Advances in Mechanics of Time-Dependent Materials 3031224000, 9783031224003

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Advanced Structured Materials

Holm Altenbach Julius Kaplunov Hongbing Lu Masayuki Nakada   Editors

Advances in Mechanics of Time-Dependent Materials

Advanced Structured Materials Volume 188

Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials are reaching their limits in many applications, and new developments are required to meet the increasing demands on engineering materials. The performance of materials can be improved by combining different materials to achieve better properties than with a single constituent, or by shaping the material or constituents into a specific structure. The interaction between material and structure can occur at different length scales, such as the micro, meso, or macro scale, and offers potential applications in very different fields. This book series addresses the fundamental relationships between materials and their structure on overall properties (e.g., mechanical, thermal, chemical, electrical, or magnetic properties, etc.). Experimental data and procedures are presented, as well as methods for modeling structures and materials using numerical and analytical approaches. In addition, the series shows how these materials engineering and design processes are implemented and how new technologies can be used to optimize materials and processes. Advanced Structured Materials is indexed in Google Scholar and Scopus.

Holm Altenbach · Julius Kaplunov · Hongbing Lu · Masayuki Nakada Editors

Advances in Mechanics of Time-Dependent Materials

Editors Holm Altenbach Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für Maschinenbau Otto-von-Guericke-Universität Magdeburg, Germany Hongbing Lu University of Texas at Dallas Richardson, TX, USA

Julius Kaplunov Department of Mathematics Keele University Keele, UK Masayuki Nakada Materials System Res Lab Kanazawa Institute of Technology Hakusan, Ishikawa, Japan

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-031-22400-3 ISBN 978-3-031-22401-0 (eBook) https://doi.org/10.1007/978-3-031-22401-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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

The book presents a variety of recent theoretical and experimental techniques for modelling of advanced viscoelastic materials. The book has a significant interdisciplinary flavour and reports on important achievements of the international academic community. This is a collection of selected chapters reporting on the current trends in Mechanics of Time Dependent Materials. The authors are renowned experts in the field affiliated with research intensive universities in Europe, North America and Far East. The book covers a number of cutting-edge themes, such as characterization of linear and nonlinear mechanical behaviour of viscoelastic materials and their composites, taking into consideration finite deformations, dynamic loading, microstructure, phased transitions, along with failure and fracture phenomena. The contributions are inspired by advanced applications in modern technologies, e.g. injection moulding, extrusion etc. A variety of theoretical and experimental aspects addressed in the book will be of interest for a broad interdisciplinary audience, including but not restricted to mechanical, civil and chemical engineers, as well as material, bio and geoscientists. Postgraduate students in related areas will also find useful many of the chapters as additional literature sources. The volume is dedicated to the 70th anniversary of Prof. Igor Emri. Prof. Emri. is an internationally leading authority in mechanics of time-dependent materials and related topics. He originated numerous influential developments in this area and co-authored around 300 publications on the subject.

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Professor Emri was born in Musrka Sobota, Slovenia in 1952. He graduated from the Faculty of Mechanical Engineering at the University of Ljubljana in 1977. He was awarded a Ph.D. from California Institute of Technology, USA, in 1981. For a long time, Prof. Emri was affiliated with the University of Ljubljana. He was promoted there to a Full Professorship in 1996. Soon after, he established the Department of Mechanics of Polymers and Composites leading it until his retirement in 2016. Professor Emri is involved in a variety of important academic activities. In particular, he is one of the founders and Editor-in-Chief of the reputable international journal »Mechanics of Time Dependent Materials«, published by Springer-Nature. His outstanding achievements received a number of major awards and honours. He is an international member of USA National Academy of Engineering (NAE) and also a member of European Academy of Sciences and Arts, European Academy of Sciences, Slovenian Academy of Sciences and Arts and several others. Professor Emri served as the Chairman of the Science Europe Scientific Committee on Engineering and Technology (ENGITECH), the Co-Chairman of the Science Europe Scientific Advisory Committee, the President of the Society of Experimental Mechanics (SEM), and the President of the International Committee on Rheology (ICR). In 1993 I. Emri jointly with his academic mentor W. G. Knauss from the California Institute of Technology, have created the new research field called “Mechanics of Time-Dependent Materials—MTDM”. MTDM was first organized as Technical Division (TD) of SEM. Another of his key contribution to the community service is related to the organization of numerous international scientific events. In particular, he launched the series of major conferences on Mechanics of Time-Dependent Materials taking place all over the world since 1995. Fruitful less formal workshops on Advances in Experimental Mechanics regularly hosted by Emri’s research group in Slovenia are also worth to be mentioned. Research activities of Prof. Emri are mainly focused on mechanics of dissipative systems with emphasis on studying the effect of the rate of changing of thermomechanical boundary conditions on processes of structure formation of polymeric materials and their macro-, micro- and nanocomposites, as well as on the behaviour of solid granular systems. He has developed a new nonlinear viscoelastic constitutive

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model, known as the so-called Knauss-Emri model, which enables modelling of nonlinear behaviour of engineering polymers and composites subject to complex time-varying thermo-mechanical loading, and prediction of their long-time behavior (durability of polymer based products and structures). Professor Emri and his group have shown that macroscopic properties of polymeric materials and their composites, can be controlled and modified either by changing material initial kinetics, i.e. molecular mass distribution and topology of molecules, or by varying thermo-mechanical conditions, including pressure and temperature, or by high rate mechanical loading. The pioneering collaborative efforts by I. Emri together with W. G. Knauss and N. W. Tschoegl from the California Institute of Technologies resulted in an innovative theoretical–experimental approach for analysing the structure formation processes of multimodal polyamide materials under the influence of complex thermomechanical conditions. Emri’s findings were patented and implemented by BASF in the production of polyamides. Later it was found that by using these materials one can manufacture osseointegrable implants with a gradient structure that mimics properties of bones and may be used in dental and orthopaedic surgery. Among more recent Emri’s scientific results there is the development of sound insulation structures involving granular materials. The underlying theory is based on the mechanism of a “force-network” formation. His invention has a substantial potential to be implemented in various modern industries including mechanical, automotive, electrical, aerospace, railway, naval and civil engineering. This piece of work has been also patented. The list of a few selected publications by Prof. Emri illustrating a broad range of his research interests is given below. • Knauss W.G., Emri I. Non-linear Viscoelasticity Based on Free Volume Consideration. Computers & Structures, 1981, 13 (1), 123–128. • Knauss W.G., Emri I. Volume Change and the Nonlinearly Thermo-Viscoelastic Constitution of Polymers. Polymer Engineering & Science, 1987, 27 (1), 86–100. • Emri I., Tschoegl N.W. Generating Line Spectra from Experimental Responses. Part I: Relaxation Modulus and Creep Compliance. Rheologica Acta, 1993, 32 (3), 311–322. • Emri I., Tschoegl N.W. Generating Line Spectra from Experimental Responses. Part II: Storage and Loss Functions. Rheologica Acta, 1993, 32 (3), 322–327. • Tschoegl N.W., Emri I., Generating Line Spectra from Experimental Responses. Part III: Interconversion between Relaxation and Retardation Behavior. International journal of polymeric materials, 1992, 18, 117–127. • Emri I., Tschoegl N.W. Generating Line Spectra from Experimental Responses. Part IV: Application to Experimental Data. Rheologica Acta, 1994, 33 (1), 60– 70. • Tschoegl N.W., Knauss W.G., Emri I. Poisson’s Ratio in Linear Viscoelasticity– A Critical Review. Mechanics of Time-Dependent Materials, 2002, 6 (1), 3–51.

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• Tschoegl N.W., Knauss W.G., Emri I. The Effect of Temperature and Pressure on the Mechanical Properties of Thermo-and/or Piezorheologically Simple Polymeric Materials in Thermodynamic Equilibrium—A Critical Review. Mechanics of Time-Dependent Materials, 2002, 6 (1), 53–99. • Nikonov A., Davies A.R., Emri, I. The Determination of Creep and Relaxation Functions from a Single Experiment. Journal of Rheology, 2005, 49(6), 1193– 1211. • Emri I., Prodan T. A Measuring System for Bulk and Shear Characterization of Polymers. Experimental Mechanics, 2016, 46 (4), 429–439. • Knauss W.G., Emri I., Lu H. Mechanics of Polymers: Viscoelasticity, Springer Handbook of Experimental Solid Mechanics, 2008, 49–96. • Gergesova M., Zupanˇciˇc B., Saprunov I., Emri I. The Closed Form tTP Shifting (CFS) Algorithm. Journal of Rheology, 2011, 55(1), 1–16. • Emri I., Gonzalez-Gutierrez J., Gergesova M., Zupanˇciˇc B., Saprunov I. Experimental Determination of Material Time-Dependent Properties. In Hetnarski, R.B. (ed.). Encyclopedia of thermal stresses. Dordrecht: Springer Reference, 2014, 1494–1510. • Emri I., Voloshin A. Statics: Learning from Engineering Examples. New York: Springer, 2016. 570 p. • Aulova A., Govekar E., Emri I. Determination of Relaxation Modulus of TimeDependent Materials Using Neural Networks. Mechanics of Time-Dependent Materials, 2017, 21(3), 331–349. • Aulova A., Oseli A., Bek M., Prodan T., Emri I. Effect of Pressure on Material Properties of Polymers. In: Altenbach H. (ed.), Öchsner A. (ed.). Encyclopedia of continuum mechanics. Berlin; Heidelberg: Springer, 2018, 1–14. • Oseli A., Aulova A., Gergesova M., Emri I. Effect of Temperature on Material Properties of Polymers. In: Altenbach H. (ed.), Öchsner A. (ed.). Encyclopedia of continuum mechanics. Berlin; Heidelberg: Springer, 2018. 1–20. Magdeburg, Germany Keele, UK Richardson, USA Hakusan, Japan

Holm Altenbach Julius Kaplunov Hongbing Lu Masayuki Nakada

Contents

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Rheological Modeling—Historical Remarks and Actual Trends in Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holm Altenbach 1.1 Rheology as a Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Development of Rheology as an Independent Scientific Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Method of Rheological Modelling of Palmov . . . . . . . . . . . . . 1.4 Two-Dimensional Rheological Modelling . . . . . . . . . . . . . . . . . . . . 1.5 Advanced Rheological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Stieltjes Continued Fractions and Their Role in Determining Viscoelastic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Russell Davies and Faris Alzahrani 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Continuous Relaxation Spectrum and Its Moments . . . . . . . . 2.3.1 Unimodal Spectra with a Finite Number of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Unimodal Spectra with an Infinite Number of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Multi-modal Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Stieltjes Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The S-Series and S-Fraction . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Dirichlet Series and Discrete Spectra . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Spectral M-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Spectral P-Sets and Stieltjes Dictionaries . . . . . . . . . . . . . 2.6 Two Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 A Theoretical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Polybutadiene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 6 8 10 13 14 14 17 17 18 20 20 23 24 25 26 28 29 30 32 32 33

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2.7 Discrete Retardation Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aleksey D. Drozdov 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fitting of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 HA Gels with Hydrazine–Aldehyde Bonds . . . . . . . . . . . 3.3.2 PEG Gels Cross-Linked by HIP and CB[7] Bonds . . . . . 3.3.3 PAAm Gels Cross-Linked by HIP and CB[7] Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Igor Emri, a Student, a Colleague and a Friend . . . . . . . . . . . . . . . . . . Wolfgang G. Knauss References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation for Tensile Failure of Fiber-Reinforced Polymer Composites Based on Viscoelastic-Entropy-Damage Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jun Koyanagi 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Application of Thermodynamic Entropy for Continuum Damage Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Simulation for Discontinuous CFRP . . . . . . . . . . . . . . . 5.3.1 Layer-Wise Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Periodic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Algorithm of Viscoelastic-Entropy-Damage Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Investigation of the Nonlinear Viscoelastic Behavior of PMMA Near the Glass Transition Using the Spectral Hole Burning Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huiluo Chen, Sadeq Malakooti, Ren Yao, Stephanie L. Vivod, Gregory McKenna, and Hongbing Lu 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Experiment Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Mechanical Spectrum Hole Burning . . . . . . . . . . . . . . . . . 6.2.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34 36 36 39 39 41 48 48 51 52 54 55 57 63

65 65 68 70 70 72 75 78 78 80 81

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Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Linear Regime Determination . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Exploration of Pump Amplitude and Frequency for PMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Experiments with Tuned Parameters . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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Accelerated Testing Methodology for Life Prediction of Unidirectional CFRP Under Tension Load . . . . . . . . . . . . . . . . . . . . Masayuki Nakada and Yasushi Miyano 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Generalization of Time–Temperature Superposition Principle for Strength of CFRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Formulations Based on ATM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 CFRP Strands Employed as Unidirectional CFRP . . . . . 7.4.2 Test Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Viscoelasticity of Matrix Resins . . . . . . . . . . . . . . . . . . . . 7.5.2 Statistical Static Strengths of CFRP Strands . . . . . . . . . . 7.5.3 Statistical Creep Strengths of CFRP Strands . . . . . . . . . . 7.5.4 Statistical Fatigue Strengths of CFRP Strands . . . . . . . . . 7.5.5 Predictions of Long-Term Statistical Creep and Fatigue Strengths of CFRP Strands . . . . . . . . . . . . . . 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of Time–Temperature Superposition Principle for Polymer Lifetime Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Takenobu Sakai and Satoshi Somiya 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Time-Influence Factor Superposition Principle . . . . . . . . . . . . . . . . 8.3 Time–Heat Treatment Conditions Superposition Principle . . . . . . 8.4 Time–Fiber Volume Fraction Superposition Principle . . . . . . . . . . 8.5 Time–Crystallinity Superposition Principle . . . . . . . . . . . . . . . . . . 8.6 Time–Crystallinity–Fiber Volume Fraction Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90 92 96 97 99 99 101 103 107 107 109 109 109 111 113 113 115 115 118 121 121 122 123 127 129 132 135 136

Viscoelastic and Viscoplastic Behavior of Polymer and Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Kenichi Sakaue 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.2 Viscoelastic–Viscoplastic Constitutive Model . . . . . . . . . . . . . . . . 140

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9.2.1

Series-Connected Model of Viscoelastic and Viscoplastic Elements . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Constitutive Equation of the Viscoelastic Elements . . . . 9.2.3 Constitutive Equation of the Viscoplastic Elements . . . . 9.3 Viscoelastic–Viscoplastic Behavior of PBT Resin . . . . . . . . . . . . . 9.3.1 Viscoelastic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Viscoplastic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Viscoelastic–Viscoplastic Behavior of Short Glass Fiber-Reinforced PBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Test Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Prediction of Behavior of PBT Composite by Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Using Asymptotic Homogenization in Parametric Space to Determine Effective Thermo-Viscoelastic Properties of Fibrous Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. N. Vlasov, D. B. Volkov-Bogorodsky, and V. L. Savatorova 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Viscoelastic Fiber-Reinforced Composites. Maxwell’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Asymptotic Homogenization of the Equations with Complex Moduli in Parametric Space . . . . . . . . . . . . . . . . . . . 10.4 The Solution of the Problem on Microscale . . . . . . . . . . . . . . . . . . 10.5 Numerical Results and Their Analysis . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Biomechanical Modeling and Characterization of Cells . . . . . . . . . . . Arkady Voloshin 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Materials and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 HMSC Cell Culturing Conditions . . . . . . . . . . . . . . . . . . . 11.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Elastic, Viscoelastic, and Tensegrity Models . . . . . . . . . . . . . . . . . . 11.4.1 Elastic and Viscoelastic Models . . . . . . . . . . . . . . . . . . . . . 11.4.2 Modeling the Viscoelastic Cell Behavior Using Tensegrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Definitions of Variables and Constants Used in this Paper . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 140 141 142 142 142 145 145 145 149 151

153 153 155 157 162 167 169 169 173 174 175 175 175 176 176 179 181 185 187 187

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12 Thermo-Rheological Analysis and Kinetic Modeling of Thermal and Thermo-Oxidative Degradation of Polyethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leslie Poh, Qi Wu, Esmaeil Narimissa, and Manfred H. Wagner 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Differential Scanning Calorimetry . . . . . . . . . . . . . . . . . . . 12.2.4 Thermogravimetric Analysis . . . . . . . . . . . . . . . . . . . . . . . 12.2.5 Rheological Characterization . . . . . . . . . . . . . . . . . . . . . . . 12.3 Kinetic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Model-Free Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Model-Fitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Oxidative Induction Time in Differential Scanning Calorimetry and Thermogravimetric Analysis . . . . . . . . . 12.4.2 Thermal and Thermo-Oxidative Degradation of Unstabilized and Stabilized LDPE in Nitrogen and Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Kinetic Analysis of Non-isothermal TG Curves of Unstabilized and Stabilized LDPE . . . . . . . . . . . . . . . . 12.4.4 Thermorheological Analyses of Unstabilized and Stabilized LDPEs Using Time-Sweep Rheometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Quantitative Characterization of Cracks and Contact Stresses Using Photoviscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satoru Yoneyama and Masahisa Takashi 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Photoviscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Evaluating Rolling Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Evaluating Fracture Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.3 Introducing Natural Crack . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 Results for Stationary Cracks . . . . . . . . . . . . . . . . . . . . . . .

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191 192 194 194 194 195 195 196 196 198 199 199 199

201 203

206 209 211 215 215 217 219 219 221 221 223 229 230 231 232 233 234 234

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13.4.6 Results for Moving Crack . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

236 243 244 245

Chapter 1

Rheological Modeling—Historical Remarks and Actual Trends in Solid Mechanics Holm Altenbach

Abstract Rheology as a science was established approximately 100 years ago, however the first rheological models were developed much earlier. Now rheological models are widely used in the modeling of complex material behavior. Starting with the pioneering contributions of Bingham, Reiner and others to the “new scientific branch” in the 20th of the last century, the method of rheological modeling was applied to different materials. With the increasing use of plastics in the 1950th the method of rheological modeling became more and more popular. Later, such models were also used in modeling of materials with complex microstructure. In all cases the main focus was on the phenological description of the material behaviour and better fitting of experimental data. Below a brief introduction to the method of rheological models with historical remarks and some new applications are presented. Keywords Rheological models · Material behaviour · Continuum mechanics · Equivalence hypotheses

1.1 Rheology as a Science The word rheology is from ancient Greek (ρεω = ˙ flow, λoγ oζ = ˙ science). The “science of flow” is the science that deals with the deformation and flow behavior of matter and it is related to fluids, “soft solids” and materials under stresses beyond the elastic limit or yield point and at moderate temperatures. Rheology therefore includes sub-areas of the theory of elasticity, the theory of plasticity, the theory of visco-elasticity, fluid mechanics among others. It deals with problems of continuum mechanics as well as with the derivation of the necessary material laws from the micro- and nanostructure of different classes of condensed matter (e.g. macromolecular systems and suspensions). The phrase “panta rhei” (ancient Greek π αντ α ρει, H. Altenbach (B) Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität, Magdeburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_1

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H. Altenbach

which means everything flows) is an aphorism traced back to the Greek philosopher Heraclit. Following Giesekus (1994) rheology can be divided into four branches: • Phenomenological rheology (macrorheology) This branch describes the deformation and flow behavior of materials without considering the material structure. • Structural rheology (microrheology) The phenomena are explained here from the microscopic structure of the substances. • Rheometry This branch deals with measuring methods for determining the rheological properties. • Applied rheology Findings about rheological behavior flow into the design and development of products, technical processes and systems. Recently, high-temperature rheology was established, which means the upper temperature limit for measurements has been raised from 1600 ◦ C in platinum-rhodium crucibles to 1800 ◦ C in ceramic crucibles. Below, the main focus will be on the first two items. The design of water clocks in ancient Egypt was maybe the first rheological problem. Around 1600 BC, it was clear that the viscosity of water depends on the temperature and this dependency influences the water clocks. During this time, there were hardly any considerations about constitutive models in mechanics. In the 17th century, first studies of rheological questions started. For example, Hooke1 established the law of linear elasticity (Hooke’s law 1676/1678) and Newton2 introduced the viscosity for liquids and assumed a proportional relation between shear stress and shear strain rate. After introduction of the normal and the shear stresses σ, τ , the normal and the shear strains ε, γ , the elasticity modulus E and the dynamic viscosity η, the two laws can be given in modern notation as follows: σ = Eε τ = ηγ˙

Hookean law, Newtonian law.

(1.1) (1.2)

The (. ˙. .) means derivative w.r.t. time. A historical survey on rheological models and rheology is presented, for example, in Tanner (1985), Doraiswamy (2002). The first two rheological models (1.1) and (1.2) describe ideal constitutive behavior: both equations are linear and the material parameters can be estimated in simple tests. Robert Hooke (∗ 18 Julyjul. /28 Julygreg. 1635 in Freshwater, Isle of Wight; †3 March 1702jul. /14 Marchgreg. 1703 in London) English polymath. 2 Isaac Newton (∗ 25 December 1642jul. /4 January 1743greg. in Woolsthorpe-by-Colsterworth, Lincolnshire; †20 March 1726jul. /31 March 1717greg. in Kensington, Middlesex) English mathematician, physicist, astronomer, alchemist, and theologian. 1

1 Rheological Modeling—Historical Remarks and Actual Trends …

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Since the 17th century, we distinguish more classes and subclasses of rheological models. The following classification can be used: • ideal material behavior with the subclasses – rigid (non-deformable) or Euclidean3 solids Non-deformability means that any two material points of the body are always the same distance apart, regardless of external forces. Such solids were, for example, studied by Newton. – elastic linear and non-linear solids This type of deformable solids was introduced by Hooke and Boyle.4 For the linear-elastic case one needs only one material parameter characterizing the individual purely elastic response of a material on a load (Young’s modulus, probably introduced by Riccati5 25 years before Young,6 and discussed by Euler7 earlier). The complete set of equations of the theory of elasticity assuming isotropy, linear behaviour and small strains was presented by Cauchy,8 de Coulomb,9 Navier,10 Poisson,11 among others. The corresponding linear-elastic isotropic three-dimensional constitutive law for small strains reads in the modern (invariant) notation σ = λtr ε I + 2μεε (1.3) with σ , ε as the stress and the strain tensor, I as the second rank unit tensor, tr denotes trace (or first invariant) and λ, μ are the Lamé’s12 parameters. It is obvious that the number of constitutive parameters is increasing with the complexity of the rheological model. 3

Euclid of Alexandria (probably in the 3rd century BC lived in Alexandria) Greek mathematician. Robert Boyle (∗ 25 January 1626jul. /4 Februarygreg. 1627 in Lismore Castle, Lismore, County Waterford; †31 December 1691jul. /10 Januarygreg. 1692 in London) Anglo-Irish natural philosopher, chemist, physicist, and inventor. 5 Giordano Riccati (∗ 25 February 1709 in Castelfranco Veneto, Trévise; † 20 July 1790 in Trévise) first experimental mechanician to study material elastic moduli, son of the mathematician Jacopo Francesco Riccati. 6 Thomas Young (∗ 13 June 1773 in Milverton, Somersetshire; † 10 May 1829 in London), British polymath. 7 Leonhard Euler (∗ 15 April 1707 in Basel; †7 Septemberjul. /18 Septembergreg. 1783 in St. Petersburg) Swiss mathematician, physicist, astronomer, geographer, logician, and engineer. 8 Augustin-Louis Cauchy (∗ 21 August 1789 in Paris; † 23 May 1857 in Sceaux), French mathematician. 9 Charles Augustin de Coulomb (∗ 14 June 1736 in Angoulême; † 23 August 1806 in Paris), French officer, engineer, and physicist. 10 Claude Louis Marie Henri Navier (∗ 10 February 1785 in Dijon; † 21 August 1836 in Paris), French mechanical engineer, affiliated with the French government, and a physicist who specialized in continuum mechanics. 11 Siméon Denis Poisson (∗ 21 June 1781 in Pithiviers, Département Loiret; † 25 April 1840 in Paris), French mathematician, engineer, and physicist. 12 Gabriel Lamé (∗ 22 July 1795 in Tours; † 1 May 1870 in Paris) French mathematician. 4

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– inviscid or Pascalian13 fluids An inviscid fluid is a fluid with the viscosity equal to zero and so this model is an analogy to the Euclidean solid. Such fluids without viscosity were in the focus of Euler, D. Bernoulli14 and Venturi.15 – Newtonian fluids introduced not only by Isaac Newton, but also by Hagen,16 Poiseuille,17 Couette,18 Navier, Stokes,19 and others. The last two presented a first set of three-dimensional equations for linear-viscous Newtonian fluids describing experimental data in a proper manner. The corresponding linearviscous isotropic three-dimensional constitutive law for small strain rates reads in the modern (invariant) notation (Altenbach 2018) σ = (− p + λV tr D ) I + 2μV D with D=

(1.4)

1 ∇v + (∇v ∇v [∇v ∇v)T ]. 2

v denotes the velocity vector, p is the hydrostatic pressure and λV , μV are the Lamé’s parameters depending on the density and temperature in the case of viscous fluids. ∇ is the Nabla-operator and (. . .)T means “transposed”. If the Stoke’s condition (3λV + 2μV = 0) is valid, Eq. (1.4) degenerates to the case of an incompressible fluid. • linear visco-elasticity, which is characterized by such material behavior like creep and relaxation. It cannot be represented by elastic or viscous models alone and one should use more complex models (Christensen 1982). Various models have been

Blaise Pascal (∗ 19 June 1623 in Clermont-Ferrand; † 19 August 1662 in Paris) French mathematician, physicist, inventor, philosopher, writer, and Catholic theologian. 14 Daniel Bernoulli (∗ 29 Januaryjul. /8 Februarygreg. in Groningen; †17 March 1782 in Basel) Swiss mathematician and physicist. 15 Giovanni Battista Venturi (∗ 11 September 1746 in Bibbiano; †10 September 1822 in Reggio nell’Emilia) Italian physicist, diplomat, and historian of science. 16 Gotthilf Heinrich Ludwig Hagen (∗ 3 March 1797 in Königsberg, Prussia; †3 February 1884 in Berlin) German specialist in hydraulic engineering. 17 Jean Léonard Marie Poiseuille (∗ 23 April 1797 in Paris; †26 December 1869 in Paris) French physicist and physiologist. 18 Maurice Marie Alfred Couette (∗ 9 January 1858 in Tours; †18 August 1943 in Angers) French physicist known for his studies of fluidity. 19 George Gabriel Stokes (∗ 13 August 1819 in Skreen, County Sligo; †1 February 1903 in Cambridge) Irish-English physicist and mathematician. 13

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suggested by Maxwell,20 Thomson (later Lord Kelvin),21 and Poynting.22 Finally, Boltzmann23 introduced the superposition principle for this material behaviour. • generalized Newtonian materials, suggested for materials with a more complex behaviour, which was discussed by Schwedoff24 (colloids) or Bingham25 (paints). • non-linear viscosity, etc. Many scientists have important contributions to rheology and it is not surprising that many rheological models have been named after these scientists. Examples are the Hookean model (ideal elastic model), the Newtonian model (ideal viscous model), the St.-Venant model26 (ideal plastic model), the Prandtl27 model (ideal elastic and ideal plastic model in series), the Kelvin-Voigt28 model (ideal elastic and ideal viscous model in parallel, first presented by Meyer29 in 1874), the Maxwell model (ideal elastic and ideal viscous model in series), the Schwedoff model (Prandtl model in parallel with an ideal elastic model and added by an ideal elastic model in series), the Bingham model (ideal plastic and ideal viscous model in parallel), the Burgers30 model (Maxwell and Kelvin-Voigt model in series) model, among others (Reiner 1960).

James Clerk Maxwell (∗ 13 June 1831 in Edinburgh, Scotland, United Kingdom; †5 November 1879 in Cambridge, England, United Kingdom) Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, magnetism, and light. 21 William Thomson, 1st Baron Kelvin (∗ 26 June 1824 in Belfast, Ireland, United Kindom; †17 December 1907 in Largs, Scotland, United Kindom) British mathematician, mathematical physicist, and engineer. 22 John Henry Poynting (∗ 9 September 1852 in Monton, Lancashire, England, United Kingdom; †30 March 1914 in Birmingham, England, United Kingdom) English physicist. 23 Ludwig Eduard Boltzmann (∗ 20 February 1844 in Vienna, Austrian Empire; †5 September 1906 in Tybein, Triest, Austria-Hungary) Austrian physicist and philosopher. 24 Fëdor Nikiforovich Schwedov (∗ 14jul. /26greg. February 1840 in Kiliya, Russian Empire; †12jul. /25greg. December 1905 in Odessa, Russian Empire) Russian physicist. 25 Eugene Cook Bingham (∗ 8 December 1878 in Cornwall, Vermont, USA; †6 November 1945 in Easton, Pensilvania, USA) American chemist and pioneer of modern rheology. 26 Adhémar Jean Claude Barré de Saint-Venant (∗ 23 August 1797 in Villiers-en-Bière, Département Seine-et-Marne, France; †6 January 1886 in Saint-Ouen, Département Loir-et-Cher, France) French engineer, mathematician and physicist. 27 Ludwig Prandtl (∗ 4 February 1875 in Freising, Upper Bavaria, German Empire; †15 August 1953 in Göttingen, West Germany) German fluid dynamicist, physicist and aerospace scientist. 28 Woldemar Voigt (∗ 2 September 1850 in Leipzig, Saxonia; †13 December 1919 in Göttingen, Germany) German physicist. 29 Oskar Emil Meyer (∗ 15 October 1834, Varel, Grand Duchy of Oldenburg; †21 April 1909, Breslau, German Empire) German physicist. 30 Johannes Martinus Burgers (∗ 13 January 1895, Arnheim, Netherlands; †7 June 1981, Washington, D.C., USA) Dutch physicist. 20

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1.2 Development of Rheology as an Independent Scientific Branch In the late 1920s, the term “rheology” for the science that deals with the flow and deformation of matter was established by Bingham and Reiner31 (Giesekus 1994). For the chemist Bingham the necessity of a branch of mechanics that deals with rheological problems was obvious. The challenges of rheology required certain intersections with natural sciences and engineering. Bingham mentioned in a discussion with Reiner (Giesekus 1994): Here you, a civil engineer, and I, a chemist, work together on common problems. With the development of colloid chemistry, such a situation will appear more and more often. We must therefore establish a branch of physics that deals with such problems. On August 29, 1929, the Society of Rheology was founded under the leadership of Bingham in Columbus, Ohio (Doraiswamy 2002). The aim of research of the new scientific branch was formulated: with the help of methods of rheology the deformation and flow of matter can be investigated. However, there were some limitations at the beginning. For example, the flow of electrons and heat should be not included. This restriction was later lifted. The new rheological models make it possible to present qualitatively and quantitatively, and predict various types of flow and deformation behaviour. In addition, with the help of rheology many new investigations are possible in various areas of technology and science, e.g., in materials science, in geology and in food technology. Rheology and continuum mechanics are in a close interaction. Constitutive equations connect all macroscopic phenomenological variables describing the behavior of the continuum (Krawietz 1986). The constitutive equations established in continuum mechanics contain parameters or parameter functions which should be identified. One possible way for formulation constitutive equations is their representation by combining rheological models. In addition, within rheology, experimental methods for characterizing the flow of the materials are developed. In Tanner (1985), the following basics of continuum mechanics for rheology are presented: 1. 2. 3. 4. 5.

conservation of mass, stress concept, symmetry of stress tensor, stress equations of motion, and energy conservation.

These statements should be discussed. At first, there is a strong discussion concerning balance equations or conservation laws. It seems that there are three items which need further discussion: • The balances are more general in comparison with conservation laws (Altenbach 2018). But there are controversy statements: instead of two terms on the right hand side responsible for the changes of the balance properties one introduces Markus Reiner (∗ 5 January 1886 in Czernowitz, Austria-Hungaria; †25 April 1976 in Haifa, Israel) Israeli civil engineer and a major figure in rheology.

31

1 Rheological Modeling—Historical Remarks and Actual Trends …

7

three terms (Hutter and Jöhnk 2004). This concept allows to take into account more effects. However, maybe it is better to have only two terms (one related to surface and one to volume effects). • In some papers the argument that if there are no actions on the right side of the balance equation the balance equation is equal to zero. This means that the first integral of the left hand side is constant and we have a conservation of the conservation law. • In contrast we have arguments in Ruggeri (1989), Boillat and Ruggeri (1998), Müller and Ruggeri (1998), Müller (2014), Müller et al. (2020) stating that conservation laws are more general. At second, taking into account the progress w.r.t. generalized continua, see for example Altenbach et al. (2011), Eremeyev et al. (2013), Altenbach and Eremeyev (2011), Altenbach et al. (2013), the symmetry of the stress tensor cannot be always guaranteed. However, the assumption that the symmetry condition for the stress tensor (this condition can proven with the balance of moment of momentum) is valid simplifies the constitutive equations significantly (less number of equations and less parameters). The assumption plays the role of a constrain and we can connect rheo logy with continua based on this assumption in a simple manner. Otherwise, even if we can establish constitutive equations for continua with symmetric and antisymmetric stress tensors the identification effort for the additional constitutive parameters is increasing dramatically. At third, the second law of thermodynamics should be taken into account—it allows to distinguish physical admissible and not admissible constitutive equations. However, from the second law we cannot find the answer to the question: “How can one reflect the individual response of a material on acting loading(s)”. For many applications five balances (mass, momentum, moment of momentum, energy and entropy) are enough. If there are models taking into account different scales the number of balances can increase (Müller et al. 2017). At present, the rheological modelling of the material behavior is successfully applied to many practical problems, particularly in mechanical and civil engineering. The method has elements of the deductive and the inductive approach (Altenbach 2018). The straightforward combination of rheological elements by connecting them in parallel or in series allows the establishment of very complex models. However, the material parameters in many cases cannot estimated in simple basic tests—they are the result of curve fitting. Furthermore, the physical admissibility of complex rheological models is guaranteed if each individual rheological element is physically admissible. The axioms of rheology (Reiner 1960) result in further simplifications of the models. For example the axiom “Under the action of hydrostatic pressure, all materials behave in the same manner as perfectly elastic body” together with isotropy assumption allows a split of the constitutive equations into hydrostatic (volumetric) and deviatoric part. Using some equivalence hypotheses (Kolupaev 2018), one-dimensional rheological models can be generalized to three dimensions.

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1.3 The Method of Rheological Modelling of Palmov In Palmov (1998),32 the basic ideas concerning Palmov’s33 rheological modeling are presented. The starting point are the simple models of ideal behavior (elastic, viscous, plastic) introduced, for example, in Reiner (1960). For the three-dimensional case isotropy for the material behaviour was assumed. In addition, the split of the stress tensor σ into its hydrostatic part σm I and its deviatoric part s = σ − σm I with the second-order unit tensor I and the hydrostatic stress σm =

1 tr σ 3

is valid. A similar split can be performed for the strain tensor ε , whose deviatoric part is denoted by 1 e = ε − εII 3 with the volumetric strain ε = tr ε . Now we can formulate constitutive equations of the rheological element α for the stress deviator, the free energy F, and the entropy S taking into account the equipresence axiom of the theory of materials (Haupt 2004) s α = s α (Θ, ∇ Θ, ε, e ),

Fα = Fα (Θ, ∇ Θ, ε, e ),

Sα = Sα (Θ, ∇ Θ, ε, e ), (1.5) whereby the subscript α denotes the corresponding rheological element and the temperature Θ, the temperature gradient ∇ Θ, the volumetric strain ε, and the strain deviator e are arguments of the constitutive relations. There are two types of connections of rheological elements • connection in parallel: s=

n 

s α, e = e1 = . . . = eα = . . . = en,

α=1

F=

n  α=1

Fα , S =

n  α=1

Sα , (1.6)

• connection in series:

32

In 1976, the book was published in Russian by Nauka publisher. Vladimir Alexandrovich Palmov (∗ 7 July 1934 in Batumi, Soviet Union; †15 October 2018 in St. Petersburg, Russian Federation), Russian professor of mechanics.

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1 Rheological Modeling—Historical Remarks and Actual Trends …

s = s1 = . . . = sα = . . . = sn, e =

n 

eα,

α=1

F=

n 

9

Fα , S =

α=1

n  α=1

Sα . (1.7)

By analogy, also mixed models can be established. For example, a three-element model is given as two elements in series and a third in parallel to this s 1 = s 2 = s 12 , e 1 + e 2 = e 12 , e 12 = e 3 = e 123 , s 12 + s 3 = s 123 or as two elements in parallel and the third in series s 1 + s 2 = s 12 , e 1 = e 2 = e 12 , e 12 + e 3 = e 123 , s 12 = s 3 = s 123 . The next step is related to the introduction of the following basic elements: • Hookean element for linear elasticity with the shear modulus μ as material parameter s = 2μee , (1.8) • Newtonian element for linear viscosity with the viscosity coefficient ν s = 2νe˙

(1.9)

• and the St. Venant element for plasticity with the yield stress σy as material parameter:  σ ) < σy , e˙ = 0 if N (σ 1 (1.10) σ ) = σy , e˙ = s if N (σ λ σ ) is the norm of the stress tensor and the variable λ is referred to as where N (σ “plasticity factor”. • In addition, for the remaining part, i.e., the relation between the hydrostatic stress σm and the volumetric strain ε (with the stress tensor σ and the strain tensor ε ), we make use of the following constitutive equation for pure elastic behaviour, where the parameter K is the bulk modulus: σm = K ε.

(1.11)

By connecting the basic elements, one get complex visco-elastic, elasto-plastic, etc. models, discussed in Sect. 1.1. In addition, generalized models can be introduced • generalized Kelvin-Voigt model: n Kelvin-Voigt elements in series, • generalized Maxwell model: n Maxwell elements in parallel, • generalized Prandtl model: n Prandtl models in parallel, etc.

10

H. Altenbach

It is obvious, that if n → ∞, models with continuous spectra of properties can be defined. Further discussions concerning the extension of Palmov’s approach to large strains are given in Palmow (1984, 1997) . There are some open questions: • One of the basic assumptions is isotropy. The split of the stress tensor and the strain tensor into volumetric and deviatoric parts is unique. How can we extend the models to the anisotropic case? • The volumetric part in the constitutive equations is assumed to be purely elastic. There are experimental data contradicting this assumption (first axiom of rheology), see, for example, Bridgman (1949). • For rheological modelling, one can use both the Langrangian or the Eulerian description. In Palmov (1998), Bruhns (2020) are arguments for the Eulerian description. Concerning these three items further research efforts are necessary to give proper answers. Many papers on these topics were published recently, among them (Shutov and Kreißig 2008; Bröcker and Matzenmiller 2013, 2015; Bröcker 2014; Kießling et al. 2016; Seifert 2022).

1.4 Two-Dimensional Rheological Modelling In the 1980th Palmov’s approach was applied to two-dimensional continua, see, for example, Altenbach (1985). The following considerations were the starting point: • the governing equations are based on the direct approach of formulation plate theories, see, for example, Palmow and Altenbach (1982, 1984), and • three basic elements (ideal elastic plate, ideal viscous plate, ideal plastic plate) The main variables of the rheological plate models were • the stress resultants, i.e., the transverse force vector F and the moment tensor M , • the strains, i.e., the transverse shear strain vector γ and the tensor of the bending and torsional strains κ , and • an energetic variable, i.e., the free energy H . Following Palmow and Altenbach (1982), a new variable can be introduced G = M × n. This is the polar moment tensor (Aßmus et al. 2017). In this case the stress resultant can be calculated by G =< a · σ z · a >, where σ is the classical symmetric stress tensor, a is the first metric tensor, z is the coordinate in the transverse direction to the plate and < . . . > denotes the integration over the thickness of the plate-like body. It is obvious that G = G T .

1 Rheological Modeling—Historical Remarks and Actual Trends …

11

The two-dimensional rheological elements can be introduced: • ideal elastic plate With the Helmholtz free energy as a function of the kinematic variables one can calculate the time-derivative as ρ H˙ =

∂ρ H ∂ρ H · γ˙ + ·· μ˙ . μ ∂γγ ∂μ

(1.12)

Then the vector of the transverse forces and the polar moment tensor are F =

∂ρ H , ∂γγ

G=

∂ρ H . μ ∂μ

(1.13)

Note that the Helmholtz free energy in the simplest case is assumed to be a quadratic form 1 1 (1.14) ρ H = γ · Γ · γ + μ ·· (4)C · μ 2 2 with the transverse shear stiffness second-rank tensor Γ and the out-of-plane stiffness fourth-rank tensor (4)C Γ = Γ0 a ,

(4)

C = C1cc + C2 (aa 2a 2 + a 4a 4 ),

(1.15)

where in the case of the two-dimensional orthonormal coordinate system e α with α = 1, 2, the following notations are used (Einstein’s sum up rule is considered) a = e α e α = a 1 , a 2 = e 1 e 1 − e 2 e 2 , c = e 1 e 2 − e 2 e 1a 3 , a 4 = e 1 e 2 + e 2 e 1 . Finally, taking into account Eqs. (1.12)–(1.15) we obtain the constitutive equations for the stress resultants F = Γ0γ , G = (C1 − C2 )(aa ·· μ )aa + 2C2μ Assuming linear-elastic isotropic material behavior and solving boundary-value problems for the two-dimensional continuum and the plate-like body (Altenbach 1987) one gets the stiffness parameters C2 =

 π 2 1+ν Gh 3 , C1 = C2 , Γ0 = C2 . 12 1−ν h

It is obvious that C = C1 + C2 =

Eh 3 12(1 − ν 2

is the classical Kirchhoff bending stiffness (Timoshenko and Woinowsky-Krieger 1959).

12

H. Altenbach

• ideal viscous plate Now we assume the following expression for the Helmholtz free energy ρ H = ρ H (γ˙ , μ˙ ).

(1.16)

The force vector and the polar moment tensor can be now estimated as F = Γ˜0γ˙ , G = (C˜ 1 − C˜ 2 )(aa ·· μ˙ )aa + 2C˜ 2μ˙ ,

(1.17)

where Γ˜0 , C˜ 1 and C˜ 2 are material parameters (“viscosities”)of the two-dimensional continuum. The time-derivative of the Helmholtz free energy in this case is ρ H˙ =

∂ρ H ∂ρ H · γ¨ + ·· μ¨ . ∂ γ˙ ∂ μ˙

(1.18)

Combining Eqs. (1.16)–(1.18) we obtain ∂ρ H 1 ∂ρ H 1 Γ˜0γ˙ · γ˙ + (C˜ 1 − C˜ 2 )(aa ·· μ˙ )2 + C˜ 2μ˙ ·· μ˙ − · γ¨ − ·· μ¨ ≥ 0 2 2 ∂ γ˙ ∂ μ˙ and, finally, the second law of thermodynamics for the two-dimensional continuum results in Γ˜0 ≥, C˜ 1 − C˜ 2 ≥ 0, C˜ 2 ≥ 0. Assuming linear-viscous isotropic behavior the constitutive parameter are  π 2 1+κ ηh 3 , C˜ 1 = C˜ 2 , Γ˜0 = C˜ 2 C˜ 2 = 12 1−κ h Note that the identification procedure of the material parameters is similar to Altenbach (1987). • ideal plastic plate The third rheological model introduced here is the perfect-plastic two-dimensional continuum. It is obvious that we have to introduce “yield conditions” for both (the transverse force vector and the polar moment tensor) ⎧ F ) < FY , γ˙ = 0 , N (F ⎪ ⎪ ⎨ F , α ≥ 0, F ) = FY , γ˙ = αF N (F G ˙ , μ = 0 , N (G ) < G ⎪ Y ⎪ ⎩ G, β ≥ 0 G ) = G Y , μ˙ = βaa ·· devG N (G N denotes the norm, which is defined for the vector and the tensor in different manner

1 F ) = |F F |, N (devG G) = G ·· devG G. N (F devG 2

1 Rheological Modeling—Historical Remarks and Actual Trends …

13

It seems that there is no solution in the general case of a yield condition. A solution for rigid-plastic materials was presented in Palmov (1982). Assuming σ = σ 0 signz + n τ + τ n with the plane stress tensor σ 0 and the stress vector τ in the “thickness” direction we can estimate h2 G = σ0 , F = σh 4 Finally, we have the yield condition h2 F )2 < G Y , γ˙ = 0 , μ˙ = 0 , N (F 16 h2 F )2 = G Y , γ˙ = AαF F , μ˙ = βaa ·· devG G. G )2 + N (F N (G 16 G )2 + N (G

It is obvious that the plastic plate model cannot be formulated in the same way as done for elastic and viscous plates. The problem is coming from “no-thickness” in the case of two-dimensional plates. However, the plastic material is developing over the thickness (exceptional case is the rigid-plastic material behavior—but this is only a rough approximation of the real material behavior). The actual state of the art is given in Aßmus and Altenbach (2020).

1.5 Advanced Rheological Models Below an advanced rheological model based on linear and non-linear rheological models is introduced. In one of the practical applications the challenge was the development of phenomenological constitutive equations that describe inelastic material behaviour at elevated temperature and stresses below the yield limit. To characterize hardening, recovery, and softening processes, a fraction model with creep-hard and creep-soft constituents was introduced (Naumenko et al. 2011). The basic idea for modelling was the consideration that the material behaves like a binary mixture and can be modeled like a composite. The total stress σ is composed of the stress σh for the hard phase and the stress σs for the soft phase: σ = ηs σs + ηh σh ,

(1.19)

where ηs and ηh are the volume fractions of the inelastic soft and hard phase, respectively. Furthermore, the iso-strain assumption is fulfilled ε˙ = ε˙ s = ε˙ h .

(1.20)

14

H. Altenbach

The further development started with the introduction of constitutive laws for elasticity and inelastic behave. The details are presented in Naumenko et al. (2011). The advantages of the given rheological model are: • the main deformation mechanisms were reflected in a satisfying manner and • the extension to the three-dimensional was possible with the use of some equivalence hypotheses. Simulations based on the identification of all parameters and response functions (Eisenträger et al. 2018, 2018) are presented in Eisenträger et al. (2018, 2019). Note that such approach is also applicable to model the creep behaviour of thermoplastics, as done in Altenbach et al. (2015).

1.6 Summary and Outlook Rheology is a powerful tool, even for the modeling of the constitutive behavior of solids. More applications (binary mixtures, plastics, …) can be established. New challenges are related to the integration of the symbolic tensor calculus, the basics of continuum mechanics and the establishment of equivalence hypotheses.

References Altenbach H (1984) Die Grundgleichungen einer linearen Theorie für dünne, elastische Platten und Scheiben mit inhomogenen Materialeigenschaften in Dickenrichtung. Technische Mechanik 5(2):51–58 Altenbach H (1985) Zur Theorie der Cosserat-Platten. Technische Mechanik 6(2):43–50 Altenbach H (1987) Definition of elastic moduli for plates made from thickness-uneven anisotropic material. Mech Solids 22(1):135–141 Altenbach H (2018) Kontinuumsmechanik - Eine elementare Einführung in die materialunabhängigen und materialabhängigen Gleichungen, 4th edn. Springer, Berlin, Heidelberg Altenbach H, Eremeyev VA (eds) (2011) Generalized continua-from the theory to engineering applications, CISM International Centre for Mechanical Sciences, vol 541. Springer, Vienna Altenbach H, Maugin GA, Erofeev V (eds) (2011) Mechanics of generalized continua, advanced structured materials, vol 7. Springer, Berlin, Heidelberg Altenbach H, Forest S, Krivtsov A (eds) (2013) Generalized continua as models for materials with multi-scale effects or under multi-field actions, advanced structured materials, vol 22. Springer, Berlin, Heidelberg Altenbach H, Girchenko A, Kutschke A, Naumenko K (2015) Creep behavior modeling of polyoxymethylene (pom) applying rheological models. In: Altenbach H, Brünig M (eds) Inelastic behavior of materials and structures under monotonic and cyclic loading, advanced structured materials, vol 57. Springer, pp 1–15 Aßmus M, Altenbach H (2020) On viscoelasticity in the theory of geometrically linear plates. In: Altenbach H, Öchsner A (eds) State of the art and future trends in material modeling, advanced structured materials, vol 100. Springer International Publishing, Cham, pp 1–22

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Aßmus M, Eisenträger J, Altenbach H (2017) Projector representation of isotropic linear elastic material laws for directed surfaces. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 97(12):1625–1634 Boillat G, Ruggeri T (1998) On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin Mech Thermodyn 10(5):292–295 Bridgman PW (1949) The physics of high pressure. G. Bell and Sons, London Bröcker C (2014) Materialmodellierung für die simultane Kalt-/Warmumformung auf Basis erweiterter rheologischer Modelle. PhD thesis, Kassel, Universität, FB 15, Maschinenbau, Kassel Bröcker C, Matzenmiller A (2013) An enhanced concept of rheological models to represent nonlinear thermoviscoplasticity and its energy storage behavior. Contin Mech Thermodyn 25(6):749– 778 Bröcker C, Matzenmiller A (2015) An enhanced concept of rheological models to represent nonlinear thermoviscoplasticity and its energy storage behavior, Part 2: spatial generalization for small strains. Contin Mech Thermodyn 27(3):325–347 Bruhns OT (2020) History of plasticity. In: Altenbach H, Öchsner A (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg, pp 1129–1190 Christensen RM (1982) Theory of viscoelasticity-an introduction, 2nd edn. Academic Press, New York et al Doraiswamy D (2002) The origins of rheology: a short historical excursion. Rheol Bull 71(1):7–17 Eisenträger J, Naumenko K, Altenbach H (2018) Calibration of a phase mixture model for hardening and softening regimes in tempered martensitic steel over wide stress and temperature ranges. J Strain Anal Eng Des 53(3):156–177 Eisenträger J, Naumenko K, Altenbach H (2018) Numerical implementation of a phase mixture model for rate-dependent inelasticity of tempered martensitic steels. Acta Mech 229(7):3051– 3068 Eisenträger J, Naumenko K, Altenbach H (2019) Numerical analysis of a steam turbine rotor subjected to thermo-mechanical cyclic loads. Technische Mechnik 19(3):261–281 Eremeyev VA, Lebedev LP, Altenbach H (2013) Foundations of Micropolar Mechanics. SpringerBriefs in applied sciences and technology-continuum mechanics. Springer, Berlin, Heidelberg Giesekus H (1994) Phänomenologische Rheologie. Eine Einführung. Springer, Berlin, Heidelberg, New York Haupt P (2004) Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin, Heidelberg Hutter K, Jöhnk K (2004) Continuum methods of physical modeling-continuum mechanics, dimensional analysis, turbulence. Springer, Berlin, Heidelberg Kießling R, Landgraf R, Scherzer R, Ihlemann J (2016) Introducing the concept of directly connected rheological elements by reviewing rheological models at large strains. Int J Solids Struct 97– 98:650–667 Kolupaev VA (2018) Equivalent stress concept for limit state analysis, Advanced Structured Mechanics, vol 86. Springer, Cham Krawietz A (1986) Materialtheorie-Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin, Heidelberg Müller I, Ruggeri T (1998) Rational Extended Thermodynamics, 2nd edn. Springer, New York Müller WH (2014) An Expedition to Continuum Theory. Springer, Dordrecht Müller WH, Vilchevskaya EN, Weiss W (2017) Micropolar theory with production of rotational inertia: a farewell to material description. Phys Mesomech 20(3):250–262 Müller WH, Rickert W, Vilchevskaya EN (2020) Thence the moment of momentum. ZAMM-J Appl Math Mecha/Zeitschrift für Angewandte Mathematik und Mechanik 100(5):e202000,117 Naumenko K, Altenbach H, Kutschke A (2011) A combined model for hardening, softening, and damage processes in advanced heat resistant steels at elevated temperature. Int J Damage Mech 20(4):578–597 Palmov VA (1982) On the theory of Cosserat plates (in Russ.). In: Trudy LPI, vol 386, Leningrad, pp 3–8

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Palmov VA (1997) Large strains in viscoelastoplasticity. Acta Mech 125(1):129–139 Palmov VA (1998) Vibrations of elastic-plastic bodies (translated by Alexander Belyaev). Foundations of Engineering Mechanics. Springer, Berlin, Heidelberg Palmow WA (1984) Rheologische modelle für Materialien bei endlichen Deformationen. Technische Mechanik 5(4):20–31 Palmow WA, Altenbach H (1982) Über eine Cosseratsche Theorie für elastische Platten. Technische Mechanik 3(3):5–9 Reiner M (1960) Deformation, Strain and Flow: an Elementary Introduction to Rheology. H. K. Lewis, London Ruggeri T (1989) Galilean invariance and entropy principle for systems of balance laws. Contin Mech Thermodyn 1(1):3–20 Seifert T (2022) Models of cyclic plasticity for low-cycle and thermomechanical fatigue life assessment. In: Altenbach H, Ganczarski A (eds) Advanced Theories for Deformation, Damage and Failure in Materials, CISM international centre for mechanical sciences courses and lectures, vol 605. Springer, pp 177–234 Shutov AV, Kreißig R (2008) Finite strain viscoplasticity with nonlinear kinematic hardening: phenomenological modeling and time integration. Comput Methods Appl Mech Eng 197(21):2015– 2029 Tanner RI (1985) Engineering Reology. Clarendon Press, Oxford Timoshenko S, Woinowsky-Krieger S (1959) Theory of Plates and Shells. McGraw-Hill, New York

Chapter 2

On Stieltjes Continued Fractions and Their Role in Determining Viscoelastic Spectra A. Russell Davies and Faris Alzahrani

Abstract Regularity conditions are given under which the Laplace transform of the relaxation modulus of a viscoelastic material can be represented by a Stieltjes continued fraction. It is shown how this fraction generates a sequence of exponential decay modes for representing experimental stress relaxation data. The article addresses the theoretical underpinning of the process whereby the continuous relaxation spectrum is replaced by a set of discrete modes, as well as demonstrating the practical applicability of the process. On the theoretical side, we appeal to the Stieltjes moment problem and its elegant solution by Stieltjes, while on the practical side we choose Riesz bases for the continuous relaxation spectrum which generate spectral sets of discrete relaxation times for fitting experimental data. The associated discrete retardation spectrum and Prony series for the creep compliance is then obtained directly from the discrete relaxation spectrum.

2.1 Introduction The problem of determining the relaxation spectrum of a viscoelastic material from a stress relaxation experiment is essentially that of evaluating an inverse Laplace transform from inexact and incomplete data. The same is true for the problem of determining the retardation spectrum from creep data. Each problem represents an ill-posed inverse problem in which small perturbations in the data can lead to large errors in the spectrum. Given good enough data, the instability can be controlled by regularizing the inversion process, regularization being a trade-off between an accurate representation of the experimental data and stability of the spectrum. A. R. Davies (B) School of Mathematics, Cardiff University, Cardiff CF24 4AX, UK e-mail: [email protected] F. Alzahrani Department of Mathematics, King Abdul Aziz University, Jeddah, Saudi Arabia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_2

17

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A. R. Davies and F. Alzahrani

The general subject area of inverting data represented by ill-conditioned linear algebraic or integral equations commands an extensive literature spanning at least four decades. An early software package, Provencher (1982), marked a significant contribution. The early books by Tikhonov and Arsenin (1977), and Morozov (1984), still contain useful material, while the books by Hansen (2010), and Asch et al. (2016), offer new approaches to, and modern perspectives on, solving inverse problems. Early reviews on inverse and ill-posed problems in rheology have been given by Honerkamp (1989), Malkin (1990) and Friedrich et al. (1996), while the more specific literature on the determination of viscoelastic spectra may be divided into three categories. We select only a cross-section of references illustrating different approaches and observations in each case. • Recovery from dynamic mechanical data (storage and loss moduli). The majority of papers address this topic. See Anderssen and Davies (2001), Baumgaertel and Winter (1989), Cho (2010), Davies and Goulding (2012), Davies et al. (2016), Davies and Douglas (2020), Honerkamp and Weese (1989), Jensen (2002), Shanbhag (2020), Stadler and Bailly (2009), Stieltjes (2013). • Recovery from stress relaxation and creep data. See Dooling et al. (1997), Emri et al. (1993), Es-haghi and Gardner (2021), Liu (2001), Loy and Anderssen (2014), Shanbhag (2019). • Interconversion between linear viscoelastic material functions. See Anderssen et al. (2008), Kwon et al. (2016), Loy and Anderssen (2014), Lv et al. (2019), Mallat (2009), Morozov (1984), Provencher (1982), Tschoegl and Emri (1992). We note that interconversion need not be ill-posed. In this article, we shall be mainly concerned with the recovery of viscoelastic spectra from stress relaxation data. The Dutch mathematician Stieltjes (1894) contributed a remarkable body of work Stieltjes (1894, 1918) related to continued fractions which has not previously been fully exploited in the context of linear viscoelasticity. In the following, we shall relate his theoretical work to the problem of determining viscoelastic spectra from experimental data, by calling upon recent results on the mathematical properties of continuous spectra, Davies and Douglas (2020), and ideas from the sparse representation of signals (Mallat 2009).

2.2 Mathematical Background In an incompressible shear deformation, Boltzmann’s general linear integral model for viscoelastic materials relates the stress σ (t) to the strain-rate γ˙ (t) in the form  σ (t) =

t −∞

G(t − t  )γ˙ (t  )dt  ,

(2.1)

where G(t) denotes the relaxation modulus, which is a positive, monotonically decreasing and continuously differentiable function of time. The memory kernel,

2 On Stieltjes Continued Fractions and Their Role in Determining …

19

˙ m(t), of the material is defined by means of the first derivative as m(t) = −G(t), which, in keeping with the principle of fading memory, Saut and Joseph (1983), is ˙ also monotonically decreasing. This means that G(t) is monotonically increasing. Bernstein’s theorem (Bernstein 1928) states that successive derivatives of G(t) of all orders are alternately monotonically increasing and decreasing if and only if G(t) is the Laplace transform of a positive measure. Under this constraint G(t) is said to be completely monotone and may be written in the form  G(t) = G e +

∞

H (τ )e− τ t

0

dτ , τ

(2.2)

where G e ≥ 0 is a material constant, given by G e = lim G(t),

(2.3)

t→∞

and H (τ ) is the continuous relaxation spectrum associated with the range of relaxation times τ, 0 < τ < ∞. Equation (2.2) serves as a mathematical definition of the continuous spectrum. In keeping with Bernstein’s theorem, it will be assumed throughout that H (τ ) ≥ 0. The function G(t) − G e tends to zero as t → ∞, and has a Laplace transform Ge ˆ = G(s) − s



∞

[G(t) − G e ]e

−st



∞

dt =

0

0

H (τ )dτ . 1 + sτ

(2.4)

Equation (2.4) may also be written in terms of the Stieltjes transform of a measure μ Ge ˆ = G(s) − s



∞ 0

dμ(τ ) , s+τ

dμ(τ ) = τ −1 H (τ −1 )dτ.

(2.5)

The measure μ is a bounded, non-decreasing function of τ , and its relationship to the relaxation spectrum H exerts a constraint on both H and G: 

τ

μ(τ ) =

ρ −1 H (ρ −1 )dρ =

0



∞

G(0) − G e =

H (ρ)

0



∞

τ −1

H (ρ)

dρ , ρ

dρ < ∞. ρ

(2.6)

(2.7)

Writing s = iω, the complex modulus, G ∗ (ω), of the material may be found from (2.4) in the form ˆ G (ω) ≡ iω G(iω) = Ge + ∗



∞ 0

dτ iωτ H (τ ) . 1 + iωτ τ

(2.8)

20

A. R. Davies and F. Alzahrani

Whereas the relaxation modulus G(t) is usually measured as a function of time in a step-strain experiment, the real and imaginary parts of the complex modulus G ∗ (ω) are usually measured as a function of frequency, ω, in an oscillatory shear experiment. In theory, the spectrum H (τ ) can be determined from the relaxation modulus or from the complex modulus, i.e., a solution of Eq. (2.2) is also a solution of Eq. (2.8) and vice versa. That the Laplace transform of G(t) may be expressed as a Stieltjes transform has far-reaching consequences. There is some discussion of these in the book by Tschoegl (1989), but in this article we delve more deeply into the connection between the theory of linear viscoelasticity and the mathematical legacy of Stieltjes. In particular we shall discuss the relevance of Stieltjes series (S-series), the Stieltjes continued fraction (S-fraction) and the Stieltjes moment problem to the determination of viscoelastic spectra. Van (1993) has given an excellent review of Stieltjes’ work. Much important material can also be found in the classical works of Wall (1929) and Widder (1941).

2.3 The Continuous Relaxation Spectrum and Its Moments In Boltzmann’s formulation of linear viscoelasticity it is an inherent premise that, for shear deformations at fixed temperature and pressure, every material has an unique continuous relaxation spectrum. This is not the case for discrete spectra, as can be discerned immediately from the fact that continuous spectra can be discretized in a number of different ways (Baumgaertel and Winter 1992). We shall treat discrete spectra in later sections, but for the present we will focus on the continuous spectrum.

2.3.1 Unimodal Spectra with a Finite Number of Moments We define the negative moments of the spectrum, H (τ ), as  μk =

∞

τ −(k+1) H (τ )dτ =

0



∞

τ k dμ(τ ),

k = 0, 1, 2 . . . ,

(2.9)

0

and the non-negative moments as 

∞

νk = 0

 τ H (τ )dτ = k

∞

τ −(k+1) dμ(τ ),

k = 0, 1, 2, . . . .

(2.10)

0

Note that all the moments are strictly positive in sign, i.e., μk > 0 and νk > 0, k = 0, 1, 2, . . . . The terms ‘negative’ and ‘non-negative’ moments refer to the powers of τ in their integrals with respect to H . Furthermore, the negative moments of H are the non-negative moments of μ, and vice versa.

2 On Stieltjes Continued Fractions and Their Role in Determining …

21

It is important to realize that not every continuous spectrum possesses an infinite number of moments. However, every spectrum is normally endowed with at least three moments, namely μ0 , ν0 and ν1 . We have 

∞

μ0 =

H (τ )

0

dτ = G(0) − G e = lim G ∗ (ω) < ∞. ω→∞ τ

(2.11)

For a viscoelastic liquid we have G e = 0, with 

∞

G ∗ (ω) < ∞, ω→0 ω 0 ∞ G ∗ (ω) ν1 = τ H (τ )dτ = lim < ∞. ω→0 ω2 0

ν0 =

H (τ )dτ = lim

and

(2.12) (2.13)

For a viscoelastic liquid, the moment ν0 is just the zero shear-rate viscosity, η0 . Finally, if G(t) is m-times differentiable at t = 0, we have μk = (−1)k G (k) (0), k = 0, 1, . . . , m,

(2.14)

ˆ is m-times differentiable at whereas, with G e = 0, if the Laplace transform G(s) s = 0, we have (−1)k ˆ (k) νk = (2.15) G (0), k = 0, 1, . . . , m. k! Regularity constraints. In addition to the conditions that H (τ ) is positive and continuous on the interval 0 < τ < ∞, we impose the constraints (Davies and Douglas 2020, Theorem 5.1)   ∞  ∞  ∞   −iξ ln τ   dξ < ∞, (2.16) H (τ )d ln τ < ∞, H (τ )e d ln τ   −∞

−∞

−∞

and denote the space of all functions satisfying (2.16) as Lw . The second constraint in (2.16) is an integrability condition on the Fourier transform of H with respect to ln τ . Under the assumption H ∈ Lw , Davies and Douglas (2020) have identified a set of subspaces in which the continuous spectrum must reside. We consider the unimodal distribution 2τ . (2.17) h(τ ) = sech(ln τ ) = 1 + τ2 Then if H (τ ) = h m (ατ ), where α > 0 is a positive parameter, and m ≥ 1 is an integer, it is easy to see that H has precisely m negative moments and precisely m − 1 non-negative moments. For each integer m ≥ 1 let M0m denote the subspace of Lw defined by M0m = span{h m (ατ ) for all α > 0}.

(2.18)

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A. R. Davies and F. Alzahrani

Under the constraint (2.16), Davies and Douglas prove that the continuous spectrum satisfying Eq. (2.8) resides in the composite space M = M01 + M02 + M03 + . . . .

(2.19)

If m ≥ 3, then every function in M0m satisfies the constraints (2.11)–(2.12), i.e., it possesses all three moments μ0 , ν0 and ν1 . This is not the case if m = 1 or m = 2. However, it is possible to find functions in M01 and M02 which possess all three moments. Let β denote a free parameter in the range 0 < β < 1. For m = 1 and 2 we then define Mβm = span{φm (ατ ) for all α > 0},

where

2(1 + β ) τ , (1 + + β 2 τ 2 )(β 2 + τ 2 ) 4(1 + β 2 )2 τ 4 [Aτ 2 + B(1 + τ 2 )2 ] φ2 (τ ) = , (1 + τ 2 )2 (1 + β 2 τ 2 )2 (β 2 + τ 2 )2 φ1 (τ ) =

A=

(2.20)

2 2 3

τ 2 )(1

(1 + β 4 )(1 − β 2 )2 , (1 + β 2 )2 + 2β 2

B=

2β 2 [(1 + β 2 )2 − β 2 ] . (1 + β 2 )2 + 2β 2

(2.21)

(2.22)

As β → 0, A → 1 and B → 0, while as β → 1, A → 0 and B → 1. The following properties may be established: • By expanding φ1 and φ2 as partial fractions it can be shown that, for all 0 < β < 1, φ1 ∈ M01 and φ2 ∈ M02 . • For every value of β in the range 0 < β < 1, both φ1 and φ2 are non-negative unimodal functions which possess all three moments μ0 , ν0 and ν1 . • At the lower limit β = 0, we have φ1 (τ ) = h(τ ) and φ2 (τ ) = h 2 (τ ). At this limit φ1 has only one moment while φ2 has two. • At the upper limit β = 1, we have φ1 (τ ) = h 3 (τ ) and φ2 (τ ) = h 4 (τ ). At this limit all three moments exist, with φ1 ∈ M03 and φ2 ∈ M04 . The profiles of φ1 and φ2 are plotted in Fig. 2.1a and b, respectively, for the values β = 0, 0.25, 0.5 and 1. We deduce from the above discussion that physically acceptable continuous spectra in L2w reside in the subspace of M given by Mβ = Mβ1 + Mβ2 + M03 + M04 + · · · .

(2.23)

This subspace contains spectra with precisely m negative moments and precisely m−1 non-negative moments for all finite values of m ≥ 3. From our experience of working with experimental data we would anticipate that most continuous spectra, whether unimodal or multi-modal, have no more than about 10 moments. However, in the next section we meet a remarkable result.

2 On Stieltjes Continued Fractions and Their Role in Determining …

23

(b)

(a)

Fig. 2.1 a The basis functions φ1 (τ ). b The basis functions φ2 (τ ). β = 0 ( ), β = 0.25 ( β = 0.5 ( ), β = 1 ( ). The greater the value of β, the more rapid the decay when τ > 1

),

2.3.2 Unimodal Spectra with an Infinite Number of Moments Every spectrum H in Mβ has a family of close neighbours, Hε , not in Mβ , each of which has an infinite number of negative moments. We shall call Hε an ε-spectrum of H . For a given value of ε > 0, H determines Hε , while Hε determines G(t), so there is a sense in which the two spectra are equivalent. An ε-spectrum of H (τ ) is defined by Hε (τ ) = e−ε/τ H (τ ), t > 0, ε > 0,

Hε (0) = 0.

(2.24)

It follows immediately from the factor e−ε/τ that an ε-spectrum has an infinite number of negative moments, μk , k ≥ 0. On the other hand, Hε has the same number of nonnegative moments as H . Figure 2.2a shows the spectrum H = h 3 (τ ) in black, with its ε-spectrum for ε = 0.1 in blue. The ε-spectrum has an ε-modulus G ε (t), defined by  G ε (t) = G e + 0

∞

Hε (τ )e− τ t

dτ = G(t + ε), t ≥ 0. τ

(2.25)

The ε-modulus is just the regular modulus advanced in time by an amount ε. The ε-modulus has a practical significance as well as a theoretical one. In a practical stepstrain experiment, with step imposed at t = 0, machine inertia prohibits the accurate measurement of G(t) close to the origin. Accurate measurements of G(t + ε) are usually possible with ε = O(10−1 ) s.

24

A. R. Davies and F. Alzahrani (a)

(b)

Fig. 2.2 a Continuous spectrum H (τ ) = h 3 (τ ) ( ). ε-spectrum Hε (τ ) = h 3ε (τ ), ε = 0.1 ( 3-mode discrete spectrum ( ). b Corresponding exact modulus G(t) (  ) and G 3 (t) ( )

).

If Hε is an ε-spectrum of H then G ε (t) is completely monotone on the interval 0 < t < ∞. Consequently it is real analytic on an interval including −ε < t < ∞. G(t) = G ε (t − ε) is therefore determined by G ε . This result, of course, assumes exact data. When Hε is found from inexact sampled data, G ε will provide an estimate for G(t). We will demonstrate this in later sections.

2.3.3 Multi-modal Spectra The word mode is commonly used to describe different features and properties, and care must be taken to infer its meaning in different contexts. For example, the continuous spectrum for polybutadiene (see Davies and Douglas 2020, Fig. 2.3) has two distinct peaks, and is referred to as a bimodal spectrum. The number of basis functions in the space Mβ required to represent this bimodal spectrum is four. In our discussion above we describe these basis functions unambiguously as unimodal. The basis functions represent resolving modes.The sum in (2.23) represents a partition of the space Lw2 into subspaces of different resolving power, the resolution increasing with m, for m ≥ 3. Each peak in the true spectrum is resolved into modes of higher resolution. In the above, each mode represents a continuous distribution of relaxation times and decay rates. In Sect. 5 we will treat discrete modes depicting single relaxation times and decay rates, associated with an approximation of the relaxation modulus by a Dirichlet series. This attributes yet another meaning to the word mode.

2 On Stieltjes Continued Fractions and Their Role in Determining …

25

(b)

(a)

Fig. 2.3 a The memory kernel m(t) (  ) and m 4 (t) ( ), over the interval 0 ≤ t ≤ 60. b The memory kernel m(t) (  ) and m 3 (t) ( ), over the interval 0 ≤ t ≤ 4

2.4 The Stieltjes Moment Problem Every ε-spectrum has an associated measure μ(τ ) which is positive, non-decreasing, and defined by  μ(τ ) =

τ

ρ −1 Hε (ρ −1 )dρ,

dμ(τ ) = τ −1 Hε (τ −1 )dτ.

(2.26)

0

The measure determines an infinite sequence of positive numbers, μk , given by its non-negative moments, 

∞

μk =

τ k dμ(τ ),

k = 0, 1, 2, . . . ,

(2.27)

0

which are also the negative moments of the ε-spectrum. The Stieltjes moment problem addresses the inverse problem. It may be expressed in two parts: (i) Given an infinite sequence of positive numbers μk , k = 0, 1, 2, . . . , under what circumstances do they determine a positive, non-decreasing function μ(τ ) such that (2.27) holds? (ii) Under what circumstances is μ(τ ) uniquely determined by its non-negative moments?

26

A. R. Davies and F. Alzahrani

Let An and Bn denote the Hankel determinants of order n given by  . . . μn  . . . μn+1  .. ..  , n = 1, 2, 3, . . . . . .  . . . μ2n−1  (2.28) Stieltjes gave necessary and sufficient conditions for the existence of a solution to Part (i) of the moment problem. These conditions are that the Hankel determinants An and Bn are all strictly positive:   μ0   μ1  An =  .  ..  μn−1

 . . . μn−1  . . . μn  .. ..  , . .  μn . . . μ2n−2  μ1 μ2 .. .

 μ1 μ2  μ2 μ3  Bn =  . ..  .. .  μn μn+1

An > 0 and Bn > 0, n = 1, 2, 3, . . . .

(2.29)

The answer to Part (ii) may be given as follows. Let A0 = B0 = 1, and let cn be positive numbers defined by c2n−1 =

2 Bn−1 A2n , c2n = , n = 1, 2, 3, . . . . An An−1 Bn Bn−1

(2.30)

Then a necessary and sufficient condition for μ to beuniquely determined by its non-negative moments, given (2.29), is that the series cn diverges. It is not immediately obvious what relevance Stieltjes’ solution to the moment problem has to the practical determination of relaxation and retardation spectra from experimental data. Its significance will be revealed in the sections which follow.

2.4.1 The S-Series and S-Fraction For convenience we let G e = 0, the case for viscoelastic liquids. For solids, the constant G e > 0 is simply added to the Dirichlet series in Eq. (2.36) below. Consider the Stieltjes transform of μ given by 

∞

S(μ, z) = 0

dμ(τ ) . z+τ

(2.31)

This may be expanded as the formal series S(μ, z) ∼

μ0 μ1 μn μ2 − 2 + 3 + . . . (−1)n n+1 + . . . . z z z z

(2.32)

When the moments satisfy the Hankel conditions (2.29), the series (2.32) is known as a Stieltjes series or S-series.

2 On Stieltjes Continued Fractions and Their Role in Determining …

27

The transform (2.31) can also be expanded as a Stieltjes continued fraction or S-fraction, which takes the form 1

.

1

c1 z +

(2.33)

1

c2 +

1

c3 z + · · · + c2n +

1 c2n+1 z + . . .

 If the conditions (2.29) and (2.30) are satisfied, with cn divergent, then the Sfraction converges to S(μ, z), which is analytic everywhere in the complex z-plane apart from the negative real axis. When z is real-valued we shall write z = s. Since S(μ, s) = Gˆ ε (s), s > 0, we see that the Laplace transform of G ε (t) is completely determined by the negative moments of Hε . These moments, therefore, completely determine G ε (t) and, consequently, G(t). The truncated S-fraction ending in the single term c2n is called the 2nth convergent of the S-fraction. With z = s, the 2nth convergent is a rational function of the form pn−1 (s) , n = 1, 2, 3, . . . , Gˆ n (s) = μ0 qn (s)

(2.34)

where pn−1 (s) is a polynomial in s of degree n − 1, while qn (s) is a polynomial of degree n. The coefficients cn in (2.30) can be found by expanding (2.34) as a formal power series in s −1 , and matching its coefficients in sequence with those of S(μ, s) in (2.32). The S-series, therefore, determines the S-fraction, and vice versa (Tschoegl and Emri 1992). Stieltjes showed that the sequence of polynomials {qn (−s)}∞ n=0 form an orthogonal system, from which it can be deduced that the rational function (2.34) has simple poles and residues. The n distinct zeros of qn (s) are real and negative, and may be associated with reciprocal relaxation times: −1 , k = 1 . . . n. qn (s) = 0 when s = −τn,k

(2.35)

From the inverse Laplace transform of Gˆ n (s) in (2.34) we find a finite Dirichlet series approximation to G ε (t) = G(t + ε) in the form G n (t) =

n  k=1

gn,k e−t/τn,k ,

gn,k = μ0

−1 pn−1 (−τn,k ) −1 qn (−τn,k )

.

(2.36)

The convergence of the S-fraction (2.33) ensures that the sequence G n (t), n = 1, 2, 3, . . . , converges to the ε-modulus G ε (t) as n → ∞. The analytic continuation of G ε (t) to G(t) is then a formality. In practice, however, we seek numerical

28

A. R. Davies and F. Alzahrani

convergence to the sampled experimental data G(t + ε) to within a prescribed tolerance. We will demonstrate that an accurate approximation can be achieved by selecting relaxation times from a set of low order convergents. The polynomials p and q are most easily generated by a three-term recurrence relation. We introduce the sequence of positive numbers κn+1 = (cn cn+1 )−1 , n = 1, 2, 3, . . . , which can be expressed in terms of the Hankel determinants (2.28), as κ2n =

An−1 Bn An+1 Bn−1 , κ2n+1 = , n = 1, 2, 3, . . . . An Bn−1 An Bn

(2.37)

The appropriate recurrence relation for p is pn (s) = (s + κ2n+1 + κ2n+2 ) pn−1 (s) − κ2n κ2n+1 pn−2 (s), n = 1, 2, 3, . . . , (2.38) with p−1 (s) = 0 and p0 (s) = 1, while that for q is qn+1 (s) = (s + κ2n+1 + κ2n+2 )qn (s) − κ2n κ2n+1 qn−1 (s), n = 1, 2, 3, . . . , (2.39) with q0 (s) = 1 and q1 (s) = s + κ2 .

2.5 Dirichlet Series and Discrete Spectra Given sufficiently large n, the relaxation modulus in (2.2) can be approximated as accurately as we wish, in the uniform norm, by a Dirichlet series of the form G n (t) = G e +

n 

gk e−t/τk .

(2.40)

k=1

We demand that gn and τn are strictly positive, so that G n (t) is completely monotone. There is no unique way of constructing such a series, and fitting the experimental data for G(t) by different methods leads to different series which can represent the data equally well (Malkin and Masalova 2001). Ignoring the constant term, the series (2.40) is associated with the n-mode discrete spectrum Hn (τ ) =

n 

ηk δ(τ − τk ),

ηk = gk τk ,

(2.41)

k=1

where δ(.) denotes the Dirac delta-distribution. Various authors have commented on the non-uniqueness of discrete spectra. Malkin (2002) states that: “The discrete relaxation spectrum is just a convenient way of representing experimental data... It has no basic physical meaning”. A prevalent view is that no line spectrum—produced by whatever method—is ever the true spectrum. Attempts have been made to relate

2 On Stieltjes Continued Fractions and Their Role in Determining …

29

discrete and continuous spectra (Bae and Cho 2016; Baumgaertel and Winter 1992; Lv et al. 2019) without any rigorous underpinning. On the other hand, the PadéLaplace method, see, for example Es-haghi and Gardner (2021), relates the Laplace ˆ transform of the series (2.40) to a formal power series of G(s) and has certain features in common with the S-fraction approach described above. Such series have limited convergence properties, but can prove useful asymptotically. We shall exploit this connection in Sect. 2.5 (see Eq. (2.45)).

2.5.1 Spectral M-Sets In dealing with experimental data it is neither necessary nor desirable to seek representations (2.40) and (2.41) with n large. In the Stieltjes theory presented above, an n-mode spectrum (2.41) requires 2n moments to determine its 2n parameters {ηk , τk }nk=1 . The moments of Hε grow rapidly in size as n increases, leading to extremely large values in the Hankel determinants (2.28), with consequent numerical instabilities. We shall evade these instabilities by employing ideas from sparse representation theory and multiresolution analysis (Mallat 2009), thereby keeping the number of terms in (2.40) and (2.41) to a minimum. This is an example of regularization by low-dimensional subspace projection. Our aim is to associate with every continuous spectrum H in Mβ , a set of relaxation times defined by the moments of its ε-spectra. Every Hε generates a sequence of polynomials, qn (s), whose n zeros define a set of relaxation times {τn,k }nk=1 as shown by (2.35). The set may be written {τ1,1 ; τ2,1 , τ2,2 ; τ3,1 , τ3,2 , τ3,3 ; . . . },

(2.42)

where the discrete spectrum  η1,1 δ(τ − τ1,1 ) has the same two moments μ0 , μ1 as those of Hε , the spectrum 2k=1 η2,k δ(τ − τ2,k ) has the same four moments μ0 , .., μ3  as those of Hε , and the spectrum 3k=1 η3,k δ(τ − τ3,k ) has the same six moments μ0 , .., μ5 as those of Hε . The parameters ηn,k = gn,k τn,k and τn,k depend on ε. −ε/τ m Riesz bases. Let h m h (τ ). Consider a continuous ε (τ ) denote the ε-spectrum e m spectrum H (τ ) = Ch ε (ατ ), where C and α are positive constants. Its relaxation times are independent of C, and may be written

{α −1 τ1,1 ; α −1 τ2,1 , α −1 τ2,2 ; α −1 τ3,1 , α −1 τ3,2 , α −1 τ3,3 ; . . . },

(2.43)

where τn,k are the relaxation times associated with h m ε (τ ). It remains to choose a basis for continuous spectra in the subspaces Mβm , m = 1, 2 and M0m , m ≥ 3. We choose a Riesz basis which plays an important role in the theory of wavelets (Mallat 2009). As a Riesz basis in M0m we choose h m (2r τ ), r = 0, ±1, ±2, . . . , while in Mβm we replace h m by φm . Finally, we may associate with the subspace M0m a spectral M-set

30

A. R. Davies and F. Alzahrani m m Mm = M m 1 ∪ M2 ∪ M3 ∪ . . . , r Mm n = {2 τn,k , k = 1, .., n; r = 0, ±1, .., ±R}, n = 1, 2, 3, . . . . (2.44)

In (2.44), with r = 0, the τn,k are the relaxation times associated with h m ε (τ ) through its negative moments, and we refer to them as basic relaxation times. All other times in (2.44) are dyadic multiples of these basic times. The integer R may be chosen to restrict the range of times within the set. Similar spectral M-sets may be associated with the subspaces Mβm .

2.5.2 Spectral P-Sets and Stieltjes Dictionaries So far we have focused on negative moments of the ε-spectrum. The non-negative moments also play a role. We expand the Stieltjes transform of Hε as a formal power series in s:  ∞ Hε (τ )dτ (2.45) ∼ ν0 − ν1 s + ν2 s 2 − ν3 s 3 + . . . , 1 + sτ 0 where νk is a non-negative moment of Hε . Even if all the moments exist, such a series will not normally converge, and should be viewed as asymptotic. A few relaxation times can be found from the matching continued fraction 1

.

s

c1 +

(2.46)

s

c2 + c3 +

s c4 + . . .

Here the coefficients cn are again given by (2.30), where the μk in (2.28) are replaced by νk . It will be enough for our current purposes to restrict attention to the relaxation time defined by the 2nd convergent, which takes the value ν1 /ν0 . Further relaxation times may be found if sufficiently many moments exist. We may now associate with the subspace M0m , m ≥ 3, the spectral P-set Pm = {2r τ p,1 , r = 0, ±1, ±2, .., ±R},

(2.47)

where τ p,1 = ν1 /ν0 , and ν0 and ν1 are the first two non-negative moments of h m ε (τ ). Spectral P-sets may be defined in exactly the same way for the subspaces Mβm , m = 1 and 2.

2 On Stieltjes Continued Fractions and Their Role in Determining … Table 2.1 Basic relaxation times in four Stieltjes dictionaries with ε = 0.1 τ S1β=1/2 S3 S2β=1/2 τ1,1 τ2,1 τ2,2 τ3,1 τ3,2 τ3,3 τ p,1

0.833110 0.199912 1.057800 0.053983 0.273825 1.193031 2.741940

0.852370 0.213641 1.058237 0.055185 0.291312 1.180573 2.440251

0.868640 0.276991 1.111878 0.080596 0.359089 1.245067 1.945150

31

S4 0.889481 0.305740 1.111113 0.084727 0.390906 1.229001 1.741297

Stieljtes dictionaries. We reiterate our basic objective, namely, to assemble a Dirichlet series (2.40) with as few modes as possible which • accurately represents a set of measurements of the relaxation modulus G(t) derived from a stress relaxation experiment; • has a discrete spectrum related to the continuous spectrum of the material under investigation. Our strategy is to select elements from a Stieltjes dictionary of exponential modes associated with the function space Mβ in (2.23) in which the continuous spectrum resides. Every subspace is attributed a separate dictionary, although elements can be selected from more than one dictionary if required. We define a Stieltjes dictionary as (2.48) Sm = {e−t/τk : τk ∈ Mm ∪ Pm }. The M- and P-sets have been constructed in such a way that every element (exponential mode) in Sm is associated with an element of a sequence which either converges to G ε (t) for all t ≥ 0, or is asymptotic to G ε (t) as t → ∞. In Table 2.1 we list basic relaxation times for four dictionaries: S1β=1/2 , S3 , S2β=1/2 and S4 . In each column, the first six basic relaxation times are listed by row in the order shown in (2.42), while the bottom row shows the basic relaxation time τ p,1 . Maximal sparsity. The Morozov discrepancy principle (Morozov 1984), in its simplest form, states that when a model is fitted to data, the residual error (measured in some convenient norm) should not be less than the noise level in the data. This places a limit on the number of modes which can be fitted to a set of stress relaxation data. For an ill-posed problem, the number of modes acts as a regularization parameter. A representation of the data by the smallest number of modes consistent with a prescribed tolerance level, for example, a rms residual error of 1%, is said to be maximally sparse. Some authors refer to this as a parsimonious representation. Maximal sparsity depends on

32

A. R. Davies and F. Alzahrani

• the prescribed tolerance; • the dictionary or dictionaries from which the modes are chosen; • the time period over which the data is collected. A maximally sparse representation over a short time interval may not be maximally sparse over a longer time interval, and vice versa.

2.6 Two Case Studies We shall illustrate the process of selecting elements from a dictionary to fit two data sets. The first data set is taken from the sampled relaxation modulus of a theoretical continuous spectrum, while the second data set is the sampled relaxation modulus for polybutadiene determined from oscillatory shear data.

2.6.1 A Theoretical Spectrum We consider the continuous unimodal spectrum H (τ ) = h 3 (τ ) with its varepsilon spectrum h 3ε (τ ), ε = 0.1, shown in Fig. 2.2a. We compute its relaxation modulus G(t) to high precision, for values of t between 0 and 5 s, with a sampling interval of 0.2 s. The 26 values are shown in Fig. 2.2b. In addition we compute 25 values of the ε-modulus G ε (t) = G(t − ε) intermediate to those shown. In this case, we know in advance the appropriate dictionary from which to assemble the relaxation modulus G n (t). It is S3 with r = 0, so we may choose from the seven basic relaxation times in the second column of Table 2.1. The following simple search algorithm determines a maximally sparse G n (t) to within a prescribed tolerance (rms error-of-fit) of 1%. • Estimate the absolute values of the gradients of ln G ε at its end points t = 0 and t = 5, and evaluate their reciprocals. These reciprocals are 0.90 and 2.42, respectively. • Select the nearest relaxation times in the dictionary S3 shown in Table 2.1. These are τ1,1 = 0.852370 and τ p,1 = 2.440251. • With these two relaxation times and n = 2, fit the series (2.40) to the G ε data by linear least-squares. This results in an rms residual error of 5.8%, which is above the prescribed tolerance level. • Add a third relaxation time from the S3 list in Table 2.1, proceeding through the list, one at a time, keeping n = 3. In each case fit the series (2.40) to the G ε data, noting the rms residual error. A residual error of 0.7% is achieved when the value τ3,2 = 0.291313 is reached. This meets the prescribed tolerance. We have thus obtained a maximally sparse representation (2.40) with n = 3. In Fig. 2.2b the approximation G 3 (t − ε) is shown in red, compared with values of the exact modulus G(t) in black. At t = 0 the extrapolated value G 3 (−ε) provides an estimate for G(0). The estimate is G 3 (−ε) = 1.5703 where the exact value is G(0) = 1.5708.

2 On Stieltjes Continued Fractions and Their Role in Determining …

33

The corresponding 3-mode discrete spectrum is shown in Fig. 2.2a. This is the discrete representation for h 3 (τ ). The spikes representing h 3 (τ ) and h 3ε (τ ) occupy the same positions but have different weights (coefficients). The height of each spike shown represents the coefficient gk eε/τk . The corresponding spike for h 3ε (τ ) has height gk .

2.6.2 Polybutadiene The relaxation spectrum of a polybutadiene blend (PBD1) has been estimated by several methods from complex modulus data published by Honerkamp and Weese (1989). For a comparison of existing results see Davies et al. (2016), Davies and Douglas (2020) and Kedari et al. (2022). All authors agree on a bimodal continuous spectrum. Davies and Douglas (2020) show that this resolves into four continuous modes, as well as a 4-mode discrete spectrum. The 4-mode discrete spectrum was determined by a Levenberg-Marquardt Nonlinear Regression (LMNR) algorithm and gives rise to the following memory kernel (scaled by 10−3 ): m(t) = 1209e−t/0.303539 + 108e−t/2.242868 + 30e−t/10.051836 + 13e−t/24.723526 . Here time is measured in millisec. The data used to determine m(t) spanned a reciprocal-frequency range of 1–400 ms. However, the level of experimental noise in the data would suggest that values of m(t) at times greater than about 60 millisec cannot be considered significant. The memory kernel m(t) is matched over the interval 0 ≤ t ≤ 60 by the 4-mode Stieltjes kernel m 4 (t) = 1207.98e−t/0.298258 + 108.58e−t/2.222226 + 30.61e−t/9.960536 + 12.83e−t/25.017984 .

The relaxation times are the following dyadic multiples from Table 2.1: { 41 × 1.193031, 2 × 1.111113, 8 × 1.245067, 64 × 0.390906}, with associated dictionaries S1β=1/2 , S2β=1/2 and S4 . The rms-discrepancy between m(t) and m 4 (t) is less than 0.5%. Sampled values of m(t) are shown in black in Fig. 2.3a, with m 4 (t) plotted in red. The values of m(0) and m 4 (0) are identical, equal to 1360. Numerical experiments suggest that both m(t) and m 4 (t) are maximally sparse over the interval 0 ≤ t ≤ 60. The memory kernel m(t) is not maximally sparse over intervals 0 ≤ t  10. For example, over the interval 0 ≤ t ≤ 4, m(t) is matched by the 3-mode Stieltjes kernel m 3 (t) = 1213.89e−t/0.298258 + 103.77e−t/1.945150 + 53.90e−t/9.960536

34

A. R. Davies and F. Alzahrani

derived from S1β=1/2 and S2β=1/2 . The rms-discrepancy between m(t) and m 3 (t) over the interval 0 ≤ t ≤ 4 is less than 0.3%. Sampled values of m(t) are shown in black in Fig. 2.3b, with m 3 (t) plotted in blue. There is nothing to choose between m and m 4 in regard to fitting experimental data. However, the LMNR-spectrum offers no information about the continuous spectrum. In contrast, the relaxation times of m 4 come from the spectral sets for S1β=1/2 , S2β=1/2 and S4 , which establishes a direct link to the continuous spectrum.

2.7 Discrete Retardation Spectra Loy et al. (2015) give formulae for the interconversion of Prony series for relaxation and creep. Whereas their formulae can be applied directly to convert the discrete relaxation spectrum to a discrete retardation spectrum, in this final section we work within the framework of the S-fraction to obtain the equivalent result. Let G(t) and J (t) denote the relaxation modulus and creep compliance of the material, respecˆ tively, with Laplace transforms G(s) and Jˆ(s). They are related as follows: 

t

ˆ Jˆ(s) = 1. G(t − t  )J (t  )dt  = t, G(0)J (0) = 1, s 2 G(s)

(2.49)

0

To proceed, we choose to treat viscoelastic liquids and viscoelastic solids separately. Viscoelastic liquids. For liquids the constant G e is zero, and the n-mode modulus G n (t) in (2.40) has a corresponding (n − 1)-mode creep compliance Jn−1 (t) satisfying 

t

G n (t − t  )Jn−1 (t  )dt  = t, G n (0)Jn−1 (0) = 1, s 2 Gˆ n (s) Jˆn−1 (s) = 1.

0

Expressing the S-fraction for Gˆ n (s) as a rational function, we have Pn−1 (s) Q n (s) and Jˆn−1 (s) = 2 , where Gˆ n (s) = Q n (s) s Pn−1 (s) Pn−1 (s) =

n  k=1

ηk

 l=k

(1 + sτl ), ηk = gk τk , and Q n (s) =

(2.50)

(2.51)

n  (1 + sτk ). k=1

(2.52) As a property of the S-fraction, the polynomial Pn−1 (s) has n − 1 simple zeros which are real and negative and distinct from the zeros of Q n (s). They provide reciprocal retardation times, i.e., Pn−1 (s) = 0 when s = −λ−1 k , k = 1, . . . , n − 1.

(2.53)

2 On Stieltjes Continued Fractions and Their Role in Determining …

35

Expanding Jˆn−1 (s) as a partial fraction, and taking the inverse Laplace transform, we may write Jn−1 (t) in the form Jn−1 (t) = J0 +

ν0−1 t

+

n−1 

jk (1 − e−t/λk ), where

k=1

J0 = 1/G n (0) = 1/



gk , ν0 =



ηk , and jk = −λ2k

Q n (−λ−1 k )

 Pn−1 (−λ−1 k )

. (2.54)

Again, as a property of the S-fraction, there exists an ordering such that the relaxation and retardation times satisfy the interlacing property τ1 < λ1 < τ2 < λ2 < · · · < τn−1 < λn−1 < τn . The coefficients jk can be shown to be positive. Viscoelastic solids. For solids the coefficient G e is greater than zero, and the creep compliance acquires an additional mode. In (2.51), Gˆ n (s) is replaced by Gˆ n (s) − G e /s and Jˆn−1 (s) by Jˆn (s), where Q n (s) , Jˆn (s) = s P˜n (s)

P˜n (s) = s Pn−1 (s) + G e Q n (s),

(2.55)

and Pn−1 and Q n are given by (2.52). It is easy to show that the polynomial P˜n (s) has n simple zeros which are real and negative and distinct from the zeros of Q n (s), The retardation times, λ˜ k , are obtained from its zeros, i.e., P˜n (s) = 0 when s = −λ˜ −1 k , k = 1, . . . , n,

(2.56)

with an ordering τ1 < λ˜ 1 < τ2 < λ˜ 2 < · · · < λ˜ n−1 < τn < λ˜ n . Finally, the n-mode creep compliance takes the form Jn (t) = J0 +

n 

˜

jk (1 − e−t/λk ), where

k=1

J0 = 1/



gk and jk = −λ˜ k

Q n (−λ˜ −1 k ) . −1  P˜n (−λ˜ k )

(2.57)

36

A. R. Davies and F. Alzahrani

2.8 Summary By using mathematical properties of the continuous relaxation spectrum, we have shown how the Stieltjes continued fraction delivers a sequence of basic discrete relaxation times whose dyadic multiples may be assembled into spectral sets. Times from these spectral sets can then be used directly to fit stress relaxation data. The corresponding relaxation spectrum may be converted into a discrete retardation spectrum and Prony series for the creep compliance, if required. Choosing relaxation times from one or more spectral sets involves a grid search and a sequence of linear leastsquares fits to the data. A number of suitable grid search algorithms are readily available in the literature on machine learning, basis pursuit, or hyperparameter tuning. A major advantage of choosing relaxation times in this way is that it establishes a direct link to the continuous spectrum via the spectral sets selected.

References Anderssen RS Davies AR (2001) Simple moving average formulae for the direct recovery of the relaxation spectrum. J Rheol 45:1–27 Anderssen RS, Davies AR, de Hoog FR (2008) On the sensitivity of interconversion between relaxation and creep. Rheol Acta 47:159–167 Asch M, Bocquet M, Nodet M (2016) Data assimilation: methods, algorithms, and applications. SIAM, Philadelphia Bae JE, Cho KS (2016) A systematic approximation of discrete relaxation time spectrum from the continuous spectrum. J Non-Newton Fluid Mech 235:64–75 Baumgaertel M, Winter HH (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol Acta 28:511–519 Baumgaertel M, Winter HH (1992) Interrelation between continuous and discrete relaxation time spectra. J Non-Newton Fluid Mech 44:15–36 Bernstein SN (1928) Sur les fonctions absolument monotones. Acta Math 52:1–66 Cho KS (2010) A simple method for determination of discrete relaxation time spectrum Cho. Macromol Res 18:363–371 Davies AR, Goulding NJ (2012) Wavelet regularization and the continuous relaxation spectrum. J Non-Newton Fluid Mech 189:19–20 Davies AR, Anderssen RS, de Hoog FR, Goulding NJ (2016) Derivative spectroscopy and the continuous relaxation spectrum. J Non-Newton Fluid Mech 233:107–18 Davies AR, Douglas RJ (2020) A kernel approach to deconvolution of the complex modulus in linear viscoelasticity. Inverse Probl 36:015001 Dooling PJ, Buckley CP, Hinduja S (1997) An intermediate model method for obtaining a discrete relaxation spectrum from creep data. Rheol Acta 36:472–482 Emri I, Tschoegl NW (1993) Generating line spectra from experimental responses. 1: relaxation modulus and creep compliance. Rheol Acta 32:311–321 Es-haghi SS, Gardner DJ (2021) A critical evaluation and modification of the Padé-Laplace method for deconvolution of viscoelastic spectra. Molecules 26:4838. https://doi.org/10.3390/ molecules26164838 Friedrich C, Honerkamp J, Weese J (1996) New ill-posed problems in rheology. Rheol Acta 35:186– 193 Hansen PC (2010) Discrete inverse problems: insight and algorithms. SIAM, Philadelphia Honerkamp J (1989) Ill-posed problems in rheology. Rheol Acta 28:363–371

2 On Stieltjes Continued Fractions and Their Role in Determining …

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Honerkamp J, Weese J (1989) Determination of the relaxation spectrum by a regularization method. Macromolecules 22:4372–4377 Jensen EA (2002) Determination of discrete relaxation spectra using Simulated Annealing. J NonNewton Fluid Mech 107:1–11 Kedari SR, Gowtham A, Vemaganti K (2022) A hierarchical Bayesian approach to regularization with application to the inference of relaxation spectra. J Rheol 66:125. https://doi.org/10.1122/ 8.0000232 Kwon MK, Lee SH, Lee SG, ChoK S (2016) Direct conversion of creep data to dynamic moduli. J Rheol 60:1181–1197 Liu YK (2001) A direct method for obtaining discrete relaxation spectra from creep data. Rheol Acta 40:256–260 Loy RJ, Anderssen RS (2014) Interconversion relationships for completely monotone functions SIAM. J Math Anal 46:2008–2032 Loy RJ, Anderssen RS (2014) On the construction of Dirichlet series approximations for completely monotone functions. Math of Comput 83:835–846 Loy RJ, de Hoog FR, Anderssen RS (2015) Interconversion of Prony series for relaxation and creep J Rheol 59:1261 https://doi.org/10.1122/1.4929398 Lv H, Liu H, Tan Y, Sun Z (2019) Improved methodology for identifying Prony series coefficients based on continuous relaxation spectrum method. Mater Struct 52(4):1–13. https://doi.org/10. 1617/s11527-019-1386-1 Malkin AY (1990) Some inverse problems in rheology leading to integral equations. Rheol Acta 29:512–518 Malkin AY (2002) The sense of a relaxation spectrum and methods for its calculation. Vysokomol Soedin Ser B 44:1598. (Polym Sci Ser B 44:247) Malkin AY, Masalova I (2001) From dynamic modulus via different relaxation spectra to relaxation and creep functions. Rheol Acta 40:261–271 Mallat S (2009) A wavelet tour of signal processing. The sparse way. Academic Press, San Diego Mead DW (1994) Numerical interconversion of linear viscoelastic material functions. J Rheol 38:1769–1795 Morozov VA (1984) Methods for solving incorrectly posed problems. (translated by Nashed M Z). Springer, New York Park SW, Schapery R (1998) Methods of interconversion between linear viscoelastic material functions. Part I-a numerical method based on Prony series. Int J Solids Struct 36:1653–1675 Provencher SW (1982) A constrained regularization method for inverting data represented by linear algebraic or integral equations. Comput Phys Commun 27:213–227 Saut JC, Joseph DD (1983) Fading memory. Arch Ration Mech Anal 81:53–95 Shanbhag S (2019) pyReSpect: a computer program to extract discrete and continuous spectra from stress relaxation experiments. Macromol Theory Simul 28. https://doi.org/10.1002/mats. 201900005 Shanbhag S (2020) Relaxation spectra using nonlinear Tikhonov regularization with a Bayesian criterion. Rheol Acta 59:509–520 Stadler FJ, Bailly C (2009) A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data. Rheol Acta 48:33–49 Stieltjes TJ (1894) Recherches sur les fractions continues. Ann Fac Sci Univ Toulouse 8:1–122; 9A 1895 1-47. (Reprinted in Mem Acad Sci Paris 33, 1–196 and in Stieltjes 1918, 402–566) Stieltjes TJ (1918) Oeuvres completes, vol 2. Noordhoff, Groningen Takeh A, Shanbhag S (2013) A computer program to extract the continuous and discrete relaxation spectra from dynamic viscoelastic measurements. Appl Rheol 23. https://doi.org/10.3933/ applrheol-23-24628 Tikhonov AN, Arsenin VY (1977) Solutions to ill-posed problems. Winston, New York Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behaviour. Springer, Berlin, Heidelberg

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Tschoegl NW, Emri I (1992) Generating line spectra from experimental responses. 3: interconversion between relaxation and retardation behaviour. Int J Polym Mater 18:117–127 Van Aasche W (1993) The impact of Stieltjes work on continued fractions and orthogonal polynomials. In: van Dijk G (ed) Collected papers, vol I. Springer, Berlin, pp 5–37 Wall HS (1929) On the Padé approximants associated with the continued fraction and series of Stieltjes. Trans Am Math Soc 31:91–116 Widder DV (1941) The Laplace transform. Princeton University Press, Princeton

Chapter 3

Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds Aleksey D. Drozdov

Abstract The viscoelastic response of supramolecular gels (whose polymer networks are formed by chains bridged by supramolecular and dynamic covalent bonds) can be tuned in a rather wide interval by changes in chemistry of the surrounding solutions. This property plays an important role in biomedical applications of these materials for stem cell therapy, regenerative medicine, and neuromodulation. A simple model is developed in finite viscoelasticity of supramolecular gels. An advantage of the constitutive equations is that they involve only three material parameters. The ability of the model to describe experimental data is confirmed by fitting observations on hyaluronic acid gels, poly(ethylene glycol) gels, and poly(acrylamide) gels with reversible bonds in uniaxial tensile and compressive tests with large strains and in small-amplitude shear oscillatory tests.

3.1 Introduction The conventional approach to tuning the viscoelastic behavior of amorphous and semicrystalline polymers consists of changes of temperature T . As rearrangement of chains in these materials is a temperature-activated process, an increase in temperature leads to a strong acceleration of the relaxation processes. Knauss and Kenner (1980) have shown that changes in the content of moisture in hydrophilic polymers provide another way for modulation of their viscoelastic properties. The presence of water in a polymer network induces breakage of hydrogen bonds between hydrophilic side groups. Adsorption of water molecules on hydration shells activates rearrangement (breakage and reformation) of temporary junctions between segments of chains. A unified theory able to predict the effects of temperature and moisture on the time-dependent response of these materials was developed by Knauss and Emri (1981) and verified by several research groups (Ishisaka and Kawagoe 2004; Fabre et al. 2018; Vieira et al. 2020). A. D. Drozdov (B) Department of Materials and Production, Aalborg University, Fibigerstraede 16, Aalborg 9220, Denmark e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_3

39

40

A. D. Drozdov

A new approach has recently been suggested for modulation of the viscoelastic response of hydrogels with supramolecular and dynamic covalent bonds (Zhang and Khademhosseini 2017). As rearrangement (dissociation of re-association) of these bonds can be activated by the addition of a catalyst or ionization of bound charges, modulation of the time-dependent behavior of these gels can be performed by changes in the chemistry of the pre-gel solutions. Unlike traditional hydrogels, where polymer chains are connected in a network by covalent cross-links, chains in supramolecular gels are bridged by non-covalent bonds formed via supramolecular interactions (hydrogen bonding, hydrophobic forces, van der Waals interactions, metal–ligand coordination complexes, π -π interactions, and electrostatic forces (Dong et al. 2015; Li et al. 2020). Due to the reversible (dynamic) nature of these bonds, supramolecular gels demonstrate strongly viscoelastic and viscoplastic responses under loading. This opens new areas for their applications in targeted drug delivery (Bernhard and Tibbitt 2021), stem cell therapy (Yeom et al. 2015), regenerative medicine (Hoque et al. 2019), tissue engineering (Saunders and Ma 2019), soft bioelectronics (Yuk et al. 2019), and neuromodulation (Liu et al. 2019). Design of compositions and preparation routes for supramolecular gels with required mechanical and physical properties have recently become a focus of attention (Shigemitsu and Hamachi 2017; Jiang et al. 2019; Zhao et al. 2021). Tuning the viscoelastic behavior of hydrogels by an appropriate choice of reversible bonds with different rates of dissociation and re-association plays the key role in biomedical applications of these materials (Rosales and Anseth 2016; Levalley and Kloxin 2019; Lou et al. 2021; Chen et al. 2021). For this purpose, simple and reliable models are required for the time-dependent response of supramolecular gels that can fit experimental data in conventional tests with the help of a small number of adjustable parameters. The objective of this study is twofold: (i) to derive a simple model in finite viscoelasticity of supramolecular gels with three material constants only and (ii) to demonstrate its ability to describe experimental data in uniaxial tensile and compressive tests with large strains, as well as in small-amplitude shear oscillatory tests, in a unified manner. Constitutive models for the elastic, viscoelastic, and viscoplastic responses of hydrogels with supramolecular and dynamic covalent bonds were recently developed in Mao et al. (2017), Drozdov and deClaville (2018a, b), Lu et al. (2018), Yu et al. (2018), Drozdov et al. (2019), Lin et al. (2020), Saadedine et al. (2021), Drozdov and deClaville (2021), see also review (Guo and Long 2020). Unlike previous studies, where the viscoelastic behavior is described by a series of relaxation times (an analog of the generalized Maxwell model), we characterize the relaxation spectrum by two parameters only (the effective rate of dissociation of bonds γ and the dimensionless measure  of inhomogeneity of the polymer network).

3 Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds

41

To validate the model, we study experimental data on (I) Hyaluronic acid (HA) gels with supramolecular hydrazine–aldehyde bonds (Lou et al. 2021). The polymer network is formed by HA chains grafted with short poly(ethylene glycol) (PEG) chains that are functionalized with hydrazine and aldehyde moieties. Two sets of observations are analyzed on gels with short (39 kDa) and long (75 kDa) HA chains prepared in aqueous solutions (with mole fractions c ranging from 0 to 300 mM) of an organic catalyst that accelerates the hydrazone exchange. (II) Four-arm PEG gels dynamically cross-linked by hexanoate isoquinoline (HIP) and cucurbit[7]uril (CB[7]) motifs (Chen et al. 2021). Due to the presence of carboxyl groups (HIP) in the structure of host–guest bonds, the rates of their association and dissociation are strongly affected by pH of pre-gel solutions (which varies in the interval between 4 and 9). (III) Poly(acrylamide) (PAAm) gels with dynamic bonds (Chen et al. 2021). The network is formed by PAAm chains grafted with linear PEG chains that are functionalized with HIP-CB[7] motifs (degree of functionalization of PAAm chains equals 12%). Modulation of the time-dependent behavior is performed by changes in the mole fraction of catalyst (HA gels) and pH of aqueous solutions (PEG and PAAm gels). The aim of this study is to examine how these factors affect the shear modulus G and the coefficients γ and . The exposition is organized as follows. A model for the viscoelastic response of supramolecular gels is derived in Sect. 3.2. This model is applied to describe experimental data in small-amplitude shear oscillatory tests and uniaxial tensile and compressive tests with large strains in Sect. 3.3. Concluding remarks are formulated in Sect. 3.4.

3.2 Constitutive Model A gel is modeled as a two-phase medium consisting of a transient network of polymer chains and water molecules. Chains in the network are connected by dynamic bonds whose rearrangement (dissociation and re-association) is driven by thermal fluctuations. The initial state of a gel coincides with its undeformed as-prepared state (where stresses in all chains of the polymer network vanish) at a fixed temperature T0 . Adopting the affine hypothesis, we presume deformations of the network to coincide with macro-deformation of the gel. Transformation of the initial state into the actual state is described by the deformation gradient F. We focus on the analysis of “rapid" deformations whose rates exceed the rate of diffusion of water molecules through the gel. Under this condition, transport of water is disregarded, and the gel is treated as an incompressible material with

42

A. D. Drozdov

J = 1,

(3.1)

where J = det F stands for the determinant of the deformation gradient. Two types of chains are distinguished in the network: (i) active (both ends of a chain are connected to separate junctions) and (ii) dangling (an end of a chain detaches from the network due to the dissociation of an appropriate bond). When an end of an active chain separates from the network at some instant τ1 , the chain is transformed into the dangling state. When the free end of the dangling chain merges with the network at an instant τ2 > τ1 , this chain returns to the active state. The network consists of meso-regions with various activation energies for dissociation of bonds. The rate of breakage (separation of active chains from their junctions) in a meso-domain with activation energy u is governed by the Eyring equation  u  ,  = γ exp − kB T0 where γ is the attempt rate (the maximum rate of dissociation of dynamic bonds), and kB is the Boltzmann constant. Introducing the dimensionless energy v = u/(kB T0 ), we present this equality in the form (v) = γ exp(−v).

(3.2)

The inhomogeneity of the network is characterized by the probability density f (v) to find a meso-region with dimensionless activation energy v ≥ 0. With reference to the random energy model (Derrida 1980), the quasi-Gaussian formula is adopted for this function  v2  (3.3) f (v) = f 0 exp − 2 , 2 where the pre-factor f 0 is determined from the normalization condition ∞ f (v)dv = 1.

(3.4)

0

An advantage of Eq. (3.3) is that the function f (v) is characterized by the only parameter  > 0 that serves as a measure of inhomogeneity of the polymer network. The current state of a network is characterized by the function n(t, τ, v) that equals the number (per unit volume) of chains at time t ≥ 0 that have returned into the active state before instant τ and belong to a meso-domain with activation energy v. In particular, n(t, t, v) is the number of active chains in meso-domains with activation energy v at time t. The number of chains that were active at the initial instant t = 0 and have not separated from their junctions until time t reads n(t, 0, v). The number of chains that were active at t = 0 and detach from their junctions within the interval [t, t + dt] is given by −∂n/∂t (t, 0, v) dt. The number of dangling chains that return into the active state within the interval [τ, τ + dτ ] reads p(τ, v)dτ with

3 Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds

p(τ, v) =

 ∂n  (t, τ, v) . t=τ ∂τ

43

(3.5)

The number of chains (per unit volume) that merged (for the last time) with the network within the interval [τ, τ + dτ ] and detach from their junctions within the interval [t, t + dt] equals −∂ 2 n/∂t∂τ (t, τ, v) dtdτ . We suppose that the number of active chains (per unit volume) in meso-domains with various activation energies v remains constant, n(t, t, v) = Na f (v),

(3.6)

where Na is the total number of active chains per unit volume. Separation of active chains from their junctions is described by the following kinetic equations: ∂ 2n ∂n (t, τ, v) = −(v) (t, τ, v). ∂t∂τ ∂τ

∂n (t, 0, v) = −(v)n(t, 0, v), ∂t

(3.7)

Equations (3.7) imply that the rate of transformation of active chains into the dangling state (the rate of dissociation of temporary bonds) is proportional to the number of active chains in an appropriate meso-region. Integrating Eq. (3.7) with conditions (3.5) and (3.6), we find that ∂n (t, τ, v) = p(τ, v) exp[−(v)(t − τ )]. ∂τ (3.8) Inserting expressions (3.8) into the equality n(t, 0, v) = Na f (v) exp[−(v)t],

t n(t, t, v) = n(t, 0, v) + 0

∂n (t, τ, v)dτ ∂τ

and using Eq. (3.6), we find that p(t, v) = Na (v) f (v).

(3.9)

Combination of Eqs. (3.8) and (3.9) implies that ∂n (t, τ, v) = Na (v) f (v) exp[−(v)(t − τ )]. ∂τ

(3.10)

It is convenient to distinguish chains connected to the network before loading (τ = 0) and those attached the network under deformation (τ > 0). For a chain with τ = 0, the deformation gradient for elastic deformation is given by f0 that reads f0 (t) = F(t).

(3.11)

44

A. D. Drozdov

Presuming stress in a dangling chain to relax entirely before this chain joins the network at instant τ > 0, we calculate its deformation gradient by the following formula: (3.12) fτ (t) = F(t) · F−1 (τ ), where the dot stands for the inner product, and F−1 is the inverse of the tensor F. The corresponding Cauchy–Green tensors are given by b0 = f0 · f0 ,

bτ = fτ · fτ ,

(3.13)

where  denotes transpose. Denote by I01 , I02 and Iτ 1 , Iτ 2 the first two principal invariants of the tensors b0 and bτ , respectively. Their third principal invariant equals unity, (3.14) I03 = Iτ 3 = 1. For definiteness, the neo-Hookean equation is adopted for the mechanical energy of an active chain attached to the network at instant τ , w=

1 μ(Iτ 1 − 3), 2

(3.15)

where μ stands for the shear modulus of a chain. More sophisticated expressions for the function w were suggested and verified by comparison with observations in Drozdov and Christiansen (2013), Drozdov and deClaville (2018c). The strain energy density of the polymer network W equals the sum of mechanical energies stored in active chains, 1 W (t) = μ 2

∞



t

dv n(t, 0, v)(I01 (t) − 3) + 0

0

 ∂n (t, τ, v)(Iτ 1 (t) − 3)dτ . (3.16) ∂τ

The first term in Eq. (3.16) stands for the strain energy (per unit volume in the initial state) of chains that have merged with the network before deformation and remain active at time t. The other term expresses the energy of chains that have merged with the network at various instants τ ≤ t and have not separated from the network within the intervals [τ, t]. The functions I01 and Iτ 1 satisfy the differential equations: I˙01 = 2b0 : D,

I˙τ 1 = 2bτ : D,

(3.17)

where the superscript dot stands for the derivative with respect to time t, the colon denotes convolution, I is the unit tensor, and D is the rate-of-strain tensor D=

1 (L + L ), 2

L = F˙ · F−1 .

3 Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds

45

It follows from Eqs. (3.16) and (3.17) that the derivative of the function W with respect to time reads W˙ = μ

∞ 0

t   ∂n (t, τ, v)bτ dτ : D − D, dv n(t, 0, v)b0 + ∂τ

(3.18)

0

where D=

1 μ 2

∞ 0

  t ∂n (t, τ, v)(Iτ 1 (t) − 3)dτ ≥ 0 (v)dv n(t, 0, v)(I01 (t) − 3) + ∂τ 0

(3.19) is the rate of viscous dissipation of energy. The Clausius–Duhem inequality for isothermal deformation of an incompressible medium is given by (3.20) Q = −W˙ + T : D ≥ 0, where Q stands for internal dissipation per unit volume and unit time, T is the Cauchy stress tensor, and the prime denotes the deviatoric component of a tensor. It follows from Eqs. (3.18) and (3.19) that inequality (3.20) is fulfilled for an arbitrary deformation program, provided that ∞

t



dv n(t, 0, v)b0 +

T = − I + μ 0

0

 ∂n (t, τ, v)bτ dτ , ∂τ

(3.21)

where  stands for the unknown pressure (the Lagrange multiplier that supports the incompressibility condition (3.1)). Substitution of Eqs. (3.8) and (3.10) into Eq. (3.21) implies that ∞ T = − I + G



t

f (v)dv exp(−(v)t)b0 + 0

(v) exp(−(v)(t − τ ))bτ dτ



, (3.22)

0

where G = μNa stands for the shear modulus. An advantage of the constitutive relations (3.2), (3.3) and (3.22) is that they involve only three material constants: (i) G stands for the shear modulus of a gel (Eq. 3.22), (ii)  is a measure of inhomogeneity of the polymer network (Eq. 3.3), and (iii) γ denotes the rate of dissociation of dynamic bonds (Eq. 3.2). Under uniaxial tension of a specimen, the deformation gradient reads 1 F = λi 1 i 1 + √ (i 2 i 2 + i 3 i 3 ), λ

(3.23)

46

A. D. Drozdov

where λ = λ(t) stands for elongation ratio, and i m (m = 1, 2, 3) are unit vectors of a Cartesian frame. It follows from Eqs. (3.11)–(3.13) and (3.23) that 1 (i 2 i 2 + i 3 i 3 ), λ(t)  λ(t) 2 λ(τ ) bτ (t) = i1i1 + (i 2 i 2 + i 3 i 3 ). λ(τ ) λ(t) b0 (t) = λ2 (t)i 1 i 1 +

Insertion of these expressions into Eq. (3.22) implies that T = T1 i 1 i 1 + T2 (i 2 i 2 + i 3 i 3 ),

(3.24)

where ∞ T1 = − + Gλ

f (v)S1 dv,

2

T2 = − + Gλ

−1

0

∞ f (v)S2 dv,

(3.25)

0

and the functions S1 and S2 read t S1 (t, v) = exp(−(v)t) +

(v) exp(−(v)(t − τ ))λ−2 (τ )dτ,

0

t S2 (t, v) = exp(−(v)t) +

(v) exp(−(v)(t − τ ))λ(τ )dτ.

(3.26)

0

Equation (3.26) imply that these functions are governed by the differential equations S˙1 = (v)(λ−2 − S1 ),

S˙2 = (v)(λ − S2 ),

S1 (0, v) = S2 (0, v) = 1. (3.27) It follows from the equilibrium equation and the boundary condition at the lateral surface of a sample that T2 = 0. Combining this equality with Eq. (3.25) and introducing the engineering tensile stress by the formula T1 σ = , λ we find that

∞ σ =G 0

 S2  f (v) S1 λ − 2 dv. λ

(3.28)

3 Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds

47

At small strains, when λ = 1 +  with ||  1, Eqsuations (3.27) and (3.28) are simplified. Setting S2 = 1 − s2 , S1 = 1 − s1 , where |s1 |, |s2 |  1, and preserving the linear terms only, we conclude that S1 λ −

S2 = 3 + (s2 − s1 ). λ2

Inserting this expression into Eq. (3.28) and using Eq. (3.4), we find that 

∞

σ = G 3 −

f (v)(s1 − s2 )dv .

(3.29)

0

Disregarding terms of higher order of smallness in Eq. (3.27), we find that the functions s1 and s2 obey the equations s˙1 = (v)(2 − s1 ),

s˙2 = −(v)( + s2 )

(3.30)

with the initial conditions s1 (0, v) = s2 (0, v) = 0. It follows from Eq. (3.30) that the function 1 s = (s1 − s2 ) 3 is governed by the equation s˙ = (v)( − s),

s(0, v) = 0.

(3.31)

The solution of Eq. (3.31) reads t s(t, v) =

  (v) exp −(v)(t − τ ) (τ )dτ.

(3.32)

0

Substitution of Eq. (3.32) into Eq. (3.29) implies that 

∞

t (v) f (v)dv

σ (t) = E (t) − 0

  exp −(v)(t − τ ) (τ )dτ ,

(3.33)

0

where E = 3G stands for the Young’s modulus. Equation (3.33) describes the viscoelastic behavior of a gel under uniaxial tension (compression) with small strains. Its response under shear is determined by the same equation where E is replaced with the shear modulus G,

48

A. D. Drozdov

∞



σ (t) = G (t) −

t (v) f (v)dv

0

  exp −(v)(t − τ ) (τ )dτ ,

(3.34)

0

It follows from Eq. (3.34) that in a shear oscillatory test with a small amplitude 0 and angular frequency ω, (t) = 0 exp(ıωt), the storage, G  (ω), and loss, G  (ω), moduli are given by Drozdov and deClaville (2021) 

∞

G (ω) = G 0

ω2 f (v) 2 dv,  (v) + ω2



∞

G (ω) = G

f (v) 0

(v)ω dv.  2 (v) + ω2

(3.35) Eq. (3.35) coincide with the governing equations for the generalized Maxwell model.

3.3 Fitting of Experimental Data To examine the ability of the model to describe experimental data on supramolecular gels under various deformation programs, three sets of observations are analyzed.

3.3.1 HA Gels with Hydrazine–Aldehyde Bonds We begin with fitting observations (Lou et al. 2021) on hyalaronic acid (HA) gels cross-linked with hydrazine–aldehyde bonds. Experimental data in shear relaxation tests with small strains (transformed in Lou et al. (2021) into the dependencies of G  and G  on angular frequency ω by means of the standard interconversion method (Emri et al. 2005) on two HA gels with small (39 kDa) and large (75 kDa) molar masses of chains are depicted in Fig. 3.1a and b together with results of numerical analysis. As the rate of dissociation of hydrazine–aldehyde bonds γ is strongly affected by 2-(2-(2-(azidoethoxy)ethoxy)ethoxy)acetaldehyde, the data are reported on the gels synthesized in aqueous solutions with various mole fractions c of this catalyst. Each set of observations is fitted separately. The parameters γ and  are determined by the nonlinear regression method to minimize the expression 

G exp (ω) − G sim (ω)

2

 2 + G exp (ω) − G sim (ω) ,

3 Modulation of the Viscoelastic Response of Hydrogels with Supramolecular Bonds

a

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b

c

Fig. 3.1 a, b Storage modulus G  (•) and loss modulus G  (◦) versus angular frequency ω. Symbols: experimental data (Lou et al. 2021) on HA gels with hydrazine–aldehyde bonds (a—39 kDa, b—75 kDa) prepared in solutions with various mole fractions c mM of catalyst (blue—c = 0, green— c = 10, orange—c = 100, red—c = 300). c The rate of dissociation of bonds γ versus mole fraction of catalyst c. Circles: treatment of experimental data on HA gels with various molar masses of chains (blue—39 kDa, red—75 kDa). Solid lines: results of numerical analysis

where summation is performed over all frequencies ω under consideration, G exp , G exp are the storage and loss moduli measured in a test, and G sim , G sim are determined by Eq. (3.35). The shear modulus is calculated by the least-squares algorithm. The following conclusions are drawn from the numerical analysis: (i) The modulus G is practically independent of c (it varies in the interval between 246 and 261 Pa for HA gel with short chains and in the interval between 855 and 917 Pa for the gel with long chains). (ii) The best-fit values of  are not affected by mole fraction of the catalyst c (they equal 0.9 for the HA gels with short chains and 1.9 for the gels with long chains). (iii) The rate of dissociation of supramolecular bonds γ increases linearly with mole fraction of catalyst c in pre-gel solution. The experimental dependencies γ (c) are presented in Fig. 3.1c, where the data are approximated by the equation

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γ = γ0 + γ1 c

(3.36)

with the coefficients determined by the least-squares technique. (iv) An increase in molar mass of HA chains (from 39 to 75 Da) results in the growth of the shear modulus by a factor of 3.5, an increase in the measure of inhomogeneity of the polymer network  by twice, and the growth of the coefficient γ1 (characterizing sensitivity of the viscoelastic properties of HA gels to the catalyst) by a factor of 5. We proceed with matching observations of HA gels (molar mass 75 kDA) prepared in solution with mole fraction of catalyst c = 100 mM in uniaxial tensile test with strain rate ˙ = 0.1 s−1 and compressive test with strain rate ˙ = 0.17 s−1 (Lou et al. 2021). The experimental data are reported in Fig. 3.2 together with results of numerical analysis. We use the values γ and  determined by fitting observations in Fig. 3.1b. The best-fit value of the shear modulus G is found from the condition of minimum for the expression 2  σexp − σsim , where σexp is the stress measured in a test, σsim stands for prediction of Eq. (3.28), and summation is performed over all strains  at which observations are reported. Our analysis leads to the following conclusions: (i) The shear moduli found by fitting observations under tension and compression with finite strains (Fig. 3.2) are close to those determined by matching experimental data in small-amplitude shear oscillatory tests (Fig. 3.1). The value of G calculated by approximation of the stress–strain diagram in Fig. 3.2b

a

b

Fig. 3.2 a Tensile stress σ versus tensile strain . b Compressive stress σ versus compressive strain . Circles: experimental data (Lou et al. 2021) on HA gel with hydrazine–aldehyde bonds (75 kDa, 100 mM) in tensile and compressive tests. Solid lines: results of numerical analysis

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(G = 918 Pa) coincides with that in Fig. 3.1b. The value of G found by matching experimental data in Fig. 3.2a (G = 1257 Pa) is slightly higher, which may be explained by the inaccuracy of measurements. (ii) Covalently cross-linked hydrogels demonstrate a strong asymmetry in their mechanical response under tension and compression with large strains (Drozdov and Christiansen 2020). Figure 3.2 shows that this asymmetry disappears in gels with supramolecular bonds, which implies that their response under uniaxial compression can be predicted by using observations under tension.

3.3.2 PEG Gels Cross-Linked by HIP and CB[7] Bonds We now fit experimental data (Chen et al. 2021) on four-arm PEG gels dynamically cross-linked with hexanoate isoquinoline (HIP) and cucurbit[7]uril (CB[7]) bonds in small-amplitude shear oscillatory tests. Modulation of the viscoelastic response was performed by changing pH of aqueous solutions under preparation of the gels. As HIP motifs contain carboxylic groups, their ionization (induced by an increase in pH) affects the rate of rearrangement of CB[7] host–guest bonds. Experimental data in shear oscillatory tests are depicted in Fig. 3.3a together with the results of numerical analyses. Material parameters in Eq. (3.35) are determined by fitting each set of observations separately.

a

b

Fig. 3.3 a Storage modulus G  (•) and loss modulus G  (◦) versus angular frequency ω. Circles: experimental data (Chen et al. 2021) in shear oscillatory tests on four-arm PEG gels cross-linked by means of HIQ and CB[7] motifs in aqueous solutions with various pH (brown—pH = 4, red—pH = 5, orange—pH = 6, yellow—pH = 7, green—pH=8, blue—pH = 9). b Parameters γ and G versus pH. Circles: treatment of experimental data. Solid lines: results of numerical analysis

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The following conclusions are drawn: (i) Inhomogeneity of the polymer network (characterized by the coefficient ) is independent of pH. The parameter  varies in a narrow interval between 1.03 and 1.17 for all pHs under consideration. (ii) The rate of dissociation of supramolecular bonds γ decreases exponentially with pH. Figure 3.3b shows that the experimental dependence γ (pH) is adequately approximated by the equation log γ = γ0 − γ1 pH,

(3.37)

where the coefficients γ0 and γ1 are calculated by the least-squares method. (iii) The shear modulus G decreases slightly with pH. This decay is described in Fig. 3.3b by the following linear equation: G = G 0 − G 1 pH.

(3.38)

It may be explained by the fact that ionization of carboxyl groups in HIP motifs resists the formation of host–guest bonds between CB[7] residues.

3.3.3 PAAm Gels Cross-Linked by HIP and CB[7] Bonds To examine the effect of chemical structure of polymer chains on the viscoelastic response of supramolecular gels, we match observations (Chen et al. 2021) on poly(acrylamide) (PAAm) gels with the same type of dynamic bonds (HIP and CB[7]). The gels were synthesized in aqueous solutions with various pH by random copolymerization of AAm monomers and macromonomers prepared by functionalization of PEG chains with HIP and CB[7] motifs. PAAm is used in preparation because the stiffness of PEG gels (with the shear modulus of 1kPa) is insufficient for biomedical applications. Experimental stress–strain diagrams in uniaxial tensile and compressive tests (with strains rates ˙ = 0.167 s−1 under tension and ˙ = 0.011 s−1 under compression) on PAAm gels prepared in aqueous solutions with various pH are reported in Fig. 3.4a and b together with results of numerical analysis. Each set of observations is fitted separately by Eq. (3.28) with two coefficients, γ and G. To reduce the number of parameters, we find  = 3.0 in the approximation of observations at pH = 9 and use the same value at smaller pH. The effect of pH on the rate of dissociation of dynamic bonds γ and the shear modulus G is illustrated in Fig. 3.4C, where data are fitted by Eqs. (3.37) and (3.38). Figure 3.4a and b demonstrate good agreement between the experimental data and results of simulation. However, the scatter of data in Fig. 3.4C is relatively large. This may be explained by the use of non-standard samples (plates instead of dog-bone

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a

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b

c

Fig. 3.4 a Tensile stress σ versus tensile strain . b Compressive stress σ versus compressive strain . Circles: experimental data (Chen et al. 2021) in uniaxial tensile (a) and compressive (b) tests on poly(acrylamide) gel cross-linked by supramonomers with HIQ and CB[7] motifs in aqueous solutions with various pH (red—pH = 5.3, orange—pH = 6, yellow—pH = 7, green—pH = 8, blue— pH = 9). c Parameters γ and G versus pH. Symbols: treatment of experimental data in tensile (◦) and compressive (•) tests. Solid lines: results of numerical analysis

specimens) in tensile tests and a small thickness of disk-like specimens (3 mm) used in compressive tests. The following conclusions are drawn from Fig. 3.4: (i) The influence of pH of aqueous solutions on the rate of dissociation of host– guest bonds γ is practically independent of the structure of polymer network (the coefficient γ1 in Eq. (3.37) equals 0.75 for PEG gel and 0.91 for PAAm gel). (ii) An increase in molar mass of polymer chains leads to the growth of inhomogeneity of the network (the parameter  equals 1.1 for short PEG chains and 3.0 for long PAAm chains). This may be explained by the influence of entanglements formed between PAAm chains under preparation. (iii) The shear modulus G decreases slightly with pH in PEG gel and increases substantially (by a factor of 5) in PAAm gel. This difference can be explained

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Fig. 3.5 Storage modulus G  (•) and loss modulus G  (◦) versus angular frequency ω. Symbols: predictions of the model for shear oscillatory tests on PAAm gel cross-linked by supramonomers with HIQ and CB[7] motifs in aqueous solutions with various pH (red—pH = 5.0, orange—pH = 6, yellow—pH = 7, green—pH = 8)

by different preparation procedures for the gels: PEG gel was prepared by the association of macromonomers, whereas PAAm gel was synthesized by random copolymerization of monomers under UV irradiation by using a photoinitiator (2-hydroxy-4 -(2-hydroxyethoxy)-2-methylpropiophenone) whose efficiency increases strongly with pH. To demonstrate the predictive ability of the model, the response of PAAm gels in small-amplitude shear oscillatory tests is calculated numerically. The effect of pH on the dependencies G  (ω) and G  (ω) is illustrated in Fig. 3.5. This figure shows that the characteristic time for relaxation of the gel prepared in solution with pH = 5 is close to 1 min. The latter value appears to be optimal for regulation of the cell behavior in hydrogel scaffolds (Rizwan et al. 2021).

3.4 Conclusions A simple model is developed for the viscoelastic response of hydrogels with supramolecular bonds. An advantage of the model is that describes adequately experimental data in rheological (small-amplitude oscillatory tests) and mechanical (tension and compression with finite strains) experiments, on the one hand, and contains only three material constants: (i) the shear modulus of a gel G, (ii) a measure of inhomogeneity of the polymer network , and (iii) the rate of dissociation of dynamic bonds γ . The ability of the model to describe observations is confirmed by the analysis of experimental data on hyaluronic acid gels with hydrazine–aldehyde bonds, PEG gels cross-linked by hexanoate isoquinoline and cucurbit[7]uril motifs, and poly(acrylamide) gels dynamically cross-linked by PEG supramonomers. For all hydrogels under consideration, good agreement is demonstrated between the data

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and results of numerical analysis. The model can be applied for “quick-and-dirty” prediction of chemical compositions of supramolecular gels with required viscoelastic properties. Acknowledgements Financial support by Innovationsfonden (Innovation Fund Denmark, project 9091-00010B) is gratefully acknowledged.

References Bernhard S, Tibbitt MW (2021) Supramolecular engineering of hydrogels for drug delivery. Adv Drug Delivery Rev 171:240–256 Chen H, Zhang J, Yu W, Cao Y, Cao Z, Tan Y (2021) Control viscoelasticity of polymer networks with crosslinks of superposed fast and slow dynamics. Angew Chem Int Ed 60:22332–22338 Derrida B (1980) Random-energy model: limit of a family of disordered models. Phys Rev Lett 45:79–92 Dong R, Pang Y, Su Y, Zhu X (2015) Supramolecular hydrogels: synthesis, properties and their biomedical applications. Biomater Sci 3:937–954 Drozdov AD, Christiansen J deC (2013) Stress–strain relations for hydrogels under multiaxial deformation. Int J Solids Struct 50:3570–3585 Drozdov AD, Christiansen J deC (2020) Tension-compression asymmetry in the mechanical response of hydrogels. J Mech Behav Biomed Mater 110:103851 Drozdov AD, deClaville CJ (2018a) Double-network gels with dynamic bonds under multi-cycle deformation. J Mech Behav Biomed Mater 88:58–68 Drozdov AD, deClaville CJ (2018b) Nanocomposite gels with permanent and transient junctions under cyclic loading. Macromolecules 51:1462–1473 Drozdov AD, deClaville CJ (2018c) Time-dependent response of hydrogels under multiaxial deformation accompanied by swelling. Acta Mech 229:5067–5092 Drozdov AD, deClaville CJ (2021) Structure-property relations in linear viscoelasticity of supramolecular hydrogels. RSC Adv 11:16860–16880 Drozdov AD, deClaville CJ, Dusunceli N, Sanporean C-G (2019) Self-recovery and fatigue of double-network gels with covalent and non-covalent bonds. J Polym Sci B: Polym Phys 57:438– 453 Emri I, von Bernstorff BS, Cvelbar R, Nikonov A (2005) Re-examination of the approximate methods for interconversion between frequency- and time-dependent material functions. J NonNewtonian Fluid Mech 129:75–84 Fabre V, Quandalle G, Billon N, Cantournet S (2018) Time-temperature-water content equivalence on dynamic mechanical response of polyamide 6,6. Polymer 137:22–29 Guo Q, Long R (2020) Mechanics of polymer networks with dynamic bonds. Adv Polym Sci 285:127–164 Hoque J, Sangaj N, Varghese S (2019) Stimuli-responsive supramolecular hydrogels and their applications in regenerative medicine. Macromol Biosci 19:1800259 Ishisaka A, Kawagoe M (2004) Examination of the time-water content superposition on the dynamic viscoelasticity of moistened polyamide 6 and epoxy. J Appl Polym Sci 93:560–567 Jiang Z, Bhaskaran A, Aitken HM, Shackleford ICG, Connal LA (2019) Using synergistic multiple dynamic bonds to construct polymers with engineered properties. Macromol Rapid Comm 40:1900038 Knauss WG, Emri IJ (1981) Non-linear viscoelasticity based on free volume consideration. Comput Sruct 13:123–128 Knauss WG, Kenner VH (1980) On the hygrothermomechanical characterization of polyvinyl acetate. J Appl Phys 51:5131–5136

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Levalley PJ, Kloxin AM (2019) Chemical approaches to dynamically modulate the properties of synthetic matrices. ACS Macro Lett 8:7–16 Li Y, Zhu C, Dong Y, Liu D (2020) Supramolecular hydrogels: mechanical strengthening with dynamics. Polymer 210:122993 Lin J, Zheng SY, Xiao R, Yin J, Wu ZL, Zheng Q, Qian J (2020) Constitutive behaviors of tough physical hydrogels with dynamic metal-coordinated bonds. J Mech Phys Solids 139:103935 Liu Y, Liu J, Chen S, Lei T, Kim Y, Niu S, Wang H, Wang X, Foudeh AM, Tok JB-H, Bao Z (2019) Soft and elastic hydrogel-based microelectronics for localized low-voltage neuromodulation. Nat Biomed Eng 3:58–68 Lou J, Friedowitz S, Will K, Qin J, Xia Y (2021) Predictably engineering the viscoelastic behavior of dynamic hydrogels via correlation with molecular parameters. Adv Mater 33:2104460 Lu H, Wang X, Shi X, Yu K, Fu YQ (2018) A phenomenological model for dynamic response of double-network hydrogel composite undergoing transient transition. Comp B 151:148–153 Mao Y, Lin S, Zhao X, Anand L (2017) A large deformation viscoelastic model for double-network hydrogels. J Mech Phys Solids 100:103–130 Rizwan M, Baker AEG, Shoichet MS (2021) Designing hydrogels for 3D cell culture using dynamic covalent crosslinking. Adv Healthcare Mater 10:2100234 Rosales AM, Anseth KS (2016) The design of reversible hydrogels to capture extracellular matrix dynamics. Nat Rev Mater 1:15012 Saadedine M, Zairi F, Ouali N, Tamoud A, Mesbah A (2021) A micromechanics-based model for visco-super-elastic hydrogel-based nanocomposites. Int J Plast 144:103042 Saunders L, Ma PX (2019) Self-healing supramolecular hydrogels for tissue engineering applications. Macromol Biosci 19:1800313 Shigemitsu H, Hamachi I (2017) Design strategies of stimuli-responsive supramolecular hydrogels relying on structural analyses and cell-mimicking approaches. Acc Chem Res 50:740–750 Vieira de Mattos DF, Huang R, Liechti KM (2020) The effect of moisture on the nonlinearly viscoelastic behavior of an epoxy. Mech Time-Depend Mater 24:435–461 Yeom J, Kim SJ, Jung H, Namkoong H, Yang J, Hwang BW, Oh K, Kim K, Sung YC, Hahn SK (2015) Supramolecular hydrogels for long-term bioengineered stem cell therapy. Adv Healthcare Mater 4:237–244 Yu K, Xin A, Wang Q (2018) Mechanics of self-healing polymer networks crosslinked by dynamic bonds. J Mech Phys Solids 121:409–431 Yuk H, Lu B, Zhao X (2019) Hydrogel bioelectronics. Chem Soc Rev 48:1642–1667 Zhang YS, Khademhosseini A (2017) Advances in engineering hydrogels. Science 356:eaaf3627 Zhao X, Chen X, Yuk H, Lin S, Liu X, Parada G (2021) Soft materials by design: unconventional polymer networks give extreme properties. Chem Rev 121:4309–4372

Chapter 4

Igor Emri, a Student, a Colleague and a Friend Wolfgang G. Knauss

It is a great pleasure to contribute “a few words” about Igor, my former student, my co-worker, and a life-long friend, on the occasion of his 70th birthday. Igor had been admitted to Caltech’s Graduate Aeronautical Laboratories (GALCIT) as a graduate student for the academic year 1979/80 on the initiative of my colleague C.D. (Chuck) Babcock to work with him because of Igor’s interest in physical properties of metals under cyclic deformation. Changes in Chuck’s responsibilities brought Igor into contact with me. Having been devoted to the study of physical properties of polymers since entering graduate school myself, I was in the process of trying to describe their nonlinearly viscoelastic constitutive behavior. The major ingredient of this development was conceived in terms of the influence of free volume which was to be considered to be a function of the mechanical stress state; this concept derived from the then accepted importance of free volume in describing the temperature dependence of creep and relaxation behavior of polymers and explained to Igor that I had no interest in metals research. Apparently, this did not make a fundamental impression on him so during our first technical discussion I explained to him that it was first important to understand the principles that are likely to govern such processes, and that, therefore, it was mandatory to understand the processes at the micro/nanoscale, which are known to have a great influence on macroscopic mechanical behavior. Igor immediately comprehended this concept and has demonstrably adhered to it throughout his career. One such process was the by then well-established time/temperature superposition principle, whereby the temperature governs the intrinsic response-time of polymers, which relation had been published by different authors over half a century earlier. Because free volume can be changed by pressure (positive and negative) as well as W. G. Knauss (B) Aeronautics and Applied Mechanics, California Institute of Technology, Pasadena, CA 91001, USA e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_4

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by infusion of liquids into polymers, I suggested to Igor that to study the related macroscopic effects, it would first be prudent to infuse controlled amounts of, e.g., water, and observe the effect of the polymer’s time-dependent properties. Water infusion was feasible through placing the polymer into environments of various saturated salt solutions at fixed temperatures until a steady state of weight gain was achieved. Though his academic background was in quite a different scientific domain he accepted this challenge with a little shake of his head and shrugging his shoulders a bit: and this acceptance was the key to his many academic and practical successes, which should, ultimately, also earn him the induction into the US Academy of Engineering, among many other honors. After this roughly half hour discussion I did not speak with Igor for about five weeks, except to arrange for his use of lab-space and a seat/desk in an office (which turned out to be mostly in the lab). And then, about five weeks after our initial discussion, he showed up with neatly plotted data of how the (steady state) relative humidity correlated closely with weight increases in polyvinyl acetate. He had asked his way around in the lab (my technician and students) on where and how to get the simple equipment together (and how to have me pay for it). In fact, everything he did was done very neatly, and that has been his hallmark throughout his career. At this seemingly modest beginning with me, this clear characteristic started our future cooperation, collegiality and friendship. As a graduate student, Igor started to work with me on problems related to nonlinear behavior of polymeric materials. This initial work executed during his first stay at Caltech as a student (1979–1980), resulted in a new nonlinear viscoelastic constitutive model involving time (or rate) dependent variations of free volume, the first phase of which was published a year later (Knauss and Emri 1981). Because at that time the rules governing academic advancement in Slovenia were still dominated by the Soviet Union system, whereby a “western” Ph.D. was not accepted in place of a Doctor of Science (Dr. Sci), Igor pursued his academic goals at home, with his research started at Caltech. His doctoral work still centered around the nonlinearly viscoelastic behavior of polymers based on the molecular structure but geared to application in structural problems. His carefully executed measurements at Caltech were matched by a surprisingly good agreement with the computational evaluation of the molecular model. After completing his doctoral work in Ljubljana in 1986, Igor faithfully returned annually for at least ten years to Caltech “to interact with us”, until he was appointed Department Head at the University of Ljubljana, and his workload needed adjustment. His first visit was mostly devoted to preparing the improved theory of nonlinearly viscoelastic mechanical behavior for publication, which happened in 1987. I use the phrase “to interact with us” in the previous paragraph, because the “individuals” involved also a professorial colleague at Caltech as well as my other students and postdocs over the years. This senior colleague was Tschoegl, Professor of Chemical Engineering, whose interest was also in the domain of mechanical properties of polymers. Professor Tschoegl’s carefully conducted work on the effect of pressure on the viscoelastic properties of polymers figured significantly in supporting Igor’s and my work in the nonlinear behavior studies. Nick and Igor also became collaborators

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on their own and were friends until Nick passed away in 2011 (2011). During this time Nick and Igor produced multiple publications, dealing with improving the mathematical representation of physical measurements and the precision of representing the physical properties over large time ranges after application of the time–temperature principle. This work ultimately culminated in an application that allows for a consistently best process for time–temperature shifting experimental data. When Professor Tschoegl retired, Igor “inherited” a considerable part of his carefully constructed laboratory equipment. Igor included this in a complete measuring system for viscoelastic property measurements (“time-dependent” tensile, shear and volume properties) named “CEM Measuring System” (Center of Experimental Mechanics at the University of Ljubljana); it is described in detail in Springer’s “Handbook of Experimental Mechanics” (Knauss et al. 2008) and is in the process of being included in the ISO standards (International Standards Organization). The same publishing house printed Igor’s book “Statics” co-authored with his colleague Arcady Voloshin from Lehigh University (Emri and Voloshin 2016). Besides these interactive and enduring connections to Caltech, Igor developed close cooperation with fellow students or later students during his many visits following his appointment as professor of Mechanical Engineering at his home Institution in Ljubljana. Foremost among these is the “life-long” connection with Bernd von Bernstorff during Igor’s initial visit as a graduate student, who was a visiting student from Karlsruhe to execute with me his Diploma Thesis, that is required of all German “Diploma” engineering students. This coalescence of interest resulted in important projects, especially after Bernd became a part of the upper echelons in BASF, where he had the power to store practically oriented research efforts into directions supported by their joint knowledge gained at Caltech. One phenomenal accomplishment in that regard was to develop a method that allowed spinning fibers of Nylon 6 (BASF product) with a multimodal molecular mass distribution such that it had properties equal to Nylon 66 (DuPont product). Normally—i.e., without this special processing—Nylon 6 was considered inferior to Nylon 66 in its physical properties. The basis for this accomplishment was to make use of hitherto unexplored ranges of temperature and high speed (time) in fiber processing such that the special processing produced new forms of molecular arrangements, thus leading to materials at the macro-scale that possessed physical properties, which were superior to its former competitor by orders of magnitude. This work was patented and implemented by BASF, though, after DuPont terminated its production of PA, BASF has not yet officially implemented the process as a replacement for the existing monomodal PA66 materials. When PolyAmid-6 (PA6) of a properly selected molecular weight distribution is exposed to a well-defined (extreme) thermomechanical deformation history that is significantly different from the previously “normal” spinning process, it will form very different microstructural forms that can result in physical properties, which are orders of magnitude different from conventional PA6, surpassing the properties of PolyAmid-66 (PA66) materials for several orders in magnitude.

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In particular it was established that by properly modifying the molecular weight/mass distribution of PA6 into a specific bimodal version, one can significantly alter its time-dependent macro-scale properties. The latter is described by the relaxation spectrum, H (τ ), or equivalently, the retardation spectrum L (τ ). These mechanical spectra define the “time-scale” of any mechanical response when exposed to a given rate (or frequency) of mechanical loading. The determination of mechanical spectra requires an inverse solution of Fredholm’s integral equation of the first or second order, which is numerically an illconditioned (ill-posed) problem. Nick and I had started to work on this problem back in 1967. Later, Igor and Nick continued this work and developed one of the most successful numerical algorithms for determining mechanical spectra from experimentally obtained creep and/or relaxation material functions. This algorithm for determining mechanical spectrum, and the CEM measuring system served in the BASF projects as a “steady tool” to study the interrelation between a material’s molecular weight distribution and its mechanical spectrum. The effect of the high rate of loading was then analyzed with the nonlinearly viscoelastic model (Knauss and Emri 1981, 1987). Later it was found that these materials could be formed into solids with gradient properties suitable as bone replacements in dental and orthopedic surgery. Currently, these are the world’s first implants that are able to complete the absorption into the surrounding tissue and represent a technological breakthrough in the field of body-part implanting (corresponding patents are pending). A further extension of that work was the implementation of these findings into the electrospinning technology with the goal to develop fibers with the thinnest diameter and, from them, membranes with the smallest porosity on earth, well below 50 nm, which is half the pore size of competitive materials. This research was conducted in collaboration with Professor Leonid Kossovich from the Saratov State University in Russia.1 Igor and Bernd started with the electrospinning of polyamides and then moved to other polymers, including biopolymers such as Chitosan and cellulose. Without going into too many details, the “trick” here was to choose polymers with a proper multimodal molecular weight distribution and expose them to a well-defined (high) rate of spinning such that the molecular structure in the fiber is prevented from recoiling before the fibers solidifies and are deposited. Igor has interpreted that this will happen at spinning rates where the dissipative part of the shear modulus dominates over the corresponding storage modulus part. Such nano-porous membranes may be used in a variety of engineering solutions, as well as in medicine. It is of interest to relate here at least some: The air-and water vapor permeability of membranes constituted from these fibers surpass all existing membranes used in the textile industry, where moisture permeability is important, including, for example, GoreTex. However, these membranes (they are, in fact, nonwoven nano-structured fabrics) offer a much more important field of application as a new kind of filter. 1

It is interesting to note that the requisie electrospinning process demands very high voltages, which are not readily permitted commercially, for example, in Germany, but they may be used without particularly restrictions in Russia.

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Though these filters would be sufficiently fine-pored to filter out the Covid-19 virus, this development was too new to encompass the scale needed to impact the current pandemic. Another very important field of application of these membranes is as separators for new generation high-capacity batteries, mainly to shorten the battery charging time. But even more important is the possibility to generate potable water by filtering sea water. For medicinal use, a most interesting application of these membranes, when made from Chitosan, are plasters and wound covering, because these membranes are air permeable, but at the same time, will mechanically stop all bacteria and viruses from passing through to attack the wound. This allows faster but safe healing of very large wounds (large parts of the human body), caused by frostbite, burn, or diabetes that might otherwise be mortal for a patient. Development and certification of such plasters and wound coverings are underway by the European Medicines Agency (~FDA) in collaboration with an industrial partner. A further development in Igor’s laboratory was the invention of granular damping units under high hydrostatic pressure. Using the CEM measuring system for studying the effect of pressure on the damping properties of polymeric materials, and the nonlinear model described in reference 5, it was found that by the proper choice of a damping material and application of hydrostatic pressure during a cyclic loading process, one can select the optimal damping properties within the frequency or rate of the applied loading. In this way one can fully utilize the damping characteristics of the selected material and maximize the energy absorption properties of a damper over several orders of magnitude. The open question was how to generate the hydrostatic pressure. This is not possible under uniaxial compression, because the critical shear (flow) stresses are reached far before one reaches pressures that would alter the properties of the selected polymeric material. Igor’s group found that granulated viscoelastic materials with properly selected multimodal size distribution exhibit fluid-like behavior, while maintaining the volumetric behavior of the material from which they were made. Hence, granular materials may be used as “pressurizing media” to impose inherent hydrostatic pressure on itself within a flexible but inextensible container, and consequently change its own damping properties. This started with enclosing granular solids in flexible containers of ultrastrong carbon- and/or basalt fibers for use in/on foundations of structures under vibration (e.g., earthquake engineering) or vibration isolation in connection with trains, especially for high speed versions. For a while Igor and Bernd worked with the German Federal Railroad to isolate the noise of underground trains that influence the foundation of the Cologne Cathedral. The execution of this project stalled only because the German Authorities were unable to achieve any results in a foreseeable future because of the myriad of permissions and conditions that need to be met from too many agencies for a project of this size. At this time this technology is being actively pursued in China in high speed rail construction, as part of the "One Belt One Road Initiative.” This development proceeds together with another one of my students whom Igor and Bernd met at GALCIT, Professor Hongbing Lu (UTexas at Dallas) who has both the technical and viscoelastic background as well as the “political” understanding of the Chinese environment.

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A more recent breakthrough invention is a new generation of sound insulation which, depending on the frequency range, and for layers of 30 mm thickness, surpasses the world’s best high-end sound insulations (offered by BASF) by about 10– 30 dB. The new sound insulation is not “material-dependent” but rather “dissipativeprocess-dependent”. Hence, this insulation can be manufactured from a variety of different cost-effective granular materials, such as natural and synthetic polymers, silica, and from a variety of metallic and even waste materials. Hence, this researchbased invention opens a completely new perspective in solving one of the world’s most critical ecological problems—the noise pollution. It is of interest to note that this new, high insulation efficiency is obtained only in the case of a multimodal granular particles size distribution that can be predicted by the model, which Igor stated that he developed from our earlier work on modelling nonlinear time-dependent behavior of polymeric materials (Knauss and Emri 1987). It was surprising to find that the behavior of granular materials on the micro–macroscale and the behavior of polymers on the molecular scale can be described by the very same mathematical model if the mechanical spectrum belonging to the selected molar mass distribution is replaced by the normalized granular particle size distribution. The physical reason for this surprising result is something that needs to be investigated in more detail. The patent for producing these ultra-high damping materials was granted just prior to the start of the pandemic and its implementation is starting up again as the pandemic moves into more tolerant phases. As one application in this context four of my former graduate students Igor Emri, Bernd von Bernstorff, and Hongbing Lu are working closely with my last Ph.D. student, Luis Gonzales, at Hyundai Helicopter, as a potentially first major industrial partner in its application. I have to say that for me it is a very warm feeling to see how four of my former students work together in bringing this research-based invention into a high-end industrial application. It is also of interest to note that Igor’s efforts were recognized at the ZGC Forum 2021 (Zhongguancun, Ministry of Science and Technology of the People’s Republic of China) where this invention was selected from more than 2800 innovations contributed to the forum from 36 countries, ranked among the 100 best innovations overall, and among the top 10 in the field of new materials. In 1993 Igor was the driver behind creating a new research field within the organization of the Society for Experimental Mechanics (SEM), which he called the “Mechanics of Time-Dependent Materials (MTDM)”. It started in SEM as a Technical Division and Igor became its first chairman. With support from Nick Tschoegl and myself, Igor initiated a larger effort by co-launching the first biannual International MTDM Conference, the first being hosted by Igor in a spectacular, pacesetting manner in Ljubljana in 1995. Two years later Igor persuaded me to join him as a senior editor of a new journal “Mechanics of Time-Dependent Materials” which since then has been published by Springer Nature. In 2000 Igor became the first non-American president of SEM. With his wide-ranging international contacts, it was very natural that Igor became heavily involved in the founding of the European postgraduate school of excellence in

4 Igor Emri, a Student, a Colleague and a Friend

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Engineering Rheology–EURHEO (http://eurheo.eu/rheology), which united 6 European Universities. Since 2002 he became a member of the Associations of University departments–WAK (Wissenschaftlicher Arbeitskreis der Universitäts-Professoren der Kunststofftechnik, http://wak.mv.uni-kl.de/), which brings together 25 institutes (24 from EU and 1 from the USA) in the field of polymer technology. Igor has lectured and still lectures at many respectable universities in Europe, Japan, USA (mostly Caltech), Canada, Russia (Bauman University in Moscow, appointed there as an invited foreign professor), China, and Kazakhstan. In 2013 he became an Honorable Professor of the Kazakh National Agrarian University in Kazakhstan and in 2018 Visiting Professor of the Nanjing Institute of Technology, Nanjing, China. For his professional achievements, he received numerous national and international awards and recognitions, and was elected to several academies, among them the Russian Academy of Natural Sciences (1997), Russian Academy of Engineering (1996), European Academy of Sciences and Arts (2006), as well as the prestigious (US) National Academy of Engineering (2020). In January 2017, Igor reached the official retirement age for professors in Slovenia and has since been heavily involved in the industrial implementation of his researchbased inventions.

References Emri I, Voloshin A (2016) Statics: learning from engineering examples. Springer, New York Knauss WG, Emri I (1981) Non-Linear Viscoelasticity Based on Free Volume Consideration. Comput Struct 13:123–128 Knauss WG, Emri I (1987) Volume change and the nonlinearly thermo-viscoelastic constitution of polymers. In: Yee A (ed) Polymer engineering and science, vol 27. pp 86–100 Knauss WG, Emri I, Lu H (2008) Mechanics of polymers: viscoelasticity; Chapter 3. In: Handbook of experimental solid mechanics. Springer Tschoegl NW (2011) See the memorial synopsys. https://www.caltech.edu/about/news/nicholasw-tschoegl-93-2015

Chapter 5

Numerical Simulation for Tensile Failure of Fiber-Reinforced Polymer Composites Based on Viscoelastic-Entropy-Damage Criterion Jun Koyanagi Abstract In this chapter, entropy-based damage criterion is introduced into prediction of long-term durability for a composite material. The entropy-based damage criterion is potentially an essential factor for discussing residual strength and lifetime because it involves stress–strain histories and temperature. The entropy can be mechanically calculated by dividing the dissipated energy by absolute temperature. The dissipated energy can be determined by inelastic constitutive equation. The entropy is also linked to material damage. This algorithm is integrated into a commercial software of finite element method. The entropy-based damage is applied to matrix resin in a composite material and through failure of composite material is predicted numerically. Also in this chapter, an efficient method of finite element analysis with periodic boundary conditions and a method to predict tensile strength of randomly oriented short fiber-reinforced plastics.

5.1 Introduction Engineering structures are increasingly composed of carbon-fiber-reinforced polymer composites owing to their promising properties. In the aerospace industry, metallic materials are replaced with carbon-fiber-reinforced plastic (CFRP) to improve their mechanical properties. This is because the strength of CFRP in the fiber direction is stronger than that of metals. Furthermore, considering the excellent advantage wherein the weight of CFRP is light, it is obvious that CFRP is superior to metallic materials in terms of specific strength and specific elastic modulus. Global regulations on CO2 emissions, soaring fuel prices, and impending introduction of electric vehicles have led to a strong demand for lighter vehicles. There are several ways to reduce the weight of automobiles, including the use of high-tensile steel plates and extensive use of aluminum plywood. However, the most efficient J. Koyanagi (B) Department of Materials Science and Technology, Tokyo University of Science, 6-3-1 Niijuku, Katushika-ku, Tokyo 125-8585, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_5

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way to reduce weight is to use considerable amounts of CFRP, which has excellent specific strength, in the structure. Currently, CFRP, which is commonly used in aircraft and other structures, is mainly composed of a thermoset resin matrix with epoxy resin as its main component; however, it has not yet been put to practical use in mass-produced vehicles owing to problems of economy and mass-production technology (Elmarakbi 2013; Boria and Belingardi 2012; Maier et al. 2014; Mainka et al. 2015; Meng et al. 2017; Sato and Kurauchi 1997; Sim et al. 2020; Zhu et al. 2017). In addition, thermoset resin matrix composites are difficult to recycle after use, and the development of low-cost technology, mass-production technology, and recycling technology for CFRP is awaited. When CFRP is applied to mass-produced automobiles, discontinuous CFRP is used as they have excellent productivity and can be molded into complex shapes (Hashimoto et al. 2012; Okabe et al. 2010). The matrix should be a thermoplastic resin with excellent properties such as impact resistance, recyclability, and short molding cycles (Sinha and Tyagi 2019; Offringa 1996). Relatively inexpensive resins such as polypropylene, polycarbonate, and polyamide have been considered as candidates. Among them, polyamide 6 (PA6) has the advantages of good adhesion to carbon fibers and excellent fabrication easiness, which may be useful in its application to the primary structure of mass-produced vehicles. Therefore, discontinuous carbonfiber-reinforced PA6 (discontinuous CF/PA6) has been widely applied to automotive components. Discontinuous CFRPs are generally manufactured by injection molding or press molding; however, in each case, the fibers are easily damaged. Furthermore, fiber orientation is non-uniform in composites because the fibers rotate with the flow of the matrix during molding (Sasayama et al. 2013). Owing to these uncertainties, ensuring the reliability of the strength of the composite material is difficult, thereby limiting the use of discontinuous CFRPs in structural components. Therefore, predicting the strength based on the microstructure of the fibers, such as their length and orientation, is crucial for the use of discontinuous CFRPs. For an accurate strength prediction, considering the fracture of the composites is necessary. The main factors in composite fractures are fiber failure and matrix damage. In general, the properties of composite materials depend on the constituent material. In particular, the mechanical and thermal properties of the matrix resin have a large influence on composite materials. Therefore, the development of good resin plays a significant role in the development of CFRPs having high strength and excellent heat resistance. Furthermore, considering that the base materials of general composite materials are polymers, their characteristics depend on time and temperature. The time-dependent properties of polymeric materials can be explained using theories such as viscoelasticity and viscoplasticity (Kim and Muliana 2009; Kontou and Spathis 2014; Marklund et al. 2008; Miled et al. 2011; Schapery 1997; Sato et al. 2021; Koyanagi et al. 2021). With regard to these viscous theories, various models have been proposed to express time-dependent characteristics of materials. Currently, the most accurate representation of polymer properties is a continuous model wherein a number of basic models are arranged in series or parallel. However, even with these models, the linear viscoelastic model cannot accurately represent

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deformation behavior under high-stress conditions. To accurately express the deformation behavior, even in a high-stress state, providing viscoelastic model nonlinearity is necessary. In addition, focusing on the thermal properties of CFRP, the long-term durability of CFRP is predicted based on the time–temperature superposition principle, which is generally useful for viscoelastic materials. Using this law, viscoelastic characteristic functions measured at different temperatures may coincide by parallel shifts along the logarithmic time or frequency axis. Thus, long-term durability was predicted from the durability results at various temperatures. As the CFRP matrix is a polymer, the long-term durability of CFRP is determined by the time–temperature dependence of the polymer material (Arao et al. 2010; Koyanagi et al. 2004, 2007, 2012). To accurately predict the stress–strain response, a constitutive model that considers time dependency and material damage are required. Many studies have been conducted on a material lifetime under specific test conditions such as a monotonic cyclic load, creep load, or constant strain rate. Therefore, a constitutive model applicable to various loads has not yet been developed. In this chapter, irreversible thermodynamic entropy is introduced to address these problems. In recent years, attempts have been made to clarify the damage evolution in solid materials using irreversible thermodynamics (Sato et al. 2021; Mohammadi et al. 2021; Naderi et al. 2010; Moghanlou and Khonsari 2021; Mehdizadeh and Khonsari 2021; Wang and Yao 2017; Mohammadi and Mahmoudi 2018; Sakai et al. 2022). Permanent deterioration is a manifestation of an irreversible process according to the second law of thermodynamics. That is, the deterioration of the material proceeds with disorder of the system. Disorder in the system, which causes deterioration of the characteristics, is caused by the deformation of the materials, and the entropy increases accordingly. When the disorder in the degraded system reaches to a critical state, material failure occurs. Therefore, the irreversible entropy in materials is expected to be a criterion for evaluating the durability of materials. The objective of this chapter is to propose a viscoelastic/plastic model that follows a nonlinear viscoelastic/plastic constitutive equation for resin based on irreversible thermodynamics and viscoelasticity theory. In addition, the tensile strength of discontinuous CFRPs is predicted to be the first step in long-time durability prediction. The material properties of viscoelasticity, viscoplasticity, and damage of PA6 were identified from creep recovery, stress relaxation, and tensile tests. The proposed viscoelastic/plastic model was applied to the base resin part, and a finite element analysis of the uniaxial tension of discontinuous CF/PA6 was performed in Abaqus/Standard. The tensile strength of the discontinuous CF/PA6 was calculated using the layer-wise method. The implementation of the numerical algorithm for the viscoelastic/plastic model was based on the Abaqus/Standard user subroutine UMAT.

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5.2 Application of Thermodynamic Entropy for Continuum Damage Mechanics Damage is a progressive degradation that occurs in a material prior to failure. Analyzing the cumulative damage has an important role in predicting the life of a structure under load. In the accumulative damage process, the entropy in the system continues to increase according to the second law of thermodynamics; therefore, using the entropy generation amount as a criterion for the damaged state is possible (Murakami 1982). In this section, we describe a method for associating entropy generation with mechanical damage based on thermo-mechanical dynamics. The entropy generation amount sf per unit volume until fracture can be obtained by integrating the entropy generation rate s˙ shown in Eq. (5.1) up to time tf until fracture occurs (Murakami 1982). Therefore, the fracture entropy sf is expressed as  tf  ˙ Wd Ak : V˙ k gradT dt, − − Jq · sf = T T T2

(5.1)

0

where Wd is the energy dissipated through non-elastic deformation, T is absolute temperature, Ak is generalized thermodynamic force associated with the internal variable, Vk is a suitable set of internal variables, and J is the heat flux at the boundary. The first term in Eq. (5.1) is the mechanical dissipation due to plastic deformation, the second term is the dissipation due to non-recoverable energy accumulated in the material due to damage and strain hardening, and the third term is the heat conduction heat dissipation. In the process accompanying large deformation, entropy generation by plastic deformation is dominant; therefore, entropy generation by heat conduction in the third term is negligible (Naderi et al. 2010). Jq ·

gradT ≈0 T2

(5.2)

Therefore, the Eq. (5.1) is shown as  tf  ˙ Wd Ak : V˙ k − dt. sf = T T

(5.3)

0

Here, consider the second term in Eq. (5.3). As mentioned, there is arbitrariness in the selection of internal and accompanying variables in the equation. Therefore, when selecting the scalar damage variable D as an internal variable, the accompanying variable Y is expressed as   1 ∂ We Y = 2 ∂ D σ =const .

(5.4)

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Fig. 5.1 Energy release by development of damage

According to this formula, Y is the release of the elastic strain energy associated with the development of damage variable D. Therefore, the damage accompanying variable Y has a role similar to the strain energy release rate in fracture mechanics. This damage accompanying variable Y is sometimes referred to as the strain-density release rate. Here, a state where it is loaded up to point A on a stress–strain map with uniaxial tension (Fig. 5.1) is considered. With the stress σ = constant, the damage progresses by d D, and along with it, the elastic strain increases by dεe and reaches point B. Triangle ABC in Fig. 5.1 shows the energy release rate Y d D associated with the evolution of damage, d D. Thus, clarifying the entropy generation due to plastic strain and damage by calculating the energy from the stress–strain relation due to tension until fracture is possible. According to the second law of thermodynamics, material fracture occurs when the entropy generation rate is zero, i.e., when the entropy generation reaches the maximum value. Therefore, the entropy generation can represent a measure of deterioration with respect to the initial state. In this section, the damage variable D is used to associate entropy generation with mechanical damage. We assumed that the damage variable D is proportional to the entropy generation amount, and defined it as follows: D = ξs ξ=

Dcr sf

(5.5)

where s is the entropy generation, ξ is a constant of proportionality, Dcr is the critical damage variable.

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5.3 Numerical Simulation for Discontinuous CFRP Owing to its complex structure, CFRP exhibits a very complicated deformation and fracture behavior compared with a homogeneous material. Generally, the base material of CFRP is resin, and its behavior depends on the temperature and speed, as well as it strongly affects the durability of the composite material. Therefore, in CFRP failure simulations, accurately predicting the behavior of the resin at different temperatures and speeds is important. Furthermore, in the case of discontinuous CFRPs, the fiber orientation angle is random; therefore, considering the fiber orientation angles is necessary. A laminate analogy approach (LAA), in which a composite is assumed to be a laminate wherein each layer has a unique fiber orientation, is also often used (Fu and Lauke 1998). In this chapter, uniaxial tensile analysis of discontinuous CF/PA6 is performed to predict the tensile strength using the proposed viscoelastic/plastic model and the determined material constants. The layer-wise method refers to the LAA and it was used to account for random fiber orientation angles. The analysis algorithm for the viscoelastic model was implemented in the Abaqus/Standard user subroutine UMAT. The results of the analysis were compared with the experimental results.

5.3.1 Layer-Wise Method The layer-wise method (LWM) was used to predict the tensile strength of discontinuous CFRPs with random fiber orientation angles (Hashimoto et al. 2012; Okabe et al. 2010). The LWM assumes that the discontinuous CFRP is identical to laminates comprising unidirectionally fiber-reinforced (UD) plies (Fig. 5.2a and b). Each ply had a unique unidirectional fiber orientation. As shown in Fig. 5.2c, the strain increment ˆε applied to the composite in the composite coordinate system O-x 1 x 2 is transformed to the ply coordinate system O-xˆ1 xˆ2 as follows: ˆε = Rε,

(5.6)

where R denotes the coordinate transformation matrix. By applying ˆε to the unit cell model of a UD ply, we obtain the stress σ (θ ) of the corresponding ply, which has a fiber orientation angle θ . where σ (θ ) is the average stress over the entire unit cell model. In this study, the fiber orientation angle was varied from 5° to 85° in 10° steps. The stress σ (θ ) of the corresponding ply in the coordinate system O-x 1 x 2 is given by σ (θ ) = R−1 σˆ (θ ).

(5.7)

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(a) SEM image of discontinuous CFRP

71

and (b) laminate of unidirectional layers.

(c) Coordinate transformation Fig. 5.2 Schematic illustration for an overview of layer-wise method. a SEM image of discontinuous CFRP. b Piled layers like a laminate. Each layer has its own fiber orientation. c Coordinate transformation between the composite and the layer coordinate systems

The stresses for fiber orientation angles from − 85° to − 5° were calculated from the 5° to 85° analysis. The stress σ of the composite laminates is given by the average of σ (θ ): π/2 σ =

f (θ )σ (θ )dθ, −π/2

where f (θ ) is a probability density function of fiber orientation.

(5.8)

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5.3.2 Finite Element Analysis A numerical analysis was conducted using the Abaqus/Standard 2019. We used a three-dimensional (3D) unit cell analysis model of the UD ply, as shown in Fig. 5.3. An representative volume element (RVE) containing 10 carbon fibers with a 6-μm diameter was surrounded by matrix resin. The fiber/matrix interface was not considered herein. The 3D unit cell size was 0.26 mm × 2.016 mm × 2 μm, resulting in a fiber volume fraction of 20%. The element type is an 8-node brick element, the number of elements is 65,520, and the number of nodes is 132,313. The carbon fiber was assumed to be an isotropic elastic body (Table 5.1), and a probability distribution of the fiber strength was introduced. We assumed that the fiber strength is determined by the maximum stress criterion based on the Weibull distribution. When a fiber segment with length L is subjected to axial stress σ , the Weibull distribution yields the cumulative failure rate Pf .

Fig. 5.3 Finite element model of UD ply

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Table 5.1 Mechanical properties of the carbon fiber Elastic modulus [MPa]

Ef

294

Poisson’s ratio

νf

0.3

Fiber radius [μm]

rf

6

Weibull scale factor [MPa]

σw

4900

Gage length of a single-fiber specimen [mm]

L0

2

Weibull modulus

ρ

10

Pf (σ ) = 1 − exp

 ρ   σ L − L0 σ0

(5.9)

where L 0 is the fiber length, σ0 is the representative strength of the fiber, and ρ is the Weibull coefficient. The fiber strength of the ith segment σi is calculated by choosing a random number Ri and solving Ri = Pf (σi ). When the fiber axial stress acting on the ith segment reaches the critical strength σi , fiber failure is expressed by eliminating the nodal force on the corresponding element. The matrix PA6 resin is assumed to be a nonlinear viscoelastic/plastic body following the proposed constitutive equation considering entropy damage (Sato et al. 2021). The existing viscoelastic model was not possible to accurately represent the deformation and failure characteristics of PA6 resin under high-stress and high temperature environment. We propose a viscoelastic model based on the Ramberg–Osgood equation that applies nonlinearity, entropy damage, and nonNewtonian behavior permanent strain to the existing generalized Maxwell model. The viscoelastic/plastic model is shown in Fig. 5.4. Fig. 5.4 Viscoelastic/plastic model

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The number of Maxwell elements is 14, with one spring added. A large number of Maxwell elements are needed to consider the long-term viscoelastic behavior of polymers. The total strain ε consists of a viscoelastic strain, εve , and a viscoplastic one, εvp , as shown in Eq. (5.10). The second term of Eq. (10), the viscoplastic strain is expressed by Eq. (5.11). ε = ε ve + ε vp t vp εi j

=

vp −1

Hi jkl

(5.10)

σkl dt.

(5.11)

0

Here, Hvp is the viscosity coefficient that induces permanent strain, which can be written in a matrix form as follows: ⎤ ⎡ ν ν 0 0 0 (1 − ν) ⎢ ν ν 0 0 0 ⎥ (1 − ν) ⎥ ⎢ ⎥ ⎢ vp η ν ν − ν) 0 0 0 (1 ⎥ ⎢ ×⎢ Hvp = ⎥. (5.12) 1−2ν ⎥ 0 0 0 0 0 (1 + ν)(1 − 2ν) ⎢ 2 ⎥ ⎢ 1−2ν ⎣ 0 ⎦ 0 0 0 0 2 0 0 0 0 0 1−2ν 2 Here, v is the Poisson’s ratio and ηvp is determined by the following:



η0 1 + e η

vp

β

vp εeqv σeqv

χ 

 . =  1 + eα(σeqv −σ0 )

(5.13)

Here, subscript eqv is the equivalent value of the current state, η0 is the initial value of α, β, and σ 0 and χ are specific constants determined based on a comparison between the experimental and analytical values. As shown in Eq. (5.13), ηvp is expressed as a function of the current equivalent viscoplastic strain and equivalent stress. The value of the denominator increases non-linearly with the current stress when it exceeds a specific stress value of σ 0 . The value of the numerator increases non-linearly with the current viscoplastic strain and decreases with the current stress. The viscoelastic constitutive equation based on the constitutive equation considering the stress history proposed by Schapery can be expressed through the following: t σi j (t) = (1 − D) 0

ve   gdεkl E irjkl t − t  dt  .  dt

(5.14)

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Here, Er is the relaxation modulus as expressed by Eq. (5.15). E n is the elastic modulus of nth Maxwell element. ηn is the viscosity coefficient of nth Maxwell element, t is the time, and t  is the time at which the strain is introduced. g is a nonlinear coefficient based on the sigmoid function, as expressed by Eq. (5.16). σ eqv is the equivalent stress, σ 0 is the specific stress, γ is a nonlinear parameter, and m is a nonlinear exponent. When σ eqv is greater than σ 0 , the strain increase is higher than that of the stress, and the nonlinearity is expressed. D is a damage variable that accounts for the entropy-damage law. The all values of the parameters are shown in Table 5.2. These values are determined by various experiments (Koyanagi et al. 2021). E irjkl (t) =

15 

E injkl e−t E

n

/ηn

(5.15)

n=1

g=

1+γ

1 

σeqv σ0

D = Dcr

s sf

m

(5.16)

(5.17)

A periodic boundary condition at all edges using the key-node method (Sato et al. 2019; Li et al. 2011; Melro et al. 2013; Li et al. 2015), as described in the following section, is applied to the numerical model. Interface failures were not included in this study. The interface strength is time- and temperature-independent (Sato et al. 2018; Niuchi et al. 2017; Koyanagi et al. 2014; Koyanagi and Ogihara 2011; Koyanagi et al. 2010), and interface does not exhibit some kind of delayed failure. Hence, it must not be modeled, especially when discussing the long-term durability of composite materials. In this section, matrix and fiber failures are modeled using finite element analysis.

5.3.3 Periodic Boundary Condition Periodic boundary conditions are imposed on the unit cell. In this section, we explain how to impose a periodic boundary condition. For convenience, the faces, edges, and vertices with periodicity are named faces f[1a]–f[3b], edges e[1,1]–e[3,4] and vertices v1–v8, respectively, as shown in Fig. 5.5. To avoid overlapping periodic boundary conditions, vertices v1–v8 are not included in edges e[1,1]–e[3,4] and edges e[1,1]–e[3,4], and vertices v1–v8 are not included in faces f[1a]–f[3b]. The displacement ui at an arbitrary point x i in the RVE is expressed using the per macroscopic strain εi j and microscopic perturbed displacement u i described by the following equation:

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Table 5.2 Mechanical properties of PA6 η1

1.0 × 10 4

η2

9.0 × 10 4

305

η3

1.1 × 10 6

E3

515

η4

5.2 × 10 6

E4

470

η5

2.7 × 10 7

E5

380

η6

8.8 × 10 7

E6

203

η7

4.7 × 10 8

E7

132

η8

1.1 × 10 9

E8

17

η9

1.3 × 10 10

E9

12

η10

9.0 × 10 10

E 10

11

η11

1.1 × 10 12

E 11

11

η12

8.0 × 10 12

E 12

8

η13

9.0 × 10 13

E 13

6

η14

1.0 × 10 15

E 14

5

Specific stress for nonlinearity (MPa)

σ n0

50

E 15

970

Nonlinear parameter

γ

0.3

Nonlinear exponent

m

7

Initial viscosity coefficient (MPa * s)

η0

5.8 × 10 10

Specific stress for viscoplasticity (MPa)

σ vp0

38

Viscoplasticity parameter

α

0.89

β

385

Initial elastic modulus (Mpa) E 0

3300

Elastic modulus of Maxwell elements (Mpa)

E1

255

E2

Viscosity coefficient of Maxwell elements (MP · s)

χ

0.43

Critical damage

Dcr

0.3

Fracture entropy (kJ/m 3 K)

Sf

102

per

u i (xi ) = εi j x j + u i (xi )

(5.18)

Here, between the corresponding nodes on faces f[1a] and f[1b] as an example, the microscopic perturbed displacements satisfy the following periodic condition.   per per u i (xi )f[1a] = u i (xi )f[1b] .

(5.19)

Substituting Eq. (5.18) into Eq. (5.19), the following periodic boundary conditions were obtained:

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Fig. 5.5 Definitions of the faces, edges, and vertices of an RVE

⎧ ⎨ u 1 |f[1a] − u 1 |f[1b] = 2aε11 , − u 2 |f[1b] = 2aε12 , u | ⎩ 2 f[1a] u 3 |f[1a] − u 3 |f[1b] = 2aε13 .

(5.20)

Similarly, the periodic boundary condition between edges e[1,1] and e[1,3] and between vertices v1 and v7 are expressed by the following equations: ⎧ ⎨ u 1 |e[1,1] − u 1 |e[1,3] = 2bε12 + 2cε13 , − u 2 |e[1,3] = 2bε22 + 2cε23 , u | ⎩ 2 e[1,1] u 3 |e[1,1] − u 3 |e[1,3] = 2bε23 + 2cε33 . ⎧ ⎨ u 1 |v1 − u 1 |v7 = 2aε11 + 2bε12 + 2cε13 , u | − u 2 |v7 = 2aε12 + 2bε22 + 2cε23 , ⎩ 2 v1 u 1 |v1 − u 1 |v7 = 2aε13 + 2bε23 + 2cε33 .

(5.21)

(5.22)

Periodic boundary conditions other than the above can be obtained in the same manner. We then introduced the key DoF method to impose periodic boundary conditions (Li et al. 2011). In addition to the DoFs belonging to the mesh of the FE model, other independent DoFs were introduced in the model. These DoFs are called the key DoFs and are regarded as equivalent to the macroscopic strain components as follows:

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J. Koyanagi key

key

key

d11 = ε11 , d22 = ε22 , d33 = ε33 , key

key

key

d12 = 2ε12 , d23 = 2ε23 , d13 = 2ε13 .

(5.23)

By introducing Eq. (5.20) into the multipoint constraint in Eqs. (5.20)–(5.22), these key DoFs are used to control periodic boundary conditions. For example, the multipoint constraint equation in the x 1 direction between vertices v1 and v7 is key

key

key

u 1 |v1 − u 1 |v7 = 2ad11 + bd12 + cd13 .

(5.24)

Consequently, macroscopic strain can be applied to the finite element model by inputting the value of the key DoFs.

5.3.4 Algorithm of Viscoelastic-Entropy-Damage Criterion The analysis algorithm for the matrix PA6 resin, which is a nonlinear viscoelastic/plastic body, was implemented using the Abaqus/Standard user subroutine UMAT. Using the calculation flow shown in Fig. 5.6, UMAT calculates the stress increase for each strain increment. The temporary stress σ temp of the entire Maxwell model was calculated from the viscoelastic strain increase without relaxation behavior due to the dampers. In this model, σ pre is the stress at the previous time step. The stress of the Maxwell model was used to calculate the dissipated energy, nonlinear coefficient, and viscoplastic strain. The dissipated energy is used to calculate the entropy; subsequently, the damage variable is calculated, and the damage variable is used to calculate the stress at the next time step, where εela denotes the elastic strain. The calculation is repeated until the damage variable reaches a critical value. The matrix components of E and H vp are expressed by Eqs. (5.15) and (5.12), respectively: We assumed that Poisson’s ratio ν is 0.3.

5.3.5 Numerical Results In this section, analytical and experimental results for the uniaxial tension of discontinuous CF/PA6 are compared. For the tensile test of discontinuous CF/PA6, a universal testing machine (SHIMADZU AG–X plus) was used, and a planar specimen (10-mm width, 4-mm thickness, and 100-mm gauge length) was used (Fig. 5.7). The test was performed on five specimens at 27 °C at a strain rate of 0.01/min. The average value and standard deviation of the tensile strength were calculated and are shown in Fig. 5.8, along with the analytical results. The experimental and analytical results were in good agreement. This indicates that the tensile strength of discontinuous CFRP can be predicted using the proposed

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Fig. 5.6 UMAT calculation flowchart Fig. 5.7 Discontinuous CF/PA6 specimen

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160 140 Tensile strength [MPa]

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120 100 80 60 40 20 0

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viscoelastic/plastic model that considers entropic damage and the layer-wise method that considers random fiber orientation angles. In addition, the prediction of tensile strength is the first step in predicting durability in the future, and the results are promising. One of the most important merits of this failure analysis is that we should be able to discuss the durability of the composite materials. However, till date, we have faced numerical difficulties regarding computational cost, and calculating many failure simulations is still not easy. This is an interesting aspect that will be studied in the near future.

5.4 Conclusion In this chapter, a deformation model of the resin based on thermodynamic entropy damage and viscoelasticity theory is investigated. The newly proposed material model is a viscoelastic/plastic model that includes the time- and temperaturedependent viscoelastic behavior, damage, and permanent strain. The material parameters required for this viscoelastic/plastic model can be determined from creep, recovery, and stress relaxation tests. The time–temperature conversion rule was used herein. The proposed viscoelastic/plastic model can simulate the stress and fracture behavior of the PA6 resin in various strain and temperature regions, except for the unloading process. Using the proposed viscoelastic/plastic model for the PA6 matrix resin of CFRPs, a uniaxial tensile simulation of discontinuous CF/PA6 was performed. The analysis algorithm for the PA6 matrix resin, which is a nonlinear viscoelastic/plastic body, was implemented using the Abaqus/Standard user subroutine UMAT. The tensile strength was calculated using the layer-wise method to

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account for random fiber orientation angles. The analytical results are in good agreement with the experimental results, indicating that the current model and layer-wise method are useful for predicting the tensile strength of discontinuous CFRP. Acknowledgements Parts of this study were financially supported by the New Energy and Industrial Technology Development Organization (NEDO), JST MIRAI grant number 19215408, the JKA Foundation, and KAKENHI grant number 21KK0063.

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Chapter 6

An Investigation of the Nonlinear Viscoelastic Behavior of PMMA Near the Glass Transition Using the Spectral Hole Burning Method Huiluo Chen, Sadeq Malakooti, Ren Yao, Stephanie L. Vivod, Gregory McKenna, and Hongbing Lu Abstract Large-Amplitude Oscillatory Shear (LAOS) spectroscopy, i.e., mechanical spectral hole burning (MSHB) was used to characterize the nonlinear viscoelastic behavior of polymethyl methacrylate (PMMA) in the glass transition region. Here, we performed MSHB experiments on this amorphous polymer at 100 °C (in the glassy state) to investigate the presence of mechanical hole. The experiments include different phases with various controlled parameters to study the effect of pump amplitude and pump frequency on PMMA. We found that the mechanical holes are visible at pump amplitude from 4.4 to 8.25% and pump frequency from 0.005 to 0.015 Hz with 0.55% of probe strain for PMMA. The trend for the intensity of the holes has an inverse relationship with pump amplitude for the PMMA polymer, which is opposite to the results obtained at ambient temperature in experiments we conducted previously.

6.1 Introduction Solid polymers play a crucial role in different industries including electronics, biomedical, automotive, and aerospace (Drobny 2011; Ulery et al. 2011; Mouritz H. Chen · R. Yao · H. Lu (B) Department of Mechanical Engineering, The University of Texas at Dallas, Richardson, TX 75080, USA e-mail: [email protected] S. Malakooti · S. L. Vivod Materials and Structures Division, NASA Glenn Research Center, 21000 Brookpark Road, Cleveland, OH 44135, USA e-mail: [email protected] S. L. Vivod e-mail: [email protected] G. McKenna Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, NC 27695, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_6

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2012). Their linear viscoelastic properties at different time scales and elevated temperatures are well studied using mechanical spectroscopies (Knauss et al. 2008). In contrast, little effort went into the investigation of their nonlinear behavior (Davis and Macosko 1978; Lu and Knauss 1998; Lee et al. 2009a; Richert 2001). Under large deformations, techniques such as large amplitude oscillatory shear (LAOS) (Mangalara and G.B., McKenna. 2022) and recently mechanical spectral hole burning (MSHB) (Mangalara and McKenna 2020) have been implemented to characterize the nonlinear behavior of solid polymers. In (2005), Shi and McKenna reported the first attempt on hole burning technique for the investigation of dynamic heterogeneity in a long chain branched polyethylene at a deep rubbery state (i.e., far above the glass transition) (Shi and McKenna 2006). Later, they showed that classical nonlinear models such as the Kaye–Bernstein, Kearsley and Zapas (K–BKZ) theory (Kaye 1962; Bernstein et al. 1963), and Bernstein–Shokooh stress clock model (Bernstein and Shokooh 1980) do not capture the holes. That means that the origin of the holes is not necessarily just related to the application of a large amplitude pump. Moreover, a series of MSHB experiments was conducted on polystyrene solutions with spatial heterogeneity (Qin et al. 2009). The heterogeneity was controlled by varying the polymer molecular weight in the solution. It turned out that the induced heterogeneity does not influence the hole burning event. Similar conclusions were made by Qian and McKenna (2018) on similar polystyrene solutions but in a stress-controlled domain. The MSHB has been extended to investigate the dynamic nonlinearity of glassy polymers. For this purpose, Mangalara and McKenna conducted MSHB on PMMA and polycarbonate (Mangalara and McKenna 2020; Mangalara et al. 2021) at their glassy states (i.e., room temperature). At room temperature, PMMA has a strong β-transition compared to polycarbonate. As the β-transition is related to the polymer backbone motions, they observed weak holes for polycarbonate compared to PMMA, and therefore, it was concluded that the presence of hole is strongly related to the strength of the β-transition in materials dynamics. With this background, it is of interest to study the nonlinear viscoelastic behavior of amorphous polymers at higher temperatures near their glass transition regime. Accordingly, in this work, we investigate the hole burning events on a PMMA material system near its glass transition regime at different pump amplitudes and frequencies. This work will help to complete our understanding on the dynamic nonlinearity of glassy polymers in the vicinity of the glass transition regime from both fundamental and practical perspectives.

6.2 Experiment Method 6.2.1 Mechanical Spectrum Hole Burning In the MSHB experiment, a large sinusoidal deformation with a specific frequency is applied to the sample as the “pump”. Two separate experiments are performed with

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the same pump modification but using either a positive or a negative small constant probe, respectively. Subsequently, the experimental results in the second experiment are subtracted from the results in the first experiment indicated in Eq. (6.1). This subtraction is referred to as “phase cycling.” By doing so, the linear aftereffect due to the pump is eliminated so that nonlinear effects are probed. For comparison, the same procedures used in the non-resonant spectral hole burning (NSHB) work (Schiener et al. 1996; Schiener 1997) are followed to obtain the modified shear relaxation modulus G mod (γ , t), which is analogous to the dielectric response in NSHB: G + (γ , t) = G pump (γ , t) + G(γ , t)mod, G − (γ , t) = G pump (γ , t) − G(γ , t)mod , G mod (γ , t) =

G + (γ , t) − G − (γ , t) , 2

(6.1)

where G + (γ , t) and G − (γ , t) are the measured overall responses due to pump ± probe; G pum p (γ , t) is the aftereffect due to the pump history only; t is the probe time. G mod (γ , t) is compared to the small undisturbed or linear probe response G l i near (t) which was obtained by applying probe strain only (Fig. 6.1). In the plot of modulus versus logarithm of probe time, the mechanical vertical hole is defined as the vertical difference between G l i near (t) and G mod : G(γ , t) = G l i near (t) − G mod (γ , t) Fig. 6.1 Schematic of the large-strain sinusoidal pump followed by positive or negative small-strain probes in the mechanical hole burning experiment. The waiting time is inserted to study the transient characteristics of the mechanical holes

(6.2)

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and the horizontal hole is defined as the horizontal difference between G l i near (t) and G mod (γ , t) at the same modulus level: l og 10 (t) = l og 10 (t)l i near − l og 10 (t)mod

(6.3)

6.2.2 Experiment PMMA rods were obtained from McMaster-Carr. The rods were machined into cylinders with a 25.4 mm gauge length (L) and a diameter of 6.35 mm (2R). The experiments were conducted on an Instron all-electric dynamic and fatigue test system (E10K) with screw-driven clamps. The specimen and the sample mounting parts are shown in Fig. 6.2. A torsion load cell with a dynamic range of ±25 N·m was used for these experiments. The shear strain was measured from the crosshead rotation angle and after applying compliance correction. An environmental chamber is used to maintain an isothermal condition in each experiment. PMMA samples were annealed at 130 °C (Tg + 8 °C) for 1 h to remove any mechanical stress history. They were subsequently cooled down to room temperature (22 °C) at a cooling rate of approximately 5 °C/min. All samples had aging times of more than 2.5 h which is 10 times the entire experimental time so that the physical aging effect will be negligible on the behavior of mechanical holes. The waiting time section (refer to Fig. 6.1) was chosen to be 10 s for all experiments. Fig. 6.2 PMMA specimen with gauge length L and diameter R gripped by wedge clamps on an Instron E10000 All Electric Materials Testing System. An environmental chamber allows reaching isothermal conditions between −100 and 300 °C

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The probe shear strain was set to 0.55% initially and the holding time is 15 min. to reach the equilibrium condition. The sinusoidal deformation, or pump, was applied in a strain-control mode. Prior to each test, the specimen was clamped and placed in the environmental chamber. Then, the chamber temperature was set to 100 °C. The axial load was closely monitored during the heating stage and the axial distance was constantly adjusted to avoid any large axial load on the sample. The experiments were started after reaching steady state conditions with no more significant axial load change. The entire heating process was taken about 1.5 h. The MSHB experiment had two phases for PMMA samples. Phase I included three pump amplitudes (2.76, 5.52, and 8.25%) and three pump frequencies (0.01, 0.1, and 1 Hz). In phase I, all three experiment steps shown in Fig. 6.1 were combined and carried out in one test. To examine the role of prior loading history on the experimental results, we also considered the effect of the experiment sequences. For example, the negative probe strain was applied first in the new sequence prior to applying the positive probe. The PMMA specimens were annealed every time prior to the new sequence. Phase II of the experiments included four pump amplitudes (4.4, 5.5, 6.6, and 8.25%) and 3 pump frequencies (0.005, 0.01, and 0.015 Hz). In phase II, all three experiments illustrated in Fig. 6.1 were conducted separately, meaning that the specimen was annealed each time prior to the next experiment. The experiment amplitude, frequency, and sequence used in phase II were chosen based on the results of phase II to achieve a clearer hole. All experiments have been conducted three times for repeatability. Digital image correlation (DIC) was also implemented to measure and validate the shear deformation within the reduced section. It was found that, at the near glass transition temperature, the unreduced section deformed and resulted in a smaller pump amplitude at the reduced section. We then conducted both Abaqus simulation and DIC analysis for all pump amplitudes used in the experiment. The corresponding pump amplitudes used in the final phase analysis have been corrected to 4.4, 5.5, 6.6, and 8.25%, and the probe shear strain has been corrected to 5.5%.

6.3 Results and Discussion 6.3.1 Linear Regime Determination Stress relaxation experiments were performed to determine whether the probe deformation used in the study (0.55% strain) falls in the linear regime for PMMA. Figure 6.3 shows the shear modulus variation with time for different strains. The modulus does not change with strain for deformations up to 1.1% which indicates the choice of 0.55% of probe strain falls into linear regime.

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Fig. 6.3 Determination of the linear regime: as the increase of the prob deformation, the stress responses decreased. a Shear modulus versus time plot for PMMA at 100 °C and b Isochronous stress–strain curve

6.3.2 Exploration of Pump Amplitude and Frequency for PMMA In the first stage of the experiments, three pump amplitudes (2.76, 5.52, and 8.25%) and three pump frequencies (0.01, 0. 1, and 1 Hz) were used to investigate the mechanical hole for PMMA. Since there is very little work conducted on MSHB on solid polymers near glass transition temperature, a wide range of amplitudes and frequencies were purposely chosen to determine the appropriate parameters for

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obtaining a clear hole. Figure 6.4 shows the graph of a controlled strain and its typical stress response. Figure 6.5 shows the typical modified response and linear response for the MSHB experimental. The initial MSHB results, as shown in Fig. 6.6, indicated that for a given pump amplitude, the hole intensity increases with the increase in pump frequency. However, it is noticed that for large frequencies 0.1 and 1 Hz, the hole did not burn clearly, and the hole intensity cannot be observed. In the case of 0.01 Hz, the hole was clearly observed with a distinctive peak intensity which is marked by the circle in Fig. 6.6. For the high pump frequencies, in this case, 0.1 and 1 Hz, shear softening dominates the short-time response resulting in a shoulder-like incomplete hole. It is also observed that the position of the hole shifted to a longer time with the increase of Fig. 6.4 A typical pump and positive strain experiment graph for 4.4% shear pump amplitude at 0.01 Hz and 100 °C. a Controlled shear strain and b Corresponding shear stress response for PMMA samples

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Fig. 6.5 The modified response and linear response (PMMA, 4.4% pump, 0.01 Hz, 0.55% prob, at 100 °C)

the amplitude. However, we found very little vertical hole difference between pump amplitude 5.52 and 8.25%, so it is necessary to add additional pump amplitude at this level to further reveal the heterogeneity. The initial MSHB results also showed inconsistent results for experiments conducted with the same amplitude and frequency but different loading sequences. Figure 6.7 shows the result of a repeat experiment (2.76% pump amplitude and 0.55% probe strain) with a different loading sequence. The intensities of holes for all three frequencies are relatively large comparing Fig. 6.6a, indicating a dependence of loading sequence and a need of additional annealing.

6.4 Experiments with Tuned Parameters Based on the results from phase I, more dedicated parameters were selected including four pump amplitudes (4.4, 5.5, 6.6, and 8.25%) and three pump frequencies (0.005, 0.01, and 0.015 Hz). These values centered around 10% of pump amplitude and 0.01 Hz frequency which can lead to clear holes. In the second phase, the sample aging process was also modified slightly where samples were stored in an enclosed desiccator with approximately 50% relative humidity produced by placing a saturated salt solution in an enclosed environment. The entire annealing process was repeated between neighboring experiments to prevent any effect from the previous mechanical history. The same heating steps were implemented to ensure reproducibility. In the second phase, experiments were grouped at the same frequency but with different amplitudes each time, and the results are shown in Fig. 6.8. For a given pump amplitude, at 100 °C, the intensity of holes increases with the increase of pump frequency. Such trend is different from PMMA specimens at room temperature (Mangalara and McKenna 2020). Qin et al. (2009) reported that the hole intensities

6 An Investigation of the Nonlinear Viscoelastic Behavior of PMMA Near … Fig. 6.6 Mechanical spectral modification. a Vertical hole for 2.76% pump amplitude, b vertical hole for 5.52% pump amplitude, and c vertical hole for 8.25% pump amplitude at 100 °C and 0.55% probe strain. It can be found that, only at 0.01 Hz a clear hole, the peak intensity was burned

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Fig. 6.7 Vertical hole for the repetition experiment with conditions of 2.76% amplitude, 0.55% probe strain, and different loading sequences

decrease with the increase of pump frequency in the Rouse and rubbery plateau regimes, whereas intensities increase in the rubbery plateau to terminal transition regime as the frequency increases. Such a reverse trend is expected as the experimental temperature is at 100 °C where the PMMA specimens are near the glass transition regime. Additionally, it is noticed that for a given pump frequency, at 100 °C, the vertical hole intensity increases with the decrease of pump amplitude, and the position of the hole shifts to a shorter time. This observation has never been seen for a PMMA hole burning test at room temperature. We suspect the pump amplitude used did not yet reach strain hardening and resulted in a reversed relationship between hole amplitude and pump amplitude, which can be evidence that the dynamics are more homogeneous after yield than before yield. A similar observation is also reported in Lee et al. (2009b) where their probe suggests a closer exponential relaxation as the magnitude of the creep strain increases in the PMMA tensile creep test. A quadratic relationship of the hole intensities, shown in Fig. 6.9a, is consistent with previous works on NSHB (Schiener et al. 1996; Schiener 1997) and MSHB (Mangalara and McKenna 2020; Shi and McKenna 2006; Qin et al. 2009; Qian and McKenna 2018; Mangalara et al. 2021). Figure 6.9b shows the frequency dependence of the vertical hole intensity as a function of pump frequency, both terms are plotted in log scale. A power law dependence is observed for all pump amplitudes with no significant dependence change for the selected amplitude range. It is noticed that vertical hole intensity at 0.005 Hz and 4.4% amplitude deviates from the quadratic behavior. This could be due to shear softening and this data was excluded in the dependence study. The horizontal hole intensity plots were not presented here because at a constant modulus, modified modulus and linear modulus have limited intercept points due to the relatively small relaxation in stress within the probe strain holding time. This can be solved in the future by implementing a longer holding time and construct a master curve of PMMA at a controlled strain and temperature.

6 An Investigation of the Nonlinear Viscoelastic Behavior of PMMA Near … Fig. 6.8 Mechanical spectral modification for PMMA. a Vertical hole for 0.005 pump frequency, b vertical hole for 0.01 pump frequency, and c vertical hole for 0.015 pump frequency at 100 °C and 1% probe strain. It is found that, with the increase in the pumping amplitude, the vertical hole intensity decreases and shifts towards a longer time

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Fig. 6.9 Vertical hole intensity dependence on amplitude and frequency. a Hole intensity as a function of square of pump amplitude and b hole intensity in log scale as a function of pump frequency in log scale

6.5 Conclusion Large-amplitude oscillatory shear and mechanical spectral hole burning have been conducted on amorphous glassy polymer PMMA to investigate the nonlinear dynamics at a temperature of 100 °C, which is near the glass transition regime. The effects of pump frequency and pump amplitude were considered. The experimental parameters were adjusted based on the results from an initial round of MSHB experiments to guarantee a formation of a clear hole. With the frequency range of 0.005–0.015 Hz and pump amplitude range of 4.4–8.25%, the holes were burned and observed clearly for PMMA. It has been shown that the intensity of the vertical holes increases with the increase in the pump frequency and decreases with the increase in the pump amplitude.

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Acknowledgements The authors acknowledge the support of NSF CMMI-1661246/1662474, and the Louis A. Beecherl Jr. Chair.

References Bernstein B, Shokooh A (1980) The stress clock function in viscoelasticity. J Rheol 24(2):189–211 Bernstein B, Kearsley EA, Zapas LJ (1963) A study of stress relaxation with finite strain. Trans Soc Rheol 7(1):391–410 Davis WM, Macosko CW (1978) Nonlinear dynamic mechanical moduli for polycarbonate and PMMA. J Rheol 22(1):53–71 Drobny JG (2011) Polymers for electricity and electronics: materials, properties, and applications. Wiley 352 Kaye A (1962) Non-Newtonian flow in incompressible fluids. Coll Aeronaut Note 134 & 149 Knauss WG, Emri I, Lu H (2008) Mechanics of polymers: viscoelasticity. Springer handbook of experimental solid mechanics, pp 49–96 Lee HN, Paeng K, Swallen SF, Ediger MD, Stamm RA, Medvedev GA, Caruthers JM (2009a) Molecular mobility of poly (methyl methacrylate) glass during uniaxial tensile creep deformation. J Polym Sci Part B: Polym Phys 47(17):1713–1727 Lee H, Paeng K, Swallen SF, Ediger MD, Stamm RA, Medvedev GA, Caruthers JM (2009b) Molecular mobility of poly(methyl methacrylate) glass during uniaxial tensile creep deformation. J Polym Sci 47(17):1713–1727 Lu H, Knauss WG (1998) The role of dilatation in the nonlinearly viscoelastic behavior of PMMA under multiaxial stress states. Mech Time-Depend Mater 2(4):307–334 Mangalara SCH, McKenna GB (2020) Mechanical hole-burning spectroscopy of PMMA deep in the glassy state. J Chem Phys 152(7):074508 Mangalara SCH, Paudel S, McKenna GB (2021) Mechanical spectral hole burning in glassy polymers—investigation of polycarbonate, a material with weak β-relaxation. J Chem Phys 154(12):124904 Mangalara SCH, McKenna GB (2022) Large-amplitude oscillatory shear to investigate the nonlinear rheology of polymer glasses–PMMA. Mech Time-Depend Mater 1–19 Mouritz AP (2012) Introduction to aerospace materials. Elsevier Qian Z, McKenna GB (2018) Mechanical spectral hole burning of an entangled polymer solution in the stress-controlled domain. Phys Rev E 98(1):012501 Qin Q, Doen H, McKenna GB (2009) Mechanical spectral hole burning in polymer solutions. J Polym Sci Part b: Polym Phys 47(20):2047–2062 Richert R (2001) Spectral selectivity in the slow β-relaxation of a molecular glass. EPL (europhys Lett) 54(6):767 Schiener B (1997) Nonresonant dielectric hole burning spectroscopy of supercooled liquids. J Chem Phys 107(19):7746–7761 Schiener B, Böhmer R, Loidl A, Chamberlin RV (1996) Nonresonant spectral hole burning in the slow dielectric response of supercooled liquids. Science 274(5288):750–754 Shi X, McKenna GB (2005) Mechanical hole burning spectroscopy: evidence for heterogeneous dynamics in polymer systems. Phys Rev Lett 94(15):157801 Shi X, McKenna GB (2006) Mechanical hole-burning spectroscopy: demonstration of hole burning in the terminal relaxation regime. Phys Rev B 73(1):014203 Ulery BD, Nair LS, Laurencin CT (2011) Biomedical applications of biodegradable polymers. J Polym Sci Part B: Polym Phys 49(12):832–864

Chapter 7

Accelerated Testing Methodology for Life Prediction of Unidirectional CFRP Under Tension Load Masayuki Nakada and Yasushi Miyano

Abstract An accelerated testing methodology (ATM) for evaluating the long-term durability of CFRP laminates has been developed by the authors during many years. In this chapter, first, the strength of PAN-based CFRP laminates conforms to the time– temperature superposition principle (TTSP) for matrix resin viscoelasticity irrespective of the structural configuration and loading style. These facts were confirmed by experimentation. Therefore, the applicability of ATM to the static, creep, and fatigue strengths of PAN-based CFRP laminates is verified based on experimentation. Second, the formulation for time-dependent and temperature-dependent statistical static, creep, and fatigue strengths for CFRP laminates was done based on the matrix resin viscoelasticity. Third, these tensile strengths for the longitudinal direction of unidirectional CFRP were predicted statistically using the formulated equations. The predicted values were compared with experimentally obtained data measured from resin-impregnated CFRP strands as specimens of unidirectional CFRP. Finally, the statistical long-term tensile creep and fatigue strengths are discussed in terms of the role of the matrix resin viscoelasticity. Keywords CFRP · Durability · Life prediction · Statistics · Viscoelasticity

7.1 Introduction In recent years, materials that possess high specific strength and specific modulus were developed to fulfill the need for advanced lightweight structures. Carbon fiberreinforced plastics (CFRP) are materials that have these properties. They are being used in structures as primary as well as secondary load-carrying members. Therefore, to develop CFRP structures, numerous material properties such as long-term creep and fatigue data as well as static strength data under various environmental M. Nakada (B) · Y. Miyano Materials System Research Laboratory, Kanazawa Institute of Technology, 3-1 Yatsukaho, Hakusan 924-0838, Ishikawa, Japan e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_7

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conditions are needed for all CFRPs considered as a candidate for structural materials. However, carrying out time-consuming creep and fatigue tests under various temperature conditions and under various loading conditions is almost impossible for all CFRPs. The life prediction method for CFRP has been studied using several approaches based on damage metrics such as strength degradation (Hahn and Kim 1975; Reifsnider et al. 2000; André Lavoir et al. 2000; Patel and Case 2000; McBagonluri et al. 2000; Caprino 2000; Schaff and Davidson 1997a, b; Bond 1999; Bond and Farrow 2000; Klemenc and Fajdiga 2000; Tovo 2000; Yao and Himmel 2000; Carmine and Himmel 1998; Himmel 2002), stiffness degradation (Philippidis and Vassilopoulos 1999, 2000; Agfajani et al. 2021), and combination of strength degradation and stiffness degradation (Shokrieh and Lessard 2000a, 2000b; Lee and Jen 2000a, b). Furthermore, fatigue behavior and its prediction have been reported in relation to the stress ratio (Beheshty and Harris 1998; Beheshty et al. 1999; Rotem and Nelson 1989; Petermann and Plumtree 2001; Schön 2001) and off-axis loading (Philippidis and Vassilopoulos 1999, 2000; Petermann and Plumtree 2001; Kawai et al. 2001a, b). The mechanical behavior of CFRP matrix resin exhibits time dependence and temperature dependence, so-called viscoelastic behavior, not only above the glass transition temperature T g but also below T g . Consequently, the mechanical behavior of CFRP presumably depends strongly on time and temperature (Aboudi and Cederbaum 1989; Sullivan 1990; Gates 1992; Kawai et al. 2013; Rojas-Sanchez et al. 2020). Therefore, CFRP durability is highly dependent on the environmental effects of temperature, water absorption, and so on. Numerous studies have examined the durability of CFRP under actual environmental conditions (Alam et al. 2019; Mourad et al. 2019; Li et al. 2019; Nash et al. 2019; Almudaihesh et al. 2020). Our many reports over the years presented an accelerated testing methodology (ATM) for the prediction of the long-term life of CFRP based on the time–temperature superposition principle, which holds for matrix resin viscoelasticity. It was cleared experimentally that our proposed ATM is applicable to CFRP, which consists of PAN-based carbon fiber and thermosetting resin with arbitrary fiber direction, laminate constitution, and loading direction. The overall results were published in an earlier report (Miyano and Nakada 2018). Next, statistical formulations (Miyano et al. 2008; Nakada and Miyano 2013, 2015) for scattered time-dependent and temperature-dependent static, creep, and fatigue strengths of CFRP were done based on Christensen’s viscoelastic crack kinetics (Christensen and Miyano 2006, 2007). Our formulation is based on a viscoelastic crack kinetics that considers not only the time and temperature dependences of strength but also the variation of strength. The long-term creep and fatigue life predictions for the longitudinal direction of unidirectional CFRP with PAN-based carbon fibers and thermosetting epoxy resins were performed successfully by our proposed formulation (Miyano and Nakada 2020). For the long-term life prediction of CFRP, we were not able to find scientific studies other than our formulation. In this chapter, ATM for evaluating the CFRP durability developed by the authors over many years is reviewed by an emphasis on creep and fatigue life prediction for

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the longitudinal direction of unidirectional CFRP with matrix resins of two kinds: thermoset and thermoplastic epoxy resins.

7.2 Generalization of Time–Temperature Superposition Principle for Strength of CFRP The most important condition for the ATM is the time–temperature superposition principle (TTSP) for the nondestructive matrix resin deformation, i.e., the viscoelasticity. The master curve of the creep compliance Dc of matrix resin against the reduced time t  at a reference temperature T 0 shown on the left side of Fig. 7.1 can be constructed by horizontally shifting Dc measured at various temperatures. The horizontal shift amount defined as the time–temperature shift factor aT0 shown on the right side of Fig. 7.1 is the acceleration rate. Figure 7.2 presents a generalization of TTSP for nondestructive matrix resin deformation to those for quasi-static, creep, and fatigue strengths of CFRP. When the same TTSP for the matrix resin deformation holds for the strengths of CFRP, the same acceleration rate aT 0 for the matrix resin deformation holds for the strengths of CFRP, as shown by the following equation:

aT0 (Ti ) =

tsi tci tfi ti = = = , i = 1, 2, 3 t0 ts0 tc0 tf0

(7.1)

where the t 0 and t i are the time of matrix resin deformation under temperature T 0 and T i , t s , t c , t f are the time to failure of CFRP for quasi-static, creep, and fatigue loadings under temperatures T 0 and T i, respectively.

Fig. 7.1 Master curve of creep compliance and time–temperature shift factor (Miyano and Nakada 2020)

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Matrix resin deformation

Strengths of CFRP

Quasi-static:

Creep:

Fatigue:

Fig. 7.2 Generalization of TTSP for matrix resin deformation to those for quasi-static, creep, and fatigue strengths of CFRP (Miyano and Nakada 2020)

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Time compression by elevating the temperature for the matrix resin deformation is also realized for the CFRP strength. Therefore, the long-term strengths of CFRP can be predicted from the measured short-term strengths of CFRP at elevated temperatures and the time–temperature shift factor aT0 (T ) for the matrix resin deformation. Table 7.1 presents experimental verification of TTSP for various FRP strengths. As shown in the table, the PAN-based CFRP strength with thermosetting resins as the matrix conforms to the TTSP, irrespective of the structural configuration and loading style. These facts were confirmed by experimentation.

7.3 Formulations Based on ATM Let us consider a CFRP structure loaded by several forces (Fig. 7.3). Assuming that the matrix resin deformation in CFRP structure during loading is constrained perfectly by carbon fiber rigidity, the time-dependent strength of CFRP structure is controlled by the matrix resin relaxation modulus. We have proposed the formulation of statistical static strength σ s of CFRP based on the matrix resin viscoelasticity, as presented in the following equation in our papers (Miyano and Nakada 2020; Nakada et al. 2019, 2022), as   ∗ 1 E s (t, T ) logσs = logσ0 + log[−ln(1 − Pf )] + n R log α E r (t0 , T0 )

(7.2)

where Pf signifies the failure probability, t denotes the failure time, t 0 represents the reference time, T stands for the temperature, T 0 stands for the reference temperature, σ 0 and α respectively, denote the scale parameter and the shape parameter on the Weibull distribution of static strength. In addition, nR is the viscoelastic parameter. In addition, E r and E s * , respectively, represent the relaxation and viscoelastic moduli of matrix resin. The viscoelastic modulus E s * for the static load with a constant strain rate is calculated as shown below. E s∗ (t, T ) = E r (t/2, T )

(7.3)

Statistical creep strength σ c can be ascertained by shifting the master curve of static strength with log A based on Christensen’s theory for viscoelastic crack kinetics (Christensen and Miyano 2006, 2007). Therefore, the master curve of creep strength σc can be represented with subscript “s” replaced with “c” in Eq. (7.2) as follows: logσc = logσ0 +

  ∗ 1 E c (t, T ) log[−ln(1 − Pf )] + n R log α E r (t0 , T0 )

(7.4)

In this equation, E c * represents the viscoelastic modulus for a constant stress load.

Pitch

Epoxy

–

–

T300/PEEK XN05/828

UD

◯

T700/VE

T300/VE

UD

NCF

PW

◯

−

×

× LT

×

×

–

◯ TB

–

◯

– ◯

◯

◯

–

–

LB

LB

LB

◯

◯

◯

◯

T800S/TR-A33 LB

◯

IM600/PIXA-M

◯

–

◯

–

◯

◯

◯

◯

◯

◯

◯

LB

LB

LB

LB

LT

UT500/#135

T800S/3900-2B

T300/828

T400/3601

Fortafil510/Cape2002

◯

–

PEEK

◯

◯

◯

–

◯

QIL

◯

–

Vinylester

G40-800/5260

QIL

◯

–

QIL

PW

–

◯

◯

SW

◯

–

–

MR50K/PETI-5

UD

–

◯

T400/828

◯

◯

UDO

◯

◯

T300/828 ◯

–

LC

◯

Creep

◯

Static

TTSP ◯

LT

Loading direction

HR40/828

–

T400/828

Fiber/matrix

◯

Type

–

E UD

Dc

Deformation

BMI

PI

Epoxy

Carbon

PAN

Matrix

Fiber

Table 7.1 Experiment-based verification of TTSP for various FRP strengths (Miyano and Nakada 2020)

(continued)

−

×

×

◯

◯

◯

–

–

–

–

◯

◯

◯

◯

◯

◯

–

–

–

◯

Fatigue

104 M. Nakada and Y. Miyano

Vinylester

Glass

UD

PW

–

◯ ◯

◯

–

– UD :

Type E

Dc

Deformation

UDO :

WE18W/VE

◯

◯

–

◯

–

◯

XN05/25P LB

–

◯

E-glass/VE

–

×

YS15/25P

–

◯

XN70/25C

Creep

Static

TTSP

×

LB

Loading direction

×

XN40/25C

XN50/828

Fiber/matrix

–

◯

◯

◯

×

×

–

Fatigue

Notice Dc : Creep compliance, E’: Storage modulus; UD: Unidirectional, Strand, Ring, SW: Satin Woven, PW: Plain Woven, QIL: Quasi-Isotropic Laminates, NCF: Non-crimp Fabric; LT: Longitudinal Tension, LB: Longitudinal Bending; LC: Longitudinal Compression, TB: Transverse Bending; ◯: Applicable, ×: Not Applicable, −: No test

Matrix

Fiber

Table 7.1 (continued)

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Fig. 7.3 Mechanical model for CFRP

E c∗ (t, T ) = E s∗ (At, T )= E r (At/2, T )

(7.5)

The shifting amount log A, as ascertained from slope kR of logarithmic static strength against the logarithmic failure time curve, is calculated using the following equation:   1 , kR = n R m R logA = log 1 + kR

(7.6)

Therein, mR represents the slope of logarithmic relaxation modulus of matrix resin against the logarithmic time curve. We proposed the formulation of statistical fatigue strength of CFRP σ f with fatigue degradation parameter Ff based on the matrix resin viscoelasticity, as shown below (Miyano and Nakada 2020):   ∗   1 E f (t, T ) − Ff log(2Nf ) (7.7) logσf = logσ0 + log[−ln(1 − Pf )] + n R log α E r (t0 , T0 ) The viscoelastic modulus E f * is calculated using the following equation for the cyclic load for the case in which the stress ratio of the minimum stress/the maximum stress is zero:      1 Nf 1 1 Er , T + Er − , T , Nf = f t (7.8) E f∗ (t, T ) = 2 4f f 4f Fatigue degradation parameter Ff , as a function of the number of cycles to failure Nf , is obtainable by the following polynomial function of log(2Nf ), which is determined based on experimentation:

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     3 2  Ff log(2Nf ) = a log(2Nf ) + b log(2Nf ) + c log(2Nf )

(7.9)

The fatigue strength at Nf = 1/2 is equal to the static strength when failure time t is equal to 1/(2f ). The dimensionless fatigue strength Sf of the CFRP strand is defined as shown by the following equation:   ∗   1 σf E f (t, T ) = log[−ln(1 − Pf )] − Ff log(2Nf ) logSf = log − n R log σ0 E r (t0 , T0 ) α (7.10) The Sf is the master curve of fatigue strength when Sf makes statistically one curve for various frequencies and temperatures.

7.4 Experiments 7.4.1 CFRP Strands Employed as Unidirectional CFRP The carbon fiber used for this study is high-strength PAN-based carbon fiber (T300-3000; Toray Industries Inc.). Its mechanical properties, as referred to catalog descriptions, are presented in Table 7.2. Figure 7.4 shows that the CF/TS strand, combined with the carbon fiber and a general-purpose epoxy resin (TS resin: jER828; Mitsubishi Chemical Corp.), were molded using a filament winding system developed by the authors (Miyano et al. 1999, 2000). Actually, 200 CFRP strand specimens were molded at one time using this winding frame. The epoxy resin composition and the CF/TS strand curing conditions are presented in Table 7.3. For comparison, we present test results for a CF/TP strand that consists of the same carbon fiber T300-3000 and thermoplastic epoxy (TP resin: XNR6850V; Nagase ChemteX Corp.) that was molded using pultrusion (Nakada et al. 2019). Table 7.3 shows the matrix epoxy composition and the cure condition for CFRP strands of these two types. The glass transition temperatures T g = 160 °C of the TS resin and 102 °C of the TP resin were ascertained from the peak of the loss tangent against temperature at 1 Hz using a dynamic mechanical analyzer (DMA) testing machine. The CFRP strands’ fiber volume fraction V f = 55% is inferred from the CFRP strand weight. Table 7.2 Carbon fiber and the mechanical properties Carbon fiber

Density (g/cm3 )

Yield (g/1000 m)

Elastic modulus (GPa)

Tensile strength (MPa)

T300-3000

1.76

198

230

3,530

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Fig. 7.4 Molding system for CFRP strands as specimens (Miyano and Nakada 2020)

Table 7.3 Composition and cure schedule of CFRP strand CFRP strand

Carbon fiber strand

Composition of resin (weight ratio)

Cure schedule

CF/TS strand

T300-3000

Epoxy resin (100) Hardener (104) Cure accelerator (1.0)

70 °C × 12 h +150 °C × 4 h +190 °C × 2 h

CF/TP strand

T300-3000

Thermoplastic epoxy resin (100) Cure accelerator (6.5)

150 °C × 0.5 h

Figure 7.5 shows that the gage lengths of CFRP strands are approximately 200 mm. The grip mechanisms of the CFRP strands were developed for static, creep, and fatigue tests conducted at various temperatures.

(1) Specimen and grips

(2) Details of grip

(3) Loading and temperature chamber

Fig. 7.5 Configuration of CFRP strand specimens (Miyano and Nakada 2020)

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(1) Static

(2) Creep

109

(3) Fatigue

Fig. 7.6 Static, creep, and fatigue tests for CFRP strands (Miyano and Nakada 2020)

7.4.2 Test Methods The static testing machine with the temperature chamber shown in Fig. 7.6 can achieve a constant elongation rate over a wide range and constant temperatures of 25–170 °C. Twenty sets of the original creep testing machines and five sets of the original fatigue testing machines with temperature chambers shown in this figure can, respectively, realize a constant load and a constant cyclic load in the maximum load 1 kN and constant temperatures of 25–150 °C.

7.5 Results and Discussion 7.5.1 Viscoelasticity of Matrix Resins The relaxation moduli at various temperatures for thermoset epoxy resin (TS resin) and thermoplastic epoxy resin (TP resin) are shown in Fig. 7.7. These relaxation moduli are determined by the storage moduli measured by the DMA in the shorttime range and creep compliances measured using the creep bending test system in the long-time range based on the linear viscoelastic theory (Christensen 1982). As a result, the relaxation moduli at various temperatures for both resins were obtained in a wide range of time for six to eight decades.

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(a) TS resin

(b) TP resin

Fig. 7.7 Relaxation moduli at various temperatures for TS and TP resins

Figure 7.8 shows master curves of the relaxation modulus for TS and TP resins at the reference temperature T 0 = 25 °C. These master curves are determined by shifting the relaxation moduli horizontally and vertically at various temperatures shown in Fig. 7.7 based on the modified time–temperature superposition principle (Nakada et al. 2011). The time–temperature (horizontal) shift factors and the temperature (vertical) shift factors for TS and TP resins are shown in Fig. 7.9. The modified time–temperature superposition principle holds for the relaxation moduli of both resins because the relaxation moduli at various temperatures superimpose upon each other and make a smooth master curve for a very wide range of time of 12–13 decades for each resin.

(a) TS resin

(b) TP resin

Fig. 7.8 Master curves of relaxation modulus for TS and TP resins

(a) TS resin

(b) TP resin

Fig. 7.9 Time–temperature shift factors and temperature shift factors for TS and TP resins

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The relaxation moduli of both resins show their respective characteristic behaviors. The relaxation moduli of both resins decrease concomitantly with increasing time. The relaxation modulus of TS resin decreases moderately with increasing time. However, that of TP resin decreases drastically with increasing time. Furthermore, it can be seen from Fig. 7.8 that the relaxation moduli measured by creep test indicated by solid symbols are larger than those measured by DMA indicated by hollow symbols. It can be considered that the differences in creep test results from DMA results on the master curves are attributable to hardening caused by physical aging during creep test.

7.5.2 Statistical Static Strengths of CFRP Strands Static tension tests for CFRP strands of two kinds, a CF/TS strand and a CF/TP strand, were conducted at several constant temperatures with cross-head speed of 2 mm/min. The tensile strengths of CFRP strands σs were obtained using the following equation: σs =

Wmax ρ te

(7.11)

Therein, W max represents the maximum load [N]. In addition, ρ and t e, respectively, denote the density of the carbon fiber [kg/m3 ] and the yield (weight per unit length) of the carbon fiber strand [g/1000 m]. Figure 7.10 shows the static strengths of CFRP strands versus temperature. The glass transition temperatures for TS and TP resins are, respectively, 160 °C and 102 °C. Consequently, the temperature dependences of tensile strength for CF/TP and CF/TS strands mutually differ.

Fig. 7.10 Static strengths of CFRP strands against temperature (Nakada et al. 2021)

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(a) CF/TS strand

(b) CF/TP strand

Fig. 7.11 Weibull distributions of static strength of CFRP strands at various temperatures (Nakada et al. 2021)

Figure 7.11 shows the Weibull distributions of these static strengths at various temperatures. Although scale parameter β s decreases according to the temperature rise, shape parameter α s maintains an almost constant value for each of the CFRP strands, as shown in the lower right portion of this figure. The β s and α s at T = 25 °C were selected as σ 0 and α in Eqs. (7.2), (7.4), (7.7), and (7.10). Figure 7.12 portrays the dimensionless static strengths of CFRP strands σ s /σ 0 against the dimensionless viscoelastic compliance of matrix resin E s∗ /E r0 . The relation of σ s /σ 0 against E s∗ /E r0 can be shown as a straight line with a slope of nR , which is the viscoelastic parameter in Eqs. (7.2), (7.4), (7.6), (7.7), and (7.10). Parameters σ 0 , α, and nR for CFRP strands CF/TS and CF/TP are presented in Table 7.4. The relations of σ s /σ 0 against E s∗ /E r0 for both CFRP strands (Fig. 7.12) are obtained as straight lines with almost equal slope of nR = 0.063 (Nakada et al. 2022),

Fig. 7.12 Static strength of CFRP strands against the viscoelastic modulus of matrix resin (Nakada et al. 2021)

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Table 7.4 Parameters for statistical fatigue strength prediction for CFRP strands (Nakada et al. 2021) CF/TS

CF/TP

Scale parameter of static strength of CFRP strand at 25 °C: σ 0 [MPa]

3721

3600

Shape parameter of static strength of CFRP strand: α s

27

16

Viscoelastic parameter of matrix resin: nR

0.060

0.064

Parameter for fatigue strength of CFRP strand: a [10−4 ]

4.28

2.02

Parameter for fatigue strength of CFRP strand: b [10−4 ]

−9.15

12.1

Parameter for fatigue strength of CFRP strand: c [10−4 ]

15.0

18.6

which shows the same failure mechanism for both CFRP strands and the same timedependent and temperature-dependent strength degradation for both CFRP strands, which are controlled by the viscoelasticity of each matrix resin based on Rosen’s model (Rosen 1964).

7.5.3 Statistical Creep Strengths of CFRP Strands Creep failure tests of CFRP strands were conducted. Results of the creep failure tests are presented in Fig. 7.13. Tensile creep strength σ c of CFRP strands was obtained using Eq. (7.11). The predicted creep failure probability against the failure time as calculated by Eq. (7.4) substituting the parameters in Table 7.4 is also shown in Fig. 7.13. The predicted statistical creep failure time agrees well with the experimentally obtained data. Therefore, results clarified that the statistical creep failure time can be predicted from the master curve of relaxation modulus of matrix resin shown in Fig. 7.8 and the parameters for statistical static strength of CFRP strands shown in Table 7.4.

7.5.4 Statistical Fatigue Strengths of CFRP Strands Fatigue tension tests for CFRP strands of two kinds, CF/TS and CF/TP, were conducted at four temperatures with frequency f = 2 Hz and the stress ratio of minimum stress/maximum stress R = 0.1. Tensile fatigue strength σ f of CFRP strands was obtained using Eq. (7.11). The number of cycles to failure N f was measured through testing. Figure 7.14 portrays the fatigue strength versus the number of cycles to failure for CFRP strands of two kinds at various maximum stresses and temperatures. The fatigue strengths of both the CFRP strands decrease markedly with an increasing number of cycles to failure and temperature. Figure 7.5 presents the relation between the dimensionless fatigue strength S f and the number of cycles to failure N f . This relation clarifies only one curve for

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(a) CF/TS strand [46]

(b) CF/TP strand [48]

Fig. 7.13 Creep strength and failure probability against failure time

(a) CF/TS strand

(b) CF/TP strand

Fig. 7.14 Fatigue strength versus the number of cycles to failure for CFRP strands of two kinds (Nakada et al. 2021)

each CFRP strand, which is independent of temperature. We defined these curves as the S–N master curves. These curves are approximated by the polynomial function of Eq. (7.9). Parameters a, b, and c for each CFRP strand are also presented in Table 7.4. Therefore, the statistical fatigue strength at an arbitrary time, temperature, and frequency under a pulsating load can be predicted clearly from the long-term relaxation modulus of matrix resin shown in Fig. 7.8. Parameters for statistical static and fatigue strengths of CFRP strands are presented in Table 7.4. Figure 7.15 also shows that the scatter of N f and the fatigue strength degradation against N f for CF/TP strand are larger than that for CF/TS strand. It can be inferred that this difference derives from the CFRP strand molding method.

7 Accelerated Testing Methodology for Life Prediction of Unidirectional …

(a) CF/TS strand

115

(b) CF/TP strand

Fig. 7.15 S–N master curves for CFRP strands of two kinds (Nakada et al. 2021)

7.5.5 Predictions of Long-Term Statistical Creep and Fatigue Strengths of CFRP Strands Prediction of the long-term relaxation modulus of matrix resin and the statistic creep and fatigue tensile strengths for CFRP strands of two kinds used for this study is performed by substituting the viscoelastic properties of matrix resin from Fig. 7.8 and the material parameters for CFRP strands from Table 7.4, respectively, intfo Eqs. (7.4) and (7.7). Figure 7.16 shows the predicted long-term modulus of resin and the respective creep and fatigue strengths of CFRP strands of two kinds at two typical temperature levels. These figures clarify that both the modulus and creep strength decrease concomitantly with increasing elapsed time and that they accelerate with increasing temperature. The fatigue strength depends strongly on the number of cycles to failure. However, the influence of temperature on fatigue strength is not so strong compared to the effect on creep strength. Figure 7.17 presents a comparison of the predicted time-dependent relaxation modulus, creep, and fatigue strengths for matrix resins and CFRP strands of two kinds at T = 80 °C. This figure clarified that the creep strength is influenced directly by the matrix resin’s viscoelasticity and that the fatigue strength is not so influenced by the matrix resin’s viscoelasticity.

7.6 Conclusions This chapter presented a review of an accelerated testing methodology (ATM), developed by the authors over many years, for measuring CFRP laminate durability. First, the applicability of time–temperature superposition principle (TTSP) for matrix resin viscoelasticity was confirmed experimentally to the strength of CFRP laminates with PAN-based carbon fiber and thermosetting resin irrespective of the structural configuration and loading style. Second, the formulation for statistical determination of timedependent and temperature-dependent static, creep, and fatigue strengths for CFRP laminates was proposed based on the matrix resin viscoelasticity. Third, the validity

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(a) CF/TS strand

(b) CF/TP strand

Fig. 7.16 Prediction of the long-term relaxation modulus of matrix resin and the CFRP strand creep and fatigue strengths (Nakada et al. 2021)

of that formulation was clarified from experimentation to assess tension loading along the longitudinal direction of two kinds of unidirectional CFRP using our developed CFRP strand system. One is the combination of PAN-based carbon fiber and thermosetting epoxy resin. The other is of PAN-based carbon fiber and thermoplastic epoxy resin. Results clarified that the long-term creep and fatigue failure time of two kinds of unidirectional CFRP under tension loading can be readily inferred, statistically, based on results of static tests conducted at various temperatures and fatigue tests conducted at a reference temperature. Furthermore, the characteristic behavior of the long-term durability of two kinds of unidirectional CFRP can be clarified based on the viscoelastic behavior of matrix resin in each unidirectional CFRP.

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Fig. 7.17 Comparison of the long-term relaxation modulus of matrix resin and the creep and fatigue strengths of CFRP strands of two kinds (Nakada et al. 2021)

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Acknowledgements The authors thank Professor Igor Emri of Ljubljana University and Professor Richard Christensen of Stanford University as kind and continuous advisors over the long period of research work conducted by the authors. The authors also thank the many researchers and students of the Kanazawa Institute of Technology who assisted as cooperative researchers.

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Kawai M, Yagihashi Y, Hoshi H, Iwahori Y (2013) Anisomorphic constant fatigue life diagrams for quasi-isotropic woven fabric carbon/epoxy laminates under different hygro-thermal environments. Adv Compos Mater 22:79–98 Klemenc J, Fajdiga M (2000) Description of statistical dependencies of parameters of random load states (dependency of random load parameters). Int J Fatigue 22:357–367 Lee C-H, Jen M-HR (2000a) Fatigue response and modelling of variable stress amplitude and frequency in AS-4/PEEK composite laminates, Part 1: Experiments. J Comp Mater 34:906–929 Lee C-H, Jen M-HR (2000b) Fatigue response and modelling of variable stress amplitude and frequency in AS-4/PEEK composite laminates, Part 2: Analysis and formulation. J Comp Mater 34:930–953 Li H, Zhang K, Fan X, Cheng H, Xu G, Suo H (2019) Effect of seawater ageing with different temperatures and concentrations on static/dynamic mechanical properties of carbon fiber reinforced polymer composites. Compos B 173:106910 McBagonluri F, Garcia K, Hayes M, Verghese KNE, Lesko JJ (2000) Characterization of fatigue and combined environment on durability performance of glass/vinyl ester composite for infrastructure applications. Int J Fatigue 22:53–64 Miyano Y, Nakada M (2020) Accelerated testing methodology for durability of CFRP. Compos B 191:107977 Miyano Y, Nakada M, Kudoh H, Muki R (1999) Prediction of tensile fatigue life under temperature environment for unidirectional CFRP. Adv Compos Mater 8:235–246 Miyano Y, Nakada M, Kudoh H, Muki R (2000) Determination of tensile fatigue life of unidirectional CFRP specimens by strand testing. Mech Time-Depend Mater 4:127–137 Miyano Y, Nakada M, Cai H (2008) Formulation of long-term creep and fatigue strengths of polymer composites based on accelerated testing methodology. J Comp Mater 42:1897–1919 Miyano Y, Nakada M (2018) Durability of fiber-reinforced polymers. Wiley-VCH Mourad AHI, Idrisi AHI, Wrage MC, Abdel-Magid BM (2019) Long-term durability of thermoset composites in seawater environment. Compos B 168:243–253 Nakada M, Miyano Y (2013) Formulation of time- and temperature-dependent strength of unidirectional carbon fiber reinforced plastics. J Comp Mater 47:1897–1906 Nakada M, Miyano Y (2015) Advanced accelerated testing methodology for long-term life prediction of CFRP laminates. J Comp Mater 49:163–175 Nakada M, Miyano Y, Cai H, Kasamori M (2011) Prediction of long-term viscoelastic behavior of amorphous resin based on the time-temperature superposition principle. Mech Time-Depend Mater 15:309–316 Nakada M, Miyano Y, Morisawa Y, Nishida H, Hayashi Y, Uzawa K (2019) Prediction of statistical life time for unidirectional CFRTP under creep loading. J Reinforced Plast Comp 38:938–946 Nakada M, Miyano Y, Kageta S, Nishida H, Hayashi Y, Uzawa K (2021) Prediction of statistical life time for unidirectional CFRTP under cyclic loading. J Reinforced Plast Comp 40:749–758 Nakada M, Miyano Y, Kageta S, Nishida H, Hayashi Y, Uzawa K (2022) Prediction of statistical life time for unidirectional CFRTP under water absorption. J Reinforced Plast Comp 41(7–8):257– 266. Nash NH, Portela A, Bachour-Sirerol CI, Manolakis I, Comer AJ (2019) Effect of environmental conditioning on the properties of thermosetting- and thermoplastic-matrix composite materials by resin infusion for marine applications. Compos B 177:107271 Patel SR, Case SW (2000) Durability of a graphite/epoxy woven composite under combined hygrothermal conditions. Int J Fatigue 22:809–820 Petermann J, Plumtree A (2001) A unified fatigue failure criterion for unidirectional laminates. Compos A 32:107–118 Philippidis TP, Vassilopoulos AP (1999) Fatigue of composite laminates under off-axis loading. Int J Fatigue 21:253–262 Philippidis TP, Vassilopoulos AP (2000) Fatigue design allowables for GRP laminates based on stiffness degradation measurements. Compos Sci Technol 60:2819–2828

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Chapter 8

Application of Time–Temperature Superposition Principle for Polymer Lifetime Prediction Takenobu Sakai and Satoshi Somiya

Abstract In this chapter, we introduce the application of time–temperature superposition principle for influence factors (crystallinity, fiber volume fraction, and physical aging) of polymers and their composites. The basic concept of time-influence factors superposition principle was explained, and the examples were shown in the following section. Consequently, the effect of influence factors can be considered by time and modulus shift factors which are the amount of horizontally and vertically shifting for making master curves. Keywords Polymer · Time–temperature superposition principle · Crystallinity · Fiber volume fraction · Physical aging

8.1 Introduction In recent times, polymers and their composites have been widely used in aerospace, energy, and automobile industries and various infrastructures. The accurate prediction of their lifetimes is important for applications; however, the lifetimes are strongly influenced by viscoelastic properties in the time domain. To predict the lifetime of polymers and their composites, the time–temperature superposition principle (TTSP) is applied to materials, and the long-term time-dependent behavior is predicted based on experiments performed at elevated temperature. Creep behavior, which is related to viscoelastic behavior, influences the reliability of structures. Polymers and fiber-reinforced polymers (FRPs) are used in various environments; thus, the factors that influence their creep behavior include time, temperature, the fiber volume fraction of composites, polymer crystal generation caused by the rearrangement of molecular chains, and physical aging related to the T. Sakai (B) Graduate School of Science and Engineering, Saitama University, Saitama, Japan e-mail: [email protected] S. Somiya Department of Mechanical Engineering, Keio University, Tokyo, Japan

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_8

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stabilization behavior of molecular chains. These factors have been studied by several researchers. Struik studied the time–temperature dependency of the physical aging of amorphous polymers and their composites (Struik 1978). Cangialosi et al. investigated physical aging by measuring the change in the free volume of polycarbonate (PC) and concluded that physical aging depends on pre-aging time and temperature (Cangialosi et al. 2003). Knauss et al. studied physical aging based on the free volume (Knauss and Emri 1987). Miyano et al. investigated the effect of physical aging on thermosetting plastics (Miyano et al. 1993). Other researchers have investigated the physical aging behavior (Hutchinson et al. 1999; Soloukhin et al. 2003; Jong and Yu 1997; Brinson and Gates 1995) and effect of the fiber volume fraction on the creep behavior of PC, glass fiber-reinforced polycarbonate (GFRPC), poly (propylene-co-ethylene) (PPE) composites, and carbon fiber-reinforced polyamide (Biswas et al. 1999, 2001; Iwamoto and Somiya 1995; Somiya 1994; Igarashi and Somiya 1996). The fiber volume fraction can help strengthen the resin matrix. Moreover, the effect of crystallinity on creep behavior has been researched (Chen 1998; Chiang and Lo 1987; Sukhanova et al. 2004); in addition, the crystal grain size of crystalline polymers is considered to be an important factor (Sukhanova et al. 2004). In our research (Sakai and Somiya 2006, 2009, 2011, 2012; Sakai et al. 2009, 2007, 2011, 2015, 2018), we investigated the influence of factors such as physical aging, fiber volume fraction, crystallinity, and crystal grain size on creep behavior. We described these effects as shift factors calculated based on the master curves of the influencing factors. These shift factors delayed the creep time based on the influencing factors. This chapter is based on our previous study (Sakai and Somiya 2011; Sakai et al. 2018) and adds more data.

8.2 Time-Influence Factor Superposition Principle This section introduces the method of applying TTSP to the influencing factors (Fig. 8.1). This figure explains how to make master curve and to obtain the shift factors. The figure on the left has red, green, and blue creep compliance curves, which correspond to the results for different parameters in any influence factor, such as temperature. If the creep curves exhibit time-influence dependency and the curves look similar to those related to the TTSP, the superposition principle can be applied to the influence factors, which are physical aging, fiber volume fraction, and crystallinity in this chapter, by shifting the creep compliance curves for each condition, and we obtained the master curve (dotted curve in graph). For example, in the case of heat treatment, the short-term side of the creep compliance curve with shorter heat treatment exhibits similar behavior as the long-term side with extended heat treatment. Young’s moduli increase with an increase in the fiber volume fraction and crystallinity of polymer composites and crystalline polymers. Therefore, the creep compliance value decreases. At this time, the creep compliance curves should be shifted toward the direction of compliance based on the law of mixtures. Please refer

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Fig. 8.1 Illustration of application of TTSP as time-influence factor superposition principle

to the following section and detail of research were written in the references (Sakai and Somiya 2006, 2009, 2011, 2012; Sakai et al. 2007, 2009, 2011, 2015, 2018).

8.3 Time–Heat Treatment Conditions Superposition Principle Struik et al. investigated the time–temperature dependency of the physical aging of amorphous polymers and their composites (Struik 1978) and described the heat treatment temperature and time (pre-aging temperature and pre-aging time) dependency. We used this dependency to estimate the creep behavior considering the physical aging effect. Below the glass transition temperature, polymers exhibit a thermal history owing to the manufacturing process, which includes molding and curing. Thus, to estimate the creep behavior of polymers, it is necessary to determine their thermal history. However, polymers are stabilized using heat treatment. This behavior is known as physical aging and exhibits the same effect as that of the thermal history. Therefore, to obtain the thermal history, the physical aging state must be determined. The effect of physical aging on polycarbonate (PC) has been reported (Sakai and Somiya 2006, 2009, 2011, 2012). To investigate the effect of physical aging, materials were heat-treated to accelerate physical aging. The creep test was performed at elevated temperatures under various physical aging conditions. Physical aging conditions were controlled with heat treatment with different time and temperature conditions, called pre-aging time and pre-aging temperature. Two experiments were conducted:

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one experiment examined the effect of the pre-aging time prior to creep tests, and the other examined the effect of the pre-aging temperature prior to creep tests. Initially, creep tests were performed at pre-aging temperatures, which were the same as the test temperatures, to study the effect of the pre-aging time on creep behavior. Figure 8.2a shows creep compliance curves for four pre-aging times (0, 100, 300, and 1000 min) at 120 °C for PC. As the pre-aging time increased, the creep compliance curve shifted toward (i.e., downward) longer time periods. The amount of shift toward longer time periods indicated that the physical aging behavior restrained the creep behavior, and the downward shift was caused by an increase in the bending modulus, which was caused by an increase in density. The amount of shift along the vertical axis was small and treated as zero. The creep compliance curve of the nonheat-treated material (i.e., shown at 0 min) was selected as the reference curve, and the other compliance curves were shifted horizontally to create master curves that included the effect of the pre-aging time on creep compliance. The master curves of the pre-aging time were obtained by replacing the real time, t, of each shifted curve with the reduced time, t  , at the reference pre-aging time. The master curve for creep compliance, including the aging effect, is shown in Fig. 8.2b. This master curve was extremely smooth, implying that the fundamental mechanism of creep deformation was the same for each pre-aging time. The shift factor, known as the aging time shift factor, μTt , was obtained by creating a master curve of the aged PC. Figure 8.3 shows the aging time shift factor for all the test temperatures versus the pre-aging time based on a logarithmic scale. The pre-aging time based on the superposition principle was confirmed by the shift factors that formed a line. As the test temperature increased, the slope of the line denoting the aging time shift factor decreased.

Fig. 8.2 a Creep compliance curves of various pre-aging times at 120 °C and the master curve of pre-aging time (reference pre-aging time: t a0 = 0 min)

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Fig. 8.3 Aging time shift factor, μTt , at various test temperatures for PC (reference pre-aging time: t a0 = 0 min)

To investigate the effect of the pre-aging temperature on creep behavior, creep tests were performed at various pre-aging times and temperatures and test temperatures. At a predetermined temperature, pre-aging possessed holding times of 0, 100, 300, and 1000 min. Then, a creep test was performed at predetermined temperatures. Figure 8.4a shows the results of PC pre-aged for 1000 min at various preaging temperatures and a test temperature of 120 °C. As the pre-aging temperature increased, the creep compliance curve shifted toward (i.e., downward) longer times. The creep compliance curves of short time intervals exhibited similar shapes (Fig. 8.4a). The creep compliance curve for pre-aging at 100 °C was selected as the reference curve, and the other compliance curves were horizontally shifted to create a master curve that comprised the effect of the pre-aging temperature on creep compliance using the same method mentioned in the aforementioned paragraph. The master curve for the pre-aging temperature was obtained at the reference pre-aging temperature of T aR = 100 °C. The master curve for the pre-aging temperature is shown in Fig. 8.4b. It was extremely smooth, implying that the fundamental mechanism of creep deformation was the same at the pre-aging temperatures. Comparable results were observed for other test temperatures and pre-aging times. The shift factor, known as the aging shift factor, μRT , was obtained by creating a master curve with regard to the pre-aging temperature of the PC. Figure 8.5 shows the aging shift factor of PC at a pre-aging time of 1000 min and the plots of the aging shift factor for all test temperatures versus pre-aging temperatures. The shift factors with regard to the test temperatures were on one line. Thus, the physical aging variation was caused by the pre-aging temperature instead of the test temperature, and the time–pre-aging temperature superposition principle was confirmed. Figure 8.6 shows the aging shift factor, μRT , for the pre-aging times and the plots of the aging shift factor for the pre-aging times versus pre-aging temperatures. As the pre-aging time increased, the slope of the aging shift factor line increased.

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Fig. 8.4 a Creep-compliance curves of PC pre-aged for 1000 min at various pre-aging temperatures and a test temperature of 120 °C and b master curve for these creep compliances (reference pre-aging temperature: T aR = 100 °C)

Fig. 8.5 Aging shift factor, μRT , at the pre-aging time of 1000 min and the plots of the aging shift factor for all test temperatures versus pre-aging temperatures (reference pre-aging temperature: T aR = 100 °C)

The acceleration of physical aging linearly varied with the pre-aging temperature and time. The relationship between the time, test temperature, and pre-aging temperature was used to analyze the effect of the pre-aging temperature on the creep behavior.

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Fig. 8.6 Aging shift factor, μRT , for the pre-aging times and the plots of the aging shift factor for the pre-aging times versus pre-aging temperatures (reference pre-aging temperature: T aR = 100 °C)

8.4 Time–Fiber Volume Fraction Superposition Principle This section introduces a method to estimate the creep behavior of polymer composites based on the time–fiber volume fraction superposition principle. To estimate the creep deformation of composites, the effect of the fiber volume fraction on the creep behavior must be studied. Previous studies (Sakai and Somiya 2006, 2011) showed the effects of time, temperature, and fiber volume fraction on GFRPC. The weight fractions (volume fractions) for the glass fiber used as test specimens were 0 (0), 10 (5), 20 (10.6), and 30% (16.8%), which are denoted hereafter as G0, G10, G20, and G30, respectively. These studies discuss the applicability of the TTSP. The materials were heat-treated for 1000 min, which is ten times the test time, to avoid the effect of physical aging during the creep test. To confirm the applicability of the TTSP, the master curves of creep compliance were based on creep tests at elevated temperatures. Figure 8.7 shows the master curves (a) and grand master curves (b) of the PC and GFRPC. The master curves were obtained by shifting toward the horizontal axes until they completely overlapped with the reference temperature curve. The grand master curve was obtained by shifting toward the horizontal and vertical axes until these curves overlapped the reference curve to form a single curve representing creep compliance. The creep compliance curve of PC was part of the grand master curve, and the shape of the grand master curve was the same as that of the creep compliance curve of PC. Three types of shift factors, namely the time–temperature shift factor, time shift factor, and modulus shift factor, were used to generate the grand master curve. Here, time–temperature shift factor is used for TTSP, time, and modulus shift factors are the amount of shifting in each master curve to make the grand master curve to horizontal and vertical directions, respectively. Figure 8.8 shows the time–temperature shift factor, which was obtained by creating master curves, and these curves are plotted as Arrhenius-type plots for each fiber volume fraction of the molded PC and

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Fig. 8.7 a Master curve of the creep compliance for PC and GFRPC and b the grand master curve of PC and GFRPC heat-treated for 1000 min (reference temperature: T R = 130 °C; reference fiber volume fraction: V fR = 0%)

GFRPC. These curves show straight lines below 130 °C for each fiber volume fraction and above 130 °C for all the materials. This result demonstrates that the creep phenomenon complies with the TTSP in the Arrhenius mode. Figure 8.9 shows the time shift and modulus shift factors. The time shift factor exhibits the amount of horizontal movement and depends on the restraint of viscosity caused by the amount of fiber. The modulus shift factor indicates the amount of movement along the vertical direction and depends on the modulus. These shift factors show that the temperature and fiber volume fraction retarded the creep behavior. Fig. 8.8 Time–temperature shift factors, aTR (T), for PC and GFRPC heat-treated for 1000 min (reference temperature: T R = 130 °C)

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Fig. 8.9 Time (red solid line) and modulus (blue dash-dotted line) shift factors for the grand master curves of PC and GFRPC heat-treated for 1000 min (reference temperature: T R = 130 °C, fiber volume fraction: V fR = 0%)

8.5 Time–Crystallinity Superposition Principle Thermoplastics exhibit amorphous and crystalline structures; this section describes the effects of polymer crystals on creep behavior (Sakai and Somiya 2011; Sakai et al. 2009, 2018). The crystals exhibited two effects on polymer crystallization: crystallinity, which is the ratio of crystal to amorphous fractions, and the crystalline state, which represents the interior morphology. We describe the effects of crystallinity and morphology on the creep behavior of polyoxymethylene (POM) and the time-crystallinity superposition principle. First, we explain the morphology of the POM crystals using microscopy. The crystal of the polymer changes based on the cooling state from the melting point to the crystallization temperature and heat treatment over the crystallization temperature. To change the crystal structure, the cooling rates of 10 and 50 °C/min were used to observe the crystal initiation behavior using a microscope. Figure 8.10 shows the crystallization behavior of POM on a glass slide. It was confirmed from Fig. 8.10a and c that large spherulites were generated when the cooling rate was slow, and in Figs 8.10b and d, many small spherulites were observed at a high cooling rate. In addition, as shown in Fig. 8.10e, spherulites filled the entire surface of the sample when cooled to room temperature. The crystal grain size at room temperature was measured based on the images. Based on these results, large (L-POM), medium (MPOM), and small (S-POM) with crystal grain sizes of ~320, ~120, and ~30 µm, respectively, were prepared. The particle size of the sample placed on the glass slide was different from the particle size inside the test specimens. After controlling the crystal grain size, the materials were heat-treated at 170 °C to control the crystallinity of POM. For the application of the TTSP, creep compliance and creep compliance master curves were generated based on the creep test results at elevated test temperatures, and the time–temperature shift factors were obtained. Figure 8.11a shows the master

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Fig. 8.10 Spherulites of POM generated at various cooling rates from the melting state

(a)

(b)

(c)

(d)

60 μm

(a): POM - 10 K/min at 153.0 ºC (b): POM - 50 K/min at 153.0 ºC (c): POM - 10 K/min at 151.5 ºC (d): POM - 50 K/min at 151.5 ºC (e): POM - 10 K/min at 25.0 ºC (e)

curves of M-POM, and Fig. 8.11b shows the time–temperature shift factors of MPOM with different crystallinities as straight lines. Thus, the TTSP was applied as an Arrhenius type to M-POM and other materials with arbitrary crystallinity, including S-POM and L-POM. This principle was applied to POM under arbitrary conditions, implying that the creep behavior could be calculated based on the principle. The slope in Fig. 8.11b represents the activation energy of the viscoelastic deformation. As the slope of crystallinity (74%) was the largest, the increased crystallization made viscoelastic movement very difficult. To examine the relationship between the crystallinity and TTSP, the grand master curve with regard to crystallinity was generated by superimposing the master curves of Fig. 8.11a. The master curves were vertically and horizontally moved. Then, the creep compliance grand master curve was obtained (Fig. 8.12a). This curve exhibited a smooth line, although the crystallinity changed. Thus, the mechanisms of viscoelastic deformation are the same. The amount of shift of the grand master curve is referred to as the modulus shift factor for crystallinity, acTDc (vertical direction), and the time shift factor for crystallinity, acTt (horizontal direction). Figure 8.12b shows the relationship between the modulus and time-shift factors for crystallinity. A comparison of the effect of crystallization on the amount of shifting demonstrated that crystallinity significantly affected the shifting of the time shift factor but not the modulus shift factor, compared to the fiber volume fraction case shown in Fig. 8.9. Consequently, S-POM and L-POM exhibited almost the same tendency as M-POM, indicating that the time–crystallinity superposition principle can be applied to POM of the same crystal grain size.

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Fig. 8.11 a Master curves of M-POM and b time–temperature shift factors with different crystallinities

Fig. 8.12 a Grand master curves of the crystallinity of M-POM and b time and modulus shift factors with regard to crystallinity. ◯: Modulus shift factor; ●: Time shift factor

Next, to observe the effect of the crystal grain size on the creep behavior, we compared two master curves with the same crystallinity. The master curves consisted of the creep compliance curves tested at 50, 70, 90, 110, and 130 °C. Figure 8.13 shows the master curves of S-POM, M-POM, and L-POM with the crystallinity of 71%. The three master curves are not the same. The crystal grain size affected the creep behavior; therefore, the behavior can be controlled by controlling the crystal grain size.

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Fig. 8.13 Master curves of all the POM with various crystal grain sizes at a crystallinity of 71%

8.6 Time–Crystallinity–Fiber Volume Fraction Superposition Principle We introduced the superposition principle combining fiber volume fraction, and crystallinity with an example in this section. Creep tests were performed at elevated temperatures to assess the effect of crystallinity on the creep behavior of polyamide (PA) and GFR polyamide (GFRPA) (Sakai et al. 2018). To confirm the effect of crystallinity on the creep behavior, the master curves of PA with crystallinities of 32, 40, and 45% are compared. Figure 8.14a shows the master curves of creep compliance based on measurements at 40, 50, 60, 70, and 80 °C. These data indicate that lower crystallinity and longer creep test duration resulted in higher creep compliance. The grand master curve of crystallinity shifted horizontally and vertically until the individual curves overlapped with the reference curve. Figure 8.14b shows the grand master curve of crystallinity. These data show that the creep compliance curves of PA specimens with different crystallinities are components of the grand master curve. This shows that the creep behavior was the same regardless of crystallinity. The shift factors along the horizontal and vertical directions are shown in Fig. 8.15. The vertical shifts show a change in modulus, whereas horizontal shifts indicate time-retardation effects. The crystalline domains delayed the creep behavior. This delay was hypothesized to be caused by the decreasing molecular mobility with an increase in crystallinity and a consequent increase in viscosity. Similar trends were observed with GFRPA, for which the creep deformation of materials with arbitrary crystallinity could only be estimated using the master curve and shift factors of each fiber volume fraction. Next, the effect of the fiber volume fraction on the creep behavior was estimated for the PA and GFRPA specimens at a fixed crystallinity of χ = 45% using the TTSP in the Arrhenius mode. The master curves of PA and GFRPA were generated

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Fig. 8.14 a Master curves of creep compliance based on measurements at different temperatures and b the corresponding grand master curve of the PA crystallinity (reference curve: χ = 32%; T 0 = 40 °C).

Fig. 8.15 Time (solid line) and modulus (dash-dotted line) shift factors with regard to the crystallinity of PA

at elevated temperatures with the TTSF of PA resin, as shown in Fig. 8.16a, by horizontally shifting the creep compliance curves until they overlapped. Figure 8.16a shows that lower fiber volume fractions and increasing creep test durations yielded higher creep compliance. To confirm this effect, a grand master curve of the fiber volume fraction, as shown in Fig. 8.16b, was generated by horizontally and vertically shifting the individual curves until they overlapped with a reference master curve at V f = 0%. These data show that the creep behavior of GFRPA was the same as that of PA and the creep behaviors of PA and GFRPA depended on the behavior of the

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PA resin. The horizontal and vertical shift factors with regard to the fiber volume fraction are shown in Fig. 8.17. Vertical shifts were caused by changes in the modulus with an increase in the fiber content. The horizontal shifts were caused by the time retardation effect. We discussed the effects of crystallinity and fiber volume fraction on creep behavior. Two effects were observed considering the following variables: changes in the material modulus and the retardation effect on time. We attempted to elucidate the relationship between these two variables. The shapes of the grand master curves in Figs. 8.14b and 8.16b are similar; therefore, we attempted to superimpose these shapes into a single great-grand master curve. To generate such a curve requires

Fig. 8.16 a Master curves created from measurements at different temperatures and b grand master curve for fiber volume fraction in PA and GFRPA with χ = 45% (reference curve: V f = 0%, T 0 = 40 °C).

Fig. 8.17 Time (dash-dotted line) and modulus (solid) shift factors with regard to the fiber volume fraction of GFRPA

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Fig. 8.18 Great-grand master curve obtained from grand master curves for crystallinity and fiber volume fraction of PA and GFRPA

the selection of a suitable reference curve. To achieve this, one must consider the relationship between the two grand master curves at χ = 32% and V f = 0%. The crystallinity of the specimens used to generate the grand master curve of the fiber volume fraction was χ = 45%. This curve was horizontally and vertically shifted using the modulus and time shift factors for crystallinity to create a grand master curve for crystallinity. Thus, we obtained the great-grand master curve, as shown in Fig. 8.18. The great-grand master curve was smooth, indicating that the effects of crystallinity and fiber volume fraction on creep behavior were similar. Furthermore, these data indicate that the creep behavior can be controlled by adjusting the crystallinity and/or fiber volume fraction based on the great-grand master curve and associated shift factors.

8.7 Conclusions In this chapter, we introduced the application of TTSP as “time–pre-heat treatment superposition principle,” “time–fiber volume fraction superposition principle,” “time–crystallinity superposition principle,” and “time–crystallinity–fiber volume fraction superposition principle.” As described in this section, the superposition principle can be applied to various influential factors. The activation energy in the TTSP with regard to various influencing factors can help quantitatively express the effects of those factors on creep behavior; this has not been explained in this chapter. Several methods can be expressed by applying the TTSP.

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References Biswas KK, Somiya S, Endo J (1999) Creep behavior of metal fiber-PPE composites and effect of test surroundings. Mech Time-Depend Mater 3:85–101 Biswas KK, Ikueda M, Somiya S (2001) Study on creep behavior of glass fiber-reinforced polycarbonate. Adv Compos Mater 10:265–273 Brinson LC, Gates TS (1995) Effect of physical aging on long-term creep of polymers and polymer matrix composites. Int J Solids Struct 32:827–846 Cangialosi D, Schut H, Veen A, Picken SJ (2003) Positron annihilation lifetime spectroscopy for measuring free volume during physical aging of polycarbonate. Macromolecules 36:142–147 Chen M (1998) Crystallinity of isothermally and nonisothermally crystallized poly(ether ether ketone) composites. Polym Compos 19:689–697 Chiang WY, Lo MS (1987) Cooling and annealing properties of copolymer-type polyacetals and its crystallization behavior. J Appl Polym Sci 34:1997–2023 Hutchinson JM, Smith S, Horne B, Gourlay GM (1999) Physical aging of polycarbonate: enthalpy relaxation, creep response, and yielding behavior. Macromolecules 32:5046–5061 Igarashi K, Somiya S (1996) Effect of fiber volume fraction on creep compliance of composites of metamorphic poly-phenylene ether with stainless steel fiber. J Jpn Soc Mech Eng 62:1761–1766 Iwamoto N, Somiya S (1995) Effect of the fiber volume fraction on creep compliance of fiber reinforced thermoplastic polyimide: #AURUM. J Jpn Soc Mech Eng 61:1951–1956 Jong SR, Yu TL (1997) Physical aging of poly (ether sulfone)-modified epoxy resin. J Polym Sci, Part b: Polym Phys 35:69–83 Knauss WG, Emri I (1987) Volume change and the nonlinearly thermo-viscoelastic constitution polymers. Polym Eng Sci 27:86–100 Miyano Y, Kasamori M, Nakada M, Tagawa T (1993) Effect of physical aging on creep behavior of epoxy resin. Jpn Soc Mater Sci 42:530–535 Sakai T, Somiya S (2006) Estimating creep deformation of glass-fiber-reinforced polycarbonate. Mech Time-Depend Mater 10:185–199 Sakai T, Somiya S (2009) Effect of Thermal History on the Creep Behavior of Polycarbonate. J Solid Mech Mater Eng 3:1193–1201 Sakai T, Somiya S (2011) Analysis of creep behavior in thermoplastics based on visco-elastic theory. Mech Time-Depend Mater 15:293–308 Sakai T, Somiya S (2012) Estimating the creep behavior of polycarbonate with changes in temperature and aging time. Mech Time-Depend Mater 16:241–249 Sakai T, Tao T, Somiya S (2007) Viscoelasticity of shape memory polymer: polyurethane series DiARY® . J Solid Mech Mater Eng 1:480–489 Sakai T, Ueno S, Yamada K, Somiya S (2009) Creep analysis of polyacetal with effect of crystal state. Exp Mech (Japanese) 9:388–394 Sakai T, Tao T, Somiya S (2015) Estimation of creep and recovery behavior of a shape memory polymer. Mech Time-Depend Mater 19:569–579 Sakai T, Hirai Y, Somiya S (2018) Estimating the creep behavior of glass-fiber-reinforced polyamide considering the effects of crystallinity and fiber volume fraction. Mech Adv Mater Mod Process 4:1–9 Sakai T, Okabe K, Yoneyama S (2011) Effect of powder contents on viscoelastic behavior of glass powder filled epoxy resin. Spec Iss J Jpn Soc Exp Mech 11:187–191 Soloukhin VA, Brokken-Zijp JCM, Asselen OLJ, With G (2003) Physical aging of polycarbonate: elastic modulus, hardness, creep, endo-thermic peak, molecular weight distribution, and infrared data. Macromolecules 36:7585–7597

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Somiya S (1994) Creep behavior of a carbon-fiber reinforced thermoplastic resin. J Thermoplast Compos Mater 7:91–99 Struik LCE (1978) Physical aging in amorphous and other materials. Elsevier Scientific Publishing Co., New York Sukhanova T, Matveeva G, Vylegzhanina M (2004) Morphology and properties of poly(oxymethylene) engineering plastics. Macromol Symp 214:135–145

Chapter 9

Viscoelastic and Viscoplastic Behavior of Polymer and Composite Kenichi Sakaue

9.1 Introduction Thermoplastics are widely used for various products due to the advantages of excellent formability and lightweight. Also, thermoplastics are used as the matrix materials for fiber-reinforced plastics. The thermoplastics are inherently viscoelastic– viscoplastic materials. Therefore, the viscoelastic–viscoplastic analysis is necessary to estimate the structural lifetime and the mechanical behavior of composites. Especially, the viscoplasticity is important to analyze the plastic deformation that occurs near the reinforcement in short fiber reinforced composites and near the fiber discontinuous part in continuous fiber-reinforced composite. Several researches used viscoelastic–viscoplastic analysis to simulate the behavior of particle-reinforced composites under loading–unloading conditions (Kim and Muliana 2009, 2010; Jeon and Muliana 2012; Jeon et al. 2013; Miled et al. 2013). These researches indicate that the viscoelastic–viscoplastic analysis is important to simulate the mechanical behavior of composites. The previous researches frequently use the series-connected model of a viscoelastic element and a viscoplastic element as the viscoelastic-viscoplastic materials (Kim and Muliana 2009, 2010; Tscharnuter et al. 2012; Abu Al-Rub et al. 2014; Tehrani and Abu 2011). The linear viscoelasticity and nonlinear viscoelasticity are expressed by linear viscoelastic constitutive equation and Shapery nonlinear viscoelastic constitutive equation, respectively (Schapery 1966, 1969). The viscoplasticity is expressed by the yield function and the viscoplastic flow rule. The viscoplastic flow rule is given by viscoplastic models such as the Perzyna model and Lemaitre–Chaboche model (Pezyna 1971; Lemaitre and Chaboche 1990). The characteristics of viscoelastic–viscoplastic materials are required to evaluate based on K. Sakaue (B) Department of Mechanical Engineering, Shibaura Institute of Technology, 3-7-5 Toyosu, Koto-ku, Tokyo 135-8548, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_9

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these material models. However, the evaluation of viscoelastic–viscoplastic characteristics requires many material tests; the linear viscoelastic characteristics are usually measured by dynamic viscoelastic measurement, stress relaxation test, and creep test, the nonlinear viscoelastic parameters are evaluated by creep-recovery after identifying linear viscoelastic characteristic, and the viscoplastic parameters are identified by tensile test. Furthermore, the numerical simulations based on the viscoelastic– viscoplasticity require to develop algorithms to be implemented in commercially available software such as finite element method because the commercially available software rarely includes viscoelastic–viscoplastic or viscoelastic–plastic analyses. Because there need numerous efforts on material tests and the development of algorithms, it is not clear which viscoplastic model is suitable to simulate the mechanical behavior of thermoplastics. In this article, polybutylene terephthalate (PBT) resin and short glass fiberreinforced PBT is used as test materials. Firstly, viscoelastic–viscoplastic mechanical model is described. Then, it is shown that the viscoelastic characteristics and viscoelastic characteristics of PBT resin are evaluated through dynamic viscoelastic measurement and tensile test. Finally, the viscoelastic–viscoplastic analysis using finite element method is described to simulate the mechanical behavior of short glass fiber-reinforced PBT.

9.2 Viscoelastic–Viscoplastic Constitutive Model 9.2.1 Series-Connected Model of Viscoelastic and Viscoplastic Elements The series-connected model of a viscoelastic element and a viscoplastic element assumes that total strain tensor ε(t) is the sum of two strains: viscoelastic strain tensor εvp (t) and viscoplastic strain tensor εve (t) (Takaoka and Sakaue 2020). This assumption is expressed as follows: ε(t) = εve (t) + ε vp (t) The viscoelastic and viscoplastic elements are subjected to the same stress σ (t). If the applied stress is smaller than the initial yield stress, the series-connected model is pure viscoelastic model and viscoplastic strain is zero.

9.2.2 Constitutive Equation of the Viscoelastic Elements Linear viscoelasticity is used as the constitutive equation in the viscoelastic element and is expressed as follows:

9 Viscoelastic and Viscoplastic Behavior of Polymer and Composite

t ε (t) =

C(t − ξ ) :

ve

141

dσ (ξ ) dξ dξ

0

where σ (t) is the stress tensor and C(t) is the creep compliance tensor. The components of the creep compliance tensor Ci jkl (t) is expressed by a constant Poisson’s ratio ν and uniaxial creep compliance D(t) as follows: Ci jkl (t) = −ν D(t)δi j δkl +

1+ν D(t)(δik δ jl + δil δ jk ) 2

where δi j is Kronecker delta. The uniaxial creep compliance D(t) is often expressed by the Prony series as follows: D(t) = D0 +

N 

  −t Di 1 − e τi

i=1

where Di and τi are coefficients and retardation times in the Prony series, respectively.

9.2.3 Constitutive Equation of the Viscoplastic Elements If yield function F(σ , σY ) is satisfied, viscoplastic strain εvp (t) appears. The viscoplastic strain rate ε˙ vp is given by the viscoplastic flow rule as follows: ε˙ vp = γ˙

∂Φ(σ , σY ) ∂σ

where Φ(σ , σY ) is a flow potential function and σY is yield stress. γ˙ is a viscoplastic multiplier. The flow potential function Φ(σ , σY ) and the viscoplastic multiplier γ˙ depend on viscoplastic models. Consequently, a viscoplastic constitutive equation requires a yield function F(σ , σY ), a flow potential function Φ(σ , σY ), and a viscoplastic model for viscoplastic multiplier γ˙ . In this article, the von Mises yield function is used as the yield function F(σ , σY ) and flow potential function Φ(σ , σY ), i.e., this flow rule is the associated flow rule that uses the yield function as the flow potential function. The von Mises yield function is expressed as follows: F(σ , σ Y ) = q(σ ) − (σY0 + hεvp ) where, σY0 is the initial yield stress, h is the parameter in linear strain hardening law, q(σ ) is the Mises equivalent stress, and εvp is the equivalent viscoplastic strain. Additionally, a Perzyna model is used for viscoplastic multiplier γ˙ and is expressed as follows:

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  γ˙ =

1 Γ

F(σ ,σ Y ) σY0

κ

0

: F(σ , σY ) ≥ 0 : F(σ , σY ) < 0

where Γ and κ are the viscosity and rate sensitivity, respectively. F(σ , σ Y ) is the von Mises yield function. If the multiplier is rate-independent, the viscoplastic flow rule is the same equation form as the plastic flow rule. In actuality, this viscoplastic model coincides with the plastic flow rule under the limit case of Γ → 0.

9.3 Viscoelastic–Viscoplastic Behavior of PBT Resin 9.3.1 Viscoelastic Characteristics The test material is polybutylene terephthalate, PBT (Sakaue et al. 2020). The viscoelastic characteristics are evaluated by dynamic viscoelastic testing. Rectangular-shaped specimens with 6 mm in width, 1 mm in thickness, and 20 mm in length are used. These specimens are cut off from the test section in ASTM D631 standard tensile specimen by mold injection. The measurement frequencies are 1, 2, and 10 Hz, and temperature is ranged from –100 to 200 °C and increases at a rate of 2 °C/min. Uniaxial tensile force is applied to the specimen, the mean strain is about 1% and the strain amplitude is about 0.1%. Stress relaxation modulus Er (t)  is calculated from the measured storage modulus E (ω) and loss modulus E  (ω) by Ninomiya and Ferry method as follows (Ninomiya and Ferry 1959): Er (t) = E  (ω) − 0.4E  (0.4ω) + 0.014E  (10ω) where ω = 1/t. Then, creep compliance Dc (t) is evaluated by Park–Schapery method (Park and Schapery 1999). Figure 9.1 shows the master curve of stress relaxation modulus Er (t) and creep compliance Dc (t). Figure 9.2 shows the shift factor for time–temperature superposition principle.

9.3.2 Viscoplastic Characteristics The viscoplastic characteristics are evaluated by uniaxial tensile test using the ASTM D631 standard tensile specimen with 3.1 mm in thickness by mold injection. The test section is 57 mm in length, 12.8 mm in width, and 3.1 mm in thickness. The tensile test is performed in a thermostatic chamber, and the test temperature is set to 15, 25, 40, 50, and 60 °C. The cross-head displacement speed is 0.1, 1, and 10 mm/min. The applied force is measured with a load cell with a capacity of 10 kN. The strain in the tensile specimen is measured by digital image correlation, DIC (Yoneyama 2016). For

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Fig. 9.1 Viscoelastic characteristics of polybutylene terephthalate resin

Fig. 9.2 Time–temperature shift factor of polybutylene terephthalate resin

the DIC measurements, the speckle-like pattern is painted on the specimen surface by using white and black spray paint. A white LED light is used to illuminate the specimen in order to prevent temperature increase of the specimen. The measurement area by DIC is 6 mm square in the test section, and the average strain is evaluated in the measurement area. The subset size for DIC is 35 pixel square (0.6 mm2 ). The plots in Fig. 9.3 are true stress-true strain curves evaluated by the uniaxial tensile testing before large deformation in necking. The arrows indicated in Fig. 9.3 mean that the fracture could not be observed in the measured strain region. Figure 9.3 represents the temperature-dependent and strain rate-dependent behavior. In order to identify viscoplastic characteristics, viscoplastic strain is separated from the measured total strain. Firstly, the viscoelastic strain history εve (t) is calculated by the viscoelastic constitutive equation in Sect. 9.2.2 and measured stress history. Then, the viscoplastic strain history εvp (t) is estimated by subtracting the viscoelastic strain history εve (t) from the measured total strain ε(t). Consequently, viscoplastic parameters σY0 , Γ , and κ are determined by least square method from the viscoplastic strain history εvp (t). Here, in order to simplify the identification of

144 Fig. 9.3 Stress–strain curves of polybutylene terephthalate resin at the crosshead speed of a 0.1 mm/min, b 1 mm/min, and c 10 mm/min

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the viscoplastic parameters, the hardening parameter h is assumed to be zero under the assumption that strain hardening and strain softening do not occur. The identified viscoplastic parameters are shown in Fig. 9.4. The solid lines in Fig. 9.3 are true stress-true strain curves simulated by viscoelastic–viscoplastic finite element analysis using identified viscoplastic parameters. In all temperature and strain rate conditions, the viscoelastic–viscoplastic finite element analysis is in good agreement with the tensile test results.

9.4 Viscoelastic–Viscoplastic Behavior of Short Glass Fiber-Reinforced PBT 9.4.1 Test Material The test material is short glass fiber-reinforced PBT composite (Sakaue et al. 2020). The fiber volume fraction of PBT composite is 8.6% and the diameter of the glass fiber is 10 µm. Figure 9.5 shows the surface image of the PBT composite specimen. Figure 9.6 shows the probability density functions for fiber length distribution and fiber orientation distribution. These density functions are obtained by measurement of length and orientation angle of total 700 fibers on two specimen surfaces and on five internal planes that are made by polished 0.5 mm increments from surface. The uniaxial tensile test for PBT composite is the same test method for PBT resin.

9.4.2 Prediction of Behavior of PBT Composite by Finite Element Analysis 9.4.2.1

Finite Element Model

Figure 9.7 shows the geometry of a two-dimensional unit cell model for finite element analysis by ABAQUS/Standard. The fiber volume fraction is 8.6%. The fiber diameter is a half of the actual fiber diameter (Hashimoto et al. 2012). The fiber length varies from 100 to 700 µm in 100 µm increments; i.e., seven-unit cells are used. Aspect ratio of the unit cell model is determined in order that the initial stiffness obtained by FE analysis is consistent with that measured by tensile test of the PBT composite. The unit cell for 100 µm-length fiber is 1,368 nodes and 2,280 elements, and that for 700 µm-length fiber is 17,752 nodes and 15,952 elements. Periodic boundary condition is applied by the degrees of freedom of key nodes (Li et al. 2011). The mechanical characteristics of the matrix part in the unit cell are described in the next subsection. The fiber part in the unit cell is considered elastic with Young’s modulus of 72 GPa and Poisson’s ratio of 0.23. The multiple fiber breaking is simulated by XFEM. The strength at the first breaking point is 1.00 GPa, 1.05 GPa at the

146 Fig. 9.4 Viscoplastic characteristics of polybutylene terephthalate resin: a initial yield stress, b viscosity, and c rate sensitivity

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Fig. 9.5 Macroscopic image on specimen surface of short glass fiber-reinforced PBT composite

Fig. 9.6 Probability density function of a fiber length and b fiber orientation angle in short glass fiber-reinforced PBT composite

Fig. 9.7 Geometry of unit cell for finite element analysis

second, and increases in 0.05 GPa increment at the following breaking parts. The crack by the fiber breaking propagates only in mode I, and the critical energy release rate is 8 J/m2 . Interfacial debonding between the matrix and the fiber is simulated by a cohesive model (Camanho and Davila 2002). The tensile strength and shear strength

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of the interface are 70 MPa, critical energy release rate is 400 J/m2 , and the stiffness between cohesive force and relative distance on the interface is 108 MPa/mm. All unit cells are subjected to two-dimensional strain in the direction of the orientation angles θ . The orientation angle is changed from 0 to 90° in a 10° increment. The input two-dimensional strain history is calculated by the coordinate transformation of the measured strain history by tensile test for PBT composite. Each analysis calculates tensile stress σ (l, θ ). Then, the macro tensile stress σ˜ is superposed based on the lamination theory and is given by the following equation: π

lmax  2

σ (l, θ ) f (l)g(θ )dθ dl

σ˜ = lmin − π2

where f (l) and g(θ ) are the density functions of length and orientation angle in Fig. 9.6, respectively. In this article, the macro tensile stress σ˜ is obtained by superposition of 70 tensile stresses σ (l, θ ): under seven-unit cells with various fiber lengths, and ten orientation angles.

9.4.2.2

Validity of Viscoelastic–Viscoplastic Analysis

Here, the behavior of PBT composites in tensile tests at a temperature of 40 °C and a crosshead displacement rate of 10 mm/min is simulated by viscoelastic–viscoplastic analysis and elastoplastic analyses. The viscoelastic–viscoplastic analysis uses the mechanical characteristic identified in Sects. 9.3.1 and 9.3.2 as the mechanical characteristics of the matrix part in the unit cell. The elastoplastic analysis uses the identified mechanical characteristics from the tensile test. Young’s modulus of 1.49 GPa, Poisson’s rate of 0.35, yield stress of 28.7 MPa, and strain hardening law σ = K εn with K = 77.3 MPa and n = 0.243 are used; this stress–strain curve approximated by elastoplastic model is indicated in Fig. 9.3c as a broken line. Figure 9.8 shows nominal stress–nominal strain curves; plots are the tensile test results. As shown in Fig. 9.8, the stress calculated by elastoplastic analysis is much smaller than that by tensile test in the strain range over 0.01. Therefore, elastoplastic analysis is not suitable to simulate the behavior of PBT composite. This result means that viscoelastic–viscoplastic analysis is required to simulate the behavior of PBT composite. The stress by viscoelastic–viscoplastic analysis without damage is much higher than that by tensile test in the strain range over 0.01 due to high strain rate at edge of the fiber. However, incorporating the damage analyses such as fiber breaking and interfacial debonding to the viscoelastic–viscoplastic analysis, the stress in the strain range over 0.01 becomes close to the value by tensile test. The stress–strain

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Fig. 9.8 Stress–strain curves of short glass fiber-reinforced PBT composite evaluated by tensile test and simulated by finite element analysis at a temperature of 40 °C

curve calculated by viscoelastic–viscoplastic analysis with fiber breaking is discontinuous due to fiber breaking. The stress–strain curve evaluated by tensile test appears between the analysis result with fiber breaking and the analysis result with fiber breaking and interfacial debonding. Therefore, when the more suitable interfacial strength is used for the analysis, the viscoelastic–viscoplastic analysis is thought to simulate the behavior in tensile test more precisely. Figure 9.9 shows nominal stress–nominal strain curves at the temperature of 15, 50, and 60 °C and a crosshead displacement rate of 10 mm/min. The plots are the tensile test results, the broken lines are viscoelastic–viscoplastic analysis with only fiber breaking, the solid lines are the viscoelastic–viscoplastic analysis with fiber breaking and interfacial debonding. The stress–strain curve evaluated by tensile test appears between two analysis results. Therefore, the viscoelastic–viscoplastic analysis can predict the temperature-dependent behavior of PBT composite.

9.5 Summary This article explained viscoelastic–viscoplasticity of PBT resin and short glass fiber-reinforced PBT composite. As mentioned, the mechanical behavior of short glass fiber-reinforced PBT can be simulated by finite element analysis using viscoelastic–viscoplastic characteristics of PBT resin. As the usage of thermoplastic composite increases, it is expected that the viscoelastic–viscoplastic analysis becomes important.

150 Fig. 9.9 Stress–strain curves of short glass fiber-reinforced PBT composite evaluated by tensile test and simulated by finite element analysis at temperatures of a 15 °C, b 50 °C, and c 60 °C

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References Abu Al-Rub R, Tehrani A, Darabi M (2014) Application of a large deformation nonlinearviscoelastic viscoplastic viscodamage constitutive model to polymers and their composites. Int J Dam Mech 25–2:198–244 Camanho PP, Davila CG (2002) Mixed-mode decohesion finite elements for the simulation of delamination in composite materials, NASA/TM-2002–211737. pp 1–37 Hashimoto M, Okabe T, Sasayama T, Matsutani H, Nishikawa M (2012) Prediction of tensile strength of discontinuous carbon fiber/polypropylene composite with fiber orientation distribution. Compos A Appl Sci Manuf 43–10:1791–1799 Jeon J, Muliana A (2012) A simplified micromechanical model for analyzing viscoelasticviscoplastic response of unidirectional fiber composite. J Eng Mater Technol 134–3:31003 Jeon J, Kim J, Muliana A (2013) Modeling time-dependent and inelastic response of fiber reinforced polymer composites. Comput Mater Sci 70:37–50 Kim J, Muliana A (2009) A time-integration method for the viscoelastic-viscoplastic analyses of polymers and finite element implementation. Int J Numer Method Eng 79–5:550–575 Kim J, Muliana A (2010) A combined viscoelastic-viscoplastic behavior of particle reinforced composites. Int J Solids Struct 47–1:580–594 Lemaitre J, Chaboche J (1990) Mechanics of solid materials. Cambridge University Press Li S, Warrior N, Zou Z, Almaskari F (2011) A unit cell for FE analysis of materials with the microstructure of a staggered pattern. Compos A Appl Sci Manuf 42–7:801–811 Miled B, Doghri I, Brassart L, Delannay L (2013) Micromechanical modeling of coupled viscoelastic-viscoplastic composites based on an incrementally affine formulation. Int J Solids Struct 50–10:1755–1769 Ninomiya K, Ferry JD (1959) Some approximate equations useful in the phenomenological treatment of linear viscoelastic data. J Colloid Sci 14–1:36–48 Park S, Schapery R (1999) Methods of interconversion between linear viscoelastic material functions. Part I–A numerical method based on Prony series. Int J Solids Struct 36–11:1653–1675 Pezyna P (1971) Thermodynamic theory of viscoplasticity. Adv Appl Mech 11:313–354 Sakaue K, Ochiai M, Endo S, Takaoka H (2020) Viscoelastic characteristics of short fiber reinforced polybutylene terephthalate. Mech Time Depend Mater 24–3:317–328 Schapery R (1966) An engineering theory of nonlinear viscoelasticity with applications. Int J Solids Struct 2–3:407–425 Schapery R (1969) On the characterization of nonlinear viscoelastic materials. Polym Eng Sci 9–4:295–310 Takaoka H, Sakaue K (2020) Evaluation of viscoelastic-viscoplastic characteristics and finite element analyses for thermoplastics. Adv Compos Mater 29–3:273–284 Tehrani A, Abu Al-Rub R (2011) Mesomechanical modeling of polymerclay nanocomposites using a viscoelastic-viscoplastic-viscodamage constitutive model. J Eng Mater Technol Trans ASME 133–134:041011 Tscharnuter D, Jerabek M, Major Z, Pinter G (2012) Uniaxial nonlinear viscoelastic viscoplastic modeling of polypropylene. Mech Time Depend Mater 16–3:275–286 Yoneyama S (2016) Basic principle of digital image correlation for in-plane displacement and strain measurement. Adv Compos Mater 25–2:105–123

Chapter 10

Using Asymptotic Homogenization in Parametric Space to Determine Effective Thermo-Viscoelastic Properties of Fibrous Composites A. N. Vlasov, D. B. Volkov-Bogorodsky, and V. L. Savatorova Abstract In this paper, we use asymptotic homogenization in parametric space to determine thermoviscoelastic properties of materials composed of elastic fibers embedded in viscoelastic matrix. In comparison with the traditional method of asymptotic homogenization, our approach allows treating temperature as a parameter and studying the response of mechanical characteristics to the change in temperature. We use Maxwell’s model of viscoelasticity and start with the Fourier transform of the governing equations. Then we perform asymptotic homogenization of the resulting equations in parametric space. We solve the problem on the periodic cell consisting of a fiber surrounded by the matrix. Then we derive macroscopic equations for the complex Young and shear moduli. For different frequencies, we determine viscoelastic properties of the material and their sensitivity to changes in temperature. The obtained results can provide information on how fibers’ concentration and properties can affect effective characteristics and strain or stress relaxation patterns. Keywords Structurally inhomogeneous thermoviscoelastic media · Parametric method of asymptotic homogenization · Cell problem · Effective characteristics of thermoviscoelastic materials · Maxwell’s model · Storage and loss moduli

10.1 Introduction Modeling of the viscoelastic and thermoviscoelastic behavior of composite materials finds its applications in civil engineering, automobile industry, medicine, land mine detection, ultrasonic imaging, etc. (Lakes 2009). The theory of viscoelastic composite materials originates in the works (Hashin 1965, 1970a, b) and (Christensen 1971) where the composite spheres assemblage model was used together with the correspondence principle between linear elastic and viscoelastic behaviors A. N. Vlasov · D. B. Volkov-Bogorodsky Institute of Applied Mechanics RAS, Moscow, Russia V. L. Savatorova (B) Central Connecticut State University, New Britain, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_10

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to predict effective moduli of linear viscoelastic composites. Among the more recent examples of micromechanical models, we would like to mention papers (Li et al. 2006) and (Muliana and Kim 2007). One may want to refer to works (Ezzat et al. 2015; Muliana and Haj-Ali 2008) and (Khan and Muliana 2010) devoted to study of thermomechanical properties of viscoelastic composites. The problem of determining mechanical and thermomechanical properties of viscoelastic composites can be computationally expensive due to the presence of heterogeneities and possibility of high contrast between the properties of the matrix and inclusions. To reduce the cost of computations, it is common to use homogenization methods which allow replacing inhomogeneous material with a homogeneous with equivalent properties. As examples of classical asymptotic homogenization, we can mention texts (Bakhvalov and Panasenko 1989; Bensoussan et al. 1978) and (Pobedrya 1984). Concerning with asymptotic homogenization techniques used to model viscoelastic materials, we would like to refer to papers (Cruz-González et al. 2017, 2020; Otero et al. 2020; Rodriguez-Ramos et al. 2020; Yu and Fish 2020) and Vlasov et al. (2022). For example, in Yu and Fish (2020), Yu and Fish considered Kelvin–Voight model and obtained a homogenized solution of the coupled thermoviscoelastic problem valid for certain relationships between temporal and spatial scales. In Rodriguez-Ramos et al. (2020), Rodriguez-Ramos et al. worked in Laplace–Carson space and obtained effective coefficients for the case of fibrous viscoelastic composites with square and hexagonal cells. In Otero et al. (2020), Otero et al. used asymptotic and numerical homogenizations to determine the effective properties of linear viscoelastic fibrous composites in the cases when the unit cell is square or hexagonal. Cruz-Gonzalez et al. (see Cruz-González et al. (2017, 2020)) calculated the effective properties of non-aging viscoelastic composites in one- and three-dimensional cases. In our recent work (Vlasov et al. 2022), we considered the case of spherical inclusions in viscoelastic matrix and used the asymptotic homogenization in parametric space to get effective relaxation functions. This work is a further development of the method of asymptotic homogenization in parametric space first proposed in Vlasov and Volkov-Bogorodsky (2014) and then expanded in Vlasov and Volkov-Bogorodsky (2018, 2021) and Vlasov et al. (2022). One of the advantages of our approach is the fact that it does not require using any specific model of viscoelasticity. It can deal with nonlinear effects, for example, with nonlinear dependences of physical properties on temperature. At the same time, the method has its limitations, especially in the context of nonlinear viscoelasticity. We need to mention that the Fourier transform needs to be applied to make the transition into the frequency domain. Also our method requires the regularity of the microstructure and cannot be used if this regularity is not preserved due to the high temperatures, high amplitudes of oscillations, or high energy dissipation on one cycle of oscillations. The novelty of the current work is that we generalized our approach on the case of orthotropic materials composed of elastic fibers embedded in viscoelastic matrix and were able to take into account the effect of temperature expansion on effective mechanical characteristics. Maxwell’s model of viscoelasticity was chosen to address the situation when the strain was a harmonic function of time. As a result,

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we derived complex elastic and shear moduli and got the effective storage and loss Young and shear moduli as functions of frequency for different temperatures. The structure of the paper is the following. In Sect. 10.2, we set up the problem in the form of the system of equations of thermo-viscoelasticity. In the frames of Maxwell’s model, viscoelastic composite behaves as linear elastic material with complex-valued moduli. In Sect. 10.3, we define and perform asymptotic homogenization of the equations with complex moduli in parametric space and derive the expressions for the effective characteristic of the material. Section 10.4 is devoted to the solution of the cell problems on the microscale. In Sects. 10.5 and 10.6 we present our numerical results and make conclusions.

10.2 Viscoelastic Fiber-Reinforced Composites. Maxwell’s Model The paper discusses a numerical-analytical method developed to determine thermomechanical behavior of composite materials. The composite is formed by filling a viscoelastic matrix with elastic fibers. Mechanical properties of the matrix and inclusions can be contrasting. Our method allows to present contact conditions on the surface of the inclusion in an accurate analytical form. Let us assume that viscoelastic characteristics of the matrix can be described by the Maxell model with temperature-dependent parameters. The constitutive relation is written as a relation between the strain tensor ε and stress tensor σ in the form: ∂εi j ∂σkl = ri jkl + ri∗jkl σkl ∂t ∂t

(10.1)

Here, ri jkl and ri∗jkl (i, j, k, l = 1, 2, 3) are the elastic and viscoelastic parts of the compliance tensor, respectively, with the components that depend on temperature T : ri jkl = ri jkl (T ), ri∗jkl = ri∗jkl (T ). Our model takes conductive heat transfer into consideration. The corresponding constitutive relations are given by Fourier’s law: qi = −κi j (T )

∂T ∂x j

(10.2)

where qi is the heat flux, and κi j (T ) are the components of the tensor of thermal conductivity, that also can depend on temperature. In case of isotropic materials, components ri jkl and ri∗jkl of the compliance tensor and components κi j of the tensor of thermal conductivity can be defined as ri jkl =

 1  δik δ jl + δ jk δil + 4μ



 1 1 − δi j δkl , κi j = κδi j E 2μ

(10.3)

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ri∗jkl

 1  δik δ jl + δ jk δil + = 2η K



 1 1 δi j δkl − 3η K ημ 3

(10.4)

where E is Young’s modulus, μ is shear modulus, κ is the coefficient of thermal conductivity, η K is volumetric viscosity coefficient, and ημ is a shear viscosity coefficient. In case of isotropic materials, the constitutive relations (10.1) can be rewritten as a system of equations for the invariants of the stress and strain tensors: 





σi j σ p ∂εi j 1 ∂σ p 1 ∂σi j ∂θ = + = + , ∂t K ∂t η K ∂t 2μ ∂t ημ

(10.5)



where σ p is the first invariant of the stress tensor, σi j = σi j − δi j σ p is deviatoric  stress, θ is volumetric deformation (the first invariant of the strain tensor), εi j = εi j − δi j isdeviatoricstrain, and K = Eμ/(9μ − 3E) is volumetric deformation modulus. We consider a Maxwell-type model of composite material with viscoelastic matrix and elastic cylindrical inclusions (fibers). The viscoelastic matrix obeys the law of deformation defined by Eq. (10.5). The law of deformation for elastic fibers can formally be written in the form of Eq. (10.5) where η K → ∞ and ημ → ∞. We are going to study viscoelastic properties of fiber-reinforced composite material for various temperatures. We are also interested in learning about composite’s damping properties at certain frequencies of steady-state harmonic oscillations. In the case of harmonic oscillations with angular frequency ω, Maxwell’s viscoelastic materials behave like a linear elastic media with complex-valued moduli defined by the formula (Bateman and Erdélyi 1954): 

∗

θ = 

   i i 1 1  ∗ ∗ σ p , εi j = σ∗ − − K ωη K 2μ ωημ i j

(10.6)



where θ ∗ , σ p∗ , εi∗j , σi j∗ are the amplitudes of the corresponding deformations and stresses. Since K and μ depend on temperature T , it is present in Eq. (10.6) as a parameter. The propagation of heat is still defined by Eq. (10.2) with a temperature-dependent thermal conductivity tensor. Inverting relations (10.6), we obtain stiffness tensor ci∗jkl for the case of harmonic oscillations in a viscoelastic medium:       1 + i K ωηk 1 + 2iμ ωημ ∗ ∗  ∗  ∗ (10.7) σp = K 2 θ , σ i j = 2μ 2 εi j     1 + K ωηk 1 + 2iμ ωημ 

ci∗jkl

     1 + 2iμ/ωημ 1 + i K /ωη K δi j δkl =μ 2 δik δ jl + δil δ jk − 2δi j δkl /3 + K  1 + (K /ωη K )2 1 + 2iμ/ωημ

(10.8)

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The real part of the complex modulus defining our tensor is called the storage modulus, the imaginary part is the loss modulus (Christensen 1971). In material science, it is generally accepted that the damping properties of a viscoelastic medium are determined by the ratio of these two modules, i.e., by the tangent of the loss angle that has the meaning of the fraction of energy dissipated during harmonic oscillations. To set up the problem, let us consider the system of equations of thermoviscoelasticity with rapidly changing coefficients that also depend on temperature: ∂ ∂ xi



∂u Ai∗j (x/ε, T )

∗

∂x j

  ∂ ∂T + F (x) = 0, κi j (x/ε, T ) + h(x) = 0 (10.9) ∂ xi ∂x j ∗

∗ ∗ ∗ Here, Ai∗j = cik jl  are complex matrix functions; u = u i  is the amplitude of the ∗ ∗ displacement for harmonic oscillations; F = F  is the amplitude of volumetric forces; h is the density of heat sources; ε is a small parameter representing the distance between fibers in a macro-sample with a characteristic size L (L  ε). The condition of ideal contact is assumed on the surface of cylindrical inclusions:



 

∂ u∗ ∂T = 0, [T ] = ni κi j (ξ, T ) = 0, x ∈ , ξ = x/ε u∗ = ni Ai∗j (ξ, T ) ∂x j ∂x j (10.10)

Here, square brackets represent the jump of the specified values on the surface of contact, n = n i  is an outward normal (with respect to inclusion), ξ = x/ε are “fast” variables (see Vlasov and Volkov-Bogorodsky (2018, 2021)). Effective thermomechanical characteristics of non-homogeneous medium can be determined using the method of asymptotic averaging of equations with rapidly oscillating coefficients (see Bakhvalov and Panasenko (1989)). In this formulation, the geometry of inclusions does not have a fundamental significance for the asymptotic averaging procedure. At the same time, it determines the possibility of obtaining analytical solutions for auxiliary problems in a convenient closed form and allows finding the effective characteristics of the material using the analytical–numerical methods for the cell problem. In the next section, we present the method of asymptotic homogenization in parametric space (see Vlasov and Volkov-Bogorodsky (2021, 2014)) and use it to obtain effective characteristics of the composite treating temperature as a parameter.

10.3 Asymptotic Homogenization of the Equations with Complex Moduli in Parametric Space Based on the structure of Eqs. (10.9) and (10.10), their solutions are considered as rapidly oscillating 1-periodic functions of fast variable ξ also depending on the slow coordinate x and on temperature T . Implementing the method of asymptotic

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homogenization, we will look for the solution for the problem (10.9) and (10.10) in the form of asymptotic series and keep only the first two terms (see Vlasov and Merzlyakov (2009), Vlasov and Volkov-Bogorodsky (2021), Vlasov et al. (2022)):   u∗ (x) = V 0 (x) + εu(1) (ξ, x, T0 ) + O ε2

(10.11)

  T (x) = T0 (x) + εT (1) (ξ, x, T0 ) + O ε2

(10.12)

Notice that the first terms of expansions V 0 (x) and T0 (x) depend only on slow variable x and have a meaning of the average of u∗ and T over one period of ξ . The second terms u(1) and T (1) in asymptotic expansions (10.11) and (10.12) are rapidly oscillating functions of the fast variable ξ , and we assume that the result of their averaging over the volume of a periodic cell is zero: 1 1 1

u (1) (ξ, x, T0 ) = ∫ ∫ ∫ u (1) (ξ, x, T0 )dξ = 0, 0 0 0

1 1 1

T (1) (ξ, x, T0 ) = ∫ ∫ ∫ T (1) (ξ, x, T0 )dξ = 0, 0 0 0

Angular brackets . . . are used as a notation for a quantity averaged over the volume of a periodic cell. The temperature T0 is fixed and treated as a parameter. The structure of functions u(1) (ξ, x, T0 ), T (1) (ξ, x, T0 ) and their dependences on fast and slow variables are determined from the asymptotic analysis of Eqs. (10.9) and (10.10). Asymptotic homogenization of equations with rapidly oscillating coefficients in parametric space requires writing the derivative of a function in the form ∂ ∂ T0 ∂ ∂ = ε−1 + Dk + ∂ xk ∂ξk ∂ xk ∂ T0

(10.13)

where D k denotes the derivative of the function with respect to slow variable xk regardless of the variable ξ and temperature T0 considered as a parameter. Using Eqs. (10.11) and (10.12) together with Eqs. (10.9), (10.10), and (10.13), we can establish the structure of the terms of the formal asymptotic series (10.11) and (10.12) (see Vlasov and Volkov-Bogorodsky (2021), Vlasov et al. (2022)). For functions of slow variables V 0 (x) and T0 (x), we derive homogenized equations that do not depend on ξ and are formally obtained by averaging the original Eqs. (10.9) and (10.10) over ξ . In order to determine effective characteristics of the medium, we are going to focus on the structure of the second terms in asymptotic series (10.11) and (10.12). Asymptotic analysis allows us to define them in the form

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u(1) (ξ, x, T0 ) = Ni1 (ξ, T0 )D i1 V 0 (x), T (1) (ξ, x, T0 ) = m i1 (ξ, T0 )D i1 T0 (x) (10.14) where Ni1 (ξ, T0 ) and m i1 (ξ, T0 ) are 1-periodic functions of fast variable ξ that also depend on parameter T0 . Substituting the asymptotic expansions (10.11) and (10.12) into Eqs. and   (10.9) (10.10) using Eq. (10.13) and collecting the terms of the order of O ε−1 , we are getting the system of equations: ∂ ∂ξi

 ∂ N +ξ I Ai∗j (ξ, T0 ) ( i1∂ξ j i1 ) = 0,

∂ ∂ξi

 ∂ m i1 +ξi1 ) κi j (ξ, T0 ) ( ∂ξ = 0, j

(10.15)

With corresponding contact conditions on the boundaries:





Ni 1 =

∂ Ai∗j (ξ, T0 )



     

Ni1 + ξi1 I ∂ m i1 + ξi1 n i = 0, m i1 = κi j (ξ, T0 ) ni = 0 ∂ξ j ∂ξ j

(10.16) Equations (10.15) and (10.16) represent the problem on a periodic cell and will be used to determine the effective characteristics of the material. Collecting the terms of the order of O(1) and performing the averaging over one period of fast variables ξ , we derive the macroscopic homogenized equations for V 0 (x) and T0 (x) in the form ∂ ∂ xi



∂V0 Ai j (T0 ) ∂x j





  ∂ ∂ T0 + F (x) = 0, κ i j (T0 ) + h(x) = 0 ∂ xi ∂x j

(10.17)

    ∂ m j + ξj ∂ Nj + ξj I

, κ i j = κi j (ξ, T0 ) ∂ξk ∂ξk

(10.18)

∗



where 

Ai j (T0 ) = Ai∗j (ξ, T0 )



Equations (10.17) are homogenized macroscopic equations of thermoviscoelasticity. Effective thermomechanical characteristics defined by Eqs. (10.18) are the result of the solution of the cell problem (10.15) and (10.16). Notice that Ai∗j are the 



complex moduli; Ai j = ci jkl  is the complex matrix of the effective stiffness tensor; κ i j are the components of effective tensor of thermal conductivity. Both Ai j and κ i j depend on parameter T0 , thereby they define an algorithm for obtaining constitutive relations for effective characteristics of thermoviscoelastic material. Effective characteristics of the material are computed as average of solutions of cell problems (10.15) and (10.16) in accordance with (10.18). Since we are interested in effective characteristics of the material composed of cylindrical inclusions embedded in the viscoelastic matrix, effective coefficients Ai j will be defined by six independent complex moduli: E , ν , G (in the orthogonal plane across the fibers) and Eˆ  , νˆ  , Gˆ  (in the longitudinal plane along fibers) representing 













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technical characteristics of composite. Additionally, for effective coefficients κ i j we have two real coefficients of thermal conductivity κ and κˆ  describing the heat transfer across and along the fibers, respectively. As a result, the constitutive relationships for an effective thermoviscoelastic material are defined as follows: 

σ1∗ −ˆν σ2∗ Eˆ

ε1∗ =

−

νˆ  ∗ σ , Eˆ  3

ε2∗ =

∗ γ12 =

q1 = −κˆ

σ2∗ −ˆν σ1∗ Eˆ

−

νˆ  ∗ σ , Eˆ  3

ε3∗ =

σ3∗ −ˆν  (σ1∗ +σ2∗ ) Eˆ 

∗ τ12 τ∗ τ∗ ∗ ∗ = 13 , γ23 = 23 , γ13 Gˆ Gˆ  Gˆ 

(10.19) (10.20)

∂T ∂T ∂T , q2 = −κˆ , q3 = −κˆ  ∂ x1 ∂ x2 ∂ x3

(10.21)

Here, ε1∗ = u ∗1,1 , ε2∗ = u ∗2,2 , ε3∗ = u ∗3,3 , γi∗j = u i,∗ j + u ∗j,i , σi∗ (i, j = 1, 2, 3) are principal components of the complex stress tensor and τi∗j are the tangent components of the complex stress tensor. Inverting relations (10.19), we obtain the components of the complex stress tensor in the form:     σ1∗ = C ε1∗ + λε2∗ + λ ε3∗ , σ2∗ = λε1∗ + C ε2∗ + λ ε3∗ , σ3∗ = λ ε1∗ + ε2∗ + C  ε3∗ (10.22) 















where the following relationships are established between the stiffness moduli and the technical characteristics: λˆ  Cˆ  λˆ − λˆ 2  , νˆ = Eˆ = Cˆ − νˆ λˆ − ν  λˆ  , Eˆ  = Cˆ  − 2νˆ  λˆ  , νˆ = , Cˆ  Cˆ − λˆ 2 Cˆ + λˆ

 λˆ  Cˆ − λˆ  (10.23) νˆ = Cˆ  Cˆ − λˆ 2 



















Here, C = c1111 , C = c3333 , λ = c1122 , λ = c1133 are the components of the stiffness tensor. The matrix form Ai j = ci jkl  of this tensor is defined by the Eq. (10.18). Note that the symmetry relationships are fulfilled in the case of cylindrical inclusions: c1111 = c2222 , c1133 = c2233 . Now we are going to show that, in case of fiber-reinforced composite, the problem of finding quantities we need to compute the moduli in Eq. (10.23) can be reduced to a problem defined on a cross section of a fiber with special conditions of a periodic jump for displacements and for temperature at the periodic cell boundaries. In the class of periodic functions, matrix Eq. (10.15) can be written as vector problems for homogeneous equations with respect to column vectors of the matrix Ni1 + ξi1 I (there are nine of them). We can interpret them as functions satisfying 











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the conditions of a “periodic jump” for certain components in a given direction (see Vlasov et al. (2022)). For the case of cylindrical inclusions, we consider the problem of a “periodic jump” in the direction of the ξ1 axis for the first and second columns of the matrix N1 +ξ1 I , which we will denote by symbols U (1) and U (2) . Both of these functions are defined by the plane strain conditions, depend only on variables  ξ1 and ξ2 , and have the  domain in the shape of the unit square |ξi | < 21 , i = 1, 2 with a circular inclusion {r < r0 }. Also, U (1) and U (2) satisfy two-dimensional Lame equations complemented by conditions of no jump of displacements and stresses at the boundaries of the inclusions: 

 ∂σi∗(k) ∂U (k) = 0, U (k) = σ i∗(k) n i = 0, σ i∗(k) = Ai∗j = σ ∗(k) i j , k = 1, 2 ∂ξi ∂ξ j (10.24) Being written in the form (10.24), the Lame equations are presented in an implicit form of equilibrium equations for the isotropic media. Being reformulated for vector functions U (k) , Eq. (10.24) have the following form:   μ∗ ∇ 2 U (k) + μ∗ + λ∗ ∇divU (k) = 0, k = 1, 2

(10.25)



   1 + 2iμ/ωημ 2μ∗ 1 + i K /ωη K ∗ μ =μ − 2 , λ = K  2 3 1 + (K /ωη K ) 1 + 2μ/ωημ ∗

(10.26)

Notice that Eqs. (10.25) and (10.26) written for inclusions and for the matrix will have a similar structure, only material coefficients will be different. According to the first Eq. (10.18), components of matrices C = c1111 and λ = ∗(1) ∗(1) c1122 are determined by complex stresses σ11 and σ22 that are derived using U (1) while components of complex shear modulus G = c1212 are determined by complex ∗(2) that are derived using U (2) : stresses σ12 

















∗(1) ∗(1) ∗(2) C = σ11

, λ = σ22

, G = σ12

(10.27)

All these quantities can be found by solving the equations of the plane theory of elasticity on a cross section of a fiber. 







Quantities C and λ are determined by the third column of the matrix N3 + ξ3 I , which we denote by the symbol U (3) and which corresponds to the problem with a periodic jump in the direction of the ξ3 axis: 







∗(3) ∗(3) C = σ33

, λ = σ11

(10.28)

The solution to this problem can be written explicitly using the function U (1) :

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   [K − 2μ/3] (1) U1 (ξ1 , ξ2 ) + U2(1) (ξ1 , ξ2 ) − ξ1 U (3) = U1(3) , U2(3) , U3(3) , U1(3) = 2[K + μ/3] (10.29)

 [K − 2μ/3] (1) U (ξ1 , ξ2 ) + U1(1) (ξ1 , ξ2 ) − ξ2 ,U3(3) = ξ3 U2(3) = (10.30) 2[K + μ/3] 2 where [K − 2μ/3] and [K + μ/3] denote the difference of respective characteristics for the matrix and inclusion.  The remaining quantity is the shear modulus along fibers G = c1313 , and it is derived using the third component U (4) (ξ1 , ξ2 ) of the third column of the matrix N1 + ξ1 I (the rest of the components are equal to zero). Nonzero component U (4) (ξ1 , ξ2 ) satisfies Laplace equation defined on the fiber’s cross section: 

   

∂U (4) ∂U (4) G = μ , ∇ 2 U (4) = 0, U (4) = μ =0 ∂n ∂n





(10.31)



Note that the coefficient of thermal conductivity κ is determined in the same manner under an assumption of a periodic jump of the function  = m 1 + ξ1 in the direction of ξ1 :   ∂ ∂ 2 , ∇  = 0, [] = κ =0 κ= κ ∂n ∂n 



(10.32)

Here function  has the meaning of “quasi-temperature” (see Pobedrya (1984)). Finally, it follows from the analysis of Eqs. (10.15) and (10.16) for the function  m 3 +ξ3 that the modulus κ is determined as an average weighted by volume fractions  κ = κ , since m 3 = 0. Thus, we could formulate an algorithm for calculating all constants involved in constitutive relations (10.19)–(10.21). We demonstrated that their calculation can be reduced to four problems in the plane of the cross section of a fiber for the Lame Eq. (10.25) with complex coefficients (10.26) and for Laplace Eqs. (10.31) and 10.32. In the next section we will develop an analytical–numerical method which allows us to obtain highly accurate approximation to the solution for these four problems written in an analytical form. 



10.4 The Solution of the Problem on Microscale Here we develop an analytical approach to the cell problem solution. To this end, we use the solutions of the generalized Eshelby problem (Christensen 1979; Eshelby 1957) as basis functions for the Trefftz method (Mikhlin 1964). The classical Eshelby problem considers an inclusion in an infinite matrix and assumes a solution to the problem (10.25) that corresponds to a homogeneous stress–strain state at infinity, i.e.

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has linear asymptotic. We propose to consider the generalized Eshelby problem with an arbitrary polynomial asymptotic behavior of the solution at infinity. This asymptotic behavior corresponds to an arbitrary stress–strain state at the cell boundary and thus provides an approximation of the periodic boundary conditions to the problem (10.25). Similar solutions with polynomial asymptotic behavior at infinity can be constructed for Eqs. (10.31) and (10.32). To solve the generalized Eshelby problem, we choose a special representation, which must satisfy the Lame Eqs. (10.25) with complex coefficients and take into account the contact conditions at the boundaries between the matrix and inclusions: u(P) =

λ∗ + μ∗ ∇(r f ) 2 f − , ∇ f (P) = 0 μ∗ λ∗ + 2μ∗ 2μ∗

(10.33)

Here λ∗ and μ∗ are complex parameters of the Lame equation and f (P) is harmonic vector-function (harmonic potential). The vector-function u(P) chosen in the form of Papkovich-Neuber representation (10.33) satisfies the Lame Eq. (10.25) automatically (Neuber 1934; Papkovich 1932). Now we need to choose an analytical form of the auxiliary potential f (P) to automatically satisfy the contact conditions (10.24) for the generalized Eshelby problem in case of cylindrical inclusions. Let us state the conditions of the generalized Eshelby problem. In contrast to the classical problem (Christensen 1979; Eshelby 1957), we consider an equation with complex coefficients and set a polynomial growth condition for its solution at infinity: u(P) →

U (0) n ,

U (0) n (P)

  λ∗M + μ∗M ∇ r f (0) f (0) n n , P(x, y) → ∞ (10.34) = ∗ − ∗ μM λ M + 2μ∗M 2μ∗M

Here f (0) n (n = 0, 1, 2, . . . ) is a vector harmonic polynomial of degree n that determines this asymptotic behavior of the generalized Eshelby problem, λ∗M and μ∗M are complex moduli for the material of the matrix. The contact conditions at the matrix-fiber boundaries include two vector equations [u] = 0 and [ p(u)] = 0,   where p(u) = σi∗j (u)n j and n j are the components of the outward normal vector n. Analyzing the contact equations, we represent p(u) in the following form p(u) =

λ∗ λ∗ + μ∗ ∂∇(r f ) ∂f + n∇ f + ∗ ndiv f − ∗ ∗ ∂n λ + 2μ λ + 2μ∗ ∂n

(10.35)

The structure of the potential f is established by representing the solutions of the Laplace equation in the form of products of homogeneous harmonic polynomials of n-th degree and radial multipliers h n (r ) = r −2n , where r is the radial coordinate in cylindrical coordinate system (see Volkov-Bogorodsky (2016)). Based on the Gaussian representation of an arbitrary polynomial in harmonic series (Sobolev and Vaskevich 1997), we can specify the harmonic potential f in the form determined by the polynomial f (0) n given by formula (10.33):

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 2n f = A j + A j r −2n f (0) + B j ∇ r −2n+2 div f (0) n n + Bj r  

    − rdiv r −2n f (0) + C j r 2n + C j ∇ r −2n r f (0) n n

   + D j r 2n+4 + D j ∇div r −2n f (0) n 

+ E j + E j r −2n+4 ∇div f (0) n 









(10.36)

The potentials (10.36) satisfy the following theorem on the representation of the contact conditions for functions (10.33) and (10.34) in the form of a canonical decomposition with radial multipliers. Theorem. Each of the relations (10.33), (10.35) (which we denote as F) with harmonic potential f in the form of Eq. (10.36), defined on a cylindrical surface, can be represented as linear of five canonical   polynomials of nth  combination (0) (0) (0) −2 (0) , rdiv f with radial multipliers , f , ∇ r f , r r r f degree: r 2 ∇div f (0) n n n n n h 0 (r ), h 1 (r ), h 2 (r ), h 3 (r ), h 4 (r )   (0)   (0) −2 + h 3 rdiv f (0) r r f (0) F(P) = h 0 r 2 ∇div f (0) n + h1 f n + h2∇ r f n n + h 2r n (10.37)  where r(x, y) is the radius-vector to the point P, and r = x 2 + y 2 is its norm. Note that the radial multipliers assume constant values on the surface of the cylinder. Relations (10.37) for the potential in the form (10.36) can be verified by direct substitution of Eq. (10.36) into Eqs. (10.33) and (10.35). Note that taking the derivative in the direction normal to the cylindrical surface is taking the derivative with respect to the radial coordinate. One should keep in mind the following relation for homogeneous polynomials of degree n: n f (0) ∂ f (0) n n = ∂r r and also use the following form of differential relations for products of radial multipliers h n (r ) and homogeneous polynomials of degree n:        h n  (0)   hn  (0) (0) div h n f (0) = h n div f (0) = h n ∇ r f (0) n n + r r f n , ∇ hn r f n n + r r r fn . We use Theorem to define the harmonic potential f (P) in the generalized Eshelby problem (10.33) written for an inclusion  −2n+2  2n div f (0) f = A0 f (0) n + B0 r ∇ r n       + C0 r 2n ∇ r −2n r f (0) − rdiv r −2n f (0) n n   + D0 r 2n+4 ∇div r −2n f (0) + E 0 ∇div f (0) n n and for the matrix

(10.38)

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  −2n (0) f = f (0) f n + B 1 ∇ r −2n+2 div f (0) n + A1 r n       − rdiv r −2n f (0) + C 1 ∇ r −2n r f (0) n n   + D 1 ∇div r −2n f (0) + E 1r −2n+4 ∇div f (0) n n

(10.39)











Notice that in Eq. (10.39) all terms except the first tend to zero at r → ∞. This assure the required asymptotic behavior of the matrix at infinity (see Eq. (10.34)). In general, Eqs. (10.38), (10.39) contain ten unknown coefficients. According to representation

(10.37) written for u and p(u) and using the contact conditions [u] = 0, p(u) = 0 at the interface between the matrix and inclusion, we equate   , rdiv f (0) , f (0) , ∇ r f (0) the relationships at the canonical polynomials r 2 ∇div f (0) n n n n   and r −2 r r f (0) n , and obtain exactly ten equations for these ten coefficients. We solve these equations and obtain the values of coefficients in representations (10.38) and (10.39), which automatically provide the necessary contact conditions at the boundary of cylindrical inclusions. Thus, in order to approximate the periodic conditions at the cell boundary, we obtained a complete system of functions {un (P)} in the form of representations (10.33). They correspond to all possible potentials (10.38) and (10.39) that are built on the basis of the harmonic polynomials f (0) n defined in Eq. (10.34). The system of functions {un (P)} approximates functions U (1) and U (2) used in Lame Eq. (10.24). The scalar functions U (4) and  satisfying Eqs. (10.31), (10.32) can be approximated by a similar system of functions which have polynomial asymptotic behavior at infinity and can be written explicitly in terms of the scalar harmonic polynomials f n(0) :  φn (P) =

A0 f n(0) ,r < r0

1 + A1r −2n f n(0) , r > r0 

(10.40)

Here r0 is the radius of an inclusion. The coefficients in representation (10.40) can be obtained from the contact conditions for the functions U (4) and . We vary all possible polynomials f n(0) and obtain a complete system of functions {φn (P)} in the form of representation (10.39). This system of functions is used to approximate periodic solutions U (4) and  of cell problems (10.31) and (10.32). Given an analytical representation (10.38) and (10.39) for the solution of generalized Eshelby’s problem, we can use Trefftz method to approximate periodic conditions for the cell problems (10.25). The main idea of an approximation (see Mikhlin (1964)) is to consider the deviation of the approximate solution u(P) from exact W u − U (k) , where solution U (k) (P) and minimize the  energy functional    W (u) = 21 G 2μ∗ εi2j + λ∗ θ 2 dG, εi j = u i, j + u j,i /2, θ = εii . For our cell problems the minimization of energy can be considered in accordance with the Castiliano principle (see Timoshenko (1970)) or the Bubnov-Galerkin method (see Ern and Guermond (2004)). That means that we need to find the

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  stationary point of W u − U (k) based on a complete system of functions {un (P)} in scalar product (bilinear form) corresponding to this energy functional:    an u n W u − U (k) , un = 0, u = n

W u, u







=

   2μ∗ εi j εi j + λ∗ θ θ dG, W (u) = W (u, u)/2

(10.41) (10.42)

G

Using Green’s integral formula, the integration over the volume of a cell in Eq. (10.42) can be reduced to integration over the cell’s boundary:

W u, u





 =





u p(u )d(∂G) = ∂G



p(u)u d(∂G)

(10.43)

G

Let us consider the periodic cell and specify the opposite faces Si+ and Si− corresponding to the directions of the coordinate axes i = 1, 2. The periodicity conditions for U (k) are equivalent to the next system of equations (see Vlasov et al. (2022)): − U (2)− = δ2 j , σi∗(k)+ − σi∗(k)− = 0, P ∈ Si+ , i = j Ui(1)+ − Ui(1)− = δ1i , U (2)+ j j j j (10.44)   Let us introduce the scalar products Wk u, u associated with the bilinear form (10.43). To this end, we consider the surface integral on the cell boundary and separate the components of the function u included in the boundary conditions (10.44) (notice that no summation over repeated indices is performed here):

 

    u i pi u + p j (u)u j d(∂G), j = i; W1 u, u = i

Si±

 

    u j p j u + pi (u)u i d(∂G), j = i. W2 u, u = i

Si±

  The bilinear forms Wk u, u , k = 1, 2 still correspond to the energy functional   W (u) = Wk (u, u)/2, and we are looking for the stationary point of W u − U (k)    for k = 1, 2 in scalar product Wk u, u in accordance with (10.41). As a result, we obtain a system of equations that is uniquely defined by the problem (10.25) and approximates periodic boundary conditions (10.44). So, the coefficients an in expansion (10.41) are determined from the system of equations ϒ X = H , where X = an ; ϒ = υnl(k) , υnl(k) = Wk (un , ul )

(10.45)

10 Using Asymptotic Homogenization in Parametric Space to Determine …

H = ηn(k) , ηn(k)

⎛ ⎞   1⎜ ⎟ = ⎝ pk (un )d(∂G) − pk (un )d(∂G)⎠ 2 S1−

167

(10.46)

S1+

Thus to approximate the periodic boundary conditions for the cell problem (10.25), we implemented the scheme (10.41) and were able to reduce it to linear system of Eqs. (10.46). Due to the analytical representation of the approximating functions our approximation scheme demonstrates an exceptional efficiency and accuracy. To solve cell problems (10.31) and (10.32), we use the similar  approach based on approximating functions (10.40) and functionals W (u) = 21 G μ∗ |∇u|2 dG and  W () = 21 G κ|∇|2 dG for cell problems (10.31) and (10.32) respectively. The Trefftz method for these problems demonstrates the same efficiency as for problem (10.25). ∗(1) in accordance with the As a test, we can give an example of calculation of σ11 formula (10.25). As a material we choose carbon fibers with μ∗I = 71.43G Pa and λ∗I = 285.72G Pa embedded in epoxy matrix with μ∗M = 1.02 + i0.37G Pa and λ∗M = 4.8 + i1.48G Pa (see Vlasov et al. (2020)). We calculate the discrepancy  in the norm L 2 between the exact and approximate solutions as the function of the maximum degree of approximation for the large volumetric filling factor equal to c0 = 65%: =

  i

Si±

1/2 %  %2 % %2 % (1)± % % % %u i − Ui % + p j (u) d(∂G)

. The result of calculations demonstrates fast convergence to the exact solution, and we obtain  = 0.0002 when N = 45. The optimal degree of approximation for volumetric filling factor c0 = 20% used in the next section is N = 10.

10.5 Numerical Results and Their Analysis As an illustrative example, we consider a material composed of carbon fibers inserted in viscoelastic epoxy matrix. The volume concentration of fibers c0 , temperature T and frequency of harmonic oscillations f ( f = ω/2π ) can vary. Below we present the results of our calculations for the volume concentration of fibers equal to c0 = 20%. Lame’s coefficients of carbon fibers are chosen to be μ I = 71.43G Pa and λ I = 285.72G Pa. We assume that the properties of carbon fibers do not change with frequency and temperature and are described by an elastic model: η K = ημ = ∞. Material coefficients for epoxy matrix at various temperatures were determined using the experimental data for L385:340 epoxy resin given in the

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work (Mc Hugh et al. 2004). At T = 72 °C (near glass transition temperature), Lame’s coefficients used in calculations are μ M = 1.014GPa, λ M = 4.057GPa and shear viscosity coefficient is η M = 1.5Gpa · s. At T = 0 °C, the corresponding values are μ M = 1.50GPa, λ M = 6.00GPa and η M = 18.7Gpa · s. The volumetric viscosity coefficient of the matrix is chosen to be η K = ∞. In Figs. 10.1 and 10.2, we plot storage Young and shear moduli and loss Young and shear moduli across the fiber as functions of frequency for two different temperatures. It can be seen from Figs. 10.1 and 10.2 that both storage Young modulus and storage shear modulus quickly reach a plateau with an increase in frequency. Loss Young and shear moduli reach a maximum at a certain frequency and then quickly decrease with a further increase in frequency. One can observe that temperature increase causes lower values of moduli.

Fig. 10.1 Effective storage (left) and loss (right) Young and shear moduli across the fibers as a function of frequency of harmonic oscillations f, H z. Temperature and volume fraction of inclusions are T = 0 °C and c0 = 20% respectfully

Fig. 10.2 Effective storage (left) and loss (right) Young and shear moduli across the fibers as a function of frequency of harmonic oscillations f, H z. Temperature and volume fraction of inclusions are T = 72 °C and c0 = 20% respectfully

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10.6 Conclusion The present study proposes the method of asymptotic homogenization in parametric space used to determine the effective storage and loss Young’s and shear moduli as functions of the frequency for different temperatures. An important advantage of the proposed homogenization in parametric space is its ability to treat temperature as a parameter to study both viscoelastic and thermoviscoelastic properties of the material. In this work, we presented the results for fixed volume fraction of fibers c0 . At the same time, our approach allows us to vary this quantity and has a potential to study the frequency dependence of the effective storage and loss modulus related to the volume fraction of fibers changing withing the range c0 = 20–65%. The method can be adopted to model not only periodic but also functionally graded composite materials where properties and location of fibers can change slowly with the coordinate. Another improvement is that in our model, fibers can be surrounded by an intermediate layer or a system of intermediate layers separating them from the matrix. For the given geometry of inclusions, we obtain exact expressions for the solutions of the problem on the microscale, and this ability has a potential of using our method in material design.

References Bakhvalov N, Panasenko G (1989) Homogenization: averaging processes in periodic media. Kluwer Academic Publishers, Dordrecht/Boston/London Bensoussan A, Lions J-L, Papanicolaou G (1978) Asymptotic analysis for periodic structures. North-Holland, Amsterdam Bateman H, Erdélyi A (1954) Tables of integral transforms. McGraw Hill, New York/Toronto/London Christensen RM (1971) Theory of viscoelasticity, An introduction. Academic Press, New York/London Christensen RM (1979) Mechanics of composite materials. John Wiley & Sons Cruz-González OL, Rodríguez-Ramos R, Bravo-Castillero J, Martinez-Rosado R, Guinovart-Diaz R, Otero JA, Sabina FJ (2017) Effective viscoelastic properties of one-dimensional composites. Am Res Phys 3(1):1–17 Cruz-Gonzalez OL, Rodrigez-Ramos R, Otero JA, Ramirez-Torres A, Penta R, Lebon F (2020) On the effective behavior of viscoelastic composites in three dimensions. Int J Eng Sci 157:103377 Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc Lond A 241:376–396 Ezzat MA, El-Karamany AS, El-Bary AA (2015) Thermo-viscoelastic materials with fractional relaxation operators. Appl Math Model 39:7499–7512 Ern A, Guermond J-L (2004) Theory and practice of finite elements. Springer Hashin Z (1965) Viscoelastic behavior of heterogeneous media. J Appl Mech Trans ASME 32(3):630–636

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Hashin Z (1970a) Complex moduli of viscoelastic composites-I. General theory and application to particular composites. Int J Solids Struct 6:539–552 Hashin Z (1970b) Complex moduli of viscoelastic composites-I. General theory and application to particular composites. Int J Solids Struct 6:797–807 Khan KA, Muliana AH (2010) Effective thermal properties of viscoelastic composites having fielddependent constituent properties. Acta Mech 209:153–178. https://doi.org/10.1007/s00707-0090171-6 Lakes R (2009) Viscoelastic materials. Cambridge University Press, New York Li K, Gao X-L, Roy AK (2006) Micromechanical modeling of viscoelastic properties of carbon nanotube–reinforced polymer composites. Mech Adv Mater Struct 13(4):317–328. https://doi. org/10.1080/15376490600583931 Mc Hugh J, Doring J, Stark W, Erhard A (2004) Characterization of epoxy materials used in the development of ultrasonic arrays. In: Proceedings, 16th World Conference on NDT. Montreal Mikhlin SG (1964) Variational methods in mathematical physics. Pergamon Press, Oxford Muliana AH, Haj-Ali RM (2008) A multi-scale framework for layered composites with thermorheologically complex behavior. Int J Solids Struct 45:2937–2963 Muliana AH, Kim JS (2007) A concurrent micromechanical model for nonlinear viscoelastic behaviors of particle reinforced composites. Int J Solids Struct 44:6891–6913 Neuber H (1934) Ein neuer Ansatz zur Lösung raümlicher Probleme der Elastizitätstheorie. ZAMM 14(4):203–212 Otero JA, Rodríguez-Ramos R, Guinovart-Díaz R, Cruz-González OL, Sabina FJ, Berger H, Böhlke T (2020) Asymptotic and numerical homogenization methods applied to fibrous viscoelastic composites using Prony’s series. Acta Mech 231:2761–2771. https://doi.org/10.1007/s00707020-02671-1 Papkovich PF (1932) Solution générale des équations différentielles fondamentales de l’élasticité, exprimeé par trois fonctiones harmoniques. CR Acad Sci Paris 195:513–515 Pobedrya BE (1984) Mechanics of composite materials. MGU Publishers, Moscow (in Russian) Rodriguez-Ramos R, Otero JA, Cruz-Gonzalez OL, Guinovart-Diaz R, Bravo-Castillero J, Sabina FJ, Padilla P, Lebon F, Sevostianov I (2020) Computation of the relaxation effective moduli for fibrous viscoelastic composites using the asymptotic homogenization method. Int J Solids Struct 190:281–290 Sobolev SL, Vaskevich L (1997) The theory of cubature formulas. In: Mathematics and its applications. Kluwer, Dordrecht Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York Vlasov AN, Volkov-Bogorodsky DB (2018) Method of asymptotic homogenization of thermoviscoelasticity equations in parametric space. (Part I). Compos Mech Comput Appl Int J 201(4):331–343 Vlasov AN, Merzlyakov VP (2009) Averaging of deformation and strength properties in rock mechanics. ASV Press, Moscow (in Russian) Vlasov AN, Volkov-Bogorodsky DB (2021) Application of the asymptotic homogenization in a parametric space to the modeling of structurally heterogeneous materials. J Comput Appl Math 390:113191 Vlasov AN, Volkov-Bogorodskii DB, Kornev YuV (2020) Influence of carbon additives on mechanical characteristics of an epoxy binder. Mech Solids 55(3):577–586 Vlasov AN, Volkov-Bogorodsky DB (2014) Parametric method of asymptotic averaging for nonlinear equations of thermoelasticity. Mekhanika Kompoz Mather Konstr 20(4):491–507 (in Russsian) Vlasov AN, Volkov-Bogorodsky DB, Savatorova VL (2022) Calculation of the effective properties of thermo-viscoelastic composites using asymptotic homogenization in parametric space. Mech Time-Depend Mater 26:565–591. https://doi.org/10.1007/s11043-021-09501-4

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Volkov-Bogorodsky DB (2016) Radial multipliers method in mechanics of inhomogeneous media with multi-layered inclusions. Mechanika kompozitzionnykh materialov i konstruktsii 22(1):19– 39 (in Russian) Yu Q, Fish J (2020) Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem. Int J Solids Struct 39(26):6429–6452

Chapter 11

Biomechanical Modeling and Characterization of Cells Arkady Voloshin

Abstract The study of the cell mechanics and its response to the external mechanical excitation has great importance in biology and medical sciences since it can help to understand and identify the causes, progression, and cures of diseases. The study of deformation and motion of cells involves the cooperation of various disciplines like biology, chemistry, and mechanics. The application of microelectromechanical systems (MEMS) in biomedical devices has expanded vastly over the last few decades, with MEMS devices being developed to measure different characteristics of cells. The study of cell mechanics offers a valuable understanding of cell viability and functionality. Cell biomechanics approaches also facilitate the characterization of important cell and tissue behaviors. In particular, understanding the biological response of cells to their biomechanical environment would enhance the knowledge of how cellular responses correlate to tissue-level characteristics and how some diseases, such as cancer, grow in the body. This study focuses on modeling the viscoelastic mechanical properties of single suspended human mesenchymal stem cells (hMSCs). Mechanical properties of hMSC cells are particularly important in tissue engineering and research for the treatment of cardiovascular diseases. We evaluated the elastic and viscoelastic properties of hMSC cells using a miniaturized custom-made MEMS device. Our results were compared to the elastic and viscoelastic properties measured by other methods such as atomic force microscopy (AFM) and micropipette aspiration. Different approaches were applied to model the experimentally obtained force data, including elastic, Kelvin, and Standard Linear Solid (SLS) models, and the corresponding constants were derived. These values were compared to the ones in the literature that were based on micropipette aspiration and AFM methods. We then utilized a tensegrity model, which represents major parts of the internal structure of the cell and treats the cell as a network of microtubules and microfilaments, as opposed to a simple spherical blob. The results predicted from the tensegrity model were consistent with the recorded experimental data. The modal analysis of the tensegrity model allowed for analysis of the cell’s natural frequencies. A. Voloshin (B) Department of Mechanical Engineering and Mechanics, Bioengineering Department, Lehigh University, Bethlehem, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Altenbach et al. (eds.), Advances in Mechanics of Time-Dependent Materials, Advanced Structured Materials 188, https://doi.org/10.1007/978-3-031-22401-0_11

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Keywords Cell biomechanics · MEMS · BioMEMS · Human mesenchymal stem cells (hMSC) · Viscoelastic properties · Standard linear solid model · Tensegrity · Modal analysis

11.1 Introduction The study of cell mechanical and viscoelastic behavior has a great importance in biology and medical sciences since it can be used to elucidate the causes, progression, and cures of diseases. Mechanical forces applied to the cell are transmitted over molecular pathways that mediate cell-to-cell and cell-to-extracellular matrix adhesion. The study of cell mechanobiology is very beneficial to tissue engineering strategies (Schumann et al. 2006). The knowledge of mechanotransduction will be advanced by studying the mechanical responses of cells. These responses may play an important role in investigating metastasis and invasiveness of cancerous cells (Sieck 2005; Guck et al. 2005; Wyckoff et al. 2000). It was observed that normal cells and cancerous cells can exhibit different stiffness values (Baker 2009) and it was hypothesized that higher deformability corresponds to more invasive cells (Darling 2007), which can be a basis for their separation and detection (Xu et al. 2012; Suresh 2007). The stiffness of cells has also an effect on disease progression. For instance, more deformable cancerous cells can propagate more easily in the body (Byun et al. 2013). Therefore, understanding the cell response to mechanical stimuli can benefit cancer diagnosis attempts. Furthermore, mechanical stimuli play an important role in basic cellular behavior, including proliferation, cell lineage, function, and differentiation (Mason et al. 2012). Since the mechanical properties of a cell are important factors in determining its response to external mechanical forces (Janmey and Weitz 2004), it is essential to characterize single cells to have a better understanding of the cellular mechanism. The stiffness of hMSC cells is a significant factor in tissue engineering and regeneration (Stok et al. 2010; Guilak et al. 2010). The viability and function of hMSC cells, which have promising use in gene therapy (Chan and Lam 2013) and cardiac tissue engineering, are influenced by the presence of mechanical stimuli similar to other types of cells (Li et al. 2011). Furthermore, the mechanobiology of differentiation can be better understood by studying mechanical properties and inter-relations of hMSC cells. There is a hypothesis that chondrogenic or osteogenic differentiation might be caused by mechanical strain in undifferentiated hMSCs (Friedl et al. 2007). Mesenchymal stem cells are multipotent stromal cells. Stromal cells are connective tissue cells of any organ. Their interaction with tumor cells is known to play a major role in cancer growth and progression (Wiseman and Werb 2002). Modeling and analysis of single-cell responses to mechanical stimuli is the first step toward gaining knowledge about mechanical responses in tissue (Leipzig and Athanasiou 2005). Several methods such as micropipette aspiration, AFM, cytoindentation, and magnetic beads have been developed to study the mechanical properties of single cells and different models were applied to single-cell responses to quantify their mechanical characteristics

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(Hochmuth 2000; Mahaffy et al. 2004; Shin and Athanasiou 1999; Bausch et al. 1998). Recent work (Wang et al. 2017) attempted to model viscoelastic cell behavior by loading a cell using an AFM, recoding the time-dependent force and deformation and use of general Maxwell model for viscoelastic materials containing a spring and n parallel spring-damping paths. They modeled the cell as a second-order system with five parameters. A miniaturized device such as a BioMEMS provides another way to characterize a single cell (Tatic-Lucic and Gnerlich 2012). This kind of device has the advantage of being smaller and easier to integrate onto a lab-on-a-chip with clinical uses in addition to the possibility of using it for multiple experiments. This study is utilizing a BioMEMS device to apply a load and measure a time-dependent deformation. The results were used to model the biomechanical properties of single human mesenchymal stem cells (hMSCs). The BioMEMS device fabrication process and characterization have been previously described. The mechanical characteristics of single hMSC cells in vitro were measured by using the BioMEMS device capable to apply a controlled force to a single cell. The goal of this study was to quantify the mechanical and viscoelastic characteristics of a single detached hMSC cell and model its behavior utilizing various elastic, viscoelastic, and tensegrity-based models.

11.2 Materials and Method 11.2.1 HMSC Cell Culturing Conditions Cell lines originated from bone marrow cells (Lonza), cryopreserved, and cultured in a monolayer in 10 ml culture dishes. The cells in the dishes were incubated at 37 °C and 5% carbon dioxide (CO2 ). Culture media consisted of low-glucose Dulbecco’s modified eagle medium (DMEM), 10% fetal bovine serum (FBS), and penicillin antibiotic at 1%. The cells were harvested when there was a 90% confluent monolayer on the culture dish. The cells used for the experiment had a passage number of n = 4. The cells were placed in a solution using trypsinization and were kept in suspension in a cell medium during the duration of the experiment ( b/2) is the initial crack length introduced into the specimen, and the elastic analysis by Knauss (Knauss 1966) shows that the stress and displacement fields of the crack tip similar to those of a strip with a semi-infinite crack can be obtained approximately. The symbol v(t) in this figure is the displacement applied to the grips on both the upper and lower ends. In the static crack experiment, to observe the stress field around the stationary crack tip, the shape of the crack tip is made into a circular hole with a diameter of 0.3 mm to prevent the crack from growing. In a crack growth experiment, since the shape and properties of the initial crack tip are known to affect the subsequent crack growth behavior, the initial crack tip should be very sharp and smooth and coplanar with the later growth crack surface.

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Fig. 13.14 Specimen geometry for viscoelastic crack growth experiment

13.4.3 Introducing Natural Crack Since the initial tips of the crack made by a blade, which are often used in experiments related to the crack growth behavior of viscoelastic bodies, are not in the same plane and are uneven, multiple partial cracks occur during crack growth. As a trace of this phenomenon, a step called a tear line is formed on the fracture surface. The tear resistance due to the mutual interference of multiple cracks is quite large, which is the main cause of disturbing the reproducibility of crack growth behavior. Therefore, to accurately study the crack growth behavior, it is indispensable to create an ideal initial crack with good reproducibility, in which the initial crack tip is coplanar and has a sharp tip. Regarding the ideal initial crack introduction method, Misawa et al. (Misawa et al. 1980) have proposed a method that utilizes the properties of the crack growth behavior of polyurethane resin, which is a linear viscoelastic material. However, this method requires considerable skill to create an ideal crack. Therefore, Takashi et al. (Takashi et al. 1982) created an ideal initial crack by applying a tensile load to a specimen whose temperature gradient was applied by liquid nitrogen and gradually developing the crack. However, this method is also complicated. In this study, therefore, the initial crack introduction method proposed by Ogawa et al. (Ogawa and Takashi 1990) is applied, which utilizes the fact that the fracture surface of the crack that has grown to some extent in the viscoelastic region is on the same plane. The introduction method is as follows. First, a crack of about 5 to 8 mm is introduced using a sharp cutter. Next, by applying a tensile displacement at a low speed at a predetermined temperature (T = 273 °K), which is a viscoelastic region, the crack grows quietly. Then, the load is rapidly unloaded when it reaches the specified length (20 mm). Finally, annealing is performed at room temperature to release the slight residual stress that may have occurred during this initial crack introduction process. When an initial crack is created by such a method and a crack growth experiment is performed, it has the advantage of having a smooth surface and a sharp tip. It is difficult to obtain an accurate initial crack length with this crack introduction method. For the semi-infinite crack used in this experiment, if the condition of C 0 >

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b/2 is satisfied, the stress fields of the crack tip are the same. Therefore, the initial crack length is adjusted within the range that can be visually measured. The load history as the crack propagates during the initial crack introduction process is removed by annealing and does not affect the subsequent crack growth behavior (Ogawa and Takashi 1990).

13.4.4 Experimental Procedure The constant rate displacement loading is applied on the lower edge of the specimen normal to the crack surface under five different rates of the extension V = 8.33 × 10–3 , 3.33 × 10–2 , 8.33 × 10–2 , 3.33 × 10–2 and 8.33 × 10–1 mm/s at the temperature T = 273 °K for stationary crack experiments. At the temperature, the material shows marked viscoelastic behavior. In crack growth experiments, T = 263, 268, and 273 °K are used so that the viscoelastic region can be widely covered. However, the experiment under V = 8.33 × 10–1 mm/s at T = 263 °K is excluded since the specimen exhibits a brittle fracture under this condition. Photoviscoelasticity with white elliptically polarized light is applied in this experiment, as described in the previous section. When the analysis target has cracks, higher-order fringes may appear at the crack tips, causing attenuation of light intensity at higher-order fringe orders peculiar to white light, making it impossible to perform fringe image analysis. Therefore, in this experiment, a three-wavelength fluorescent lamp is used as the light source. The three-wavelength fluorescent lamp has three emission line spectra. Therefore, it is considered that three types of monochromatic light with different wavelengths are used, and the attenuation of higher-order photoviscoelastic fringe patterns can be reduced. The photoviscoelastic fringe patterns, which change from moment to moment after the start of loading, are photographed using a camera and recorded on a color reversal film. The fringe patterns on the film are captured on a computer as 24-bit digital images of 1344 × 896 pixels and 256 gradations of each color using a film scanner for image analysis.

13.4.5 Results for Stationary Cracks Figure 13.15 shows an example of the time variation of the photoviscoelastic fringe pattern around the crack tip at a temperature T = 273 °K and a loading rate V = 8.33 × 10–3 mm/s. It can be seen that the fringe pattern showing the mode I type changes with time, and the fringe order increases. In addition, because of the influence of the elliptically polarized light, the light intensity is attenuated in the left part around the crack tip. In this experiment, the crack does not grow; that is, it is a proportional loading problem. Therefore, the stress field at the crack tip can be expressed in the same way as in the case of an elastic body and the stress intensity factor can be theoretically

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Fig. 13.15 Examples of the time-dependent photoviscoelastic fringe pattern around the stationary crack tip (T = 273 °K, V = 8.33 × 10–3 mm/s)

determined. Therefore, the stress intensity factors calculated from the principal stress difference obtained in the experiment are compared with the theoretical values. First, the distributions of the fringe order and the birefringence direction are determined at each time and the time variations of the fringe order N(t) and the birefringence direction α (t) are obtained. Then, the principal stress difference, the principal strain difference, and their directions are calculated from N(t) and α(t) using the photoviscoelastic constitutive equation. After that, the stress intensity factor is calculated from the principal stress difference using the relation between the stress distribution at the crack tip and the stress intensity factor (Irwin 1958). The relationship between the principal stress difference and the stress intensity factor can be expressed by the following equation. (σ1 − σ2 )2 =

K I∗ 2 2 2σox K I∗ θ sin θ + √ sin (1 + 2cosθ ) + σox 2 2πr 2 2πr

(13.3)

Here, K I * shows the viscoelastic stress intensity factor, σox represents the nonsingular term, and r and θ are the polar coordinate system at the crack tip. Many methods have been proposed to evaluate the stress intensity factor from the distribution of the principal stress differences around the crack tip (Ramesh et al. 1997). In the study, a least-squares method proposed by (Sanford and Dally 1979; Sanford 1980) is used. The theoretical value of the stress intensity factor in the specimen used in this experiment can be obtained by the following equation using the correspondence principle (Misawa 1987), assuming that Poisson’s ratio ν is constant.    t dv0 (τ ) dτ E(t − τ ) K I (t) =   v0 (0)E(t) + dτ 0 1 − ν2 b 1

(13.4)

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Fig. 13.16 Stress intensity factor for stationary crack at a temperature of 273 K

Figure 13.16 shows the variation of the stress intensity factor with respect to the displacement v(t) of the upper and lower edges of the specimen obtained at each loading rate. In the case of an elastic body, there is a linear relationship between the displacement and the stress intensity factor, but in this experiment, the nonlinear relations are obtained as shown in this figure. In addition, it is observed that the stress intensity factor increases as the loading rate increases. Both are phenomena that reflect the properties of viscoelastic materials. In the figure, the solid curves are the theoretical values obtained by Eq. (13.4). The experimental values agree well with the theoretical values, although the temperature differences greatly affect the response. This result shows that the stress intensity factor for a stationary crack can be calculated using this method. The stress intensity factor is evaluated in the same way for the crack growth problem later.

13.4.6 Results for Moving Crack When the ideal initial crack is adopted, the crack grows straight in the specimen and a very reproducible crack growth curve can be obtained. Figure 13.17 shows the crack growth curves at the temperature T = 273 K. Similar results are obtained at other temperatures. The crack growth length a(t) is determined by observing the photoviscoelastic fringe image shown later. As shown in this figure, the cracks grow smoothly until the final fracture. It has already been reported that a linear relationship is observed when the length of crack growth a(t) and the time t are plotted on a double logarithmic scale (Takashi et al. 1983; Ohtsuka and Takashi 1984). Therefore, the following relation is assumed between the crack growth length a(t) and the time t.

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Fig. 13.17 Crack extension curves at a temperature of 273 K

a(t) = αt β

(13.5)

Here, α and β are constants for each crack growth curve. The crack growth curves obtained under all experimental conditions are approximated by the method of leastsquares. Figure 13.18a shows the double logarithmic plots that approximate the crack growth curves obtained by the experiment at the temperature of T = 273 °K. Since all the coefficients of determination when approximated are 0.995 or more, it can be seen that Eq. (13.5) can express the time variation of the crack growth length very well. From Fig. 13.18a, it can be seen that the exponent β of each crack growth curve at the temperature T = 273 °K is not so dependent on the loading rate. As an example showing the temperature dependence of crack growth behavior, the crack growth curve under each temperature condition for a loading rate V = 8.33 × 10–3 mm/s is shown in Fig. 13.18b. As is clear from the figure, in the early stage of crack growth, the growth rate increases as the temperature decreases, and conversely, the higher the temperature, the higher the growth rate when the crack grows considerably. That is, the lower the temperature, the less likely the crack is accelerated, and the higher the temperature, the faster the crack is accelerated. The power index β has a small load velocity dependence, but a considerably large temperature dependence. Although there is a power law in the crack growth curve, the power law cannot determine the start time of a crack. In some of the fracture criteria proposed so far for crack growth conditions, the crack growth start time is implicitly included as a parameter representing the critical condition for the onset of crack growth. Therefore, the inability to determine the crack onset time remains an unsolvable problem in the viscoelastic crack growth problem (Ogawa et al. 1998).

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Fig. 13.18 a Loading rate dependence and b temperature dependence of the crack extension curve

(a)

(b)

Figure 13.19 shows an example of the time variation of the photoviscoelastic fringe pattern around the crack tip at a temperature T = 273 K and the loading rate V = 8.33 × 10–3 mm/s. A typical mode I fringe pattern varies as the crack grows. On the other hand, reflecting the viscoelastic behavior of the material, it is observed that photoviscoelastic fringes remain on the initial crack tip and the fracture surface. Figure 13.20 shows the loading rate dependence of the photoviscoelastic fringe pattern, which is the fringe pattern around the crack tip at a temperature of

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T = 273 °K and the displacement of the upper and lower ends of the specimen v(t) = 2.5 mm. Similarly, Fig. 13.21 shows the temperature dependence of the fringe pattern at the loading rate of V = 8.33 × 10–3 mm/s at the time of t = 600 s. It can be seen that even if the displacements of the upper and lower ends of the specimen are the same, the crack positions and the fringe patterns differ depending on the loading rate and the temperature. It is observed that the higher the loading rate and the lower the temperature, the higher the fringe order at the crack tip. In order to evaluate the stress intensity factor, it is necessary to evaluate the principal stress difference around the crack tip using photoviscoelasticity, as in the case of the stationary crack problem. Regarding the fracture criteria of viscoelastic cracks, COD fracture criteria (Schapery 1975a), displacement field fracture criteria M Ic (Misawa et al. 1984), fracture criteria based on conservation law J Ic ’ (Misawa et al. 1990), and others (Bradley et al. 1998) have been proposed. Many of them sought to find fracture mechanics parameters that were independent of the loading rate and temperature. In those fracture criteria, the crack growth onset time is implicitly included as a parameter representing the critical condition of the crack growth start, but as shown in Fig. 13.18, no critical phenomena such as discontinuous point are observed in the crack growth curve. Therefore, in this study, another evaluation method is proposed. The expression that represents the stress field around the tip of the propagating crack in the viscoelastic body is still unknown, unlike the case of the elastic body and the stationary viscoelastic crack. Therefore, the method used in the analysis of stationary cracks cannot determine the exact stress intensity factor. However, at present, it is not possible to express the stress field at the tip of the crack that propagates in the viscoelastic body in a general form, so in this study, the stress intensity factor obtained in the same way as in the case of a stationary crack is used as the stress intensity factor K I * extended to the viscoelastic material. Figure 13.22 shows the variation of the stress intensity factor K I * at each temperature as a function of the displacement v(t) at the upper and lower ends of the specimen. In these figures, the theoretical values for the stationary cracks in Eq. (13.4) are also shown by solid curves. In the initial stage of the experiment, the values of the stress intensity factor K I * increase as the loading rates increase under each temperature condition. The lower the temperature, the higher the values of the stress intensity factor K I * . These are the phenomena that reflect the viscoelastic behavior of the material. At the initial stage of this experiment, the theoretical values of the stress intensity factor and the experimental values are almost coincident. The theoretical values for the stationary crack increase monotonically until the end, but the experimental values suddenly become constant or decrease at certain points even though the cracks accelerate and grow. Therefore, when the experimental values do not match the theoretical values, that is, when the experimental values change discontinuously, the effect of crack growth appears in the results obtained by photoviscoelasticity as optical information. Judging from the crack growth curves shown in Fig. 13.18, the cracks have already grown by about 0.9 to 1.8 mm when the experimental values of the stress intensity factor K I * change discontinuously, depending on the loading rate and the temperature. However, considering the results of these experiments,

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Fig. 13.19 Examples of the time-dependent photoviscoelastic fringe pattern around the propagating crack tip (T = 273 °K, V = 8.33 × 10–3 mm/s)

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Fig. 13.20 Loading rate dependence of the photoviscoelastic fringe pattern around the propagating crack tip (T = 273 °K, v(t) = 2.5 mm)

Fig. 13.21 Temperature dependence of the photoviscoelastic fringe pattern around the propagating crack tip (V = 8.33 × 10–3 mm/s, v(t) = 2.5 mm, t = 600 s)

some change appears in the stress field around the crack tip at this point. In other words, the stress intensity factor K I * extended to the viscoelastic body at this point may be evaluated as a value related to the start of crack growth or the crack growth resistance obtained as optical information. Therefore, crack growth after K I * varies discontinuously is called rapid crack growth. The discontinuous change in the stress intensity factor K I * , that is, the value of * K I when the experimental value does not match the theoretical value, is evaluated as the critical value K c * for the start of rapid crack growth, as shown in Fig. 13.19. It is expressed as a function of the speed of crack extension da(t)/dt. Here, the error bars represent the maximum and minimum values of K I * that fluctuate around the discontinuous change. The speed of crack extension da(t)/dt is determined by differentiating the approximate curve of crack growth in Fig. 13.18. The critical value K c * for the start of the rapid crack growth increases as the crack growth rate increases. However, since the crack growth speed here is faster as the loading rate is faster, it

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Fig. 13.22 Extended stress intensity factor for viscoelastic materials at the temperature of a 263 K; b 268 °K; c 273 °K

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Fig. 13.23 Critical stress intensity factor for rapid crack growth K c * as a function of the crack growth rate da/dt

can be said that the value of K c * increases as the loading rate increases. It can also be seen that these values depend on the temperature and increase as the temperature decreases. Thus, the K c * value has the rate- and temperature-dependence. Therefore, it is considered that K c * can be represented by a single master curve in the same way as the property functions of the material such as the relaxation modulus. Each value shown in Fig. 13.23 is shifted in the horizontal axis direction using the same WLF time–temperature shift factor when the master curves of the mechanical and optical properties of the material are created. Figure 13.24 shows a logarithmic representation of K c * . In this figure, the solid curve is an approximation obtained using a least-squares method. In this way, K c * can be approximated by a single master curve. Therefore, it is expected that the critical value K c * at the start of the rapid crack growth can be treated as one of the material properties for monotonically increasing displacement load.

13.4.7 Summary Using photoviscoelasticity with white elliptically polarized light, the stress field around the tip of a semi-infinite stationary crack, which enables theoretical analysis based on the correspondence principle, is evaluated as the first step. As a method for extracting fracture mechanics parameters, a method based on the least-squares used in photoelasticity is used. The effectiveness of the fracture mechanics parameter extraction method is shown by comparing the theoretical solution with the results obtained from the experiment. The stress intensity factor extended to the viscoelastic

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Fig. 13.24 Master curve of the critical stress intensity factor for rapid crack growth K c*

body of the growing crack is then evaluated. Here, a method for evaluating the critical point at the start of crack growth is investigated by comparing the fracture mechanics parameters of an extending crack with the theoretical fracture mechanics parameters of a stationary crack. As a result, the following findings are obtained. • The value of the stress intensity factor K I * extended to the viscoelastic material is in good agreement with the theoretical value until the crack begins to grow, that is, in a stationary crack. • The value of K I * changes discontinuously at some stage after the crack begins to grow and then becomes almost constant. • The discontinuity phenomenon that could not be found in the conventional study of viscoelastic crack growth could be captured as the critical value K c * of the rapid crack growth starting obtained as optical information. In addition, the timetemperature superposition principle holds for the value of K c * as with other property values of the material.

13.5 Conclusions Photoviscoelasticity is applied to the viscoelastic rolling contact problem and the crack growth problem, which are typical examples of viscoelastic nonproportional load problems. In this study, the fringe pattern analysis method using white elliptically polarized light, which is proposed to obtain the fringe order and the birefringence direction from a single image, is used. By applying this method to the consecutive photoviscoelastic image data, the time variation of both the fringe order and the birefringence direction can be obtained. Then, the use of the photoviscoelastic constitutive equations gives the principal stress difference and the principal strain

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difference, and their directions. The experimental results for both the rolling contact problem and the crack growth problem show the effectiveness of photoviscoelasticity in the mechanics of time-dependent materials.

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