing approach is as follows: conducting different duration creep tests at fixed temperature and stress level, determination of strains at the end of each test and corresponding residual strains after 24 h of free recovery at room temperature. For

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Fig. 2.3  Calibration results for D13 and D23 for the creep tests performed at 5 MPa for machine direction (MD) (left) and transverse direction (TD) specimens (right), initial loading portions correspond to constant strain rate

a constant temperature and stress level, residual strains and strains at the end of the tests increase with increased test duration. For a given temperature and stress level, residual strains can be seen as an injective function of maximum applied strain (strain at the end of the test). Therefore, it is possible to map residual strains to unique strain values. Previously, Quinson et al. [3] proposed a back-­extrapolation for residual strains to zero in order to find the corresponding applied strain which they called the yield strain. Other researchers have employed back-extrapolation for its convenience and practicality [12] for the loading cases where load-drop point is not observed. In this paper, instead of back-extrapolation of residual strains, a limiting value of residual strain after 24 h of free recovery is considered to be determinant of yielding. Recovering for 24 h at room temperature is considered to be a good indicator of the residual strain considering the loading durations of the tests [12]. The limiting value is chosen as 0.5% due to the experimental and measurement errors. Consequently, a specimen is considered to be yielded if the residual strain after 24 h of free recovery at room temperature is equal to or larger than 0.5%. The applied strain at the end of the test resulting in 0.5% strain is considered to be the yield strain for that stress level and temperature. Due to the small strain rate of 0.001% s−1, the time needed to reach a specified stress level increases with increasing stress and temperature. However, at lower stress levels, the time needed in creep conditions increases drastically. Hence, an exploration plan is adopted. In order to implement this plan, two initial creep durations are needed corresponding to one yielded and one unyielded specimen for a specific stress level and temperature. Exact determination of the creep duration can be iteratively achieved by bisecting the two creep durations and carrying out another creep test with calculated duration. The procedure is illustrated in Fig. 2.4. In order to find time-to-yield to the finest degree, next test needs to be carried out to explore between 1 and 30 min creep duration tests, for example, 15 min creep duration (bisecting 1 and 30 min tests). For engineering stresses lower than 4 MPa, curves follow nearly identical paths, some discrepancy can be observed as stresses become larger.

2.4  Results and Discussion Figure 2.5 displays the best estimates for the yield strains at specific stress levels and temperatures. Up to this point, SF420 creep yielding is observed for two temperatures (10  °C, 22  °C) at different stress levels. Stress levels are increased by 0.5 MPa to see the effect of the stress level. In Fig. 2.5, the upper and lower intervals are given for observed experimental data where yielding is observed and not observed, respectively. It can be seen that the yield strain varies with stress level as well as temperature. It is important to note that yield strain values increase with the increasing stress levels. However, as Fig. 2.6 illustrates, time-to-yield actually decreases drastically with increasing stress levels. For example, creep duration needed for specimen to yield at 6.5 MPa is estimated to be 1 min, but at 5.5 MPa it is estimated to be 480 min.

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11

Fig. 2.4  Yield point determination procedure, presented for 6.5 MPa tests. Creep durations are 1, 30, and 60 min. On right y-axis corresponding residual strains after 24 h are plotted

Fig. 2.5  Best estimates for yield strains for various stress levels and temperatures. Blue and red dots indicate results for 10 °C and 22 °C results, respectively. No yielding has been observed for 5 MPa for creep duration of 14 h (shown blank red dot). Blank blue point is used to illustrate the observation of yielding for 7 MPa

Considering the effect of temperature, lower temperatures result in lower yield strain value. This can be attributed to modulus change due to temperature. Colder temperature results in a stiffer response of the material, i.e. the material deforms less for the same stress level compared to higher temperature, but the strain value at yielding decreases. Creep tests at 6.5  MPa and at 10  °C and 22  °C show that the time to reach specified stress level decreases with decreasing temperature, however, time-to-yield in creep duration is larger for colder temperature. Hence, it can be concluded that time-­ t o-­ y ield in creep conditions increases with decreasing stress levels or decreasing temperatures. Determination of yield strains at different stresses and temperatures is essential to developing a yield criterion for more general cases. In practical applications, stress and temperature can vary with time; therefore, yield strain approach may not be suitable in practice. Therefore, an energy-based approach is explored in this section. Development of an energetic criterion will be preferred for its simplicity and generality.

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Fig. 2.6  Comparison of stress vs. time plots (left) for best estimates for the yield points at different stress levels at room temperature. No yielding is observed for 5 MPa for the plotted test. Comparison of strain measurements for various creep tests (right)

For the conducted creep experiments, it is considered useful to compare the absorbed mechanical energy by the material for the span of loading. The total work of deformation (absorbed mechanical energy) per unit volume is given by Eq. (2.10) [23]. t



W ( t ) = ∫S ( ρ ) E ( ρ ) d ρ 0



(2.10)

where S is the second Piola–Kirchhoff stress, E is the rate of Green–Lagrange strain. Eq. (2.10) refers to the area under the curve of stress–strain diagram for combined constant strain rate and creep tests for the viscoelastic material. The second Piola–Kirchhoff stresses are required to evaluate the integral since Green–Lagrange strains are used for the analyses. Experimentally, the engineering stress is achieved by dividing the load to the initial cross-sectional area of the SF420. After calculating the second Piola–Kirchhoff stress, total work of deformation for some sample tests are calculated. Figure 2.7 shows the total work of deformation history for the current best estimates for creep yield points at different stress levels. It can be seen that constant strain rate portion of the tests follows identical paths, diverging at the stress level specified. In the creep portion, work done on the material increases due to creep deformation, and the slope of energy is considerably higher at higher stress levels, both due to accelerated response of the material and stress level. There is no unique value of energy that defines the yield point for all these tests. Total work by deformation contains the stored and dissipated energies. Previously mentioned Reiner–Weissenberg criterion assumes that yielding of polymer material is dependent on the deviatoric part of the stored energy term and it is a material constant [24]. Referring to the generalized Kelvin model in Fig. 2.8, the stored energy can be thought of as energy stored in the springs, and dashpots are acting as a dissipative mechanism. Generalized Kelvin model is chosen because of its easier formulation for creep condition. To be able to extract the stored energy, experimental data needs to be in conjunction with the nonlinear free volume model as opposed to total absorbed energy which can be directly calculated from experimental data. The stored energy can be further divided into volumetric and deviatoric parts. Noting that SF420 at room temperature is in rubbery state, from SF420 tests carried out at room temperature (well above its glass transition temperature), it is observed that the volumetric part of the energy can be neglected [25] since the material undergoes fully distortional deformation with near-zero volume change. Therefore, in this study, the stored energy is assumed to be only distortional. The assumption is validated with strain field readings from DIC. The in-plane Poisson’s ratio is approximately 0.5 for room temperature tests. Generalized Kelvin model assumes that for any stress σ(t), the stress carried by each spring-dashpot unit is equal to σ(t). Considering a spring-dashpot unit, undergoing the same deformation, the following relation can be written:

σ = σ dashpot ,i + σ spring,i For a step loading condition σ = σ0H(t), the stress carried by the spring can be evaluated as

(2.11)

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13

Fig. 2.7  History for the total absorbed energy for creep tests at different stresses at 22 °C

Fig. 2.8  Generalized Kelvin Model for viscoelastic materials



t −  σ spring,i = σ 0  1 − e τ i  

   

(2.12)

which shows that the stress carried by the spring will increase over time and reach a limiting value of the externally applied stress. Considering the spring is linear elastic, the stored energy in each spring can be written as

1 1 2 Uspring,i = σ spring,ispring,i = σ spring ,i Di 2 2

(2.13)

And combining Eqs. (2.12) and (2.13), and including the effects of free spring and other units, total stored energy can be shown to be [26] D n D Ustored ( t ) = σ  0 + ∑ i  2 i =1 2  2 0



t −  τi 1 − e  

   

2

   

(2.14)

Equation (2.14) is valid for step loading condition. In this study, it is assumed that the applied stress in the experiments can be approximated as a combination of step loadings at discrete times. Numerical integration is carried out to calculate the stored energy as a function of time. Also, the retardation times in Eq. (2.14) are modified with shift factors to include the effects of temperature and deformation. For the experimental data, stress history is discretized with a constant time step of 50 s. It is important to note that different time steps will result in different numerical values for the stored energy using the aforementioned formulation for step

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Free volume model

, Stress history from experiment

Preprocessing and discretization

Convolution for free energy due to Δ at

Free energy history

Fig. 2.9  Calculation of free energy history from experimental data with the aid of nonlinear free volume model

Fig. 2.10  Free energy history for the selected 22 °C experiments, normalized with the maximum free energy of the 6.5 MPa test

loading, however, for the initial quantitative comparison of free energies, this approach is found to be satisfactory. Discretized stress history is used in the nonlinear free volume model to calculate the shift factors over the duration of the experiment. The contribution of the step loadings on the free energy is then calculated using Eq. (2.14) with the shift factors obtained from the nonlinear free volume model. The approach can be illustrated in Fig. 2.9, and the free energy history for the selected experiments is given in Fig. 2.10. From Fig. 2.10, it can be observed that free (stored) energy increases in the creep portion of the experiments, indicating that stress carried by the springs increases as the dashpots continue to deform. For the current best estimates for yield points, Reiner–Weissenberg approach performs satisfactorily but fails to obtain a universal free energy for different stress levels.

2.5  Conclusion In this work, time-dependent yielding of linear low-density polyethylene based polymer thin film, StratoFilm 420, is investigated. A testing procedure for effective determination of exact yield points at different stress levels and temperatures is presented. To determine the onset of the permanent deformation, the strain recovery method is employed where specimens are allowed 24 h at room temperature to recover free from external loads. In this work, 0.5% residual strain after 24 h is

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15

chosen as the indicator for yielding. Estimates for the yield strains are presented and trends are noted. The findings show that time-to-failure in creep condition is drastically shorter for higher stress levels. Also, at colder temperatures, time-to-failure increases for the same stress level. An energetic approach developed by Reiner and Weissenberg is employed together with the nonlinear free volume model to unify the experimental findings. Nonlinear free volume model for creep conditions is calibrated from experimental data obtained at four different temperatures. Findings show that the Reiner–Weissenberg criterion may not be used to obtain the limiting free energy value which was assumed as a material constant. In the future, to be able to predict the time-to-failure of the SF420 in super-pressure balloon operational conditions, colder temperature creep tests will be carried out. The estimations for the yield strains will be improved with the proposed experimental procedure. Acknowledgements  This research is supported by a NASA grant (80NSSC18K0913) from the Goddard Space Flight Center. The authors thank the NASA Balloon Program Office and Tensys Ltd. for helpful comments and discussion.

References 1. Rand, J.: A nonlinear plasto-viscoelastic constitutive equation for balloon films. In: AIAA Balloon Systems Conference, Seattle, Washington (2009). https://doi.org/10.2514/6.2009-2812 2. Rand, J.L., Wakefield, D.: Studies of thin film nonlinear viscoelasticity for superpressure balloons. Adv. Space Res. 45(1), 56–60 (2010). https://doi.org/10.1016/j.asr.2009.09.004 3. Bosi, F., Pellegrino, S.: Molecular based temperature and strain rate dependent yield criterion for anisotropic elastomeric thin films. Polymer. 125, 144–153 (2017). https://doi.org/10.1016/j.polymer.2017.07.080 4. Donato, G.H.B., Bianchi, M.: Pressure dependent yield criteria applied for improving design practices and integrity assessments against yielding of engineering polymers. J. Mater. Res. Technol. 1(1), 2–7 (2012). https://doi.org/10.1016/S2238-7854(12)70002-9 5. Christiansen, A.W., Baer, E., Radcliffe, S.V.: The mechanical behaviour of polymers under high pressure. Philos. Mag. J. Theor. Exp. Appl. Phys. 24(188), 451–467 (1971). https://doi.org/10.1080/14786437108227400 6. Radcliffe, S.V.: Effects of hydrostatic pressure on the deformation and fracture of polymers. In: Kausch, H.H., Hassell, J.A., Jaffee, R.I. (eds.) Deformation and Fracture of High Polymers, pp. 191–209. Springer, Boston, MA (1973) 7. Raghava, R.S., Caddell, R.M.: A macroscopic yield criterion for crystalline polymers. Int. J. Mech. Sci. 15(12), 967–974 (1973). https://doi. org/10.1016/0020-7403(73)90106-9 8. Raghava, R., Caddell, R.M., Yeh, G.S.Y.: The macroscopic yield behaviour of polymers. J.  Mater. Sci. 8(2), 225–232 (1973). https://doi. org/10.1007/BF00550671 9. Xu, M., Huang, G., Feng, S., McShane, G.J., Stronge, W.J.: Static and dynamic properties of semi-crystalline polyethylene. Polymers. 8(4), 77 (2016). https://doi.org/10.3390/polym8040077 10. Freire, J.L.F., Riley, W.F.: Yield behavior of photoplastic materials: paper provides information on the yield behavior of polyester mixtures which appear suitable for model studies of manufacturing processes such as rolling and extruding. Exp. Mech. 20(4), 118–125 (1980). https:// doi.org/10.1007/BF02321292 11. Brooks, N.W., Duckett, R.A., Ward, I.M.: Investigation into double yield points in polyethylene. Polymer. 33(9), 1872–1880 (1992). https:// doi.org/10.1016/0032-3861(92)90486-G 12. Jin, T., Zhou, Z., Shu, X., Wang, Z., Wu, G., Zhao, L.: Investigation on the yield behaviour and macroscopic phenomenological constitutive law of PA66. Polym. Test. 69, 563–582 (2018). https://doi.org/10.1016/j.polymertesting.2018.06.014 13. Quinson, R., Perez, J., Rink, M., Pavan, A.: Yield criteria for amorphous glassy polymers. J. Mater. Sci. 32(5), 1371–1379 (1997) 14. Ward, I.M., Sweeney, J.: Mechanical Properties of Solid Polymers. Wiley, Hoboken (2012) 15. van Dommelen, J.A.W., Poluektov, M., Sedighiamiri, A., Govaert, L.E.: Micromechanics of semicrystalline polymers: towards quantitative predictions. Mech. Res. Commun. 80, 4–9 (2017). https://doi.org/10.1016/j.mechrescom.2016.01.002 16. Naghdi, P.M., Murch, S.A.: On the mechanical behavior of viscoelastic/plastic solids. J.  Appl. Mech. 30(3), 321–328 (1963). https://doi. org/10.1115/1.3636556 17. Crochet, M.J.: Symmetric deformations of viscoelastic-plastic cylinders. J.  Appl. Mech. 33(2), 327–334 (1966). https://doi. org/10.1115/1.3625045 18. Li, J., Kwok, K., Pellegrino, S.: Thermoviscoelastic models for polyethylene thin films. Mech. Time Depend. Mater. 20(1), 13–43 (2016). https://doi.org/10.1007/s11043-015-9282-8 19. Williams, M.L., Landel, R.F., Ferry, J.D.: The temperature dependence of relaxation mechanisms in amorphous polymers and other glassforming liquids. J. Am. Chem. Soc. 77(14), 3701–3707 (1955). https://doi.org/10.1021/ja01619a008 20. Knauss, W.G., Emri, I.J.: Non-linear viscoelasticity based on free volume consideration. Comput. Struct. 13(1), 123–128 (1981). https://doi. org/10.1016/0045-7949(81)90116-4 21. Kwok, K., Pellegrino, S.: Large Strain Viscoelastic Model for Balloon Film (2011). https://doi.org/10.2514/6.2011-6939 22. Bosi, F., Pellegrino, S.: Nonlinear thermomechanical response and constitutive modeling of viscoelastic polyethylene membranes. Mech. Mater. 117, 9–21 (2018). https://doi.org/10.1016/j.mechmat.2017.10.004 23. Tschoegl, N.W.: Energy storage and dissipation in a linear viscoelastic material. In: Tschoegl, N.W. (ed.) The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction, pp. 443–488. Springer, Berlin, Heidelberg (1989)

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Chapter 3 Non-Newtonian Fluid-Like Behavior of Poly(Ethylene Glycol) Diacrylate Hydrogels Under Transient Dynamic Shear K. Upadhyay, K. Luo, G. Subhash, and D. E. Spearot

Abstract  Hydrogels exhibit a fluid-like viscous response under high shear strain rates with significant rate- and microstructure-­ dependent rheological properties. In this study, the transient shear response of poly(ethylene glycol) diacrylate (PEGDA) hydrogels is characterized by application of the power-law fluid model to incompressible Navier–Stokes equation of start-up planar Couette flow. To impart the necessary boundary conditions for model calibration, a split-Hopkinson pressure bar based single-pulse dynamic simple shear experiment is developed, in which unsteady momentum diffusion between two shear plates is measured using two-dimensional digital image correlation (DIC). The measured shear profiles and their characteristic self-similarity under these boundary conditions are utilized to calculate power-law exponent and transient-state viscosity via finite difference simulations. Keywords  Poly(ethylene glycol) diacrylate (PEGDA) · Hydrogels · Simple shear · Power-law viscoelasticity Split-­Hopkinson pressure (Kolsky) bar

3.1  Introduction Hydrogels are polymeric materials that swell in water to form physically or chemically cross-linked three-­dimensional networks. Many applications of hydrogels stem from their characteristic biocompatibility, biodegradability, and a porous network structure similar to that of many tissues. They have been employed as tissue engineering scaffolds [1], drug delivery systems [2], wound dressings [3], and as surrogate materials [4] to study ballistic impact, crash, and traumatic brain injury. In these applications, hydrogels are subjected to a variety of three-dimensional (3D) deformation modes and loading rates. Therefore, an accurate understanding of both the 3D quasi-static and dynamic mechanical behavior of these materials is important for an optimal design of these devices and to attain high-fidelity predictive capability of their mechanical response. Experiments simulating the three primary deformation modes of compression, tension, and shear are generally required to obtain hyper/viscoelastic constitutive relationships [5–7]. Out of these three deformation modes, compression is the most convenient and commonly conducted experiment for soft materials in both quasi-static conditions and in dynamic conditions with appropriate modifications of the traditional split-Hopkinson pressure bar (SHPB) test for incorporating these low impedance materials [8, 9]. Tension and shear are difficult to implement experimentally primarily because of the difficulties related to specimen mounting and strain measurement [10, 11]. Because of the soft, wet, and fragile nature of hydrogel K. Upadhyay (*) Department of Mechanical and Aerospace Engineering, Herbert Wertheim College of Engineering, University of Florida, Gainesville, FL, USA Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, MD, USA e-mail: [email protected]; [email protected] G. Subhash Department of Mechanical and Aerospace Engineering, Herbert Wertheim College of Engineering, University of Florida, Gainesville, FL, USA K. Luo Department of Materials Science and Engineering, Herbert Wertheim College of Engineering, University of Florida, Gainesville, FL, USA D. E. Spearot Department of Mechanical and Aerospace Engineering, Herbert Wertheim College of Engineering, University of Florida, Gainesville, FL, USA Department of Materials Science and Engineering, Herbert Wertheim College of Engineering, University of Florida, Gainesville, FL, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Silberstein, A. Amirkhizi (eds.), Challenges in Mechanics of Time Dependent Materials, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-59542-5_3

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specimens, even small mounting/gripping pressures tend to damage the specimens. This also makes it unfeasible to use traditional contact-based strain measurement methods such as strain gages and extensometers on these materials, which are required to attach firmly to the specimen during the experiment. Shear deformation at dynamic strain rates is particularly challenging to characterize because the very slow momentum diffusion in hydrogels results in a transient deformation whose duration is comparable with the total duration of the experiment, thus requiring separate analysis and modeling of the transient and steady-­state deformation stages [12]. Interestingly, under transient shear deformation at high strain rates, hydrogels behave in a fluid-like manner [13]. The aim of this study is to present a method of characterizing the fluid-like response of hydrogels under transient shear deformation at high strain rates using a power-law fluid model. To this end, poly(ethylene glycol) diacrylate (PEGDA) hydrogel is chosen as a model material. A modified SHPB-based shear experiment is used to impart the appropriate boundary conditions. In this experiment, the shear deformation of the specimen is captured using a high-­speed camera and is analyzed using digital image correlation (DIC) technique. The constitutive model is implemented using numerical finite difference method (FDM) to capture the viscoelastic material properties from the experimental data. This results in the shear thickening exponent and the transient-state viscosity of PEGDA.

3.2  Materials and Methods To prepare the hydrogel specimens, PEGDA (Mn ≈ 700; Sigma-Aldrich, Milwaukee, WI) is dissolved in ultrapure water to obtain a 20 wt% polymer concentration. This mixture is sonicated for 1 h, following which free-radical initiator ammonium persulfate (APS) and accelerator N,N,N′,N′-tetramethylethylenediamine (TEMED) (Sigma-­Aldrich, Milwaukee, WI) are added to this solution in 0.1 wt% concentrations each. This triggers cross-linking in the polymerization mixture, which is then poured into custom-­ made rectangular molds of poly(methyl methacrylate) material (dimensions: 45 mm × 30 mm × 3.5 mm). It takes ~5 h for a complete polymerization of the samples, following which they are stored in ultrapure water until testing. Figure 3.1 illustrates the different stages of dynamic shear deformation in a rectangular prism hydrogel sample. During transient shear, the momentum diffuses between the moving wall (shearing plane) and the fixed wall. Once the momentum diffusion is completed, the steady-state deformation begins, during which a uniform shear strain-field exists in the sample. A power-law viscoelastic model is a popular model for characterizing the non-Newtonian viscoelastic response of materials,

τ =α



∂u ∂z

n −1

∂u ∂z

(3.1)

where τ, u = ∂x/∂t and z represent the shear stress, material point velocity, and direction normal to the shearing plane (see Fig. 3.1), respectively. α and n are model parameters called the viscous coefficient and the shear thickening exponent. The latter defines the degree of non-Newtonian behavior, with n  1 being the shear thinning, Newtonian, and shear thickening responses, respectively. For a power-law material, the transient shear (also called start-up planar Couette flow) shown in Fig. 3.1 is described by the following incompressible Navier–Stokes equation of start-­up planar Couette flow,

ρ

Fig. 3.1  Illustration of shear deformation in a hydrogel

n −1 u ∂  ∂u ∂u  =α   ∂t ∂z  ∂z ∂z 

(3.2)

3  Non-Newtonian Fluid-Like Behavior of Poly(Ethylene Glycol) Diacrylate Hydrogels Under Transient Dynamic Shear

19

subjected to boundary conditions



u ( 0,z ) = 0 u ( t ,0 ) = U ( t ) u ( t ,h ) = 0

(3.3)

where ρ is the material density, t is time, U is the velocity of the shearing plane (boundary velocity), and h is the thickness of the sample in the z-direction. At any given time, the extent of shear momentum diffusion between the two plates during transient shear is characterized by the momentum diffusion length,



δz (t ) =

h

2 xd ( t ,z ) dz xd ( t ,0 ) ∫0

(3.4)

where xd(t, z) is the material point displacement in the x-direction. Kwon et al. [13] showed that for a linearly rising boundary velocity (U(t) = bt; b being the constant a­ cceleration value), by utilizing the self-similarity of the displacement and velocity profiles during transient shear, the following relation can be derived between the momentum diffusion length and time:

δ z ( t ) = C2 t

n / ( n +1)

(3.5)



where C2 is a fitting parameter. Eq. (3.5) is used in this study to compute the shear thickening exponent of the hydrogel from experimentally captured δz– t response. Luo et al. [14] showed that out of the two model parameters in Eq. (3.2), only the shear thickening exponent n is a true material property independent of the shear acceleration. To characterize the viscous properties in acceleration-­independent conditions, the following equation of transient-­state viscosity was derived



µ=

α b n −1t0( ) ( C2n −1

n −1 / n +1)



(3.6)

where μ is the transient-state viscosity, and t0 is the duration of the transient shear stage. Notice that for an ideal Newtonian fluid, the transient-state viscosity is equal to the viscous coefficient. However, for a non-Newtonian fluid, the viscous coefficient is highly acceleration-dependent. The two material properties n and μ completely characterize the fluid-like response of hydrogels under transient shear deformation. In this study, an SHPB-based high strain rate shear experiment is used to experimentally measure these two properties of 20 wt.% PEGDA hydrogel. In a previous study [15], the authors developed a modified SHPB-based shear experiment to obtain the stress–strain response of polydimethylsiloxane (PDMS) elastomers under steady-state deformation at high strain rates. This experiment is adopted in this study to capture the viscoelastic material parameters under transient shear deformation (see the setup in Fig. 3.2). Similar to the traditional compression SHPB, it consists of two long and cylindrical maraging steel pressure bars called the incident and transmission bar, and a shorter striker bar. The incident and transmission bars have centrally bonded strain gages. A single-lap shear fixture is housed between the two pressure bars by inserting the incident bar and transmission bar ends to the Lap-1 and Lap-2 shear plates, respectively. The hydrogel specimen (45 mm × 30 mm × 3.5 mm) is mounted between the two lap shear plates using mounting plates. A low clamping force is maintained by controlling the screw thread rotation to avoid any damage to the specimen. The specimen gage surface (45 mm × 4.5 mm) is speckled using fine black spray paint (Krylon Products Group, Cleveland, OH). This speckled surface is imaged during the experiment using a highspeed camera (Phantom v710, Vision Research, Inc., Wayne, NJ). Compressed nitrogen in a gas chamber is used to propel the striker bar toward the incident bar. Upon impact, a characteristic trapezoidal stress/strain wave is generated in the incident bar, travelling at the speed of ~5000 m/s. On reaching the incident bar-Lap-1 interface, as the impedance of the hydrogel specimen is negligible as compared to the bar material, almost the entire incident strain signal is reflected (εI = εR). The incident trapezoidal wave is composed of a linear rise followed by a flat portion. As the velocity of incident bar end (interface between incident bar and Lap-1) is directly proportional to the strain signals (v = 2cbεI, where cb is the 1D stress wave velocity in bar), the Lap-1 is propelled with a linearly rising velocity for several microseconds, causing shear deformation in the sample. This kind of shear input provides the necessary constant acceleration boundary conditions of the start-up planar Couette flow problem in Eqs. (3.2) and (3.3). Because of a sliding fit between the incident bar and the Lap-1, the two separate after the initial acceleration phase, and Lap-1 is propelled with roughly a constant velocity for several 100 μs. In this study, only the initial constant acceleration (linear velocity rise) phase is of interest, during which the material undergoes transient shear deformation. Note that Lap-2 remains stationary because

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Fig. 3.2  Setup of the modified SHPB-based dynamic shear experiment

only a negligible portion of the incident stress wave is transmitted through the specimen. To capture the transient shear deformation, the high-speed camera is manually triggered using a 5 V transistor-transistor logic pulse. The camera captures the deformation at 97,000 frames per second (fps; one image every 10.31 μs); captured images are analyzed in the commercial Vic-2D 6 digital image correlation (DIC) software (Correlated Solutions, Inc., Irmo, SC). The use of a single camera and 2D DIC for this experiment is justified because under simple shear deformation, the sample undergoes no out-of-plane deformation (theoretically), leading to a strictly 2D deformation. The pixel resolution in DIC is 512 × 128, and the spatial resolution is approximately 21 pixels/mm. Three experiments are conducted to study repeatability of measurements.

3.3  Analytical Procedure and Results Three individual vertical line segments (L0, L1, and L2) in the specimen gage section displacement field (measured using DIC) are analyzed for each test using Eqs. (3.1)–(3.6). The snapshots of the specimen at different time steps with superposed x-displacement fields during transient shear deformation in a typical test are shown in Fig. 3.3a. During this time, the shear momentum is diffusing from top to the bottom end of the segment. Figure 3.3b shows the corresponding velocity versus time plot of the top end of the line segments (boundary velocity). Clearly, the velocity input is nearly linear (slope of the linear fit is equal to b (Eq. (3.6))), which concurs with the boundary conditions of transient shear in Eqs. (3.2) and (3.3). Thus, the equivalence of Eqs. (3.4) and (3.5) is valid. For the typical result shown in Fig. 3.3b, the constant acceleration b is 71,140 ms−2. Once the boundary condition is validated, the momentum diffusion lengths at different time steps can be calculated using Eq. (3.4) from the displacement profiles shown in Fig. 3.4a. Note that although the displacement profiles are obtained at every 10.31 μs time interval, only every third profile is shown in the figure for clarity. For computational convenience during trapezoidal numerical integration in Eq. (3.4), a hyperbolic tangent function is fit to the displacement profiles. This is particularly useful in cases where signal-to-­noise ratio is low. Note, the length of the line segment being analyzed is substituted for h, and xd(t, z) at any time step and location z with respect to the top end of the segment is the x-displacement of that datapoint. Figure 3.4b plots the so-­calculated momentum diffusion length versus time in a log–log scale. From Eq. (3.5), the slope of the fitted line of this response is n/(n + 1). In other words, the shear thickening exponent is



n=

mδ − t 1 − mδ − t 

(3.7)

where mδ − t is the slope of the momentum diffusion length-­time data (log–log) averaged across the three line segments. From Eqs. (3.6) and (3.7), the slope of momentum diffusion length-time data of 20% PEGDA is obtained as 0.79  ±  0.07 (p value