Computational Modeling of Intelligent Soft Matter: Shape Memory Polymers and Hydrogels 0443194203, 9780443194207

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Computational Modeling of Intelligent Soft Matter

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Computational Modeling of Intelligent Soft Matter Shape Memory Polymers and Hydrogels

Mostafa Baghani School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Majid Baniassadi School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Yves Remond University of Strasbourg, ICube/CNRS, Alsace, France

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-443-19420-7 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Mason Malloy Production Project Manager: Surya Narayanan Jayachandran Cover Designer: Greg Harris Typeset by MPS Limited, Chennai, India

Dedication

To our wives Roshanak, Maryam, Marie-Claude for their love, endless support, and encouragement.

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Contents

About the authors Preface Acknowledgments 1

2

Intelligent soft matters: need for numerical modeling in design and analysis 1.1 Introduction 1.2 Applications of shape memory polymers 1.2.1 Biomedical applications 1.2.2 Aerospace application 1.2.3 Textile application 1.2.4 Automotive application 1.2.5 Other applications 1.3 Smart hydrogel applications 1.3.1 Tissue engineering 1.3.2 Drug delivery 1.3.3 microfluidic valves 1.3.4 Hydrogels for wound dressing 1.3.5 Hydrogel and cancer therapy 1.3.6 Hydrogels and water treatment 1.3.7 Hydrogels and contact lens products 1.3.8 Hydrogels and agriculture 1.3.9 Hydrogels and biosensors 1.3.10 Hydrogels and hygiene products 1.4 Numerical modeling in design and analysis of intelligent soft matters References A detailed review on constitutive models for thermoresponsive shape memory polymers 2.1 Introduction 2.2 Classification of temperature-dependent polymers 2.2.1 Thermoset and thermoplastic polymers 2.2.2 The effect of temperature on thermoset and thermoplastic polymers 2.3 The molecular structure of shape memory polymers and their classification

xiii xv xvii

1 1 2 2 3 4 4 4 5 6 6 7 7 8 8 8 9 9 9 10 12

15 15 18 18 19 20

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Contents

2.3.1 Chemical structure of shape memory polymers 2.3.2 Classification of shape memory polymers 2.4 Modeling thermoresponsive shape memory polymers 2.4.1 Modeling of conventional thermally activated shape memory polymers 2.4.2 Modeling of two-way thermally activated shape memory polymer 2.4.3 Modeling of thermally activated multishape memory polymer 2.5 Statistical analysis of available shape memory polymer models 2.6 Summary and conclusion References 3

4

20 21 29 30 54 55 58 60 61

Experiments on shape memory polymers: methods of production, shape memory effect parameters, and application 3.1 Introduction 3.2 Methods of shape memory polymer production 3.2.1 Melt mixing 3.2.2 Solution mixing 3.2.3 Additive manufacturing 3.2.4 Shape memory characterization in combined torsion
tension loading 3.3 Investigation on structural design of shape memory polymers 3.3.1 Structural (geometrical) design 3.3.2 Method of sample production 3.3.3 Characterization of printed material 3.3.4 Thermomechanical shape memory tests 3.4 Shape memory polymer stent as an application 3.4.1 Materials 3.4.2 Stent fabrication 3.4.3 Stent radial compression 3.5 Summary and conclusion References

79 90 90 92 94 94 104 105 107 116 122 124

Shape memory polymers: constitutive modeling, calibration, and simulation 4.1 Introduction 4.2 Macroscopic phase transition approach 4.2.1 Strain storage and recovery 4.2.2 Thermodynamic considerations 4.2.3 Extension of the model to the time dependent regime 4.2.4 Numerical solution of the constitutive model 4.2.5 Consistent tangent matrix 4.2.6 Hughes
Winget algorithm: large rotation effects

127 127 128 130 132 133 135 137 138

77 77 78 78 78 79

Contents

4.2.7 Material parameters identification 4.2.8 Material model predictions 4.3 Shape memory polymer constitutive model through thermo-viscoelastic approach 4.3.1 Strain-dependent part of the stress 4.3.2 Time-dependent part of the stress 4.3.3 Temperature-dependent modification of the stress 4.3.4 Solution of shape memory polymer’s response in a shape memory polymer path 4.3.5 A time-discretization scheme for constitutive equations 4.3.6 Material parameters identification 4.3.7 Solutions development for torsion
extension of SMP 4.4 Summary and conclusion References 5

6

Shape memory polymer composites: nanocomposites and corrugated structures 5.1 Introduction 5.2 Modeling and homogenization of shape memory polymer nanocomposites 5.2.1 Constitutive equations for shape memory polymer based on phase transition 5.2.2 3D modeling and numerical considerations 5.2.3 Numerical results 5.3 Numerical homogenization of coiled carbon nanotube-reinforced shape memory polymer nanocomposites 5.3.1 Constitutive model of shape memory polymer based on thermo-viscoelasticity 5.3.2 Finite element model 5.3.3 Numerical results and discussion 5.4 Thermomechanical behavior of shape memory polymer beams reinforced by corrugated polymeric sections 5.4.1 Shape memory polymer constitutive model based on phase transition concept 5.4.2 Bending of a reinforced shape memory polymer beam 5.4.3 Numerical results and discussion References Shape memory polymer metamaterials based on triply periodic minimal surfaces and auxetic structures 6.1 Shape memory polymer metamaterials based on triply periodic minimal surfaces 6.1.1 Introduction 6.1.2 Materials and methods 6.1.3 Results and discussion

ix

139 141 144 144 144 145 146 148 149 150 156 156

159 159 160 161 163 166 173 174 178 182 187 189 192 196 205

209 209 210 212 217

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Contents

6.2

Numerical investigation of smart auxetic 3D metastructures based on shape memory polymers via topology optimization 6.2.1 Introduction 6.2.2 Geometrical modeling of an representative volume element 6.2.3 Finite element analysis of shape memory polymer microstructure 6.2.4 Results and discussion 6.3 Summary and conclusions References

225 225 227 229 234 238 240

7

A review on constitutive modeling of pH-sensitive hydrogels 7.1 Introduction 7.2 Applications of pH-sensitive hydrogels 7.2.1 Drug delivery 7.2.2 Soft actuators 7.2.3 Control of microfluidic flow 7.3 Swelling/deswelling phenomena 7.3.1 Conservation of mass 7.3.2 Chemical reaction 7.3.3 Ion transfer 7.3.4 Electrical field 7.3.5 Fluid flow 7.3.6 Mechanical field 7.4 Swelling theories 7.4.1 Monophasic models 7.4.2 Multiphasic models 7.5 Numerical implementation 7.6 Experiments 7.7 Summary and conclusions References

245 245 249 249 249 249 250 250 250 252 252 253 253 255 256 258 258 260 262 263

8

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels: constitutive modeling and FEM implementation 8.1 An equilibrium thermodynamically consistent theory 8.2 Transient electro-chemo-mechanical swelling theory 8.2.1 Large deformation theory 8.2.2 Chemical field 8.2.3 Electrostatic field 8.2.4 Continuity of ions 8.2.5 Mechanical field 8.2.6 Initial and boundary conditions 8.3 Numerical solution procedure 8.3.1 Development of weak form 8.3.2 Development of time and space discretization 8.3.3 Residuals and tangent moduli

271 271 279 279 279 281 282 283 286 286 287 287 288

Contents

8.4

Results and discussion 8.4.1 Equilibrium swelling 8.4.2 Inhomogeneous deformations 8.4.3 Analytical solution 8.4.4 Numerical results for transient swelling response of pH-sensitive hydrogels 8.4.5 Numerical predictions of the visco-hyperelastic constitutive model for tough pH-sensitive hydrogels 8.5 Summary and conclusion References

9

Structural analysis of different smart hydrogel microvalves: the effect of fluid
structure interaction modeling 9.1 Introduction 9.2 Different constitutive models’ description 9.2.1 Swelling theory of thermal-sensitive hydrogels 9.2.2 A stationary swelling theory for pH-sensitive hydrogels 9.2.3 A transient theory for pH-sensitive hydrogels 9.2.4 Coupled fields in fluid
structure interaction modeling 9.3 Results and discussion 9.3.1 Results for fluid
structure interaction analysis of temperature-sensitive hydrogel valves 9.3.2 Results for stationary response of pH-sensitive hydrogels 9.3.3 Transient results of the pH-sensitive hydrogel valve 9.4 Summary and conclusions References

Index

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289 289 293 293 298 303 308 309

313 313 314 314 316 320 320 320 320 329 340 345 346 349

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About the authors

Dr. Mostafa Baghani received his PhD in mechanical engineering from Sharif University of Technology in 2012. He received his Master’s degree from the same university in 2008, and his BSc from the University of Tehran in 2006. His research activities are focused on synthesis, design, and constitutive modeling of smart materials in the framework of continuum mechanics. He has also worked on computational modeling of the proposed constitutive models often developed via nonlinear finite element method. Dr. Baghani is also collaborating with ICube Laboratory as a permanent-invited professor at the University of Strasbourg with activities in computational mechanics in materials science. He has published more than 190 journal papers and reviewed for more than 30 international journals. Dr. Majid Baniassadi received his PhD degree in mechanics of materials from the University of Strasbourg (2011). He received his Master’s degree from the University of Tehran (2007) and his undergraduate degree from Isfahan University of Technology (2004), in mechanical engineering. His research interests include multiscale analysis and micromechanics of heterogeneous materials, numerical methods in engineering, and electron microscopy and image processing for microstructure identification. Dr. Baniassadi is also collaborating with ICube Laboratory as permanent-invited professor at the University of Strasbourg with activities in computational mechanics, multiscale modeling, and image processing applied to materials science. Thus far, he has published more than 120 scientific journal papers in prestigious international journals and he is often contacted for peer reviewing the submitted papers. Prof. Yves Remond is currently a Distinguished Professor (Exceptional Class) at the University of Strasbourg in France. He is working at ICube—The Engineering Science, Computer Science and Imaging Laboratory—which belongs to both the University of Strasbourg and the CNRS. His teaching activity is conducted at the European Engineering School of Chemistry, Polymers and Materials Science (ECPM) in the field of continuum mechanics, mechanics of polymers, composite materials and mechano-biology. He graduated in mechanics from Ecole Normale Supe´rieure of Cachan (now Paris-Saclay) and received his PhD degree from the University Paris VI in 1984 (Pierre et Marie Curie). Since 2012, he held a position of scientific deputy director at the CNRS headquarter in Paris—INSIS, Institute for Engineering and Systems Sciences. He was also distinguished as Officer in the

xiv

About the authors

Order of Academic Palms. He is a member of the International Research Center for Mathematics & Mechanics of Complex Systems, at the Universita dell’Aquila (Italy) and was the president of the French Association for Composite Materials (AMAC). He advised about 30 PhD students and habilitations and published about 150 scientific papers in the field of mechanics of composite materials, polymers, and bioengineering.

Preface

This book aims to cover different types of smart polymer materials, that is, temperature-sensitive shape memory polymers and temperature/light/chemo-sensitive hydrogels. Practical applications for these materials require constitutive models that describe the three-dimensional strain
stress response. Therefore, in this book, different thermo
chemo
mechanical constitutive models are presented for these smart polymers. To employ these constitutive models in applied real-world problems, for example, in cardiovascular stents, smart microvalves and actuators, their time-discrete version should be developed, for example, in nonlinear finite element framework. For each introduced constitutive model, its numerical counterpart is also reported. Using these constitutive models, some interesting applications are presented which are followed by some structural analysis developed for structures made of smart polymers. The major features of the book that make it unique are listed as follows: G

G

G

G

G

G

G

G

Providing a review on constitutive models for shape memory polymers Giving a review on constitutive modeling of pH-sensitive hydrogels Examining some experimental aspects of shape memory polymers Development of some constitutive modeling of shape memory polymers Shape memory polymer nanocomposites and corrugated structures Numerical investigation of smart auxetic 3D meta-structures based on shape memory polymers via topology optimization Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels Fluid
structure interaction analysis of different smart hydrogel microvalves

Finally, we should mention that this book has been reproduced based on the research of the authors, which were published in prestigious journals in recent years.

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Acknowledgments

We would like to thank our collaborators, Mohammad Reza Bayat, Mohammad Shojaeifard, Ebrahim Yarali, Mahdi Baniasadi, Mehdi Ansari, Nima Roudbarian, Hashem Mazaheri, Mohammad Mehdi Rafiei, Nasser Arbabi, Elham Khanjani, Mehrzad Taherzadeh, and Fatemeh Sadeghi. They helped us over the years to publish enriched manuscripts in prestigious journals and finally we had a chance to arrange them as this book. We also would like to thank Mrs. Fatemeh Jarahi for designing the front and back cover of this book. The authors would be grateful for the reports of typographical and other errors to be sent electronically via the following email: [email protected]

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Intelligent soft matters: need for numerical modeling in design and analysis

1

Chapter outline 1.1 Introduction 1 1.2 Applications of shape memory polymers 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5

1.3 Smart hydrogel applications 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7 1.3.8 1.3.9 1.3.10

2

Biomedical applications 2 Aerospace application 3 Textile application 4 Automotive application 4 Other applications 4

5

Tissue engineering 6 Drug delivery 6 microfluidic valves 7 Hydrogels for wound dressing 7 Hydrogel and cancer therapy 8 Hydrogels and water treatment 8 Hydrogels and contact lens products 8 Hydrogels and agriculture 9 Hydrogels and biosensors 9 Hydrogels and hygiene products 9

1.4 Numerical modeling in design and analysis of intelligent soft matters 10 References 12

1.1

Introduction

In almost all previous books on smart polymers, attention is mostly paid to microstructural phenomena responsible for the smartness of the material. There is no book available in the literature that is fully devoted to continuum-mechanics and numerical aspects of different types of smart polymeric materials. It is aimed in this book to introduce constitutive models for thermo-responsive shape memory polymers and smart hydrogels developed on the basis of continuum thermodynamics. Calibration methods for identifying material model parameters as well as finite element implementation of the presented models are discussed. With the aid of these tools, one is able to tackle practical problems, which require thermomechanical response of these soft materials, and design or analyze real-world structures made of these smart polymers, for example, in cardiovascular stents, smart micro-valves and actuators. It should be noted that the time-discrete counterpart of these Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00008-2 © 2023 Elsevier Inc. All rights reserved.

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Computational Modeling of Intelligent Soft Matter

constitutive models should be developed for example in nonlinear finite element framework. For each introduced constitutive model, its numerical implementation is also reported. Using these constitutive models some interesting applications are presented. The major features of this book that make it unique are listed as follows: G

G

G

G

G

G

G

G

providing a review on constitutive models for shape memory polymers; examining some experimental aspects of shape memory polymers; development of some constitutive modeling of shape memory polymers; shape memory polymer nano-composites and corrugated structures; numerical investigation of smart auxetic 3D metastructures based on shape memory polymers via topology optimization; giving a review on constitutive modeling of pH-sensitive hydrogels; equilibrium and transient swelling of soft and tough pH-sensitive hydrogels; and fluid—structure interaction analysis of different smart hydrogel microvalves.

It should be noted that almost in most cases, the authors of this book have published the reference articles during the last decade. In the following, some applications of these materials are briefly introduced.

1.2

Applications of shape memory polymers

Shape memory polymers are a class of smart materials that are able to recover their primary shape after applying large deformation through an external stimulus, such as light, chemical, electricity, and temperature. Due to properties, such as high elastic strain (more flexible in processing), the biodegradability of some SMPs, and especially economical aspects, SMPs have attracted a great deal of interest among researchers. Already, there are a wide range of applications in biomedical, aerospace, automotive, and textile in use in real-world settings and the range of applications will probably continue to grow. In the following, some applications of SMP in different fields are introduced with a few examples (Table 1.1).

1.2.1 Biomedical applications The potential for active deformation has increased the applications of SMP in biomedical fields, such as vascular and surgery applications for the replacement of handheld surgical instruments. The use of the shape memory effect for cardiovascular stent treatments has been presented in [1] as a way to minimize the catheter size, while also providing highly regulated and customized deployment at body temperature. To achieve control over the thermomechanical response of the system, SMP networks were created by photopolymerizing tert-butyl acrylate and poly(ethylene glycol) dimethacrylate. Shape recovery response of stents was studied at body temperature as a function of crosslink density, glass transition temperature (Tg), deformation temperature, and geometrical perforation, while all of these parameters can be independently controlled.

Intelligent soft matters: need for numerical modeling in design and analysis

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Table 1.1 Some examples of SMPs application in different fields. Biomedical

Aerospace

Cardiovascular applications Orthopedic devices Ophthalmic applications Orthodontic application Suture

Deployable structures Morphing structures Solar panels Truss

Tissue engineering Dialysis needle

Radar

Drug delivery

Antennas

Reflector

Textile Life jacket Floating wheels Finishing fabrics Breathable fabrics Damping fabrics

Automotive

Others

Self-healing composite Seat and adaptive lens assemblies Reconfigurable storage bins Airflow control devices

Food packaging Flexible Electronics Wearable electronics Soft Robotics Actuators

Intelligent bumpers Smart energy storage textiles Intelligent textile

4D-printing Origami structure Arrays structures Shape memory gripper

Zhang et al. [2] cured a novel sustainable soybean oil epoxy acrylate into a highly biocompatible and smart scaffold capable of supporting the development of human bone marrow mesenchymal stem cells using 3D fabrication techniques based on 3D laser printing. Experiments on SMPs demonstrated that the scaffold fixed a temporary shape at 218 C and entirely regained its original shape at 37 C, this result indicates that 4D printing applications have a great potential in tissue-engineering field. Sutures should be tied extremely carefully in general. If the threads are sewn too firmly, they may injure the skin cells; on the other hand, if the threads are stitched too loosely, the function could be compromised. Suturing success is determined on the surgeon’s ability and experience. These issues are effectively addressed by the self-tightening SMP suture. SMP sutures can be stitched loosely in their temporary form, and when they come into contact with patient’s body temperature, they shrink and can then be tightened as needed. In this situation, it will use the least amount of force possible to avoid cell injury. Sutures that are biocompatible and biodegradable would be more prominent in wound closure devices. The 14th chapter of [3] focuses on applications of self-reinforcing shape-memory sutures.

1.2.2 Aerospace application SMPs were originally employed in aerospace applications with cross-linked polyethylene in the 1960s and are still used today [4]. Weight and volume are the most

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Computational Modeling of Intelligent Soft Matter

important parameters to consider in aerospace applications, especially when sending the equipment into orbit. Deployment structures are common in spacecraft to keep compacted large devices in launch vehicles. In 2005 CTD has developed a shape-memory hinge called the elastic memory composite hinge for the deployment of reflectors, radiators, and solar arrays [5]. These hinges had four parts, two end-fittings for connection to the device body and two carbon-fiber EMC components with internal heaters. On the International Space Station, a six-hinge type with different length was experimented. The ISS crew used 7 W to deploy these hinges over 5 min and monitored their precision.

1.2.3 Textile application A thermo-responsive SMP with a glass transition temperature near to the body’s external temperature was obtained from blending of linear low-density polyethylene and different polycyclooctene (PCO) with 2 wt% of dicumyl peroxide (DCP) [6]. The synergistic combination of shape-memory capabilities of PCO and excellent mechanical and processing performance of polyethylene allows these polymeric blends to tune characteristics. Thermal stability, thermal properties, and moisture permeability of these blends were considered, and it was observed that these PCOLLDPE blends could be employed to develop application of thermo-active membranes for textile [6]. Deng et al. [7] created a fiber-shaped supercapacitor (FSSC) by using carbon nanotube sheets on an SMPU substrate. The constructed FSSC could be changed and “frozen” into a user-specified size and shape; once the temperature reaches above the thermal transition point, it will automatically reshape to its original shape and size. The FSSC may also be woven into electronic textiles to create smart clothing for flexible electrical devices.

1.2.4 Automotive application General Motors Corporation (GMC) and BMW (Bayerische Motoren Werke AG) are working on redesigning cars. SMPs provide new potential for automobiles to have new opportunities. The revolutionary materials improve vehicle performance, while adding new high-tech features at a cheaper price. Also, SMPs allow for design flexibility as well. GMC is presently applying smart materials to the test on prototype cars. GMC has over 250 US patents pending and patents for “smart” materials for its vehicles and plans to introduce smart materials applications in its cars to develop potential automotive applications for novel smart materials. GMC is collaborating with the Smart Vehicle Concepts Center at Ohio State University (HRL Laboratories LLC, a corporate R&D laboratory owned by GMC/Boeing; the University of Michigan; and the Smart Vehicle Concepts Center at Ohio State University) [8].

1.2.5 Other applications Shape Memory Gripper is a small soft robot that can grift and lift objects. This instrument can work easily by programmed recovery and bending behavior of polymer and

Intelligent soft matters: need for numerical modeling in design and analysis

5

Figure 1.1 (A) A shape memory stent with an example of its free recovery measurement [1]. (B) An example of breathable and waterproof intelligent textile [6]. (C) A self-deployment origami 4D-printed honeycomb sandwich structure with a large shape transformation [11]. (D) 4D-printed scaffolds fabricated with laser frequency 12,000 Hz and printing speed 10 mm/s with different infill density [2]. (E) A versatile gripper equipped with three fastresponse and stiffness-tunable actuators with ability of good shape adaptivity and high load capacity [12]. (F) Some potential applications of SMP in automobile.

also has a simple manufacturing process by using a range of SMPCs with specified mechanical characteristics. Wei et al. [9] used direct 3D printing to create an electroactive hybrid gripper. The gripper was able to provide quick reactions at varying voltages according to printable ink (combination of PLA/ CNT/Ag) with high electricity. Designing a quick response gripper, Zhang et al. [10] embedded SMP composite layers into a pneumatic soft actuator fabricated by 3D printer. This gripper has the ability to set up stiffness and could raise objects weighing between 10 g and 1.5 kg within 32 s. Xin et al. [11] presented a spatially self-expanding structure with an active sandwich structure with various honeycombs (i.e., hexa-chiral, star, double arrow, and tetra-chiral) fabricated by 4D printing for practical engineering applications. The shape memory properties of the active sandwich structure were studied, and the result indicated that it has an extremely good shape memory performance, with 99% and 98% shape recovery and shape fixity, respectively, high precision selfdeployment capability, and also large area change rate. A schematic of discussed applications of SMPs are shown in Fig. 1.1.

1.3

Smart hydrogel applications

In this section, main applications of smart hydrogels are briefly discussed. A schematic representation of some of these applications is illustrated in Fig. 1.2.

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Computational Modeling of Intelligent Soft Matter

Figure 1.2 (A) A shape memory stent with an example of its free recovery measurements [13]. (B) An example of breathable and waterproof intelligent textile [14]. (C) A selfdeployment origami 4D-Printed honeycomb sandwich structure with a large shape transformation [15]. (D) 4D-printed scaffolds fabricated with laser frequency 12,000 Hz and printing speed 10 mm/s with different infill density [16]. (E) A versatile gripper equipped with three fast-response and stiffness-tunable actuators with ability of good shape adaptivity and high load capacity [17]. (F) Some potential applications of SMP in automobile [18].

1.3.1 Tissue engineering Today, tissue engineering is a fascinating area for hydrogels application. The basic concept that has laid down on tissue-engineering applications at hospitals. Different types of cells can be separated from a patient, while a bunch of them extended in a cell culture and then seeded onto a carrier. Then, targeted structure grafted back into the location target of the same patient and replaced as an early tissue. In this approach, an architecture scaffold is needed to accommodate cells and guide their growth through providing appropriate nutritional conditions and spatial organization and help to the regeneration of tissues in the three dimensions. Until now, fabrication of tissues has carried out for some parts of the body like liver and cartilage [19]. The high number of research clearly specifies that hydrogels are ideal candidate materials to be applied in the field of tissue engineering due to their unique characteristics. Such ideal tissue-engineering scaffold made of hydrogels must combine the mechanical tailoring possibilities of synthetic polymers with the biomimetic properties of natural materials [20].

1.3.2 Drug delivery Recent developments in biomedical field address the use of water-swollen materials, such as hydrogels, as carriers for the development of novel pharmaceutical formulations and for the delivery of drugs, peptides, and proteins, as targeting agents for site-specific delivery [21,22]. A number of strategies have been developed to

Intelligent soft matters: need for numerical modeling in design and analysis

7

achieve efficient drug delivery systems (DDSs). Such devices can be used for oral, rectal, ocular, epidermal, and subcutaneous application. Treatment through oral cavity route can be used for mouth diseases. The gastrointestinal tract is the next route of drug delivery because of the facility of administration of drugs and its large surface area for systemic absorption. The rectal route has been used to deliver many types of drugs. Ocular drug delivery route is another cure method, which confronts with different challenges. Drug delivery to the skin is the last one [23].

1.3.3 microfluidic valves There is an extended range of applications for valves in microscale, especially in medicine. For some diseases, such as diabetes, regulating blood glucose levels through autonomous insulin delivery is a challenging process. On the other hand, traditional pharmaceuticals unable to withstand the acidic environment of the digestive tract or penetrate the dermis need to be injected. Hence, smart materials, such as responsive hydrogel materials, can make the cure process easier, especially when these materials are fabricated by in situ photopolymerization. Although unconventional microfluidic systems do not require external power for operation, they should be used in microscale due to their relatively long response times. At these scales, stimuli-responsive hydrogels could enhance the capabilities of microfluidic systems by allowing self-regulated flow control. Such designed systems could serve as devices in self-regulated drug delivery or even biosensors. By tailoring the chemical composition of the hydrogel, modifying of output response of the system is accessible. These self-regulated flow control systems undergo volumetric swelling in response to chemical changes in their local environment. For example, of this application, the novel medicine treatment of diabetes can be mentioned, which consists of a glucose-sensitive flow control system, which actuates in the designed way [14,24].

1.3.4 Hydrogels for wound dressing Skin provides a vital natural barrier against the harmful aspects of environment. When a defect or a break in the skin happens, the original layers of the skin protected mechanism damages; thus additional protection on the injured skin is required. Modern medical technologies instead of traditional ones have proposed some compatible candidate biomaterials, for example, functional hydrogels for rapid wound healing. Because of their natural porous structure and high-water content, they could absorb abundant exudates or blood, act as a barrier to microorganisms, preserve the wound from external sources of infection, and maintain a comparable moist environment at the skin defect site and maintain good O2 and water permeability. Also, injectable hydrogels can be another useful method thanks to their especial benefits, such as filling wound sites completely, in situ encapsulating bioactive molecules, and adhering to wounds [15]. In the process of wound healing, endothelial cells grow rapidly and induce the formation of blood vessels in the granulation tissue, the damaged tissue is gradually replaced by epithelial cells

8

Computational Modeling of Intelligent Soft Matter

and fibroblasts, and the wound surface is gradually filled with granulation tissue. At the final stage, a connective tissue is formed and the new epithelium is strengthened. An ideal hydrogel wound dressing should have the characteristics, such as biocompatibility and ability, to stop bleeding immediately, and stimulate hemostasis-related factors to accelerate wound healing [25].

1.3.5 Hydrogel and cancer therapy The dynamic cancer microenvironment causes changes in extracellular matrix properties over a long period. In recent years numerous hydrogel-based reconstructive tumor models have been introduced to be used for cancer investigations. Findings indicate that researchers are close to designing a comprehensive tumor model. Also, cancer DDSs have extensively tended toward hydrogel-based approaches. Hydrogel-based DDSs may improve chemotherapy results and gene therapy efficacy by enhancing the drug half-life, facilitating controlled and adjustable drug release, and subsequently diminishing nontargeted exposure. All these characteristics lead to a wide range of possibilities to optimize cancer DDSs [16]. Recent decades have witnessed numerous successful examples of hydrogel applications as anticancer drug delivery vehicles. In terms of cancer treatment, in situ hydrogels may not only serve as drug carriers, loading drugs that otherwise suffer from poor solubility and low stability, but can also offer a localized depot for sustained delivery of multiple drugs. Compared with other nanoparticles, nanogels enjoy better solubility and biocompatibility in a host and can avoid clearance after chemical modification [26].

1.3.6 Hydrogels and water treatment The expansion or contraction state of hydrogel is suitable for adsorption, retention, or separation of various target substances. All the compatible characteristics have exhibited a notable performance of hydrogels as a promising material in separation engineering [27]. To be specific, utilizing hybrid hydrogels to remove metal cations, radionuclides, dyes, anions, and other miscellaneous pollutants from water has becoming a hot field of study among researchers. Although they possess poor mechanical and low kinetics properties, which are challenging, they are good at water-based treatment due to their hydrophilic and porous network structure. Thus they could be used as a cheap, recyclable, and effective adsorption method of separation. Fortunately, hybrid structure of such hydrogels can be designed for various situations for water treatment [28].

1.3.7 Hydrogels and contact lens products Hydrogel can be used as a synthetic biocompatible material useful in contact lens applications. Contact lenses are mainly classified as “hard” or “soft” according to their elasticity. Even though hard lenses are longer lasting, they tend to be poorly accepted by the wearers and may require a lengthier adaptation period. Hard contact

Intelligent soft matters: need for numerical modeling in design and analysis

9

lenses are primarily based on hydrophobic materials, whereas soft lenses are based on hydrogels. Soft contact lenses can be produced with different techniques, such as spin-casting, mold-casting, and lathe-cutting. A polymeric hydrogel should have some important physical properties for utilizing as a contact lens material, such as luminous transmittance, refractive index, sufficient oxygen permeability, wettability and permeability to water, stability, excellent mechanical properties, and biocompatibility [29].

1.3.8 Hydrogels and agriculture Some sorts of hydrogels like superabsorbent hydrogels can be beneficial in agricultural use. agricultural applications of SAHs can be categorized under two broad segments namely water reservoir and nutrient carrier. they can eliminate the soil aridity via moisturizing and improve water retaining capacity of the soil for an extended period of time. SAHs save the reserved water in their structure from evaporation and drainage. Stimulus-responsive hydrogels can provide functional services. They also can be improved with respect to the controlled release of agrochemicals (drugs, herbicides, or pesticides) [30]. SAHs employ in the fabrication of synthetic soil conditioners, which add to soil of cultivation areas with weak water resources.

1.3.9 Hydrogels and biosensors In recent years biosensors have been developed for miscellaneous applications, such as detecting contagious diseases and contributing to the fields of clinical diagnostics. A biosensor is a device capable of providing specific analytical information (about target molecule) using a biological recognition element. 2D bio-probes are traditional devices of biosensing in which possess many disadvantages, such as insufficient accessibility and capacity for loading. One of the structures, which improves and optimizes biosensors, is the hydrogel. Compared to 2D types of biointerfaces, the 3D porous structure of hydrogels increases the surface area of the material, allowing the immobilization of orders of magnitude larger amounts of recognition elements, ranging from small molecules to proteins and even cells. Main factors in biosensing are sensitivity, specificity, response time, selectivity, prevention of nonspecific binding, and reproducibility [31].

1.3.10 Hydrogels and hygiene products In last decades superabsorbent polymers (SAPs) have been introduced into the diaper industry. They have extended use thanks to their excellent water retention. Therefore hydrogels are widely applied for designing disposable diapers, napkins, adult urinary incontinence products, and personal safety products. Moreover, SAPs have a better performance than multilayered diapers and synthetic over pants providing protection against diaper rash, germs colonization, leakage of diapers for minimizing fecal contamination, and gastrointestinal illnesses. After developments,

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it was possible to develop diapers with leakage levels below 2% and the standard weight of a medium size diaper could be reduced by 50%, which is important in terms of environmental and manufacturing issues.

1.4

Numerical modeling in design and analysis of intelligent soft matters

In this book, it is aimed to present required elements to numerically simulate smart soft matters, that is, thermo-responsive shape memory polymers and pH-sensitive hydrogels. Chapters 2
6 are devoted to shape memory polymers, while Chapters 7
9 are devoted to pH-sensitive hydrogels. For this purpose, in Chapter 2, to predict the thermomechanical response of SMPs, several constitutive relations have been introduced over the past two decades. The focus of this chapter is to give a review on structures, categorization, applications of SMPs, and necessary constitutive models. First, a detailed review on properties, structure, and classifications of SMPs is carried out. In next sections, the developed models in literature are discussed that, particularly, are focused on the phase transformation and thermo-viscoelastic points of view for conventional, twoway, as well as multi-SMPs. A statistical analysis on constitutive models of temperature-dependent SMPs is also conducted. Finally, some conclusions are drawn, which are useful in the selection of appropriate SMP constitutive equations. In Chapter 3, attention is paid to experimental aspects of SMPs’ response, which are required in the development of SMPs constitutive equations. Along with this purpose, some production methods are discussed. Besides, parameters playing key roles on shape memory effect of SMPs are investigated. In the last section of this chapter, as an applied example, an SMP stent is designed and its performance in radial force and shape recovery and recovery start temperature in a constrained state are evaluated. In Chapter 4, we deal with constitutive modeling, calibration, and simulation of SMPs. In the first section, a 3D constitutive model for SMPs under time-dependent multiaxial thermomechanical loadings is presented. The derivation is based on an additive decomposition of the strain and satisfying the second law of thermodynamics. The evolution laws for internal variables are derived during both cooling and heating thermomechanical loadings, where the viscous effects are also fully accounted for. Furthermore, the time-discrete form of the evolution equations is presented. In the next section, a constitutive model from the thermo-visco-hyperelasticity point of view is briefly discussed. Employing this model, a general solution is presented for the thermomechanical behavior of SMP in large deformation. The proposed formulation is suitable for describing shape memory behaviors with one or more temporary shapes in uniaxial and combined extension-torsion problems. It is also suitable for calibrating the SMP material parameters. Chapter 5 is devoted to SMP composites, that is, nanocomposites and corrugated structures. The main drawback for SMPs is their poor elastic properties, which

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impedes the applications of SMPs when large recovery stress is required, for example, in actuators. Thus a variety of reinforcements are used in SMP composites. In this chapter, we first deal with modeling and homogenization of SMP nanocomposites with perfectly dispersed graphene nanoplatelets. In the second part, numerical homogenization of coiled carbon nanotube reinforced SMP nanocomposites is carried out to reduce the damage generated in interphases of the SMP and inclusions. At the third part, we examine thermomechanical behavior SMPs reinforced by corrugated polymeric sections to enhance the structural force recovery capacity of the SMP composite. Metamaterials, a spotlight of researchers nowadays, are special types of designer materials having unusual mechanical properties and advanced functionalities usually not found in nature. In Chapter 6, we study SMP metamaterials based on triply periodic minimal surfaces (TPMS) and auxetic structures. They are two well-known classes of metamaterials, introduced to develop new structures. In this chapter, it is aimed to employ these two types of structures where the material is SMP, to enhance and control, for example, shape recovery, shape fixity, and force recovery of the designed metastructure. In Chapter 7, a review is given on constitutive models of pH-sensitive hydrogels, available in the literature. In response to the external pH variation, these materials span the volume of an aqueous solution and swell without dissolution. They have applications in various fields, such as drug delivery, tissue engineering, and soft actuators, which makes the simulation of their behavior crucial. The essential prerequisite for this end is the swelling theory. In this chapter, a review is presented on the fundamental phenomena in swelling theories of the pH-sensitive hydrogels. We also classify the swelling theories into two groups including the monophasic and multiphasic theories. Then, we briefly examine the most highlighted swelling theories in each group. In Chapter 8, equilibrium and transient swelling of soft and tough pH-sensitive hydrogels are discussed, where their constitutive modeling and FEM implementation are reported. In this chapter, as the first step, a stationary (equilibrium) model is developed to continuously predict swelling response of pH/temperature-sensitive PNIPAM hydrogels. In the second part, a coupled electro-chemo-mechanical transient large deformation swelling theory is developed for pH-sensitive hydrogels. In the third section, we deal with tough pH-sensitive hydrogels and develop a transient electro-chemo-mechanical theory for swelling and mechanical behavior of tough pH-sensitive hydrogels. All these theories are implemented into finite element frameworks, and experimental data are used to validate the relevant theory. To showcase the capabilities of the proposed constitutive equations in latter chapter for hydrogels, in Chapter 9, as some complex examples, structural analysis of some smart hydrogel microvalves is presented, considering the effect of fluid— structure interaction (FSI). In this chapter, we aim to examine the effects of FSI on the behavior of microvalves made of both temperature and pH-sensitive hydrogels. We first analyze the thermomechanical equilibrium response of temperaturesensitive hydrogel microvalves in a T-junction flow sorter accounting for FSI effects. Then, we study stationary (equilibrium) response of pH-sensitive hydrogel

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microvalves through an FSI approach. Finally, a fully coupled fluid-solid interaction study on the transient behavior of pH-sensitive hydrogel-based microvalves to show the importance of transient analysis.

References [1] C.M. Yakacki, R. Shandas, C. Lanning, B. Rech, A. Eckstein, K. Gall, Unconstrained recovery characterization of shape-memory polymer networks for cardiovascular applications, Biomaterials 28 (14) (2007) 2255
2263. [2] S. Miao, W. Zhu, N.J. Castro, M. Nowicki, X. Zhou, H. Cui, et al., 4D printing smart biomedical scaffolds with novel soybean oil epoxidized acrylate, Scientific Reports 6 (1) (2016) 1
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300. [4] D.L. Safranski, J.C. Griffis, Applications of shape-memory polymers, Shape-Memory Polymer Device Design, Elsevier, Amsterdam, 2017, pp. 189
217. [5] S.C. Arzberger, M.L. Tupper, M.S. Lake, R. Barrett, K. Mallick, C. Hazelton, et al., Elastic memory composites (EMC) for deployable industrial and commercial applications, Smart Structures and Materials 2005: Industrial and Commercial Applications of Smart Structures Technologies, SPIE, 2005. [6] A. Reizabal, J.M. Laza, J.M. Cuevas, L.M. Leo´n, J.L. Vilas-Vilela, PCO-LLDPE thermoresponsive shape memory blends. Towards a new generation of breathable and waterproof smart membranes, European Polymer Journal 119 (2019) 469
476. [7] J. Deng, P. Ren, D. Deng, X. Bao, Enhanced electron penetration through an ultrathin graphene layer for highly efficient catalysis of the hydrogen evolution reaction, Angewandte Chemie International Edition 54 (7) (2015) 2100
2104. [8] D. Rosato, Smart polymer applications entering mainstream markets, Multibriefs, 2015. ,http://exclusive.multibriefs.com/content/smart-polymer-applications-entering-mainstream-markets/engineering.. [9] H. Wei, X. Cauchy, I.O. Navas, Y. Abderrafai, K. Chizari, U. Sundararaj, et al., Direct 3D printing of hybrid nanofiber-based nanocomposites for highly conductive and shape memory applications, ACS Applied Materials & Interfaces 11 (27) (2019) 24523
24532. [10] Y.F. Zhang, N. Zhang, H. Hingorani, N. Ding, D. Wang, C. Yuan, et al., Soft robots: fast-response, stiffness-tunable soft actuator by hybrid multimaterial 3D printing, Advanced Functional Materials 29 (15) (2019) 1970098. [11] X. Xin, L. Liu, Y. Liu, J. Leng, Origami-inspired self-deployment 4D printed honeycomb sandwich structure with large shape transformation, Smart Materials and Structures 29 (6) (2020) 065015. [12] B. Zhang, W. Zhang, Z. Zhang, Y.-F. Zhang, H. Hingorani, Z. Liu, et al., Self-healing four-dimensional printing with an ultraviolet curable double-network shape memory polymer system, ACS Applied Materials & Interfaces 11 (10) (2019) 10328
10336. [13] P. Chaudhari, V.M. Ghate, S.A. Lewis, Next-generation contact lenses: towards bioresponsive drug delivery and smart technologies in ocular therapeutics, European Journal of Pharmaceutics and Biopharmaceutics 161 (2021) 80
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[15] J. Qu, X. Zhao, Y. Liang, Y. Xu, P.X. Ma, B. Guo, Degradable conductive injectable hydrogels as novel antibacterial, anti-oxidant wound dressings for wound healing, Chemical Engineering Journal 362 (2019) 548
560. [16] M. Sepantafar, R. Maheronnaghsh, H. Mohammadi, F. Radmanesh, M.M. Hasanisadrabadi, M. Ebrahimi, et al., Engineered hydrogels in cancer therapy and diagnosis, Trends in Biotechnology 35 (11) (2017) 1074
1087. [17] Z. Wu, P. Zhang, H. Zhang, X. Li, Y. He, P. Qin, et al., Tough porous nanocomposite hydrogel for water treatment, Journal of Hazardous Materials 421 (2022) 126754. [18] K.T. Nguyen, J.L. West, Photopolymerizable hydrogels for tissue engineering applications, Biomaterials 23 (22) (2002) 4307
4314. [19] L.K. Yang S, Z. Du, C.K. Chua, The design of scaffolds for use in tissue engineering. Part I. Traditional factors, Tissue Eng (2001) 679
689. [20] S. Van Vlierberghe, P. Dubruel, E. Schacht, Biopolymer-based hydrogels as scaffolds for tissue engineering applications: a review, Biomacromolecules 12 (5) (2011) 1387
1408. [21] Z. Gharehnazifam, R. Dolatabadi, M. Baniassadi, H. Shahsavari, A.M. Kajbafzadeh, K. Abrinia, et al., Computational analysis of vincristine loaded silk fibroin hydrogel for sustained drug delivery applications: multiphysics modeling and experiments, International Journal of Pharmaceutics 609 (2021). [22] Z. Gharehnazifam, R. Dolatabadi, M. Baniassadi, H. Shahsavari, A.-M. Kajbafzadeh, K. Abrinia, et al., Multiphysics modeling and experiments on ultrasound-triggered drug delivery from silk fibroin hydrogel for Wilms tumor, International Journal of Pharmaceutics 621 (2022) 121787. [23] N.A. Peppas, P. Bures, W. Leobandung, H. Ichikawa, Hydrogels in pharmaceutical formulations, European Journal of Pharmaceutics and Biopharmaceutics 50 (1) (2000) 27
46. [24] D. Beebe, J. Moore, J. Bauer, et al., Functional hydrogel structures for autonomous flow control inside microfluidic channels, Nature 404 (2000) 588
590. [25] J. Su, J. Li, J. Liang, K. Zhang, J. Li, Hydrogel preparation methods and biomaterials for wound dressing, Life (Basel) 11 (10) (2021). [26] Z. Sun, C. Song, C. Wang, Y. Hu, J. Wu, Hydrogel-based controlled drug delivery for cancer treatment: a review, Molecular Pharmaceutics 17 (2) (2020) 373
391. [27] G. Pan, K.-I. Kurumada, Y. Yamada, Application of hydrogel for the removal of pollutant phenol in water, Journal of the Chinese Institute of Chemical Engineers 39 (4) (2008) 361
366. [28] G. Jing, L. Wang, H. Yu, W.A. Amer, L. Zhang, Recent progress on study of hybrid hydrogels for water treatment, Colloids and Surfaces A: Physicochemical and Engineering Aspects 416 (2013) 86
94. [29] E. Calo´, V.V. Khutoryanskiy, Biomedical applications of hydrogels: a review of patents and commercial products, European Polymer Journal 65 (2015) 252
267. [30] S.M.F. Kabir, P.P. Sikdar, B. Haque, M.A.R. Bhuiyan, A. Ali, M.N. Islam, Cellulosebased hydrogel materials: chemistry, properties and their prospective applications, Progress in Biomaterials 7 (3) (2018) 153
174. [31] A. Herrmann, R. Haag, U. Schedler, Hydrogels and their role in biosensing applications, Advanced Healthcare Materials 10 (11) (2021) e2100062.

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A detailed review on constitutive models for thermoresponsive shape memory polymers

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Chapter outline 2.1 Introduction 15 2.2 Classification of temperature-dependent polymers

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2.2.1 Thermoset and thermoplastic polymers 18 2.2.2 The effect of temperature on thermoset and thermoplastic polymers 19

2.3 The molecular structure of shape memory polymers and their classification

20

2.3.1 Chemical structure of shape memory polymers 20 2.3.2 Classification of shape memory polymers 21

2.4 Modeling thermoresponsive shape memory polymers

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2.4.1 Modeling of conventional thermally activated shape memory polymers 30 2.4.2 Modeling of two-way thermally activated shape memory polymer 54 2.4.3 Modeling of thermally activated multishape memory polymer 55

2.5 Statistical analysis of available shape memory polymer models 2.6 Summary and conclusion 60 References 61

2.1

58

Introduction

Smart materials are a class of materials that are programmable and their properties change in reaction to external stimuli, for example, magnetic field, electrical field, moisture, ultrasound exposure [1,2], pH, temperature, and light. Shape memory polymers (SMPs), shape memory alloys, hydrogels, smart fluids, for example, electrorheological fluid, magnetorheological fluid, and ferrofluids, and self-repairing materials are some well-known examples of smart materials [3,4]. Among these smart materials, during the past two decades SMPs have drawn widely attention from mechanical point of view, and many experimental and modeling efforts have been made on these materials. SMPs can preserve their temporary shapes for a long time and recover their original (permanent) shape upon applying an external stimulus such as heat, magnetic field, electrical field, light, pH, moisture or water, radiation, dissolution, focused ultrasound, external infrared (IR) light, microwaves or laser light. Thanks to the shape memory effect (SME) and recovery properties of SMPs, they have wide range of potential applications in many fields, in particular, biomedical devices [5,6]. It should be noted that external stimuli can applied to SMPs either direct (such as direct heat, magnetic field, and electric fields) or indirect, such as applying magnetic field, to generate an induced heat in SMPs. Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00010-0 © 2023 Elsevier Inc. All rights reserved.

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Polymers commonly have no intrinsic considerable SMEs. In other words, a single polymer does not show significant SME by itself, but this effect is achieved by combining different polymers with different morphologies in a specific loading procedure called polymer functionalization [7
9]. Most of SMPs are temperature sensitive, which means they can recover their permanent (initial) shape while applying an external trigger, that is, by heating beyond their transition temperature (Ttrans) (referred to Thigh in this text) or switching temperature. The thermomechanical cycle in thermally activated SMPs commonly involves the following steps. When the SMP is in its permanent shape, it is subjected to mechanical loading at one or more high temperatures Thighs to take one or multidesirable temporary shapes (loading step). Then, during the cooling process, the material is cooled down to below Ttrans (namely, Tlow here) to stabilize a temporary shape arising from occurrence of a solidification phenomenon (cooling step). After cooling, the applied loadings are removed (unloading step). These three steps together form the shape programming. It should be noted that shape programming might involve several temporary shapes. In the last step, by reheating the SMP, the permanent shape is retrieved and the SMP returns to its original state (heating step). According to the cyclic behavior view, SMPs can be divided into two different categories: (1) one-way and (2) two-way. In the first type, SMPs work in a single thermomechanical cycle. In fact, they have a nonreversible feature, and they are thus appropriate in applications where only one single fixing-recovery cycle is needed. In contrast, two-way SMPs have the capacity to work between two temporary shapes under on-off stimulus in several cycles due to a reversible transition. For thermally triggered two-way SMPs, if the SMP is cooled again to Tlow after the first cycle, SMP retrieves its temporary shape (with or without external load), and this process could continue in many cycles. From an application point of view, two-way SMPs due to their reversible feature, are more interesting and have more applications than one-way SMPs. But from the material processing and fabrication angle, it is more complicated. For further information about two-way SMPs (see Section 2.3.2.2). It is important to know that SMP programming for fixing the temporary shape is not necessarily carried out at above Ttrans, but also it could be done through coldcompression-programming in thermoset SMPs [10
13] as well as cold-drawingprogramming in semicrystalline thermoplastic SMPs [14]. Usually, in order to study the behavior of SMPs by empirical testing and mathematical modeling, they are examined in two thermomechanical shape and force recovery steps. To this end, if the SMP is free in its final step of recovery (reheating step), the original shape of SMP is retrieved. This is referred to as stress-free strain-recovery process. In addition, if at this state, the SMP is heated up under a constraint (fixed strain), the stress or force is recovered which is called fixed-strain-stress-recovery process. A successful constitutive equation should be able to predict both the shape and stress recovery behaviors of an SMP as the primary governing effects. Fig. 2.1 schematically illustrates the thermomechanical cycle of a conventional1 thermally activated SMP.

1

In this chapter, conventional SMP is referred to one-way or dual-shape memory polymer.

A detailed review on constitutive models for thermoresponsive shape memory polymers

(A)

a

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(B)

b

c

d

(C) Figure 2.1 Thermomechanical cycle of a typical thermally activated shape memory polymer. (A) Stress
strain
temperature. (B) The variations in temperature, strain, and stress in both stress-free-strain-recovery and fixed-strain-stress-recovery processes. (C) The morphology and structure of shape memory polymers in each thermomechanical steps. Red rectangles denote fixed phases or primary bonds. Black ovals denote reversible phases or secondary bonds [15].

As one may observe from Fig. 2.1, when SMP is at Thigh and its permanent shape (state A), the network has a series of primary bonding or fixed phases (covalent or crystalline phases). The SMP is loaded to reach point B in which the molecular chains are extended. Then, maintaining this deformation and cooling the SMP to below Ttrans, it reaches point C, in which the SMP takes the desired temporary shape (programming stage of SMPs). In this step, due to the cooling, a series of secondary bonds (or crystalline phases) or reversible phases are being created. Then, for conventional SMPs, the solidification phenomenon occurs; thus when the applied load is removed, a slight portion of the stored strain is recovered at point D. In the final step, in two different recovery steps, DB or DA, respectively, the stress in the SMP or equivalently the temporary shape is retrieved. From material point of view, an ideal SMP should be able to recover 100% of its shape and force. In other words, its recovery and fixity ratio should be 100%.

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Regarding the available researches on SMPs, there is a need for a comprehensive review on the mathematical constitutive modeling of SMPs. The available review papers on SMPs currently are more focused on the biomedical applications of SMPs. One of the most important tools for realizing and utilizing the structure and applicability of SMPs is mathematical and physical modeling of these materials behaviors. The main motivation for this section is that, if an engineer or researcher aims to analyze the SME of polymers and SMP-based 4D printing in various applications, for example, biomedical, aerospace, and oil industry, or develop a new constitutive model, there is lack of a relevant comprehensive review paper specifically focused on SMP constitutive models. To settle this issue, we provide an up-to-date comprehensive review on the mechanisms, classifications, novel 4D printing applications and thermomechanical constitutive modeling of SMPs (conventional SMPs, two-way SMPs and multi-SMPs). The structure of this chapter is as follows. At first, a detailed explanation is given about the molecular structure of the SMPs, their classifications, and the mechanisms responsible for their SMEs. Then, in the next two sections, two-way SMPs and multi-SMPs are described, respectively. In the next section, we discuss different types of constitutive models for thermo-responsive SMPs (different modeling perspectives for conventional, two-way and multi-SMPs), and in the last section, a statistical analysis of the constitutive models for thermally activated SMPs is provided. We finally present a summary and present some concluding remarks.

2.2

Classification of temperature-dependent polymers

Before addressing the molecular microstructure of SMPs, first in two sections, a short description on thermosets and thermoplastics SMP, as well as their behavior in the presence of temperature is presented.

2.2.1 Thermoset and thermoplastic polymers From structural and molecular points of view, polymers are typically divided into thermoplastic and thermoset families. Thermoplastic materials only have secondary bonds (hydrogenic, van der Waals, polar) or physical cross-links among the molecular chains, but thermosets, in addition to the secondary bonds, have primary bonds (covalent) between the molecular chains (see Fig. 2.2). In thermosets, cross-links are formed chemically under thermal or compressive catalytic conditions. Typically, in terms of the molecular structure, thermoplastics are known also as linear, and thermosets known as cross-linked. It could be mentioned that photoplasticity is usually used to characterize thermoplastic materials, while thermoset materials are commonly studied by photo-viscoelasticity to identify their behavior [16]. Also, thermoplastics are highly soluble in solvent solutions. Moreover, polymeric materials are divided into two types of amorphous and crystalline in terms of phases and the order of molecules. Thermoplastics can be either amorphous or crystalline. Crystalline thermoplastics are usually denser than

A detailed review on constitutive models for thermoresponsive shape memory polymers

(A)

19

(B)

Figure 2.2 The molecular structure of thermosets and thermoplastics. (A) Thermoplastic polymers. (B) Thermoset polymers [15].

Figure 2.3 Shear modulus for three different amorphous thermoplastic, crystalline thermoplastic, and thermoset polymers versus temperature [15].

the amorphous ones. Due to the closing packing of their long molecules, they are hard and exhibit anisotropic properties. In contrary, amorphous polymers have irregular and isotropic structure, with no sharp melting point. Polypropylene and polyethylene (PE) are common examples of crystalline thermoplastics, polyvinyl chloride is a common amorphous thermoplastic, and epoxy, rubbers, and polyester are common examples of thermoset materials. In general, thermoplastics have a much simpler production process and a lower cost than thermosets [8,16,17]. Thermoset polymers are usually amorphous because there is almost no possibility to arrange some parts of the network structure due to the constraints created by the presence of cross-links. A schematic drawing of molecular structure of thermoplastic and thermoset materials are shown in Fig. 2.2.

2.2.2 The effect of temperature on thermoset and thermoplastic polymers At high temperatures, thermosets and thermoplastics are softer and have a lower elasticity modulus. Here, Fig. 2.3 schematically describes the temperature-

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dependent behavior of the thermosets, amorphous thermoplastics as well as the crystalline thermoplastics. As shown in Fig. 2.3C, in thermoset materials, by heating up to a certain temperature, the material is softened, but at temperatures higher than Ttrans, the material does not melt and eventually burns. In the amorphous thermoplastics, when the temperature rises, the material becomes softer and softens until it finally melts at the melting temperature (Fig. 2.3B). On the contrary, in crystalline thermoplastics, due to the regular arrangement of molecular chains, the material is harder than amorphous thermoplastics; thus its melting temperature is higher (Fig. 2.3A) [16].

2.3

The molecular structure of shape memory polymers and their classification

In this section, the chemical characteristics of SMPs are pointed out. Most of this section is a routine concept, which could be found in a standard review. In the first subsection, the chemical mechanisms of SMPs (e.g., amorphous, semicrystalline, thermoplastic, and thermoset) are reviewed briefly. In next section, classification of SMPs (e.g., conventional, two-way, and multi-SMPs), and the relevant definitions are described.

2.3.1 Chemical structure of shape memory polymers From Ttrans definition point of view, SMPs can be divided into two categories. Glass transition temperature (Tg) is normally used as Ttrans for amorphous SMPs, while melting temperature (Tmelt) is used for crystalline SMPs [8,18]. The melting point provides a relatively sharp transition, while the glass transition temperature extends over a quite large temperature span [19]. The Ttrans in SMPs, determining their SME, depends on the type of polymer and the temperature range of the SMP application [19]. In both amorphous and crystalline states, the SME is the result of the phase transition from the rubbery phase (the temperature is higher than Ttrans) to the glassy or crystalline phase (the temperature is lower than Ttrans). Thus it can be assumed that SMP is at least a combination of two different phases (active, frozen), where the SME (temporary or permanent) is due to conversion of these phases [8,20]. Hard phases (or fixed phases or monomers) form the netpoints that link the soft segments and may have a chemical or physical nature (covalent bond or crystalline and glassy phase). In contrast, soft segments (cross-link chains or curing agents) as a reversible phase determine the SMP temporary shape fixation and act as shape switcher. During the thermomechanical programming of SMPs, the hard segment remains unchanged and provides the shape-memorizing capability of the SMP, while the soft segment at above Tg is softer in the heating stage and is harder at the cooling stage. The soft phase is responsible for the elastic recovery response of the SMP [18,21].

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It should be noted that Tobushi et al. [22] for the first time considered SMP as a combination of two soft and hard segments. When the SMP is at a higher temperature than Ttrans, it behaves like a rubber with entropic conformational motions (i.e., quasielastic with an entropic elasticity) where its macromolecules can move under high strains. In other words, the chains are more flexible, and so their relaxation time and stiffness are small. However, when the temperature goes below Ttrans, the material behaves like a viscoelastic semisolid, and a part of the deformation in the material is stored due to formation of additional polymer chain interactions. In this situation, the movement of macromolecular chains declines, so that only local intermolecular interactions remain and prevents from large-scale conformational motion. At this stage, the crystallizable portion of the material becomes crystalline. To be specific, when the hard or frozen section is formed, the SMP becomes stiffer with a larger relaxation time. In this case, since only nonconformational motion associated with the internal energy change may occur, the material is dominated by internal energy. SMEs in amorphous SMPs are usually created due to the simultaneous presence of two invariant phases (to fix the original shape of SMP), and a softening and hardening reversible phases in SMPs. One might as well say the storage and recovery properties of SMP materials are due to the same phase transition with temperature changes, as a temperature-dependent change in the molecular chains. When SMP is at Tlow, the reversible phase becomes solid. In semicrystalline SMPs, the SME is due to the crystallization process during temperature changes. Crystallization direction depends on the deformation of SMP during the crystallization process. Crystallization plays a key role in characterizing the mechanical properties of semicrystalline polymers. The crystallization process is not always complete, which means that some chains remain amorphous, that’s why they are called semicrystalline [8,19,23
25]. According to Leng et al. [18], both thermoplastic and thermoset SMPs can be amorphous or crystalline. From bonding and chemical structure points of view, SMPs can be classified into four categories: amorphous thermoplastic, crystalline thermoplastic, amorphous thermoset, and crystalline thermoset [7,26]. SMPs with chemical bonds in comparison with physical bonds, commonly have more improved mechanical, thermal, chemical, and shape memory properties. Based on chemical architectures, a thermoset epoxy-SMP acts as a two-phase structure, that is, a hard phase and a soft phase. When cross-links added to the base polymers (e.g., polyurethanes or polyolefine-based SMPs), improve thermal stability and recovery properties of SMPs; and reduce creep behavior and role of the hard phase on SMEs. The base materials of SMPs are usually polyolefine (semicrystalline), polyurethanes (semicrystalline), acrylic (poly ((meth) acrylates)) (amorphous), polystyrene (amorphous), and polysiloxanes (amorphous).

2.3.2 Classification of shape memory polymers In this section, conventional, two-way, and multi-SMPs are presented. After the description of their structures and mechanisms, they are investigated form the experimental point of view.

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2.3.2.1 Conventional shape memory polymers In Section 2.1, conventional SMPs have been descripted in detail. This kind of SMPs mostly has been studied in the literature and most of the proposed constitutive models have been developed for this type of SMPs. The main feature of this type of SMPs is nonreversible. Meanwhile, they only work in a single cycle, which this feature may restricts them under many applications in soft actuators and sensors. In Table 2.1, several conventional SMPs used in different researches are presented. Here, from the experimental point of view, in literature, many experiments have been carried out on conventional SMPs to develop SMP constitutive models, as well as calibration or validation of the proposed models. For thermomechanical analysis of SMPs, like any other polymers, in addition to classic features (e.g., thermal expansion coefficient and Tg) its mechanical properties (including hyperplastic properties and viscoelastic properties) should be established from experimental tests. Typically, from the viscoelastic point of view, dynamic mechanical thermal analysis (DMTA) test is also performed to identify the stored and loss modulus of polymer used in characterization of the viscoelastic properties of polymers. Besides, from the phase transition view, classical thermo-mechanical tests, such as uniaxial tension or compression, are necessary. In our group, some experimental studies in this regard have been carried out recently [126
129]. Here, a sample of the acrylate polymer network test with 90% benzyl methacrylate molecular weight and 10% molecular mass of poly(ethylene glycol) dimethacrylate (PEGDMA) and 0.5% 2,2-dimethoxy-2-phenylacetophenone is used from test results of Arrieta et al. [57] in Fig. 2.4. The results from DMTA test are shown in Fig. 2.4A, where master curve for the shear storage modulus and tan(δ) are presented. It is clear that the SMP is softer at higher temperatures. In addition, in Fig. 2.4B, both stress-free-strain-recovery and fixed-strain-stress-recovery thermomechanical cycles of the acrylate-based SMP in two temperature rates 1 and 5 C/ min are represented. It is worthwhile to mention that keeping the strain fixed during the heating step (recovery), there are two competing phenomena, that is, thermal expansion and phase transition. At lower temperatures, the thermal expansion is governing, while at higher ones, phase transition is the governing phenomenon. This is shown in the stress recovery in Fig. 2.4B.

2.3.2.2 Two-way shape memory polymers As was briefly pointed out in the introduction section, two-way SMPs are sometimes referred to as reversible SMPs. The two-way SMP properties are intrinsic and are related to the intrinsic properties of material. Two-way SMPs thermomechanical path is as follows: High-temperature loading by applying an external stress (applying a temporary deformation in SMP), and then cyclic cooling and heating the material (with or without applied load on the SMP). They are referred to as twoways SMPs, because if the cycle is repeated, that is, the material is cooled down again, it takes the temporary shape, and then if it is re-heated, it remembers its permanent shape. To put it simply, the SMP, like an actuator, can switch between the

Table 2.1 Thermally activated conventional shape memory polymers in literature and its characteristics [15]. Material

Type

Structure

Tt ( C)

References

Polyurethane (PU) of the polyester polyol series Crosslinked ester-type polyurethane Oligo-(ε-caprolactone) (PCL) dimethacrylate as crosslinker and n-butyl acrylate as comonomer Epoxy resin families Tert-butyl acrylate (tBA) monomer and crosslinker poly (ethylene glycol) dimethacrylate PU 1 PCL A cyclo-olefin polymer (Zeonex-690R) Polycarbonate (PC) Poly(methyl methacrylate) Veriflex (an styrene-based SMP) Syntactic foam (Veriflex 1 glass microballoons) (Veriflex 1 plainweave carbon fiber) Poly(D,L-lactide-co-glycolide) Polyether block amid elastomer (Pebax 7233) A crosslinked epoxy network An epoxy network (Epoxy 12DA3) Silicone rubber (Sylgard 184) 1 PCL fiber

Thermoset Thermoset BThermoplastic

N.S. BA S

54 .135 B40
70

[22,27
41] [42] [26,43]

Thermoset B Thermoplastic

A A

.62 .32

[21,24,28,35,40,44
56] [17,23,57
71]

BThermoplastic Thermoplastic Thermoplastic Thermoplastic Thermoset Thermoset Thermoset Thermoplastic Thermoplastic Thermoset Thermoset Thermoset 1 Thermoplastic Thermoplastic Thermoplastic Thermoplastic Thermoset Thermoplastic

S A A A A A A A S A A S

0
50 136 145 115 B62 62 B65 37 57 35 B50 -114 And ,37 N.S. .48 ,0 95 35

[72,73] [74] [74] [74] [10,11,14,55,75
86] [10,55,78,84,87] [79,83,88] [89] [90,91] [90,91] [92] [93
95]

Polyethylenes (PEs) MHI Diaplex (urethane-based SMP) Polyether urethane (PEU) Veriflex E Ether-based polyurethane SMP (MM3520)

S N.S. BA A A

[96] [97
99] [100,101] [29,102
106] [107,108] (Continued)

Table 2.1 (Continued) Material

Type

Structure

Tt ( C)

References

An epoxy-based SMP A polyurethane-based SMP (Diaplex) 1 plain-weaveT300 carbon fiber Low-density polyethylene and high-density polyethylene Aliphatic urethane networks (polyurethane) Poly(cyclooctene)/dicumyl peroxide Poly(para-phenylene) An epoxy-based SMP 1 carbon nanotube (CNT) A polyurethane-based SMP An epoxy SMP (epoxy E44) and Poly(propyleneglycol) bis(2aminopropyl) ether (D230) Poly(dimethylsiloxane) (Sylgard) and poly(vinyl acetate) fiber

Thermoset Thermoplastic

A N.S.

43 48

[109,110] [111,112]

Thermoplastic Thermoset Thermoset Thermoplastic Thermoset Thermoplastic Thermoset

S A S A A S A

.110 .42 43 .160 B44 N.S 48.8

[113] [25,114,115] [116] [117] [118] [119] [120,121]

Thermoset 1 Thermoplastic Thermoplastic Thermoset

A

-114 And 42 B103 .47

[122]

Polystyrene (PS) (Grafix) Water-borne epoxy 1 silica particles

A A

N.S. means not stated in the relevant references. Also, “A” and “S” indicate amorphous and semicrystalline, respectively.

[123] [124,125]

A detailed review on constitutive models for thermoresponsive shape memory polymers

(A)

25

(B)

Figure 2.4 The experimental data from Arrieta et al. [57]. (A) The storage modulus and tan (δ) of the acrylate-based shape memory polymer. (B) The stress-free-strain-recovery and fixed-strain-stress-recovery thermomechanical cycles of the acrylate-based shape memory polymer in different rates. In (B), the scale factor for blue, red, green, and black lines is 0.18, 0.19, 1.05, and 0.99 MPa, respectively [15].

permanent and temporary shapes several times. Researchers such as [130
138] investigated on the topic of two-way SMPs. Some authors have argued that the two-way property is specific to semicrystalline materials [139]. Cyclic thermomechanical testing is used to check the performance of two-way SMPs. The two-way properties, apart from the base polymer, depend on the percentage of the crosslinks; it means at some percentages, two-way properties may not be observed, while in some other percentages it could. Usually, in three fashions, the two-way property can be observed in polymers. Of the first two-way SMPs were liquid crystal elastomers (LCEs) that inherently possessed two-way property [140]. The second way is use of semicrystalline SMPs under external load [132], and the third one is use of laminated composite SMP [141]. According to Chen et al. [138], they used polymer laminated composites as an alternative to LCEs due to the high cost of manufacturing as well as the instability of SME in LCEs (in such case, a semicrystalline SMP with a typical elastomer base matrix with a Tg lower than the working temperature range of SMP is employed [134,135]). It is noteworthy that the two-way property of poly(cyclooctene) (PCO) reported by Chung et al. [132] was raising from the stretch-induced crystallization (SIC). It also had the one-way SME, that is, by removing the external load at the cooling and reheating step, the permanent shape is remembered. As already mentioned, the two-way SME can be created either with application of external load throughout the cycle (see [130,132,133] among others) or without any external loading. To this end, Chen et al. [136,138] and Westbrook et al. [135], introduced an SMP composite that had free-standing two-way shape memory behavior. In the absence of applied stress during the next cycles, the two-way SME was achieved. They examined this for some applications in actuators that required two-way SME. Table 2.2 provides some examples of existing SMPs in literature with two-way SME.

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Table 2.2 Two-way shape memory polymer samples in literature [15]. Material

References

PCO 1 dicumyl peroxide PCL LDPE (LD100BW) pellets (lPE) HDPE Poly(ethylene-co-vinyl acetate) (EVA) Poly(tetramethylene oxide) glycol-based networks EVA/PCL Bilayer polymeric laminate composed of an elastic polymer and a crystallizable PU PCO 1 PEGDMA/tBA

[132,133,135] [137] [131] [142] [139,143,144] [145] [146] [136,138] [134,135]

PCO, polyethylene, and EVA are a subset of crystalline polyolefins.

(A)

(B)

Figure 2.5 Two-way behavior of two-way shape memory polymer [15]. (A) Thermomechanical cycle of poly(cyclooctene)
dicumyl peroxide-based two-way shape memory polymer from experiments of Westbrook et al. [135]. (B) Digital photographs illustration of PCL/Fe3O4-based two-way shape memory polymer with repetitive cooling/ heating steps from experiment of Du et al. [147].

As a sample of experiments conducted in literature for two-way SMP behavior, could refer to the tests carried out by Westbrook et al. [135]. In this work, they used PCO and dicumyl peroxide (DCP) in an acrylate-based polymer elastomer. Then, they performed the cyclic free-standing tests on a strip as shown in Fig. 2.5A. In Fig. 2.5B, the digital photographs of a two-way SMP based on PCL/ Fe3O4 under a constant stress is shown. In this sample, by applying a magnetic field and raising the temperature, the SMP works in a reversible cycle under a constant load.

2.3.2.3 Multishape memory polymers Multi-SMP polymers are SMPs that can retrieve different temporary shapes in a thermo-mechanical cycle. As previously stated, conventional SMPs or dual-SMPs

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27

are in fact the result of combination of a reversible phase and a freezing mechanism. While in order to achieve, for instance a triple-SMP, it at least requires another reversible phase [148]. One way to produce multi-SMP is to use two or more Ttrans in multiphase SMPs. In fact, the use of an SMP with more than one Ttrans lets us have different temporary shapes during the programing step. In the heating step, passing through any Ttrans, its permanent shape is retrieved. This can be conducted by combining amorphous and crystalline SMPs with Ttrans of Tg and Tm, respectively, or with SMP with different Ttrans [148]. In other words, one may adjust the ratio between the two transition phases. For example, Xie et al. [149] analyzed the effect of multi-SMP on a bi-layer cross-linked polymer, which contained two separate phase transitions, or Bellin et al. [148,150] who experimentally studied the multi-SMP. It is noteworthy that in multi-SMP, the number of temporary shapes during the thermodynamic cycle is important. For example, triple-SMP implies a fixed shape and two temporary ones [148]. However, one-way SMPs, in contrary to multi-SMPs, could be considered as inherent properties of some polymers. Behl and Lendlein [7] stated that the dual-SMP is not related to the inherent properties of the material. However, Xie et al. [149] argued that the SME is not a material property, but is the ability and capacity of material, because it requires proper combinations of intrinsic properties to arrive at the desired SMP. Another way to create a multi-SMP is to control the recovery and programming requirements for an SMP with a very wide Ttrans range. These two techniques are often performed by means of copolymerization, grafting, blending, or interpenetrating polymer networks [151]. Given that Tg of amorphous SMPs is wider than that of the semicrystalline SMP, it can be said that amorphous SMPs, according to this approach, have a more ideal triple-SMP. For example, Xie et al. [152,153] employed thermoplastics Nafion perfluorosulfonic acid ionomer (PFSA), its Tg range is between 55 C and 130 C. First, they loaded SMP at a high temperature, and then conducted the cooling and unloading process. Again, they loaded the SMP, and then they cooled down and loaded the SMP for the second time. At the recovery step, by heating, two different temporary shapes were retrieved. From another viewpoint, this phenomenon can be called temperature memory effect. Because the temperature associated with the maximum recovery stress in the recovery cycle, or the maximum strain recovery in the strain recovery cycle, is equal to the temperature of the deformation [154,155]. Fig. 2.6 shows graphically two-way SMPs and triple-SMPs. Fig. 2.6A illustrates two-way or reversible SMP under repetition of cooling and heating steps. Fig. 2.6B shows the triple-SMP, which in regions P1 and P2, SMP was programmed and in regions R1 and R2, both stored strains are released. For more detailed studies on multi-SMP, one may refer to [100,148
150,152,153,155
159]. Some examples of multi-SMPs are presented in Table 2.3. In this section, as an example of experiments on a triple-SMP, we briefly discuss Xie et al. [149] test results. They performed experiments on an epoxy thermoset SMP called Diglycidyl ether bisphenol A epoxy monomer (EPON 826) and poly (propylene glycol) bis (2-aminopropyl) ether (Jeffamine D-230) and neopentyl glycol diglycidyl ether. Using a combination of epoxy constituents, they constructed

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Computational Modeling of Intelligent Soft Matter

(A)

(B)

Figure 2.6 Schematics of thermomechanical cycles for (A) two-way Shape memory polymer and (B) triple-shape memory polymer (P1 and P2 indicate the first and second programming stages, repectively, and R1 and R2 indicate the first and second recovery stages, respectively) [15].

Table 2.3 The multi-shape memory polymer samples used in the literature [15]. Material

Type

References

Aliphatic PEU PFSA

Amorphous, thermost semicrystalline, thermoplastic Semicrystalline, thermoset 1 thermoplastic Semicrystalline, thermoplastic Amorphous, thermoset Amorphous, thermoplastic Semicrystalline, thermoplastic Amorphous, thermoset

[100] [57,152,153,159
161]

An epoxy-PCL fiber Poly(ω-pentadecalactone) (PPD) 1 PCL An epoxy-based SMP PVA 1 CNT Poly(lactic acid) (PLA) and PCL 1 graphene An styrene-based SMP

[157,162] [156,163,164] [149,165,166] [167] [151] [166,168]

two types of epoxy H and L (with different Ttranss). They produced a multilayer and carried out DMTA and thermomechanical tests to report two stored modulus curves and thermomechanical cyclic diagrams as illustrated in Fig. 2.7. As shown in Fig. 2.7A, it could be concluded that the SMP has two trends of storage modulus, which leads to creation of triple-SMP response. Fig. 2.7B shows thermomechanical cycle test of the triple-SMP. As one may observe, in two various areas of the tripleSMP, the SMP is loaded in two different prestrains and consequently, will retrieve its temporary shapes in two steps. These results later were employed in modeling the triple-SMP [159,160,165]. In Fig. 2.7C, thermomechanical cycle of these triple SMPs is illustrated. In this sample, first, the sample deforms to its first temporary shape at 90 C and then cools down to 56 C. Then, the sample at 56 C deforms into its second temporary shape and then cools down to 22 C. Finally, in two corresponding steps, the sample retrieves to its permanent shape.

A detailed review on constitutive models for thermoresponsive shape memory polymers

(A)

29

(B) 1. Fixing A

B

Deform at 90 °C ; Cool to 56 °C

2. Fixing B

C

Deform at 56 °C ; Cool to 22 °C

3. Recovery C

B

Heat to 56 °C

4. Recovery B

A

Heat to 90 °C

(C)

Figure 2.7 Triple behavior of multilayer epoxy L- and H-based shape memory polymer. (A) Indicates the storage modulus, E of the shape memory polymer from dynamic mechanical thermal analysis test. (B) The thermomechanical triple cycle of the triple-shape memory polymer [149]. (C) Loading and recovery steps of the triple multilayer epoxy L- and H-based shape memory polymer [149]. The scale factor for temperature, stress, and strain is 90.4076 C, 3.8833 MPa, and 5.8119%, respectively [15].

2.4

Modeling thermoresponsive shape memory polymers

As stated before, due to their unique properties, SMPs have potential for use in many industrial and research fields. To predict and describe the behavior of SMPs in these applications, introduction of mathematically developed mechanical models are necessary. These models should account for different 3D loading conditions and large deformations and have a successful performance in strain recovery and stress recovery processes. To employ these constitutive models in engineering applications they should be implemented in numerical tools. Given that the purpose of this chapter is to review and present existing mechanical models for temperature-responsive SMPs, so

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other types of SMPs are beyond the scope of this chapter (e.g., light-responsive or pH-sensitive SMPs). For instance, for details on light-sensitive SMPs one may refer to Refs. [169
171]. Reference [117] developed a constitutive model for an SMP responsive to dissolution. Moreover, recently, a chemo-thermo-mechanically coupled 3D viscoelastic model for water/temperature trigger polyimine-based covalent network polymers (MCNPs) was proposed by Mao et al. [172] under infinitesimal strains. For SMP modeling, governing phenomena in polymer behavior should be accounted for at temperatures lower and higher than Tg, such as crystallization or melting [26]. In general, for modeling SMPs according to available models in literature, there are two main points of view of thermoviscoelastic and phase transition, while there are a few other works based on approaches such as meso-scale, micromechanics, molecular dynamics analysis and quantum mechanics analysis. Given the frequency of models in the literature, thermoviscoelastic and phase transition models are the most common techniques employed in development of constitutive models for SMPs. Other methods have not been much considered by researchers because of their complexity and often high computational costs. Each of these viewpoints will be discussed in detail in the following.

2.4.1 Modeling of conventional thermally activated shape memory polymers 2.4.1.1 Constitutive models of shape memory polymer under thermoviscoelastic approach The physically based view, or thermoviscoelastic or rheological [99], is referred to SMP models that take into account the inherent properties of material such as cross-links, molecular chain motions, movement of contact surfaces and intermolecular chains interactions, and relaxation time. In this view, the material coefficients commonly have a physical meaning. These models have a high ability to accurately describe the SMP behavior. Since microstructural concepts are used in development of these models, they are apparently more appropriate to be utilized in modeling the multi-SMPs or two-way SMPs. One of the advantages of this approach is that it considers underlying polymer physics in amorphous SMPs so that other phenomena, for example, aging or solution-driven SMEs can also be taken into account in development of constitutive models. In this approach, normally two important parameters of mobility and relaxation are considered. Before having a closer look into the proposed models based on this approach, we first present an introduction to the modeling in this approach by describing the standard linear solid (SLS) rheological model as shown in Fig. 2.8.

Figure 2.8 A 3-parameter rheological model [15].

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When SMP is at a temperature higher than Ttrans, the viscosity of the damper drops sharply, and as a result, with loading the spring in series with the dashpot in the Maxwell branch has a very small displacement. Therefore most share of the displacement is given to the spring in the equilibrium branch, and so most part of the energy is stored in the equilibrium branch (entropic energy). Then, when the material is cooled down under a fixed strain, the viscosity of the damper element rises sharply. Now, when it is unloaded, since the spring constant in the Maxwell nonequilibrium branch is generally higher than that of the spring in the equilibrium branch, consequently, a slight deformation is developed in the nonequilibrium branch. Thus the temporary deformation is being fixed in the nonequilibrium branch. In the final step, by heating the material, the viscosity diminishes in the dashpot element. Therefore the force stored in the spring in series with the dashpot, returns back the dashpot, and SMP eventually returns to its original shape [159]. The first constitutive model for SMPs was introduced by Tobushi et al. [38]. Employing the viscoelastic macroscopic approach, they introduced a four-element linear model for polyurethane of the thermoset polyester polyol series. The rheological schematics of Tobushi et al.’s model [38] and its corresponding creep and creep recovery responses are shown in Fig. 2.9. To account for the irreversible strain at T , Ttrans , they introduced a slip element (blue element in Fig. 2.9A). Employing this concept, other researchers developed SMP constitutive models [33,36,37,40,99]. In fact, this element should account for the internal friction, direction of the molecular chains and the decoupling of crosslinks. When the force overcomes the internal friction in this element, some part of the irreversible strain εs remains. They proposed a relationship for the irreversible strain in form of εs 5 C ðεc 2 εl Þ, in which C is a temperature-dependent parameter, εc denotes the creep strain and εl stands for a reference strain defined at low temperature Tlow. At high temperature, εs is small and εl is high, and vice versa at low temperature. A one-dimensional form of this constitutive model is as follows: ε_ 5

σ_ σ ε 2 εs 1 2 1 αT_ E μ λ

(A)

(2.1)

(B)

Figure 2.9 The schematic drawing of Tobushi et al.’s model [38]. (A) The proposed 4parameter rheological model. (B) The creep and creep recovery of the proposed model [15].

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where E; μ; λ , and α are material model parameters. Schematics of the creep and irreversible strains are depicted in Fig. 2.9B. To calibrate the model, by performing several DMTA tests at different temperatures, the elastic modulus was found as a temperature-dependent function. It was observed that all model coefficients, except for the thermal expansion coefficient were strongly dependent to the temperature. Carrying out four DMTA tests, thermal expansion coefficient, creep and relaxation coefficients were identified. Performing two thermomechanical stress and strain recovery tests, they verified the model predictions. In this model, assuming rheological linear viscoelastic elements limits its range of validity to small strains. In this model, there was no explicit relationship for the relaxation time and viscosity in terms of temperature, which was subsequently modified by other researchers [65,67,68,71]. Tobushi et al.’s model [38] was the starting point for thermomechanical modeling of SMPs. Then, Lin et al. [42], based on Tobushi et al.’s model [38], proposed a linear model for the behavior of an ester-type polyurethane-based SMP (Fig. 2.10). In their model, they employed two parallel branches of Maxwell (a branch to represent a reversible phase and a branch for the fixed phase). The cross-linked ester-type PU used by Lin et al. [42] was an amorphous thermoset polymer. In this model, they did not consider the stress recovery, which could be a limitation for this model. According to available literature, this model cannot predict SMP behavior well. Then, Bhattacharyya and Tobushi [173], in another work, developed analytical solution of the isothermal process for the four-element model presented by Tobushi et al. [38]. They examined parameters, such as constant stress, constant strain, constant stress rate, constant strain rate, and the periodic strain load for isothermal process on PU, which, according to literature, is not much a constitutive model for SMP. As mentioned, Tobushi et al.’s model [38] was a linear model operating within the range of small strains (in strains less than 3% SMP usually has a linear behavior [22]). Inspired by its previous model, Tobushi et al. [22] proposed a nonlinear constitutive model for polyurethane polyol series-based SMP with the same viscoelastic macroscopic approach in the range of large strains. They were able to extend the model to large strains by adding a series of powers to their previous model. The governing equation for the model is as follows:
σ2σ m21 σ σ_ 1 y ε_ 5 1 m 1 1 E μ b k



σ 21 σc

n 2

ε 2 εs 1 αT_ λ

(2.2)

so that σc and σy are time-independent proportional limits of the stress in the viscous element. In this model, six tests conducted for calibrating the linear model

Figure 2.10 The linear viscoelastic model proposed by Lin et al. for shape memory polymers [42].

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33

were used along with a uniaxial tension test. Parameters m, n, α and b, in comparison to other parameters, are less dependent on temperature and therefore they are assumed to be constant and their values were computed as an average in the desired temperature range. Inspired by the Tobushi et al.’s model [38], Morshedian et al. [174], and Abrahamson et al. [175], respectively, proposed a three-element linear model (two dashpots and a spring) for PE under strains up to 200% (in uniaxial loading) and a four-element model (two springs, one dashpot and a sliding element). Bonner et al. [176] based on the Tobushi et al.’s model [38] presented a Kelvin
Voigt linear model and calibrated the model coefficients using experiments for an amorphous lactide-based copolymer with 35% weight of calcium carbonate. It is noted that they examined only the stress relaxation behavior for SMPs. Employing a modified Maxwell model with two branches and the SLS model in an isothermal process, Heuchel et al. [100] examined the relaxation behavior of polyamide polyurethane (PEU). Then, Wong et al. [89] presented a one-dimensional model for poly(D,L-lactide-co-glycolide) as a biodegradable amorphous polymer similar to that of Morshedian et al. [174]. In this model, they predicted SMP behavior in the strain recovery step by suggesting a one-dimensional flow rule and assuming the ArrudaBoyce eight-chain models [177] for the network element as well as for the elastic element in the equilibrium branch. Li et al. [34] using the same approach as Tobushi et al. [38], developed a threeelement rheological model (a Maxwell branch in series with a linear spring) for 3D small strains. Using the experimental results of Tobushi et al. [38], they calibrated their model. Writing a UMAT subroutine in ABQUS, they predicted Polyurethane behavior in two thermo-mechanical stress and strain recovery processes. Shi et al. [37], Balogun and Mo [36] and Pan et al. [30] developed 3D-modified versions of the Tobushi et al.’s model [38], and then, by writing UMAT, they were able to implement their models in ABAQUS. Balogun and Mo, in another work [41], considering the binding factor, extended the linear SLS model into 3D loadings. Providing a UMAT subroutine they calibrated the model for experiments reported by Tobushi et al. [38]. Further, taking the basis from the theory of solid mechanics and viscoelasticity in the range of small strains, Zhou et al. [83], assumed the strain as sum of three thermal, viscoelastic, and elastoplastic strains, and finally derived the 3D form of the constitutive equations. Performing tests on a thermoset styrenebased SMP, they calibrated the model coefficients. In a different work, using the Helmholtz potential, Ghosh and Srinivasa [33] predicted the shape recovery response of Polyurethane of the polyester polyvel. The same authors in another work [40], applying the same Helmholtz potential approach, proposed a 1D four-element (including a slip-element parallel to a Kelvin element and a spring in series with this set) thermoviscoelastic model in the range of large strains based on the Tobushi et al.’s model [38]. In this model, they used Tobushi et al.’s results on a PU-based SMP [38] and Liu et al. on an epoxy resin [24] to find model coefficients. It is noted that this model was not examined in the stress recovery. This model is provided for two-network polymers in the form of state space, which considers the yield stress of SMP, and shows that most of the

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SMP features are tied to its yield stress. In this model they made use of a multinetwork theory proposed by Tobolsky and Andrews [178] and Rajagopal and Wineman [179]. Motivated by Tobushi et al.’s model [38], they employed a friction dashpot element with yield stress k. Then Ghosh and Srinivasa extended their model to the range of 3D loadings [27]. Following the same approach [99], they further extended their model to the range of finite strains. They applied the QR or upper triangular decomposition technique [180] as well as the Rajagopal and Srinivasa method [181] to successfully predict both the shape and stress recoveries. The first nonlinear 3D model of SMP in finite deformation range was proposed by Diani et al. [182] for a cross-linked SMP networks (an epoxy thermoset resin). In this model, they used the results of experiments by Liu et al. [24] to calibrate the model coefficients. According to Fig. 2.11, in this model, they used an entropic equilibrium branch based on the neo-Hookean model and a nonequilibrium internal energy branch. They reported the elastic modulus and the thermal expansion coefficient at temperatures higher and lower than Tg. However, not much detail on the theory and model calibration was given. Nguyen et al. [67] provided a quite comprehensive 3D model in compared to previous models. Employing the experimental results of Qi et al. [66] for an acrylate-based SMP (tBA/PEGDMA), they calibrated their finite strain thermoviscoelastic model. To model the mechanical part, they applied the theory of nonlinear viscoelasticity of Reese and Govindjee [183]. They introduced the concept of mobility of molecular chains due to the glassy phase transition by introducing relaxation time and stress relaxation concepts. For the first time, they linked the effect of the structural relaxation and the viscoplastic flow at a temperature below Tg. Structural relaxation is a time-dependent process in which molecular chains are reordered in a new equilibrium configuration in the presence of temperature variation. In their model for simplicity, they ignored the effect of heat conduction and pressure on the structural relaxation as well as the inelastic behavior. One of the main features of this model is linking of the Adam-Gibbs model to a finitedeformation thermo-elastic model for amorphous SMPs. To account for the structural relaxation, they used Tool
Narayanaswamy
Moynihan model [184,185]. They predicted the nonequilibrium behavior by introducing an internal variable called fictive temperature. For simplicity, only one relaxation process is assumed to describe the structural relaxation and stress relaxation mechanisms. However, given that amorphous polymers have a spectrum of structural relaxation, Castro et al. [65] applied the Kovacs
Aklonis
Hutchinson
Ramos (KAHR) model, which is actually a parameter for the deviation of enthalpy from the equilibrium state.

Figure 2.11 The rheological scheme of Diani et al.’s model [182].

A detailed review on constitutive models for thermoresponsive shape memory polymers

35

Considering 33 parameters for the structural relaxation, they accounted for a spectrum of relaxation time. In their model, following Nguyen et al. [67], they took into account both the structural relaxation (for describing the glass transition kinematics) and the temperature-dependent viscoelastic (to describe the stress-strain variation). In this model, they only examined the one-dimensional stress recovery in the range of small deformations. To calculate the stress relaxation they applied Kohlrausch, Williams and Watts (KWW) method [186]. Moreover, for the first time, the effect of temperature rate was investigated both experimentally and theoretically for the same SMP used by Qi et al. (tBA/PEGDMA) [66]. Subsequently, Ge et al. [71] employed Castro et al. [65] approach (to use the modified SLS model and KWW method) to study the shape recovery behavior for an acrylate-based SMP (tBA/PEGDMA). At the same time, Nguyen et al. [70], to improve their previous model, in addition to considering several Maxwell branches parallel to the equilibrium elastic branch (multiple branches for the stress relaxation), scrutinized a wide range for the structural relaxation using KAHR theory. For different volume fractions of the PEGDMA based on the tBA, the behavior of this SMP was assessed in the shape recovery. It should be noted that in this model, Haupt et al. [187] method was utilized to obtain relaxation spectra from standard time temperature-superposition. Later, using the model in [70], Choi et al. [188] examined the effect of physical aging on an acrylate-based SMP (tBA/PEGDMA). Also, Chen and Nguyen [189] used the Nguyen et al.’s model [67,70] to determine the effect of parameters such as heating and cooling rate, strain rate, annealing time and temperature on the thermomechanical behavior in shape and stress recoveries for an acrylate-based SMP (tBA/PEGDMA). Further studies of structural relaxation can be found in Nguyen et al. [67], Castro et al. [65], and Xiao et al. [23]. Since in previous models the programming stage was being carried out at a temperature higher than Ttrans, to improve this, Li and Xu [10,11] proposed a 3D model in the range of large strains, similar to that of Nguyen et al. [67,70]. They tested and calibrated the model for a polystyrene-based thermoset SMP resin system (Veriflex) and a syntactic foam containing Veriflex and glass hollow microspheres in recovery stages. They described cold-compression programming using the Narayanaswamy-Moynihan model [184,185], which provided structural relaxation at a temperature lower than Ttrans, as well as a branch for describing the viscous flow. Motivated from Li and Xu’s approach [10,11], for modeling particle reinforced SMPs nanocomposite, Yin et al. [124] proposed a 3D thermo-viscohyperelastic SLS model to predict shape recovery and force recovery of particle reinforced water-borne epoxy SMPCs of different silica weight fractions. Along with the work of Li and Xu [10,11], Xiao et al. [23] employing a multiprocess model introduced by Nguyen et al. [70], as well as using a thermodynamics framework for thermo-mechanical coupling theory, SMP behavior was studied in cold and hot programming for both the shape and force recoveries. They applied a coupled formulation to account for a multiple stress-activated flow process to model the yielding, so that the flow process is assumed to depend only on an activation stress [190]. They finally calibrated the model for experiments on a tBA/ DEGDMA-based SMP with different weight percentages of the cross-linker

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Computational Modeling of Intelligent Soft Matter

solution in two stress and strain recovery processes. Then, Zeng et al. [62], similar to the two models presented by Nguyen et al. [67,70], introduced a 3D thermomechanical uncoupled structural and stress relaxation in the range of finite strains in both strain recovery and strain recovery steps for a tBA-based SMP for experimental results reported by Xiao et al. [23]. In this model, they used a nonlinear SLS model, which employed the modified neo-Hookean model for its equilibrium part and the Arruda and Boyce model for its nonequilibrium branch. For its thermal part (structural relaxation), they used the internal variable δ from the KAHAR model, and for the stress relaxation, modifying the Eying model they introduced a new model for the stress relaxation time. In line with the work of Li and Xu [10,11], Gu et al. [87] to predict the behavior of syntactic foam from Li and Nettles’s experiments [78], developed a rheological model with some elastic viscoplastic branches and a hyperelastic branch. The proposed 3D model in large deformations was able to provide successful predictions for both the stress and strain recovery steps. It is worth mentioning that this syntactic foam was a combination of glass spherical microparticles in a styrene-based thermoset SMP resin matrix. In the same direction in the modeling of SMP composites, Mao et al. [122] presented a 3D model for anisotropic SMP composites which was capable of prediction of cold programming effect in finite strains. Their SMP composite consists of a Sylgard elastomer as base matrix and fibers made of amorphous thermoplastic PVA. In their rheological model, an elastic spring (neoHookean model) was assumed for the matrix, a Maxwell element for the isotropic fiber and another Maxwell model for the anisotropic part of the fiber, all in parallel. They assumed there is a perfect bonding between the fibers and the matrix. Employing the same approach used in [67], Chen et al. [21], and Gu et al. [107] developed relatively simple 3D models based on the SLS model in the range of large deformations for an epoxy-SMP and an ether-based polyurethane SMP, respectively (experiments were borrowed from Yang et al. [108]) in both strain and stress recovery paths. Chen et al. [21] used the Mooney-Rivlin hyperelastic model for its equilibrium part, a nonlinear Hencky function for its nonequilibrium part, and the Erying method for their viscous flow. A UMAT subroutine was also developed in ABAQUS to implement this model in real world boundary value problems. To model SMP nanocomposites, including CNT nanoparticles, Abishera et al. [118], using the same SLS model, considered structural relaxation and stress relaxation (similar to [21,67,107]). The developed nanocomposite model was examined both numerically and experimentally in strain recovery step. Srivastava et al. [17,74] provided a thermomechanical strain recovery model using a nonlinear viscoplastic model for thermoplastic SMPs. Their model contains 45 parameters related to the effect of heat generation and heat conduction on the plastic deformation and the strain rate associated with Ttrans. Westbrook et al. [68] presented a 3D finite strain thermo-viscohyperelastic model similar to models such as Nguyen et al. [70], except that they considered different branches for both the rubbery and glassy phases for nonequilibrium part. They used tests on an acrylate-based SMP (tBA/PEGDMA) as an amorphous polymer to calibrate the model coefficients. They assumed a rheological model similar

A detailed review on constitutive models for thermoresponsive shape memory polymers

37

to the generalized Maxwell
Wiechert model, which includes an equilibrium branch and two sets of nonequilibrium branches. One branch to represent the SMP behavior at temperatures lower than Tg, and multiple branches for determining the Rouse modes in the rubbery phase at temperatures higher than Tg. They employed the hyperelastic Arruda
Boyce model [177] for the equilibrium branch, where its material parameters were calibrated through conducting an isothermal tensile test at high temperature. For the nonequilibrium branches, they used viscous flow rules. The KAHAR model with 33 parameters was assumed to account for the thermal part. To take into account the effect of temperature on nonequilibrium branches, they took advantage of the thermo-rheological simplicity principle introduced by Rubinstein and Colby [191], which states that the relaxation time in each individual branch at any temperature is equal to the relaxation time at the reference temperature multiplied by a time-temperature shift factor. Shift-factor depends on Ttrans of the material and its mechanical properties. Commonly, for a temperature higher than Ttrans, the Williams
Landel
Ferry (WLF) and for a temperature lower than Ttrans, Arrhenius equation is applied to identify the shift-factor. Selecting the reference temperature equal to Tg, they put to use William et al.’s results [192]. In this model, they considered two nonequilibrium rubbery branches, where this number of branches was identified by the number of the Kuhn segments in the macromolecular chains. To find the coefficients of the nonequilibrium branch, tensile and compression test results at different rates were used. They also implemented the presented model in ABAQUS through developing a UMAT subroutine. Then, Yu et al. [104] presented a model similar to Westbrook et al. [68], but with the difference that they used multiple branches for the nonequilibrium for the glassy phase, as well. They calibrated the model for the test results on a Veriflex-E epoxy SMP (presented by McClung et al. [102]) and successfully predicted the results in recovery regimes. Then, Cui et al. [193] studied thermomechanical behavior of crystallizable shape memory polymers (CSMPs). In their model, they utilized an SLS model for the equilibrium part and several parallel branches, including a damper for the nonequilibrium part. Based on the model presented by Westbrook et al. [68], first, Diani et al. [92] employed the generalized Maxwell thermoviscoelastic model and used the DMTA tests on an epoxy-based SMP to identify model coefficients. They simulated the torsion of a rectangular bar in a shape recovery stage. Yu et al. [85,106], Wang et al. [101,194], and Mailen et al. [123] employed the same linear model of the generalized Maxwell thermoviscoelastic model, which was actually a simplified version of the nonlinear model of Westbrook et al. [68] in different problems. It is worth noting that Yu et al. [106] examined the cyclic behavior of an amorphous thermoset epoxy putting to use the same generalized Maxwell-Wiechert model. Arrieta et al. [57] extended their model to the range of finite strains, and provided a thermoviscoelastic 3D model using the same generalized Maxwell-Wiechert model. They obtained model coefficients for the acrylate network and presented results in two stress and strain recovery processes. One of the main advantages of Arrieta et al.’s model [57] is its simple use in FEM software. They conducted a high-temperature tensile or compression test and assumed that the SMP behavior was completely

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Computational Modeling of Intelligent Soft Matter

rubbery to find hyperelastic coefficients. Viscoelastic coefficients are identified using DMTA tests. WLF parameters should also be computed through performing some additional tests. Based on the proposed model by Arrieta et al. [57], Yarali et al. [195] examined homogenization of SMP nanocomposite. They used Arrieta et al.’s model [57] and numerically investigated large deformation of coiled carbon nanotube reinforced SMPs. Here, the Arrieta et al.’s model [57], developed based on thermoviscoelastic point of view, is chosen to be discussed in more detail. It is well established that the viscoelastic material includes two features: straindependent response and time-dependent response. In this research, as mentioned previously, since the model tolerates large deformations, a hyperelastic model is used for the strain dependent part (elastic part). Also, the neo-Hookean strain energy function is employed for the hyperplastic model. In general, the linear viscoelastic theory could be expressed as follows [196]: σ ðε; tÞ 

ðt

dσ 0 ðξÞ gðt 2 ξÞdξ 5 gðε; tÞσ 0 dξ 0

(2.3)

in which, σðε; tÞ is the total stress and σ 0 is the strain-dependent stress, g is a dimensionless function and commonly represents in Prony series as below. gðε; tÞ 5 gN ðεÞ 1

XN

g ðεÞexp i51 i



2t τi

 (2.4)

where gN and gi are dimensionless material constants corresponding to the equilibrium and instantaneous (viscous) part, respectively. Furthermore, the values of gN P and gi are between 0 and 1 and also the constraint gN 1 Ni51 gi 5 1 should be satisfied. τ i is the corresponding relaxation time or retardation time in ith branch (i 5 1, 2, 3, . . ., N). For the hyperelastic part of the model, as mentioned before, the neo-Hookean model has been applied. For an isotropic, incompressible and homogenous elastomer, the Cauchy stress tensor could be written as follows [197]: 

σ

Hyper

 @ψ @ψ @ψ 2 ðεÞ 5 gN ðεÞσ0 ðεÞ 5 2 pI 1 2 1 I1 B B22 @I1 @I2 @I2

(2.5)

where B 5 FFT is the left Cauchy
Green deformation tensor, p is a hydrostatic pressure. ψ is the strain energy function and Ii ði 5 1; 2; 3Þ are the invariants of B where (Holzapfel, 2002): I 1 5 trðBÞ   I 2 5 1=2 ðtrðBÞÞ2 2 tr B2 I 3 5 detðBÞ

(2.6)

In neo-Hookean model, ψ is defined as ψ 5 C10 ðI1 2 3Þ, where C10 is material parameter.

A detailed review on constitutive models for thermoresponsive shape memory polymers

39

To considering the temperature effects, the time-temperature superposition (TTSP) is used. Based on this theory, the temperature and the timescale are linked. For applying this phenomenon in the above relation, it just replaces t by t’ where: 0

t 5

ðt

dξ 0 aðT Þ

(2.7)

in which aðT Þ is called shift factor. For determination of this parameter, the glass transition of the material should be specified. In the above and near Tg, the horizontal shift factor could be expressed by WLF equation as follows: logðaðT ÞÞ 5

2 C1 ðT 2 Tr Þ C2 1 ðT 2 Tr Þ

(2.8)

which C1 and C2 are material constants and Tr denotes the reference temperature. In contrast, for above Tg, the horizontal shift factor could be found by the Arrhenius-type equation as follows:   1 1 1 2 lnðaðT ÞÞ 5 2 C T Tr

(2.9)

in which C is constant. It is noted that the results from Arrieta et al.’s model [57] were presented in Fig. 2.4. Afterwards, Fan et al. [120], employing the same model, only performed several relaxation tests at different temperatures to calibrate the model coefficients simultaneously through an optimization procedure. They obtained the model parameters for an epoxy-based SMP and presented the results in two strain and stress recoveries. Similar to Arrieta et al.’s model [57], Xiao et al. assumed only a few parallel nonequilibrium branches [117] to capture the strain recovery behavior of thermoplastic poly (para-phenylene) within the range of large strains. They assumed a polynomial function (temperature dependent) to calculate the shift factor. To improve the model of Arrieta et al. [57], Gu et al. [88], by writing the storage and loss modulus in terms of temperature, optimized them according to DMTA test results at a constant frequency. This way they were able to identify viscoelastic coefficients and TTSP parameters, simultaneously. They calibrated the model for a styrene-based thermoset resin and syntactic foam-based SMP. Moreover, by performing a high temperature tensile test, they obtained Arruda-Boyce coefficients. This model can reproduce the stress and strain recoveries. In line with new rheological models for SMPs, Li et al. [29], developed a new 3D finite strain rheological model that was a combination of springs, and linear and nonlinear dashpots. This model was calibrated for the experimental results of Tobushi et al. [22,38] and McClung et al. [102]. They verified the model in uniaxial and biaxial thermo-mechanical strain recovery scenarios. Saleeb et al. [31] presented a new 3D large strain model based on the viscoelastic approach for the strain

40

Computational Modeling of Intelligent Soft Matter

recovery of a thermoset polyurethane. The proposed model was to some extent similar to their model on SMAs which had already been presented in Ref. [198]. One of the advantages of this model for SMP is the low number of material parameters necessary for calibration. They also investigated the behavior of SMP under cyclic loading. On the other hand, Pulla et al. [109] applying a simple approach, assumed SMP as a hyperelastic material and obtained model coefficients at different temperatures. They were able to capture the shape recovery response of an epoxybased SMP. In a couple of proposed models for SMPs, from the same thermoviscoelastic point of view, instead of classic viscoelastic models, fractional viscoelastic constitutive models were put to use. In these models, instead of a common dashpot, a fractional dashpot is used. They believed that this method gives more accurate results, in addition to lower number of material parameters [86,199
201]. So far, about six models have been developed with this method [35,64,75,86,160,201]. In the same way, Fang et al. [64] predicted an acrylate-based SMP (tBA/PEGDMA) response in the strain recovery considering the viscoelastic fractional approach as well as a three-element linear fractional-order Zener model. It is noteworthy that in this work, only the relaxation behavior is examined, and the shape and force recovery responses were not accounted for. Then, Li et al. [201], taking advantage of the same approach and a linear SLS model, studied the shape recovery behavior of SMP. To consider the recovery response, Pan and Liu [86] employing the generalized Maxwell
Wiechert model with two nonequilibrium branches, modeled the SME in styrene-based thermoset polymer (the model was 1D and limited to small strains). Zeng et al. [75], applying the same approach used generalized Maxwell
Wiechert model with four branches to more accurately describe the behavior of a Veriflex-epoxy as a thermoset SMP (experimental results of Yu et al. [106]) in shape recovery. Zeng et al. [35], following the same idea, and turning to account the generalized fractional Maxwell model (a backbone network in series with a transient network composed of n parallel Maxwell branches), used multiple structural relaxation in 1D to successfully reproduce the shape and force recovery in SMP. It is noteworthy that they used the KAHAR model with 33 parameters for the thermal part. Finally, the model was calibrated for experimental results reported by Tobushi et al. [38] and Liu et al. [24]. It is noted that applying the same technique, Fang et al. [160] have examined the multi-SMP, which will be discussed in section 5.3. In Table 2.4, the rheological schematics of the discussed conventional SMP models from the viscoelastic point of view are illustrated (It is noted that due to the simplicity, mechanical parts of the models are presented merely).

2.4.1.2 Constitutive models of shape memory polymer based on phase transition approach In this approach, SMP is assumed as a combination of two active (rubbery) and frozen (glassy) phases [207], which describes how the phase transfer occurs upon temperature changes commonly by defining a frozen phase volume fraction. In this view, contrary to the viscoelastic view, based on the observed phenomena, the SMP

Table 2.4 The proposed rheological models based on viscoelastic modeling approach for conventional shape memory polymers [15]. Rheological schematic

References

Rheological schematic

References

[29]

[119]

[42,117]

[41]

[17]

[35,57,58,70,75,85,86,88,92,101, 105,106,120,122,160,194,202]

(Continued)

Table 2.4 (Continued) Rheological schematic

References

Rheological schematic

References

[34,89,110]

[68,159]

[40]

[87]

[174]

[10,11,21,30,59,61,62,64,65,67, 71,107,118,124,182,203,204]

[175]

[22,36
38,173,205,206]

[35]

44

Computational Modeling of Intelligent Soft Matter

behavior is simulated employing proper terms and models. Usually, this view is more efficient for semicrystalline SMPs, which include distinct phases [25,26,116,133,208,209]. In such case, they are often developed based on physical concepts [59]. It is noteworthy that this view is also applicable to amorphous SMPs. But given that these amorphous materials normally do not have a real phase transition, if this approach is used for amorphous SMPs, they are often referred to as a phenomenological point of view [56,66]. Some authors refer to the concept of phase transition as a macro perspective. However, in this chapter, we do not attempt to classify SMP material models from a micro or macro modeling point of view, and simply divide them into the phase transition and viscoelastic points of view. In general, in this approach, while the SME is characterized by parameters such as stored strain and frozen phase volume fraction, these parameters give a nonphysical description of the phase transition phenomenon [11,118]. In other words, this view does not take into account the microscopic thermoviscoelastic phenomena occurring during the phase transition [210]. In contrast with the viscoelastic models, which considers the fundamental mechanisms responsible for the SME based on the temperature and time dependence of the molecular mobility, the phase transition view models the SMP behavior macroscopically [25]. According to Boatti et al. [25], the phase transition idea has more adaptability in modeling the behavior of SMP and it is usually easier to obtain its model coefficients, and usually because of lower computational costs, engineering may be more likely to use this perspective. After giving this brief introduction to the phase transition approach, in the following, we review the phasetransition-based constitutive models for SMPs available in the literature. Liu et al. presented a phase-transition-based phenomenological constitutive model for amorphous SMPs for the first time [47] and then [24]. In this model, which is one of the most basic models presented in this field, a concept called the stored strain was introduced. This model employs two internal variables of frozen phase volume fraction and the stored strain to describe the transition phenomenon in the microstructure. In this model, there are two active and frozen phases and the transition between these two phases acts as a factor for development of a stored strain and the recovery mechanism (Fig. 2.12A). It was assumed that the stress in two frozen and glassy phases is equal. This model is the basis for many other models presented thereafter. Among the features of Liu et al.’s model [24], one may point out the following: 1. This model does not account for time dependent effects. 2. Employing the Reuss approximation (i.e., uniform stress in both phases) as the main approximation for homogenization of the phase composition. 3. Considers the SME as a particular elastic problem. 4. SMP material as an epoxy resin, is assumed to be composition of two phases (rubbery and glassy phases). 5. The model is proposed for one-dimensional small strain deformations. 6. An evolution law is presented only for the cooling process, and there is no separate equation for the heating stage.

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Figure 2.12 Liu et al.’s model [24]. (A) Schematic drawing of micromechanical structure of the SMP-based RVE (Representative Volume Element). (B) Recovery strains from the prediction of the model and experimental data. 7. An empirical relationship is proposed for the volume fraction of each phase using the performed experiments. 8. In this model, the volume fraction of the frozen and active phases is defined as follows.

ϕa 5

Va Vf ; ϕ 5 ; ϕa 1 ϕf 5 1 V f V

(2.10)

where ϕa and ϕf , respectively, are the volume fraction of the active and frozen phases. V, Va, and Vf denote the total volume, active phase volume and frozen phase volume in a representative volume element, respectively. In this model, the strain is assumed to consist of three parts: ε 5 εs 1 εm 1 εT

(2.11)

in which, ε represents the total strain. εs is the stored strain, εm denotes the elastic mechanical strain, and εT stands for the thermal strain. The stored strain in the cooling process is assumed to be computed from the following equation. εs 5

ð ϕf 0

εef dϕ

(2.12)

where εef denotes an entropic frozen strain. The mechanical elastic strain is found from the following equation: h
 i εm 5 ϕf Si 1 1 2 ϕf Se :σ

(2.13)

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Computational Modeling of Intelligent Soft Matter

in which σ is the stress tensor and: indicates the double contraction of tensors. As mentioned before, there is a basic assumption that the stress is equal in two frozen and active phases. Fourth-order compliance tensor Si is related to the internal deformation energy and the compliance tensor Se stands for the entropic deformation. The thermal strain is identified from the following equation: εT 5

ð T h i 
 ϕf αf ðθÞ 1 1 2 ϕf αa ðθÞ dθ I

(2.14)

T0

where αa and αf are the thermal expansion coefficients of the active and frozen phases, respectively. Identity tensor is denoted by I. According to this model, the strain
strain relation is recast as follows: h
 i21 σ 5 ϕf Si 1 12ϕf Se :ðε 2 εs 2 εT Þ

(2.15)

The strain recovery prediction of the model for the SMP during heating step along with the experimental data are shown in Fig. 2.12B. Since Liu et al. [24] had presented their model in a one-dimensional fashion and did not propose constitutive relations for the heating stage, Chen and Lagoudas [19], extended the Liu et al.’s model [24], and making use of the experimental observations of Liu et al.’s model [24], introduced a 3D finite strain model. This model was based on the nonlinear thermo-elastic theory, which ignores any hysteresis. neo-Hookean model was employed to account for large deformations. Similar to the model of Liu et al. [24], homogenization of different phases was carried out based on the Reuss assumption. Finally, using the same experimental results of an epoxy resin reported by Liu et al. [24], they calibrated the model coefficients and validated it in two stress and strain recovery paths which gave more successful results than those of Liu et al.’s model [24]. Chen and Lagoudas in the second paper, with the assumption of small deformations, a reduced linearized version of the model was derived [54]. Along these works, Volk and Lagoudas [81], performed two experimental sets of tests on a Veriflex-based SMP. In another work [80], they calibrated the model proposed by Chen and Lagoudas [19,54]. At the same time, Qi et al. [66] applying the same phase transition approach, introduced a new function for the volume fraction of the frozen phase and used a Voigt-type assumption (equal strain in both phases for homogenization). With the aid of this new finite strain constitutive model, they were able to regenerate both the strain and stress recovery scenarios for an acrylatebased SMP (tBA/PEGDMA). They synthesized test samples of the SMP in their experiments according to Yakacki et al. [211]. Dividing the SMP into three phases during the temperature change, they developed the constitutive equations for each of the three phases of the rubbery, the initial glassy and the nucleated glassy phase. They treated the rubbery phase as a hyperelastic eight-chain Arruda-Boyce material. The initial glassy phase was modeled through a viscoelastic model, which consisted

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47

of two nonequilibrium (viscoelastic) and equilibrium (hyperelastic eight-chain Arruda-Boyce model) parallel branches (a three-element SLS model). In the glassy phase, they took advantage of the same equations except that they used a thermal deformation gradient instead of the total deformation gradient. They introduced the frozen phase to avoid the introduction of equivalent stored strain in 3D. Developing a UMAT subroutine, they implemented this model in ABAQUS. In the same line, Wang et al. [50] argued that the Liu et al.’s model [24] has some limitations, such as it does not account for the rate dependent effects, and it is not a much realistic assumption to derive equivalent Young’s modulus of SMP from a mixture rule. They presented a new one-dimensional model to address these issues. In their model, using the Ozawa [212] modified version of the Avrami equation [213], they proposed a new rate-dependent function for the frozen phase volume fraction. Pan et al. [60] based on the model provided by Liu et al. [24], and considering a rate dependent Ttrans, developed a new constitutive model and calibrated the model for the polystyrene and an acrylate-based SMP (tBA/PEGDMA), for test results of Arrieta et al. [57]. In line with work of Wang et al. [50], Guo et al. [28] introduced another function for the frozen phase volume fraction. Then, employing a fourelement model for the active phase and a three-element model for its frozen phase, they were able to model and calibrate the experimental results of Liu et al. [24] and Tobushi et al. [38] to predict the SMP strain recovery stage. Gilormini and Diani [214] inspired by Liu et al. [24], Chen and Lagoudas [19,54], and Qi et al. [66] models, by proposing new functions for the volume fraction of the frozen phase, employed various forms of homogenization approximations, that is, Reuss, Voigt type, Hashin
Shtrikman upper and lower bounds, and self-consistent to examine their potential in reproductions of SME. They utilized the experimental results of Liu et al. [24] and reported that the Voigt model had a higher accuracy. Kim et al. [72] assumed that the SMP consists of three phases, two soft phases, the so-called frozen and active phases, and a hard phase. They introduced a new 3D finite strain model based on the three-phase nature of the Polyurethane, taking into account two hyperelastic soft-phase and a viscoelastic hard phase. For soft phases, they applied Mooney
Rivlin model and used the SLS model for its viscoelastic part. In line with this model, Gu et al. [73] employed the affine network model (presented by Flory [215]) instead of Mooney-Rivlin model for the two phases of the soft segment. Considering microstructure changes during the recovery processes and a three-element rheological component, they proposed a new 3D model. Since the Liu et al.’s model [24] had provided the evolution law only for the cooling process in uniaxial loading, proposing a new evolution law for the heating step, Baghani et al. [91] modified the Liu et al.’s model [24]. They put to use Diani et al.’s experiments [90] to calibrate the model. Writing a UMAT, they examined twisting of a rectangular beam in strain recovery process. Sujithra et al. [44] presented a similar model to Baghani et al. model [91], which obtained model coefficients based on experimental results of Liu et al. [24] for an epoxy thermoset and they also simulated a multi-SMP. Taking the basis from the models provided by Liu et al. [24] and Baghani et al. [91], Roh et al. [111] developed a model for a

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composite matrix. The main difference is that they used a generalized Maxwell model with an equilibrium branch and three nonequilibrium branches for the rubbery and glassy phases. In this way, motivated from Liu et al. [24] and Baghani et al.’s [91] model, Arvanitakis [216] presented a constitutive equation for an epoxy thermoset SMP based on the level-set approach. In this model, the constitutive equation for stain decomposition are similar to Liu et al. [24]. In contrast, the evolution law and frozen volume fraction function are different and defined based on level-set approach. It is noted that the presented model is implemented for both conventional SMP and triple-SMP. Reese et al. [208] introduced a 3D model for application in an SMP stent by introducing a new frozen volume fraction function. This 3D form was later employed by Boatti et al. [25] for a polyurethane-based SMP (experiments provided by Volk et al. [115]) and an epoxy (experiments reported by Gall et al. [217]) in strain and stress recovery processes. In order to account for the time-dependent effects, Xu et al. [55] improved the model of Liu et al. [24]. To have physically meaningful coefficients, they introduced a 3D model for syntactic foam in thermodynamic processes of strain and stress recovery. In their model, they used the frozen phase volume fraction function proposed by Qi et al. [66]. Satisfying the second law of thermodynamics in the form of Clausius
Duhem inequality, Baghani et al. [48] presented a 3D model for a thermoset SMP under time-dependent, multiaxial thermomechanical loading in the range of small strains, where the strain has been decomposed into six components. In this constitutive model, the evolution equations for the internal variables are proposed both during the cooling and heating stages. Time-dependent effects are also accounted for. The implicit time-discrete form of governing equations is reported. They calibrated the model for three different sets of experiments of Liu et al. [24], Volk et al. [80,81] and Li and Nettles [78]. Then, Implementing the time-discrete form of the model in FEM, two boundary value problems (a 3D beam and an SMP-based medical stent) have been analyzed to show the generality of the proposed formulation. In this model, the strain is decomposed as follows: ε 5 φp εp 1 φh εh 1 εi 1 εT

(2.16)

in which φp and φh indicate volume fraction of the SMP segment and the hard segment, respectively. It is noted that the hard segment stands for the dispersed microballoons in the material to enhance the mechanical strength. εi shows the irreversible strain. A schematic representation of this constitutive model is shown in Fig. 2.13. In Eq. (2.16), the SMP segment strain εp is further decomposed as follows: εp 5 ϕr εr 1 ϕg εG

(2.17)

in which εG is total strain in the glassy phase. Subscripts “r” and “g” represent the rubbery and glassy phases, respectively. It is noted that in this model, it is

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49

Figure 2.13 The rheological schematic illustration of Baghani et al.’s model [48].

assumed that φp and φh are constant parameters but ϕr dependent ones. Term ϕg εG could be expressed as:

and ϕg are temperature

ϕg εG 5 ϕg εg 1 εis

(2.18)

Ð in which, in the Ðcooling step, we have εis 5 εr dϕr , and in heating step, it is pror posed that εis 5 ϕε dϕg . Thus one may write: g

εis ε_ is 5 ϕg T_ ks1 εr 1 ks2 ϕg

!

0

(2.19)

and for other internal variables, evolutions laws are proposed as: ε_ ir 5

1 @Ψr neq ig 1 @Ψg neq ih 1 @Ψh neq i 1 _ _ ; ε 5 ; ε 5 ;ε_ 5 σ ηr @εer ηg @εeg ηh @εeh ηi

(2.20)

where Ψ is free energy density function and η is viscosity coefficient. Then, in another work, Baghani et al. [51], extended the model to the range of finite strains. In another paper taking advantage of a logarithmic strain, as a more physical measure of strain [46], they presented a large strain 3D mechanical model based on the classical thermodynamic frameworks. In deriving this constitutive model, the deformation gradient tensor was divided multiplicatively into two, elastic and stored parts (where the stored part includes two rubbery and glassy parts as well). The evolution equations for the internal variables are presented for both the cooling and heating processes. Based on Baghani et al.’s model [48,91], Taherzadeh et al. [218] examined homogenization of SMP nanocomposite. They used Baghani et al.’s model [48,91] and numerically examined large deformation of the graphene-reinforced SMPs. Guo et al. [82] presented a new function for the frozen phase volume fraction. Following the same phase transition approach, they presented a 3D small strain model which was successful in prediction of shape and force recovery processes of a styrene-based epoxy-type amorphous SMP. In this model, in addition to the temperature as the main parameter, the stress was assumed to play an important role in the phase transition. Inspired by the Liu et al.’s approach [24], they assumed an

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Computational Modeling of Intelligent Soft Matter

elastic linear element for the frozen phase, a three-element for the active phase and a strain storage element, all in series. For the viscoelastic strain, they employed a 3D version of the rheological stress-strain relationship in Tobushi et al.’s model [38]. In another work [56], using strain rate parameter, they scrutinized the strain hardening behavior of an epoxy type SMP. Moreover, from a phase transition point of view, considering a linear spring for the frozen phase, and a Kelvin element for the active phase in series presented by Rault [219], they were able to model the 3D strain hardening behavior of the material. In another work, the same authors [52], employing a new function presented for the volume fraction of the frozen phase, examined the strain recovery behavior of SMP by proposing a new mechanical model for the rubbery phase and generalized Kelvin elements for the frozen phase. Furthermore, as mentioned in previous section, in thermoviscoelastic point of view, Gu et al. [107] proposed large deformation constitutive equations for an SMP. In this way, in phase transition approach, Zeng et al. [204] by using the concept of frozen phase volume fraction in complementation of viscoplastic and hyperelastic stresses, investigated thermomechanical behavior of a thermoset epoxy. Pieczyska et al. [220] following the same phase transition perspective and the volume fraction of the frozen phase provided by Qi et al. [66], considered two parallel rubbery and glassy phases in their model. The rubbery phase was assumed as a hyperelastic Arruda
Boyce material and the glassy phase was treated as a threeelement hyperelastic-viscoplastic (Zener) model. This 3D finite strain model was evaluated for shape recovery process of polyurethane. It seems they focused more on the hysteresis behavior of SMP as well as the impact of loading rates. Yang and Li [45] presented a temperature- and rate-dependent constitutive model for amorphous SMPs. They considered the SMP material as a composite material which its matrix is the inactive phase and the dispersed particles play the role of the active phase. Then, they applied the theory of Mori-Tanaka [221] to arrive at the effective mechanical properties of SMP. Employing the relationship provided by Levin [222] and modified by Rosen and Hashin [223], they obtained the equivalent thermal expansion coefficient for SMP. They also proposed a new function for the volume fraction of the frozen phase. In this model, for the programming and cooling steps, the formula provided by Gilormini and Diani [214], and for the force recovery, Westbrook et al. [68] equations were exploited. They computed the coefficients from the experimental results of Liu et al. [24]. Contrary to previous models, an evolution law is presented which is consistent with experimental data and has physically meaningful parameters. To put it simply, this model depends on the measurable mechanical properties of the material. They evaluated the model in both the strain and stress recovery processes. Park et al. [116] proposed a 3D finite deformation two-phase constitutive model which is suitable for multiaxial loading (up to about 200% strains). In this model, they presented a new function for the volume fraction of the frozen phase. They first assumed that both phases are connected in parallel, then for the rubbery part, a three-element model was used and for the glassy part, and they employed a threeelement Kelvin model. The total deformation gradient has been decomposed into hyperelastic, viscoelastic, viscoplastic, and the shape recovery components.

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51

As mentioned, they put to use the Poynting
Thomson three-element model for the rubbery part and, a specific rheological model with six elements for the frozen part. To develop the formulation, they defined the Helmholtz free energy function and applied the Clausius
Duhem inequality to properly account for the second law of thermodynamics. Finally, the model was calibrated for experiments on a PCCbased SMP. It is noteworthy that the number of material coefficients in this model are relatively high, which makes it hard to calibrate. This model only predicts the strain recovery process. Then, these authors in another work [224] by using their previous model and considering residual stresses, simulated woven fabricreinforced SMP composites. They used classical anisotropic hyperelastic theorems to model the orthotropic properties of the woven fabric reinforced SMP. Finally, the total stress is complemented with the stress generated in the SMP constitutive equation, anisotropic hyperelastic model and thermal residual stress by the Eshelby’s inclusion theorem. Following the same viewpoint, Li et al. [49] by dividing the rubbery and frozen phases into a large number of rubbery and frozen subphases that are connected in series, and taking into account a linear elastic behavior for each phase, anticipated the shape and force recovery behavior of the SMP. In this model, they used experimental results from Liu et al. [24] and Arrieta et al. [57,225]. The same authors in another work [32], based on their previous approach, assuming a three-element viscoelastic SLS model for the rubbery phase and an elastic spring for the glassy phase together with the use of the frozen phase volume fraction of Qi et al. [66], presented a new small strain model for SMP recovery processes. They calibrated their model with experimental results of Tobushi et al. [22,38], Liu et al. [24], and Li et al. [49]. Dong et al. [97] following the same approach and using the volume fraction of the frozen phase introduced by Qi et al. [66], proposed a nonlinear three-element model for both the frozen and active phases. The equations were developed in onedimension, and performing tests on a Polyurethane-based SMP, they predicted SMP behavior in a strain recovery step. In a slightly different approach, Gu et al. [59] similar to their previous model [87], but from a phase transition standpoint, using a three-element model, introduced relevant internal state variables for an acrylatebased SMP (tBA/PEGDMA) in the strain recovery path. For the equilibrium part, they used the Arruda-Boyce model and for the nonequilibrium part, a flow rule theory was employed. In addition, they took into account the structural and stress relaxation, and finally, to find the SMP Poisson’s ratio, they made use of the frozen phase volume fraction function provided by Qi et al. [66]. They used experimental data from Westbrook et al. [68]. Following this approach, Su et al. [61], decomposed the free energy based on the assumptions of Reese et al. [208]. They proposed a 3D constitutive model containing a three-element viscoelastic model (the Rivlin model for the equilibrium part and a flow rule theory similar to the Nguyen et al.’s model [67] for the nonequilibrium part) which was able to reproduce the shape recovery process. For its thermal part a fictive temperature was assumed as an internal variable, similar to those introduced in [67,68]. Then, utilizing the phase transition and frozen volume fraction concepts, they defined material parameters in terms of the frozen and active

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Computational Modeling of Intelligent Soft Matter

phase volume fractions. Finally, taking advantage of some tests borrowed from [17,66,68], they calibrated the model. Then, in another work, the same authors analyzed an anisotropic SMP composite using the same model [59], as well as the method of continuum mechanics composite formulations [226], in the stress and strain recovery processes. They found the model coefficients for the SMP-based matrix based on the results of Westbrook et al. [68] on an acrylate-based SMP (tBA/PEGDMA). The SMP composite in this work was a combination of an acrylate-based SMP (tBA/PEGDMA) with carbon fiber, where the volume averaging theory was employed to find the effective properties. It should be noted that Kazakevic˘i¯ut˙e-Makovska et al. [20], briefly examined the evolution laws for the volume fraction of the frozen phase presented in literature [24,50,66,80,208]. In Table 2.5, it is attempted to gather all forms of volume fraction of the frozen phase available in the literature. Table 2.5 The represented frozen and crystalline volume fraction functions in phase transition approach [15]. Frozen phase or crystalline phase volume fraction (φ, ξ c )

References

φ512

[24,47,227]

φ5

1 1 1 cf ðTh 2T Þn

1 1 1 expð 2T 2a Tt Þ



Tt m

φ 5 αexp 2 T 1  φ5 1 1 exp

φ5

[66,228] β

 2n

[50] [208]

2w T 2 Tt

b 2 tanhððT 2 AÞ=BÞ b2a

[80]



m n T2Tmin φ 5 12 Tmax 2Tmin   2 ÐT ðT2Tg Þ dT φ 5 Sp1ffiffiffiffi exp 2 2 2S 2π

[214] [82]

Ts

ξg 5

2d !  

Δt N Ð τ0 ΔHa ðT Þ pðr Þdr 3 1 2 12exp 2 kB T

b 1 1 expðcðT 2 Ttr ÞÞ

φ512

[116] [45]

r c ðT Þ

φf 5 1 2

12b 1 1 expð 2 a1 ðT 2 Tr ÞÞ

2

b 1 1 expð 2 a2 ðT 2 Tc ÞÞ

 1 T 2 Ttr ðT_ Þ 1 1 exp 2 b

[119]

φ512 8 1 > >    ; T_ # 0 > > < 1 1 exp β cool T 2 Tc;eff ξc 5 1 1 > >       ; T_ $ 0 > > : 1 1 exp β cool TEND 2 Tc;eff 1 1 exp β heat T 2 Tm;eff

[60]

tanhðγ T 2 γ TÞ 2 tanhðγ T 2 γ T Þ φ 5 1 1 tanh γ 1T g 2 γ 2T 2 tanh γ1 Tg 2 γ2 Th ð 1 g 2 hÞ ð 1 g 2 lÞ   Ð  φ 5 V1 Ω 12 1 1 tanh ψ=s dV

[91,229]

[130]

[216]

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53

Crystalline SMPs possess a real phase transition. Following the same perspective, they are of interest to some authors such as [26,94,96,113,209]. Rao et al. [230] presented a constitutive model for crystallizable SMPs, in which the crystalline phase is the main driver for the shape fixation in SMP. They assumed that SMP undergoes four processes, rubbery, semicrystalline, crystallization process, and melting process. Barot and Rao [26,209] presented a fully coupled thermomechanical model for semicrystalline SMPs based on the theory of Rajagopal et al. [231,232]. In this model, they divided the SMP into two solid crystalline phase and a compliant rubbery phase, assuming that they are in parallel. Therefore the energy density function considered here was a combination of two amorphous and crystalline energies (introducing the crystallinity function). Finally, the governing stressstrain-temperature equations were proposed in a 3D shape-retrieval process. Ge et al. [94] for an SMP composite consisting of an elastomeric matrix (Silhouette rubber (Sylgard 184)) and a semicrystalline fiber PCL (as a multiphase homogeneous system), applying the rule of mixture and accounting for the stress concentration, for the matrix and fibers, proposed a finite strain 3D model. They employed the theory provided by Avrami [233] to take into account the crystal-melt phase transition. Finally, they presented the results for two thermomechanical stress and strain recovery processes. Then, the same authors in another work [95], presented a production technique as well as a constitutive model for anisotropic SMPs. In this finite strain 3D model, connecting the phase evolution theory for the active soft materials to the reinforced composites theory with anisotropic thermal strains, they could regenerate the shape recovery experiments. Along with the modeling of a semicrystalline PE-based SMP, Kolesov et al. [96] based on the same phase transition viewpoint, predicted the shape recovery response of a PE-based SMP by a three-element viscoelastic model. To simulate semicrystalline SMPs from this approach, Scalet et al. [113] proposed a one-dimensional finite deformation model which was able to predict the recovery behavior of low-density and high-density PE. They developed constitutive laws, similar to de Souza Neto et al. [234]. The model included 14 material parameters. Finally, Bouaziz et al. [119] proposed a 3D model in the range of large strains to predict the behavior of the semicrystalline thermoplastic polyurethane in different recovery scenarios. They implemented this model in COMSOL Multiphysics©. They presented a new function in this model for the frozen phase volume fraction. In constitutive relations, they put to use a rheological model similar to generalized Maxwell model with n nonequilibrium branches (to account for the viscoplastic behavior of SMP) and an equilibrium branch containing a slipping element, and a spring to represent the hyperelastic and inelastic behavior, respectively. They also employed the theory provided by Simo [235] to develop finite strain viscoelastic relations.

2.4.1.3 Constitutive models of shape memory polymer under other approaches From another point of view modeling, Kafka et al. [39], utilizing the mesoscale concept as a different approach from the phase transition and viscoelastic

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Computational Modeling of Intelligent Soft Matter

approaches, were able to model the shape recovery response of SMP for experimental results of Tobushi et al. [22,38]. In this model, they applied the theory and concept of mesoscale described in detail in their previous work [236]. In this perspective, they modeled SMP introducing internal variables without employing any basic assumptions such as rheological elements in parallel or series, or assuming homogeneous stress (Reuss’s model) or homogeneous strain (Voigt’s model). Diani and Gal [53] studied properties of a Polyisoprene-based amorphous SMP through molecular dynamics (Full-atomistic molecular dynamics simulations) method. The whole model was simulated under applied tensile and compressive strain in the range of large strains. Finally, they obtained the polyisoprene internal energy cycle during its thermomechanical process. Applying the density functional theory, Zhang et al. [237] examined the hydrogen bond between the soft and hard segments of Polyurethane block copolymers as an SMP under quantum mechanics principle. Based on the same approach, other researchers such as Yang et al. [238], Davidson and Goulbourne [239], Santiago et al. [240], Hu et al. [241], Yang et al. [242], and Uddin et al. [243] investigated an epoxy SMP, cross-linked SMPs, hyperbranched poly(ethyleneimine)-modified epoxy thermosets, polyurethane, an epoxy SMPs, and thermoplastic polyurethane, respectively. However, this method has not much been discussed by other researchers. Shojaei and Li [14] employing a different approach, and based on statistical mechanics, developed a large strain one-dimensional thermomechanical model for polystyrene including mechanical damage and continuum functional effects. They employed a new viscoplastic model based on the statistical mechanics. From a statistical mechanics standpoint, they examined the viscoelastic properties of the material, such as relaxation time and shear modulus. It is noteworthy that their model is successful in prediction of both the strain and stress recovery responses.

2.4.2 Modeling of two-way thermally activated shape memory polymer To provide a constitutive model that can predict two-way response of SMPs, Westbrook et al. [133], for the first time, using Chung et al. [132] experiments on an SMP composite containing DCP as the elastomeric-based matrix (cross-linker), and semicrystalline PCO, proposed a large strain 1D constitutive model for the two-way SMP property (under applied stress) from a phase transition approach. They assumed that the SMP consists of a rubbery phase and an SIC that tolerate equal stresses. They applied the modified Avrami theory [244] to develop the phase transition constitutive relationships. Later, to overcome the problem of acquisition of a two-way property in the absence of an applied load, the same group produced an SMP composite (PCO/ DCP with base matrix of a PEGDMA/tBA-based SMP) and examined the two-way SMP in the absence of an external load. They analyzed the bending of a multilayer beam with a simple beam theory [135]. Furthermore, Ge et al. [134], in the same group, following the phase transition approach, studied the behavior of the same

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55

composite by the model reported in Westbrook et al. [133]. Then, applying the phase transition viewpoint, Dolynchuk et al. [142] presented a 1D model assuming a three-viscoelastic-element model for a semicrystalline high-density PE as a thermoset SMP that can predict its two-way behavior under applied stress. Recently, Scalet et al. [130] in a similar phase transition view of Baghani et al. [91], presented a 1D model for a two-way semicrystalline PCL-based thermoset SMP under external stress. They used experimental results borrowed from Pandidi et al. [137].

2.4.3 Modeling of thermally activated multishape memory polymer The first attempts made to examine the quantitative behavior of the multi-SMP was reported by Sun and Huang [158] for both the stress and strain recoveries using the stored and released energy concept. They considered the SME assuming two parallel, elastic and transitional parts. Then, following the viscoelastic approach of Westbrook et al. [135], Yu et al. [159] predicted the triple shape recovery of an PFSA for the experimental results of Xie [153] through an 1D linear thermoviscoelastic model. Thereafter, as mentioned before, Arrieta et al. [57] proposed a thermoviscohyperelastic model in large deformation regime which calibrated the model for an acrylate-based SMP as a conventional SMP and for an PFSA as a triple-SMP. To examine the model capability, similar to Yu et al. [159], they made use of the test results from Xie [153]. Then, Ge et al. from the viscoelastic viewpoint presented a one-dimensional model [157] and then extended the model to 3D deformations [162] in the range of large strains for an SMP composite as a combination of an amorphous epoxy matrix and crystallizable PCL fiber networks. Providing a UMAT subroutine, they validated their model against experimental results. They used an epoxy to fix a temporary shape and used PCL for other temporary shapes. In this model, they applied multibranch techniques to describe the viscoelastic behavior of an amorphous epoxy-based SMP. They proposed constitutive equations with differently deformed crystalline phases to elucidate the behavior of crystallizable fiber networks. Then, based on the carried out experiments on an amorphous epoxy matrix and crystallizable PCL fiber networks from Ge et al. [162], a small strain model was introduced by Arvanitakis [216]. This model is similar to Liu et al. [24] and Baghani et al.’s model [91], despite the evolution law and frozen volume fraction function are different and defined based on the level-set approach. Sujithra et al. [44] proposed a 3D model similar to that of Baghani et al. [91] using the experimental results of Liu et al. on an epoxy [24]. They expressed the multiSMP numerically in the range of small strains based on the phase transition approach. Then, Xiao et al. [161] for the Nafion as an amorphous SMP, which has a wide glassy transition temperature range, presented a 3D model in the strain recovery process (similar to generalized Maxwell
Kelvin’s model). In this model, assuming parallel multibranch equilibrium and nonequilibrium elements, they developed 3D constitutive equations. They considered the modulus of relaxation according to Schwarzl and Staverman [245] and identified the model coefficients exploiting the DMTA tests.

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Fang et al. [160], employing the generalized Maxwell fractional elements, predicted the shape retrieval behavior of an amorphous PFSA with a large glassy transition temperature range. It is noteworthy that Fang et al. [64] used the experiments by Xie [153], Westbrook et al. [68], Nguyen et al. [70], and Gu et al. [88], respectively, for an PFSA, acrylate-based network polymer, tBA-co-PEGDMA networks, and styrene-based thermoset resin (Syntactic foam-based SMP). Then, following the phase transition approach, similar to the approach provided by Barot and Rao [26] for semicrystalline SMPs, Moon et al. [156] modeled a triple-SMP based on poly(ω-pentadecalactone) (PPD) and PCL in 3D large deformations. Afterwards, employing the same phase transition view, and according to the theory of multiple natural configurations, they developed a more accurate 3D model. They used experimental results from Zotzmann et al. [163]. Subsequently, based on the same approach, they extended their model into 3D loading regime [164]. Then, Lu et al. [165] proposed a constitutive model for multi-SMPs from the same standpoint, taking into account the soft and hard segments, and using the Takayanagi [246] relationship in both series and parallel phases arrangement. This way, they were able to predict the multi-SMP behavior. Their model was applicable to the strain recovery process. They calibrated the model for the experimental results of Xie et al. on a triple-SMP [149] and Arrieta et al. [57]. Through the phase transition point of view, Du et al. [168] similar to Liu et al.’s model [24] and by using the generalized Maxwell model, investigated triple-SME of styrene-based SMP under 20% uniaxial tensile test. At the end, a summary of the models presented in literature to predict the behavior of multi-SMPs, is presented in Table 2.6. Based on Table 2.6, it could be found that: (1) mathematical modeling of multiSMP has been just on amorphous thermoset and semicrystalline thermoplastic polymers; (2) thermoviscoelastic point of view was just used for semicrystalline thermoplastic polymers; and (3) phase transition point of view was used for both thermoset and thermoplastic polymers. Table 2.6 A summary of the presented models on multi-SMP. Author, references

Year

Material

Point of view

Description

Sun and Huang [158] Yu et al. [159]

2010

N.S.

2012

PFSA

Physically based T.V.

Ge et al. [157]

2013

An epoxy-PCL

P.T.

Force and shape recovery, 1D, small deformation Shape recovery, 1D, small deformation semicrystalline SMP, thermoplastic Force and shape recovery, 1D, large deformation, semicrystalline SMP composite, thermoset 1 thermoplastic (Continued)

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57

Table 2.6 (Continued) Author, references

Year

Material

Point of view

Description

Arrieta et al. [57]

2014

PFSA

T.V.

Ge et al. [162]

2014

An epoxy-PCL

T.V.

Sujithra et al. [44]

2014

An epoxy

P.T.

Moon et al. [156]

2015

PPD and PCL

P.T.

Xiao et al. [161]

2015

Nafion

T.V.

Moon et al. [247]

2016

PPD and PCL

P.T.

Fang et al. [160]

2018

PFSA

T.V.

Lu et al. [165]

2018

An epoxy

P.T.

Moon et al. [164]

2019

PPD and PCL

P.T.

Du et al. [168]

2019

A Styrene

P.T.

Arvanitakis [216]

2019

An epoxy-PCL

P.T.

Shape recovery, 3D, large deformation semicrystalline SMP, thermoplastic Force and shape recovery, 3D, large deformation, semicrystalline SMP composite, thermoset 1 thermoplastic Shape recovery, 3D, small deformation, amorphous, thermoset Shape recovery, 3D, large deformation, semicrystalline SMP, thermoplastic Shape recovery, 3D, large deformation, semicrystalline SMP, thermoplastic Shape recovery, 3D, large deformation, semicrystalline SMP, thermoplastic, programmed through UMAT subroutine Shape recovery, 1D, small deformation, semicrystalline SMP, thermoplastic Shape recovery, 1D, small deformation, amorphous, thermoset Shape recovery, 3D, large deformation, semicrystalline SMP, thermoplastic Shape recovery, 1D, small deformation, amorphous SMP, thermoset Shape recovery, 3D, small deformation, semicrystalline SMP composite, thermoset 1 thermoplastic

N.S, T.V., and P.T. stand for not-stated in the relevant references, thermoviscoelastic point of view, and phasetransition point of view, respectively [15].

58

2.5

Computational Modeling of Intelligent Soft Matter

Statistical analysis of available shape memory polymer models

In this section, a statistical analysis is carried out for constitutive models existing in literature for different types of SMPs, that is, conventional, two-way, and multiSMPs. Here, we show graphically the proposed models in different point of views including, their deformation regimes (large or small deformation model), the types of their base materials (thermoset, thermoplastic, amorphous, semicrystalline, etc.), modeling point of views (thermoviscoelastic or phase transition), the number of constitutive models for conventional, two-way, multi-SMPs, etc. To analyze and visualize the constitutive models provided for SMPs in literature, CitNetExplorer’s analytic software is employed to evaluate the main articles as well as the highly cited ones, together with how they are connected (Core Publications). In total, for available SMP models in the literature, the following results are obtained as illustrated in Fig. 2.14. According to the database collected in this chapter and shown in Fig. 2.15, up to now more than 100 constitutive models have been proposed to predict the behavior of temperature-sensitive SMPs and their composites, of which about 98 constitutive

Figure 2.14 Citation network and connections of core publication of constitutive modeling of shape memory polymers [15].

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59

Figure 2.15 The number of presented constitutive equations in literature per year for thermally activated shape memory polymers. Blue bar, red bar, and black bar indicate the conventional shape memory polymer, multishape memory polymer, and two-way shape memory polymer, respectively [15].

models are devoted to conventional thermally activated SMPs. There are about 14 different models for multi-SMP and about five constitutive models for two-way SMPs. If we consider the modeling approach to model the two-way SMP behavior, it is noted that almost all constitutive models are presented from the phase transition point of view for semicrystalline SMPs. To model the multi-SMP behavior, attention is paid to both amorphous SMPs and semicrystalline SMPs. Eight models are presented in the phase transition framework and about five models are developed based on the viscoelastic approach. Finally, for conventional SMPs and their composites, about 50 models are proposed from the viscoelastic viewpoint and 38 models are presented based on the phase transition idea, while few models are presented from other perspectives. Finally, as state of the art of this chapter, one may refer to Fig. 2.16. This figure, graphically, demonstrates the different connections between thermally activated SMPs macroscopic types, modeling points of view, SMPs micromolecular types, strain range, and relevant authors. According to this figure, one may conclude that: (1) It is clear more attention has been paid to conventional SMPs, consequently most of available constitutive equations are devoted to conventional SMPs. (2) For two-way SMPs, constitutive equations have been developed only for semicrystalline SMPs as well as their composites. (3) Also for multi-SMPs, constitutive equations have been reported only for amorphous thermoset, semicrystalline thermoplastic polymers as well as their composites. (4) For conventional SMPs, up to now, proposed constitutive equations for amorphous thermoset and semicrystalline thermoset are the most and least, respectively. (5) Semicrystalline polymers either thermoset or thermoplastic have been modeled only through the phase transition approach.

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Computational Modeling of Intelligent Soft Matter

Figure 2.16 Shape memory polymers classification and available constitutive equations. P.T. and T.V. mean phase transition and thermoviscoelastic point of views [15].

2.6

Summary and conclusion

In this chapter, a comprehensive review of SMPs, their mechanism, and constitutive modeling of thermally activated SMPs were presented. First, the definitions, micromechanical structure, and classifications of SMPs were discussed. Afterward, a brief review was given on mathematical modeling of thermosensitive SMPs (including conventional SMP, multi-SMP and two-way SMP) and their modeling’s point of views. Finally, a statistical analysis on the available constitutive equations of thermally activated SMP was carried out. Besides thermally activated SMPs, there are a lack of modeling of magnetic activated SMP, which recently are being used in the biomechanics engineering. For thermally activated SMP there are a lot of constitutive models developed upon various points of view, particularly, thermoviscoelastic and phase transition. A number of these models predicts the conventional, multi or two-way SMP behavior under different conditions (i.e., multiaxial loading, large deformations, and thermally rate dependent). Besides, one of the key steps in this area is the calibration and application of the constitutive equations of SMPs in the biomedical applications (for example, temperature range of SMP in the range of body temperature). From material point of view, the acrylate-based SMPs are common biocompatible ones which mostly have been modeled based on the viscoelastic

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approach and in contrast, the epoxy’ family-based SMPs are common thermoset SMPs, which mostly have been studied through the phase transition approach. At the end, it could be stated that an ideal model for describing the behavior of temperature-responsive SMPs would have the following characteristics: 1. The model calibration method should be simple and straightforward, as far as possible. 2. Model calibration should not involve so many empirical experiments to find the model coefficients (higher errors and costs). 3. Most of the proposed models, even developed in a 3D framework, are not calibrated and verified in 3D regime of loading. Therefore the model should have the ability to predict SMP behavior in other complicated loading conditions rather than just uniaxial ones. 4. The time-discrete counterpart of the SMP constitutive model should not be so complicated that its numerical implementation be computationally expensive which precludes it from wide applications. 5. Due to the large application of SMPs in various engineering and biomechanics fields, it is preferred to be developed in a large strain framework. 6. SMP constitutive models should be capable of prediction of both thermomechanical stress and strain recovery processes. 7. The capability of model to predict the conventional, multi- and two-way behavior of SMP. 8. It is clear more attention has been paid to conventional SMPs, consequently most of available constitutive equations are devoted to conventional SMPs. 9. For two-way SMPs, constitutive equations have been developed only for semicrystalline SMPs as well as their composites. 10. Also, for multi-SMPs, constitutive equations have been reported only for amorphous thermoset, semicrystalline thermoplastic polymers as well as their composites. 11. For conventional SMPs, up to now, proposed constitutive equations for amorphous thermoset and semicrystalline thermoset are the most and least, respectively. 12. Semicrystalline polymers either thermoset or thermoplastic have been modeled only through the phase transition approach.

References [1] Z. Gharehnazifam, R. Dolatabadi, M. Baniassadi, H. Shahsavari, A.-M. Kajbafzadeh, K. Abrinia, et al., Multiphysics modeling and experiments on ultrasound-triggered drug delivery from silk fibroin hydrogel for Wilms tumor, International Journal of Pharmaceutics 621 (2022) 121787. [2] Z. Gharehnazifam, R. Dolatabadi, M. Baniassadi, H. Shahsavari, A.M. Kajbafzadeh, K. Abrinia, et al., Computational analysis of vincristine loaded silk fibroin hydrogel for sustained drug delivery applications: Multiphysics modeling and experiments, International Journal of Pharmaceutics 609 (2021). [3] E. Yarali, A. Mohammadi, S. Mafakheri, M. Baghani, H. Adibi, Mathematical modeling and experimental evaluation of a prototype double-tube magnetorheological damper, SN Applied Sciences 1 (11) (2019) 1341. [4] M. Bayat, R. Dolatabadi, M. Baghani, Transient swelling response of pH-sensitive hydrogels: a monophasic constitutive model and numerical implementation, International Journal of Pharmaceutics (2020) 119030.

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Experiments on shape memory polymers: methods of production, shape memory effect parameters, and application

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Chapter outline 3.1 Introduction 77 3.2 Methods of shape memory polymer production 78 3.2.1 3.2.2 3.2.3 3.2.4

Melt mixing 78 Solution mixing 78 Additive manufacturing 79 Shape memory characterization in combined torsion
tension loading 79

3.3 Investigation on structural design of shape memory polymers 3.3.1 3.3.2 3.3.3 3.3.4

90

Structural (geometrical) design 90 Method of sample production 92 Characterization of printed material 94 Thermomechanical shape memory tests 94

3.4 Shape memory polymer stent as an application

104

3.4.1 Materials 105 3.4.2 Stent fabrication 107 3.4.3 Stent radial compression 116

3.5 Summary and conclusion References 124

3.1

122

Introduction

Smart materials are a type of materials that can respond to their environment and surroundings. In other words, smart materials can be defined as materials that can remember configurations by a proper stimulation. This property of smart materials has attracted a great deal of interest in various applications, such as automotive, aerospace, textile, robotics, and medical [1] industries. Shape memory polymers (SMPs) are a division of smart materials capable of deforming to their permanent shape [2,3]. When these materials are heated up to a temperature higher than the glass transition temperature (Tg), polymeric chains of the material weaken and can easily slide on each other. Therefore by applying a low force, material can be deformed into a desired shape. Then, by cooling to a temperature lower than Tg, the temporary shape is maintained and by increasing the temperature higher than Tg, the material automatically restores its initial shape. Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00003-3 © 2023 Elsevier Inc. All rights reserved.

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SMPs with different glass transition temperatures are introduced during last decades. This chapter first describes the method of melting, dissolving, and printing for the production of SMPs. Afterwards, parameters, which are effective on the behavior of SMPs, are studied. According to the reports, 50%
70% of applications of SMP are in medical fields [4]. One of the most practical ones relevant to this area is medical coronary stents, which will be discussed in the following.

3.2

Methods of shape memory polymer production

SMPs can be manufactured in different ways according to the sample shape, application, and material type. Each of these methods have their own advantages. In the following, three methods of producing SMP are briefly discussed.

3.2.1 Melt mixing Mixing via melt blending is one of the most privileged methods for producing nanocomposite polymers, which are based on thermoplastic and elastomer materials. The exact ratio of the primary materials and the homogeneity of the materials after mixing are two main factors affecting the quality of polymer mixing so they should be accurately considered. Furthermore, internal or Banbury mixers could be used for polymeric mixing. These mixers are used widely in the polymers engineering where high-power polymer mixers are indisputable for blending and breaking chains. This method is successfully employed in chemical systems, such as polymers that have high and low viscosity. Using mixers for blending solid additives in plastic or elastic matrixes or homogeneous mixture of several polymer materials here is the main goal [5]. This manufacturing method has merits that overweighs additive manufacturing and solvent methods. The melt mixture is environmentally friendly due to the absence of organic solvents. By molding and using the plastic injection method, these materials could be mass-produced with desired dimensions. This method is not feasible enough in the fabrication of polymeric materials for laboratory samples.

3.2.2 Solution mixing Mixing through the solvent method is based on polymer mixing via chemical solvents. At the beginning, polymers are dissolved in solvents, such as water, chloroforms, toluene, and other matters of the kind. When polymers and nanoparticles available in the solution are mixed, polymer chains will connect. After removing the solvent, we have the remaining mixed polymer [6]. Mixing polymeric solution by energetic agitation, such as magnetic stirring, shear mixing, reflux, and widely common sonication, is feasible. At the next step, the solution is poured into

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laboratory containers and molds and then they will be placed in an oven. After remaining the solution in the oven for a specified time at a high temperature, the solution in the container or mold evaporates and the polymer remains with a new chain bond. This method is economical and suitable for producing laboratory samples. Besides, samples from this method have fine surface quality. Manufacturing through this method is time-consuming and therefore it is considered a challenging drawback [5].

3.2.3 Additive manufacturing Additive manufacturing, that is, 3D printing is a procedure in which it fabricates the sample layer by layer [7]. 3D printers were first used in 1980s to threedimensionally repair the polymer layers. Currently, there are various types of commercial 3D printers available, which include material extrusion additive manufacturing, fused deposition modeling (FDM), laminated object modeling, and selective laser sintering. 3D printers have been widely welcomed to different places, such as industrial plants, homes, offices, and scientific institutions. As was earlier mentioned, various factors influence the behavior of SMPs, including geometric design, molecular bonds, and addition of nanoparticles. In this section, the impacts of these parameters on SMPs will be experimentally examined. 4D printing is a fabrication process for active materials deformed by different stimulations such as heat and humidity [8]. 4D printing is primarily introduced by Tibbits in 2013 [9]. Hitherto, hydrogels and SMPs are used as 4D printing materials. Nonswelling polymers or filaments are used in production of hydrogels using 4D printing. When the printed structure is submerged in a solvent, the hydrogel swells up, which leads to the deformation of the specimen [10
12]. SMPs can be deformed using external stimulations such as heat. Hydrogels require no postproduction programming, which is an advantage over SMPs. But there are three vital issues with hydrogels. First, hydrogels are soft materials, and the corresponding printed structure may not have the desired stiffness. Second issue is about the low speed of deformation of gels upon swelling, especially in large deformations. Third issue is possible instability of the activated swollen shape. For instance, a reduction in solvent leads to a different environmental condition, which may substantially change the hydrogel shape [13]. Considering the advantages of SMPs over hydrogels, researchers are striving to improve the shape memory effect of SMPs in recent years [14]. In the following sections, the effect of different parameters on the behavior of SMPs is investigated experimentally.

3.2.4 Shape memory characterization in combined torsion
tension loading In this section, shape recovery of SMP in a combined tension
torsion loading is studied and compared with uniaxial tension and pure torsion loading. Then, SMP

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stress recovery is reported in different prestretch and deformation temperatures (Td). The result can be used to develop the constitutive equation in combined loadings and describe the performance of SMP structures such as helical stents or springs. Thermoplastic polyurethane (PU) and poly(ε-caprolactone) (PCL) are blended by solution method. Free length recovery of PU/PCL melt-blend and cell culture have been previously presented in literature [15]. For the characterization of PU/PCL, differential scanning calorimetry (DSC), dynamic-mechanical thermal analysis (DMTA), uniaxial tensile, stress recovery, and combined shape recovery test were used. Shape recovery and stress recovery versus temperature in different states of deformation are discussed in the following [16].

3.2.4.1 Materials Thermoplastic PU (LARIPUR 2102-85AE; COIM Co.) that is PCL ester based is used as a hard segment. PU consists of hard and soft segments respectively based on 4,40 methylenediphenyl diisocyanate, 1,4-butanediol, and PCL with a weightaverage molecular weight of 2000 g/mol. PCL (704105; Sigma Aldrich Co.) was used as a soft segment which also plays the role of switching phase in SMP. The molecular weight-average is 42,500 g/mol, density is 1.145 g/cm3, and melting point is 60 C. For blending, N,N-dimethylformamide (DMF: 1030532500; Merck Co.) is used. The blend ratio of PU/PCL is 7/3 by weight [15].

3.2.4.2 Sample preparation The temperature of DMF reached to 70 C and then PU was added to the solvent and mixed about 12 h. In the next step, PCL was added to the solution and it took about 30 min until the solution became homogeneous. The solution was placed in the vacuum oven for 18 h at 70 C. After evaporation of DMF, PU/PCL was formed as a sheet. Finally, for the shape recovery, DMTA, and tensile/stress recovery test, PU/PCL sheet was cut into rectangular samples. The samples dimension of the shape recovery test was 60 mm 3 5 mm 3 0.4 mm, tensile/stress recovery test was 50 mm 3 5 mm 3 0.4 mm, and for DMTA was 10 mm 3 4 mm 3 0.4 mm. Fig. 3.1 shows the microscopic image of PU/PCL solutions at 25 C in three conditions: (1) after 1 h (homogenous), (2) after 1 week, and (3) after 3 weeks. If the solvent does not evaporate, after a long time, the phases will be separated (soft and hard segment as shown in Fig. 3.1B and C).

3.2.4.3 Differential scanning calorimetry Thermal properties of PU/PCL were determined by DSC according to ASTM D3418-99 standard. Before the test, to eliminate thermal history, samples were heated and cooled from 0 C to 200 C with the rate of 20 C/min. DSC heating rate was 10 C/min and samples were tested at the temperature ranging from 260 C to 220 C.

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81

(B)

(C)

Figure 3.1 Polyurethane/poly(ε-caprolactone) solution at room temperature of 25 C. (A) Solution after 1 h (homogenous), (B) solution after a week (nonhomogeneous), and (C) solution after three weeks (nonhomogeneous) [16].

Figure 3.2 Differential scanning calorimetry thermo-grams of the polyurethane, poly (ε-caprolactone), and polyurethane/poly(ε-caprolactone) (7/3) blend [16].

Fig. 3.2 shows PU, PCL, and PU/PCL DSC test results in the heating cycle where the PCL melting temperature is about 60 C. After blending PU and PCL a transformation behavior can be seen during the heating cycle approximately between 42 C and 68 C. This effect is related to melting PCL crystals inside the PU structure. There are two peaks in the DSC curve of PU/PCL (about 45 C and 60 C) known as melting temperature of PCL crystals.

3.2.4.4 Dynamic mechanical thermal analysis DMTA (Netzsch 242 C) was used to determine the thermo-viscoelastic properties such as storage modulus (E0 ), loss modulus (Ev), and tan delta. Samples (prepared by solution and melt process) were tested in tensile resonant mode at frequency of 50 Hz and heating rate of 5 C/min from 2100/ 2 40 to 100 C. As shown in

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Figure 3.3 Thermo-viscoelastic behavior of polyurethane/poly(ε-caprolactone) by solution and melting method. (A) Storage modulus. (B) Loss modulus [16].

Fig. 3.3, at a low temperature, the storage and loss modulus for solution samples are much higher than the melting samples; however, by heating these values are getting closer. Around 64 C, storage and loss modulus coincide, and heating does not affect the results. Indeed, according to Fig. 3.2, the difference between storage and loss modulus is attributed to the melting point of PCL crystals. It seems that high temperature in the melting process (200 C [15]) affects the thermoviscoelastic behavior of PU/PCL blend. Based on DMTA results, all samples were produced by solution method. Because of the effects of biological environment on the polymeric scaffold, there is a possibility of changing the physical, mechanical or viscoelastic properties of

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Figure 3.4 Dynamic-mechanical thermal analysis results for polyurethane/poly (ε-caprolactone) after immersion in water for 140 h and dry condition. (A) Storage modulus and loss modulus versus temperature. (B) Tan delta versus temperature [16].

the scaffold. Given that the moisture may affect the properties of PU/PCL, for example, scaffold in the blood, it is necessary to examine the moisture effect on the SME of PU/PCL. The effect of moisture on the thermal, mechanical, and viscoelastic behavior of polymers is different. Fig. 3.5 shows the effect of moisture on the storage modulus, loss modulus and tan delta for PU/PCL that were immersed in water from 0 (dry) to 140 h at 25 C. As depicted in Fig. 3.4A, due to the moisture absorption the storage and loss modulus decreased and shifted to the left. These changes decreased at higher temperatures and became very low about 60 C. Despite this fact, water as a plasticizer moves the pikes of tan delta curve (Tg) about 2 C at 60 C and 6 C at 26 C to the left. Tan delta behavior of PU/PCL is not highly dependent on the moisture.

3.2.4.5 Shape memory behavior Fig. 3.5 illustrates the stress recovery for prestretches of 10%, 15%, and 25%. The heating rate was set to 1.5 C/min and Td was 60 C. As shown, recovery start temperature (RST) decreased by increasing the prestretch, but recovery finish temperature (RFT) in all three samples was almost equal. The elastic normal strain after unloading was 3.1%, 2.9%, and 3.4% for samples with prestretch of 10%, 15%, and 25% respectively. Fig. 3.6 shows the relation between the PU/PCL tensile stress (Fig. 3.3), stress recovery, and RST (Fig. 3.5). As demonstrated in Fig. 3.6, by increasing the prestretch from 10% up to 25%, the stress recovery raised 86% and also RST decreased 7.6 C. The decrease in RST in SMPs is favorable in the biological application. The maximum stress recovery in prestretch of 10%, 15%, and 25% are, respectively, 34%, 40%, and 44%. Also, the maximum stress recovery is less than the maximum stress in the uniaxial tensile test. In other words, stress recovery increases at the lower rate than prestretch. Low-stress recovery is an important challenge in SMPs [9]. When Td , Tg, the stress recovery curve is far from sigmoidal function. Fig. 3.7 shows the stress recovery at 25% prestretch for different Td during the heating cycle. As shown in

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Figure 3.5 Stress recovery response as a function of temperature for 10%, 15%, and 25% prestretch and Td of 60 C [16].

Figure 3.6 Comparison between the maximum stress in the tensile test, maximum stress recovery and RST in 10%, 15%, and 25% prestretch (Td 5 60 C) [16].

Fig. 3.7, maximum stress recovery belongs to Td of 30 C and 40 C. Maximum stress recovery in Td of 30 C and 40 C increased approximately 100% compared to Td 5 60 C. Also, minimum of both stress recovery and stress recovery rate belongs to Td 5 20 C. According to Fig. 3.7, by lowering Td from 60 C to 30 C, RST was shifted to the smaller temperature. This behavior is important in biomedical applications. In biomedical applications, control of the maximum temperature of SMPs is necessary because of the vulnerability of tissues in high temperatures. For PU/PCL with

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Figure 3.7 Stress recovery response as a function of temperature for 25% prestretch and Td of 60 C, 50 C, 40 C, 30 C, and 20 C [16].

Figure 3.8 Images of stimulation of polyurethane/poly(ε-caprolactone) which shows angle recovery process. (A) The deformed shape at 15 C. (B
E) Shape recovery process by heating to 60 C [16].

deformation temperature of 50 C, 40 C, and 30 C, maximum stress recovery occurs at about Td. Furthermore, for T . Td by heating, stress recovery decreased and became stable about 60 C until the PCL crystals completely melted. Therefore since the body temperature is 37 C, the most applicable Td for PU/PCL in the in vivo application is 40 C. Fig. 3.8 demonstrates shape recovery of the pretorsion sample at the temporary twist angle of 720 degrees. A straight-strip sample (original shape) was deformed into temporary shape at 60 C and fixed and cooled to 15 C (Fig. 3.8A). The sample

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Figure 3.9 Shape recovery versus temperature in a noncombined loading. (A) Free angle recovery for pretorsion of 720 degrees. (B) Free length recovery for prestretch of 25% [16].

was quickly immersed in water at a specific temperature (25 C, 35 C, 45 C, 55 C, and 60 C) and the deformed strip returned to its original shape (see Fig. 3.8B
E). Fig. 3.9 illustrates free angle and free length recovery in noncombined loading condition for 720 degrees pretorsion, 25% prestretch, and Td of 60 C. It is obvious that the shape recovery behavior in tension and torsion is almost the same so that the RST and RFT in torsion/tension is about 34/40 C and 58/58 C, respectively. As already mentioned, the control of RST and RFT is significantly important because of the body’s biologic constraints and the goal of researchers to approach to the body temperature and close RST and RFT to each other. As shown in Fig. 3.9A and B, a finite elastic strain has been released after unloading (step 3) and a finite deformation remains after heating cycle (step 4).

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Figure 3.10 Series of images showing combined tension
torsion shape recovery process. (A) The deformed shape at 15 C, prestretch and pretorsion are 720 degrees and 25%, respectively. (B
E) Shape recovery process by heating to 60 C [16].

The difference between smart materials behavior in a one-dimensional stress condition with a two-dimensional stress condition is an important point. In the following, the recovery behavior of SMP in noncombined and combined tension
torsion loading is examined. As an example, the combined SME for the pretorsion of 720 degrees and the prestretch of 25% is visualized in Fig. 3.9. PU/PCL in its permanent shape was deformed to temporary shape with the pretorsion of 720 degrees and the prestretch of 25% at 60 C. The deformed shape (Fig. 3.10A) was fixed and then the sample cooled to 15 C rapidly. Similar to the experiment in Fig. 3.8, by heating to 60 C, the deformed strip returned to its original shape approximately (Fig. 3.10B
E). The result of angle recovery in torsion and combined torsion
tension loading, including 720, 540, 360, and 180 degrees pretorsions and 25% of prestretch were examined and are presented in Fig. 3.11A and B. The result showed that angle recovery is affected by prestretch in low pretorsion. By increasing pretorsion, angle recovery behaves almost independently of the prestretch. As shown in Fig. 3.11B, elastic recovery (shear strain) and angle recovery were affected by the normal strain in a low pretorsion. Also, the elastic shear strain in unloading (step 3) decreased by increasing the pretorsion. Fig. 3.12 illustrates the recovery of prestretch in combined and noncombined loading condition. The result reveals that the RST and RFT for the sample in noncombined loading were equal to the samples with pretorsion of 180 and 360 degrees. At higher pretorsions (540 and 720 degrees), the RST shifted to the lower temperatures (about 4 C). In other words, in high pretorsion, the free length recovery is affected by the torsion slightly. Also, increasing the pretorsion to 720 degrees, the elastic and unrecovered strain are much reduced compared to other loading condition. The effect of heating rate on RST and RFT of PU/PCL in pretorsion of 720 degrees is shown in Fig. 3.13. The angle recovery was tested at heating rate of

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Figure 3.11 (A) Angle recovery and (B) normalized angle recovery versus temperature of polyurethane/poly(ε-caprolactone) that submitted to 720, 540, 360, 180, and 0 degrees pretorsions and 25% and 0% prestretch [16].

3.41, 1.68 and 0.84 C/min. In higher heating rates, angle recovery curve shifted to the left and recovery occurred faster. The fastest angle recovery occurs when the PU/PCL immediately immersed in hot water and its temperature increased suddenly (see Figs. 3.8 and 3.10). For example, in Fig. 3.10, at 45 C, angle recovery was about 540 degrees, but according to Fig. 3.11, angle recovery of 220 degrees was carried out (angle recovery increased 245%). This feature can be used to compensate the temperature limitation of SMPs in biological applications. In other words,

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Figure 3.12 Recovery of prestretch versus temperature of polyurethane/poly(ε-caprolactone) that submitted to 720, 540, 360, 180, and 0 degrees pretorsions for prestretch 25% and 0% [16].

Figure 3.13 Angle recovery versus temperature of polyurethane/poly(ε-caprolactone) at 720 degrees prestretch as a function of heating rate [16].

at low temperature, by increasing heating rate, more shape recovery can be obtained. The effect of heating rate on SME can be attributed to the PCL crystallization. In low heating rate, the PCL crystals will have more time to form and crystallization percentage will be increased. By increasing the PCL crystallization percentage, the melting point increased (more energy is needed to melt the PCL crystals) and angle recovery curve shifted to the right.

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Investigation on structural design of shape memory polymers

Interesting structural behaviors could be attained with the aid of the geometrical design of smart materials. Design and manufacturing of materials with special structures can produce structural zero Poisson ratio, negative Poisson ratio, variable stiffness, and negative variable stiffness, which are some examples of the geometrical design of smart materials. As mentioned, there are several parameters to optimize smart materials. In this section, structural design is considered as an effective procedure to arrive at desirable properties. Beams with three different cell types are considered: honeycomb cells, rounded rectangle cells and diamond cells. These beams are produced with constant dimensions and mass. Using a system of nonlinear equations and definite constraints (constant density, dimensions, and mass and different geometrical cells), these beams dimensions have been identified. To examine the effects of different geometries on the shape memory behavior (force recovery and shape recovery) and to show the importance of the geometrical design, thermomechanical tests of three-point bending, and uniaxial tension are conducted. Therefore different structures are considered to manufacture SMPs depending on the desired applications and restrictions of dimensions and mass, which leads to the control of the SMP behavior using the geometrical design, with respect to the governing conditions [15].

3.3.1 Structural (geometrical) design Manufacturing using 4D printing not only can raise the production capacity via the geometrical design, material selection range, and processing design, but also can produce different complex structural components. Smart polymers can be designed with specific behavior and actuation forces, to eventually demonstrate expected structural stiffness [17]. In this section, the effects of geometrical parameters are investigated on the thermomechanical behavior of SMPs. To this end, three beams have been designed and compared with different geometrical cells, namely honeycomb (I), diamond (II), and rounded rectangle (III) under defined constraints (similar external dimensions (length, width, and thickness) and mass). First, a beam with dimensions of 60 3 21 3 6 mm, specific mass, and honeycomb cells (perfect hexagon) is considered (type I) with respect to the dimensional constraints of the thermomechanical test equipment. Stablishing a set of nonlinear equations, dimensions of the geometrical cells for beam type II and III are identified in accordance to beam type I. If a, n1, n2, n3, and n4 are the diamond cell length, the number of rectangles in columns d and e and rows f and g (Fig. 3.14), with the same procedure, one may find dimensions of the perfect diamond cells (II). Thus n1 and n2 are the number of full diamonds with the length of a, which are placed in d and e columns of a beam with the width of 21 mm, defined by mathematical relationships described in Eqs. (3.1) and (3.2). Correspondingly, n3 and n4 stand for the numbers of full

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(I) de f g

85

(II)

5.67

1

3.

4

1

d e f g

(III)

.2 R1

Figure 3.14 2D schematic of beams I, II, and III geometrical cells with dimensions (in millimeter) obtained by solving the systems of nonlinear equations [15].

diamonds with 60 mm of length, which are placed in f and g columns and are written with respect to Eqs. (3.3) and (3.4). Since beams I and II should have equal length, width, and thickness as well as mass, the area of the solid surface of beam I (795.7 mm2) must be equal to the area of the solid surface of beam II, as described in Eq. (3.5). According to the mentioned constraints, a system of nonlinear equations is constructed by Eqs. (3.1)
(3.5) [15]. pffiffiffi pffiffiffi pffiffiffi n1 2a 1 ðn1 2 1Þ 2 1 2 5 21; n1 AN

(3.1)

pffiffiffi pffiffiffi n2 2a 1 ðn2 2 1Þ 2 5 21

(3.2)

pffiffiffi pffiffiffi n3 2a 1 ðn3 2 1Þ 2 5 60

(3.3)

pffiffiffi pffiffiffi n4 2a 1 ðn4 2 1Þ 2 5 60

(3.4)

ðn1 n3 1 n2 n4 Þa2 5 795:7

(3.5)

To obtain the length of sides of the rounded rectangle cells of beam-type III, we 0 0 0 0 0 consider a ; b; r; n1 ; n2 ; n3 ; and n4 as the length, width, fillet radius of corners, number of rectangles in d and e columns and f and g rows, respectively (Fig. 3.14). Equations of beam III are derived following the same idea for the beam II equations, where the only difference is that r and b are defined by parameter of a0 .

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In light of these mentioned constraints, a system of nonlinear equations can be arranged to compute dimensions of beam-type III with the wall thickness of 1 mm as following. The area of the blank surface of beam III is 795.7 mm2 which is equal to the area of the blank surface of beam II or 795.7 mm2. Since the height of all three beam types is 6 mm2, with an equal surface area for three beam types, the total mass will be equal [15].  0 0 0 n1 a 1 1 5 20; n1 AN 0

0

(3.6)

0

n2 a 1 n1 5 22 0

0

0

0

(3.7) 0

n3 b 1 n4 ðb 1 2Þ 5 58; a 5 1:4b

(3.8)

n4 b 1 n3 ðb 1 2Þ 5 58 

0

0

0

n1 n3 v 1 n2 n4

0



0

a b 2 r 2 ð4 2 π

(3.9) 

0

5 795:7; a 5 4:7r

(3.10)

Solving these systems of nonlinear equations, geometrical cells of beams II and III are found. 2D schematics of geometrical cells of beams I, II, and III with computed dimensions are shown in Fig. 3.14.

3.3.2 Method of sample production Material, sample production method, and the effect of the production method on the shape memory behavior are explored in this section.

3.3.2.1 Material As mentioned, 50%
70% of patents in SMPs technology are related to the biomedical field [4]. In this research, we use poly lactic acid (PLA) filament with the commercial brand of “Z.F. Filament Private Limited.” This material is an aliphatic polyester, which is a biodegradable and bioactive thermoplastic. This material consists of renewable resources, such as starch, corn, and sugarcane, which makes it a suitable and ubiquitous material in the medical applications [18]. Zhang et al. [19] in 2016 proposed a technique to produce a lightweight structure with thin composite layers using 3D printing; this material was used as fiber on a thin layer of PLA where its glass transition temperature Tg was reported as 60 C.

3.3.2.2 Additive manufacturing Beam types I, II, and III with the computed dimensions, are built in CATIA. Dimensions of each beam type are assumed to be 60 3 21 3 6 mm. Samples are made with FDM method and using “Quantum 2035” 3D printer. Simple and more advantageous production, short time, and low cost are the main reasons to choose

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Table 3.1 Summary of 3D printer settings [15]. Printer setting

Value

Layer height Shell thickness Fill density Print speed Nozzle size Nozzle temperature Print-bed temperature Filament diameter

0.2 mm 0.4 mm 100% 20 mm/s 0.4 mm 200 C 50 C 1.75 mm

Figure 3.15 The samples produced by FDM method [15].

this method of production. Whereas the initial printing conditions can affect the mechanical properties, all samples are printed with the same conditions. 3D printer specifications are listed in Table 3.1. Printed parts using additive manufacturing are depicted in Fig. 3.15. As one may observe, beams are printed using the FDM method with high quality, while each one has a total mass of 4.8 g.

3.3.2.3 The effect of 3D printing on shape memory polymer response The FDM manufacturing method has a layered structure, which should be considered in the printing of SMP structures. Samples produced using this method normally do not have a homogeneous structure. As shown in Fig. 3.15, layers are distinguishable. This layered structure affects the thermomechanical properties of samples. To eliminate these effects, Raasch et al. [20] conducted some analogous thermomechanical experiments on several additives manufactured samples, before and after the annealing process. Shape memory properties of the annealed samples were improved in some annealing temperatures. Therefore to assure compatibility between the experiments and model predictions, sample production conditions should be the same in both the final structures and those prepared for calibration tests. For example, dynamic mechanical analysis (DMA) tests should be carried out on the printed samples.

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0

00

Figure 3.16 Diagram of E ; E ; and tan δ versus the temperature [15].

3.3.3 Characterization of printed material DMA is a thermomechanical technique to identify the time and temperature dependent mechanical properties of materials. DMA can be conducted with applying a fluctuating force on a sample. In this technique, a sample is placed in a DMA machine with a specific temperature interval, stress frequency, and stress domain. Then, the machine applies sinusoidal stress to the sample within the desired temperature interval. Mechanical response of the material is measured in a controlled tem0 perature E , waste modules  00  environment. Properties such as storage modules E , tan ðδÞ, and material glass transition temperature are achieved in this procedure. This test is done in a temperature range between 220 C and 90 C and the frequency range of 0.5
20 Hz. The goal of DMA test is to find Tg. There are three common approaches to calculate Tg [21]:  0 E onset: it happens in the lowest temperature and is related to the debilitation of the mechanical properties.  00  E peak: it happens in the median temperature and is mostly related to the physical changes. Tan ðδÞ peak: it happens in the highest temperature where the material is in a rubbery-glassy state. Tg measured  0  based on different definitions is illustrated in Fig. 3.16. The storage modulus E
temperature diagram is used to find Tg, which is above 60 C. In this temperature, the mechanical properties of the material are abated.

3.3.4 Thermomechanical shape memory tests According to the studies on SMPs, changes in temperature lead to development of thermal stresses and strains in beams, which affects the behavior of SMPs. However, this effect is much more significant in uniaxial tests than bending [22
25]. Therefore to study this effect experimentally, bending and tensile tests are

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conducted in the present work. In this section, we discuss the conditions in which thermomechanical three-point bending and uniaxial tensile tests are performed.

3.3.4.1 Bending and tensile shape memory test Thermomechanical three-point bending and uniaxial tensile tests are conducted with a loading rate of 10 mm/min for each sample, in temperature interval between 25 C and 60 C. This device measures the required force for each sample in an applied displacement. To determine the shape recovery and force recovery quantitatively, beams are installed in device, constrained or unconstrained, to measure the displacement and force during the shape recovery as well as the force recovery.

3.3.4.2 Shape memory effect in water The shape recovery and force recovery processes are conducted in water. Since the water temperature propagates evenly to every point of the structure, it is reasonable to conduct these thermomechanical tests in water. In these tests, boundary conditions of the system are varying with time, there is a time-dependent temperature distribution in the structure. Therefore the required time for a temperature equivalence of the entire system should be determined. To ensure this, Biot number (Bi) is calculated for the most critical section (i.e., longest face of each geometry which has the minimum contact with the fluid) of the structure. Biot number is a dimensionless number used in transitional heat transfer analyses, defined as the ratio of the internal heat resistance and the surface (boundary layer) heat resistance of a body. Bi 5

hLc Kb

(3.11)

In Eq. (3.11), h; Lc , and Kb are convection heat transfer coefficient, characteristic length, and conduction heat transfer coefficient of the material, respectively. Bi⟩0:1 means that the conduction heat transfer inside the body occurs much faster than the convection heat transfer on the surface of the body. Therefore it is ensured that the temperature is almost uniform throughout its entire structure [26]. Since in the performed experiments Biot number is lower than 0.1, the temperature equivalence is always established in these systems. Thermomechanical shape recovery and force recovery tests steps in three-point bending and uniaxial tensile tests are presented in the following (Fig. 3.17): 1. Heating: the fluid temperature is raised to above the programing temperature (Tp60 C). 2. Loading: an external load is applied to the beams in this temperature (bending load, tensile load). 3. Cooling (shape fixing): the fluid temperature is lowered to the environment temperature (25 C) while the external load still exists. 4. Unloading: unloading occurs at the previous step temperature. In this step, a slight amount of the elastic strain is released.

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Figure 3.17 A schematic of the programming process: (A) shape memory polymer shape recovery in bending and tensile tests and (B) shape memory polymer force recovery in bending and tensile tests [15]. 5. Four initial steps of the shape and force recovery are identical, the fifth step for each of these two processes is, respectively, described as: a. Reheating: with the rise of the temperature with a specific rate from the environment temperature to the programing temperature, the sample remembers its initial shape. This process is called shape recovery. b. Constraining: in this step, the sample is constrained by a pin attached to a load cell, with a predefined configuration. By heating with a specific rate from the environment temperature to the programing temperature, the sample intends to recover its initial shape. Since the sample is constrained via the load cell pin, upon heating, the force exerted to the load cell rises. This process is called force recovery.

3.3.4.3 Shape recovery tests Shape recovery cycle for the three-point bending and the uniaxial tensile tests under 7 mm displacement (different maximum principal strain) for each beam is shown in Fig. 3.17A. Tests are done in a water vessel and the temperature is controlled with warm and cold-water flow. Each sample is placed individually in the vessel containing 60 C water (step 1). 7 mm of displacement is applied through the attached pin of the load cell in the programming temperature (step 2). Then, cold water flow is entered by the inlet pump and the warm water flow is drained by the outlet pump (step 3). In this step, the beam has retained the planned shape and the pin is removed from the loading point (step 4). In the final step, heating the water up to the programming temperature causes the beam to recover its initial shape (step 5a).

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Figure 3.18 Temperature
time diagram from the programming cycle to shape recovery process in (A) the three-point bending test and (B) the uniaxial tensile test [15].

Shape recovery cycle is shown in Fig. 3.18 as time
temperature diagrams for the three-point bending and the uniaxial tensile tests. A microscopic camera and image processing method are used to measure beams deflection and to find the shape recovery of each structure. Fig. 3.19 shows shape recovery versus temperature diagrams for each beam. As shown in Fig. 3.19, in both the bending and tensile tests with 7 mm of applied displacement, beam I has the most shape recovery ratio compared to other two beam types. Shape recovery for all three beams in bending and tensile tests commences at 50 C and it reaches its maximum value at 60 C. Eq. (3.12) is used to compute the shape recovery ratio. In this relation, Xdeformed is the maximum deformation and Xun-recovered is the unrecovered value of the sample deflection at 60 C.   Xdeformed 2 Xun2recovered Shaperecovery 5 3 100% (3.12) Xdeformed

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Figure 3.19 The diagram of shape recovery versus the temperature in equal deflection of 7 mm in (A) three-point bending test and (B) uniaxial tensile test [15]. Table 3.2 Shape recovery in an equal deflection of 7 mm in the bending and the tensile tests [15]. Test

Beam

Xrecovered (mm)

Shape recovery (%)

Bending

I II III I II III

5.52 5.80 6.51 6.00 5.30 6.15

78.89 82.86 93.03 85.71 75.71 87.86

Tensile

The shape recovery ratio for all three beams in bending and tensile tests are calculated and shown in Table 3.2. Xrecovered is the recovered deflection of the sample at 60 C. According to Table 3.2, beam III has the highest shape recovery ratio in bending (93.03%) and tensile (87.86%,) tests. The maximum principal strain in the tensile test for beams I, II, and III is equal to 27.50%, 31.58%, and 20.91%, and in the bending test is equal to 15.53%, 12.81%, and 17.79%, respectively. Comparing the

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Figure 3.20 The shape recovery process for beam III at 25 C, 50 C, 55 C, and 60 C in bending mode [15].

maximum principal strains and the recovery ratio in tensile test, one may find out that lower maximum principal strain leads to a higher recovery ratio. Comparing these values in bending test, it is observed that beam III, despite its higher maximum principal strain, has a higher shape recovery than beams I and II. Hence, one may conclude that there are other factors that affect the shape recovery ratio. The shape recovery process in the bending test for beam III at 25 C, 50 C, 55 C, and 60 C is shown in Fig. 3.20.

3.3.4.4 Force recovery tests Force recovery paths for the bending and tensile tests in the same applied displacement and the same maximum principal strains are shown in Fig. 3.17B for each beam. Procedures of the force recovery test is identical to the shape recovery test up to step 4. The cooling process is done with a certain temperature rate until the temperature reaches to 25 C. In this step, the beam retains its memorized shape. To determine the stress recovery, the attached pin of the load cell is removed from the loading point to unload the elastic strain (step 4). Then, the pin is attached to the loading point (step 5b). In the final step, with heating the water to the programming temperature, the beam intends to restore its initial shape, but the pin impedes the beam deflection. The force that the beam exerts on the load cell to restore its initial shape is called recovered force. The shape memory response is tightly dependent on the applied prestrain. This fact can be validated by performing the force recovery test in two configurations: the same applied displacement (different maximum principal strain) and the same maximum principal strain (10%). Shape memory behavior results in three-point bending and uniaxial tensile tests are shown in Figs. 3.21 and 3.22, respectively. To have a better comparison of the shape memory response in thermomechanical tests, we can analyze the data using the force-temperature diagram, as depicted in Fig. 3.23 and 3.24. Fig. 3.23A and C describes the cooling process of the threepoint bending test (step 3). The largest force value in Fig. 3.23A and C at 60 C is

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Figure 3.21 The complete thermomechanical cycle of the three-point bending test in (A) equal maximum principal strain and (B) equal applied deflection [15].

the preforce required to produce the desired shapes. The force-temperature diagram for re-heating in the three-point bending test is illustrated in Fig. 3.23B and D (step 6). Peak values in Fig. 3.23B and D are the maximum force recovery values. Even though the force recovery value at the start of the heating process is insignificant, at higher temperatures, the force recovery phenomenon begins approximately between 33 C and 37 C. This phenomenon soars to the maximum force recovery value at 40 C
43 C, but with the raise in temperature because of reaching around the glass transition region, the force recovery declines.

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Figure 3.22 The complete thermomechanical cycle of the uniaxial tensile test in (A) equal maximum principal strain and (B) equal applied extension [15].

To calculate the maximum force recovery ratio to the applied preforce in bending and tensile tests, Eq. (3.13) is used. α5

Maximumforcerecovery Preforce

(3.13)

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(B)

Force (N)

Force (N)

(A)

Temperature ( °C)

Temperature ( °C)

(D) Force (N)

Force (N)

(C)

Temperature ( °C)

Temperature ( °C)

Figure 3.23 The force
temperature diagram in the three-point bending thermomechanical test: (A) cooling in equal maximum principal strain; (B) heating in equal maximum principal strain; (V) cooling in equal applied deflection; and (D) heating in equal applied deflection [15].

(B) Force (N)

Force (N)

(A)

Temperature (°C)

Temperature (°C) (D)

Force (N)

Force (N)

(C)

Temperature (°C)

Temperature (°C)

Figure 3.24 The force
temperature diagram in the uniaxial tensile thermomechanical test: (A) cooling in equal maximum principal strain; (B) heating in equal maximum principal strain; (C) cooling in equal applied extension; and (D) heating in equal applied extension [15].

where α . 1 indicates a maximum force recovery higher than the applied preforce. The aim from defining α coefficient is to compare the maximum force recovery (below 50 C) of each beam with the preforce (in 60 C) required to program the corresponding beam. Consequently, the ratio of the recovered force to the programming preforce of each beam can be compared. According to Table 3.3, the value of α in

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Table 3.3 The force recovery and preforce values in the three-point bending test [15]. Type

Beam

Preforce (N)

Maximum force recovery (N)

α

Equal maximum principal strain

I II III I II III

4.19 2.89 1.50 5.46 2.95 2.01

4.47 3.12 2.39 4.77 2.50 1.98

1.07 1.08 1.59 0.87 0.85 0.98

Equal applied deflection

Table 3.4 The force recovery and preforce values in the uniaxial tensile test [15]. Type

Beam

Preforce (N)

Maximum force recovery (N)

α

Equal maximum principal strain

I II III I II III

19.55 5.69 5.60 27.63 18.64 12.97

2.27 1.51 1.53 12.75 6.21 5.66

0.12 0.26 0.27 0.46 0.33 0.44

Equal applied extension

equal maximum principal strain (10%) is larger than the value of α in equal applied deflection (7 mm) for all beam types. This fact demonstrates that the force recovery ratio increases in lower maximum principal strains. Force recovery value for beam I is higher. However, beam III has the highest α and shape recovery values due to the definition of α, which indicates the ratio of maximum force recovery point (at temperatures below 50 C) and the preforce required to program beams (at the temperature of 60 C). By applying higher strains, similar to the “maximum applied deflection” configuration of Table 3.4, higher damage is applied to the polymeric chains and the α value is decreased in comparison to “maximum principal strain” configuration. This fact indicates that force recovery power is abated due to the increase in strain. Like the three-point bending test, the cooling and heating stages in the uniaxial tensile test are shown in Fig. 3.24. The largest force value in Fig. 3.24A and B at 60 C is the preforce required to apply the desired deflections. In the cooling process, as in the bending test, temperature reduction leads to force shrinkage. However, the force recovery is inclined to increase around 45 C. Thermal stress is the main cause of this change in behavior compared to the bending test. In bending and tensile tests, thickness and length of the sample are aligned with the load direction, respectively. Temperature reduction results in a lower change in the sample thickness than in the sample length. Since the tensile test is confined, the thermal stress only appears in the tensile test, which causes the force to increase. Thermal stress in the tensile test affects all steps in the shape memory process, and influences force recovery results.

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The raise in temperature causes thermal stress to drive a tensile force, while the shape memory property persists to recover the sample back to its initial shape. In other words, the aforementioned factors act against each other which results in a lower force recovery in comparison with the bending test. The rise of temperature in the three-point bending test led to the start of the force recovery process in the interval of 33 C
37 C. This process inclines up to the interval of the maximum force recovery, around 40 C
43 C, and then declines as it gets closer to Tg. While, all three beams in three-point bending test start the shape recovery at almost 50 C, they reach to their maximum shape recovery ratio at around 60 C. The difference between the temperature intervals of the start and finish of the shape and force recovery indicates that the required force for the shape recovery is larger than the maximum force in the force recovery in temperatures between 40 C and 43 C. The reduction of the force recovery after the interval of 40 C
43 C is arising from the constraints of the pins attached to the load-cell on the samples. The difference in the temperature interval is caused by the thermal stress in the bending test. This process is delayed to around 48 C and 51 C due to the mentioned factors, as shown in Fig. 3.24. The force recovery inclines with the raise in temperature to the interval of 55 C
57 C, which reaches its maximum, then starts to decline as the sample getting closer to Tg, as in the bending test. With a higher change in the sample length in the programming step, the SME is more dominant than the thermal stress. In the tensile test, the applied displacement in equal maximum principal strains is equal to 3.85 mm, which is lower than the equal maximum displacement configuration (7 mm). Hence, the force recovery is higher in the equal displacement configuration (7 mm). In contrast to the bending test and according to Table 3.4, in the tensile test, we have α , 1 in equal maximum principal strain of 10% for all beams which are lower than the equal applied displacement cases (7 mm). Due to the constraints of the tensile test clamps exerted on the sample in the axial direction, thermal stress is generated by changes in temperature. In larger deformations, SME is more dominant than thermal stress [24], hence, the α value is higher in Table 3.4. In the tensile test, the force caused by the thermal stress acts against the force generated by the SME, which results in the reduction of the maximum force recovery and α values. Therefore if the compression test, instead of the tensile thermomechanical test, had been performed on the beams, forces due to the shape memory behavior and the thermal stress would be aligned, and it was expected α to be enlarged. The value of α in bending and tensile modes of beam I in equal maximum principal strains and deflections is higher than the two other beam types. Since beam II has higher strength due to its honeycomb cell structure, it requires a larger preforce to program, which results in a relatively larger maximum force recovery.

3.4

Shape memory polymer stent as an application

The aim of this experimental section is to investigate the radial compression and radial force recovery of PU/PCL shape memory tubular coronary stents. In this

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part, reviewing the requirements of the fabrication method as well as a suitable selection of materials are the subject of discussion. Two approaches have been examined in the fabrication of the SMP blend, melt-mixing and solutionmixing methods. Furthermore, two types of PU are used as the hard segment in the polymeric blend. By choosing the best fabrication method and the suitable PU, shape memory blend is fabricated, and shape recovery experiment on the polymeric samples is studied. Finally, the radial compression testing and force recovery of the stents are evaluated [27].

3.4.1 Materials The connection of long interwoven linear chains, in two modules (soft and hard), form the molecular structure of SMPs. The presence of side-by-side soft and hard molecular chains having different thermal properties lay the foundation for the phenomenon of having the memory effect. Two types of thermoplastic PU known commercially as LARIPUR 107-93A and LARIPUR 2102-85AE have been used as the hard segment (COIM company). Young’s modulus of PUs LARIPUR 107-93A and LARIPUR 2102-85AE to 50% elongation is 9 and 4.3 MPa, respectively. PCL or PCL (Sigma-Aldrich, Mw 5 90,000
48,000 g/mol, Mn 5 45,000 g/mol) is used as the soft segment in the blending of SMPs. PCL is both biocompatible and biodegradable. To dissolve PU and PCL, N, N-Dimethylformamide (Sigma-Aldrich) solvent is used. Fig. 3.25 shows the schematic molecular structure of PU/PCL SMP.

3.4.1.1 Blending and forming Two methods of melt-mixing and solution-mixing have been utilized for the blending process. The selected weight ratio of PU to PCL in the blend is PU/PCL 5 70/30. In the melt-mixing approach, Brabender W-50-EHT is used as the mixing device. The mixer’s compartment temperature for LPR10793A and LPR210285E-3 are 220 C and 200 C, respectively. After pouring PU into the mixer, the device’s torque became steady in about 3 min. Then, PCL with the weight ratio of PU/PCL 5 70/30 was

Figure 3.25 Schematic structure of polyurethane/poly(ε-caprolactone) shape memory polymer blend. Blue lines indicate polyurethane as a soft segment and red lines show polyurethane as a hard segment [27].

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added to PU and after about 2 min, device’s torque became steady. After blending, a hot press device was used to shape the blend into sheets. The hot press applied a maximum temperature of 250 C and a maximum pressure of 90 bar. The blend was poured inside a metallic molding between two press plates. The range of temperature, time and press pressure for producing the samples are 190 C
245 C, 0.5
7 min, and 0
90 bar, respectively. Four temperatures of 215 C, 225 C, 235 C, and 245 C, seven-time limits (being 1 min apart) and two pressures of 50 and 90 bar were selected for the hot press process. In the solvent method, PU was first added to the solvent DMF in 90 C in a gradual fashion and in 1-min time intervals so that the concentration of the solution reaches 0.083 g/mL. Throughout the dissolution process, the magnet had been stirring the solution. Heating the solution beyond the limit causes PU to burn and change color. After 12 h, PU was completely dissolved in the solution. In the next stage, PCL was gradually added to the solution in 1-min intervals. The dissolution of PCL was completed in about 20 min. The prepared solution was poured into a circular mold with a diameter of 140 mm and to expedite the solution’s evaporation process, the solution was placed inside a vacuum oven. The temperature of the vacuum oven was adjusted to 65 C and the pressure inside to 90 mbar. After about 18 h, the solution was completely evaporated, and polymeric memory sheets were fabricated. Fig. 3.26 represents a blend surface image resulting from the hot press process. According to Fig. 3.26, blend’s surface suffered cracks after the hot press process. The surface image was prepared with a Dino-Lite digital microscope (Fig. 3.26A) and Olympus-GX53 light inverted microscope (Fig. 3.26B). Preparing a piece of PU/PCL blend devoid of the defect through hot press procedure is difficult. The high temperature of the melt-mixing method in relation to the solution-mixing as well as the evaporation of polymer’s absorbed moisture can be counted as main reasons for the low quality of the fabricated samples in the hot press process.

Figure 3.26 The polyurethane/poly(ε-caprolactone) surface image after the hot press. Blends fabricated by the melt-mixing method. (A) Digital microscope image. (B) Light inverted microscopes image [27].

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The quality of the blend samples through the solution-mixing method is far better than the ones generated through the melt-mixing method. The solution-mixing method samples are homogeneous and without any type of defects. Furthermore, loss in the material is much less in the solvent approach than the melt-mixing method. The downside of the solvent approach would be its time-consuming process and the toxicity of the solvent. To do away with problems arising from the hot press shaping, the solution-mixing method was chosen to fabricate the blends.

3.4.2 Stent fabrication PU/PCL/DMF solution were employed as the components of the stents. At first, the solution was brought to 0.15 g/mL concentration at 90 C and then was put into a vacuum oven with 90 mbar pressure for 5 min to get all the vapor out. The presence of may create some voids within the stent in next stages of fabrication. To fabricate stents, solution coated over a cylindrical molding. The process continued by placing the mold inside the solution where the cylinder wall was soaked by the solution. Then, the mold was removed and exposed to 90 C so that DMF evaporate. After 20 min, a high percentage of DMF was evaporated, and SMP took a jelly form. Each stage in the process of the coating causes the formation of an SMP film over the molding 0.08 mm in thickness. By repeating the coating process, the stent’s desirable thickness can be achieved. In the final stage of coating, the mold was placed inside the oven for 12 h at 60 C and 90 mbar. After DMF evaporated completely, SMP had shrunk and its separation from the mold became difficult. To facilitate the removal of the stent, a film of tissue paper and a film of polypropylene with thicknesses 0.2 and 0.08 mm, respectively, were pasted over the molding. The pasting process was done before the process of coating. To remove the stent, the mold was immersed in water, and after the tissue paper had absorbed the moisture, the stent easily came out of the mold. Finally, three stents were made with specifications shown in Table 3.5. The letters L, M and S, are used as abbreviations for the stents. For better comparison of the results, the length of three stents (L, M, and S) is assumed to be the same (25 mm). After selecting the proper material and suitable fabrication process in case of the tubular stent, the geometrical dimensionless parameter of d/t can be considered as an effective factor for a fixed length of the stents. Fig. 3.27A represents the schematic of the fabrication process of the tubular stent and Fig. 3.27B shows the fabricated stent.

Table 3.5 Dimensional and physical specifications of fabricated stents L, M, and S [27]. Stent type

d (mm)

d/t

w (g)

l (mm)

L M S

6.8 6.2 4.2

10 12 15

0.420 0.290 0.118

25 25 25

d 5 internal diameter; l 5 length; t 5 thickness; w 5 weight.

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Figure 3.27 (A) Fabrication process schematic. (B) Fabricated stent after removal from the mold [27].

Figure 3.28 Differential scanning calorimetry results of polyurethane (LARIPUR 210285AE, LARIPUR 107-93A) and poly(ε-caprolactone) [27].

3.4.2.1 Differential scanning calorimetry Thermal properties of the polymers were measured in DSC testing through Netzsch DSC 200F3 device. Thermal analysis was carried out (ASTMD 3418-99) based on the nonisothermal and go-and-return process. 7 mg samples with the rate of 10 C/min in the range of 60 C
220 C heat were tested. Before conducting the experiments, PU and PCL were heated up to 200 C and 100 C, respectively and then cooled down to eliminate the thermomechanical history. Fig. 3.28 depicts the results of DSC experiment for LARIPUR 107-93-A, LARIPUR 2102-85AE, and PCL. The comparison of DSC results indicates that the absorbed energy by 1 g of polymer in 40 C
90 C for LARIPUR 107-93A,

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Figure 3.29 Differential scanning calorimetry result of polyurethane (107)/poly (ε-caprolactone) and polyurethane (2102)/poly(ε-caprolactone) blends [27].

LARIPUR 2102-85AE, and PCL are 101.16 and 1.94 j/g, respectively. In PU LARIPUR 107-93A, the energy absorption peak was not observed in 40 C
90 C. This indicates the significant crystallinity of PCL compared to PU. Therefore it is expected that by adding PCL to PU, the crystallinity of the blend would be raised. Fig. 3.29 represents the results of the DSC experiment for blends of PU (107)/PCL and PU (2102)/PCL. In the heating cycle, the endothermic reaction occurs in 30 C
70 C with two peaks at 48 C and 60 C. Considering the presence of PCL in the PU structure, the endothermic reaction corresponding to the melting crystals of PCL lies in the substrate of PU.

3.4.2.2 Dynamic-mechanical thermal analysis The behavior of thermoviscoelastic blends was checked through DTMA testing. The storage modulus, loss modulus and tan delta as functions of temperature in different frequencies were calculated. DMTA testing was done through the Netzsch 242C device. The loading process took place in a tensile fashion in 40-μm amplitude. The determination of temperature range for the blend stimulation comes from the DMTA testing results. The testing frequencies of 0.5, 1, 10, 20, and 50 Hz and the rate of 5 C/min for the temperature changes were considered. The dimensions of the test sample (width 3 thickness 3 length) are equal to 4 3 0.5 3 10 mm. The results of the storage modulus, loss modulus and tan delta for the fabricated blends PU (107)/PCL and PU (2102)/PCL through the melt-mixing method, are illustrated in Figs. 3.30
3.32, respectively. According to Fig. 3.30, in frequencies over 10 Hz and below 1 Hz, the storage modulus results are too close in a way that

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(A)

(B)

Figure 3.30 The storage modulus of polyurethane/poly(ε-caprolactone) obtained through melt-mixing method. (A) Polyurethane (2102)/poly(ε-caprolactone) blend. (B) Polyurethane (107)/poly(ε-caprolactone) blend [27].

they seem to be independent of the frequency. Likewise, PU (107)/PCL and PU (2102)/PCL in temperatures over 25 C and 50 C, respectively, show similar amounts of the storage modulus and therefore independent of the frequency. Moreover, the storage modulus of PU(2102)/PCL is proportionally far less than PU (107)/PCL, where the maximum value of the storage modulus of PU (107)/PCL is about twice as much as the maximum for the storage modulus of PU(2102)/PCL. As shown in Fig. 3.30, in 40 C
60 C, a sudden decrease has been observed in the storage modulus that can be related to the phase change by way of the endothermic reaction where PCL crystals are melted in the substrate of PU (Fig. 3.29). The loss modulus results for the fabricated blends through the melt-mixing (Fig. 3.31) indicates that at higher frequencies, the loss modulus grows and at lower temperatures, the growth rate of the loss modulus intensifies. For PU (107)/PCL and PU (2102)/ PCL in temperatures exceeding 50 C and 80 C, respectively, changes in the frequency almost do not affect the loss modulus. Weight ratio and the elastic moduli of PU and PCL affect the quality of the shape memory behavior. By melting the available PCL in PU/PCL, a change from the elastic to hyperelastic response is observed. The temperature crossing from the

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(A)

(B)

Figure 3.31 The loss modulus of polyurethane/poly(ε-caprolactone) blend produced by the melt-mixing method. (A) Polyurethane (2102)/poly(ε-caprolactone) blend. (B) Polyurethane (107)/poly(ε-caprolactone) blend [27].

elastic to the hyperelastic state is Tg which plays a key role in the stimulation of SMPs. Therefore the exploration of Tg for the purpose of studying shape memory behavior is of high importance. The peak in the loss tangent graph (Fig. 3.32) corresponds to Tg. According to Fig. 3.32, considering the existence of more peaks in the delta tangent graph of PU (2102)/PCL blend (Fig. 3.32A), it reveals that this blend possesses a better shape memory response compared to PU (107)/PCL blend (Fig. 3.32B). Regarding PU (2102)/ PCL blend, Fig. 3.32A indicates that by increasing the frequency from 1 to 50 Hz, the loss tangent is increased and Tgs have surged by 5 C (charts shift toward the right). Based on Fig. 3.32B, the lack of peak in the frequencies lower than 50 Hz is due to a higher storage modulus in PU LARIPUR 107093A compared to LARIPUR 2102-85AE. Hence, the ratio PU/ PCL 5 70/30 for PU LARIPUR 107-93A does not give a good shape memory behavior, while PU LARIPUR 2102
85AE with the weight ratio of PU/PCL 5 70/ 30, has a suitable shape memory behavior in PU/PCL blend. On the basis of these experimental observations, PU LARIPUR 2102-85AE is used in the following. To observe the effect of the solution-mixing method on the thermo-viscoelastic properties, DMTA test was applied to fabricate PU (2102)/PCL blends via the

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(A)

(B)

Figure 3.32 The loss tangent of the blend produced by the melt-mixing method. (A) polyurethane (2102)/poly(ε-caprolactone). (B) polyurethane (107)/poly(ε-caprolactone) [27].

solution-mixing method. Fig. 3.33 plots the results of the storage modulus and tan delta of the blends. The results indicate a uniform trend regarding the overall behavior of storage modulus as well as the tan delta graph for the fabricated blends by means of the melt-mixing and solution-mixing approaches. Fig. 3.34 provides a comparison between the melt-mixing and solution-mixing methods in 240 C to 100 C with a frequency of 0.5 and 50 Hz for the storage and loss modulus of the fabricated blends. The outcome reveals that the storage and loss modulus of fabricated blends through the solution-mixing method, are much higher than those fabricated by the melt-mixing method. At higher temperatures, the storage and loss modulus graph of the fabricated blends produced by the melt-mixing and solutionmixing methods, approach one another. Moreover, it is observed as a general rule that the graph arising from the results of DMTA testing on the fabricated blends employing the solution-mixing method, is less uneven than the melt-mixing

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Figure 3.33 The storage modulus and loss tangent of polyurethane (2102)/poly (ε-caprolactone) blend by the solution-mixing method [27].

Figure 3.34 Comparison of the storage and loss modulus of polyurethane (2102)/poly (ε-caprolactone) by the melt-mixing and solution-mixing methods at 50 and 0.5 Hz frequency [27].

method’s graph. This is representative of continuous changes of the viscoelastic properties. One reason for differences as well as swinging changes of the viscoelastic curves is the homogeneous distribution of PCL in the substrate of PU via the solution-mixing method compared to the melt-mixing method. Also, there is a

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probability of damage growth in PCL due to the high temperature of the meltmixing approach. Based on the mentioned results, it is concluded that the fabricated blends by the solution-mixing method, have better thermo-viscoelastic and shape memory properties. Thus in the following, the solution-mixing method has been chosen to fabricate stents.

3.4.2.3 Shape memory effect SME of PU/PCL was tested in bending shape recovery and stress recovery in uniaxial tension. In a bending deformation state, shape recovery for the fabricated samples has been performed. Since stent’s two side walls are being exposed to the bending under radial loading, the bending regime has been chosen. Firstly, in 45 C over a pin having 1-mm diameter, the sample was bent down 180 degrees. Then, by preserving the deformation, the sample was immersed at once into 10 C water and held for 30 s. Eventually, the sample was put in a free state and the angle recovery was measured at 25 C, 37 C, 40 C, and 45 C. The dimensions of the samples used here are (5 3 1 3 30) mm. In the stress recovery test, the 25% prestretch was applied to the SMP strip at the deformation temperature of 45 C. Strain rate was applied at 10 mm/min. A setup with the ability to change the temperature of SMP in the range of 5 C
70 C was attached to the tensile machine. The schematic image of the stress recovery test set-up is illustrated in Fig. 3.35. First, the sample was placed in water at 45 C. Then, the strain was applied to the sample and the temperature was brought to 10 C. In the next step, the sample was unloaded, and at the last step, the sample was heated up to 60 C in the two end constraint conditions. By heating to 60 C, SMP stress is recovered. Further details of the stress recovery test and set-up can be found in [16]. The sample dimension for the stress recovery test was (50 3 5 3 0.4) mm. Enhancing the force/stress recovery and control the shape/strain recovery are essential required characteristics of the SMP in specific applications. In this section, the shape memory effect in bending-shape recovery and tension
stress recovery were investigated. Fig. 3.36 presents the angle recovery results for the samples fabricated via the melt-mixing and solution-mixing methods under bending. After exposing the

Figure 3.35 Schematic image of the stress recovery test set-up of shape memory polymer.

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(A)

(B)

Figure 3.36 The results of the angle recovery for the melt-mixing samples and the solutionmixing samples. (A) Pictures of samples taken in 25 C, 37 C, 40 C, and 45 C. (B) Recovered angle and normalized angle recovery versus the temperature [27].

samples to temperatures of 25 C, 37 C, 40 C, and 45 C (Fig. 3.36A), they begin to recover their original shape. The detail of the angle recovery for the melt-mixing sample (MMS) and the solution-mixing sample (SMS) are depicted in Fig. 3.36B. According to Fig. 3.36B, the results indicate that in different temperatures, the angle recovery of MMS is 14 degrees higher than that of SMS. In 37 C, for MMS and SMS 50% and 44%, respectively, were recovered from the originally stored strain. Eventually, by heating up to 45 C, the amounts of angle recovery for MMS and SMS were 77% and 70%, respectively. For stress recovery test, SMP sample was immersed in water. The advantage of the immersing in water is important from two points of view: first, considering the

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Figure 3.37 Stress recovery of polyurethane/poly(ε-caprolactone) blend at prestretch of 25% and deformation temperature of 45 C [27].

water effect on the SMPs in bio-applications and providing uniform and stable heat transfer. According to Fig. 3.37, by heating in constrained condition, the PU/PCL was being stimulated and then stress was recovered. As the temperature increases, the stress recovery increases and then decreases. The maximum stress recovery belongs to about 45 C and equals to 1 N. The reason for the decrease in stress recovery is to apply the strain at a temperature lower than Tg (Td 5 45 C). Therefore the operating range of the stent is at temperatures below 45 C. Because in addition to high temperatures causing damage to the tissue, the force begins to decrease at higher temperatures.

3.4.3 Stent radial compression Radial compression experiments were done over the stents under water-immersed conditions in two temperatures of 37 C and 45 C. Before the experiment, the stents were exposed to the intended temperature for 3 min. Loading and unloading took place with a rate of 0.125 mm/min for up to 50% of the stents’ outer diameters. The loading of the stents has been carried out in three steps. In the first step, the loading process has been executed in two temperatures of 37 C and 45 C with a rate of 0.125 mm/min. The amount of deformation for all three types of the stent has been 50% of the outer diameter. The second step consisted of the stent’s relaxation test in 40 min, while preserving the deformation in the first step. In the last step, unloading was carried through with the same rate as the loading. The loading step is important in examining the stent’s ability regarding opening blood vessels as well as evaluating its performance in a short time. Relaxation process surveys the changes in the stent’s force capability over a period of time assessing its

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Figure 3.38 Three steps of the stent radial compression for stent type L at 37 C that includes loading, relaxation, and unloading [27].

performance in a long time. Fig. 3.38 provides details of the loading, relaxation and unloading process for type L stent in 37 C. According to Fig. 3.38, by imposing a 3.4 mm displacement radially, a force of 7.426 N is applied to the stent. The forcedisplacement graph of the stent is totally linear up to 1 mm displacement. In the continuum of the deformation, however, the stent exhibits slightly nonlinear behavior. In the relaxation test after 40 min, the tolerated force by type L stent dropped from 7.426 N to 6.453 N, a reduction of 13%. After the unloading process, an irreversible deformation of 0.93 mm was generated in the stent. δi Has been created due to the relaxation of PCL chains in the substrate of PU. By increasing the temperature to 60 C, PCL crystals are melted completely and δi is being eliminated. Fig. 3.39A illustrates the stents’ force-normalized displacement graph. The horizontal axis (displacement) has been normalized to the maximum deformation for each stent. Stents’ radial stiffness has been computed with respect to the results of the first and last 25% of the force-displacement graph. Based on the results in Fig. 3.39A, the radial stiffness has been calculated in two temperatures of 37 C and 45 C (Fig. 3.39B). According to Fig. 3.39B, increasing the d/t ratio would result in

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Figure 3.39 Radial compression test for stent types L, M, and S at 37 C and 45 C temperatures. (A) Force versus normalized displacement in three steps of loading, relaxation, and unloading process. (B) The radial stiffness of stents. Radial stiffness has been calculated based on 25% of the imposed displacement [27].

the scaling down of the radial stiffness. The effect of the temperature increase in reducing radial stiffness is also noteworthy. A rise in temperature from 37 C to 45 C would reduce the radial stiffness of stents L, M, and S by 51%, 39%, and 45%, respectively.

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Fig. 3.40A reports the relationship between the maximum force (force at the start of the relaxation) and the force at the end of the relaxation for the stents in temperatures of 37 C and 45 C. Based on the results, by increasing the temperature from 37 C to 45 C, the maximum radial force for stents L, M, and S have been reduced by 43%, 44%, and 51%, respectively. Furthermore, for all the three stents the percentage of the force reduction in the relaxation step at 37 C, is compared to 45 C.

Figure 3.40 (A) The relation between the maximum applied force in the radial compression test and the force after the relaxation. (B) Irreversible deformation ratio to the maximum deformation ðδi =δm Þ for stent types L, M, and S at 37 C and 45 C [27].

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The results of the irreversible deformation ratio to the maximum deformation has been shown in Fig. 3.40B. Given that δi is related to the relaxation of PCL chains, it is observed that the amount of δi for all three stents in 37 C temperature, is higher than 45 C. The least amount of δi belongs to stent M in 45 C and also the lowest relaxation belongs to M as observed from Fig. 3.40A. Comparing Fig. 3.39B and Fig. 3.40B, it is observed that at higher temperatures, the highest percentage of the radial stiffness reduction is proportional to the lowest percentage of δi =δm . In other words, when through the temperature increase the most reduction occurs in the radial stiffness, it is expected that δi would have a little reduction in the wake of the temperature increase (hence, stent type L).

3.4.3.1 Stent force recovery This test’s target is to measure the force recovery of the stent upon heating in a constrained state. The first step consisted of completely deforming the stent in 45 C under the radial compression loading. The loading was swift and took 1 s. Next, by preserving the deformation, the stent was immersed all at once in 6 C water for 1 min. Last, in the free-state, the stent was exposed to the room temperature (25 C) for 5 min. To measure the force recovery, the stent was placed into a set-up with force-and-temperature control capability. The specifics of the set-up have been mentioned in [16]. Increasing the temperature, the stent is inclined to get back to its permanent shape and as a result, begins to regenerate a force. The rate of the temperature increase was set to 1.5 C/min. Fig. 3.41 shows a schematic of the force recovery experiment. Force recovery and RST are important factors in performance evaluation of the SMP stents. Force recovery determines the ability of the stent in opening the clogged blood vessels. In some studies, 37 C temperature and in some others, higher than 37 C has been mentioned as the stimulating temperature for the SMP stents. Therefore this section has focused on the force recovery of stent types L, M, and S in the temperature range of 20 C
45 C (Fig. 3.42A). According to Fig. 3.42A, the rate of the force recovery versus the temperature is the highest for the type L stent (minimum d/t), and the lowest for the stent type S (maximum d/t). RST and force recovery at 37 C
45 C for the three stent types of L, M, and S have been shown in Fig. 3.42B. The lowest and highest RST belong to stent L (equal 25 C) and S

Figure 3.41 Steps of force recovery test of the tubular stent [27].

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Figure 3.42 (A) Radial force recovery for tubular stent versus temperature for three stent types L, M, and S. (B) Force recovery at temperatures of 37 C and 45 C and recovery start temperature for stent types L, M, and S [27].

(equal 34 C), respectively. Regarding stent L, at 34 C the rate of the force recovery growth is higher compared to the temperatures lower than 34 C whereas, in stents M and S, no major changes have occurred. The results show that the maximum force recovery and minimum RST belongs to the stent with a minimum d/t ratio, that is, stent type L.

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Summary and conclusion

1. Combined loading PU/PCL thermoplastic SMP was blended and shape recovery of thin strips in tension
torsion combined loading and also stress recovery in a large uniaxial tension were investigated. In this section, after blending the PU/PCL (7/3), experiments including DSC, DMTA, uniaxial tensile test, stress recovery, and combined shape recovery were carried out. The research aimed for better perception the SMP performance for development of constitutive equation in end-used biomedical applications. A lot of research has been done on uniaxial loading (i.e., tensile, compression and torsion test), but the high-stress recovery and the shape recovery behavior in combined torsion
tension deformation are the important demands that are addressed in this study. Some key conclusions are summarized as follows: Materials and methods: Thermoviscoelastic behavior shows the blend prepared by solution method has higher storage and loss moduli than samples made with the melt. Also, due to moisture absorption the storage and loss moduli decreased and shifted to the left, but the variations of Tg (the peak in tan delta) is not notable. Increasing the temperature has a significant effect on the stress
strain behavior of PU/PCL. Shape recovery: In combined torsion
tension loading, angle recovery is affected by high prestretch, while normal strain recovery is affected by high pretorsion. Minimum elastic recovery belongs to prestretch and pretorsion at 720 and 25%, respectively. Also, in thin PU/PCL strip, increasing heating rate has the greatest effect on reducing RFT (increasing the recovery ratio). Stress recovery: Prestretch and Td are two important parameters in controlling RST and maximum stress recovery. By increasing the prestretch from 10% to 25%, the stress recovery increased 86% and the RST decreased 20%. Also, by setting Td to 40 C, stress recovery increased 100% compared to that of 60 C. The results of this section can be used to develop the SMPs in engineering structures and biomedical sensitive applications such as the stent. In future works, the SME of helical stent experimentally will be investigated and also FEA will be presented. Also, it is attractive to study the SME behavior and biological response of the PU/PCL stent in ex vivo and in vivo conditions. 2. Structural design This section aimed to investigate the effect of geometrical design on the behavior of SMPs. Three beams with honeycomb (I), diamond (II), and rounded rectangle (III) cells were designed with certain constraints such as equal length, width, thickness, and density applied through nonlinear equations. The effect of geometry on the behavior of SMPs (force and shape recovery) in bending and tensile modes was evaluated by performing thermomechanical tests. This effect was determined by producing beams with disparate structures using additive manufacturing method and performing thermomechanical tests to capture different shape and force recovery results. Force recovery, which is reported in this section, is actually the ratio of the recovered force at the final step of the shape memory cycle to the preforce required to deform SMP at the programming step. This ratio is introduced as the α coefficient for each pattern. The definition and comparison of this parameter for different geometries and conditions, under which the research is performed, can provide a clearer quantitative understanding. Hence, it is concluded that how much of the preforce applied to program the beam, has been recovered. Practically, the value of α for beam III is higher than beams I and II.

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123

This fact indicates that beam III has recovered a higher percentage of the preforce exerted for programming compared to beams I and II. While, the magnitude of the recovered force is higher for beam I, according to the results. According to the results, the beam with rounded rectangle cells (III) had the highest shape recovery and force recovery ratio compared to its applied preforce. Shape recovery of beam III in bending and tensile tests was 93.03% and 87.86%, respectively. The beam with honeycomb cells (I) required more preforce in bending and tensile modes to program initially, due to its higher strength, which resulted in a relatively higher force recovery. Different structural designs can be implemented on SMP production to obtain the appropriate shape memory behavior concerning design requirements, depending on the desired applications, and restrictions of dimensions and mass. For instance, if the shape recovery is important or the force recovery is equal or more important than the preforce, or there is confinement on the preforce for the shape programming of SMP, the beam with rounded rectangle cells (III) could be a proper choice for the structural design. If there are no restrictions on the preforce for the initial shape programming of SMP and a large force recovery is required, the beam with honeycomb cells (I) could be a suitable choice for the structural design, due to its strong structure. The aim of this section was to introduce a method to constraint geometric parameters and to examine the effect of structural design on the behavior of SMPs. In conclusion, geometrical design is a vital and effective parameter in the behavior of SMPs. Thus the behavior of SMPs can be controlled by using a suitable geometrical design, according to expected applications. 3. SMP stent as an application

In this section, shape memory behavior of the stents based on PU was examined. In the beginning, the effect of the hard phase and fabrication process regarding the defects in the sample were explored, as well as thermal behavior, thermoviscoelastic behavior and shape recovery of SMP. Finally, three tubular stents were fabricated in varying diameter to thickness ratio (d/t) and their force recovery were inspected. A slight share of the literature and experimentation have been allocated to SMPs stents in comparison with other types of stents. In addition, the lack of experimental measurement for the force recovery in smart polymeric stents has been the downside of previous researches. The investigation of the relaxation behavior, recovery starting temperature (RST) and radial force recovery are the highlighting points of this research. The outcomes of this section are mentioned below: Materials and fabrication: The results of DSC indicated that by adding PCL to PU, an increase is observed in the crystallinity of PU/PCL blend. This phenomenon is related to the melting of PCL crystals in the blend but is not sufficient for realization of shape memory properties. Based on the loss tangent results, the blend that was made from LARIPUR 2102-85AE exhibited shape memory behavior in all frequency range of 0.5
50 Hz (the presence of “peak” in the loss tangent chart). Although the structural similarity of two types of PUs, shape memory properties was not observed in the blend fabricated through LARIPUR 107-93A. The reason being, PU LARIPUR 107-93A has twice as much elastic modulus than PU LARIPUR 2102-85AE. Therefore it is hypothesized that shape memory behavior in the blend PU (LARIPUR 107-93A) /PCL, would arise in higher percentages of

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PCL. The comparison between the melt-mixing and solution-mixing methods revealed that the fabrication approach has a very strong effect on the thermoviscoelastic properties. Thermo-viscoelastic curves for the processed samples prepared by the solution mixing were smooth in contrary to the samples prepared by the melt-mixing method. The smoothness of the thermo-viscoelastic curves is due to the homogeneous mixing and absence of cracks in the solution-mixing method. Shape memory stent: The diameter to thickness ratio (d/t) of the stent play an important role in the relaxation, irreversible deformation and shape memory behavior. Therefore it is expected that mechanical responses and shape memory behavior of the stent to be nonidentical in various blood vessels. The inner diameter of the stent is a function of the vessel diameter, and also, the thickness of the stent is a function of the vessel type and plaque size. The results showed that a decrease in d/ t ratio brought out an increase in the force recovery and a reduction in the stimulation temperature of the tubular stent. Reducing the d/t from 15 to 12 has lowered the force recovery from 0.069 to 0.422 N at 37 C (512% increase) and 0.299 to 1.55 N at 45 C (418% increase). Also, the recovery starts temperature (RST) shifted from 34 C to 25 C. Anyway, the geometrical changes have a great impact on the recovery response of the tubular stents. For different d/t ratios, the tubular stents exhibit different stress and strain distributions which affects the performance of the stents.

References [1] S. Chaterji, I.K. Kwon, K. Park, Smart polymeric gels: redefining the limits of biomedical devices, Progress in Polymer Science 32 (8
9) (2007) 1083
1122. [2] N. Roudbarian, M. Baghani, M. Baniassadi, D. George, A. Mohammadi, An experimental investigation on the energy storage in a shape-memory-polymer system, Energy Equipment and Systems 7 (4) (2019) 309
316. [3] M. Abbasi-Shirsavar, M. Baghani, M. Taghavimehr, M. Golzar, M. Nikzad, M. Ansari, et al., An experimental
numerical study on shape memory behavior of PU/PCL/ZnO ternary blend, Journal of Intelligent Material Systems and Structures 30 (1) (2019) 116
126. [4] B. Dietsch, T. Tong, A review: features and benefits of shape memory polymers (SMPs), Journal of Advanced Materials 39 (2) (2007) 3
12. [5] A.V. Rane, K. Kanny, V. Abitha, S. Thomas, Methods for synthesis of nanoparticles and fabrication of nanocomposites, Synthesis of Inorganic Nanomaterials, Elsevier, 2018, pp. 121
139. [6] A. Gurses, Introduction to Polymer-Clay Nanocomposites, CRC Press, 2015. [7] B. Akhoundi, M. Nabipour, F. Hajami, D. Shakoori, An Experimental study of nozzle temperature and heat treatment (annealing) effects on mechanical properties of high-temperature polylactic acid in fused deposition modeling, Polymer Engineering & Science 60 (5) (2020) 979
987. [8] M. Zarek, M. Layani, I. Cooperstein, E. Sachyani, D. Cohn, S. Magdassi, 3D printing of shape memory polymers for flexible electronic devices, Advanced Materials 28 (22) (2016) 4449
4454.

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[9] Tibbits, S. The emergence of “4D printing.” in TED Conference. 2013. [10] M. Shojaeifard, M.R. Bayat, M. Baghani, Swelling-induced finite bending of functionally graded pH-responsive hydrogels: a semi-analytical method, Applied Mathematics and Mechanics (English Edition) 40 (5) (2019) 679
694. [11] M.R. Bayat, K. Wang, M. Baghani, Visco-hyperelastic swelling and mechanical behavior of tough pH-sensitive hydrogels: theory development and numerical implementation, International Journal of Engineering Science 152 (2020) 103294. [12] M.R. Bayat, R. Dolatabadi, M. Baghani, Transient swelling response of pH-sensitive hydrogels: a monophasic constitutive model and numerical implementation, International Journal of Pharmaceutics 577 (2020) 119030. [13] M.R. Bayat, M. Baghani, Fully-coupled transient fluid-solid interaction simulation of the pH-sensitive hydrogel-based microvalve, International Journal of Applied Mechanics 11 (7) (2019) 1950071. [14] M. Baghani, R. Dolatabadi, M. Baniassadi, Developing a finite element beam theory for nanocomposite shape-memory polymers with application to sustained release of drugs, Scientia Iranica 24 (1) (2017) 249
259. [15] N. Roudbarian, M. Baniasadi, M. Ansari, M. Baghani, An experimental investigation on structural design of shape memory polymers, Smart Materials and Structures 28 (9) (2019) 095017. [16] M. Ansari, M. Golzar, M. Baghani, M. Soleimani, Shape memory characterization of poly (ε-caprolactone) (PCL)/polyurethane (PU) in combined torsion-tension loading with potential applications in cardiovascular stent, Polymer Testing 68 (2018) 424
432. [17] Z. Ding, C. Yuan, X. Peng, T. Wang, H.J. Qi, M.L. Dunn, Direct 4D printing via active composite materials, Science Advances 3 (4) (2017) e1602890. [18] A.J. Lasprilla, G.A. Martinez, B.H. Lunelli, A.L. Jardini, R. Maciel Filho, Poly-lactic acid synthesis for application in biomedical devices—a review, Biotechnology Advances 30 (1) (2012) 321
328. [19] Q. Zhang, K. Zhang, G. Hu, Smart three-dimensional lightweight structure triggered from a thin composite sheet via 3D printing technique, Scientific Reports 6 (2016) 22431. [20] J. Raasch, M. Ivey, D. Aldrich, D.S. Nobes, C. Ayranci, Characterization of polyurethane shape memory polymer processed by material extrusion additive manufacturing, Additive Manufacturing 8 (2015) 132
141. [21] E. Turi, Thermal Characterization of Polymeric Materials, Elsevier, 2012. [22] M. Molaaghaie-Roozbahani, N. Heydarzadeh, M. Baghani, A.H. Eskandari, M. Baniassadi, An investigation on thermomechanical flexural response of shape-memorypolymer beams, International Journal of Applied Mechanics 8 (05) (2016) 1650063. [23] M. Baghani, A. Taheri, An analytic investigation on behavior of smart devices consisting of reinforced shape memory polymer beams, Journal of Intelligent Material Systems and Structures 26 (11) (2015) 1385
1394. [24] M. Baghani, H. Mohammadi, R. Naghdabadi, An analytical solution for shapememory-polymer Euler
Bernoulli beams under bending, International Journal of Mechanical Sciences 84 (2014) 84
90. [25] M. Baghani, R. Naghdabadi, J. Arghavani, A semi-analytical study on helical springs made of shape memory polymer, Smart Materials and Structures 21 (4) (2012) 045014. [26] J. Holman, Heat Transfer, McGraw-Hill, New York, 2010. [27] M. Ansari, M. Golzar, M. Baghani, M. Abbasishirsavar, M. Taghavimehr, Force recovery evaluation of thermo-induced shape-memory polymer stent: material, process and thermo-viscoelastic characterization, Smart Materials and Structures 28 (9) (2019).

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Shape memory polymers: constitutive modeling, calibration, and simulation

4

Chapter outline 4.1 Introduction 127 4.2 Macroscopic phase transition approach 128 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7 4.2.8

Strain storage and recovery 130 Thermodynamic considerations 132 Extension of the model to the time dependent regime 133 Numerical solution of the constitutive model 135 Consistent tangent matrix 137 Hughes
Winget algorithm: large rotation effects 138 Material parameters identification 139 Material model predictions 141

4.3 Shape memory polymer constitutive model through thermo-viscoelastic approach 144 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7

Strain-dependent part of the stress 144 Time-dependent part of the stress 144 Temperature-dependent modification of the stress 145 Solution of shape memory polymer’s response in a shape memory polymer path 146 A time-discretization scheme for constitutive equations 148 Material parameters identification 149 Solutions development for torsion
extension of SMP 150

4.4 Summary and conclusion References 156

4.1

156

Introduction

In recent years many researchers have attempted to describe the thermomechanical behavior of shape memory polymers (SMPs). Though there are various approaches for describing SMPs behavior, such as mesoscale method and molecular dynamics simulation procedure, because of high computational cost and complexity, they are not very common in modeling SMPs. On the other hand, the phase transition (macroscopic phenomenological) [1,2] and the thermovisco-elastic approaches are popular in literature. An important feature of different constitutive model approaches is the procedure required to calibrate material model parameters [3]. Identifying material properties in phenomenological models involves calibration of the model with standard tests of shape memory effect (both in shape and force recovery), while this is not necessary for model developed in the framework of viscoelasticity. Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00002-1 © 2023 Elsevier Inc. All rights reserved.

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In this chapter, in addition to development of constitutive equations based on major conventional methods for modeling SMPs (i.e., phase transition and thermovisco-hyperelasticity), the necessary calibration steps have been studied and some examples have been presented. Although FEM has been used in most simulations of smart polymer materials, some especial cases can be solved using analytical methods at much lower computational costs. Hence, analytical and semianalytical methods for simulating materials and their formulation have been discussed. Macroscopic phase-transition approach constitutive models are commonly implemented in finite element framework and require subroutine coding. On the other hand, there are some constitutive models based on thermo-viscoelastic theories which are already available in commercial FEM software. In this chapter, one constitutive model based on the phase transition approach and the other one based on the thermo-visco-hyperelasticity are introduced.

4.2

Macroscopic phase transition approach

Macroscopic models are built on phenomenological thermodynamics and/or directly curve fitting experimental data. Most of them are based on the common phasetransition phenomenon in SMPs (glassy to rubbery phases and/or vice versa). Researchers typically use the glassy volume fraction as an internal variable and different mathematical functions to describe a smooth transition. These constitutive models are generally more suitable for engineering applications due to their simplicity and fast computations, but they can only describe the global mechanical response, while all the microscopic details are usually ignored; since the phasetransition is built on experimental data, the models are also quite simple and accurate. One can summarize the advantages and disadvantages of the different approaches by the following general remarks. The models developed with the macroscopic approach are generally easy to use and allow quick computations, but they are less predictive. The micro-level approach is more suitable for the development of fundamental studies than for the quantitative description of macroscopic behaviors. From a macroscopic point of view, shape memory effect can be characterized in a stress
strain
temperature diagram as illustrated in Fig. 4.1. The thermomechanical cycle starts at a strain- and stress-free state while the temperature is Th (high temperature) (point a, permanent shape). At this point, a purely mechanical loading is applied to SMP and the material demonstrates a rubbery behavior up to point b. At point b, strain is held fixed and the temperature is decreased until the rubber-like polymer drastically turns into a glassy polymer at the low temperature Tl (point c, fixed shape). In fact, in the neighborhood of the transition temperature Tg, SMP exhibits a combination of rubbery and glassy behaviors. Subsequently, the material is unloaded. Regarding much higher stiffness of the glassy phase in comparison to the rubbery phase, after unloading, strains change slightly (point d). Finally, the temperature is increased up to Th. It is seen that the strain will relax, and the

Shape memory polymers: constitutive modeling, calibration, and simulation

129

Figure 4.1 Stress
strain
temperature diagram illustrating the thermomechanical behavior of a pretensioned shape memory polymer under different strain or stress recovery conditions [1].

Figure 4.2 Equivalent representative volume element. (A) At T 5 Tl , dominant glassy phase. (B) At T 5 Tg , combination of both phases. (C) At T 5 Th , dominant rubbery phase [2].

original permanent shape can be recovered (point a). It is noted that based on the quality of the SMP, in some practical cases, some residual strain may remain in the SMP (point a0). This cycle is called stress-free strain recovery in SMP applications. In practice, other types of recoveries may happen. If at point d, the strain is fixed and the temperature is increased, the fixed-strain stress recovery (point e) happens. The dotted line in Fig. 4.1 illustrates the mentioned behavior schematically. In this section, as an example of constitutive models developed based on the concept of the phase transition, we briefly introduce the model proposed by Baghani et al. [2] and then extended to the time-dependent regime in Ref. [1]. An equivalent representative volume element (RVE) of the material composed of a glassy and a rubbery phase is illustrated in Fig. 4.2. Assuming small strains, the mixture rule can be used in the RVE to decompose additively the total strain as: ε 5 ϕr εr 1 ϕg εG 1 εT

(4.1)

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where εr and εG stand for the elastic strain in the rubbery and glassy phases, respectively, while εT denotes the thermal strain and is defined by αT ðT 2 Th Þ where αT is the effective thermal expansion coefficient [4]. Also, ϕr and ϕg are volume fractions of the rubbery and glassy phases, respectively with constraint ϕr 1 ϕg 5 1. It is assumed that ϕr and ϕg are only functions of temperature [5,6].

4.2.1 Strain storage and recovery We now consider the phase transformation (rubbery to glassy and vice versa) in the RVE. During cooling, the strain in the newly generated glassy phase, already been in the rubbery phase, had experienced the εr previously. Then ϕg εG is defined as: ϕg εG 5 ϕg ðεg 1 εg Þ 5 ϕg ðεg 1

1 Vg

ð εr dνÞ 5 ϕg εg 1 Vg

1 Vp

ð εr dν

(4.2)

Vg

where Vg and Vp are volumes of the glassy phase and the SMP segment, respectively. In Eq. (4.2), strain in the glassy phase is divided into two parts: strain in the old glassy phase, εg , and strain in the newly generated glassy phase, ε g . It can be reformed Eq. (4.2) as: ð ϕg εG 5 ϕg εg 1

εr dϕg 5 ϕg εg 1 εis

(4.3)

Consequently, in a cooling process εis is defined as: ð εis 5

εr dϕg

(4.4)

Such a strain storage in the cooling process has previously been introduced by Liu et al. [7]. In contrast to the cooling process, in a heating process, the stored strain in the glassy phase should be relaxed. This can be mathematically shown as: ϕg εG 5 ϕg ðεg 1 εg Þ 5 ϕg ðεg 1

1 Vg

ð

εis 1 dνÞ 5 ϕg εg 1 V ϕ p Vg g

ð

εis dν Vg ϕg

(4.5)

Eq. (4.5) can be written in a more compact form as: ð ϕg εG 5 ϕg εg 1

εis dϕ 5 ϕg εg 1 εis ϕg g

As a result, in a heating process εis is defined as: ð is ε εis 5 dϕ ϕg g

(4.6)

(4.7)

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131

Note that the strain storage/release occurs only in the glassy phase. However, by definition, εis is assigned to the whole RVE. Hence, a division by ϕg appears in the integrals of Eqs. (4.5)
(4.7). From Eqs. (4.4) and (4.7), it is found that εis is a fully thermal-driven variable. Moreover, Eqs. (4.4) and (4.7) can be combined as [8,9]: ð

ð

εis 5 k1 εr dϕg 1 k2

ε dϕ ; ϕg g is

8 < k1 5 1; k2 5 0; k 5 0; k2 5 0; : 1 k1 5 0; k2 5 0;

T_ , 0 T_ . 0 T_ 5 0

(4.8)

where ð Þ 5 @t@ represents the derivative with respect to time. Also, k1 and k2 are constants used to identify the heating and cooling processes. Evolution Eq. (4.8) can be recast in the rate form as: εis ε_ 5 ϕg T_ k1 ε 1 k2 ϕg is

0

!

r

(4.9)

@ denotes the derivative with respect to temperature. A schematic rheowhere ð0 Þ 5 @T logical illustration is shown in Fig. 4.3 to follow the derivation of the equations in this section. It is worth mentioning that the total strain is the weighted summation of the strain in each phase (weights are shown under each element in Fig. 4.3). The convex free-energy density function ψ for an SMP is expressed here. Based on the mixture rule, the following form for the energy function can be introduced:

ψðε; T; ϕg ; εr ; εg ; εis Þ 5 ϕr ψr ðεr Þ 1 ϕg ψg ðεg Þ 1 ψλ ðε; T; ϕr ; ϕg ; εr ; εg ; εis Þ 1 ψT ðTÞ

(4.10) where ψr and ψg denote Helmholtz free-energy density functions of the rubbery and glassy phases, respectively. ψT represents the thermal energy and ψλ enforces the kinematic constraint (4.1) as: h
 i ψλ ðε; T; ϕr ; ϕg ; εr ; εg ; εis Þ 5 λ: ε 2 ϕr εr 1 ϕg εg 1 εis 2 εT in which λ is a (tensorial) Lagrange multiplier.

Figure 4.3 Schematic rheological illustration of the proposed constitutive model.

(4.11)

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Computational Modeling of Intelligent Soft Matter

4.2.2 Thermodynamic considerations The second law of thermodynamics is applied in the sense of Clausius
Duhem inequality to derive constraints on the evolution equation [10] as:   Dmech 5 σ:_ε 2 ψ_ 1 ηT_ $ 0

(4.12)

where η represents the entropy. Substituting Eq. (4.10) into Eq. (4.12), the following inequality can be obtained as: @ψg @ψg 0 @ψr r :_ε 2 ϕg g :_εg 2 ϕ T_ r @ε @ε @ϕg g h
 i 0 2 ψT T_ 2 ε 2 ϕr εr 1 ϕg εg 1 εis 2 εT :λ_
 h i 0 0 2 ε_ 2 ϕr ε_ r 1 ϕg ε_ g 1 ϕg ðεg 2 εr ÞT_ 1 ε_ is 2 εT T_ :λ 2 ηT_ $ 0 σ:_ε 2 ϕr

(4.13)

Inequality (4.13) must be fulfilled for arbitrary thermodynamic processes, that is, _ and T_ . For arbitrary choices of the variables ε_ , ε_ r , ε_ g , λ, _ for arbitrary ε_ , ε_ r , ε_ g , λ, and T_ can be arrived at [11]: σ5λ5

@ψg @ψr 5 g r @ε @ε

(4.14a)

ε 5 ϕr εr 1 ϕg εg 1 εis 1 εT T0

η 5 σ: ε

1 ϕ0g

(4.14b)

εis ε 1 ðk1 2 1Þε 1 k2 ϕg g

r

!! 0

2 ψT 2

@ψ 0 ϕ @ϕg g

(4.14c)

Eqs. (4.14a) and (4.14b) are consequences of the basic assumption of existence of the rubbery and glassy phases, simultaneously, together with satisfying the second law of thermodynamics. Employing the rule of mixture, the Helmholtz free energy density function is defined. This definition requires that the stress in both phases to be the same. In the following, quadratic forms are used for the freeenergy density functions as: ψr ðεr Þ 5

1 r 1 ε :kr :εr ; ψg ðεg Þ 5 εg :kg :εg 2 2

(4.15)

where kr and kg are fourth-order positive definite elasticity tensors of rubbery and glassy phases, respectively. In the case of isotropic materials, it is only required to know elastic modulus E, and Poisson’s ratio ν, of each phase. A numerical integration scheme is utilized in an implicit form to solve the nonlinear system of Eqs. (4.9) and (4.14). It is quite important to remark that small-strain constitutive models can be successfully applied to the solution of large displacement

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133

problems, where strains are small though rotations can be arbitrarily large. The numerical manipulation of this procedure follows a standard approach in the literature [12], thanks to the well-known Hughes
Winget algorithm [13]. Thus to capture such effects in the presented model, the Hughes
Winget algorithm can be employed.

4.2.3 Extension of the model to the time dependent regime In this section, the model proposed in latter section is extended to the timedependent regime, based on the continuum thermodynamic considerations [1]. We focus on a phenomenological or macromodeling approach, which is able only to give an average representation of the phenomena occurring at the material micromechanical level. With the aim of improving the SMP mechanical properties, for example, strength and stiffness, hard particles (e.g., glass microballoons) can be dispersed in SMP matrices [14
18]. To capture the behavior of this family of composites, we assume the material as a mixture of a hard segment and an SMP matrix. Using the rule of mixture, we then incorporate the effect of hard segment in the constitutive model. As the stiffness of the hard segment is much higher than the stiffness of the rubbery or glassy phases, the obtained composite generates a higher value of stress in the fixed-strain stress recovery which is a desirable feature in engineering applications. Moreover, taking into account the importance of timedependent behavior in modeling of polymer materials, three tensorial internal variables (corresponding to the rubbery, glassy, and hard phases) are incorporated into the presented constitutive model. The material domain is separated into SMP and hard segments. Furthermore, the SMP segment is divided into glassy and rubbery phases and introduce the equivalent RVE of the material as schematically illustrated in Fig. 4.4. The total strain is additively decomposed into four parts: SMP, hard, irreversible and thermal parts. ε 5 φp εp 1 φh εh 1 εi 1 εT

(4.16)

Figure 4.4 Equivalent representative volume element for shape memory polymer and hard segments: Dots represent the hard segment in all cases. Left: at T 5 Tl , dominant glassy phase. Middle: at T 5 Tg , combination of all phases. Right: at T 5 Th , dominant rubbery phase [1].

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where superscripts p and h stand for the SMP and hard segments, respectively. Also, εi is the irreversible strain, while εp describes the strain in SMP segment of the RVE. φp and φh denote the volume fractions of the SMP and hard segments of the RVE, respectively, with the constraint φp 1 φh 5 1. Furthermore, as discussed in Eqs. (4.1) and (4.3), the strain in the SMP segment is decomposed into two components: εp 5 ϕr εr 1 ϕg εG

(4.17)

Based on the experimental observations, time-dependent effects are of importance in constitutive modeling of polymeric materials. It should be noted that εis is employed only to reproduce storage/release phenomenon and is not capable of capturing the time-dependent behavior of SMPs observed in experiments. In fact, to capture the viscoelastic behavior of SMPs, another decomposition should be employed. Utilizing the small strain assumption, strains are additively decomposed in the glassy, rubbery, and hard phases as: εβ 5 εeβ 1 εiβ ; β 5 r; g; h

(4.18)

where the superscripts eβ and iβ denote the elastic and inelastic (viscous) parts of the strain in all phases, respectively. For instance, εer and εir denote elastic and inelastic (viscous) strains in the rubbery phase. Note that the viscoelastic internal variables εiβ ðβ 5 r; g; hÞ, are employed to describe just the material time-dependent behavior and are not able to describe the storage and release mechanisms in SMPs. A schematic rheological illustration as shown in Fig. 4.5 could be helpful to follow the derivation of the equations in this section. The convex free-energy density function for an amorphous SMP material is expressed. Based on the mixture rule, the following form for the energy function is expressed as: ψðε; T; φp ; φh ; ϕg ; εer ; εg ; εeg ; εh ; εeh Þ 5 φh ψh ðεh ; εeh Þ 1 ψT ðTÞ h i 1 φp ϕr ψr ðεr ; εer Þ 1 ϕg ψg ðεg ; εeg Þ

(4.19)

where ψr , ψg , and ψh are Helmholtz free-energy density functions of the rubbery, glassy and hard phases, respectively. In order to enforce the kinematic constraints

Figure 4.5 Schematic rheological illustration of the proposed constitutive model [1].

Shape memory polymers: constitutive modeling, calibration, and simulation

135

(4.16) and (4.17) in the formulation, the method of Lagrange multipliers is used and the following term ψλ is added to the free energy (4.19): ψλ ðε; T; φp ; φh ; ϕr ; ϕg ; εr ; εg ; εis ; εh ; εi Þ 5 h
 i λ: ε 2 φp ϕr εr 1 ϕg εg 1 εis 2 φh εh 2 εi 2 εT

(4.20)

Therefore the free energy function is re-expressed as follows to consider the kinematic constraints (4.16) and (4.17) in the formulation: h i ψ 5 ψ 1 ψλ 5 φh ψh 1 ψT 1 φp ϕr ψr 1 ϕg ψg 1 ψλ

(4.21)

It is also defined as: neq eβ β ψβ ðεβ 1 εeβ Þ 5 ψeq β ðε Þ 1 ψeβ ðε Þ; β 5 r; g; h

(4.22)

where eq and neq stand for equilibrium and nonequilibrium parts of  the superscripts  ψβ εβ 1 εeβ . In the present model, it is emphasized that the internal variables are ϕg , εir ; εig ; εih ; εi , and εis . Thus evolution equations should be defined for the internal variables in the context of continuum thermodynamics. It is noted that a prescribed evolution equation for ϕg is used. After considering the thermodynamic effects can be written as: @ψeq @ψneq @ψeq @ψneq @ψeq @ψneq g g r r h h 1 5 1 5 1 @εr @εer @εg @εeg @εh @εeh
 ε 5 φp ϕr εr 1 ϕg εg 1 εis 1 φh εh 1 εi 1 εT

σ5λ5

0

0

0

η 5 2 ψT 1 σ: εT 1 φp ϕg

εis εg 1 ðk1 2 1Þεr 1 k2 ϕg

!! 2

(4.23a) (4.23b) @ψ 0 ϕ @ϕg g

(4.23c)

and the evolution laws for the viscous internal variables can be chosen as: ε_ ir 5

neq 1 @ψneq 1 @ψg 1 @ψneq 1 ig ih r h _ _ ; ε 5 ; ε 5 ; ε_ i 5 σ er eg eh ηr @ε ηg @ε ηh @ε ηi

(4.24)

in which ηr , ηg , ηh , and ηi are positive viscous coefficients of the rubbery, glassy, hard, and irreversible parts, respectively.

4.2.4 Numerical solution of the constitutive model The main task is to apply an appropriate numerical time-integration scheme to the evolution equations of the internal variables. It is noted that, in general, implicit

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schemes are preferred because of their stability at larger time step sizes. Moreover, the present section provides some details about the stress update and the computation of the consistent tangent matrix. These are two main points where the material model is directly connected to the finite element solution procedure. Accordingly, the time interval of interest ð0; tÞ is subdivided in sub-increments and the evolution problem is solved over the generic interval ðtn ; tn11 Þ with tn11 . tn . To simplify the notation, an indication is used with the subscript n a quantity evaluated at time tn and with no subscript a quantity evaluated at time tn11 . Furthermore, the increment of time is shown by Δt. Assuming to know the solution and the strain εn at time tn as well as the strain ε at time tn11 , the stress and the internal variables should be updated from the deformation history. The quadratic forms are introduced for the free-energy density functions as follows:  β  1 β eq β  eβ  1 eβ neq eβ ψeq ε 5 ε :kβ :ε 5 ε :kβ :ε ; ψneq β ε β 2 2

(4.25)

neq where keq β and kβ are fourth-order positive definite tensors of equilibrium and nonequilibrium parts of each phase. In the case of isotropic materials, it is only required to know elastic modulus of equilibrium part, Eβeq , elastic modulus of nonequilibrium part, Eβneq , and Poisson’s ratio, ν β ,of each phase. Note that the same Poisson’s ratio is used for equilibrium and nonequilibrium tensors. A backward-Euler integration algorithm is applied to the model: β β εis 5 Sβ :εiβ n 1 J :ε

Δt neq k S 5 I1 ηβ β β

(4.26)

!21 ; Jβ 5

Δt β eq S :kβ ; β 5 r; g; h ηβ

(4.27)

where I is the symmetric fourth-order identity tensor. Moreover, εi 5 εin 1

Δt σ ηi

(4.28)

Regarding Eq. (4.9), the discrete form of the evolution equation for the stored strain is obtained as: εis 5 εisn 1 Δϕg ðk1 εr 1 k2

εis Þ ϕg

where Δϕg 5 ϕg 2 ϕgn . Consequently, it is obtained that: !21 n o k 2 εisn 1 Δϕg k1 εr εis 5 12Δϕg ϕg

(4.29)

(4.30)

Shape memory polymers: constitutive modeling, calibration, and simulation

137

Finally, it is concluded that: ε

is

5 k3 εisn

1 k4 ε ; r

k2 k3 5 12Δϕg ϕg

!21 ;

k4 5 ks1 k3 Δϕg

(4.31)

4.2.5 Consistent tangent matrix In this section, the construction of the tangent matrix has been reported. Using the discrete form of Eq. (4.23), it can be written:  neq  β β iβ ; β 5 r; g; h σ 5 keq β :ε 1 kβ : ε 2 ε

(4.32)

Moreover, it can be written as: β σ 5 keq β :ε 1

 
eq ηβ β  β ηβ  β ηβ  iβ ε 2 εiβ J :ε 1 S 2 I :εiβ n 5 kβ 1 n Δt Δt Δt

(4.33)

Eq. (4.33) is recast in a more compact form as: σ 5 ℍβ :εβ 1 Qβn ℍβ 5 keq β 1

 ηβ β ηβ  β J ; Qβn 5 S 2 I :εiβ n ; ðβ 5 r; g; hÞ Δt Δt

(4.34) (4.35)

Also, the strains εg , εh and εi can be written in terms of εr as:   21  21  εg 5 ℍg : ℍr :εr 1 Qrn 2 Qgn εh 5 ℍh : ℍh :εr 1 Qrn 2 Qhn εi 5 εin 1

 Δt  r r ℍ :ε 1 Qrn ηi

(4.36) (4.37)

Substituting Eqs. (4.36) and (4.37) into Eq. (4.23b), it would be obtained:
  21  ε 5 φp ϕg ℍg : ℍr :εr 1 Qrn 2 Qgn 1 ϕr εr 1 ks3 εisn 1 ks4 εr   Δt  r r 21  1 ϕh ℍh : ℍr :εr 1 Qrn 2 Qgn 1 εT 1 εin 1 ℍ :ε 1 Qrn ηi

(4.38)

Solving Eq. (4.38) for εr , the following relation is obtained: A:εr 5 b

(4.39)

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Computational Modeling of Intelligent Soft Matter

where  Δt r  ℍ 1 ϕr 1 ks4 I; ηi
   Δt r 21  21  Q b 5 ε 2 φp ϕg ℍg : Qrn 2 Qgn 1 ks3 εisn 2 ϕh ℍh : Qrn 2 Qhn 2 εT 2 εin 2 ηi n 21

21

A 5 φp ϕg ℍg :ℍr 1 ϕh ℍh :ℍr 1

(4.40) Now, an expression is found for stress using Eq. (4.33). It can be written:     21 21 21 r σ 5 ℍr : φp ϕg ℍg :ℍr 1ϕg ℍg :ℍr 1 Δt I ℍ 1 ϕ 1k : s4 r ηi 0 1
  21  ε 2 φp ϕg ℍg : Qrn 2 Qgn 1 ks4 εisn . . . B C B  r  1 Qrn Δt r C h h21 T i @ Qn A ? 2 φh ℍ : Qn 2 Qn 2 ε 2 εn 2 ηi

(4.41)

Also, the consistent tangent matrix is calculated as follows: CTan 5

   21 dσ Δt r  21 21 5 ℍr : φp ϕg ℍg :ℍr 1ϕh ℍh :ℍr 1 ℍ 1 ϕr 1ks4 I dε ηi

(4.42)

4.2.6 Hughes
Winget algorithm: large rotation effects In most engineering applications, SMP components are used to provide a force over some large displacement via the shape memory effect, for example, in artery stents. Structural components, such as beams and torque tubes, usually exhibit large global displacements with small strains. This should be remarked that small-strain constitutive models can be successfully applied to the solution of such large displacement problems, where strains are small though rotations can be arbitrarily large. From theoretical point of view, it is done by replacing the Cauchy stress, σ, and infinitesimal strain, ε, tensors by the second Piola
Kirchhoff stress, and Green
Lagrange strain tensors, respectively, in the formulation [19]. However, the numerical manipulation of this procedure follows a standard approach in the literature [12], thanks to the well-known Hughes
Winget algorithm [13]. In fact, for the large displacement problems (where the updated Lagrangian formulation is employed) the timediscretization of the rate form equations should satisfy the objectivity requirements (such algorithms are called incrementally objective). For a typical Jaumann objec tive rate form equation A 5 Yðt; T; A; . . .Þ the incrementally-objective time-discrete form via Hughes
Winget algorithm is obtained as [13]: An11 5 QAn QT 1 ΔtYðtn11 ; Tn11 ; An11 ; . . .Þ

(4.43)

Shape memory polymers: constitutive modeling, calibration, and simulation

139

where A and Y are second-order tensors in the current configuration. Also, Q is the incremental rotation tensor defined as [13]: 

1 Q 5 12 ω 2

21 

1 11 ω 2

 (4.44)

where w is the vorticity tensor (asymmetric part of the velocity gradient tensor). Finally, it should be highlighted that, to use a small-strain constitutive model for solution of large displacement problems in software ABAQUS/Standard, the user should activate the option NLGEOM. However, in this case, only stress and strains are automatically rotated incrementally by the software.   The rotation of userdefined tensorial internal variables εir ; εig ; εih ; εi and εis are left to the user, while the tensor Q is passed through the UMAT as a 3 3 3 matrix Drot. Also for the analysis of finite strain problems and formulation based on macroscopic approach, Baghani et al. (2012) model [20] can be useful.

4.2.7 Material parameters identification Some guidelines on material parameters identification is presented here. 1. Characteristic temperatures Tl , Tg , and Th are measured using DMA (dynamic mechanical analysis) tests. 2. It is observed that the thermal strain (under no external load) exhibits a nonlinear behavior as the temperature traverses through the glass transition region [7]. Different nonlinear relations have been utilized in the literature to capture this effect. Commonly, a secondorder polynomial expression for the thermal strain gives a good approximation. As an example, such a curve fitting has been performed in the following form for experimental data reported by [7] as follows: εT 5 α1 ðT 2 Th Þ 1 α2 ðT 2 2 Th2 Þ

(4.45)

where α1 and α2 are material parameters and have been calculated using a curve fitting method. Fig. 4.6 (left) shows the fitted curve and the experimental data reported in [7] for the thermal strain. 3. Volume fractions φh and φp are constant parameters. To specify the evolution equation for the volume fraction of the glassy phase ϕg as a function of temperature, Eq. (4.9) during heating in a 1D stress-free strain recovery (while the stored strain after unloading is ε0 ) reduces to: ε_ is 2

ϕ_ g is ε 5 0; εis jϕg 51 5 ε0 .εis 5 ε0 ϕg ϕg

(4.46)

4. Eq. (2.52) shows that the expression for ϕg should follow the same trend as stored strain εis . Thus using a curve fitting method, a function which fits the experimental data in the best way is defined. A combination of exponential or power terms normally leads to a successful curve fitting. For instance, such a curve fitting has been utilized for the

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(A)

(B)

Figure 4.6 (A) Thermal expansion strain as a function of temperature. (B) Stored strain versus temperature in a free-stress strain recovery process (required for calibration of the volume fraction of the glassy phase, ϕg ) [1].

Table 4.1 Material parameters adopted for experiments done by [1,7]. Parameters

Values

Ereq ; Egeq νr ; νg Tl ; Tg ; Th εT ϕg

8:8; 813 ½MPa 0:4; 0:3 273; 343; 358 ½K    2 3:14 3 1024 ðT
Th Þ 1 0:7 3 1026 T 2 2 Th 2 1 1 1 1 2:76 3 1025 ðTh
T Þ4

experimental data reported by [7] in Fig. 4.6 (right). In this case, a relation for ϕg in the following form is used: 1  ϕg 5  1 1 c1 ðTh 2T ÞC2

(4.47)

where c1 and c2 are calculated applying a curve fitting method. 5. Elastic moduli and viscosity coefficients of each phase is calculated using onedimensional stress
strain curves. The stress
strain responses of a pure SMP (φh 5 0) at Th as well as Tl give the elastic moduli and viscosity coefficients of the rubbery and glassy phases, respectively. These parameters for the hard segment are calculated using stress
strain response of the pure hard segment (before producing the composite and mixing the hard segment in the SMP matrix). Due to the lack of experimental data, the values of 0.4, 0.3, and 0.3 are assumed for ν r , ν g , and ν h , respectively.

By performing aforementioned procedure, the descriptive properties for the two different materials introduced by Refs. [7,21], are extracted and presented in Tables 4.1 and 4.2.

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141

Table 4.2 Material parameters adopted from experiments reported by [1,21]. Parameters

Values

Ereq ; Egeq ; Eheq ; Erneq Egneq ; Ehneq

1:3; 15; 70000; 0:2 ½MPa 247; 2000 6 5; 30 3 10 ; 30 3 109 ½MPa min 0:4 0:4; 0:3; 0:3 296; 344; 353½K   0:554 3 1023 T 2 0:0108 2 7 3 1027 T 2 1 1 1 2 1 1 exp 2 0:66 ð ðT 2 Tg ÞÞ

ηr ; ηg ; ηh ϕh νr ; νg Tl ; Tg ; Th εT ϕg

(A)

(B)

Figure 4.7 Reproduction of the shape memory effect in A) stress-free strain recovery, B) fixed-strain stress recovery [1].

4.2.8 Material model predictions Experiments carried out by Li and Nettles [21] are used to evaluate the validity and accuracy of the model in time-dependent regimes of thermomechanical loadings. An SMP-based syntactic foam sample is compressed under a constant stress σ0 5 2 263kPa at Th , held for 30 min. Then the sample is cooled down to Tl while the stress is kept fixed. The cooling rate is determined by Newtons law of cooling: dT=dt 5 4:6 3 1025 ðT 2 293Þ. Once the temperature reaches Tl (at t 5 1256 min), the load is removed, and the sample is heated up to Th with heating rate 5 0:3K=min. Corresponding material parameters are presented in Table 4.2. The strain
time behavior is illustrated in Fig. 4.7A. These results show good correlation between experimental results and numerical simulation. Now, if the recovery happens in a fixed-strain stress recovery, the stress
temperature behavior of the material will be obtained as shown in Fig. 4.7B. As results show, in spite of some discrepancy, the overall trend is in good agreement with the experimental data. In this section, a multiaxial loading path is simulated. The material parameters reported in Table 4.1 except that ϕh 5 0.33 are used. First, at Th , a strain, ε0 5 0:1

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for Δt1 5 50 min is applied. Then, the normal strain is held constant and a shear strain, ε12 5 ε13 5 0:15 for Δt2 5 50 min is applied. Keeping the strains fixed, material is allowed to relax the viscous effects for Δt3 5 150 min. Now the temperature is lowered to Tl for Δt4 5 100 min. Then, the stresses are unloaded and the system is allowed to relax for Δt5 5 100 min. In this step, the material is in temporary shape. Finally, the temperature is increased to Th for Δt6 5 50 min. Heating leads to recovery of the strain; thus the material recovers its initial shape. Fig. 4.8A
D presents the time and temperature histories of stress and strain components, respectively. As one may observe, due to time-dependent behavior of SMPs, applying a strain produces an over-stress in the material as depicted in Fig. 4.8A. Besides, holding strains fixed during the cooling leads to an increase in normal stresses, which occurs due to negative thermal strains and glassy-rubbery phase transformation. Moreover, even after heating and recovery, some irreversible strains remain in the material as shown in Fig. 4.8B and D. Another multiaxial loading path is simulated to give a better understanding of the proposed model. The structure is initially under an external load σ33 5 500 kPa.

(A)

(B)

(C)

(D)

Figure 4.8 Reproduction of the shape memory effect stress-free strain recovery: digrams of A) stress vs time, B) strain vs time, C) stress [1].

Shape memory polymers: constitutive modeling, calibration, and simulation

143

First, at Th , a strain, ε11 5 0.1 in Δt1 5 50 min is applied. Then, the normal strain ε11 is held constant and strain ε22 5 0 is applied in Δt2 5 50 min. Keeping the strains (ε11 5 0.1, ε22 5 0) and stress (σ33 5 500 kPa) fixed, the structure is allowed to relax the viscous effects for Δt3 5 150 min. the temperature is deceased to Tl in Δt4 5 100 min. Subsequently, the stresses are unloaded (by removing the external constraints and loads) and the system relaxes for Δt5 5 100 min. In this stage, the system is in a temporary shape. Finally, the structure is heated up to Th in Δt6 5 50 min. Fig. 4.9A
D present the time and temperature history of stress and strain components, respectively. The material properties reported in Table 4.1 except that ϕh 5 0.33 are used. Similar to previous example, because of time-dependent behavior of SMP, applying a strain produces an over-stress in the material as shown in Fig. 4.9A. Also, holding strains fixed during the cooling leads to a rise in normal stresses which happens due to negative thermal strains and glassy-rubbery phase transformation. Furthermore, after shape recovery, some irreversible strains remain in the material as illustrated in Fig. 4.9B and D.

(A)

(B)

(C)

(D)

Figure 4.9 Reproduction of the shape memory effect: stress-free strain recovery, diagrams of A) stress vs time, B) strain vs time, C) stress vs temperature, and D) strain vs temperature [1].

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4.3

Computational Modeling of Intelligent Soft Matter

Shape memory polymer constitutive model through thermo-viscoelastic approach

In this section, a constitutive model for SMPs based on the thermo-viscohyperelasticity is presented. Mechanical response of viscoelastic materials is a combination of elastic and viscous properties [22]. At finite strains, the elastic and viscous parts can be described by a hyperelastic model and Prony series, respectively [23,24]. ðt dσ0 dξ (4.48) σðε; tÞ 5 gðt 2 ξÞ dξ 0 where σ0 is the strain-dependent part of stress, and gðtÞ represents time-dependent part coefficients of the stress. In the following, the method to calculate σ0 and gðtÞ are presented.

4.3.1 Strain-dependent part of the stress To calculate the strain-dependent part of the stress, at the small strain can be used from hook’s law and also for the finite strain can be written as: σ0 5 2 pI 1 2

@W @W 21 BM 2 2 B @I1 @I2 M

(4.49)

where p is a Lagrangian multiplier, W is strain energy function, and B stands for left Cauchy
Green deformation tensor. Also, M index denotes the mechanical part of B as: BM 5 FM FTM

(4.50)

in which F is the deformation gradient tensor. By definition of deformation tensor for thermal expansion effects, it is concluded that: 1

FT 5 J Th3 I 5 ð1 1 αT ðT 2 T0 ÞÞIDΛT I

(4.51)

FM 5 FF21 T

(4.52)

where αT is the thermal expansion coefficient, which depends on the temperature and temperature rate.

4.3.2 Time-dependent part of the stress Time-dependent part is calculated by the storage modulus and Prony series as: m X t GðtÞ 5 GN 1 Gi e2τi (4.53) i51

Shape memory polymers: constitutive modeling, calibration, and simulation

145

Employing the relationship between storage modulus and GN from (4.54): m X

G0 5 GN 1

Gi

(4.54)

i51

one may write: GðtÞ 5 G0 2

m X

2τt

Gi ð1 2 e

i

Þ

(4.55)

i51

gðtÞ 5

m X t GðtÞ 512 gi ð1 2 e2τi Þ G0 i51

(4.56)

4.3.3 Temperature-dependent modification of the stress Moreover, to appropriately account for the temperature changes in an SME path, the time-temperature superposition principle can be applied. In other words, changing in time scale, the effects of temperature variation can be taken into account. To compute the total stress, it can be written: σðε; t; TÞ 5

ðt

gðtr 2 ξr Þ

0

dσ0 dξ dξ

(4.57)

where tr is the reduced time. Introducing the reference temperature Tref and parameter CA as constant in Arrhenius equation, the shift factor values can be computed through [25]    1 1 1 2 αðT Þ 5 exp 2 CA T Tref

(4.58)

or by WLF equation [26] ð

C1 T ðtÞ2Tref

2C

αðT Þ 5 10

Þ

21T ðtÞ2Tref

(4.59)

where C1 and C2 are material coefficients. Thanks to calculations of shift factors (AT ) in Eq. (4.58) or (4.59) and through Eq. (4.61), time-temperature superposition is identified. ξðtÞ 5

ðt

dξ α 0 TðξÞ

(4.60)

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4.3.4 Solution of shape memory polymer’s response in a shape memory polymer path The first three steps for the force recovery and shape recovery paths are similar. Thus first, equations are presented for three primary steps for both the shape and force recovery. In the following, based on the type of recovery, the equations of final stage (heating) are presented. The temperature variation in an SME path is depicted in Fig. 4.10. As shown in Fig. 4.10, at first, deformation is initiated at a temperature above the glass transition temperature (Tg ) and is completed by reaching point (a). Subsequently, by lowering the temperature to TC , the cooling step ends in point (b). At constant temperature (TC Þ, by gradual removal of loading, constraints are finally released at point (c). The applied temperature cycle is completed by heating to a temperature higher than Tg at point (d).

4.3.4.1 Applying the desired deformation (loading step) The loading step is conducted at constant temperature (TH ) during tL . For this step, the reduced time parameter is equal to the actual time using a linear function. Thus total stress is found as:  ! ðt m tr 2ξ r X dσ0L 2 Lτ L i σ5 dξ; 0 , t , tL 12 gi 1 2 e (4.61) dξ 0 i51

4.3.4.2 Temporal shape fixation (cooling step) In the cooling step, considering Eqs. (4.59) and (4.60), the relationship between the reduced time and actual time is strongly nonlinear. Therefore for the total stress,

Figure 4.10 Temperature history in an SME path [27].

Shape memory polymers: constitutive modeling, calibration, and simulation

147

it can be written:  ! m tr 2ξr X dσ0L 2 Cτ L i dξ 12 gi 1 2 e σ5 dξ 0 i51  ! ðt m tr 2ξr X dσ0C 2 Cτ C i dξ; tL , t , tC 1 12 gi 1 2 e dξ tL i51 ð tL

(4.62)

It is noted that if in this step, the temperature is kept fixed, stress relaxation would be calculated and, therefore, in such case the second term in Eq. (4.62) would be vanished.

4.3.4.3 Constraint removal (unloading step) Unloading step is conducted in constant temperature (TC ) before tUN . Again, in this step, the relationship between the reduced time and actual time is linear. The difference between unloading step and previous step is that in unloading step the value of stretches is unknown. Since loading constraints are removed in this step, a nonlinear solving technique should be applied to find the unknown deformations.  ! ð tL m tr 2ξr X dσ0L 2 UNτ L i σ5 dξ 12 gi 1 2 e dξ 0 i51  ! ð tC m tr 2ξr X dσ0C 2 UNτ C i dξ 1 12 gi 1 2 e (4.63) dξ tL i51  ! ðt m 2ξr tr X dσ0UN 2 UN τ UN i dξ; tC , t , tUN 1 12 gi 1 2 e dξ tC i51

4.3.4.4 Shape and force recovery (reheating step) For both force recovery and shape recovery paths, this step has similar governing equations and reduced times; however, boundary conditions as well as input variables are different. Constitutive equations for the heating stage are as below:  ! ð tL m tr 2ξr X dσ0L 2 Hτ L i σ5 dξ 12 gi 1 2 e dξ 0 i51  ! ð tC m tr 2ξ r X dσ0C 2 Hτ C i dξ 1 12 gi 1 2 e dξ tL i51 (4.64)  ! ð tUN m tr 2ξr X dσ0UN 2 H τ UN i dξ 1 12 gi 1 2 e dξ tC i51  ! ðt m tr 2ξr X dσ0H 2 Hτ H i dξ; tUN , t , tH 1 12 gi 1 2 e dξ tUN i51

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According to the above equations, by passing time, governing equations are becoming more complex, particularly in the final steps. In addition, according to Eqs. (4.59) and (4.60), it is realized that analytical calculation for reduced time is complicated. Moreover, solving a nonlinear system of equations for both unloading and shape recovery steps requires an approximation using a high order function. This is another problem in analytical calculations. Although finite element methods are highly efficient for various geometries and loads, by increasing the number of elements and also for processes associated with optimization, finite element methods would have high computational costs, so analytical or semianalytical solutions would dramatically compensate this issue. In the following the description of a semianalytical method based on thermo-viscoelastic models used for modeling SMPs has been examined. This model is used for modeling linear viscoelastic behavior of both finite and small strain. Formulation for tensorial form is presented in the following.

4.3.5 A time-discretization scheme for constitutive equations Here, the governing equations are discretized in a generic time interval [tn 21; tn ]. Considering Eq. (4.57) at tn , this is obtained as: σ 5

ð tn

12

n

0

m X

tn2ξr

2 rτ

gi ð1 2 e

i

i51

! Þ

dσ0 dξ dξ

(4.65)

In other words, it can be written: σn 5 σn0 2

m X

gi σni

(4.66)

i51

σni

5 σn0

2

ð tn

trn2ξ r τi

e2

0

dσ0 dξ 5 σn0 2 Ψni dξ

(4.67)

in which: Ψni 5

ð tn

tn2ξ r

2 rτ

e

i

0

Δtn dσ0 2 r dξ 5 e τi Ψn21 1 i dξ

ð tn

tn2ξ r

2 rτ

e tn21

i

dσ0 dξ dξ

(4.68)

Since the time step from tn21 to tn is small, assuming a linear approximation for the hyperelastic stress variations in each time increment, this is obtained as follow: σ

n

5 σn0

12

m X i51

! gi 1

 Δtn  ð tn m X tn2ξ r σn0 2 σn21 2 rτ r 0 i dξ gi e2 τi Ψn21 1 e i Δtn tn21 i51 (4.69)

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149

Thanks to Eq. (4.69), the stress in each step is found using its value in the previous step, while Ψn21 acts as a history variable on the amount of stress. Unlike i Eqs. (4.63) and (4.64), computing the unknowns by Eq. (4.69), in unloading and shape recovery stage is straightforward. It is worthwhile to mention that since there is no complicated integrations, semianalytical approach calculations speed is much higher than FEM. We now present time-discretized form of TTSP equation. According to Eq. (4.58) or (4.59) when the temperature is changing with time, solving equations is too complicated, and has high computational costs. Hence, employing a suitable linear approximation, reduced time for cooling step and heating step can be computed as:   c ðTðtÞ2Tref Þ 2 1 hT 5 2 lnðATðtÞ Þ 5 2 ln 10 c21ðTðtÞ2Tref Þ 5 a 1 bt

(4.70)

a1bt A21 TðtÞ 5 e

(4.71)

where a and b parameters are as follows: an 5

 1  n n21 t hT 2 tn21 hnT ; n Δt

(4.72)

bn 5

 1  n h 2 hn21 T Δtn T

(4.73)

trn

5 trn21

1

ð tn ea1bη dη

(4.74)

tn21

4.3.6 Material parameters identification The required properties in these constitutive equations consists of thermal expansion, viscous properties of material, elastic describing parameters, and also material coefficients to consider temperature effect. The following steps can be considered for obtaining material parameters: 1. Constant frequency DMA experiment in variable temperature to extract Th ; Tg ; and TC . 2. DMA experiment in several constant temperature Ts (Th , Ts , TC ) to derive the master curve and TTSP coefficients. 3. Fitting the achieved storage modulus and tan(δ) master curves with Prony series equations in frequency domain [Eqs. (4.75) and (4.77)] G0 ðf Þ 5 GN 1

m X k51

G00 ðf Þ 5

m X k51

Gk

Gk

ð2πf τ k Þ2 1 1 ð2πf τ k Þ2

2πf τ k 1 1 ð2πf τ k Þ2

(4.75)

(4.76)

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tanðδÞ 5

G00 G0

(4.77)

4. Performing uniaxial tensile test experiment at temperatures above Th with relatively low strain rate and fitting the stress
strain curve with the appropriate equation (Hook’s law for linear elastic or appropriate equations for hyperelastic behavior). 5. Performing dilatometry experiment and deriving thermal expansion coefficients.

In the following, the properties for two various polymer materials available in literature have been derived:

4.3.7 Solutions development for torsion
extension of SMP In this section, an analytical solution is developed for an SMP cylinder under simultaneous torsion and extension in both shape and force recovery paths. Employing cylindrical coordinates for the reference and current configurations in the form of ðR; Θ; ZÞ and ðr; θ; zÞ, respectively [28], it can be written as: r 5 CR θ 5 Θ 1 βγZ z 5 γZ

(4.78)

where γ and β denote the cylinder axial stretch, and the amount of twist per unit stretched length, respectively. Applying the incompressibility constraint, it would be: detðFM Þ 5 1

(4.79)

detðFÞ 5 detðFT Þ 5 Λ3T

(4.80)

Therefore C, F, and BM are found as: 3

C 5 Λ2T γ22 0

1

(4.81) 1

3 1 2 22

BΛ γ B T F5B B @ 0 0

0 ΛT γ 0 0 ΛT

3 1 2 2

0

γ

B B B B B BM 5 FM FTM 5 B 0 B B B @ 0

C C C C ΛT γ Rβ A γ fr;θ;zg 0

3 1 2 22

ΛT γ

0

(4.82)

1

C C C Rβffi C γpffiffiffiffi C ΛT C C C C 2 A γ 3 2

1ΛT γβ 2 R2 3 2

γpffiffiffiffi Rβffi ΛT

Λ2T

(4.83)

fr;θ;zg

Shape memory polymers: constitutive modeling, calibration, and simulation

151

Due to the symmetry, the only nonzero equilibrium equation is along the radial direction as: σ0θθ 2 σ0rr 5 r

@σ0rr @r

(4.84)

where σ0θθ and σ0rr are radial and tangential stresses. Bearing in mind that the radial stress vanishes at the outer radius, one may find the Lagrangian multiplier through integrating Eq. (4.84). In other words, the nonzero strain dependent stress components are computed along with calculation of Lagrange multiplier employing Eq. (4.84). The corresponding stress components for the extension-torsion problem are: σ0rr 5 C10 γβ 2 ΛT ðR2 2 R2out Þ

(4.85)

σ0θθ 5 C10 γβ 2 ΛT ð3R2 2 R2out Þ

(4.86)

 2

σ0zz 5 C10 ΛT γβ R2 2 R2out



γ2 ΛT 12 2 22 γ ΛT

! (4.87)

3

γ 2 Rβ σ0zθ 5 2C10 pffiffiffiffiffiffi ΛT

(4.88)

where σ0zθ is the shear stress. The required torque and required force to maintain the applied deformation are computed as: ðð M5

rσzθ dA 5 2π

ð rout

r σzθ dr 5 2π 2

0

ð Rout

σzθ Λ2 γ22 R2 dR 9

3

ðð ð rout ð Rout N 5 σzz dA 5 2π rσzz dr 5 2π σzz Λ3 γ21 RdR 0

(4.89)

0

(4.90)

0

Once again, σij should be found through Eq. (4.65), while σ0ij has the form of Eqs. (4.87) and (4.88). The controlling parameters in the torsion
extension problem are given in Table 4.5. The schematic shape recovery process of the cylinder is illustrated in Fig. 4.11. In this problem, parameters γ and β are, respectively, conjugated with parameters N and M. As shown in Table 4.5, the input in loading, cooling, and force recovery steps are either the controlling deformation parameters (γ), for the tensile problem or γ and β for the extension-torsion problem. The total stress can be calculated using the above-mentioned parameters and semianalytical solution can be evaluated using Eq. (4.69). However, controlling variables change in the unloading and shape

Table 4.3 Material parameters adopted for experiments done by [27,29]. Thermal properties T( C) αT 3 1024 (1/ C)

20 1.84

25 1.82

30 1.78

35 1.73

40 1.67

45 1.59

50 1.5

55 1.41

60 1.32

65 1.26

70 1.22

75 1.17

Viscoelastic properties τ 1 ðsÞ 1.26e 2 6 τ 11 ðsÞ 6.08e 2 3 g1 0.187 g11 0.00583

τ 2 ðsÞ 2.94e 2 6 τ 12 ðsÞ 0.0142 g2 0.156 g12 0.00319

τ 3 ðsÞ 6.87e 2 6 τ 13 ðsÞ 0.0332 g3 0.138 g13 0.00185

τ 4 ðsÞ 1.6e 2 5 τ 14 ðsÞ 0.0775 g4 0.124 g14 0.00105

τ 5 ðsÞ 3.75e 2 5 τ 15 ðsÞ 0.181 g5 0.113 g15 5.64e 2 4

τ 6 ðsÞ 8.75e 2 5 τ 16 ðsÞ 0.423 g6 0.0987 g16 2.86e 2 4

τ 7 ðsÞ 2.04e 2 4 τ 17 ðsÞ 0.988 g7 0.0758 g17 1.44e 2 4

τ 8 ðsÞ 4.78e 2 4 τ 18 ðsÞ 2.13 g8 0.0475 g18 7.42e 2 5

τ 9 ðsÞ 1.12e 2 3 τ 19 ðsÞ 5.39 g9 0.0245 g19 3.91e 2 5

τ 10 ðsÞ 2.61e 2 3 τ 20 ðsÞ 12.6 g10 0.0117 g20 2.07e 2 5

Hyperelastic properties Strain energy potential Neo Hookean

C10 ðMPaÞ 109.73

Moduli time scale Instantaneous

C1 6.9

C2( C) 87.9

WLF TTSP parameters Tref ( C) 80

Table 4.4 Material parameters adopted for experiments done by [27]. Viscoelastic properties τ 1 ðsÞ 0.011497 τ 11 ðsÞ 4650.151 g1 0.048275 g11 0.003702

τ 2 ðsÞ 9.817593 τ 12 ðsÞ 7036.142 g2 0.27215 g12 2.14E 2 07

τ 3 ðsÞ 0.684301 τ 13 9013.388 g3 0.285528 g13 0.007206

τ 4 ðsÞ 0.000588 τ 14 ðsÞðsÞ 65,995.31 g4 0.074514 g14 0.001215

τ 5 ðsÞ 71.82592 τ 15 40,5070 g5 0.206132 g15 0.000451

τ 6 ðsÞ 114.9392 τ 16 ðsÞðsÞ 2,336,204 g6 2.10E 2 06 g16 2.84E 2 05

τ 7 ðsÞ 102.5128 τ 17 ðsÞ 3,049,147 g7 2.99E 2 06 g17 1.43E 2 11

τ 8 ðsÞ 382.6162 τ 18 ðsÞ 4,679,727 g8 0.069412 g18 6.53E 2 12

τ 9 ðsÞ 1437.412 τ 19 ðsÞ 6,980,122 g9 0.031036 g19 1.55E 2 11

τ 10 ðsÞ 1984.592 τ 20 ðsÞ 7,721,512 g10 5.00E 2 06 g20 0.000161

Hyperelastic properties C10 ðMPaÞ 465.7

Strain energy potential Neo Hookean

Moduli time scale Instantaneous

TTSP parameter Arrhenius ðT , Tref ) AK 258.99

Tref ( C) 51.56

WLF ðT . Tref Þ C1 11.49

C2 ( C) 30.195

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Computational Modeling of Intelligent Soft Matter

Table 4.5 history of SME for the torsion
extension problem [27] Heating Step

Loading

Time (s) 0 ! tL T ( C) TH Input Output

(A)

γ ðtÞ and β ðtÞ NðtÞ and MðtÞ

(B)

Cooling

Unloading

tL ! tC TH ! TC

tC ! tUN TC

γ L and β L NðtÞ and MðtÞ

NC and MC ! 0 γ UN and β UN γ ðtÞ and β ðtÞ NðtÞ and MðtÞ

(C)

Force recovery

Shape recovery tUN ! tH TC ! TH

(D)

NðtÞ and MðtÞ 5 0 γ ðtÞ and β ðtÞ

(E)

Figure 4.11 Shape memory stages of the shape memory polymer cylinder under simultaneous torsion M and extension N, A) initial state, B) loading step, C) cooling step, D) unloading step, E) heating step [27].

recovery steps. Also, the corresponding values for γ and β must be calculated by specifying the force or torque values. It can be assumed that N and M vanish linearly. Hence, knowing the values of forces by solving the integral equation, the correspondent stretches are computed. As an example, an axial strain of 0.25 and a torsion of 4.71 rad are applied to the polymeric sample made of material introduced in Table 4.4 at above Tg. After completing the shape recovery process, the sample deformation is identified at different temperatures. As shown in Fig. 4.12, the results reveal the compatibility of different methods for describing the shape memory effect path in the problem of torsion
extension. Another example is presented here to illustrate the ability of this constitutive equation to display the force recovery response. In this example, an axial stretch and a torsion on a polymeric cylinder made of material introduced in Table 4.3, with a radius of 0.5 m are applied at the same time. Generated axial force and moment to keep the temporary shape of the cylinder fixed during heating, are presented in Fig. 4.13A and B, respectively. It is noted that loading time (tL ) is 40 s.

Shape memory polymers: constitutive modeling, calibration, and simulation

(A)

155

(B)

Figure 4.12 Validation of (A) axial strain and (B) rotation angle in SME path for cylinder under torsion
extension [27].

5

5

Semi-Analytical

FEM

Semi-Analytical FEM

(A)

(B)

Figure 4.13 (A) Force recovery and (B) moment recovery curves for SME path of the cylinder under torsion
extension [27].

Through the loading stage 10% axial strain and a 1.57 rad twist are applied to the cylinder. Recovery of both applied force and applied moment are depicted in Fig. 4.13 for four steps. First, at high temperature (65o C), the loading step is conducted by increasing the force and moment to 230 kN and 280 Nm, respectively. During the cooling, due to the developed thermal stress, the axial force rises, while the moment remains almost unchanged. After cooling to 25o C, the unloading step is carried out slowly, where the force and moment vanish. By constraining the axial deformation and rotation angle of the cylinder, and increasing the temperature, an axial force together with a moment start to recover. Due to the thermal expansion, specimen tends to expand and elongate at the beginning of heating. Therefore a compressive axial load is generated. Raising the temperature and considering predominance of SME, and tendency of the cylinder to recover its initial shape, a tensile stress starts to develop, which justifies the pattern of the recovered force. The thermal strain is volumetric and hence, does not have a significant impact on the change of the angle during heating. Therefore, the moment change and moment recovery occur only due to the shape memory phenomenon, while thermal expansion effects do not play an important role here.

156

4.4

Computational Modeling of Intelligent Soft Matter

Summary and conclusion

In this chapter, two different 3D constitutive model for SMPs were presented which reasonably capture the essential features of the shape memory behavior under timedependent multiaxial loadings. The first model was based on the phase transition concept, while the second one takes its basis from thermovisco-hyperelasticity theory. In the first constitutive model, assuming small strains, applying a first order mixture rule an additive decomposition of the strain into six parts was used. The material was considered as a mixture of rubbery, glassy, and hard phases. It was assumed that the rubbery and glassy phases were able to be transformed to each other through external stimuli of heat. The evolution laws for internal variables were derived in an arbitrary thermomechanical loading. Moreover, the polymer in each phase was considered a viscoelastic material. For the sake of completeness, free energy function of the model is introduced in compatibility with the second law of thermodynamics, in the sense of the Clausius
Duhem inequality. The time-discrete form of the evolution equations was presented in an implicit form. It was shown that the model is capable of capturing the main features reported in experimental observations. In the second constitutive model, after describing the major features of the model, force and multiple shape recovery in SMPs under finite deformation were proposed based on the thermovisco-hyperelastic model. Also, along with two traditional shape memory scenarios (force and shape recovery), a semianalytical solution for multiple shape memory effect of torsion
extension problem was developed. The proposed equations can be employed as an efficient method to calibrate the material or optimize geometrical parameters, as well as to analyze various case studies. It is noted that the results of the analytical solution were verified by comparing with both experimental and FEM results.

References [1] M. Baghani, R. Naghdabadi, J. Arghavani, S. Sohrabpour, A thermodynamicallyconsistent 3D constitutive model for shape memory polymers, International Journal of Plasticity 35 (2012) 13
30. [2] M. Baghani, R. Naghdabadi, J. Arghavani, S. Sohrabpour, A constitutive model for shape memory polymers with application to torsion of prismatic bars, Journal of Intelligent Material Systems and Structures 23 (2) (2012) 107
116. [3] M. Baghani, R. Dolatabadi, M. Baniassadi, Developing a finite element beam theory for nanocomposite shape-memory polymers with application to sustained release of drugs, Scientia Iranica 24 (1) (2017) 249
259. [4] M. Baghani, J. Arghavani, R. Naghdabadi, A finite deformation constitutive model for shape memory polymers based on Hencky strain, Mechanics of Materials 73 (2014) 1
10. [5] M. Baghani, A. Taheri, An analytic investigation on behavior of smart devices consisting of reinforced shape memory polymer beams, Journal of Intelligent Material Systems and Structures 26 (11) (2015) 1385
1394.

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157

[6] N. Roudbarian, M. Baniasadi, M. Ansari, M. Baghani, An experimental investigation on structural design of shape memory polymers, Smart Materials and Structures 28 (9) (2019). [7] Y. Liu, K. Gall, M.L. Dunn, A.R. Greenberg, J. Diani, Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling, International Journal of Plasticity 22 (2) (2006) 279
313. [8] M. Baghani, R. Naghdabadi, J. Arghavani, A semi-analytical study on helical springs made of shape memory polymer, Smart Materials and Structures 21 (4) (2012). [9] A.H. Eskandari, M. Baghani, S. Sohrabpour, A time-dependent finite element formulation for thick shape memory polymer beams considering shear effects, International Journal of Applied Mechanics 10 (4) (2018). [10] P. Haupt, Continuum Mechanics and Theory of Materials, Springer Science & Business Media, 2013. [11] B.D. Coleman, M.E. Gurtin, Thermodynamics with internal state variables, The Journal of Chemical Physics 47 (2) (1967) 597
613. [12] D. Hartl, D. Lagoudas, Constitutive modeling and structural analysis considering simultaneous phase transformation and plastic yield in shape memory alloys, Smart Materials and Structures 18 (10) (2009) 104017. [13] T.J. Hughes, J. Winget, Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, International Journal for Numerical Methods in Engineering 15 (12) (1980) 1862
1867. [14] S. Akbari-Azar, M. Baghani, M.R. Zakerzadeh, H. Shahsavari, S. Sohrabpour, Analytical investigation of composite sandwich beams filled with shape memory polymer corrugated core, Meccanica 54 (10) (2019) 1647
1661. [15] M. Baghani, R. Naghdabadi, J. Arghavani, A large deformation framework for shape memory polymers: Constitutive modeling and finite element implementation, Journal of Intelligent Material Systems and Structures 24 (1) (2013) 21
32. [16] E. Yarali, M. Baniassadi, M. Baghani, Numerical homogenization of coiled carbon nanotube reinforced shape memory polymer nanocomposites, Smart Materials and Structures 28 (3) (2019). [17] A. Jamalimehr, M. Baghani, M. Baniassadi, M. Zakerzadeh, Thermomechanical behavior of shape memory polymer beams reinforced by corrugated polymeric sections, Meccanica 52 (8) (2017) 1947
1962. [18] M. Taherzadeh, M. Baghani, M. Baniassadi, K. Abrinia, M. Safdari, Modeling and homogenization of shape memory polymer nanocomposites, Composites Part B: Engineering 91 (2016) 36
43. [19] R. De Borst, M.A. Crisfield, J.J. Remmers, C.V. Verhoosel, Nonlinear Finite Element Analysis of Solids and Structures, John Wiley & Sons, 2012. [20] M. Baghani, R. Naghdabadi, J. Arghavani, A large deformation framework for shape memory polymers: constitutive modeling and finite element implementation, Journal of Intelligent Material Systems and Structures 24 (1) (2012) 21
32. [21] G. Li, D. Nettles, Thermomechanical characterization of a shape memory polymer based self-repairing syntactic foam, Polymer 51 (3) (2010) 755
762. [22] D. Gutierrez-Lemini, Fundamental aspects of viscoelastic response, Engineering Viscoelasticity, Springer, 2014, pp. 1
21. [23] H.F. Brinson, L.C. Brinson, Polymer Engineering Science and Viscoelasticity, Springer, 2015. [24] M.H.R. Ghoreishy, Determination of the parameters of the Prony series in hyperviscoelastic material models using the finite element method, Materials & Design 35 (2012) 791
797.

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[25] E.A. Di Marzio, A.J. Yang, Configurational entropy approach to the kinetics of glasses, Journal of Research of the National Institute of Standards and Technology 102 (2) (1997) 135
157. [26] M.L. Williams, R.F. Landel, J.D. Ferry, The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, Journal of the American Chemical Society 77 (14) (1955) 3701
3707. [27] M. Baniasadi, M.A. Maleki-Bigdeli, M. Baghani, Force and multiple-shape-recovery in shape-memory-polymers under finite deformation torsion-extension, Smart Materials and Structures 29 (5) (2020). [28] C.O. Horgan, J.G. Murphy, Extension and torsion of incompressible non-linearly elastic solid circular cylinders, Mathematics and Mechanics of Solids 16 (5) (2011) 482
491. [29] S. Arrieta, J. Diani, P. Gilormini, Experimental characterization and thermoviscoelastic modeling of strain and stress recoveries of an amorphous polymer network, Mechanics of Materials 68 (2014) 95
103.

Shape memory polymer composites: nanocomposites and corrugated structures

5

Chapter outline 5.1 Introduction 159 5.2 Modeling and homogenization of shape memory polymer nanocomposites

160

5.2.1 Constitutive equations for shape memory polymer based on phase transition 161 5.2.2 3D modeling and numerical considerations 163 5.2.3 Numerical results 166

5.3 Numerical homogenization of coiled carbon nanotube-reinforced shape memory polymer nanocomposites 173 5.3.1 Constitutive model of shape memory polymer based on thermo-viscoelasticity 174 5.3.2 Finite element model 178 5.3.3 Numerical results and discussion 182

5.4 Thermomechanical behavior of shape memory polymer beams reinforced by corrugated polymeric sections 187 5.4.1 Shape memory polymer constitutive model based on phase transition concept 189 5.4.2 Bending of a reinforced shape memory polymer beam 192 5.4.3 Numerical results and discussion 196

References

5.1

205

Introduction

Shape memory materials are a subset of smart materials, capable of recovering a permanent shape after deforming to a temporary shape. An external stimulus such as light, heat, electricity, magnetic field, pH, and chemicals are required to bring about the recovery. Mostly the recovery process accompanies a transformation in the microstructure. A temperature sensitive material is one for which the transformation is triggered by introducing the heat. Temperature induced shape memory effect has been observed in metals, ceramics, and polymers. Among numerous temperature sensitive materials, shape memory polymers (SMPs) have received considerable attention during the past decade thanks to their unique characteristics, such as light weight, ease of manufacturing, the ability to recover large strains, low energy consumption for shape memory programming, biocompatibility, and biodegradability. Despite all impressive characteristics of SMPs, the most noticeable drawback for SMPs is their poor elastic properties, which impedes the applications of SMPs when large recovery stress is required, for example, in actuators. To overcome such limitations, a variety of reinforcements is used in SMP composites. Needless to say, Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00004-5 © 2023 Elsevier Inc. All rights reserved.

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Computational Modeling of Intelligent Soft Matter

recent developments in the fabrication of nanoscale materials, such as graphene, graphite, carbon nanotubes (CNTs), and clay, also increased academic and industrial interests on using these nanomaterials as reinforcement in other materials including SMPs. In an experimental effort, Guo et al. [1] added short carbon and alumina nanoparticles to SMP for improving the mechanical properties. The experimental results revealed that the mechanical and thermal properties of nanocomposite SMP are significantly higher than those of pure SMP. They observed that increasing the inclusion nanoparticles led to improving SME as well as the mechanical strength. Chen et al. [2] experimentally studied the effect of nanohydroxyapatite (nano-HAp) and the porous structure on the SME of poly D-L-lactide. Adding graphene oxide (TrGo) to epoxy-based SMP, Chen et al. [3] examined the mechanical, thermal, and shape memory characterizations of the nanocomposite. Arrieta et al. [4] studied both experimentally and numerically the thermomechanical response of SMPs reinforced by spherical glass beads and short and long fiber nanoparticles in large deformations. Conducting some experiments, Bouaziz et al. [5] reported the thermomechanical behavior of halloysite nanotubes reinforced SMP nanocomposite. Yang et al. [6] presented a micromechanical model for inhomogeneity of nanocomposite SMP reinforced by CNT. They used model proposed in Ref. [7] for the constitutive equation of SMP and studied the aggregation of nanoparticles. To consider the effect of glass bead nanoparticles in SMP nanocomposite, Pan et al. [8] used a three elements linear viscoelastic model in small strains. Wang et al. [9] in their experiments used a novel digitally controlled spray-evaporation deposition modeling process to investigate the electrical actuation and SME of polyurethane composite reinforced by printed CNT layers. Due to difficulties in experimental characterizations, other methods such as analytical and numerical simulations may be employed as an alternative to experimental methods. Mori
Tanaka (MT), and the Halpin
Tsai are two of the most popular analytical mechanic-based composite stiffness models that are widely used for modeling composite structures. These methods are not able to precisely account for the interaction between adjacent particles. In addition, the microstress that is involved with each individual inclusion cannot be evaluated by these methods. In last decades, numerical methods, for example, FEM has attracted a great deal of interest to be utilized in simulation as well as design of composite structures. Statistical continuum mechanics techniques also offer useful tools for characterization and reconstruction of heterogeneous materials based on statistical correlation function [10].

5.2

Modeling and homogenization of shape memory polymer nanocomposites

In this section, 3D representative volume elements (RVEs) are generated to simulate the stress
strain
temperature relationship of randomly distributed graphene nanoplatelets (GNPs)-reinforced SMP composite. To evaluate thermomechanical

Shape memory polymer composites: nanocomposites and corrugated structures

161

response of the composite, the model has been assumed as a mixture of elastic GNPs inclusions and SMP matrix. SMP matrix itself is divided into two phases, rubbery and glassy, and the fraction of each phase is directly related to the temperature. To capture the thermomechanical response of SMP, a thermodynamically consistent constitutive model has been developed by Baghani et al. [11]. The stiffness tensor of the elastic GNP is identified by ultrasonic and static tests [12]. In all cases, 7% axial strain is applied and the effects of aspect ratio of the inclusions and volume fraction on the effective elastic properties are presented. Two different thermomechanical cycles are applied on the model, stress-free strain recovery cycle and fixed-strain stress recovery cycle. In the stress-free strain recovery cycle, the model returns from deformed state (temporary shape) to their original (permanent) shape due to rising the temperature. However, in the second cycle, the model is prevented to return to its original shape during the heating the materials. This section also examines the effect of imperfect GNP/SMP interfaces on SMP nanocomposites. Cohesive surface energy between GNP nanoparticles and a polymer reported elsewhere [13] is introduced into FE models to capture GNP/polymer debonding and its subsequent effect on the mechanical properties of the nanocomposite system under investigation.

5.2.1 Constitutive equations for shape memory polymer based on phase transition Fig. 5.1 schematically illustrates a typical thermomechanical cycle of an SMP. In a macroscopic point of view, it consists of four stages. Starting from a high temperature (Th ) in stress and strain free condition (point A), the material is elastically loaded to point b. At this point the specimen is held fixed while the temperature is decreased to Tl (point c). As a result, the SMP transforms from a rubbery phase (active phase with a low stiffness) to a glassy phase (frozen phase with a high stiffness). The aforementioned phase transformation is accompanied by an increase in the stress magnitude due to the strain constraint imposed on the specimen. Strain slightly changes as a consequence of the low temperature unloading (point d). Now,

Figure 5.1 Stress
strain
temperature diagram showing the thermomechanical behavior of shape memory polymer under strain or stress recovery processes [14].

162

Computational Modeling of Intelligent Soft Matter

the structure is in its temporary shape. The thermomechanical cycle is completed by a heating process. Based on the constraint applied at this point (geometric or force constraints) different recovery cycles may be achieved. If the structure is free at this point, the initial shape is recovered, that is, “stress free strain recovery” cycle. While “fixed strain stress recovery” cycle (dotted line in Fig. 5.1) occurs as a result of geometric constraint on the material during the heating process. Now, a brief explanation of the small strain constitutive model proposed in [11] is provided. In this model an equivalent RVE of SMP consisting of a frozen and an active phase has been used. According to the mixture rule, the total strain is given in the following relation: ε 5 ϕa εa 1 ϕf εF 1 εT

(5.1)

where εa and εF denote elastic strain in the active and frozen phase, respectively and εT denotes the thermal strain which is evaluated by αT dT and αT is the effective thermal expansion coefficient. In this relation, ϕa and ϕf denote the volume fraction of the active and frozen phase, respectively. Both are function of temperature. The strain in the frozen phase εF is decomposed into two parts such that ϕf εF 5 ϕf εf 1 εis

(5.2)

where εf is the elastic strain in the frozen phase and εis is the inelastic stored strain. As a result, the total strain is the weighted summation of the strains in each phase. Assuming temperature decreasing, the strain in the newly generated glassy phase, already been in the rubbery phase, had experienced εa previously. Then, ϕf εF is defined as: ϕf εF 5 ϕf ðεf 1 εf Þ 5 ϕf ðεf 1

1 Vf

ð εa dvÞ 5 ϕf εf 1 Vf

1 V

ð εa dv

(5.3)

Vf

where Vf and V are volume of the frozen phase and the total volume of the RVE, respectively. In Eq. (5.3), strain in the frozen phase is divided into two parts: strain in the old frozen phase, εf , and strain in the newly generated frozen phase, ε f . The term “ϕf εf ” is called as the stored strain and it is denoted by εis . We recast Eq. (5.3) to ϕf εF 5 ϕf εf 1 εis

(5.4)

In the cooling process, εis is defined by ð ε 5 is

εa dϕf

(5.5)

Shape memory polymer composites: nanocomposites and corrugated structures

163

In the heating process, the strain stored in the frozen phase should be relaxed. This is mathematically expressed by ϕf εF 5 ϕf ðεf 1 εf Þ 5 ϕf ðεf 1

1 Vf

ð Vf

εis 1 dvÞ 5 ϕf εf 1 V ϕf

ð Vf

εis dv ϕf

(5.6)

which in a more compact form is ϕf εF 5 ϕf εf 1 εis

(5.7)

Thus the total strain could be recast to ε 5 ϕa εa 1 ϕf εF 1 εT 1 εis

(5.8)

In addition, the stored strain obeys the following evolution law #8 is < k1 5 1; k2 5 0; T_ , 0 ε ε_ is 5 ϕ_ f k1 εa 1 k2 ; k 5 0; k2 5 1; T_ . 0 ϕf : 1 k1 5 0; k2 5 0; T_ 5 0 "

(5.9)

Now, based on a first order rule of mixture, the convex free-energy density function Ψ is defined by
   Ψ ε; T; ϕf ; ϕa ; εa ; εf ; εis 5 ϕa Ψa ðεa Þ 1 ϕf Ψf εf 1 ΨT ðT Þ

(5.10)

where Ψa and Ψf stand for the Helmholtz free-energy density function of the active and frozen phases, respectively. Also, ΨT denotes the thermal energy. Applying the second law of thermodynamics in the sense of the Clausius
Duhem inequality, the following equation is obtained σ5λ5

@Ψa @Ψf 5 @εa @εf

(5.11)

Eq. (5.11) is consequence of the basic assumption of simultaneous existence of the rubbery and glassy phases.

5.2.2 3D modeling and numerical considerations Because of high computational costs and modeling complexities, a perfect realistic modeling of a nanocomposite is extremely challenging or might be impossible. One way to bypass some of these limitations in the simulation of composite materials is to exchange the macroscopic model with a large enough RVE.

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Computational Modeling of Intelligent Soft Matter

In this section, FEM has been utilized for the evaluation of effective elastic properties of two-phase SMP composites filled with randomly distributed and oriented GNPs. The geometry of inclusions plays an important role on the overall effective mechanical properties. The effect of inclusion geometry in FEM model is also studied where the aspect ratio of inclusions is changed with platelet geometry. For a platelet, by aspect ratio the ratio between the diameter and thickness is denoted. In the modeling procedure, particles are subject to hard-core constraint where we avoid intersection or contact between them. To reconstruct the RVE similar to experimentally fabricated random composites, 3D inclusions are randomly distributed and oriented in the RVE. In Fig. 5.2, samples of 3D cubic RVE with different aspect ratio of inclusions have been shown. To automate the model generation process, RVEs are constructed in ABAQUS using an in-house Python script. Also, the position of the center of particles and their orientation are first computed using an in-house code [15]. To simulate SMP matrix in the model, the introduced 3D constitutive equations are implemented in ABAQUS/Standard, through a user-defined material subroutine (UMAT). The material parameters used in all simulations are listed in Table 5.1. The details of material model used here are given where the model has been verified with several experimental data sets. It is noted that the elastic properties of GNPs are listed in Table 5.2 [12]. To account for the effect of size dependency and determine the proper size for RVEs, several simulations are carried out initially. In these preliminary simulations, different RVE sizes with the same inclusion volume fraction were created

Figure 5.2 Different sizes of representative volume elements used in the size dependency checking simulations. The volume fraction for all of the above models was 0.005. There were 10, 20, 40, 80, 240, 320, 400, and 600 inclusions in models (A), (B), (C), (D), (E), (F), (G), and (H), respectively [14].

Shape memory polymer composites: nanocomposites and corrugated structures

165

Table 5.1 Material parameters of SMP modeling [11,14]. Material parameters a

Values

f

E , E (MPa) va ; vf Tl, Tg, Th ( C) ρ (kg/m3) ϕf

15.2, 2600 0.49, 0.4 25, 46, 62 1100 Th 2 Tg T 2 Tg b Þ 2 tanhð b Þ Th 2 Tg Tl 2 Tg tanhð b Þ 2 tanhð b Þ

tanhð

; b 5 4:8173 C

Table 5.2 The elastic constants of graphene nanoplatelet reported by [12,14].

GNP elastic constants (GPa)

C11

C12

C13

C33

C44

C66

1060

180

15

36.5

4

440

(see Fig. 5.2) to measure the size dependency of calculated results and determine an acceptable RVE size. The intension of this section is to study the effect of the aspect ratio and the volume fraction on the mechanical behavior of SMP nanocomposite. In all numerical studies on composite reinforced with platelet shaped inclusions, such as graphene, GNP, and clay nanoplatelet [16,17], inclusions have been modeled as thin circular disks and in a few cases as rectangular thin disks. This simplistic model could provide useful information about the above-mentioned parameters on the overall stress
strain behavior of the composite materials. The interface between nanoparticles and the matrix are assumed to be perfect. In practice, interface between nanoparticles and GNPs may undergo failure and debonding. To investigate the potential effect of interface debonding on the mechanical properties of the nanocomposite, we introduce imperfect interfaces with associated surface energy into our finite element model for one of the cases. For this purpose, we use cohesive zones to the interfaces between GNPs and matrix and assume a bilinear tractionseparation law [18]. The cohesive model used here is parameterized by an initial stiffness, a peak traction value, and a critical surface energy for failure. For GNPs/ SMP system, these cohesive zone parameters can be extracted from molecular dynamics studies or by experimental measurements, which is beyond the scope of this section. An alternative is to find a material, which has a similar structure to SMPs with known cohesive zone parameters. Using molecular dynamics, Awasthi et al. [13] have studied the interfacial interactions between graphene and polyethylene. Molecular structure of polyethylene is quite close to that of SMP; therefore, here we assume that the interaction between GNPs and SMP closely match that of GNP and polyethylene reported in this study as listed in Table 5.3. The RVEs are meshed using four-node linear tetrahedral elements (C3D4 elements). To evaluate the elastic modulus, a small uniform strain (7%) is applied on one side of the RVE cube where the opposite side is kept fixed in its normal

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Table 5.3 Cohesive zone model parameters for interfacial interaction between GNP and polyethylene [13,14]. Fracture mode

Fracture energy (mJ/m2)

Peak traction (MPa)

Shear mode Normal mode

331.650 246.525

108.276 170.616

direction and on the other side is allowed to move freely. The corresponding stresses are calculated using reaction forces on each side. The effective elastic modulus and Poisson’s ratio are estimated employing the Hooke’s law for isotropic materials. To ensure that the selected RVE obeys Hooke’s law for isotropic materials, the homogenized stiffness tensor of the RVE is extracted. An RVE loaded with 0.5% of GNPs with aspect ratio 10 is subjected to six different load cases and each time only one normal or pure shear strain is applied to the faces. The effective stiffness tensor for the described RVE extracted by volumetric homogenization of the resultant stresses is examined as measure of isotropy.

5.2.3 Numerical results To check the convergence of the finite element results, the dependency of the reported results to the RVE size are considered (Fig. 5.3A). The size of RVE is closely related to the dimension and geometry of inclusions. The size of RVE must be large enough eliminate the effect of boundary distribution of inclusions [19]. Fig. 5.3A shows, the finite element results for elastic modulus of the SMP composite at Th for different RVE sizes including GNP inclusions with the aspect ratio of 20. The volume fraction of particles is 0.5% for all cases. As an overall trend, an increase in the number of inclusions enhances the accuracy of the FE results. Nevertheless, utilizing larger RVE sizes is limited by the difficulties in mesh generation step and it is also associated to higher computational costs. Thus a maximum acceptable relative error was adopted in the simulation to determine the acceptable size of the RVE. As shown in Fig. 5.3A, increasing the size of RVE or equally increasing the number of inclusions, the estimated composite to matrix material elastic modulus ratio (Ec =Em ) converges to a constant value. Assuming this constant value as the accurate result, the relative error is calculated. In addition, we assume the relative error to be less than 4%, and the results from a series of simulations show that the minimum size for RVEs with the volume fraction of 0.5% should be above of the demonstrated line depicted in Fig. 5.3B. According to these results, an RVE with volume fraction of 0.5% and including 50-nm diameter inclusions is an acceptable RVE. Using cohesive zone model with parameters listed in Table 5.3, the behavior of a SMP/GNP nanocomposite system is computationally investigated under a simple tension test. The average stress/strain diagram obtained is shown in Fig. 5.3B. Based on this diagram, a linear response for stress
strain behavior is observed for strains less than 4%. At low strain levels, SMP/GNP interfaces show negligible

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(A)

(B)

Figure 5.3 (A) Effect of the representative volume element size on the shape memory polymer nanocomposite effective elastic modulus (Ec/Em) for different graphene nanoplatelet inclusions. The volume fraction of inclusions is 0.5%. (B) Homogenized stress/strain response for a representative volume element of the graphene nanoplatelet/shape memory polymer nanocomposite system loaded with 0.5% graphene nanoplatelets under simple tension test (the temperature is set to Th and remains constant) [14].

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failure, mostly limited to the stress concentration sites; therefore the assumption of perfect bonding for the interfaces is valid. For strain levels larger than 4%, the stress
strain response starts to deviate from linearity, however the extent of nonlinearities remains small for up to 6% strains. It is worth noting that for all modeling efforts reported here, the tensile strains never exceed 7%, therefore the assumption of perfect interface should provide tight estimates for the properties reported. However, for strains larger than 7%, initial damage sites start to propagate along the interfaces and the total strain energy stored in the system will be reduced by the formation of free (debonded) GNP/SMP surfaces in the system as shown in Fig. 5.4. As a result, the total elastic stiffness of the GNP/SMP nanocomposite system diminishes. Based on the initial observation presented, for the rest of this section, a perfect bonding for interfaces is assumed and a proper RVE size is utilized. As mentioned

S, Mises [Pa] + 2.569×10^07 + 3.271×10^05 + 3.152×10^05 + 3.034×10^05 + 2.915×10^05 + 2.797×10^05 + 2.678×10^05 + 2.560×10^05 + 2.442×10^05 + 2.323×10^05 + 2.205×10^05 + 2.086×10^05 + 1.968×10^05 + 1.849×10^05 + 1.731×10^05 + 1.612×10^05 + 1.494×10^05 + 1.375×10^05 + 1.257×10^05 + 1.139×10^05 + 1.020×10^05 + 9.017×10^04 + 7.832×10^04 + 6.648×10^04 + 5.463×10^04 + 4.278×10^04 + 1.929×10^04

Figure 5.4 Partial debonding of interfaces in the graphene nanoplatelets/shape memory polymer nanocomposite system. Plot shows distribution of effective stress (in von Mises) over a cross-section of the representative volume element undergoing strain levels above 7% [14].

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before, a specified strain is applied to the rubbery SMP sample. Then, the temperature is lowered to transform SMP to the glassy phase. In the next step, the constraints are removed and the sample is unloaded. During this process, SMP releases a negligible elastic strain. Finally, the temperature is increased and SMP recovers its permanent shape. To apply this cycle on SMP nanocomposite and observe effects of inclusions on the recovery behavior of SMP, three different RVEs with inclusion’s aspect ratio of 20 and volume fraction of 1%, 2%, and 3% are considered. The results of these simulations are shown in Fig. 5.5 in the form of three stress-free strain recovery cycle diagrams. The results are presented as a function of the normalized temperature (T ) which is defined by T 5

T 2 Tl Th 2 Tl

(5.12)

where Tl and Th are identified in Table 5.1. The diagrams illustrate that the permanent shape is completely recovered after a stress-free strain cycle. The applied strain is in the range of small strains and the elastic behavior of GNPs and SMP are therefore considered. Furthermore, the results prove that by increasing the volume

Figure 5.5 Temperature
strain
stress diagram of free-stress strain recovery cycles for three different shape memory polymer nanocomposite (volume fractions of 1%, 2%, and 3%) and pure shape memory polymer [14].

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fraction of inclusions, the effective elastic modulus of the composite increases. In addition, this result is obtained in fixed-strain stress recovery cycle. To simulate the SMP nanocomposite in fixed-strain stress recovery cycle, three configurations are considered for which RVEs with different volume fraction of inclusions are presented. For the first configuration, five composites with inclusion’s aspect ratio of 10 and volume fractions of 0.5%, 1%, 1.5%, 2%, and 2.5% are assumed. For the second configuration, the aspect ratio of inclusions is 20 and there are six different composites models (volume fractions of 0.5%, 1%, 1.5%, 2%, 2.5%, and 3%). For the third configuration, four different composites with inclusion’s aspect ratio of 40, and volume fractions of 0.5%, 1%, 1.5%, and 2% are considered. For each of the above composites, a fixed-strain stress recovery cycle is applied to RVE. In this cycle, the temperature in the deformed RVE is decreased and the strain is stored in the generated glassy phase. Then, the RVE is unloaded to release the stored elastic strain. In the final step, the strain is remained fixed and then the temperature is increased. In the last step, a recovery stress is generated in the sample, which imposes a recovery force on the supports. The results of FEM simulation for the first, second and third configurations are presented in Fig. 5.6. As shown in these figures, the recovery stress increases due to an increase in the volume fraction of GNP inclusions. Moreover, by comparing the reported results for different values of inclusions aspect ratio, it can be concluded that both the recovery stress and the elastic modulus increase as the aspect ratio of the particles increases for the same volume fraction. A series of simulations were carried to ensure that the reconstructed RVE for the nanocomposite obeys the Hooke’s law for isotropic materials. Eq. (5.13) shows the estimated stiffness tensor for a sample RVE. Comparing the estimated stiffness tensor with the stiffness tensor of an isotropic material, it is clear that for the selected RVE size, the assumption of isotropic material property for the RVE/SMP nanocomposite system studied here is valid. 2 6 6 6 C56 6 4

2:7212 261:56 261:41 0:02 0:02 0:01

2:6163 272:25 261:66 0:01 0:02 0:02

2:6128 0:0040 261:53 0:21 272:47 0:02 0:11 10:68 0:02 0:01 0:00 0:07

0:0012 0:23 0:04 0:01 10:67 0:02

0:0004 0:20 0:24 0:07 0:01 10:68

3 7 7 7 7MPa 7 5

(5.13)

Fig. 5.7 summarizes all the modeling results obtained. This figure illustrates the effect of the volume fraction and aspect ratio of inclusions on the elastic modulus of SMP nanocomposite. From this figure, it is clear that an increase in either values of the aspect ratio or the volume fraction enhances the elastic properties of SMP nanocomposites. For instance, by increasing the volume fraction of a GNP, with an aspect ratio of 20, up to 3% the elastic modulus of the composite rises by twofold. The results of all modeling are listed in Table 5.4. In this Table, ratios of the elastic modulus and the effective recovery stress of the composite, to those for the pure SMP are reported for a variety of SMP composites.

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Figure 5.6 Temperature
strain
stress diagram of fixed-strain stress recovery cycles for different shape memory polymer nanocomposites (different volume fractions) and pure shape memory polymer. (A) Aspect ratio of 10, (B) aspect ratio of 20, and (C) aspect ratio of 40 [14].

In this section, a finite element modeling for the characterization of the elastic properties of SMP nanocomposites was presented to take into account the effects of volume fraction and aspect ratio of nanofillers on the composite material. A UMAT was used to implement 3D constitutive behavior of SMP into nonlinear finite element software ABAQUS/Standard. Linear elastic orthotropic and isotropic stiffness were considered for GNPs inclusions and SMP matrix, respectively. To account for the limited size effects and possible boundary condition effects, a series of finite element simulations were carries out to determine suitable RVE size. The effect of imperfect GNP/SMP interface was also investigated, and it is concluded that for strain levels utilized in the current study the assumption of perfect bonding between GNP nanoparticles and SMP polymer matrix is acceptable. In the next step, as series of RVEs with varying inclusion volume fraction were created to apply free-stress strain recovery cycle and observe the shape recovery behavior. Finally, three classes of GNPs with different aspect ratio for inclusions (10, 20, and

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Figure 5.7 Effects of aspect ratio and volume fraction of inclusions on the elastic modulus [14]. Table 5.4 Elastic property and recovery stress of SMP nanocomposite obtained by finite element simulations [14]. 

RSC =RSSMP

b

1.0838 6 0.0034 1.1528 6 0.0061 1.2586 6 0.0103 1.3584 6 0.0143 1.4485 6 0.0179 1.1176 6 0.0047 1.2504 6 0.0100 1.4455 6 0.0178 1.6301 6 0.0252 1.8541 6 0.0342 2.0359 6 0.0414 1.2312 6 0.0092 1.5321 6 0.0213 1.9488 6 0.0380 2.3059 6 0.0522 a



EC =ESMP

a

1.0498 6 0.0020 1.1169 6 0.0047 1.2199 6 0.0088 1.3171 6 0.0127 1.4048 6 0.0162 1.0827 6 0.0033 1.2119 6 0.0085 1.4020 6 0.00161 1.5820 6 0.0233 1.8009 6 0.0720 1.9787 6 0.0391 1.1933 6 0.0077 1.4865 6 0.0195 1.8939 6 0.0358 2.2433 6 0.0497

Ratio of elastic modulus of the composite to that of the pure SMP. Ratio of recovery stress of the composite to that of the pure SMP.

b

Volume fraction (%)

Aspect ratio of inclusions

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2

10

20

40

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40) were considered. For each aspect ratio, several RVEs with different volume fractions (from 0.5% up to 3%) were modeled. The results indicated that elastic properties and recovery stress increase with an increase in either volume fraction or aspect ratio of inclusions. The recovery stress increases by about 100% when using 1.7% or 3% GNP with aspect ratio of 40 or 20, respectively.

5.3

Numerical homogenization of coiled carbon nanotube-reinforced shape memory polymer nanocomposites

As discussed before, addition of nano and microparticles to SMP, due to the high surface to volume ratio for the additive, improves the major disadvantage of pure SMPs. These particles include graphene, single and multi-wall CNT, coiled carbon nanotube (CCNT), nanoclay, coiled graphene, etc. Carbon fiber and coiled carbon fiber due to their relatively high aspect ratio, enhance the mechanical, electrical, and thermal properties of nanocomposites. Helical carbon fibers (HCFs) or CCNTs have a high surface area, high Young’s modulus, large deformation strain and low damage. For the first time, CCNT was studied by Itoh and Dunlap [20,21]. In the early 1950s an unusual form of carbon as the structure of CCNT was reported by Ref. [22]. In 1994 by Amelinckx et al. [23] introduced a formulation mechanism for catalytically grown of helix-shaped graphite nanotubes. The geometric structure of CCNT as shown in Fig. 5.1 is like a spring with some parameters such as pitch (P), helix angle (γ), total length (L), spring length (Ls), outer and inner diameter (Do, Di) and CNT diameter (dc) (Fig. 5.8). Some recent researches on this topic are discussed in the following. Khani et al. [25] analyzed the CCNT reinforced elastic polymer nanocomposite. They investigated the geometrical parameters and volume fraction of CCNT on the results. To properly account for the rate-dependent response, they developed a thermo-visco-hyperelastic constitutive model [26] of SMP in large deformation. Yousefi et al. [27] proposed a multiscale model to study the damage and energy absorption in polyethylene reinforced by CCNT and CNT. Using FEM, they examined the effect of geometrical parameters and volume fraction, the orientation of nanoparticles in a two-phase cohesive zone, and the perfect bond between the nanoparticles and matrix.

Figure 5.8 The geometrical parameter of coiled carbon nanotube particle [24].

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In this section, a numerical simulation for large deformation of SMP/CCNT nanocomposite is presented where CCNTs play the role of reinforcement nanoparticles in a proposed RVE. It is attempted to investigate the effects of different parameters, for example, volume fraction, aspect ratio, pitch, helix angle, and distribution of the helical nanoparticles on the thermomechanical response of SMP in an FE approach. To capture the shape memory of the SMP matrix a thermovisco-hyperelastic constitutive model is employed [28]. In all cases, 5%, 10%, and 20% unidirectional strains are applied to the presented RVE. Two different thermomechanical paths (stress-free strain-recovery and fixed-strain stress-recovery) under 1 and 5 C/min are studied. A perfect bond between the matrix and nanoparticles is considered while the isotropic SMP is assumed to be nearly incompressible. CNTs and CCNTs are assumed to be isotropic with a linear elastic response [27].

5.3.1 Constitutive model of shape memory polymer based on thermo-viscoelasticity Arrieta et al. [28] represented a thermo-visco-hyperelastic constitutive equation for SMPs at large deformations. This model consists of three parts: viscoelastic part with Maxwell
Wiechert components, an elastic part, and a thermal part [29]. The total deformation gradient tensor can be multiplicatively decomposed into elastic, viscoelastic and thermal parts as: F 5 Fhe FTh 5 Fv FTh

(5.14)

where Fhe , Fv , and FTh represent the elastic, viscoelastic and thermal parts of the deformation gradient tensor, respectively. The total Cauchy stress tensor is given as: σ 5 σv 1 σe

(5.15)

where σv is the Cauchy stress in the viscoelastic part and σ e is the Cauchy stress in the elastic part. For a linear isotropic viscoelastic material, based on the Boltzmann superposition theory, the constitutive equation in the integral form can be recast as [30]: σ i v ðt Þ 5

ðt

 0 0 2G τ 2 τ e_ dt 1 I

0

ðt

 0 _ 0 K τ 2 τ Φdt

(5.16)

0

so that e and Φ are the mechanical deviatoric and volumetric strains, respectively, 0 dot stands for the differentiation with respect to t . G and K denote the timedependent shear and bulk modulus associated with the relaxation time τ. Applying the integration by parts, Eq. (5.16) can be rewritten as: σ i v ðtÞ 5 2G0 eðtÞ 1

ðτ 0

  ðτ  0   0  0 0 0 0 2G_ τ e t 2 t dτ 1 I K0 ΦðtÞ 1 K_ τ Φ t 2 t dτ 0

(5.17)

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in which G0 and K0 are the instantaneous shear and bulk modulus. In this model, elastic dilatation is considered, that is, K ðtÞ 5 K0 H ðtÞ where K is a bulk modulus and H (t) is Heaviside function; however, it is assumed that the bulk modulus is time-independent. To account for the nonisothermal process, using the time
temperature superpo0 sition principle (TTSP), the temperature affects the relaxation time τ updated through the Williams
Landel
Ferry (WLF) equation as [31]:   2 C1 T 2 Tref dτ 1 5 ! logðaT Þ 5 dt aT ð T ð t Þ Þ C2 1 T 2 Tref

(5.18)

where aT is the shift factor of TTSP, Tref is the reference temperature, C1 and C2 are the empirical constants, and T stands for the temperature. To determine the material parameters of the viscoelastic part, considering the Maxwell
Wiechert model, the shear modulus equation represented by Prony series is employed: G ðτ Þ 5 G N 1

Xn i51

τ Gi expð 2 Þ τi

(5.19)

where GN represents the long-term shear modulus (t ! N), n is the number of transient branches assumed in Maxwell
Wiechert model (i.e., n 5 20 here). To calibrate the viscoelastic parameters experimental DMTA data in a frequency domain are needed using the Fourier transformation as: 0

G ð f Þ 5 GN 1

Xn

G i51 i

ð2πf τ i Þ2 1 1 ð2πf τ i Þ2

(5.20)

0

so that G is the storage modulus, and f is the frequency. The viscoelastic parameters borrowed from [28] are listed in Table 5.5. To compute the Cauchy stress of the elastic part, the Neo-Hookean strain energy function is used for the hyperelastic model as [32]: W 5 C10


   1  he 2 he I1 2 3 1 J 21 D1

(5.21)

 he where C10 ; D1 are the model coefficients and, I 1 ; J he are the first invariant of deviatoric he left Cauchy-Green deformation tensor (B ) and the elastic volume ratio, respectively  he 21=3 he he (B 5 J B ). The elastic volume ratio J e follows from the total volume ratio J and the thermal volume ration J Th with the relation J he 5 J=J Th ; and the equivalent Cauchy stress is: ! @W 2 @W he σ he 5 he I 1 he DEV  he B (5.22) @J J @ I1

Table 5.5 Material parameters of SMP (acrylate polymer network) and nanoparticles [25,27]. Viscoelastic material parameters of SMP gi (
) 0.18706 ❶

τ i (s)

0.1584 ❷

0.13761 ❸

0.12378 ❹

0.11283 ❺

0.098684 ❻

0.005827 ⓫

0.003194 ⓬

0.0018477 ⓭

0.0010527 0.00056387 0.00028586 ⓮ ⓯ ⓰

1.26e 2 6 ❶

2.94e 2 6 ❷

6.87e 2 6 ❸

1.6e 2 5 ❹

3.75e 2 5 ❺

0.00608 ⓫

0.0142 ⓬

0.0332 ⓭

0.0775 ⓮

0.181 ⓯

0.075819 ❼

0.04752 ❽

0.024477 ❾

0.011651 ❿

0.00014378 7.4217e 2 5 ⓱ ⓲

3.9113e 2 5 ⓳

2.0746e 2 5 ⓴

8.75e 2 5 ❻

0.000204 ❼

0.00047 ❽

0.0011 ❾

0.00261 ❿

4.23 ⓰

0.988 ⓱

2.31 ⓲

5.39 ⓳

12.6 ⓴

Thermal material parameters of SMP at α_ 5 1 C/min α (1/ C) T ( C) α (1/ C) T ( C)

9.17e 2 005 20 0.00013 50

All ki s are assumed to be zero [24].

9.56e 2 005 25 0.000139 55

9.94e 2 005 30 0.000147 60

0.000105 35 0.000153 65

0.000112 40 0.000157 70

0.000121 45 0.000159 75

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DEV denotes the deviatoric part of the argument. To find the material parameters of this hyperelastic model, assuming a nearly incompressible response, a quasi-static uniaxial test is carried out at 65 C to calculate C10 . For the thermal part, assuming an isothermal process, we may write: 1   FTh 5 J Th 3 I 5 1 1 εTh I  ½1 1 αðT 2 T0 ÞI

(5.23)

where α is the thermal expansion coefficient. In Ref. [28], α is determined for 1 and 5 C/min heating rates. All material parameters of pure SMP based on Acrylate polymer network composition (90% molar mass of benzyl methacrylate with 10% molar mas of poly (ethylene glycol) dimethacrylate) given by [28] are listed in Table 5.5. Besides, we have C10 5 109.73 MPa, and D1 5 1:745e 2 9=Pa for the Instantaneous part of the energy. For the WLF coefficients, C1 5 6.9, C2 5 87.9 C and Tref 5 80 C. Density is ρ 5 1200 kg/m3 . Elastic material parameters of CNT and CCNT are ECNT 5 1000 GPa, ECCNT 5 27.2 GPa, and υ 5 0.3. For the nanoparticle, it is assumed that the CCNT is isotropic [33] with solid fiber model [34]. To find the elastic modulus of CCNT, an equivalent CNT is considered and apply the same strain. The elastic modulus of CCNT is related to the actual cross-section area of CCNT as follow [34]:   Eeff 5 ACNT =Aeff ECNT

(5.24)

in which ACNT and Aeff are CNT and effective cross-section, respectively. Eeff and ECNT represent the effective and CNT elastic modulus, respectively. The elastic modulus of CNT is assumed to be 1 TPa [35].

5.3.1.1 Representative volume element construction A cubic RVE including the matrix (SMP) and nanoparticles (CCNT) is proposed here. The RVE is the minimum size of subvolume of a heterogeneous material, which can be employed to measure the effective properties [36]. To arrive at the desired size of RVE, by keeping volume fraction and aspect ratio fixed, different dimensions of RVE are examined. Comparing the uniaxial stress
temperature curves, the RVE length is selected as 1000 nm and the maximum length of RVE per spring length of CCNT is 7.3. After fixing the size of RVE, the influence of parameters such as the type of nanoparticles CCNT, volume fraction, aspect ratio, pitch, and helix angle are investigated. One of the important problems in RVE construction is the distribution of nanoparticles. For distribution of the nanoparticles in RVE, Mote-Carlo algorithm is used to determine the direction of nanoparticles. The longitudinal vector of the nanotube is defined by the random homogeneous MoteCarlo algorithm as follows [37]:

θ 5 2πa ϕ 5 cos21 ð2b 2 1Þ

(5.25)

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Figure 5.9 The spherical angles utilized in Monte-Carlo technique [24,38].

where θ and ϕare the spherical angle ð0 , θ , 2π; 0 , ϕ , πÞ shown in Fig. 5.9 a and b are the random constants that generate the distributionð0 , a; b , 1Þ. At first, in a MATLAB code, the coordinate of each CCNT in a.txt file by employing the Monte-Carlo approach is generated. Then, using a PYTHON script, the set of nanoparticles and matrix in the RVE construction are implemented in ABAQUS. It is assumed that to model the percolation of CCNTs in RVE, the hardcore model is used in which CCNTs are modeled as impenetrable surroundings [36]. Also, a perfect bond is assumed between the SMP and nanoparticles. It is noted that the mesh generation is carried out manually.

5.3.2 Finite element model To apply the boundary conditions and loads, stress-free strain-recovery as well as the fixed-strain stress-recovery paths are simulated. The whole steps of these thermomechanical cycles are shown in Fig. 5.10. Details of the stages are given as follows (four steps): Step 1: Loading step: a uniaxial tensile deformation (displacement on the top Face) is applied at a high temperature (65 C) at a rate of 0.4% strain/s (in 50 s). Step 2: Cooling step: keeping fixed the applied deformation, the RVE in an interval (2400 s) is cooled down at a rate of 1o C=min to 25 C (temporary shape fixation). Step 3: Unloading step: in this step, at a constant temperature 25 C, the external load is set to zero in a short time (50 s). Step 4: Heating step: the sample is submitted to the heating at a rate of 1 C/min up to 65 C in a long time (2400 s). In this step, if no stress is applied to the material, we have the stress-free strain-recovery (red-diamonds in Fig. 5.10). In contrary, if one does not allow the material to recover the initial shape, we have fixed-strain stress-recovery (green circles in Fig. 5.10).

The initial boundary conditions applied to the RVE are shown in Fig. 5.11, while Table 5.6 lists BCs in different steps of the thermomechanical cycle. Some morphologies of CCNT nanoparticles are summarized in Fig. 5.12. They are just two types of CCNT where for instance, in case b, the pitch is infinity that means an equivalent CNT and case a [25] is a CCNT particle which other used

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1

0.8

0.6

0.4

T (S= 65oC) (A = 2.3 MPa) Free

0.2

Fixed

0

Free

(A = 2.3 MPa)

(A = 0.2 )

-0.2 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Time (s)

Figure 5.10 The thermomechanical cycle of shape memory polymer at 20% prestrain and temperature rate of 1 C/min, parameter “A” denotes the scale factor [24].

Figure 5.11 Identify faces of representative volume element for the definition of boundary condition [24].

Table 5.6 The initial and boundary conditions applied to the RVE [24]. Steps

Top Plane (zx)

Bottom plane (zx)

Front plane (yx) and right plane (zy)

Back plane (yx)

Left Plane (zy)

RVE temperature ( C)

Initial Loading

None U2 6¼ 0 U1 5 U3 5 0 U2 6¼ 0; fixed U1 5 U3 5 0 U2 6¼ 0; release U1 5 U3 5 0 U2 6¼ 0; released U1 5 U3 5 0 U2 6¼ 0; fixed U1 5 U3 5 0

U2 5 0 U2 5 0

U2 U1 U2 U1 U2 U1 U2 U1 U2 U1

U3 5 0 U3 5 0 U1 6¼ U2 U3 5 0 U1 6¼ U2 U3 5 0 U1 6¼ U2 U3 5 0 U1 6¼ U2 U3 5 0 U1 6¼ U2

U1 5 0 U1 5 0 U2 6¼ U3 U1 5 0 U2 6¼ U3 U1 5 0 U2 6¼ U3 U1 5 0 U2 6¼ U3 U1 5 0 U2 6¼ U3

65 65

Cooling Unloading Heating (shape recovery) Heating (force recovery)

U2 5 0 U2 5 0 U2 5 0 U2 5 0

6¼ 0 6¼ U3 6¼ 0 6¼ 0; fixed 6¼ U3 6¼ 0 6¼ 0; released 6¼ U3 6¼ 0 6¼ 0; released 6¼ U3 6¼ 0 6¼ 0; fixed 6¼ U3 6¼ 0

6¼ 0 6¼ 0 6¼ 0 6¼ 0 6¼ 0

6¼ 0 65 to 25 6¼ 0; fixed 25 6¼ 0; released 25 to 65 6¼ 0; released 25 to 65 6¼ 0; fixed

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Figure 5.12 Some coiled carbon nanotube cases: (A) P 5 70 nm, Di 5 30 nm, Ls 5 283.2 nm and (B) P 5 infinity, Di 5 0, Ls 5 200 nm [24].

Figure 5.13 Coiled carbon nanotube distribution with randomly and unidirectional distributed nanoparticles type “a” and relevant representative volume element: (A) unidirectional distribution of coiled carbon nanotubes and (B) random distribution of coiled carbon nanotubes [24].

CCNTs particles are similar to type a with different geometrical parameters. The geometric parameters of these particles are given in caption of Fig. 5.13. The distribution of dispersed CCNTs is shown in Fig. 5.13. The RVE and distribution of CCNTs for unidirectional nanoparticles as well as the randomly distributed nanoparticles are shown in different volume fractions and geometric parameters.

5.3.2.1 Determination of representative volume element size According to the definition of RVE, the acceptable size of the RVE plays a key role in validity of the numerical results. If the RVE size is smaller than the proper size, the RVE cannot properly represent the mechanical response of the nanocomposite.

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Figure 5.14 The size dependency checking analysis of representative volume element, the stress of force recovery versus time for four sizes of representative volume element [24].

Therefore the results are not valid; on the other hand, if the size of RVE is larger than the proper size, we may have a higher computational cost. For this aim, a convergence simulation is carried out. Different RVE sizes with the same volume fraction and geometrical parameters are created. For four sizes of RVE, preliminary simulation of RVE is performed. Fig. 5.14 depicts FEM results for the stress in a fixed-strain stress-recovery thermomechanical cycle in four sizes of 908.6, 1000.0, 1100.7, and 1205.1 nm. As shown in Fig. 5.14, increasing the size of RVE from 908.6 to 1205.1 nm, the stress as one may observe increases except for the RVE size of 1100.7 nm. The stresses converge to the same amount when the RVE size with 1100.7 nm is selected, thus in the following, the results are reported for this RVE size.

5.3.3 Numerical results and discussion 5.3.3.1 The effect of volume fraction of coiled carbon nanotube on SMPC In this section, the effect of volume fraction of CCNT on the mechanical properties of SMPC is examined. To this end, three types of RVE with different volume fractions from zero (pure SMP) to 0.6% inclusion are generated. The results are reported for both fixed-strain stress- and stress-free strain-recovery paths in Fig. 5.15. Fig. 5.15 illustrates the influence of the volume fraction on the effective stress or equivalently the strength of SMPC. As one may observe, adding 0.62% CCNT to the pure SMP, the effective stress is increased about 15%, which agrees with

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Figure 5.15 The effect of volume fraction on the thermomechanical properties of SMPC under 20% prestrain. (A) Stress
strain
temperature diagram for the stress-free strainrecovery of SMPC. (B) Stress
strain
temperature diagram for fixed-strain stress recovery of SMPC [24].

[5,14,25]’s results. As shown, the volume fraction does not affect the recovery strain of SMPC in shape recovery path.

5.3.3.2 The effect of spring length of coiled carbon nanotube (or aspect ratio) As one of the major geometric parameters, the spring length Ls of CCNT (or the aspect ratio) is studied on the mechanical properties of SMPC. Along with this purpose, four types of RVE with CCNTs by spring length 150, 250, 300, and 374 nm are generated. The results have been reported for both the fixed-strain stress- and stress-free strain-recovery paths. Fig. 5.16 depicts the effect of spring length of CCNT (or aspect ratio) on the effective stress or the strength of SMPC. It is shown that increasing the spring length from 150 to 374 nm can increase the effective stress about 7% due to the specific surface increment of CCNT. Unlike to the volume fraction, the spring length has an impressive effect on the strain recovery whereby increasing the spring length of CCNT from 150 to 374 nm, the strain recovery increases about 8%, which agrees with references’ [15,27] results.

5.3.3.3 Effect of the pitch of coiled carbon nanotube (or the number of coils) In this section, the effect of the pitch of CCNT (or the number of coils) on the mechanical properties of SMPC is investigated. Four types of RVE including CCNTs by the pitch of 90, 120, and 150 nm and infinity (with the same length CNT) are created. Same as the latter section, numerical results are given for both the fixedstrain stress- and stress-free strain-recovery paths.

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Figure 5.16 The effect of spring length of coiled carbon nanotube (or aspect ratio) on the thermomechanical properties of shape memory polymer under 20% axial prestrain. (A) Stress
strain
temperature diagram of shape recovery of SMPC and (B) stress
strain
temperature diagram of stress recovery of SMPC [24].

Figure 5.17 The effect of the pitch of coiled carbon nanotube (or the number of coils) on the thermomechanical properties of shape memory polymer. (A) Stress
strain
temperature diagram of shape recovery of SMPC and (B) stress
strain
temperature diagram of force recovery of SMPC [24].

Fig. 5.17 reveals that the effect of number of coils on the mechanical properties of SMPC (while keeping the spring length, volume fraction and other geometric parameters constant) is not significant, by increasing the pitch from 90 to 150 nm, the effective stress increases about 4.4% which is in agreement with results in Ref. [39]. However, for an infinite pitch, the effective stress decreases which means that

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the SMPC reinforced by CCNT has a higher strength than the SMPC reinforced by CNT (with the same free length).

5.3.3.4 The effect of orientation of coiled carbon nanotube An important concept in the nanocomposite is the orientation and distribution of nanoparticles. Ref. [40] used experimentally floating catalyst method to large-scale synthesis of helical CNT (CCNT or HCNT). Ref. [41] experimentally investigated the effect of CCNT-coated carbon fiber on viscoelastic property of epoxy in three different modes of deformation. In this section we aim to numerically investigate this effect. Three types of RVE are constructed: two unidirectional CCNTs and one randomly distributed CCNTs. The orientation of CCNTs in unidirectional CCNTs are selected to be long the loading direction for one type and perpendicular to the loading direction for the other type. The computed results for these three types are shown in Fig. 5.18. Fig. 5.18 shows that for SMPC, the effective stress and strength along the stretching direction are higher than perpendicular to it, but for the shape recovery, the strain recovery for unidirectional and randomly nanoparticles is the same, but the strain recovery for the perpendicular loading is less than the unidirectional and randomly distributions.

5.3.3.5 The effect of heating rates and prestrain Finally, the performance of the model is checked in another heating rate (5 C/min) and in two other prestrains 5% and 10%. Along with this view, an RVE involving

Figure 5.18 The effect of orientation and distribution of coiled carbon nanotubes in three types of SMPCs: randomly distributed coiled carbon nanotubes, unidirectional coiled carbon nanotubes along the loading direction, and unidirectional coiled carbon nanotubes perpendicular to the loading direction. (A) Stress
strain
temperature diagram for shape recovery of SMPC and (B) stress
strain
temperature diagram for the stress recovery of SMPC [24].

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2.5

(A)

= 20% = 10% = 5%

Stress (MPa)

2

(B)

0.2

= 20% = 10% = 5%

0.15

Strain (-)

1.5 1 0.5

0.1

0.05

0 -0.5

0 0

1000

2000

3000

4000

0

5000

1000

2000

Time (s)

3000

4000

5000

Time (s)

1.5

(C)

0.2

(D) Filled, Model, 1o C/min

0.5

Filled, Model, 5o C/min Pure, Model, 1 o C/min

0

Pure, Model, 5 o C/min Pure, Exp, 1o C/min

-0.5

0.15

Strain (-)

Stress (MPa)

1

Filled, Model, 1oC/min

0.1

0.05

Pure, Exp, 5o C/min

Filled, Model, 5oC/min Pure, EXP, 1 oC/min Pure, EXP, 5 oC/min Pure, Model, 1oC/min Pure, Model, 5oC/min

0

-1 30

40

50

Temperature ( o C)

60

30

40

50

60

Temperature ( o C)

Figure 5.19 The effect of prestrain (5%, 10%, and 20%) and heating rate (1 C/min, 5 C/ min) on the thermomechanical response of SMPC in 0.67% inclusions. (A) Stress in a fixedstrain stress recovery, (B) strain in a stress-free strain recovery, (C) recovery stress versus the temperature in different heating rates under 20% prestrain, and (D) strain recovery versus the temperature in various heating rates under 20% prestrain. The “Exp” in legends denotes experimental data adopted from [24,28].

0.67% CCNT type “a” (but with different pitch, spring length and internal diameter) are created in four situations; same as previous simulations, a 20% uniaxial strain is applied to RVE, while the heating rate is 1 C/min. Then, the same RVE is simulated in 5 and 10% strain where the numerical results are shown for both the fixedstrain stress- and stress-free strain-recovery paths in Fig. 5.19A and B, respectively. To observe the effects of heating rate on the thermomechanical response of SMPC, the RVE is simulated in the heating rate of 5 C/min as shown in Fig. 5.19C and D. In Fig. 5.19A, the effects of prestrain (5%, 10%, and 20%) on the thermomechanical stress response of SMPC are illustrated in a fixed-strain stress recovery. In Fig. 5.19B, the same effect on the strain in a stress-free strain recovery is shown. Fig. 5.19C reveals that increasing the heating rate the amount of stress in a fixedstrain path increases. In comparison to pure SMP, SMPC involving CCNTs have high strength. Fig. 5.19D shows as heating rate increases, the strain recovery will increase, and finally, the SMPC has a higher strain recovery in comparison to the pure SMP. The main reason for this effect is similar to the viscoelastic behavior of materials, which by increasing the loading rate, the effective load increases. In here, at higher heating rates, the polymeric crosslinks do not have enough time for the relaxation. Therefore the amount of the effective force increases. This

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phenomenon is depicted in Fig. 5.19C and D in the form of a shift of the curves to the right at a faster heating. In this section, the homogenization and the finite element analysis of SMP reinforced by CCNT in large deformations were investigated. For modeling of SMP, a thermo-visco-hyperelastic constitutive equation proposed by [4] was employed. Then, based on the Monte-Carlo algorithm, the nanoparticles of CCNT were dispersed (randomly and unidirectional) in RVE. At first, the acceptable size of RVE in the stress recovery path of SMPC was determined. Secondly, the effects of the volume fraction of nanoparticles, orientation, and geometric parameters of CCNT were investigated on the thermomechanical properties of SMPC. And finally, the validity of the presented approach was calculated for different heating and strains rates. Conclusion of this research can be summarized as follows: Increasing the volume fraction to 0.62% leads to an increase in the stresses of SMPC up to 15%. In contrast, the volume fraction has no significant influence on the thermomechanical properties of SMPC. G

G

G

The spring length of CCNT as a geometric parameter has a significant effect on the strain recovery as well as the stress recovery of SMPC, so that by increasing the spring length of CCNT from 150 to 374 nm, the strength of SMPC improves about 8% and the strain recovery increases about 7%. The pitch or the number of the coil of CCNT has a slight impact on the thermomechanical properties of SMPC. Increasing the pitch of CCNT, enhances the strength and strain recovery of SMPC. But for a large magnitude of the pitch (CCNT tends to CNT in the same free length) the SMPC reinforced by CCNT has a higher strength than the SMPC reinforced by CNT (with the same free length). The orientation and distribution of the nanoparticles are important parameters. It was shown that when the mechanical loading is along the CCNTs orientation, the effective stress was higher than when the loading was perpendicular to the unidirectional CCNTs. But for the shape recovery, the strain recovery for the unidirectional and randomly nanoparticles are the same. But the strain recovery for the perpendicular loading is less than the unidirectional and randomly distributions.

5.4

Thermomechanical behavior of shape memory polymer beams reinforced by corrugated polymeric sections

To overcome the low stiffness of SMPs, researchers proposed several technics for reinforcing the SMPs with high modulus materials. By reinforcing SMPs not only the load carrying performance is improved, other desirable characteristics are introduced, the possibility of nonthermal stimuli, conductivity, and so on. The conductive SMPs are promising substitutes for devices which are exploited in microsensors and microactuators. Many researches have been dedicated to design and analysis of smart actuators based on SMPs with the ever-increasing application of them in space structures, medicals, microelectromechanical systems (MEMS) and oil exploration industries.

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Zhang and Ni [42] studied the effect of carbon fiber fabric reinforcement on the response of SMP sheets. They showed that the shape recoverability of SMP-based laminates is larger than that of the SMP sheets. Lu et al. [43], investigated the sensing and actuating characteristics of an SMP composite with hybrid fillers. They designated several potential applications of the proposed composite SMP. A comprehensive review of fabrication and modeling technics of SMPs is presented in [44]. They studied various actuation methods and multifunctional properties of SMP composites and addressed several potential applications of SMP based structures. A study of literature reveals that although SMPs are utilized in applications involving bending, the flexural study of these materials is limited. Baghani et al. [45
47] proposed analytical formula for SMP beams based on the constitutive equations in Refs. [11,48]. Takeda et al. [49] studied the flexural behavior of a hybrid composite beam, both experimentally and numerically and showed that by controlling the temperature in the SMP layer via electric resistive layers, the flexural behavior can be controlled and reported a potential in morphing structures in aerospace. It is established that the polymers have poor thermal conductivity. In order to overcome this shortage, the SMP core can be covered with a veneer of a thermally conductive material. Fillers are another potential candidate. This way, SMPs can be stimulated employing conductive fillers (see [50,51] among others). Corrugated structures have long been utilized in aerial vehicles thanks to their high flexural stiffness to weight ratio, good manufacturability, and ease of maintenance. In this section, the flexural behavior of SMP beams reinforced with polymeric corrugated structures is investigated. A polymer with high mechanical performance is used for the reinforcing corrugated structure. Euler
Bernoulli beam theory (EBBT) in conjunction with the SMP model in [52,53] are employed to study the effect of reinforcing SMPs with different corrugated patterns. It is shown that the finite difference procedure employed here is capable of rendering accurate results in a few steps which proves to be computationally efficient. Numerical results show a great enhancement in load capacity of the reinforced beam at a small decrease of shape fixity. In case of single cell asymmetric reinforced composite sections, it was observed that a temperature change leads to a small curvature due to the displacement of the neutral axis. It is established that the SMP-based actuators have a limited loading capacity and they generate small loads in their thermomechanical cycle. Further has been shown by several researchers that introducing a reinforcing agent improves the load capacity dramatically. Bearing this in mind the purpose of the current work is to introduce a new, efficient and economical solution to improve the load capacity of the SMP beams. Beams are perfect candidates for generating force in smart actuators. Moreover, considerations about position of the neutral axis as a result of transformation and its effect on bending behavior are of importance. Further, the idea presented in this section is quite affordable and feasible when compared to other reinforcing technics. This section is organized in the following form. Firstly, a constitutive model for SMPs originally proposed in [52,53] is reviewed briefly. Based on the physical

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189

grounds and EBBT, the governing equations of the composite beam are formulated. Further, a finite difference scheme is employed to solve the governing equations for different corrugated sections. Subsequently, the numerical results are presented and conclusions are stated.

5.4.1 Shape memory polymer constitutive model based on phase transition concept In this section, the constitutive theory developed by Chen and Lagoudas [52,53] is adopted to model the thermomechanical behavior of SMP. The thermoelastic constitutive equation is given by E 5 E^ ðS; θÞ

(5.26)

with E being the strain tensor, E^ ðS; θÞ the constitutive function, S the second Piola
Kirchhoff stress and θ the temperature. Eq. (5.26) can be approximated by the first order expansion (assuming infinitesimal strains) about the stress-free condition, that is,

E 5 E^ S50

^

@E 1 @S

½S 5 Eθ ðθÞ 1 M ðθÞS

(5.27)

S50

^ in which Eθ is the thermal strain and MðθÞ represents the first order derivative of E with respect to S. Here since the behavior of the SMPs in infinitesimal strain range is studied, the deformation gradient F and the strain E are related via: E5

 1 T F 1F 2I 2

(5.28)

where I stands for the second order identity tensor. Assuming continuous deformation during the cooling process and instantaneous transformation at the transition temperature, the constitutive behavior of SMPs can be expressed in the following form [52,53]: ^ a ðSðtÞ; θðtÞÞ 1 FðtÞ 5 ½1 2 φðθðtÞÞF

ðt 0

0

21 ^ a ðSðtÞ; θðtÞÞφ0 ðθðτ ÞÞθ~ ðτ Þdτ F^ f ðSðtÞ; θðtÞÞF^ f ðSðtÞ; θðtÞÞF

(5.29) ^ a ðS; θÞ and F^ f ðS; θÞ are with φðtÞ being the volume fraction of the frozen phase, F the constitutive functions of the active and frozen phases, respectively, and θ~ ðtÞ is called the "net cooling history", which can be found from the actual thermal history through replacing local heating portions with a constant temperature (Fig. 5.20). Finally, a prime denotes differentiation with respect to appropriate variable, either θ or t. In Eq. (5.29), Fig. 5.20 and hereafter in this section t represents time variable; While τ in Eq. (5.29) is the integration variable.

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Figure 5.20 Net and actual cooling history [54].

Using Eq. (5.28) and rewriting Eq. (5.27) for active and frozen phase, we obtain: i 1h^ ^T 2I5E ^ a 5 Ea 1 M a S Fa 1 F θ a 2

(5.30)

i 1h^ T Ff 1 F^ f 2 I 5 E^ f 5 Efθ 1 M f S 2

(5.31)

Combining Eq. (5.28) to Eq. (5.31), and regrouping terms, it is possible to write: E 5 Ee 1 Eθ 1 Es

(5.32)

where Ee , Eθ , and Es are elastic, thermal, and the stored strain tensors, respectively, which are defined as follows: Ee 5 ½1 2 φðθÞM a ðθÞ½S 1 φðθÞM f ðθÞ½S

(5.33)

Eθ 5 ½1 2 φðθÞEaθ 1 φðθÞEfθ

(5.34)

Es 5

ðt n o 0 0 ðEaθ 2 Efθ Þ 1 ðM a ðθÞ 2 M f ðθÞÞ½S θ~ φ dτ

(5.35)

0

in which Eaθ and Efθ are the active and frozen phase thermal strains, respectively. Moreover, M a ðθÞ and M f ðθÞ are respectively, the derivatives of constitutive equation in Eq. (5.27) for fully active and frozen phases. In case of isotropic SMPs, the tensorial variables expressions in Eqs. (5.33)
(5.35) are simplified further as Eaθ 5 εaθ I;

Efθ 5 εfθ I

(5.36)

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191

M a ðθÞ½S 5

 1  ½1 1 ν a ðθÞS 2 ν a ðθÞðtrSÞI Ea ðθÞ

(5.37)

M f ðθÞ½S 5

 1  1 1 ν f ðθÞ S 2 ν f ðθÞðtrSÞI Ef ð θ Þ

(5.38)

with εaθ and εfθ the nonzero components of thermal strain tensors. Ea ðθÞ and Ef ðθÞ are the elastic moduli of the active and the frozen phases, while ν a ðθÞ and ν f ðθÞ are the Poisson’s ratios. Further trS denotes the trace of the second Piola
Kirchhoff stress tensor. For infinitesimal strains, it can be shown that the second Piola
Kirchhoff and Cauchy stress are the same. According to the foregoing, (see [52,53] for details) the nonzero components of the aforementioned tensorial variables can be defined as follows: εe ðθÞ 5 C ðθÞσðθÞ

(5.39)

εθ 5 ½1 2 φðθÞεaθ 1 φðθÞεfθ 1 εs 5

ðt

ðT

αðθÞdθ

(5.40)

Tl

0

Dðθðτ ÞÞσθ~ dτ

(5.41)

0

in which σ and θ stand for the stress and temperature, respectively. Cθ ; αθ and Dθ represent instant elastic compliance function, coefficient of thermal expansion and distributed elastic compliance, respectively, which are defined in the following form [52,53]: C ðθÞ 5

1 2 φðθÞ φðθÞ 1 Ea ðθÞ Ef ðθÞ

(5.42)

αðθÞ 5 ð1 2 φðθÞÞαa 1 φðθÞαf  DðθÞ 5

 1 1 dφ 2 Ea ðθÞ Ef ðθÞ dθ

(5.43) (5.44)

where φðθÞ serves as an internal variable defining the volumetric fraction of the frozen phase, which is function of temperature according to: ÐT


2

φðθÞ 5 1 2 e

Th

DðθÞ CðθÞ2C ðθl Þ

 dθ

(5.45)

in which Th and Tl are the higher and the lower limit of the temperature. To sum up, the constitutive model of SMPs can be expressed in the following form: ðT

ε 5 C θÞσ θÞ 1 εθ 1 DðθÞσðθÞdθ~ (5.46) Tl

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This relation reveals that the mechanical behavior of SMPs, in a shape memory effect cycle, both for the strain and stress recovery, is dependent on the history of the applied temperature and stress and/or strain. To calibrate the parameters in the model, Ref. [53] employed the experimental results of Ref. [7], which are obtained from a commercial thermoset epoxy resin. The history of thermal strain can be found from the stress-free cooling/heating regime. Furthermore, the distributed thermal compliances CðθÞ and DðθÞ may be obtained as: C ðθÞ 5

εpre 2 ε ðθÞ σ ðθ Þ

(5.47)

DðθÞ 5

1 d ½ε ðθÞ 2 εθ  σ ðθÞ dθ

(5.48)

with εpre being the strain at which the specimen is held while cooled down, ε ðθÞ and σ ðθÞ the strain and stress history during the cooling process. For the reinforcing material linear elastic behavior is considered according to σ 5 Er ðε 2 αr ΔθÞ

(5.49)

Since the deformations are small and the strains are infinitesimal, linear elastic behavior accurately describes the behavior of the reinforcing material.

5.4.2 Bending of a reinforced shape memory polymer beam As stated before, SMP structures are usually reinforced in real world applications. Recently numerous reinforced SMP structures have been proposed. However corrugated structures that are filled with SMP, provide an efficient way for reinforcing SMP structures. This section is devoted to developing formulation of bending of a reinforced SMP beam based on the general assumptions of EBBT. In order to obtain the flexural behavior of a beam the theory assumes linear displacement across the thickness of the beam neglecting lateral deformation and change of thickness. Moreover, the effect of transverse shear is not taken into account by this theory. The EBBT is only applicable to problems with infinitesimal strains and small rotations. It is wellestablished that this theory predicts the behavior of beam in small deformation range with great accuracy. The EBBT is chosen to study the flexural behavior of the reinforced beam since the numerical investigations of the present work lies within the range of applicability of the theory. Fig. 5.21 depicts different corrugated patterns that are investigated in this section. According to EBBT, Eq. (5.46) is rewritten as: ðT

ε 5 2 κy 5 C θÞσ θ; yÞ 1 εθ 1 DðθÞσðθ; yÞdθ~ Tl

(5.50)

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Figure 5.21 Different corrugated patterns: (1) reinforcing material and (2) shape memory polymer core (reinforcing cover has been cut for better clarification) [54]. (A) Trapezoidal; (B) Sinusoidal; (C) Triangular; (D) Rectangular (Square); (E) Multi-cell corrugated section.

where κ is the beam curvature and the beam axis is located along the x-axis, while the transverse direction lays on y-axis (Fig. 5.21E). Here it is noted that the model proposed in [53] holds in integral form. In other words, it cannot be expressed in a rate form since the differential in (5-50) is the differential of net cooling history, while the distributed thermal compliance DðθÞ and stress σðθÞ are functions of the cooling history. As a result, the governing equations should be solved in a step-wise manner such as finite difference scheme. As the stress profile prior to the analysis is unknown, here a discrete form is investigated. Accordingly, we subdivide the time interval of interest [0, t] in subincrements and solve the evolution problem over the generic interval [tn, tn11] with tn11 . tn. Therefore Eq. (5.50) is discretized by the so-called finite difference method as: 2κn11 y 5 Cn11 ðθÞσn11 ðθ; yÞ 1 εn11 1 θ

1 ðDn11 σn11 ðθ; yÞ 1 Dn σn ðθ; yÞÞδθ~ 1 Sn 2 (5.51)

where Sn is the accumulated sum of previous steps from Tl to Th . During the first stage of the thermomechanical cycle, the last two terms on the right side of (5.51) have no contribution to the strain, since the temperature is kept fixed, therefore the flexural behavior of the beam falls within the composite section theory; consequentially the governing equations can be written as follows: Ms 1 Mr 5 M

(5.52)

Also, the curvature may be obtained from κ5

M Es Is 1 Er Ir

(5.53)

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in which M is the total moment applied on the section. It is noted that properties of the SMP and the reinforcing material are denoted by subscripts “s” and “r”. Moreover, I stands for the moment of inertia about z-axis. Also Es and Er represent the current elastic moduli of SMP and the reinforcing material, respectively. It should be emphasized that during the phase transformation, the neutral axis of the composite section is displaced as a function of temperature, since the effective elastic modulus of SMP varies from Ea to Ef and vice versa. From the theory of composite sections [55], the location of neutral axis may be found from the following equation: y cs 5

E s As y s 1 Er Ar y r Es As 1 Er Ar

(5.54)

where y r , y s and y CS are the locations of neutral axes for the reinforcing material, the SMP, and the composite section, respectively (Fig. 5.22). During the cooling (or heating) process the modulus of elasticity of the SMP changes. As a result, the location of the neutral axis of the composite section varies. At each temperature step, it can be assumed the bending occurs about the instantaneous neutral axis, whose location can be calculated from Eq. (5.54). At the second stage, the mechanically constrained specimen cools down to Tl . At each step, there are three unknown variables, that is, MS , MR , and κn11 . Multiplying Eq. (5.51) by (2y) and integrating over the cross-sectional area of the SMP about the current neutral axis, one obtains: s κn11 Is ðθÞ 5 Cn11 ðθÞMn11 ðθÞ 1

 1 s s s Dn11 Mn11 1 Dn Mns δθ~ 1 M^ n 1 Mth 2

(5.55)

where the moments in Eq. (5.55) are defined as follows:


ð  s s Mn11 ; Mns ; M^ n 5 2 yðσn11 ; σn ; Sn ÞdA

(5.56)

AS

ð s Mth

52 AS

yεn11 θ dA

(5.57)

Figure 5.22 Neutral axis of the shape memory polymer core and the reinforcing material in symmetric and asymmetric sections [54].

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It is stressed that since the neutral axis of the composite section does not generally lay on the neutral axis of each portion, the moment of thermally induced stresses does not vanish. Therefore a moment due to presence of thermal stresses is introduced in Eq. (5.55). IS ðθÞ is the second moment of the SMP portion of the beam section as a function of temperature. All the moments and moments of area are calculated with respect to the instantaneous neutral axis of the composite section. Moreover, Eq. (5.52) must hold, assuming an elastic behavior for the reinforcing material. Combining Eqs. (5.52) and (5.55), the governing equation of composite beam with the effects of shape memory is recast in the following form: 

   1 s s r Er Ir 1 E s Is κn11 5 M 1 E s Dn Mns δθ~ 1 M^ n 1 Mth 2 Mth 2

(5.58)

in which E s is the effective instantaneous modulus of elasticity for the SMP which is defined as follows:  21 1 ~ E s 5 Cn11 1 Dn11 δθ 2

(5.59)

r is the moment due to the thermal stresses in the reinforcing mateMoreover, Mth rial, which is defined as: r Mth 5Er αr QΔθ

(5.60)

in which Q and Δθ are, respectively, the first moment of the reinforcing material about the neutral axis of the composite section and the total temperature change (i.e., Δθ 5 Th 2 T). When the curvature is computed for the composite section, it can be substituted back into Eqs. (5.51) and (5.55), in order to find the moment exerted on the SMP part and the stress distribution. Within the third stage of the thermomechanical cycle, the conventional theory of composite sections may be employed to describe the flexural behavior of the composite beam. In other words, replacing the modulus of elasticity to the value pertaining to the end of transformation temperature, the elastic unloading stage could be interpreted similar to the loading process in the reverse direction. The governing equation on the behavior of the SMP for the heating process in the present theory is expressed as in the following equation: ðT

ε 5 2 κy 5 C θÞσ θ; yÞ 1 εθ 1 DðθÞσðθ; yÞdθ

(5.61)

Tl

or in the discrete form of 1 2κn11 y 5 Cn11 ðθÞσn11 ðθ; yÞ 11 εn11 θ

1 ðDn11 σ n11 ðθ; yÞ 1 Dn σ n ðθ; yÞÞδθ 1 S n 2 (5.62)

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and upon integration, s κn11 Is ðθÞ 5 Cn11 ðθÞMn11 ðθÞ 1

1 s s s s Dn11 M n11 1 Dn M n δθ 1 M n 1 Mth 2

(5.63)

Here it is noted that δθ 5 2 δθ~ and barred parameters are known from the cools ing stage, leaving κn11 and Mn11 the unknowns to be determined. Moreover, since the external moment is absent, the following equation holds for the composite section: Ms 1 Mr 5 0

(5.64)

Following the same discussion as in cooling process, one can write: ðEr Ir 1 Es Is Þκn11 5

1 s s s s r Dn11 M n11 1 Dn M n δθ 1 M n 1 Mth 2 Mth 2

(5.65)

where in contrast to the cooling stage, Es appears on the right-hand side of the governing equation [compare Eqs. (5.58) and (5.65)]. Based on the formulation developed in this section, both the shape memory and stress recovery regimes can be described.

5.4.3 Numerical results and discussion In this section, for the sake of discussion, beams of L/h 5 10 (Fig. 5.23) are investigated. As stated before the SMP is a commercial thermoset epoxy resin for which the tensile and compression results are reported in [7]. Further, Chen and Lagoudas [53] employed the experimental results to calibrate their model (Table 5.7). The reinforcing material is considered to be polycarbonate with thickness of 0.5 mm. Moreover, it is assumed that the reinforcing material undergoes linearly elastic deformation since the strains during deformation remain below εyield (Table 5.7). Polycarbonates have long been employed for their acceptable mechanical properties. In the thermomechanical cycle of the SMPs, the polymer-based structure undergoes a large temperature change; as a result, the thermal stability of the reinforcing material is of great importance. Polycarbonates maintain their mechanical

Figure 5.23 Geometry and the loading conditions of the investigated composite beam.

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197

Table 5.7 Mechanical properties of composite beam constituents [54]. SMP mechanical properties [53] Ea 5 ðCðTh ÞÞ21 5 8:8MPa

Ef 5 ðCðTl ÞÞ21 5 750MPa

Tl 5 270K

αf 5 0:9 3 1024 =K

αa 5 1:8 3 1024 =K Tg 5 343K

Th 5 358K

Polycarbonate mechanical properties [56] αr 5 6:6 3 1025 =K

Er 5 2:1GPa

Tglassy 5 423K

εyield 5 6%

s

T = 27 oC

y/h

0.5

1

0.6

T = 56 oC

-0.5

0

0.2

0.4

0.6

0.8

1

0.5 y/h

(B)

(A)

/

max

1 0.8

0

0.8

0.6

%

0.2

0

0

0.4 -0.5

0 0.95

1 T/Tg

-0.2 0

0.2

0.4

0.6

0.8

1

0.5

-0.4 -0.6

T = 85 oC

1.05

y/h

0.2

Elastic loading (Stage 1) Constrained cooling (Stage 2) Elastic unloading (Stage 3) Free heating (Stage 4)

0.4

0

-0.5

-0.8 0

0.2

0.4

x/L

0.6

0.8

1

-1

Figure 5.24 (A) Variation of fixed end curvature in a cantilever beam in shape memory cycle (κmax 5 0.109/m). (B) Distribution of the stored strain in shape memory cycle [54].

properties over a wide range of temperature, thanks to their high Tg [56]. The mechanical properties of the constituent materials are tabulated in Table 5.7. Following the procedure outlined in the previous section, a cantilever SMP beam (Fig. 5.23) is considered to undergo shape memory cycle. Fig. 5.24A depicts the variation of curvature at the fixed end with respect to the temperature, in the thermomechanical cycle. It is observed that the curvature does no change appreciably during the cooling process. Most of strain is stored in the SMP, while it is being cooled down; therefore only a small portion of the strain is recovered during the unloading. At the completely unloaded state, the temperature is increased and as a result the SMP composite beam recovers its initial shape (shape memory cycle). Distribution of the stored strain during shape memory cycle in the SMP beam is shown in Fig. 5.24B. It can be seen that increasing the temperature, the stored strain is gradually released and therefore the composite beam recovers its original shape. From Eq. (5.51), the instantaneous increment of stored strain is proportional to the exerted stress. In addition, during the cooling process, the moment along the cantilever beam remains constant, having this in mind the distribution of the stored strain in Fig. 5.24B is justified.

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Figure 5.25 Variation of the fixed end curvature in a cantilever composite beam in the partial stress recovery cycle (F0 5 1 N, κmax 5 0.109/m) [54].

If the mechanical load is not completely removed upon the heating, a different path is observed in the curvature-temperature diagram. Fig. 5.25 illustrates the effect of partial load removal on SMP composite behavior, which is called partial stress recovery regime. It is assumed that initially the beam deforms elastically under the action of force F0 at the tip further this force is maintained during cooling. However, the beam is partially unloaded to F. It can be seen that after the heating process, the curvature pertaining to each fraction of load is recovered. Mainly, utilizing the SMPs, two purposes are sought, shape memory effect and force generation. The former is the main concern in self-deployable structures [57]; however, in the SMP-based actuators, the force generation capability is the principal challenge. It is stated before, that the reinforced SMPs are more practical in real world applications, since SMPs possess a limited loading capacity. A practical solution to this shortcoming is reinforcing these materials. Among all possible solutions, it seems that corrugated structures filled with SMP are the easiest way. Here, for the sake of comparison, different corrugated patterns are considered to be made of equal SMP content. Although it can be shown that even with the same SMP content, the sections geometries can be changed in order to make the neutral axes coincide (e.g., isosceles triangle), usually, in design, it is required that only one of the dimensions is to change at a time, while the other one is kept constant (either height or width), to meet a possible constraint of design. Hence, in each study only one of the dimensions is changed, that is, either height of the section or its width. From now on by equal height (or width) we mean that the height (or width) of section along y-axis (z-axis) in Fig. 5.21 is the same as the height (or width) of the rectangular section while the base (or height) is changed. Imposing this constraint in conjunction with equal SMP content constraint, the dimensions of

Shape memory polymer composites: nanocomposites and corrugated structures

(A)

(B)

(C)

(D)

199

Figure 5.26 Effect of reinforcing the shape memory polymer beam on the thermomechanical cycle of each corrugated pattern: (A) rectangular, (B) sinusoidal, (C) triangular, and (D) trapezoidal [54].

nonsquare sections is computed, that is, height of the sinusoidal and triangular sections, and height and smaller base of the trapezoidal section. The shape memory cycle of different corrugated sections is depicted in Fig. 5.26. To make the results comparable and for better clarification, all curvatures of a specific section are divided by the maximum curvature of the section. Since in SMP-based actuators the shape prior to the heating process is of seminal importance in Fig. 5.26 it is assumed that all sections are bent to have the same curvature. As a result, κmax is different for each section, however κu which is the curvature after unloading, is considered to be 0.1077/m for all sections. In Fig. 5.26A, the thermomechanical cycles of the unreinforced and the corrugated, rectangular sections are compared. It can be seen that the shape fixity (i.e., the ratio of the curvature after unloading to the curvature prior to unloading) is reduced by introducing the reinforcing material. However, this is the common trend in reinforcing SMPs; in other words, at a small expense of shape fixity, other mechanical properties are enhanced. The curvature-temperature diagrams for other corrugated sections are depicted in Fig. 5.26B
D. As stated before, in asymmetric corrugated sections due to the mismatch of coefficient of thermal expansion a moment is exerted on the section, which leads to variation of the curvature during the cooling process. It can be seen that, in contrast to the equal height case, thermally induced moments tend to increase the curvature in case of sinusoidal and triangular in case of equal width. However, for the trapezoidal section, in both cases, the thermal moments raise the curvature value. This is due to the relative position of neutral axes for the reinforcing cover and the SMP core. The neutral axis of the reinforcing material lays at

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the same side of the neutral axis of the SMP core; on the contrary, the relative positions of the axes alternates in case of the triangular and the sinusoidal sections, which accounts for decreasing (increasing) curvature in case of equal height (equal width) during the cooling process. The numerical results for the performance of different corrugated sections which are studied in Fig. 5.26, are tabulated in Table 5.8 and 5.9, with εmax representing the maximum strain in the beam during the loading to ensure that the infinitesimal assumption is held valid. wtip and φtip respectively denote the maximum deflection and slope of the beam at the free end (i.e., the tip), which obviously satisfy the range of the applicability of EBBT. κc =κur c is the ratio of the maximum curvature of each section to that of unreinforced rectangular section. The value in parenthesis corresponds to the unreinforced rectangular section. It can be seen that in order to obtain the same curvature after unloading the sections should be bent more than the unreinforced section. Further, κh =κc represents the shape fixity of the section. It is observed that the shape fixity reduces as a result of introducing the reinforcing material, since a portion of curvature elastically retracts upon the unloading. Finally, Fl =Flur is the ratio of force required to bend the reinforced beam to that of the unreinforced rectangular beam. In order to verify the results obtained by the procedure of the present work and for checking the applicability of the EBBT, three-dimensional finite element models Table 5.8 Numerical results for the performance of different corrugated sections with equal SMP content and height in a shape memory cycle [54]. Section

Rectangular (unreinforced) Rectangular (reinforced) Sinusoidal (reinforced) Triangular (reinforced) Trapezoidal (reinforced)

εmax

0.0027 0.0034 0.0031 0.0038 0.0035

wtip (mm)

φtip ( )

E-B

FE

E-B

FE

9.1 11.2 8 9.4 10

9.2 11.3 8.3 9.6 10.2

1.56 1.93 1.38 1.62 1.71

1.57 1.93 1.41 1.64 1.74

κc =κur c

κh =κc

Fl =Flur

1 (0.109) 1.209 1.1946 1.2267 1.2123

0.988 0.7995 0.8274 0.8045 0.8142

1 24.82 17.5 16.64 20.51

Table 5.9 Numerical results for the performance of different corrugated sections with equal SMP content and width in a shape memory cycle [54]. Section

Rectangular (unreinforced) Rectangular (reinforced) Sinusoidal (reinforced) Triangular (reinforced) Trapezoidal (reinforced)

εmax

0.0027 0.0034 0.0092 0.0108 0.0049

wtip (mm)

φtip ( )

E-B

FE

E-B

FE

9.1 11.2 12.2 13.1 11.5

9.2 11.3 12.5 13.4 11.7

1.56 1.93 2.11 2.25 1.97

1.57 1.93 2.14 2.28 1.99

κc =κur c

κh =κc

Fl =Flur

1 (0.109) 1.209 1.3067 1.3328 1.216

0.988 0.7995 0.7563 0.7416 0.7823

1 24.82 108.44 96.8 37.96

Shape memory polymer composites: nanocomposites and corrugated structures

201

Figure 5.27 The output results and the contours of the deformation of the composites beams [54].

of different sections have been studied in ABAQUS (Fig. 5.27). Here only the deflection and slope at the free end prior to the cooling process are studied. Both the SMP (in active/rubbery phase) and the reinforcing material are assumed to behave linearly elastic with the properties presented in Table 5.7. The initial bending load is applied at the free end uniformly as a traction. Both the SMP core and the reinforcing thin layer are meshed with fully integrated cubic elements. The results of free end deflection and slope are presented in Tables 5.8 and 5.9. It can be seen that the theory employed here is capable of predicting the behavior of the composite beam with great accuracy. The output results and the contours of the deformation of the composites beams are shown in Fig. 5.27 for each section. As listed in Tables 5.8 and 5.9 and shown in Fig. 5.27, the EBBT slightly underpredicts the deformations. For better clarification, the elements are hided in Fig. 5.27.

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It has been noted that the main purpose of reinforcing SMP is increasing their load capacity. It is assumed that upon heating, a mechanical constraint is introduced, which prevents the complete shape recovery from happening. The reaction force is studied for the aforementioned sections. Force history in the heating process as a function of the temperature is depicted in Fig. 5.28A. It can be observed that with the same SMP content, the sinusoidal section has the largest load capacity, provided that the sections have the same base dimension (Fig. 5.28A). However, in cases with equal-height sections, according to (Fig. 5.28B), the maximum load capacity belongs to the rectangular section. Here, it should be noted that in order to obtain the results of Fig. 5.28A, it is assumed that the beams with different crosssections at the fixed end undergo the same initial curvature prior to the heating process in correspondence to the numerical results of Tables 5.8 and 5.9. An important consideration in designing the SMP-based actuators is the ratio of the generated force to that of the maximum for each section at each temperature step (i.e., load generation history). Referring to Fig. 5.28A, it can be seen that this ratio does not change by introducing different corrugated sections. In other words, it can be concluded that the corrugated sections do not affect the force generation capability of the SMP structures. The numerical details for bending of different corrugated sections in comparison with the unreinforced SMP beam are tabulated in Tables 5.10 and 5.11. Pl =Pur l , ur Pg =Pur , and κ=κ are the ratios of the maximum force in the loading stage, the g generated force in the heating stage, and finally the curvature after elastic unloading stage to those pertaining to the unreinforced SMP section. The values in the parentheses in each table represent the numerical values for the unreinforced SMP beam. It is shown that because of reinforcing the sections, both the loading and the generated forces increases; however, in most actuators, the generated force is the top priority in design. As mentioned earlier the shape fixity reduces inevitably, but in consequence to reinforcing the section it can be seen that the generated force in the heating stage have been increased to a great extent. Specifically, the reinforced sections with equal width are less flexible that equal height sections, however, they generally possess greater loading capacities thanks to the greater moments of inertias. It can be seen that the ratio of the maximum load to the generated load does not vary appreciably for the two studied cases. In this section, a finite difference scheme is employed to obtain numerical solution of the governing equation. Although this scheme is unconditionally convergent, it is important to have an approximation of the minimum number of steps which render accurate results. Fig. 5.28B shows the thermomechanical cycle of the triangular section, which is plotted for different number of steps. It can be observed that a few steps are needed to obtain accurate results. This, on the other hand, confirms the efficiency of the employed numerical scheme. In this section, the effect of reinforcing the SMP beams with corrugated polymeric structures was studied. Employing a widely accepted one-dimensional SMP model and EBBT in conjunction with the theory of composite sections a finite difference scheme was developed to solve the governing equations. A polymeric reinforcement was chosen with a wide range of thermal service to ensure that the

Shape memory polymer composites: nanocomposites and corrugated structures

203

(A)

(B)

Figure 5.28 (A) Force generated due to mechanical constraint in heating process: (A) equalur width sections and (B) Equal-height sections (Fmax 5 0.9877 N). (B) Comparison between accuracy of the results of different number of temperature steps (κmax 5 0.295/m) [54].

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Table 5.10 Performances of different corrugated sections with equal width [54]. Section

Pl =Pur l

Pg =Pur g

Pg =Pl

Rectangular (SMP) Rectangular (rein.) Sinusoidal Trapezoidal Triangular

1 (1) 24.82 108.44 37.96 96.8

1 (0.9877) 21.0313 84.2285 31.2433 76.2844

0.9877 0.8369 0.7671 0.8129 0.7784

Table 5.11 Performances of different corrugated sections with equal height [54]. Section

Pl =Pur l

Pg =Pur g

Pg =Pl

Rectangular (SMP) Rectangular (rein.) Sinusoidal Trapezoidal Triangular

1 (1) 24.82 17.5 20.51 16.64

1 (0.9877) 21.0313 13.921 16.5592 13.0691

0.9877 0.8369 0.7857 0.7974 0.7757

material properties of the reinforcing material does not change appreciably during the thermomechanical cycle of the SMP core. Numerical results showed a great enhance in load capacity of the reinforced beam at a small decrease of shape fixity. It was observed that corrugated patterns have different load capacities, which should be considered in design of the SMPbased actuators and other reinforced structures. It is shown that for the same SMP content, equal width sections possess larger load capabilities in comparison with those of equal height section. The generated force has shown to be highly dependent on the moment of the inertia of the section, since equal width sections have larger moments of inertia, when compared with their equal height counterparts. However, no such dependency was observed for the ratio of the generated force to the initial loading force. Furthermore, it can be seen that the introduction of the corrugated sections does not reduce the force generation capability of the SMP, that is, by studying the ratio of the generated force to the maximum achievable force for each thermal step it can be seen the same load generation ratio is obtained for different sections. In case of single cell asymmetric reinforced composite sections, it was seen that a temperature change leads to a small curvature. In addition to the properties enhanced here, other desirable properties may be achieved by reinforcing SMPs. For instance, good impact resistance and flame-retardancy. Finally, it was shown that the finite difference procedure employed in the present investigation was capable of rendering accurate results in a few steps which proves to be computationally efficient. A full three-dimensional interaction study is not possible in the framework of a one-dimensional model (i.e., current SMP model and engineering beam models

Shape memory polymer composites: nanocomposites and corrugated structures

205

such as EBT or even higher order theories). Here, it is noted that only global behavior of the composite beam is of interest here. It is common practice to distinguish between global and local behaviors of the structure. It is established that while the global behavior (static deformations, critical buckling load, principal natural frequency, etc.) can be obtained by simple classic theories, in order to obtain local behaviors (de-laminations, etc.) more sophisticated theories should be employed. It should be noted that in designing actuators the most important concerns are deflection and force capability. As far as these issues are of interest, the numerical results of 3D finite element in comparison to those of the EBBT suggest that a 1D model (i.e., EBT) is sufficient. It is noteworthy that the procedure outlined in this section is only valid provided that there is no de-cohesion between the SMP and the reinforcing material; otherwise the EBBT is not applicable. However, since the deformations in the present problem are infinitesimal these considerations are not pursued. It should be noted here, that in the present analysis it is assumed that every section of the beam is isothermal at each thermal step. If the exact temperature distribution is required, a simultaneous thermal analysis should be performed along with an appropriate SMP model. However, these discussions were not followed here and are out of the scope of this section.

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1778. [54] A. Jamalimehr, M. Baghani, M. Baniassadi, M. Zakerzadeh, Thermomechanical behavior of shape memory polymer beams reinforced by corrugated polymeric sections, Meccanica 52 (8) (2017) 1947
1962. [55] E.P. Popov, Engineering Mechanics of Solids, Prentice Hall., 1990. [56] J. Brandrup, E.H. Immergut, Polymer Handbook, Vol. 1., Wiley-Interscience, 1989. [57] S. Akbari-Azar, M. Baghani, M.R. Zakerzadeh, H. Shahsavari, S. Sohrabpour, Analytical investigation of composite sandwich beams filled with shape memory polymer corrugated core, Meccanica 54 (10) (2019) 1647
1661.

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6

Chapter outline 6.1 Shape memory polymer metamaterials based on triply periodic minimal surfaces 209 6.1.1 Introduction 210 6.1.2 Materials and methods 212 6.1.3 Results and discussion 217

6.2 Numerical investigation of smart auxetic 3D metastructures based on shape memory polymers via topology optimization 225 6.2.1 6.2.2 6.2.3 6.2.4

Introduction 225 Geometrical modeling of an representative volume element 227 Finite element analysis of shape memory polymer microstructure 229 Results and discussion 234

6.3 Summary and conclusions References 240

6.1

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Triply periodic minimal surface (TPMS) metamaterials and shape memory polymer (SMP) smart materials are known for their beneficial attributes in novel scientific and industrial fields. Through TPMS designs, low weight accompanied by high surface area is achievable, which is known as crucial parameter in many fields, such as tissue engineering. Moreover, SMPs are well-suited to generate force or to recover their permanent shape by means of an external stimulus. Combining these properties is possible by fabricating TPMS-based metamaterials made out of SMPs, which can be applicable in numerous applications. By considering different level volume fraction of four types of TPMS-based lattices (diamond, gyroid, IWP, and primitive), we focus on the effect of microarchitecture on shape-memory characteristics (i.e., shape recovery, shape fixity, and force recovery) as well as mechanical properties (elastic modulus and Poisson’s ratio) of these smart metamaterials. For this purpose, shape-memory effect (SME) is simulated employing thermoviscohyperelastic constitutive equations coupled with the time
temperature superposition principle. It is observed that by increasing the level volume fraction of each Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00001-X © 2023 Elsevier Inc. All rights reserved.

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lattice type, the elastic modulus, shape fixity, and force recovery increase, while the shape recovery diminishes. Such behaviors can be attributed to different deformation modes (flexural or uniaxial) in SMP metamaterial TPMS-based structures. Furthermore, it is shown that the Poisson’s ratio has a nonlinear behavior in these structures. The smart metamaterials introduced in this section have the advantage of providing the possibility of designing implants, especially in bone defects tailored with different microarchitectures depending on each patient’s specific need.

6.1.1 Introduction Metamaterials, a spotlight of researchers nowadays, are special types of designer materials having unusual mechanical properties and advanced functionalities usually not found in nature. The term “metamaterial” was firstly used in optics and electromagnetism fields [1]; however, they have found their way into mechanical aspects of materials, and therefore a new branch called “mechanical metamaterials” has emerged. On the contrary to conventional materials, mechanical metamaterials can exhibit properties, such as negative Poisson’s ratio (NPR) [2], negative elasticity [3], negative compressibility [4], fluid-like behavior [5], and photonic lattices [6]. To achieve these special mechanical and physical properties, repeatable microstructural topologies should be designed in nano- and microscales. By recent developments in additive manufacturing (AM) technologies, fabricating complicated structures in these scales has become possible [7]. This matter has fascinated many manufacturers and scientists to the concept of mechanical metamaterials [4,8]. In consequence, studying the relation between the topology in micro- and macroscale properties in all mechanical materials seems to be indisputable. There are a few well-known strut-based open-cell lattice structures, such as rhombic dodecahedron, truncated octahedron, BCC, and truncated cube. However, strut-based lattice structures have the disadvantage of having low surface area to volume fraction and relatively low curvature in their internal surfaces, which are both crucial parameters in applications including tissue regeneration. Nature has resolved the problem of low weight accompanied by high surface area and curvature with creating specific microgeometries known as TPMS. TPMS structures can be found in butterfly wings [9], sea urchin [10], block copolymers [11], soap films [12], and so on. The unit cells inspired by the noted natural structures, which are also the most favorable structures among the researchers in the field [13
15], will be implemented for constructing shape-memory TPMS-based metamaterials (Fig. 6.1). For the first time, Schwarz [18] implemented nature-based TPMS diamond and primitive topologies in 1865. Despite the great benefits they offer, it was not possible to create intricate TPMS structures fast and accurately in the 20th century due to limitations in manufacturing. With recent advances in AM, many new TPMS microstructures, such as gyroid [14], primitive [19], diamond [20], IWP [15], Neovius-CM [15], Schoen-FRD [21], and Fisher-Koch C [22], have been produced and their mechanical properties, physical characteristics, and fluid permeability have been studied in interpenetrating phase composites (IPCs), cocontinuous

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Figure 6.1 Natural creatures inspiring the four main unit cells implemented in this section. The top row demonstrates the macroscale visualization of the natural creature, the mid-row demonstrates the micro- or nanoscale visualization of the natural creatures, and the bottom row depicts the nature-inspired unit cells. First column: (A) butterfly wings [9], (B) microstructure of butterfly wing [9], and (C) gyroid triply periodic minimal surfaces-based unit cell. Second column: (D) the sea urchin, (E) microstructure of a cross-section through a sea urchin skeletal plate [10], and (F) primitive triply periodic minimal surface-based unit cell. Third column: (G) a diamond, (H) schematic representation of molecular structure of diamond, and (I) diamond triply periodic minimal surface-based unit cell. Fourth column: (J) the formation of a soap film with lamella supporting by the wireframes [16], (K) a soap film between two metal wires based on IWP geometry [16], and (L) IWP triply periodic minimal surface-based unit cell [17].

composites [23], tissue-engineering scaffolds, and other engineering applications. In this section, the four most popular topologies have been considered for study. Recent advances in AM, and in particular, in technologies, such as stereolithography, digital light processing, and liquid crystal display, have made it possible to use SMP-based liquid resins to make SMP-based metamaterials and SMP-based origami structures. In recent years, SMP- and TPMS-based metamaterials have been widely considered by researchers, separately. However, no extensive research has been dedicated to SMP metamaterials based on TPMS to systematically examine their shape-memory behavior, which can exhibit a wide range of exotic and marvelous behaviors. Design and development of SMP metamaterials based on TPMS creates a lot of possibilities in different research and industrial fields including tissue engineering, a few examples of which will be given later in this section. In this section, the SMP-based metamaterials are studied on four TPMS-based structures that can be conveniently generated using mathematical equations and be easily fabricated by AM technologies. The responses of these metastructures are

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tunable to provide SME and unusual mechanical properties. SMP properties are assigned to TPMS-based metamaterials by applying thermovisco-hyperelastic constitutive models coupled with the time-temperature superposition principle (TTSP) to reconstruct the shape-memory behavior. This section compares the trend of changes in the shape-memory characteristics (shape recovery, force recovery, and shape fixity) and mechanical properties (Poisson’s ratio and elastic modulus) for different TPMS-based structures, level volume fractions, and filling status (filled or core
shell). Eventually, the applications and advantages of SMP metamaterial based on TPMS are stated in tissue engineering field.

6.1.2 Materials and methods At the first step, the surfaces of metamaterials are constructed by MathMod mathematical software. The volumetric and surface discretization of both the core
shell and filled models are performed in 3-MATIC. The numerical simulation of material behavior is done using finite element (FE) analysis.

6.1.2.1 Generation and modeling of microstructures Metamaterials based on TPMS microstructures are chosen here. A TPMS unit cell, which is selected to be the base of the structures, is a minimal surface with translation symmetries in all three independent directions. Minimal surfaces are defined as constant and vanishing curvature H 5 ðk1 1 k2 Þ=2 5 0. In this equation, k1 and k2 are two principal curvatures in two mutually orthogonal directions that have equal magnitude but opposite signs. TPMS topologies based on primitive, gyroid, diamond surfaces divide space into two congruent regions, whereas the IWP triply periodic minimal surface form a bicontinuous structure with each separate domain possessing different volumes [13]. Their surfaces could be readily produced by their specified mathematical functions. Moreover, their continuous surface geometries make them manufactural, especially through AM methods. The main function of a TPMS geometry is Fðx; y; zÞ 5 ξ in which ξ is the level cut [12,24,25]. The governing equation of each TPMS-based structures chosen here is given in the following. Primitive:
F x; y; zÞ 5 a½cosðxÞ 1 cosðyÞ 1 cosðzÞ 1 b½cosðxÞcosðyÞ 1 cosðyÞcosðzÞ 1 cosðzÞcosðxÞ

(6.1)

where ða; bÞ 5 ð10; 2 5:0Þ Gyroid: F ðx; y; zÞ 5 a½cosðxÞsinðyÞ 1 cosðyÞsinðzÞ 1 cosðzÞsinðxÞ 1 b½cosð2xÞcosð2yÞ 1 cosð2yÞcosð2zÞ 1 cosð2zÞcosð2xÞ where ða; bÞ 5 ð10; 2 5:0Þ

(6.2)

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Diamond:
F x; y; zÞ 5 a½sinðxÞsinðyÞsinðzÞ 2 cosðxÞcosðyÞcosðzÞ 1 b½cosð4xÞ 1 cosð4yÞ 1 cosð4zÞ

(6.3)

where ða; bÞ 5 ð10; 2 0:7Þ IWP:
F x; y; zÞ 5 a½cosðxÞcosðyÞ 1 cosðyÞcosðzÞ 1 cosðzÞcosðxÞ 1 b½cosð2xÞ 1 cosð2yÞ 1 cosð2zÞ

(6.4)

where ða; bÞ 5 ð10; 2 5:0Þ MathMod software is implemented to produce surfaces using the above functions. Each ξ would create different volume contained in each level cut surface ðφξ Þ. The level cuts of 30, 35, 40, 45, and 50 have been considered. In the next step, CAD models are generated based on the surfaces through 3-Matic software. Two types of CAD models, filled and core
shell, are generated for each TPMSbased structure. In the filled model, the internal regions of the surfaces are filled, while in the core
shell ones, the surfaces are given a specific thickness. The phase volume fraction φβ of the core
shell models can be calculated from φβ 5

hρ υ

(6.5)

so that h is the surface thickness, ρ is the mid-surface area of a single unit cell, and υ is the volume of the unit cell. In all the TPMS-based core
shell models, a set value for phase volume fraction of φβ C30% is considered. Only for the filled structures φβ 5 φξ . The values of parameters ξ and h can be found in Table 6.1 and Fig. 6.2.

6.1.2.2 Shape-memory effect modeling using thermoviscohyperelastic model As the SMPs show complicated SME, the models which are able to predict the shape-memory behavior by taking mobility, intermolecular chain interactions, and relaxation time into account are of interest. In this section, a large deformation nonlinear thermovisco-hyperelastic model coupled with TTSP is used. For this purpose, the model proposed by Diani et al. [26] and Arrieta et al. [27] is utilized. Details of this model formulation are reported in Section 4.3.

6.1.2.2.1 Geometry generation The level cut ξ and the wall thickness h values of the developed unit cells based on triply periodic surfaces are listed in Table 6.1. The relationship between ϕξ and ξ is defined as ξðφξ Þ 5 γ 1 ϕ2ξ 1 γ 2 ϕξ 1 γ 3 , in which γ 1 5 2 1:466, γ 2 5 2 25:36, and

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Table 6.1 The parameters used for generating TPMS-based structures [13]. Diamond φξ ½% 30 35 40 45 50

Gyroid φβ ½% 30 30 30 30 30

ξ 4.920 3.572 2.240 0.930 20.338

h 0.838 0.805 0.784 0.771 0.767

φξ ½% 30 35 40 45 50

IWP φξ ½% 30 35 40 45 50

φβ ½% 30 30 30 30 30

ξ 8.082 6.467 4.770 2.993 1.157

φβ ½% 30 30 30 30 30

ξ 6.090 4.519 2.942 1.371 20.188

h 1.043 1.007 0.985 0.972 0.967

Primitive h 0.918 0.883 0.861 0.848 0.843

φξ ½% 30 35 40 45 50

φβ ½% 30 30 30 30 30

ξ 9.275 7.886 6.396 4.816 3.153

h 1.370 1.132 1.283 1.262 1.253

Figure 6.2 3D model of primitive, gyroid, diamond, and IWP microstructures based on triply periodic minimal surfaces. At the center of each diagram, the final filled lattice structure is shown, and the unit cells with different level volume fractions ðφξ 5 30%; 35%; 40%; 45%; and 50%Þ are demonstrated in the surrounding of each diagram [17].

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γ 3 5 12:66 are fitting parameters for diamond; γ 1 5 2 2:54, γ 2 5 2 29:49, and γ 3 5 15:17 are fitting parameters for gyroid; γ 1 5 2 19:54, γ 2 5 2 19:19, and γ 3 5 15:6 are fitting parameters for IWP; and γ 1 5 2 21:25, γ 2 5 2 13:75, and γ 3 5 15:32 are fitting parameters for primitive topologies. The material parameters including viscoelastic, hyperelastic, thermal, TTSP, and general parameters are provided as shown in Table 6.3. An acrylate network composition proposed and implemented in [27,28] was employed for the bulk material. The polymer is prepared by the copolymerization of benzyl methacrylate (BMA) with poly(ethylene glycol) dimethacrylate of molar weight 550 g=mol. Photopolymerized materials synthesized from a tert-butyl acrylate monomer with a moderate amount of a diethyleneglycol diacrylate crosslinker show good shape recovery. Due to their biocompatibility and their capacity for photopolymerization, these materials have significant potential for biomedical applications [29]. Three parameters of shape recovery, force recovery, and shape fixity are usually implemented by researchers to examine the SME in SMP. In this part, SME cycle of metamaterials in tension mode will be studied. The schematic of the shape recovery process, force recovery process, strain
time diagram, and temperature
time diagram is shown in Fig. 6.3. The common steps of both the shape recovery and force recovery processes are described in the following sentences. In the zero step, the specimen is preheated to a high temperature TH above Tg , which is required for experimental works, whereas in the numerical simulations the sample is placed at a high temperature from the beginning. In the first step, the specimen is kept at TH to arrive at an isothermal condition, and a determined tensile load is applied to the specimen. In the second step, the specimen is cooled down to a low temperature TL , which is lower than Tg , while it is constrained. In the third step, the specimen is unloaded, which would cause a little elastic spring back strain, known as shape fixity (its percentage can be calculated relative to the initial displacement). Finally, the fourth step is heating to TH or the recovery process. The difference in the shape recovery and force recovery processes are explained in the following. As for the shape recovery test, after unloading, the specimen is reheated to TH , because of which the specimen is prone to be transformed into its original shape. The ratio of shape transformation extent to the original length is called shape recovery (step 4i): R5

Xd 2 Xnr 3 100% Xd

(6.6)

where R is shape recovery in percentage, Xd is the extent of maximum displacement applied to the structure, and Xnr is the nonrecovered length at TH after shape recovery. Another critical parameter in SMPs is Shape fixity defined as the capability of SMP in keeping its temporary shape during the programming process. The percentage of shape fixity RF in the thermomechanical cycle can be calculated by RF 5

LT 3 100% LD

(6.7)

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Figure 6.3 Shape-memory behavior of shape memory polymer: (A) schematic of shape recovery cycle in compression mode, (B) schematic of force recovery cycle in compression mode, (C) the strain
time diagram, (D) the temperature
time diagram, and (E) 3D diagram of shape memory polymer shape-memory cycle in terms of stress, strain, and temperature [17].

in which LT is the length of the structure (in direction of loading) after the unloading step, and LD is the length at the end of the cooling process. On the other hand, during the force recovery process, the specimen is held fixed and is heated up to TH . By changing the temperature and phase transition, the specimen tends to regain its original shape, but since the specimen is constrained, it starts to develop a force. The ratio of the force generated after heating in the final step FFR , to the force required to program the initial deformation in the specimen FPre , is defined as the force recovery ratio FFR (step 4ii): RFR 5

FFR 3 100% FPre

(6.8)

For a better understanding of the shape-memory cycle, a diagram in terms of three affecting parameters related to the shape-memory behavior (stress, strain, and temperature) is depicted in Fig. 6.3E.

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6.1.3 Results and discussion In this section, the results from the shape and force recoveries, shape fixity, and some mechanical properties (elastic modulus and Poisson’s ratio) are examined for different level volume fractions, topologies, and filling status (filled or core
shell). In Fig. 6.4, the simulation steps and results for one of the specimens (IWP geometry with φξ 5 50%) are presented. The simulation steps (four steps in total), strain contours in each unit cell at each state at the end of each step, as well as the strain contours in the lattice structure at the end of each step are illustrated in this figure. In the first state (1), the lattice has its initial shape with the initial length of L0 . During the loading (step 1), the structure is under 5% tensile strain in the Z-direction at the XY plane at 65 C until t 5 100s, and at the end of this step, in state (2), the length is L0 1 ΔLloading . In the cooling (step 2), until t 5 580s, the temperature is reduced to 25 C at a rate of 5 C=min while the structure is constrained, so the length at state (3) is the same as the previous state. During the unloading (step 3), up to t 5 680s, a tiny amount of the strain is recovered elastically and the new length at state (4) is equal to L0 1 ΔLloading 2 ΔLunloading . Finally, during heating (step 4), the lattice is heated up at a rate of 5 C=min, which triggers the shape recovery due to the increase in temperature to T . Tg , and at t 5 1160s in state (5) the final length is

Figure 6.4 Thermomechanical cycle, (A) applied strain versus time plot in the shapememory effect path, (B) total strain variations in the same path of a unit cell located in the back side of the IWP lattice structure, and (C) total strain variations in the same path for the whole lattice, which recovers its initial shape after a cycle [17].

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Lfinal , and the lattice recovers its initial shape. It is shown in the strain
time diagram (Fig. 6.4A) that a shape recovery up to 89% occurs. The state of the lattice structure at the end of different steps are demonstrated in Fig. 6.4B and C.

6.1.3.1 Shape recovery The shape recovery percentage for each geometry and for both filled and core
shell unit cells in the last step (step 5i in Fig. 6.3), is calculated at different volumes fractions (30% , φξ , 50%, see Fig. 6.5). Shape recovery ratio has a reverse correlation with φξ in all topologies except for the core
shell primitive model at φξ . 35%, where an increase in φξ enhances the shape recovery. It is observed that an SMP material provides a higher shape recovery when loaded in flexural mode as compared to when it is under uniaxial loading [30]. For the case of force recovery, the opposite holds true [30]; in other words, a higher force recovery is observed in uniaxial deformation mode compared to flexural mode. In the filled TPMS-based lattices, as the level volume fractions increase, the flexural stress contribution in the total stress distribution in the walls decreases, which lowers the shape recovery. This is due to the higher percentage of the material bulk, which is aligned with the direction of the applied load. On the other hand, in the core
shell TPMS-based lattices, since φβ C30% is the same for all the level volume fractions ðφξ Þ, the thickness of TPMS-based structure is decreased by increasing φξ . Under an equal applied strain, a decrease in the thickness of TPMS-based structure will lead to higher local strains and stresses in the core
shells, which results in lower shape recovery ratios for higher φξ .

Figure 6.5 Shape recovery percentage for various level of φξ for filled and core
shell models [17].

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The maximum shape recovery occurs at the lowest φξ (i.e., 30%) in all geometries. The maximum shape recovery belongs to the core
shell diamond model (with 98% shape recovery). In the diamond and IWP geometries, the core
shell models have higher shape recoveries compared to the filled models. However, in the gyroid geometry, the filled models have higher shape recoveries in comparison to core
shell ones. On the other hand, for the primitive geometry, up to φξ 5 40%, the filled structures have higher shape recoveries as compared to core
shell ones, whereas after φξ 5 40%, an opposite trend is observed.

6.1.3.2 Shape fixity Shape fixity is the ability of SMP in keeping its length after removing the applied stress (Fig. 6.3, step 4). In most cases, SMPs experience a reduction in length when the programming stress is removed. Complete (i.e., 100%) shape fixity, the ideal situation for an SMP, occurs only when there is no length reduction in the SME cycle. The shape fixity for different geometries and φξ , and for both the filled and core
shell configurations, are depicted in Fig. 6.6. The difference of shape fixities (due to spring back effect) between the filled and core
shell specimens is very small, except for the IWP and primitive at high level volume fractions ðφξ 5 50%Þ. Nonetheless, the maximum difference in the shape fixity of the core
shell and filled specimens is very small (i.e., ,4%). Unlike the shape recovery curves (Fig. 6.5), shape fixity increases as a result of increase in φξ , and the stability of the temporary shape enhances. Based on the aforementioned reasons explained for the increase in the shape recovery, an opposite behavior occurs for the shape fixity. Strictly speaking, superiority in the shape recovery contributes to diminished shape fixity. Among all the specimen types, the highest shape fixity extents belong to the filled IWP specimen with a maximum shape fixity of 98%, and core
shell primitive specimen with a maximum shape fixity of 97% (Fig. 6.6).

6.1.3.3 Force recovery In many applications, force recovery magnitude is of more importance than the shape recovery extent. A good example of such applications is actuation by force in bounded systems. An SMP with high shape recovery but low force recovery is not useful in such instances. The behavior of the force recovery ratio is similar to that of shape fixity, means it has a positive correlation with φξ in all geometry types, and in both the filled and core
shell configurations. The reason why force recovery increases by increase in the level volume fraction can be explained as follows. By increasing the level volume fraction, a higher percentage of the bulk material is aligned with the external uniaxial load applied. As it is observed before, an SMP material gives a higher force recovery ratio when it is loaded in uniaxial mode as compared to when it is loaded flexural. This leads to higher force recoveries in the filled TPMS-based lattices when the level volume fraction of the material increases. On the other hand, in the core
shell TPMS-based lattices, higher local stresses (due to decrease in the

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Figure 6.6 Shape fixity percentage for various level of φξ for filled and core
shell models [17].

thickness) lead to better force recoveries when the level volume fraction of the material increases. The highest force recovery ratio belongs to the core
shell primitive and gyroid geometries with a force recovery of B94%, followed by the filled diamond geometry with about 93% force recovery (Fig. 6.7).

6.1.3.4 Mechanical properties The mechanical properties, including the Poisson’s ratio and the elastic modulus, are obtained at the end of loading step where T . Tg and applied strain is 5%. The Poisson’s ratio in the gyroid and IWP structures (in both the filled and core
shell configurations) decreases continuously by increasing φξ . The same results apply to the filled diamond case. However, in the core
shell diamond structure, by increasing φξ , the Poisson’s ratio first decreases and then rises. On the contrary, the Poisson’s ratio has a negative correlation with φξ in the case of primitive geometry (in both the core
shell and filled configurations). The Poisson’s ratio value in the filled and core
shell configurations of gyroid and IWP geometries remain close. Additionally, change in φξ value does not make a significant difference in the Poisson’s ratio of these geometries. However, in the diamond and primitive core
shell geometries, a completely nonlinear behavior for the Poisson’s ratio is observed, whereas their filled structures have linear curves. The highest value of the Poisson’s ratio belongs to the filled and core
shell IWP geometry, which is around 0.5, and the lowest Poisson’s ratio value is associated with the filled

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Figure 6.7 Force recovery percentage for various level of φξ for filled and core
shell models [17].

primitive geometry with φξ 5 30% and core
shell diamond geometry with φξ 5 45%, both being around 0.1 (Fig. 6.8). Increasing level volume fraction changes the surface topology as the surface functions of TPMS-based structures are highly dependent on ξ. The effect of this change in topology is different for different unit cell types. In gyroid and IWP unit cells, increasing the level volume fraction does not significantly change the topological configurations. In other words, in the gyroid and IWP topologies, the orientation of the surfaces varies only slightly as a result of change in the level volume fraction. That is why these two structures do not demonstrate a significant change in the Poisson’s ratio with respect to the level volume fraction variations. However, in the diamond structure, the fraction of oblique surfaces decreases as a result of increase in level volume fraction. In other words, the main mode of deformation in the walls changes from flexure to axial. This leads to an exponential decrease in the Poisson’s ratio in the diamond core
shell structure with respect to the volume fraction increase. In the primitive structure, however, the opposite holds true, means increasing the volume fraction leads to a structure with a higher fraction of oblique walls. This enhances the contribution of flexure in total deformation of the structure, which means it magnifies the lateral displacement of the whole structure. That is why in the primitive structure, by increasing the volume fraction, unlike the other structures, the mentioned mechanism (i.e., change in the deformation mode from axial to bending) increases the Poisson’s ratio. The normalized elastic modulus (elastic modulus of the lattice divided by the elastic modulus of the constituent material, E0 5 5:74MPa) variations of different topologies at high-temperature are shown in Fig. 6.9, in which the stiffness has a

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Figure 6.8 Poisson’s ratio for various level of φξ for filled and core
shell models [17].

Figure 6.9 Nondimensional elastic modulus for various levels of φξ for filled and core
shell models [17].

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continuous direct correlation with the material level volume fraction, regardless of being core
shell or filled. At the level volume fraction of φξ 5 50%, the lowest and highest normalized elastic moduli for the filled-configurations belong to the diamond ðE=E0 5 0:61Þ and primitive ðE=E0 5 0:46Þ topologies, respectively. On the other hand, in the core
shell models ðφξ 5 50%Þ, the minimum and maximum normalized elastic moduli are associated with primitive ðE=E0 5 0:31Þ and IWP ðE=E0 5 0:41Þ unit cells, respectively. The reason behind different trends in change of elastic modulus can be described by the same mechanism explained for the Poisson’s ratio. Surface topologies are highly dependent on ξ. Increasing the level volume fraction has different effects on the topology variation in different unit cell types. In two of the unit cells, that is, gyroid and IWP, increasing the level volume fraction does not significantly affect the topological configurations. In other words, in the gyroid and IWP topologies, the angles of the surfaces vary only slightly upon a change in the level volume fraction. However, in the diamond structure, the fraction of oblique surfaces is reduced as a result of increase in level volume fraction, meaning that the main mode of deformation in the walls transforms from flexure to axial. This leads to a sharp rise in the elastic modulus of the diamond structure in terms of level volume fraction increase. In the primitive structure, however, the opposite holds true, that is, increasing the level volume fraction gives a structure with a higher fraction of oblique walls. This enhances the contribution of flexure in total deformation of the structure, which means it diminishes the stiffness of the whole structure. Another mechanism also contributes to the elastic modulus of most of topologies. When the level volume fraction grows, the pore sizes obviously increase in the IWP and primitive structures. Increase in pore sizes amplifies the flexural stiffness of the elements and thus the whole structure. That is why in the primitive structure, by increasing the level volume fraction, even though the stiffness is enhanced due to a higher flexural stiffness, the first aspect (i.e., change in the deformation mode from axial to bending) slows down the increase.

6.1.3.5 Potential application areas Extensive commercial applications of the SMPs have attracted the attention of various industries, but according to Dietsch and Tong [31], 50%
70% of patents filed related to SMP are in the field of medical science. Researchers have suggested many biomedical applications for these materials. For instance, SMP sutures could be designed in such a way that they contract after being applied on an injured tissue and then rising the temperature locally, which leads to quick closure of wounds. The temperature increase could be applied by the surgeon when implanting the SMP in the body and by means of a device that would only influence the SMP and not the surrounding tissues/bone. For instance, such location-specific heating could be accomplished by mixing elements like nanoparticles in the SMP, which are sensitive to external stimuli, such as an electromagnetic field. In addition, attention to smart materials in recent years has led to the tendency of using these materials in the manufacturing of smart stents. Increase in the temperature of smart stents would

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cause stimulation in the stent and therefore eliminates the need for traditional complex stimuli, such as balloons [32,33]. Another main application of SMPs, that has received a lot of attention and efforts recently, is tissue engineering and repairing bone defects with critical size [34]. Critical bone defects, which may be caused by injury, tumor excision, and congenital defects, do not improve without intervention. For orthopedic surgeons, reconstruction of a large bone defect has been a major challenge [35]. Bone tissue engineering is one of the most critical methods for treating bone defects, especially relatively large bone defects. In particular, the porosity of biological materials plays an essential role in osseointegration and osteoconduction and supports the migration of cells, capillary ingrowth, and the transport of nutrients to the cells [36]. The advantage of the metamaterials introduced here is that they can be designed for specific bone defects and be tailored with different porosities depending on each patient’s specific need. Here, metamaterials can play a key role in improving the functionality of an implanted scaffold designed for providing bone regeneration in bone defects. Fig. 6.10 shows a schematic of a particular bone defect: cavities with critical size in the patient’s bone. The patient’s bone defect can be identified by 3D image scanning. According to the patient’s condition and requirement, a special type of TPMS-based lattice structure made from SMP can be designed and printed via AM. Fig. 6.10 also indicates a cross-sectional view of the patient’s humerus. After placing the damaged sections of the two bone segments on top of each other, a thin layer of microsized lattice plate made of SMP metamaterials TPMS-based microstructures is rolled and attached around the fractured area. After the SMP lattice plate is placed inside the body, its temperature is increased, which as

Figure 6.10 Schematic of application of self-fitting shape memory polymer metamaterial triply periodic minimal surface-based lattice structure and bone regeneration triply periodic minimal surface-based shape memory polymer scaffolds in human skeleton [17].

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preprogrammed, make it shorten axially (self-fitting). This transformation allows the two damaged bone segments fit together well and remain in such way preventing loosening and speeding up bone fracture repair. Another type of bone defect that may occur as a result of congenital disease is the presence of cavities in the patient’s pelvis (Fig. 6.10). In this type of bone defect, for bone remodeling, the SMP must behave in the opposite way to the one described above. In other words, the SMP lattice structure should be programmed in such a way that after being placed in the cavity, its size increases due to the temperature rise. This leads to a good self-fitting behavior for the scaffold implanted in the bone defect, which would in turn let the cavity be filled completely. The topology of TPMS-based structure, as it is porous and as it has a lot of curvatures, is highly suitable for bone and muscle cell growth. Besides, the Poisson’s ratio of the structure must be chosen to be close to the Poisson’s ratio of the bone surrounding it. Otherwise, differences in the lateral motion of the bone and the scaffold can lead to their deboning upon application of an external load.

6.2

Numerical investigation of smart auxetic 3D metastructures based on shape memory polymers via topology optimization

Today, the human being endeavors to manufacture devices and materials capable of doing something in an intelligent way. SMPs are a series of smart materials, capable of retrieving their original shape from a temporary form by applying external stimuli, for example, heat, electricity, magnetism, light, pH, and humidity. In this research, the behavior of temperature-sensitive SMP-based structures with positive and NPR has been analyzed. The purpose is the material design of smart structures with tunable Poisson’s ratio using topology optimization. In this section, a metastructure is designed, which is made by a smart material. Not only does this structure have SMEs, but also it has NPR, which can be used in new sensors, actuators, and biomedical applications. After creation of the unit cell and the representative volume element (RVE) and formation of final 3D structure, FE modeling is conducted based on a thermoviscohyperelastic constitutive model at large deformations. Examining the behavior of structures in tensile prestrains of 20%, 10%, and 5%, it is observed that prestrain has no considerable effect on Poisson’s ratio, but under compressive strain of 20%, it is concluded that the type of loading is effective on the Poisson’s ratio and the results are different in tension and compression modes. Finally, the influence of temperature rate on the behavior of structures is inspected, and it is concluded that the more slowly the temperature changes, the more strain or shape recovery is accomplished at a specific temperature.

6.2.1 Introduction Pure SMPs have low deformation stiffness and low recovery stress. To overcome these limitations required in some applications, the study of shape memory polymer

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composite (SMPC) has attracted a great deal of interest [37
39]. Auxetics are structures are artificial and engineered structures with unique properties beyond natural materials which are used in myriad applications; these properties are due to structure and not a material composition [40,41]. They are usually arranged in repeating patterns and made from assemblies of multiple elements and materials like polymers and metals. Auxetics are structures with a NPR, that is, when stretched, they become thicker, and when compressed, they become thinner in the direction perpendicular to the applied force. They may have unique industrial applications, such as new sealants and cushions [42,43]. Planar auxetic structures have the potential to influence a wide range of applications. Rossiter et al. showed that an SMP auxetic hexachiral 2D structure could be tailored to prepare a tunable stiffness behavior in its fully deployed state by changing the angle of interhub connections [44]. In another work, Jacobs et al. [45] described the design and manufacturing of a deployable antenna for deepspace missions based on hybrid honeycomb truss made of a shape memory alloy. The deployable characteristics were enhanced by the equivalent auxetic behavior of the cellular configuration. Scarpa et al. [46] illustrated various dynamic features of open cell compliant polyurethane foam with an auxetic behavior. Foam samples had been tested in a viscoelastic analyzer tensile test machine to determine Young’s modulus and loss factor for small dynamic strains. Shim et al. [47] designed a class of 2D soft auxetic metamaterials whose architecture could be changed in response to an external stimulus. Clausen et al. [48] worked on topology optimized architectures with programmable Poisson’s ratio over large deformations to design their structure with 3D printing and in order to fabricate the designs and validate the numerical behavior. They created architected materials with programmable Poisson’s ratios between negative and positive ranges over large deformations of up to 20% or more. Since Bendsoe and Kikuchi introduced a general computational material distribution method, topology optimization has been growing both in academic research and industrial applications [49,50]. Over the past decade, several similar plans with slightly different attributes have been developed in the field of topology optimization. These attributes include density-based methods [51], evolutionary procedures [52], and level set methods [53], which their goal is to find optimal structural topologies or to optimize the layout of the material under the constraints and boundary conditions (BCs). Bendsoe [54] presented a density-based method that was associated with simplification assumptions and known as the solid isotropic material with penalization (SIMP) method. In this section, it is aimed to present a numerical investigation on smart auxetic 3D metastructures made from SMPs, constructed through topology optimization and the SIMP method. A typical programming path of an SMP material is shown in Fig. 6.11, which consists of four steps: (1) loading at above Tg , where SMP is in the rubbery phase; (2) cooling the structure below Tg while the external strain has been preserved (shape fixation), which SMP enters the glassy phase; (3) unloading while SMP remains in the glassy phase; and (4) heating (strain recovery), so that SMP returns to the rubbery phase, and the original shape is recovered and (40 ) heating (stress recovery) at constrained strain, where SMP comes back to the rubbery phase, but due to the external constraint a recovery force is generated.

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Figure 6.11 Stress
strain
temperature diagram depicting the thermomechanical behavior of shape memory polymer in strain and stress recovery conditions.

6.2.2 Geometrical modeling of an representative volume element Currently, one of the most widely used techniques for topology optimization based on FEM is the SIMP method with optimality criteria algorithm (OCA), developed in the late 1980s. For the use of the SIMP-OCA method in the field of material design, the density of each element ρe is defined as the amount of material to be placed within the volumetric space of that element. In a two-phase (solid-void) microstructural topology, ρe 5 0 means the element is formed from the void (empty) and ρe 5 1 indicates that the element is composed of the solid (filled) base material. For interpolation function, a relation exists between stiffness and density of elements. The Young’s modulus of solid phase defined by this method is Ee ðρe Þ 5 Eep ðE0 2Emin Þmin , where E0 is Young’s modulus of the base material, Emin indicates Young’s modulus of void phase, taken to be a small positive number to avoid singularity problems in the numerical calculation of the stiffness tensor, and the exponent p $ 1 is the parameter of penalization. The densities between zero and one are subjected to a penalization process, in order to eventually determine their status as solid and voids. An essential part of the SIMP method is to select the penalty factor to express the physical values of the problem (such as stiffness) as a function of continuous design variables. According to Refs. [50,55,56], the range of the penalty factor is obtained as the following equation and is based on the Poisson’s ratio of the base material ν 0 , which is equal to 0.435 for BMA in pure SMP.

 1 2 ν0 3 1 2 ν0 ; p $ max 15 7 2 5ν 0 2 1 2 2ν 0

(6.9)

So far, for structures made of history-dependent materials, such as SMPs, topology optimization has not been carried out, since these categories of materials are

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structurally related to the loading path and thus practically cannot be optimized uniquely. Therefore in this section, an appropriate approximation is employed to simplify the relations to achieve an approximate optimized geometry and topology. It is considered that the material behaves almost linearly in the loading step. The basis of the relationships, coding, and the creation of the topology in this section is based on [56
58] to develop RVEs. Unit cells are generated using a code. The problem-solving algorithm in this section consists of four main steps: (1) obtaining the stiffness matrix of each unit element using FEA, (2) homogenizing based on asymptotic method to get the effective elasticity matrix, (3) optimizing the topology using the SIMP method based on minimizing the function of elasticity matrix arrays, and (4) filtering the results and determining the structure. the asymptotic homogenization theory is implemented to determine the effective elasticity tensor of the optimized RVEs. The theory of asymptotic homogenization makes it possible to predict the effective properties of the periodic microstructures using FEA of the periodic unit cell. In fact, the obtained elasticity tensor of the optimized RVEs can be generalized to the overall properties due to applying the double-scale asymptotic expansion and adopting the assumption of periodicity. Therefore the elasticity tensor of the periodic unit cell is similar to the elasticity tensor of the stacked microstructure for the periodic unit cell [59]. It is noteworthy to say that the applied filtration scheme produces efficient mesh-independent designs. In other words, the filtering approach eliminates the problem of mesh dependency for the optimized RVE [60,61]. Eventually, to decrease the computational cost, the minimum allowable size of the samples has been used for the topology optimization process and construction of the unit cell. In this work, Poisson’s ratio is chosen as the target function. The constraint is the volume fraction of the structure, and the BCs are considered periodically. The flowchart in Fig. 6.12 depicts a rational explanation of the steps involved. For more details on the SIMP-OCA method, one may refer to [54,62]. Initially, as shown in Fig. 6.13, softer materials (elements with lower volume fraction) are located at the corners of the unit cell (consist of 1/8 of a sphere with the radius equals to 1/3 of the unit cell edge) to avoid uniformly distributed sensitivity field and to update the element’s densities [56,62]. According to the type of application, different Poisson’s ratios can be obtained by changing the volume fraction as a constraint and filtration parameters. Setting 0.75 as the volume fraction and performing the topology optimization procedure with the objective function which is defined based on the arrays minimization of the effective compliance tensor, then, considering the effective Poisson’s ratio [62], smart SMP structures with negative and positive Poisson’s ratio (PPR) of
0.184 and 10.213 were produced, respectively. NPR and PPR values of
0.184 and 10.213 are the results of the numerical topology optimization algorithm, and cannot be interpreted as predetermined or special values. The generated unit cells and their middle sections are illustrated in Fig. 6.14. In the next section, we compare the pure SMP structure with positive and NPRs and examine the effect of different mechanical and thermal parameters on the results, as well.

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Figure 6.12 The flowchart of the solid isotropic material with penalization methodology for the optimal design of a unit cell [2].

6.2.3 Finite element analysis of shape memory polymer microstructure FE simulation is employed to implement the described SMP model and to study the results of the topology optimization in ABAQUS. Then, implementing the discussed SMP constitutive model, we properly account for the thermomechanical behavior of SMP. The whole steps of the thermomechanical cycle are explained in Table 6.2. The standard dynamic implicit solver with backward Euler time

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Figure 6.13 The illustration of the initial guess of topology to generate unit cells [2].

integration method is used while the nonlinear geometry effects are considered. Regarding the periodic pattern of the main structure, the unit cell, is repeated in three main directions to get RVE size. To construct the final 3D periodic cell (based on the asymptotic homogenization theory), cell dependency is examined for 2 3 2 3 2 to 8 3 8 3 8 cells, and results revealed that 6 3 6 3 6 RVE cells could have enough accuracy. One-eighth of the whole structures is shown in Fig. 6.15. Thus the final structure built from the repetition of the unit cell contains 6 3 6 3 6 cells. Here, for simplicity, BCs applied to only one sample unit cell (single element) are shown in Fig. 6.16, while for the whole structure, the same BCs are assumed. According to Fig. 6.16, in all simulation steps in the plane with normal vector (21, 0, 0), BCs are U1 5 UR2 5 UR3 5 0, in the plane (0, 21, 0), BCs are U2 5 UR1 5 UR3 5 0, and finally, in the plane (0, 0, 21) BCs are U3 5 UR1 5 UR2 5 0. In the plane (0,0,1),

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Figure 6.14 The unit cell of (A) the conventional structure (positive Poisson’s ratio) and (B) the auxetic structure (negative Poisson’s ratio). The middle sections of the unit cell of (C) the positive Poisson’s ratio structure and (D) the negative Poisson’s ratio structure [2].

a tensile strain is applied along the normal vector in the loading step, while it is removed in the unloading step. Thermal BCs are explained in Table 6.2. The structures are meshed using structured 8-node linear hexahedral elements with reduced integration mode (C3D8R elements), as shown in Fig. 6.17.

6.2.3.1 Determination of elements size To find the appropriate size of elements and optimal mesh, the final structure should be analyzed with different mesh grids. As shown in Fig. 6.18, the mesh dependency analysis for the final auxetic structure at the heating step of the strain recovery cycle at 20% prestrain and 5 C=min temperature rate is performed for four different meshes. Results for the stress recovery cycle are almost similar and follow

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Table 6.2 Thermomechanical programming of shape memory polymer. Step

Condition

Temperature ð CÞ

Time intervals ðsÞ at T_ 5 1or5 C=min

(1) Loading

Applying a uniaxial tensile or compressive displacement Cooling the structure while maintaining the prestrain Unloading occurs suddenly Heating the structure (strain or stress recovery)

65

50|20

65
25

2400|480

25 25
65

50|20 2400|480

(2) Cooling (3) Unloading (4) Heating

Figure 6.15 The representative volume element structures (A) with positive Poisson’s ratio 5 10.213 and (B) with negative Poisson’s ratio 5
0.184 [2].

Figure 6.16 The boundary conditions applied to the sample unit cell (single element) [2].

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Figure 6.17 The structured mesh for the (A) positive Poisson’s ratio representative volume element and (B) negative Poisson’s ratio representative volume element [2].

Figure 6.18 The mesh dependency analysis for the auxetic structure, the strain versus time in the heating step of the strain recovery cycle [2].

the same pattern (the structure geometry, loading, BCs, and all of the steps except the heating step are the same). The total number of elements is doubled in each mesh, and it is revealed that the mesh grid in the first and the second stage does not have enough accuracy, but the third one is accurate enough. Therefore the total number of elements in the final auxetic structure is 328016, and so on for the conventional structure is 289456.

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6.2.4 Results and discussion 6.2.4.1 Comparing positive Poisson’s ratio and negative Poisson’s ratio structures Here, the effect of different parameters on structures is investigated. It is noted that for all results, the xyz- or 123-coordinate system is used. Furthermore, thermomechanical programing of structures, including loading sequences, related temperatures, and time intervals, are according to Fig. 6.11 and Table 6.2, which are determined by numbers 1 to 4. First, a comparison between the PPR and the NPR structures behavior under 20% tensile prestrain and at 5 C=min temperature rate is carried out. As mentioned before, the applied load is along the z-axis, and the structures has almost the same behavior along the x-axis and y-axis due to the symmetry. The strain along the z-axis, the strain along the x-axis, the Poisson’s ratio, and the stress along the z-axis versus time are shown in Fig. 6.19 in both the strain and

(B)

(A)

(C)

(D)

Figure 6.19 Comparing positive Poisson’s ratio and negative Poisson’s ratio structures with pure solid shape memory polymer in the strain recovery and stress recovery paths, (A) the strain along the loading ε33 , (B) the strain perpendicular to the loading ε11 , (C) the Poisson’s ratio ν 31 , and (D) the stress along the loading σ33 [2].

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Table 6.3 Results of the average Poisson’s ratio of FE simulation and topology optimization for PPR and NPR structures in the strain recovery and stress recovery [2]. Strain recovery Structures NPR PPR Pure SMP

Topology optimization 2 0.184 1 0.213 _

FE simulation 2 0.166 1 0.234 1 0.475

Relative error (%) 9.78% 9.86% _

Stress recovery FE simulation 2 0.167 1 0.233 1 0.467

Relative error (%) 9.24 9.39 _

stress recovery paths. Fig. 6.19 reveals that the strain along the loading, that is, ε33 is almost the same for all the structures, but the strain perpendicular to the loading, that is, ε11 is different. The reason is that both structures are restricted along the loading due to external displacement; however, they are free perpendicular to the loading, and because of their different Poisson’s ratio, their behavior must be different in this direction. It is noted that the Poisson’s ratio defined as ν 31 5 2 ε11 =ε33 is selected as the optimization objective to be different for the structures. The average Poisson’s ratio is calculated and compared to that of the topology optimization as shown in Table 6.3, and it is observed that all of the relative errors are less than 10%. The trend of the stress along the loading, that is, σ33 is the same for all cases, but values are different, arising from the nonlinearity in the geometries and structures, and the stiffness matrices.

6.2.4.2 The effect of prestrain on the response of negative Poisson’s ratio structure From this section on, for the sake of briefness, the results are reported only for NPR structure. Here, the behavior of NPR structure under 20%, 10%, and 5% prestrains at 5 C=min temperature rate is scrutinized. The applied load and BCs are the same as before. ε33 , ε11 , ν 31 , and σ33 versus time are depicted in Fig. 6.20 in the strain recovery and stress recovery paths as well. This figure reveals that the amplitude of ε33 , ε11 , and σ33 is declined with almost the same ratio by reducing the prestrain. The more the amplitude of prestrain is decreased, the more the amplitude of ε33 , ε11 , and σ33 is reduced. It is clear that the prestrain almost does not affect the Poisson’s ratio, because as the amount of prestrain along the loading is declined, the amount of strain perpendicular to the loading is also decreased proportionally, and this causes the Poisson’s ratio to be unchanged during the path while the prestrain values are changing.

6.2.4.3 The effect of loading type on negative Poisson’s ratio structure In this section, a comparison between the behavior of NPR structure under 20% compressive loading and 20% tensile loading at 5 C=min temperature rate is

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(A)

(C)

(B)

(D)

Figure 6.20 The effect of prestrain on negative Poisson’s ratio structure in the strain and stress recovery paths, (A) the strain along the loading ε33 , (B) the strain perpendicular to the loading ε11 , (C) the Poisson’s ratio ν 31 , and (D) the stress along the loading σ33 [2].

conducted. The applied load and BCs are the same as before. ε33 , ε11 , ν 31 , and σ33 versus time are depicted in Fig. 6.21 in the strain recovery as well as stress recovery. As one may observe from Fig. 6.21A, ε33 for the compressive loading is symmetric compared to the tensile one, but this symmetry does not occur in ε11 as shown in Fig. 6.21B. The reason is that, in the cooling step, since the structure is not bounded in the x-direction and the temperature decreases simultaneously, the structure is contracted and ε11 is reduced in both cases. In the heating step, the temperature reaches above Tg , and ε11 returns to zero in the strain recovery path, so the strain and the shape of the structure recover in both cases. Moreover, the stress and force are recovered in the stress recovery path. As a result, the Poisson’s ratio behavior becomes different, as illustrated in Fig. 6.21C. According to Fig. 6.21D, in the cooling step, as the structure is bounded in the loading direction, by reducing the temperature, a positive stress is developed in the structure to counteract the decrease in the temperature and stands for noncontraction of the structure in this direction, while the other steps are like before. Thus we can have smart SMP structures that reach a maximum and minimum volume in their temporary shape in

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Figure 6.21 Effect of loading type on negative Poisson’s ratio structure in the strain and stress recovery paths, (A) the strain along the loading ε33 , (B) the strain perpendicular to the loading ε11 , (C) the Poisson’s ratio ν 31 , and (D) the stress along the loading σ33 [2].

tension and compression, respectively, which can be used for a variety of potential applications, especially in biomedical devices.

6.2.4.4 The effect of temperature rate on negative Poisson’s ratio structure Finally, to study the effect of temperature rate, the behavior of NPR structure under 20% prestrain at 5 and 1 C=min temperature rates is studied. As it is noted, thermal expansion coefficients depend on temperature and temperature rate, according to Ref. [27], and data for thermal expansion for other temperature rates were not available. Again, the applied load and BCs are the same as before. ε33 and ε11 versus the temperature, ν 31 versus the normalized time, and σ33 versus the temperature are shown in Fig. 6.22 in both the strain and stress recovery paths. According to Fig. 6.22A, in the heating step of the strain recovery, at lower temperature rates, the structure has a lower strain at a certain temperature, and the diagram is shifted to the left, because a higher strain recovery is achieved at a certain temperature. This event occurs in

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(A)

(B)

(C)

(D)

Figure 6.22 Effect of temperature rate on negative Poisson’s ratio structure in the strain and stress recovery paths, (A) the strain along the loading ε33 , (B) the strain perpendicular to the loading ε11 , (C) the Poisson’s ratio ν 31 , and (D) the stress along the loading σ33 [2].

Fig. 6.22B, as well. As illustrated in Fig. 6.22C, the temperature rate does not affect the Poisson’s ratio, since the ratio of the strain perpendicular to the loading to the strain along the loading is unchanged. As shown in Fig. 6.22D, in the cooling step, as the temperature changes more slowly, the structure has lower stress at a certain temperature, because it senses the temperature changes faster, so smaller stress is developed in the structure. All these discussions stand correct for the stress recovery. For example, in Fig. 6.22D, in the heating step, as the temperature rises more slowly, less amount of stress is applied to the structure.

6.3

Summary and conclusions

In Section 6.1, four types of widely used TPMS-based geometries (diamond, gyroid, IWP, and primitive) were employed to construct SMP lattice structures in different

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volume fractions. A thermovisco-hyperelastic constitutive model was utilized to model the mechanical behavior of SMP. The TPMS-based geometries were produced in two configurations, filled and core
shell. The trends of the shape recovery, shape fixity, Poisson’s ratio, and force recovery were examined for different volume fractions. Shape recovery generally had a negative correlation with growth in the volume fraction (except the case of core
shell primitive model for φξ . 35%), while shape fixity and force recovery for all the geometries (in both filled and core
shell configurations) had a positive correlation with φξ . Being filled or core
shell did not have a significant effect on the shape fixity of the TPMSbased lattices with their maximum difference being 4%. However, being filled or core
shell had a significant impact on the Poisson’s ratio in diamond and primitive geometries. The variations of Poisson’s ratio with respect to the volume fraction in the filled diamond and filled primitive structures were linear, whereas they were nonlinear in the corresponding core
shell geometries. The results demonstrated the high influence of geometric design and volume fraction on the shape-memory behavior in SMPs and their Poisson’s ratio. In some applications, the capability of the SMP in recovering its initial shape, that is, high shape recovery, is of great importance for programming. On the other hand, in some applications, such as actuators and sensors, the force recovery capability and/or displacement extents in other axes are of more importance. According to the applications and requirements in different fields, a specific geometry and volume fraction can be chosen. The wide range of SMP TPMS-based topologies, volume fractions, and configurations (core
shell or filled) introduced for metamaterials can be advantageous for all the above-mentioned requirements. The other benefit of the chosen geometries is their high production capability, as these geometries are constructed by surfaces based on continuous mathematical equations, which can be easily tailored according to the need. In Section 6.2, the material design and FEA of smart auxetic 3D metastructures based on SMPs were numerically examined. These shape memory metamaterials can be used in biomedical devices and structures, in which, two rational designing and SME have been coupled. It means the auxetic or conventional properties of structures can be controlled via an external applied thermal source. For instance, these structures could be used as bone repairing tools or deployable scaffold in the body. In this work, first, tunable PPR and NPR unit cells were created by optimizing the topology using the SIMP method. Then, RVEs and the final 3D structures were generated by repeating the unit cells. Thereafter, FE modeling was done based on the thermovisco-hyperelastic constitutive model at large deformations for SMPs. Results of topology optimization and FE simulation were compared, and the relative differences were less than 10%. The behavior of PPR and NPR smart structures were studied and evaluated. Also, the effect of mechanical and thermal parameters was explored which could be summarized as follows: (1) The amplitude of the strain and the stress is lowered with almost the same ratio by reducing the applied prestrain; however, prestrain nearly does not affect the Poisson’s ratio. (2) Different types of loading may result in a different strain, stress, and the Poisson’s ratio due to the history-dependent thermomechanical behavior of the SMP-based structures.

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(3) The temperature rate influences the behavior of smart structures, and the more slowly the temperature changes, the more strain and shape recover at a specific temperature.

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A review on constitutive modeling of pH-sensitive hydrogels

7

Chapter outline 7.1 Introduction 245 7.2 Applications of pH-sensitive hydrogels 249 7.2.1 Drug delivery 249 7.2.2 Soft actuators 249 7.2.3 Control of microfluidic flow 249

7.3 Swelling/deswelling phenomena 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6

250

Conservation of mass 250 Chemical reaction 250 Ion transfer 252 Electrical field 252 Fluid flow 253 Mechanical field 253

7.4 Swelling theories

255

7.4.1 Monophasic models 256 7.4.2 Multiphasic models 258

7.5 Numerical implementation 258 7.6 Experiments 260 7.7 Summary and conclusions 262 References 263

7.1

Introduction

Three-dimensional hydrophilic polymeric networks, which span the volume of an aqueous solution and ensnare it through surface tension effects without dissolution, are called hydrogels [1,2]. Even under pressure, the absorbed solution cannot be removed from the swollen hydrogels. In the polymeric network of hydrogels, there exist cross-links (i.e., junctions, tie-points, entanglements) which avoid them from being dissolved in the solution. Therefore the hydrogels contain properties of both solids and fluids. The fascinating properties of the hydrogels make them very attractive for researchers. Querying scopus.com, one of the enormous scientific databases for the word “hydrogel” in the title of documents, reveals a growing trend in the number of the published works, particularly a quick ascent from 1988. The query results are illustrated in Fig. 7.1A. Also, the subject of the area in which the documents are published are depicted in Fig. 7.1B, according to which most of the works on hydrogels are in the area of material science, chemistry, and engineering. Moreover, we compare the number of published works on hydrogels in different countries in Fig. 7.1C. The result shows that the pioneering countries are the United Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00007-0 © 2023 Elsevier Inc. All rights reserved.

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(A)

(B)

(C)

Figure 7.1 (A) Number of documents containing “hydrogel” in the title for the last 55 years, indicating an increase in the publications, (B) the clustered bar chart of the most favorable areas of which hydrogel works are published during the last 55 years, and (C) the global trend on the published works related to hydrogels.

States and China, followed by Japan, South Korea, India, Germany, and the United Kingdom. It should be noted here that in the present analysis it is assumed that every section of the beam is isothermal at each thermal step. If the exact temperature distribution is required, a simultaneous thermal analysis should be performed along with an appropriate SMP model. However, these discussions were not followed here and are out of the scope of this section. Hydrogels can be classified in various ways. The classification of hydrogels based on the common parameters is shown in Fig. 7.2. One of the critical parameters in the classification of hydrogels is their responsiveness. The material characteristics of a hydrogel such as the swelling ratio, permeability, and shear modulus, change in response to the external stimuli, such as pH, temperature, electrostatic field, and ionic strength of the surrounding solution. This exciting property classifies hydrogels as smart materials and makes them attractive for applications, such as in drug delivery, tissue engineering, soft actuators, and chemical microvalves.

A review on constitutive modeling of pH-sensitive hydrogels

247

Figure 7.2 The complete classification of hydrogels based on various parameters [3].

Figure 7.3 Schematic representation of swelling behavior of anionic, cationic, and ampholytic pH-sensitive hydrogels in various pH values [3].

As classified in Fig. 7.2, hydrogels are either neutral or ionic. The ionic hydrogels contain ionic (i.e., acidic and/or basic) pendant groups on their polymeric network and are categorized into three categories: the anionic, the cationic, and the ampholytic. The ionic ones are pH sensitive and suddenly change volume under pH variation of the environment. The swelling response of ionic hydrogels is schematically shown in Fig. 7.3. The anionic hydrogels contain acidic groups on their polymeric network and swell in the pH values beyond the acidic dissociation constant (i.e., Ka) of the acidic groups. However, cationic hydrogels swell at pH values lower than the basic dissociation constant (i.e., Kb) of their basic groups fixed on the

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polymeric network. On the other hand, the ampholytic hydrogels contain both the acidic and basic groups and swell at the pH values between the acidic and basic dissociation constants. The swelling mechanism of the pH-sensitive hydrogels is explained in detail in Section 7.3. Due to their attractive properties and dual attributes of a solid and a liquid, pHsensitive hydrogels are extraordinary candidates for applications such as drug delivery, tissue engineering, soft actuators, and chemical microvalves. More discussion on the applications of pH-sensitive hydrogels is given in Section 7.3. Optimum utilization of hydrogels in such applications requires precise simulating and predicting their stimuli-sensitive behavior in equilibrium and transient conditions concerning environmental conditions. A crucial prerequisite for the simulation of a hydrogel is employing a swelling theory coupled to its mechanical response. In this chapter, we review the most important swelling theories of pH-sensitive hydrogels. The most effective keywords used to gather the articles for this chapter and their relationship is shown in Fig. 7.4, where the publication date of the belonging articles is illustrated with rainbow colors. This chapter is organized as follows: in Section 7.2, the most highlighted applications of the pH-sensitive hydrogels are briefly introduced. Then, in Section 7.3, the swelling/deswelling mechanism of the pH-sensitive hydrogels are explained, and the phenomena and their governing equations are presented. In Section 7.4, the swelling theories of pH-sensitive hydrogels are classified and the most important proposed theories for each category are listed. After that, the more commonly applied methods for the numerical simulation of hydrogels are briefly introduced in Section 7.5. We also introduce and examine the most highlighted experiments on the swelling and deswelling behavior of the pH-sensitive hydrogels in Section 7.6. Finally, we present a summary and draw conclusions in Section 7.7.

Figure 7.4 The predominant keywords used to gather the articles for this review and their relationship.

A review on constitutive modeling of pH-sensitive hydrogels

7.2

249

Applications of pH-sensitive hydrogels

Hydrogels have numerous applications in drug delivery, tissue engineering, soft actuators, control of microfluidic flow, agriculture, diagnostics, pharmaceuticals, separation of biomolecules or cells, bio-medication, wound healing, food industry, etc. The following subsections briefly deal with the most highlighted applications of pH-sensitive hydrogels.

7.2.1 Drug delivery The pH-sensitive hydrogels have been broadly studied and utilized in biomedical applications particularly in drug delivery systems making use of pH variation in gastrointestinal tract. An ideal drug delivery system should be biocompatible and biodegradable and also it should be able to load the drug in high amount and release it in a controlled manner in the required location. pH-sensitive hydrogels are extraordinary candidates for drug delivery since there is a wide gradient of pH along the gastrointestinal tract [4,5].

7.2.2 Soft actuators Due to the sudden reversible swelling of pH-sensitive hydrogels at a specified pH, these materials are suitable candidates for soft sensors and actuators. As observed in recent studies, utilizing bending structures is a typical approach in the design of sensors and actuators. One of the pioneers of multilayered structures is Timoshenko [6]. In flexible bilayer structures, usually one layer is made of stimuli-responsive hydrogel, and the other layer is made of elastomer [7
11]. As the hydrogel layer swells the structure bends [12
14]. The schematic representation of the hydrogel bilayer structure is shown in Fig. 7.5.

7.2.3 Control of microfluidic flow The autonomous flow control is the most common application of stimuli-responsive hydrogels in microfluidic systems. Functional hydrogel structures for flow control inside microfluidic channels originate from the pioneering works of Beebe et al. by employing the effective photo-polymerization fabrication method [15
17]. Several designs have been presented for the microfluidic flow control. These devices also work based on the swelling behavior of the pH-sensitive hydrogels. At the desired pH value, the microvalve, which is made of pH-sensitive hydrogel, swells and blocks the fluid flow in the microchannel [18
22]. Hydrogel Elastomer

Swelling induced large deformation

Figure 7.5 Schematic representation of the swelling induced bending of the hydrogel bi-layered structure.

250

7.3

Computational Modeling of Intelligent Soft Matter

Swelling/deswelling phenomena

The most exciting property of hydrogels is their swelling behavior. When a pHsensitive hydrogel is immersed in a solution, the solution diffuses into the hydrogel polymeric network and acidic/basic groups bonded to the network dissociate in the basic/acidic solution. Therefore an H1 =OH2 ion comes off which can form an H2 O in combination with the OH2 =H1 . The basic/acidic conjugate of the acidic/basic groups constitutes an electrostatic field. Therefore the co-ions and counter-ions of the external solution migrate into the hydrogel not only because of the convection and the diffusion, but also due to the electrostatic field. The difference of the concentration of ionic species inside and outside of the hydrogel leads to the osmotic pressure, which causes the hydrogel to swell/deswell. The pH-sensitive hydrogels also swell/deswell due to the tendency of the polymeric network to mix with the external solution. However, the effect of the osmotic pressure typically is larger than the mixing force [23]. On the other hand, the elastic/viscoelastic force of the network arises against swelling of the hydrogel. Thereafter, the equilibrium of the pH-sensitive hydrogels occurs when the elastic/viscoelastic force balances the osmotic pressure, solvent pressure, and the electrostatic forces. In this section, the physical and chemical laws involved in swelling theories of the pH-sensitive hydrogels are introduced.

7.3.1 Conservation of mass The law of conservation of mass or principle of mass conservation states that the mass change of each species inside the hydrogel attributes from the migration of the corresponding species through the surface boundaries as well as the mass consumption or production of each species due to the chemical reactions. Thus the mass balance is stated as, c_ m 1 r 3 ðjm Þ 5 Rm ;

(7.1)

in which cm refers to the concentration, jm denotes the flux, and Rm is the reaction rate of the mth species (including ions and solvent molecules), also dot indicates time derivative. Hong et al. [24] and Duda et al. [25] have argued that it is more appropriate to use mass balance equation in the current configuration. The challenging part of the mass balance equation in the swelling theories is the determination of the flux (i.e., jm ). The flux of ions can be found using nonequilibrium thermodynamics [24] or utilizing the Nernst
Planck equation [23].

7.3.2 Chemical reaction As shown in Fig. 7.6, the polymer network of pH-sensitive hydrogels bears acidic/ basic groups, which we denote them as AH/BOH. When the solvent diffuses into the hydrogel, some of the acidic/basic groups dissociate into the H1 =OH2 and conjugate

A review on constitutive modeling of pH-sensitive hydrogels

251

Figure 7.6 Schematic representation of the reference and current configurations of a pHsensitive hydrogel with an arbitrary shape undergoing deformation due to external stimuli [26].

base/acid A2 =B1 . We call the conjugate base/acid the fixed charge because it gives rise to a network-bound charge. The chemical reaction is reversible, that is, AH"A2 1 H1 ;

(7.2)

BOH"B1 1 OH2 :

(7.3)

The chemical species AH/BOH, A2 =B1 , and H1 =OH2 are in chemical equilibrium as soon as their concentration does not change with passing the time. The dissociation constant is defined as, Ka 5

cH cA ; cAH

(7.4)

Kb 5

cOH cB ; cBOH

(7.5)

in which cm (m 5 H, A, AH, OH, B, and BOH) is the concentration of hydrogen ion, basic conjugate fixed charges, acidic groups, hydroxide, acidic conjugate fixed charge, and basic groups, respectively. On the other hand, the sum of the concentration of associated ionic groups and fixed charges equals the total concentration of ionic groups CfAH =CfBOH , that is, cAH 1 cA 5

CfAH ; J

cBOH 1 cB 5

CfBOH : J

(7.6)

(7.7)

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Combining Eqs. (7.4) and (7.6) as well as Eqs. (7.5) and (7.7), the concentration of fixed charges can be determined for anionic and cationic pH-sensitive hydrogels, respectively, as follow, cA 5

CfAH Ka ; J K a 1 cH

(7.8)

cB 5

CfBOH Kb : J Kb 1 cOH

(7.9)

From now on, without losing the generality, we write the equations for anionic hydrogels (i.e., pH-sensitive hydrogels with acidic groups) for the sake of briefness.

7.3.3 Ion transfer The flux of the ionic species is generally due to the diffusion, convection and electrical migration; the Nernst
Planck equation describes the flux as [26
28], jm 5 φð2 Dm rcm 2 zm μm Fcm rψÞ 1 cm v;

(7.10)

where φ is the porosity of the hydrogel, Dm refers to the diffusion coefficient of mth species, the charge number of the ionic species is denoted by zm, μm stands is the ionic mobility of mth species, F stands for Faraday’s constant, ψ is to the electric potential, and v refers to the fluid velocity relative to the polymer network. The diffusivity is related to the ionic mobility with the Einstein relation, that is, Dm 5

RTμm ; F

(7.11)

in which R is the ideal gas constant and T denotes the absolute temperature. In pH-sensitive hydrogels, typically the most highlighted term of the Nernst
Planck equation in the absence of the external electric field is the gradient of the concentration of species. In other words, the convection and electrical migration effect are negligible compared to the diffusion because the osmotic pressure is the most important reason for the swelling [23,29].

7.3.4 Electrical field The spatial electric field, which is needed in Eq. (7.10), can be determined employing Poisson’s equation as [30], r:ðε0 εr rψÞ 5 2 FðzA cA 1

X

z c Þ: i i i

(7.12)

A review on constitutive modeling of pH-sensitive hydrogels

253

In this equation, ε0 and εr , respectively, refer to the permittivity of the vacuum and the hydrogel. The normalized form of the Poisson’s equation is as follows: P    eψ zA cA 1 i zi ci P 2 ; κ2D r 3 r 52 kB T i zi ci

(7.13)

in which κD denotes the Debye length, defined as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε 0 ε r kB T κD 5 : (7.14) 2eFI P where I 5 12 i z2i ci is the ionic strength of the external solution. The Debye length indicates how far the electrostatic effect persists and is a measure of the effect of the electrostatic charge. In other words, outside the Debye length the electroneutrality is almost maintained both inside the hydrogel and in the external solution at the interface of the hydrogel with the solvent. The Debye length is usually in the order of nanometer for hydrogels [31]. For hydrogels with characteristic size much larger than the Debye length, the left-hand side of Eq. (7.9) vanishes over the volume of the hydrogel except inside the double layer. The double layer refers to two parallel layers of charge surrounding the hydrogel. However, the assumption of the electro-neutrality should be rechecked when the size of the hydrogel is comparable to the Debye length [32].

7.3.5 Fluid flow Darcy’s law describes the flow of a fluid through a porous medium and is extended to prescribe the fluid mass flux through the porous polymeric network of the hydrogel [33,34] as, rp 5 2

μf kp ρf0

q;

(7.15)

in which p denotes the total intrinsic fluid pressure inside the hydrogel, μf refers to the dynamic viscosity of the fluid, kp stands for the polymer phase permeability, ρf is the true density of the fluid phase, and q denotes the Eulerian relative flow vector of the fluid phase with respect to the polymer phase. It should be noted that Darcy’s law only holds for laminar flows with Reynolds numbers smaller than one, which is consistent with the assumption of Stokes flow on the microscale.

7.3.6 Mechanical field Hydrogels are classified as soft materials and typically can change volume up to 1000 times of their initial state. Therefore a large deformation theory is required to

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formulate the swelling behavior of the hydrogel accurately. As depicted in Fig. 7.6, typically the dry state of the hydrogel is considered as the initial configuration denoted by B0 . Moreover, in some theories, the equilibrium state of the free swollen hydrogel is taken as an intermediate configuration in order avoid the numerical difficulties [35,36]. Following the continuum theory, the hydrogel is assumed as a set of continuous points with the coordinate X in the initial (undeformed) configuration. When the hydrogel swells and deforms, each point travels to a new place with a coordinate x in the current (deformed) configuration introduced by B. The deformation is described with the deformation gradient tensor, that is, F5

@x @X

(7.16)

Also, the determinant of the deformation gradient tensor specifies the volume ratio of the current to the initial state, J 5 det ðFÞ:

(7.17)

The concentration of species m in initial configuration is defined as Cm where the same concentration in the current configuration is denoted by cm. These two variables are related by the following equation, Cm 5 Jcm

(7.18)

The osmotic pressure that causes the hydrogel to swell consists of two parts: the osmotic pressure due to the difference of the concentration of ions in the hydrogel and in the external solution and the osmotic pressure due to mixing the solvent and the polymeric network [33,37], that is, Π 5 Π mix 1 Π ion

(7.19)

Both terms of the osmotic pressure can be found using the thermodynamics laws [31,37]. As Horkay et al. [38] discussed, the ionic osmotic pressure is the dominant term in Eq. (7.15) and is expressed as [23], Π ion 5 RT

X

ðck 2 c0k Þ

(7.20)

k

in which c0k is the concentration of the kth ion in the external solution. Although some researchers treated the hydrogel as an elastic material [23,29,39], the hydrogel is typically taken as an isotropic hyperelastic material defined with a free-energy function, W, which is assumed to be a function of the deformation gradi~ and the concentration of ions ck [24,33,37,40], as, ent F, the electric displacement D, ~ c1 ; c2 ; :::Þ WðF; D;

(7.21)

A review on constitutive modeling of pH-sensitive hydrogels

255

Small changes in the independent variables cause the free-energy to change as, δW 5

X @W @W @W ~ δF 1 δck δD 1 ~ @F @ck @D

(7.22) .

In thermodynamically-consistent theories, the constitutive equations are the consequence of the second law of thermodynamics. The hydrogel is assumed as a thermodynamic system where the thermodynamics states that the Helmholtz free energy of the system never increases [24], Ð Ð Ð δWdV- Bi δxi dV- Ti δxi dA- ψδqdV Ð PÐ PÐ 2 ψδωdA 2 μk δr k dV 2 μk δik dA # 0

Ð

(7.23)

in which B and T denote the body and traction forces, respectively. Also, δq and δω are the electronic changes on a volume and interface element, respectively. Finally, μk refers to the electrochemical potential of kth mobile species with rk and ik number of species on a volume and interface element, respectively. As presented in [24] the constitutive equations can be found by substituting Eq. (7.22) in Eq. (7.23) and applying the divergence theorem and then forcing each term to be less than or equal to zero. Once the osmotic pressure and external loads on the hydrogel are known the deformation of the hydrogel can be found using the balance of linear momentum as, ρ_v 2 r:σ 2 ρb 5 0;

(7.24)

where ρðx; tÞ refers to the mass density, σ is the Cauchy stress, and ρb denotes the force density. The swelling of the hydrogel is a diffusion limited process [34], therefore the inertia term in Eq. (7.24) can be neglected (i.e., the deformation process is quasistatic). Typically, hydrogels are treated as an isotropic hyperelastic material [24,37,40,41]. Therefore free energy is needed to determine the stress field. In 1943, Flory and Rehner [42] provided the free energy function using the statistical method. For more details on the determination of the stress field using the free energy function see [24].

7.4

Swelling theories

In modeling the swelling/deformation response of ionic hydrogels, the crucial challenge is whether the system is monophase or multiphase. The appropriate answer cannot be underestimated. In the monophasic scenario, the hydrogel is taken as a single-phase matter including the polymeric network, the solvent, and the ions. This approach is suitable for hydrogels with nano/microporosity. On the other hand, in the multiphasic approach, the polymeric network, the solvent and the ions are

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considered separate phases, which can exchange momentum. This approach is more convenient for the hydrogels with macroporosity. However, the multiphasic approach is more complicated than the monophasic one because each phase has its governing equation. Multiphasic theories provide a highly detailed analysis of hydrogels behavior; however, due to its simplicity, the monophasic approach is more common for simulating the behavior of the pH-sensitive hydrogels.

7.4.1 Monophasic models The monophasic approach is based on the poroelasticity theory founded by Biot [43]. In this approach, the coupling mass transport and large deformation, as well as the nonequilibrium thermodynamics theory, are applied to the hydrogel. De et al. [23], Baek and Srinivasa [44], Marcombe et al. [37], Chester and Anand [41], Duda et al. [25], and Dehghany et al. [31] are just some examples of this line in a quite extensive literature. Eqs. (7.1), (7.8), (7.9), (7.12), and (7.24) are the governing equations for this family of theories. The list of the most important monophasic swelling theories for pH-sensitive hydrogels is presented in Table 7.1. In this section, we explain the swelling mechanism and phenomena involved in monophasic theories. In monophasic theories, the osmotic pressure causes the hydrogel to swell. To find the osmotic pressure, the concentration of the mobile ions as well as the fixed charge is required. The conservation of mass as given in Table 7.1 determines the concentration of mobile ions. However, the challenging part of the story is the determination of the ionic fluxes, which are required in the conservation of mass equation. Generally, there are two approaches for ionic flux determination: (I) the Nernst
Planck equation and (II) the gradient of the chemical potential. The Nernst
Planck equation is given in Eq. (7.10) in which the concentration of mobile ions, the electric potential, and the fluid velocity are involved. In this step, the Debye length should be calculated as given in Eq. (7.14). In the absence of the external electrical field, if the Debye length is very smaller than the hydrogel size, the hydrogel and the external solution can be assumed to be electroneutral, and the electric potential can be neglected in the Nernst
Planck equation. On the other hand, if the Debye length is comparable to the hydrogel size, the Poisson’s equation given in Eq. (7.12) gives the electric potential. As the fluid velocity has a negligible effect in the Nernst
Planck equation [29], this term is typically neglected in monophasic theories [23,24,37]. The second approach in the determination of ionic flux is the gradient of the chemical potential. This approach usually is used in thermodynamically consistent monophasic theories [24,31,37,61]. In this approach, the ionic flux is derived from thermodynamics law Eq. (7.23) and is expressed as the gradient of the electrochemical potential of each species. Once the concentration of the mobile species is known, the concentration of the fixed species can be found using Eqs. (7.8) and (7.7). Then, Eq. (7.20) gives the osmotic pressure which dictates the swelling behavior of the hydrogel. It should be noted that the concentration of ionic species is dependent on the deformation.

Table 7.1 List of the most important monophasic swelling theories for pH-sensitive hydrogels. Ionic flux Work

Citations

Large deformation

[45] [39] [23] [29] [44] [46,47] [37] [37] [48] [49] [50] [51] [52
58] [59] [60] [61] [62] [31] [63] [64] [65] [26] [66]

531 745 434 117 36 109 267 196 27 15 56 23 76 5 6 21 13 19 7 2 0 21 9

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Nernst
Planck

1 1 1

1

1 1 1 1

Simulation

Gradient of chemical potentials

Electroneutrality

1

1

Thermodynamically

1 1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1 1

1 1 1 1 1

1

1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

Equilibrium 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Transient

Experimental verification

1 1 1

1 1 1 1

1

1 1 1

1 1

1 1 1 1 1

1 1 1 1

1 1 1

Geometry

1D 2D 1D 1D 1D 1D 2D 1D 2D 2D 2D 1D 1D 1D 1D 2D 2D 3D 2D 1D 1D 3D 3D

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Therefore the governing equations of the swelling behavior of the hydrogel are fully coupled.

7.4.2 Multiphasic models The multiphasic approach is developed based on the mixture theory and typically is more popular between biomechanics studies because of the complex composition of the biological tissues. The multiphasic theory has a long history and dates back to Truesdell and Toupin [67]. Later Bowen [68] developed the theory further with the application in the porous medium with one solid phase and multiple fluid phases. Then, Lai et al. [69] presented a tri-phasic theory with the solvent, the network, and the ions as distinct phases and individual velocity fields. Recent works along this line are Li et al. [70], Del Bufalo et al. [71], and Yu et al. [33]. The multiphasic theories provide a very detailed analysis of hydrogel behavior. However, the main drawback of these theories is that they are limited by the relative difficulties in implementing in numerical simulations. The most important multiphasic swelling theories for pH-sensitive hydrogels and their specifications are listed in Table 7.2. In this section, we summarize the governing equations and briefly explain the phenomena involved in multiphasic theories. In multiphasic theories, conservation of mass Eq. (7.1) is solved for a multiphasic mixture comprising the solid, fluid, and the ionic phase utilizing the mixture rule. The conservation of momentum is also considered in terms of generalized Darcy’s law given in Eq. (7.15) for a moving porous medium and balance of linear momentum as given in Eq. (7.24) with finite deformation for each phase employing the mixture rule [78]. The ionic transfer is taken into account utilizing the Nernst
Planck equation as given in Eq. (7.10) and the electro-neutrality is assumed if the hydrogel size is very larger than the Debye length [70]. The main difference between multiphasic theories and monophasic ones is that the former take into account the relative motion of the solvent and the polymer through the Darcy’s law.

7.5

Numerical implementation

After developing the swelling theory, numerical implementation is required to simulate the behavior of hydrogels accurately. Commonly the numerical method is used to predict the swelling behavior of hydrogels because the analytical methods are usually not applicable for complex geometries. Typical simulation tools, which are used to study the behavior of hydrogels are finite element method, meshless method, and molecular dynamics (MD) simulations. Finite element simulation of hydrogel swelling has been expedited with the development of monophasic swelling theories [79
85]. Various finite-elementbased methods have been proposed in the literature, such as in-house codes [86
88], implementation through user-defined subroutines of ABAQUS (e.g., including user-defined hyperelastic material [35,37,89
91], user-defined element

Table 7.2 List of multiphasic swelling theories for pH-sensitive hydrogels. Ionic flux Work

Citations

[69] [72] [73] [70] [71] [74] [75] [76] [77]

1272 309 97 77 103 23 7 4 4

Large deformation

Nernst
Planck

1 1 1 1 1 1

Simulation

Gradient of chemical potentials

Electroneutrality

Thermodynamically consistent

Equilibrium

1 1 1

1 1 1

1 1

1 1 1 1 1

1 1 1 1 1

1 1 1

1 1 1 1

1 1 1

Transient

1 1 1

Experimental verification

Geometry

1

1D NA 2D 2D 2D NA 1D 1D 1D

1 1

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[92], and user-defined heat transfer with the analogy between the mass and heat transfer phenomena), and implementation using COMSOL platform [36,93,94]. COMSOL Multiphysics is a powerful software for analysis of coupled nonlinear physical material models. Therefore the package has been used for transient swelling analysis of neutral hydrogels [93], temperature sensitive hydrogels [78] and viscoelastic polymeric gels [95,96]. Also, a chemo-electro-mechanical finite element model is developed by Wallmersperger et al. [81,97] to solve for the equilibrium swelling of pH-sensitive hydrogels. Moreover, to simulate the inhomogeneous swelling of neutral hydrogels in equilibrium state, Liu et al. [98] presented a finite element algorithm based on the multiplicative decomposition of the deformation gradient. It should be noted that the extended finite element method is also useful in simulating the behavior of hydrogels [99
101]. Beside the finite element method, meshless methods are also used for simulation of hydrogels behavior [102]. However, there is a wide variety of approaches for meshless methods which has been used in the modeling of hydrogels including the improved complex variable element-free Galerkin [103], strong-form meshless random differential quadrature method [86], finite cloud method [23,29], and Hermite
Cloud method [103
121]. MD simulations are also employed for simulation of pH-sensitive hydrogels at smaller length scales [122
129]. Quesada-Perez et al. [130] simulated the steadystate swelling behavior of gels in response to the temperature and pH changes using coarse-grained MD. The thermodynamic and mechanical properties of polymeric hydrogel networks are investigated by Jaramillo-Botero et al. [131] utilizing the atomistic level MD.

7.6

Experiments

In simulating the swelling response of the pH-sensitive hydrogels, the experiments are of high importance because of two reasons: (I) finding the necessary parameters for simulation and (II) verification of the theory and the numerical implementation. Therefore several experimental studies have been performed on examining the swelling response of the pH-sensitive hydrogels [23,132
140]. Although some experiments on mechanical properties of pH-sensitive hydrogels such as Young’s modulus and Poisson’s ratio have been performed [23], both equilibrium and transient swelling and deswelling are the most important experiments, which can be used for parameter estimation as well as verification of employed constitutive model. As an example of swelling and deswelling experiment, De et al. [23] considered a cylindrical pH-sensitive hydrogel constrained in a glass channel with a rectangular cross-section of 1000-μm width by 180-μm height, which restricts the top and bottom displacement of the hydrogel as shown in Fig. 7.7. The swelling behavior of four cylindrical pH-sensitive hydrogels with different diameters in various pH values reported by De et al. [23] is shown in Fig. 7.8. As the pH

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261

Figure 7.7 Schematic representation of De et al. [23] test setup: a cylindrical pH-sensitive hydrogel embedded in a rigid glass channel [26].

Figure 7.8 Experimental data of De et al. [23] on equilibrium swelling behavior of pHsensitive cylindrical hydrogel with four various diameters in different pH values [3].

reaches to the dissociation constant, the fixed ionic groups begin to dissociate, which leads to an increase in the osmotic pressure and finally causes the hydrogel to swell. De et al. [23] also reported the transient swelling and deswelling behavior of the cylindrical pH-sensitive hydrogel as shown in Fig. 7.9. In this figure, the transient swelling and deswelling behavior of the pH-sensitive cylindrical hydrogel with three various diameters is depicted for a pH change of 6 to 3 and then 3 to 6. It can be seen that the deswelling process is approximately ten times faster than the swelling. The reason for this phenomenon is that during the swelling, the volume of the hydrogel increases, therefore the diffusion path becomes longer for the water; also, the concentration of the fixed acidic group which is dependent to the deformation decreases, which results in a lower osmotic pressure. In contrast, the volume of the hydrogel is reduced during the deswelling process, which lowers the diffusion path and raises the concentration of fixed acidic group, which in turn intensifies the osmotic pressure and causes the hydrogel to deswell much faster. Also, the

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Figure 7.9 Experimental data of De et al. [23] on transient swelling and deswelling behavior of the pH-sensitive cylindrical hydrogel with three various diameters for a pH change of 6
3 and then 3
6 [3].

hydrogels with smaller diameter deform faster than ones with a larger diameter. The reason is that in larger hydrogels it takes longer for the solvent to penetrate. Generally, the deformation time for the hydrogel is proportional to the square of hydrogel’s characteristic length (i.e., hydrogel’s diameter in our example) [34].

7.7

Summary and conclusions

Current research in the field of pH-sensitive hydrogels as smart materials with amazing applications was focused on the fundamental phenomena in studying the swelling behavior of the pH-sensitive hydrogels. The vital challenge in modeling the swelling response of these attractive materials was whether the system is monophase or multiphase. The answer to this question leads to two different swelling theories namely the monophasic and the multiphasic theory. The monophasic theories were based on the poroelasticity theory and were suitable for hydrogels with nano/ microporosity. On the other hand, the multiphasic approach was developed based on the mixture theory and typically is more popular between biomechanics studies because of the complex composition of the biological tissues. The main difference between multiphasic theories and monophasic ones was that the former take into account the relative motion of the solvent and the polymer through the Darcy’s law. Multiphasic theories provide a highly detailed analysis of hydrogels behavior; however, due to its simplicity, the monophasic approach is more popular for simulating the behavior of the pH-sensitive hydrogels.

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Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels: constitutive modeling and FEM implementation

8

Chapter outline 8.1 An equilibrium thermodynamically consistent theory 271 8.2 Transient electro-chemo-mechanical swelling theory 279 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.2.6

Large deformation theory 279 Chemical field 279 Electrostatic field 281 Continuity of ions 282 Mechanical field 283 Initial and boundary conditions 286

8.3 Numerical solution procedure 286 8.3.1 Development of weak form 287 8.3.2 Development of time and space discretization 287 8.3.3 Residuals and tangent moduli 288

8.4 Results and discussion 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5

289

Equilibrium swelling 289 Inhomogeneous deformations 293 Analytical solution 293 Numerical results for transient swelling response of pH-sensitive hydrogels 298 Numerical predictions of the visco-hyperelastic constitutive model for tough pH-sensitive hydrogels 303

8.5 Summary and conclusion References 309

8.1

308

An equilibrium thermodynamically consistent theory

In this section, the pH/temperature sensitive hydrogel network, that is, NIPAM is composed of polymeric chains bearing acidic groups AH. Immersing in an aqueous solution, the network imbibes the solution and swells. Due to existence of ions in the solutions and acidic groups on the network, the swelling of the network is affected by the external solutions conditions, especially its pH and ionic strength. Also, due to the existence of semi-NIPAM chain in the network, the network is capable of responding to the temperature changes. A schematic draw of this material structure beside the external solution is shown in Fig. 8.1. Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00005-7 © 2023 Elsevier Inc. All rights reserved.

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Figure 8.1 Schematics draw for structure of a pH/temperature sensitive hydrogel beside the external solution with salt contents. Due to pH changes AH groups dissociate into A2 and H 1 [1].

Diffusion of the external solvent molecules and ions through the hydrogel, results in deformation in the network. Considering an element of the network in reference (dry) and deformed states, the coordinates are denoted by X and xðXÞ, respectively. Employing the Lagrangian approach, the current coordinate, xðXÞ, is a function of its reference one which describes the network deformation. In terms of xðXÞ, the deformation gradient tensor, F, and consequently, the right Cauchy-Green deformation tensor, C, can be defined as: F5

@x ; C 5 FT F @X

(8.1)

The region occupied by the network in the references state and its boundary are denoted by V and Γ, respectively. As shown in Fig. 8.1, in the equilibrium state, the external solution is composed of four species, that is, solvent molecules, counter-ion, co-ion and hydrogen ion which their number in the external solution are n s , n 1 , n 2 , and n H1 , respectively. Assuming small variations in the species numbers, the change in the free energy of the external solution at a specified temperature, T, is: μs δn s 1 μH1 δn H1 1 μ1 δn 1 1 μ2 δn 2

(8.2)

where μs , μ1 , μ2 , and μH1 are electrochemical potential of the related species, respectively and δ denotes the variational operator. Also, neglecting body forces, the amount ofÐpotential energy change in the network due to the mechanical force is equal to 2 Γ ti δxi dΓ where t 5 Pn^ is the traction force on the boundary of the

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

273

network. P and n^ are the nominal stress tensor and the boundary unit normal Ð vector, respectively. The change in the free energy of the network is equal to V δWdV in which W is the free energy density of the network expressed in the reference state. Considering both the hydrogel and the external solution as a system, the condition of the equilibrium is [2]: ð

ð δWdV 1 μs δns 1 μH1 δnH1 1 μ1 δn1 1 μ2 δn1 2 ti δxi dΓ 5 0:

(8.3)

Γ

V

On account of conservation of any species in the system, we may write: ð δCs ðXÞdV 1 δns 5 0

(8.4)

δC1 ðXÞdV 1 δn1 5 0;

(8.5)

δC2 ðXÞdV 1 δn2 5 0

(8.6)

V

ð V

ð V

ð

ð δCH1 ðXÞdV 1 δnH1 nH1 5

V

δCA2 ðXÞdV

(8.7)

V

where Ci is nominal concentration of species i in the network and is related to the true concentration, ci , via Ci 5 ci detðFÞ. Also, employing the electro-neutrality both in the network and the external solution we arrive at: CH1 ðXÞ 1 C1 ðXÞ 5 CA2 ðXÞ 1 C2 ðXÞ

(8.8)

n1 1 nH1 5 n2

(8.9)

A variational statement for the free energy change of the system is attained as: ð V

ð

ð

ð

Γ

V

V

  δWdV 2 ti δxi dA 2 δCs ðXÞμs dV 2 δC2 ðXÞ μ2 1 μH1 dV ð

  2 δC1 ðXÞ μ1 2 μH1 dV 5 0

(8.10)

V

During the acidic dissociation, the number of the fixed acidic groups on the network remains constant and only the hydrogen ions can freely move between the

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external solution and the network. Thus we have [2]: CA2 ðXÞ 1 CAH ðXÞ 5 f =ν

(8.11)

where f and ν are number of the acidic groups attached to the network per monomer and volume of a monomer, respectively. It is concluded from Eqs. (8.8) and (8.11) that both the CA2 ðXÞ and CAH ðXÞ are not independent variables and can be stated in terms of CH1 ðXÞ, C2 ðXÞ, and C1 ðXÞ. Thus considering the free energy function as W 5 WðF; Cs ; CH1 ; C2 ; C1 Þ, its variation statement is calculated as @W @W @W @W 1 δW 5 @W @F :δF 1 @Cs δCs 1 @C 1 δCH 1 @C2 δC2 1 @C1 δC1 . Now, considering the H

free energy variation and employing divergence theorem, we arrive at: 8 > > > > > ð> < V

> > > > > > :

0

1

0

1

9 > > @W @W @W > @ > 2 PA:δF 1 @ 2 μs AδCs 1 δCH1 > > = @F @Cs @CH1 0 1 0 1 dV 5 0 > >     @W @W > > 1@ 2 μ2 1 μH1 AδC2 1 @ 2 μ1 2 μH1 AδC1 > > ; @C2 @C1 (8.12)

Therefore following the standard arguments, the constitutive laws for the network are expressed as: P5

@W @W @W ; μs 5 ; μ2 1 μH1 5 ; @F @Cs @C2 @W @W ; 50 μ1 2 μH1 5 @C1 @CH1

(8.13)

Neglecting the volume of the other species in the network due to their low concentration, the change in the volume of the network originates from the fluid molecules diffusion through the network. Thus we have [3,4]: 1 1 ν s Cs 5 detðFÞ 5 J

(8.14)

where ν s and J are the volume of a molecule of the fluid and deformation gradient determinant, respectively. To implement this constraint on the system, employing the Lagrange method, we should modify the free energy as W~ 5 W 1 Πð1 1 ν s Cs 2 J Þ. In terms of the modified free energy density, the constitutive laws for the nominal stress and chemical potential of solvent molecules are recalculated as: P5

@W~ @W @J @W 5 2 Π ; μs 5 1 Πν s @F @F @F @Cs

(8.15)

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

275

where Π is the Lagrange multiplier and its elimination results in: P5

  @W~ @W 1 @W @J 5 2 μs 2 @F @F νs @Cs @F

(8.16)

Considering a small density of crosslinking for the hydrogel, we assume an additive decomposition of the free energy density in the form of [2,5]: W 5 Welastic ðF; TÞ 1 Wmix ðCs ; TÞ 1 Wion ðF; C1 ; C2 ; CH1 ; TÞ 1 Wdiss ðCAH ; CA2 ; TÞ (8.17) where Welastic , Wmix , Wion , and Wdiss are the free energy density due to the elastic deformation, mixing between the solvent and the network chain, mixing of ions and dissociation of the acidic groups, respectively. We use a neo-Hookean model to describe the elastic deformation of the network where it is expressed in terms of the deformation as [6,7]: Welastic 5

1 NKT ðI1 2 3 2 2logðJ ÞÞ 2

(8.18)

in which N and K are density of the polymer chains in the reference state and Boltzmann constant, respectively. In addition, I1 is the first invariant of C. Also, based on the Florry- Huggins theory, the mixing part of the free energy density is [8]:   KT ν s Cs ν s Cs χ ν s Cs lnð Þ1 Wmix 5 νs 1 1 ν s Cs 1 1 ν s Cs

(8.19)

where χ 5 χ0 1 φχ1 5 A0 1 B0 T 1 φðA1 1 B1 TÞ is the interaction parameter of the network assumed to be a function of the temperature and the polymer volume fraction, φ 5 1=J, for the temperature sensitive PNIPAM hydrogel. In addition, A0 , B0 , A1 , and B1 are material parameters adopted for the experiments reported by Afroze et al. [9]. Considering the swelling constraint, we rewrite Eq. (8.19) as:   KT J 21 χ Þ1 ðJ 2 1Þ lnð Wmix 5 νs J J

(8.20)

However, as discussed by Mazaheri et al. [10] this part of the free energy density encounters a numerical instability in the vicinity of phase transition temperature due to multiple solution problems. It is worth to mention that some techniques, such as Riks methods, can be used to overcome this instability however, this is computationally expensive in a numerical scheme [11,12]. To avoid such numerical difficulties, we should modify the mixing energy statement so that the instability is removed while the model accuracy is preserved. To this end the method presented

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by Mazaheri et al. [10] is utilized. This instability is eliminated via polynomial expanding of the ill-posed logarithmic term as [10]: Wmix 5

    KT J 21 χ KT 1 1 1 χ Þ1 ðJ 2 1Þ lnð ðJ 2 1Þ 2 2 2 2 3 1 C νs J J νs J 2J 3J J (8.21)

Employing this free energy, a stable model is developed which is capable of predicting the transition behavior of the hydrogel for temperature changes. As a result, we can use this model in the inhomogeneous problems without encountering any inherent instability of the model with a lower computational cost. Considering a low concentration of the ions, the change in the free energy density of the system due to the ions mixing is entropic and expressed as [2]: " Wion 5 KT CH1 ln

C H1 cref J H1

!

!

! !#     C2 C1 2 1 1 C1 ln ref 2 1 1 C2 ln ref 21 c2 J c1 J

(8.22) where cref is a reference concentration for the species d. Also, the free energy d change of the acidic dissociation which originates from the entropy change due to mixing and enthalpy change for the acidic dissociation, is stated as [2]:   2 Wdiss 5 KT CA ln

   CA2 CAH 1 CAH ln CA2 1 CAH CA2 1 CAH      CA2 CAH 2 2 1 γCA 5 KT CA ln 1 CAH ln 1 γCA2 f =ν f =ν

(8.23)

where γ is the enthalpy change due to the dissociation of one acidic group. Employing the defined free energy density and constitutive laws, we arrive at: @W @Wion @Wdiss 5 1 50 @CH1 @CH1 @CH1

(8.24)

Keeping in mind that both CA2 ðXÞ and CAH ðXÞ are not independent, through applying the chain rule in Eq. (8.24), we have:
γ  exp 2 NA Ka 5 cref 1 H KT

(8.25)

Now, based on Eq. (8.13), the chemical potential of the co-ion is: μ2 1 μH1 5

@W @Wion @Wdiss 5 1 @C2 @C2 @C2

(8.26)

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

277

Substituting Eq. (8.25) in Eq. (8.26) and after some mathematical manipulation we obtain: μ2 1 μH1 5 KTln

cH1 cref H1

!

!   c2 cH1 c2 1 KTln ref 5 KTln ref ref c2 cH1 c2

(8.27)

Similarly, for the chemical potential of counter-ion we deduce: @W c1 5 KTln ref μ1 2 μH1 5 @C2 c1

! 2 KTln

cH1 cref H1

! 5 KTln

c1 cref H1 ref c1 cH1

! (8.28)

Assuming the dilute solution, the chemical potential of species d in the solution is [13]: ! μd 5 KTln

cd cref d

(8.29)

in which cd is the true concentration for the species d. Substituting Eq. (8.29) in Eqs. (8.27) and (8.28) yields: c2 c H1 c1 cH1 5 ; 5 c2 cH1 c 1 c H1

(8.30)

which are well-known Donnan equations. It is concluded that the co-ion and counter-ion true concentrations in the network are dependent on the true concentration of the hydrogen ion in the network and external solution conditions. Thus once the external solution conditions and hydrogen ion concentration in network are known, dependent values of the co-ion and counter-ion true concentrations in the network are determined. The acidic dissociation obeys a relation as below: 

H 1 ½A2  5 Ka ½AH 

(8.31)

where ½i represents the molarity of species i in the network which is equal to ci 5 ½iNA while NA 5 6:023 3 1023 is the Avogadro number. Now, substituting Eqs. (8.8) and (8.11) in Eq. (8.31) and expressing the concentration in the current state we arrive at: cA2 cH1 cH1 ðcH1 1 c1 2 c2 Þ 5 NA Ka 5 f cAH 1 vJ 2 ðcH 1 c1 2 c2 Þ

(8.32)

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Also, replacing Eq. (8.30) in Eq. (8.32) results in a cubic equation for cH1 as below:  A1 ðνcH1 Þ3 1 NA Ka νA1 ðνcH1 Þ2 1

2 A2 2 N A K a

 A3 ðνcH1 Þ 2 NA Ka νA2 5 0 J (8.33)

where the coefficients Ai s are:   c1 A1 5 1 1 ; A2 5 c H1 c 2 ν 2 ; A3 5 f ν c H1

(8.34)

As concluded from Donnan equations, the co-ion and counter-ion concentrations in the network are dependent on the cH1 which could be determined from Eq. (8.33) and are functions of the deformation due to presence of J 5 detðFÞ in the coefficient of this cubic equation. Thus all variables can be expressed as functions of the deformation field and external solution concentrations. It is assumed that the external solution is large enough so that the salt concentration and pH are assumed to be constant during the deformation of the hydrogel network. The  hydrogen ion concentration in the external solution is calculated as: c H1 5 NA H 1 5 NA 102pH . Also, having the counter-ion concentration in the external solution, the related coion concentration can be computed from the electro-neutrality assumption as: c 2 5 c 1 1 c H1 . Now, having the external concentration, the hydrogen ion, the coion and the counter-ion concentrations in the hydrogel network can be determined from the Donnan equation (8.30) and the cubic equation (8.33). Thus the complicated multifield swelling problem is reduced to a deformation field problem. In order to investigate the deformation field, a statement for the stress is necessary. Here, we use the nominal stress determined from the constitutive laws. After some simplifications, we have:   Pij ν 5 Nν Fij 2 Hij 1 KT



   2 1=2 1 χ0 2 χ1 2 1=3 1 2χ1 1 1 2 JHij J4 J2 J3

2 ðcH1 1 c1 1 c2 2 cH1 2 c1 2 c2 ÞJHij (8.35) @J where H 5 1J @F 5 F2T : It is worth to mention that in this Section a constitutive model for pH and temperature sensitive hydrogels is present. To account for the pH sensitivity, the well-known model of Marcombe et al. [2] is employed. But, to capture the temperature-sensitive part, the mixing parameter function in the free energy is implemented. Also, to achieve a stable behavior of the model especially in the vicinity of the phase transition temperature, the mixing function is modified. Thus the model is capable of simulating the behavior of the pH and temperature sensitive hydrogels.

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

8.2

279

Transient electro-chemo-mechanical swelling theory

When a pH-sensitive hydrogel is immersed in an aqueous solution with a prescribed pH and salt concentration, the acidic groups bonded to the polymer network, dissociate either fully or partially depending on the pH, and the fixed charges on the polymer backbone. On the other hand, the cations and anions diffuse into the hydrogel. The increased concentration of ions inside the hydrogel leads to an osmotic pressure, which makes the hydrogel swell or deswell. Also, pH-sensitive hydrogels may swell or deswell due to the propensity of the polymer network to mix with the solution as well as the electrostatic forces. Typically, the osmotic pressure is the more dominant reason in the absence of an external electric field [14,15]. On the other hand, the elastic/viscoelastic force of the polymer network arises against the swelling of the hydrogel. Thereafter, the equilibrium of the pH-sensitive hydrogels occurs when the elastic/viscoelastic force balances the osmotic pressure, solvent pressure, and the electrostatic forces. In this theory, we compute the concentration of mobile and fixed ions using the mass conservation law. Then, the ionic osmotic pressure is computed as a function of the concentration of ions. Finally, the osmotic pressure is treated as an external stress to the hydrogel domain. The hydrogel and the solvent are assumed as a single material.

8.2.1 Large deformation theory Hydrogels are classified as soft materials and typically can change volume up to 1000 times of their initial state [16,17]. Therefore a large deformation theory is required to formulate the swelling behavior of hydrogel accurately [18]. As depicted in Fig. 8.2, we take the dry state of the hydrogel as the initial configuration denoted by B0 . Following the continuum theory, the hydrogel is assumed as a set of continuous points with the coordinate X in the initial (undeformed) configuration [4]. When the hydrogel swells and deforms, each point travels to a new place with a coordinate x in the current (deformed) configuration introduced by B. The concentration of species α in initial configuration is defined as Cα where the same concentration in the current configuration is denoted by cα . These two variables are related by the following equation: Cα 5 Jcα

(8.36)

8.2.2 Chemical field As argued in the literature [5,20], it is more appropriate to define the flux of species in the current configuration. The Nernst
Planck equation describes the flux of an ionic species which is generally due to the diffusion, convection and electrical migration [21], that is,  jα 5 φ 2 D α gradðcα Þ 2 zα μα Fcα gradðψÞ 1 cα v

(8.37)

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Figure 8.2 Schematic representation of the reference (i.e., dry state) and current (i.e., swollen and deformed state) configurations of a pH-sensitive hydrogel with an arbitrary shape undergoing deformation due to external stimuli, acid groups, fixed charge, counter-ion, and co-ion is shown by pink square, green triangle, orange circle, and yellow pentagon [19].

where jα stands for the flux vector of the ion α in the current configuration, φ is the porosity of the hydrogel and is given by φ 5 1 2 J 21 [15], Dα refers to the diffusion coefficient of species α in the initial configuration, zα denotes the charge number of the ionic species, μα is the ionic mobility of species α, F is Faraday’s constant, ψ refers to the electric potential, and v is the fluid velocity relative to the polymer network. The Einstein relation connects the effective diffusivity to the ionic mobility, that is, Dα 5

RTμα F

(8.38)

in which R is the ideal gas constant, and T denotes the absolute temperature. Inside the hydrogel, the ions transport in the regions containing the fluid. The polymer network is impenetrable to mobile ions; thus it increases the path for an ion travel, giving a slower diffusion rate. In this study, we relate the diffusion rate inside the hydrogel to the diffusion in aqueous solution through an obstruction model [15]   Dα J21 2 5 J11 Dα

(8.39)

As shown in Fig. 8.2, the polymer network of pH-sensitive hydrogel bears acidic groups, which we denote them as HA. When the solvent diffuses into the hydrogel, some of the acidic groups dissociate into the hydrogen ions H1 and conjugate bases A2 . We call the conjugate base the fixed charge because it gives rise to a networkbound charge. The chemical reaction is reversible, that is, HA"A2 1 H 1

(8.40)

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

281

The chemical species HA, A2 and H1 are in chemical equilibrium as soon as their concentration does not change with passing time. We define the dissociation constant as, Ka 5

cH cA cHA

(8.41)

in which cH ; cA , and cHA refer to the concentration of hydrogen ion, fixed charges, and acidic groups, respectively. On the other hand, the sum of the concentration of associated acidic groups HA and fixed charges A2 equals the total concentration of acidic groups Cf , that is, cHA 1 cA 5

Cf J

(8.42)

Combining Eqs. (8.41) and (8.42), we determine the concentration of fixed charges as follows: cA 5

Cf Ka J Ka 1 cH

(8.43)

8.2.3 Electrostatic field It can be observed from Eq. (8.37) that apart from the concentration of charged species, the spatial electric field should also be discovered in order to obtain the ionic flux. Poisson’s equation best describes the electrostatic field as follows [22]: div½ε0 εr gradðψÞ 5 2 F zA cA 1

X

! zi ci

(8.44)

i

in which, ε0 and εr , respectively, describe the permittivity of the vacuum and hydrogel. The Poisson’s equation is normalized as,  κ2D div



eψ grad kB T





 P zA c A 1 i zi c i P 2 52 i zi c i

(8.45)

in which κD is the Debye length, defined as, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εr ε0 kB T κD 5 : 2eFI

(8.46)

P where I 5 12 i z2i ci defines the ionic strength of the external solution. The Debye length is a measure of the charge electrostatic effect which indicates how far the electrostatic effect persists. In other words, outside the Debye length at

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Figure 8.3 The Debye length which indicates the length of the double-layer in pH-sensitive hydrogels for different values of pH and salt concentration of external solution [19].

the interface of the hydrogel with the solvent, the electro-neutrality is almost maintained both inside the hydrogel and in the external solution. The Debye length is plotted for different pH and salt concentration in Fig. 8.3. It is evident that the Debye length is in the order of nanometer, therefore for macroscopic hydrogels (i.e., several micrometers in the size or larger) the left-hand side of Eq. (8.45) can be neglected over the volume of the hydrogel except inside the double layer. The double layer refers to two parallel layers of charge surrounding the hydrogel. Hence, inside the macroscopic hydrogel Eq. (8.45) can be simplified as zA c A 1

X

zi ci 5 0:

(8.47)

i

Finally, taking for granted that the activity coefficient of the mobile ions inside the hydrogel is the same as the equilibrium solution, the concentration of co-ions and counter-ions inside the hydrogel are determined using Donnan equation [19], that is, c6 5

1 2

 7 cA 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcA Þ2 1 4c2salt

(8.48)

8.2.4 Continuity of ions The concentration of each species inside the hydrogel changes due to two reasons: (1) the flux of each chemical species through the boundaries of the hydrogel and (2) the mass production or consumption of each species due to a chemical

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283

reaction. Thus the mass balance with respect to the current configuration is defined as below: 8 9 ð ð i = @< h b cj ðx; tÞ 1 cj ðx; tÞ dv 5 2 Jj ðx; tÞUnda ; @t :

(8.49)

@B

B

in which cbj refers to the concentration of the ion j in the current configuration that can reversibly bound to the fixed charge groups on the polymer network where n is the normal unit vector of the hydrogel surface in the current configuration. In the presence of chemical reactions, the concentration of ion j reversibly bound to the polymer network can be computed as, cbj

Cf 5 J



cj Ka 1 cj

 (8.50)

Utilizing the divergence theorem, Eq. (8.49) is simplified as  @
cj 1 cbj 5 2 rJ j @t

(8.51)

Due to the electro-neutrality condition, once the concentration of hydrogen ion is known, the concentration of other ions can be found using Eqs. (8.43) and (8.48). Therefore Eq. (8.51) specifically for the hydrogen ion is recast, that is,      @ Cf 1 cH 1 1 5 2 div φ½ 2DH gradðcH Þ @t J Ka 1 cH

(8.52)

The fluid flow has a negligible effect [23]; hence, it is neglected in this model. It is found from free swelling experiments that hydrogels swell faster in the presence of buffered solutions. Following Ref. [15], we introduce the buffer’s effect on the ion transfer by including additional terms in Eq. (8.52) as,

      @ Cf 1 cT DHB cT ½DH gradðcH Þ cH 1 1 5 div φ 1 1 1 @t J K a 1 cH K B 1 cH DH KB 1 cH

(8.53) where KB denotes the dissociation constant of the buffer and cT refers to the total concentration of the buffer.

8.2.5 Mechanical field In this theory, the hydrogel is assumed as an isotropic compressible hyperplastic material in isothermal condition. Three energy sources drive the swelling kinetics

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of a pH-sensitive hydrogel: energy related to the mixing of the polymer chains with the solvent molecules, energy due to the ion osmosis, and elastic energy of the polymer network [16,24]. Thus the total free energy of the hydrogel is represented as: W 5 Wmix 1 Wion 1 Wnet

(8.54)

The first two terms of the free energy (i.e., Wmix and Wion ) tend to expand the hydrogel, while the crosslinks on the polymer network avoid hydrogel from being dissolved in the solvent and arises the network energy (Wnet ) to balance out the expanding forces. Consequently, the hydrogel reaches its swelling equilibrium state, when the total free energy is minimum. Consequences of the first and second laws of thermodynamics give the following relation for the first Piola
Kirchhoff stress [1,2], P5

@Fnet 2 ΠJF2T @F

(8.55)

in which Π refers to the total osmotic pressure: Π 5 Πmix 1 Πion

(8.56)

in which the ionic osmotic pressure (i.e., Πion ) is the dominant term according to Ref. [14] and can be determined as a result of the thermodynamics laws [2,25], that is, Π  Πion 5 RT

X

ck 2 c0k



(8.57)

k

where c0k c0k is the concentration of the kth ion in the external solution. The elastic part of the energy is taken from Wilson et al. [26] as, Wnet 5

i 1 1 h Kln2 ½detðCÞ 1 G tr ðCÞ 2 3detðCÞ1=3 8 2

(8.58)

in which K stands for the bulk modulus and G is the shear modulus. The deformation of the hydrogel is much slower than the ion transfer [27] such that we can treat the deformation process as quasi-static. Thus we write the balance of the linear momentum as, DivðPÞ 1 b 5 0;

(8.59)

where b is the body force vector. Tough hydrogels can be modeled with a large strain viscoelasticity theory employing the derivation presented by Holzapfel [28]. We use the generalized

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285

Maxwell model, which is based on the decomposition of the strain energy density into volumetric, isochoric, and contribution due to the viscoelastic branches as follow, N N W 5 Wvol 1 Wiso 1

m X

ϒ α;

(8.60)

α51

The strain energy in the main hyperelastic branch is denoted with the superscript N to indicate the long-term equilibrium. The second Piola
Kirchhoff stress is computed as follows: S52

m X @W N 5 SN Qα vol 1 Siso 1 @C α51

(8.61)

where Qα refers to the auxiliary second Piola
Kirchhoff stress tensors defined as, Qα 5 2

@ϒ @C

(8.62)

In each viscoelastic branch, the time evolution of the auxiliary stress tensor Qα is found by,  d   Qα d Qα 1 Siso;α ; α 5 1; . . .; m 5 dt dt τα

(8.63)

here, the relaxation time corresponding to viscoelastic branch α is denoted with τAð0; NÞ and the isochoric second Piola
Kirchhoff stress tensor in the branch α is by Siso;α which can be derived from the energy factors β α A½0; N and the strain energy density in the main hyperelastic branch as follows: Siso;α 5 2

@Wiso;α @Wiso 5 2β α 5 β α Siso @C @C

(8.64)

Using the convolution integrals, a fairly closed form solution for the linear evolution Eq. (8.63) in time interval tA0; T can be found as follows:  t 2 ω dSiso;α dω; α 5 1; . . .; m Qα 5 exp 2 τα dω 0 ðt



(8.65)

The second Piola-Kirchhoff stress at tn11 is given as Sn11 5

N SN vol 1Siso 1

m X α51

! Qα jn11

(8.66)

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where the internal variable Qα at tn11 can be found using Eq. (8.65) as Sn11 5

N SN vol 1Siso 1

m X α51

! Qα jn11

(8.67)

8.2.6 Initial and boundary conditions To complete the theory, the initial and boundary conditions for the chemical and the mechanical fields should be introduced. the boundary @B of the body B is divided into two complementary subsurfaces namely Su (displacement) and St (traction force) (i.e., Su , St 5 @B and Su - St 5 [). Moreover, let Sc (concentration) and Sj (fluid flux) similarly be complementary subsurfaces in a sense Sc , Sj 5 @B and Sc - Sj 5 [. Consider a pair of boundary conditions in which the displacement u is prescribed on Su and the surface traction on St St in the time interval of tA½0; τ ,

u 5 u~ onSu 3 ½0; τ  PN 5 ~t onSt 3 ½0; τ 

(8.68)

and a pair of boundary conditions where the concentration is specified on Sc and fluid flux on Sj :

cH 5 c~H 2jHU n 5 j~H

onSc 3 ½0; τ  onSj 3 ½0; τ 

(8.69)

~ ~t; c~H and j~H as prescribed functions of space and time, as well as N the norwith u; mal unit vector of the hydrogel surface in the initial configuration. The initial conditions are also defined as,

uðX; 0Þ 5 u0 ðXÞ inB cH ðx; 0Þ 5 c0H ðxÞ inB

(8.70)

The coupled Eqs. (8.53) and (8.59) together with Eqs. (8.68)
(8.70) constitute an initial nonlinear boundary value problem for the displacement uðx; tÞ and hydrogen ion concentration cH ðx; tÞ.

8.3

Numerical solution procedure

In this section, the electro-chemo-mechanical swelling theory is implemented in a finite element framework. First of all, the weak form of the coupled set of nonlinear governing equations are derived. Then, the weak form of the equations is discretized both in time and space. A Lagrangian formulation is presented for the

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

287

equilibrium equation and an Eulerian formulation for the chemical species continuity. A standard Galerkin approach is employed to find the residual of displacement and ion concentration. In addition to the residuals, the tangents moduli required for the iterative Newton method are developed.

8.3.1 Development of weak form The strong form of the coupled nonlinear partial differential equations of the proposed theory in the absence of body forces is summarized here, 8 < DivðPÞ 5 0 in B Balance of momentum u 5 u~ on Su : PN 5 ~t on St 8 2 0 13 > > @ C 1 c T > 4 cH @ 1 1 f A5 5 > 1 > > > J Ka 1 c H K B 1 cH @t > > > 8 0 9 1 < Balance of the hydrogen < = D c HB T A½DH gradðcH Þ ion concentration > div φ@1 1 > > : ; D H K B 1 cH > > > > > > cH 5 c~H > : 2 jHUn 5 j~H

in B

on Sc on Sj (8.71)

The weak form of the governing equations given in Eq. (8.71) is constructed by introducing w1 and w2 as two weighting (or test) fields which respectively disappear on Su and Sc . The weak forms are presented as: Ð Ð ~ w1 dA 5 0 B0 PUGradðw1 ÞdV 2 St tU1 8 2 0 39  Ð B St 1 0 8 2 0 > 39 > > > = > Ð T > 4cH @1 1 f A5 PA dv 5 ~ < B 1 Sj P jH da :@t ; J Ka 1 cH K B 1 cH 8 1 2 0 39 > > > < = >   Ð > D c HB T > A 4 @ AðDH gradðcH ÞÞ5 dv > U φ 1 1 grad P 2 > B: : ; DH KB 1 cH (8.76) An iterative Newton procedure is employed to solve this system of coupled equations; therefore the following element-level residual for the displacement and hydrogen ion concentration are defined: 8   Ð Ð > ðRu ÞA 5 B0 PUGrad QA dV 2 St QA~tdA > > 8 2 0 13 9 > > > < >  Ð Ð  @4 @ Cf 1 c T A5 = A > > < ð Rc Þ A 5 B cH 1 1 P dv 2 Sj PA j~H da 1 :@t ; J Ka 1 cH KB 1 cH 8 0 1 2 39 > > > < = >   Ð > D c HB T > A 4 @ AðDH gradðcH ÞÞ5 dv > U φ 1 1 1 grad P > B: > ; DH KB 1 cH : (8.77)

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

289

In addition to residuals, we require tangents for the iterative Newton procedure; thus we define the tangents as: ðK uu ÞAB 5 2 ðK cu Þ

AB

@ðRu ÞA @ðRu ÞA ; ðK uc ÞAB 5 2 ; B @u @cBH

(8.78)

@ð R c Þ A @ðRu ÞA AB 52 ; ð K Þ 5 2 uc @uB @cBH

For more information about the nonlinear finite element procedure, readers are referred to [29]. The global system matrix is determined as,  K5

K uu K cu

K uc K cc

 (8.79)

where K uu corresponds to the quasistatic mechanical field, K uc and K cu are the tangents of the coupled time-dependent diffusion and deformation, and K cc is the timedependent diffusion tangent.

8.4

Results and discussion

8.4.1 Equilibrium swelling In this section, the first proposed model is employed to investigate some benchmark problems involving homogeneous and/or inhomogeneous swelling of the pH/ temperature sensitive hydrogels. The material parameters of the network are adopted for experiments available in literature and shown in Table 8.1. Based on the other works [2,5,30], we assume that ν 5 ν s 5 10228 ,KT 5 4 3 10221 J, and Ka 5 1024:7 . First, some homogeneous examples are solved and the results are compared to the experimental data available in the literature. Then, inhomogeneous swelling of the hydrogel is studied for a spherical shell of the pH/temperature sensitive hydrogel on a hard core for variation of the both pH and temperature. This problem is solved both analytically and numerically. To solve boundary value problems, the proposed model is implemented in the finite element framework through developing a user-defined subroutine in ABAQUS. In this section, we investigate the free swelling of pH/temperature sensitive hydrogels to validate the model with the experimental results available in the literature.

Table 8.1 Material parameters for the network [9]. A0

B0

A1

B1

212.947

0.04496/K

17.92

20.0569/K

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Due to the homogenous deformation in the free swelling problem, we have: 

V λ1 5 λ2 5 λ3 5 λ 5 V0

1=3 (8.80)

where VV0 is the swelling ratio defined as the ratio of the current volume V, to its value at the dry state, V0 . In the free swelling problem, the network experiences a free-stress state. Thus to determine the equilibrium state for the free swelling state, equating the stress equation, namely Eq. (8.35), to zero, the value of swelling ratio at the equilibrium is identified and the results are shown in Fig. 8.4 for both the temperature and pH variations. For comparison the results of the free swelling of the neutral PNIPAM hydrogel are presented where Nv 5 0:04 and f 5 0:00 in Fig. 8.4A, which reveals that the neutral PNIPAM has almost no sensitivity to the pH variation but in contrary is sensitive to the temperature variation. In Fig. 8.4B and C, the results are shown for f 5 0:05 and f 5 0:10, respectively, while Nv 5 0:04 and the salt molarity is 0:001M for both cases. The results demonstrate that increasing the acidic group density on the network, makes it more sensitive to the pH variations. To examine the effect of salt concentration, the free swelling results are presented for the external solutions with salt molarity of 0:001M and 0:050M in Fig. 8.4B and D, respectively, in which Nv 5 0:04 and f 5 0:10. As observed from Fig. 8.4B and D, the swelling ratio decreases at higher molarity of the external solution due to the diffusion of more ions in the network space and consequently, repulsion of some solvent molecules from the network. To study the effect of the crosslinking density, namely Nv, the relevant results are shown for Nv 5 0:03 and Nv 5 0:02 in Fig. 8.4E and F, respectively, where the salt molarity and acidic group density are constant and equal to 0:001M and f 5 0:10 for both cases. As illustrated in Fig. 8.4B
F, the network is more sensitive to pH/temperature for the smaller values of Nv and larger volume change is observed due to the less resistance of the network against deformation. As depicted in all parts of Fig. 8.4, the swelling ratio decreases as the temperature increases due to PNIPAM content of the network. But, increasing the pH value results in a decrease in the hydrogen ion and, as a result, more acidic groups are going to be dissociated and consequently more hydrophilic regions are developed and the chains are stretched due to electrical repulsion of the negative dissociated acidic groups. Thereupon, more fluid molecules are absorbed in the network and larger amounts of the volume change are observed. The attained results are in agreement with the experimental data available in the literature [30]. To validate the model with the experimental data, the experiments reported by Guo and Gao [30] are considered. The values of Nν, f and the external solution molarity should be fit for the experiments. First, the value of Nν 5 0:04 is determined for the swelling behavior of neutral PNIPAM when f 5 0:00. Then, f and the external solution molarity should be found through fitting for pH values of 2.1 and 7.4 as reported by Guo and Gao [30]. The fitted value of the external solution molarity that is kept constant in all experiments is equal to 0:001M. The value of f

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

(A)

(B)

(C)

(D)

(E)

(F)

291

Figure 8.4 he free swelling results for the neutral PNIPAM hydrogels and the pH/ temperature sensitive PNIPAM hydrogels with different values of cross-linking density Nν, acid group number f and salt molarity M 5 0.001M, A) Nν 5 0.04, f 5 0.00, B) Nν 5 0.04, f 5 0.001M, C) Nν 5 0.1, f 5 0.001M, . The related values these parameters are shown in the bottom of each figure. The sensitivity of the model to both pH and temperature variations is depicted in this figure [1].

which should be fitted for every experimental data set, is computed as f 5 0:06; 0:12, and 0:18 for the best-fit. As shown in Fig. 8.5, the presented model predicts qualitatively, the swelling behavior of the pH/temperature sensitive hydrogel. Also, below the phase transition

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35

(A)

Model : f=0.00 Model : f=0.06 Model : f=0.12 Model : f=0.18 EXP EXP EXP

30

Swelling ratio [-]

25 20 15 10 5 0 290

295

300

305

310

315

Temperature [K]

Swelling ratio [-]

(B)

15

Model : f=0.00 Model : f=0.06 Model : f=0.12 Model : f=0.18 EXP EXP EXP

10

5

0 290

295

300

305

310

315

Temperature [K]

Figure 8.5 Comparison between the presented model results and the experiment reported by Guo and Gao [30], (A) for pH 5 7.4 and (B) for pH 5 2.1. For these results, the fitted values of the material parameter and salt concentration are Nν 5 0:04, and 0:001M, respectively. Also, the value of f is determined from curve fitting as shown in figures [1].

temperature, a good agreement is observed between the model results with those of experiments. As one may see from this figure, at the small values of pH, the network is not that sensitive to the acidic group density due to the large amount of hydrogen ion in the external solution which prevents from the acidic groups’ dissociation. Thus the acidic group density has almost no effect on the swelling behavior of the network as predicted by the model and confirmed by the experiments. But, for the larger values of pH, scarcity of the hydrogen ion results in the acidic group dissociation and the larger values of swelling ratio. This phenomenon is more

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293

prominent for larger amounts of the acidic group density as shown in Fig. 8.5A. In this case, a network with more amounts of the acidic groups produces more hydrogen ions which results in more fluid uptake in the network and consequently more swelling. As depicted in Fig. 8.5, the presented model can predict the pH/temperature behavior qualitatively both in the small and large values of pH. Also, the model is capable of predicting the hydrogel behavior for the temperature changes.

8.4.2 Inhomogeneous deformations In this constitutive model, a modified version of mixing part of the free energy is used and as a result, a continuous model is developed which is numerically stable and can be implemented for solving inhomogeneous swelling problems [10]. To show the capability of the model in the inhomogeneous swelling, the solution of the inhomogeneous swelling of a pH/temperature sensitive hydrogel spherical shell on a rigid core is presented. The hard core constrains the hydrogel shell swelling in its inner radius which results in the inhomogeneous deformation of the shell. Analytical and numerical solutions are derived and compared. Thereafter, the numerical framework which is validated with the analytical results is employed to simulate the complicated contact problem of the mentioned hydrogel shell with a flat channel.

8.4.3 Analytical solution Motivated from Zhao et al. [31], a bench-mark problem in the hydrogel inhomogeneous swelling is a spherical shell on a hard core whose volume changes due to the environmental conditions, for example, pH and temperature changes [32
35]. The schematic draw of this problem is shown in Fig. 8.6 in which the shell is shown at a stress-free state (reference state). In Fig. 8.6, Ri and Ro are inner and outer radii of the spherical shell in the stress-free state, respectively.

Figure 8.6 Schematic draw of the spherical hydrogel shell on a hard core inserted in a channel with rigid flat walls [1].

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For the spherical shell, all field variables are functions of R, which is the radius of an element in the stress-free state. All energy statements are recast with respect to the reference state through substituting J 5 λ0 3 J; I1 5 λ0 2 I 1 in the free energy statements in which J and I 1 are corresponding values of J and I1 with respect to this state. The temperature and pH at the stress-free state are 290K and 2, respectively. Thus the corresponding value of λ0 is obtained as 2.190. Due to the symmetry, the equilibrium equation in the radial direction is: dPr 2ðPr 2 Pθ Þ 50 2 R dR

(8.81)

where Pr and Pθ are the nominal radial and hoop stresses, respectively. These stress components are obtained from the free energy expressions as: 0 1 0 1   2 1=2 1 χ 2 χ Pr ν 1 2 1=3 1 2χ 1 0 1 1 A1@ 1 2 10 6 4 A 5 Nν @λ0 λr 2 KT λ0 λr λ0 7 λθ 4 λr 3 λ0 4 λθ 2 λr 2 λ0 λθ λr 2 c2 Þλ0 2 λθ 2 ; 2 ðcH1 1 c0 1 1 c2 2 cH1 2 1 c1 0 1   2 1=2 1 χ0 2 χ1 Pθ ν 1 2 1=3 1 2χ 1 1 A1@ 5 Nν @λ0 λθ 2 1 2 10 7 3 A KT λ0 λθ λ0 4 λθ 3 λr λ0 λθ λr λ0 7 λθ 5 λr 2 2 ðcH1 1 c1 1 c2 2 cH1 2 c1 2 c2 Þλ0 2 λr λθ

(8.82) Also, the stretch components in terms of the current element radius are: λr 5

d rðRÞ r ðRÞ; λθ 5 dR R

(8.83)

in which rðRÞ is the radius of the element of the hydrogel in the current state. Substituting (8.82) and (8.83) in (8.81), a differential equation in terms of rðRÞ and cH1 is obtained, and accompanying (8.33), a system of two nonlinear equations established. One method of dealing with this problem is elimination of cH1 from the two equations and solving the problem for the displacement field rðRÞ. The relevant boundary conditions for this problem are then expressed as: r 5 Ri at R 5 Ri ;

Pr 5 0 at R 5 Ro

(8.84)

The material parameters are selected as Nν 5 0:04 and f 5 0:10. Also, the external solution molarity is 0:001M: To solve the present boundary value problem, a finite difference technique with Richardson extrapolation is used [36]. The calculated results as depicted in Fig. 8.7, are reported for different values of thickness ratio, namely Ro =Ri . The normalized outer radius of the hydrogel shell is plotted versus the pH and temperature in Fig. 8.7A and B, respectively. The normalized outer radius is defined as ratio of the current outer radius of the hydrogel shell

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

(A)

295

1.15

r(Ro)/R o [-]

1.1

1.05 Ro/Ri = 1.5 1

Ro/Ri = 2.0 Ro/Ri = 2.5 Free Swelling

0.95 2

4

6

8

10

12

pH [-]

(B) 1

r(Ro)/R o [-]

0.9 0.8 0.7 0.6 0.5 290

Ro/Ri = 1.5 Ro/Ri = 2.0 Ro/Ri = 2.5 Free Swelling 295

300 Temperature [K]

305

310

Figure 8.7 Analytical results for the inhomogeneous swelling of a spherical shell on a rigid pillar: A) at different pH’s, and B) at different temperatures [1].

rðRo Þ, to its value at the reference point Ro . As shown in Fig. 8.7A, increasing the pH results in an increase in the normalized outer radius. As the thickness ratio increases, the added material in the outer radius experiences a low level of confinement and as a result, can swell more freely with increase in pH. Therefore increasing the thickness ratio, the behavior of the shell approaches to the free swelling state as understood from Fig. 8.7A. The effect of temperature increase is also illustrated in Fig. 8.7B in which the hydrogel experiences shrinkage due to an increase in the temperature. In contrast to Fig. 8.7A, in Fig. 8.7B the normalized outer radius is larger for smaller values of the thickness ratio. Again, the behavior of the shell approaches to the free swelling state as the thickness ratio increases.

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8.4.3.1 Numerical implementation In the last section, the model was successfully used for solving an inhomogeneous swelling problem. Implementing the constitutive models of smart hydrogels as UHYPER in ABAQUS is popular in the literature because of the vast facilities of ABAQUS for FEM simulations [2]. Following other researchers, the proposed model is implemented in an ABAQUS user subroutine, namely UHYPER, to obtain a numerical tool for solving more complicated boundary value problems. To show the capability of the numerical scheme, inhomogeneous swelling of a hydrogel shell on a rigid pillar with boundary conditions identical to the analytical solution, presented in the previous section, is solved and the results are compared in Fig. 8.8 for

(A)

1.15 Analytical Numerical

r(Ro)/R o [-]

1.1

1.05

1

0.95 2

4

6

8

10

12

pH [-]

(B)

Analytical Numerical

1

r(Ro)/R o [-]

0.9

0.8

0.7

0.6 290

295

300 Temperature [K]

305

310

Figure 8.8 Comparison between the analytical and the numerical solution of inhomogeneous swelling of a pH/temperature sensitive hydrogel shell on a rigid pillar for (A) pH and (B) temperature variations [1].

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

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Ro =Ri 5 2. The material parameters and the external solution molarity are the same as those of the presented analytical solution. As shown in Fig. 8.8, the numerical and analytical results are in excellent agreement and the validity of the FEM solution is successfully checked. In addition, for Ro =Ri 5 2 inhomogeneous swelling of a hydrogel shell on a rigid pillar is solved which may experience a contact with rigid flat walls of a channel as shown in Fig. 8.6. The half-width of the channel, D, is selected such that D=Ri 5 2:25. Due to the symmetry, simulations are carried out for a quarter of the real spherical shell and the results are shown in Fig. 8.9. In this figure the columns from left to right show the results at temperatures 290K, 300K, and 310K, respectively and also, the top-down rows are the results for pH values of 2, 5, 7, and 10, respectively. The material parameters, the stress-free condition and the external solution molarity are identical to the previous numerical problem. The inner radius of the hydrogel shell is constrained to the rigid pillar while the outer radius of the shell experiences a contact with the walls. Thus this problem contains a complicated coupling between the temperature changes, pH changes, large deformations and a mechanical contact. As shown in Fig. 8.9, a

Figure 8.9 Deformation contours of the spherical pH/temperature sensitive hydrogel at various conditions. Columns from left to right show the contours at 290K, 300K, and 310K, respectively. Rows top-down show the contours at pH equal to 2, 5, 7, and 10 [1].

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temperature decrease or a pH increase may result in contact between the shell and the flat walls. For example, when pH 5 10, the shell is initially in contact with the wall at T 5 290K. Any increase in the temperature or decrease, in pH lead to shrinkage of the shell and as a result they aren’t in contact anymore as observed at T 5 300 and 310K. For the case that no contact occurs at the low temperature, an increase in the channel opening eventuates from increasing in the temperature or decreasing in pH. These observations are reasonable and the validated numerical scheme derived in this section can be employed to simulate boundary value problems for the swelling of the pH/temperature sensitive hydrogels.

8.4.4 Numerical results for transient swelling response of pHsensitive hydrogels In this section, the numerical implementation of the proposed electro-chemomechanical theory is validated utilizing the experiments by De et al. [15]. We then proceed with some boundary value problems on free swelling and constrained swelling of the pH-sensitive hydrogel. De et al. [15] have considered a cylindrical pH-sensitive hydrogel constrained in a glass channel with a rectangular crosssection of 1000-μm width by 180-μm height, which restricts the top and bottom displacement of the hydrogel as shown in Fig. 8.10. Then, they examined the equilibrium and transient swelling of the hydrogel in different pH values with various hydrogel diameters (i.e., D 5 150, 200, and 300 μm). FEM is used to find the swelling degree and rate, and the numerical results will be compared with those of experiments. The material properties and other constants used for simulation are listed in Table 8.2. The shear modulus of the hydrogel is also needed for the numerical implementation. To find the shear modulus, the equilibrium swelling of the hydrogel is solved both numerically and analytically and as shown in Fig. 8.11, fit the results to those of experiments. The concentration of salt in the external solution is csalt 5 0:3M. We find G 5 1:67 MPa and K 5 2:33MPa. When a pH-sensitive hydrogel containing acidic groups bound to its polymer chain is immersed in a solution with a predefined pH value, the H 1 comes off in

Figure 8.10 Test setup used by De et al. [15]: a cylindrical pH-sensitive hydrogel embedded in a rigid glass channel which is synthesized with photo-polymerization [19].

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Table 8.2 Chemical parameters used for simulation of cylindrical pH-sensitive hydrogel [19]. DH DHB Cf

9:3 3 1029 m2 =s 8:79 3 10210 m2 =s 1800 mM

Ka KB ν

1022:0 mM 6:2 3 1025 mM 0:5φs

Figure 8.11 The FEM and analytical results of swelling ratio of the cylindrical hydrogel in different pH values fitted to the experimental results to find the shear and bulk modulus [19].

pH values higher than Ka . Cations enter the hydrogel and compensate the charge. Therefore charge neutrality is maintained. However, the increased cation concentration leads to osmotic pressure which makes the hydrogel to swell. This phenomenon is depicted in Fig. 8.11. In this problem, the dissociation constant of acidic groups is Ka 5 1022:0 mM, therefore the equilibrium swelling of the hydrogel experience a transition zone in pH 5 5 and suddenly swells as the pH value passes 5. As the material properties and shear modulus of the hydrogel are determined, the transient swelling and deswelling of the hydrogel can be simulated and the results can be compared with experimental ones as depicted in Fig. 8.12. In the simulation, the hydrogel initially is in the mechanical and chemical equilibrium state with the external solution (pH 3 for swelling and pH 6 for deswelling), where the salt concentration of the external solution is csalt 5 0:2M. The swelling/deswelling process is initiated by applying the appropriate hydrogen ion concentration to the boundaries of the hydrogel, which are in contact with the external solution. The simulation matches the experimental data closely; this shows that the electro-chemomechanical swelling theory is capable of precisely predicting the swelling and deswelling behavior of pH-sensitive hydrogels. Turning our attention towards the kinetics of the swelling and deswelling of the pH-sensitive hydrogel as illustrated in Fig. 8.12, one may find that the deswelling

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(A)

(B)

Figure 8.12 The experiment versus numerical simulation of the transient swelling of cylindrical hydrogel with three different diameters, (A) the hydrogel is initially equilibrated in a solution with pH 5 3 and then is placed in a solution with pH 5 6 and (B) the hydrogel is initially equilibrated in a solution with pH 5 6 and then is placed in a solution with pH 5 3 [19].

process is approximately ten times faster than the swelling. The explanation for this phenomenon is that during the swelling, the volume of the hydrogel increases which causes a longer diffusion path for the water; also, the concentration of fixed acidic group decreases, which results in lower osmotic pressure. In contrast, the volume of the hydrogel decreases during the deswelling process which reduces the diffusion path and raises the concentration of fixed acidic group, which in turn intensifies the osmotic pressure and causes the hydrogel to deswell much faster. Also, as shown in

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301

Fig. 8.12, hydrogels with smaller diameter deform faster than ones with a larger diameter. The reason is that in larger hydrogels it takes longer for the solvent to penetrate. Generally, the deformation time for the hydrogel is proportional to the square of hydrogel’s characteristic length (i.e., hydrogel’s diameter in our problem) [27].

8.4.4.1 Free swelling The most straightforward feasible experiment for the hydrogels is the free swelling, which is performed by immersing the dry polymeric network in water with prescribed values of ions concentration; nonetheless, the results are of paramount interest in various applications, such as drug delivery systems. In this section, we simulate the free swelling of a 200-μm square pH-sensitive hydrogel which is initially equilibrated in a solvent with pH 5 2. The solution then is flushed and replaced with a solution of pH 5 12. Due to symmetry arguments, only a quarter of the square is modeled, and appropriate boundary and initial conditions are applied. As illustrated in Fig. 8.13, the free swelling process is not homogeneous, and the corners swell faster because of the more considerable contact with the external solution in comparison with the other parts of the hydrogel. This causes the edges of the hydrogel to form a bowl-like shape during the transient swelling. Moreover, Achilleos et al. [37] have observed this behavior experimentally. It is also noticed that as the hydrogel reaches its equilibrium state, not only the stress decreases during

Figure 8.13 The results of the simulation of free swelling of a square hydrogel: contour of von Mises stress (left side) and concentration of hydrogen (right side) at four different times [19].

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the swelling process, but also the fully swollen hydrogel recovers the shape similar to the original one and becomes entirely homogeneous. It should be noted that the stress at the edges of the hydrogel during the swelling is compressive and may cause the buckling and surface instabilities [38] which is neglected in this simulation.

8.4.4.2 Constrained swelling As a constrained swelling example, the presented swelling theory is employed in FEM to examine a microfluidic valve [39]. As shown in Fig. 8.14, the microvalve is a ring made of the hydrogel in a microfluidic channel. Plane strain condition is assumed for the problem. We take csalt 5 0:2M for the concentration of the salt in the external solution. The hydrogel is in the free stress state when pH 5 2 and the channel is open. When the pH value of the external solution changes to 12, the hydrogel gradually swells to push against the channel wall and closes the channel. One may observe from Fig. 8.14 that it approximately takes 1000 s for the hydrogel to touch the channel wall. This time can be lowered by reducing the size of the valve. Also, we plot the radial and tangential stress distribution of the hydrogel during the swelling process; these values are of great interest because the hydrogel may fracture during the swelling.

Figure 8.14 The results of the simulation of constrained swelling of a microfluidic pHsensitive hydrogel-bases microvalve: contour of radial (left side) and tangential (right side) stress at four different times [19].

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As depicted in Fig. 8.14, before hydrogel reaches the channel wall, the radial stress at the outer surface of the hydrogel is zero. The constraint applied at the inner surface of the hydrogel in order to prevent the radial movement at this area causes high stress. When the hydrogel closes the channel, the contact pressure and length are two important parameters which are used to define the safety factor of the microvalve [40]. Finally, after 15,000 s the hydrogel reaches its equilibrium state with maximum contact pressure and length.

8.4.5 Numerical predictions of the visco-hyperelastic constitutive model for tough pH-sensitive hydrogels In this section, the experimental results reported by [41] are used to assess the proposed model for tough hydrogels. Six branches are used for the viscoelastic part, which are defined with twelve parameters, that is, β i and τ i with i 5 1; :::; 6. Along with the swelling parameter χ and three hyperelastic constants C10 , C20 , and C30 , the proposed theory involves a total of sixteen parameters. These parameters are identified by fitting the theoretical results to the experimental ones. The results of the dynamic mechanical analysis (DMA) test can be used to find the viscoelastic constants. However, the DMA results reported by [41] correspond to a hydrogel with a different molar concentration. Thus the results of the uniaxial tension test with various loading rates were used to calibrate the visco-hyperelastic constants. The isochoric and volumetric parts of the second Piola-Kirchhoff stress are found as

        1 2 2=3 1=3 C10 2 2C20 1 9C30 I3 1 C20 2 6C30 I3 1 I1 C30 I1 ; Siso 5 2I321 3I 2 I1 C21 3 3  pffiffiffiffi
pffiffiffiffi Svol 5 I3 κ I3 2 Js 2 1 :

(8.85) The deformation gradient tensor in uniaxial test is written as: 2

λ1 F54 0 0

0 λ2 0

3 0 0 5 λ2

(8.86)

where λ1 and λ2 denote the axial and lateral stretch ratio, respectively. The second Piola
Kirchhoff stress tensor can be found. One may calculate the λ2 by setting lateral stress to zero. To fit the theoretical solution to the experimental data, an error function is defined and the material parameters are computed with minimizing the error function with an appropriate optimization technique. The error function is introduced as O5

:Sth 2 Sexp2 : :Sexp2 :

(8.87)

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where Sth and Sexp are the theoretical and experimental stress data, respectively, and the L2 norm is shown with the symbol ::2 . The experimental data are plotted for the uniaxial tension test of tough pHsensitive hydrogel with various loading rates reported by [41] along with the fitted theoretical solution in Fig. 8.15. The test pieces were cut from a hydrogel in a dumbbell shape with a length of 12 mm and a width of 2 mm by Ref. [41]. It is found that the model predictions fit well to the experimental data in the low loading rates. However, at loading rates of 300 and 500 mm/min, the theory does not fit well with the experimental data. This may be due to the experimental error. Since the hydrogel is highly stretched, it is possible for the hydrogel to slip on the jaw of the tensile test device, causing error in the measured results. Fitting the proposed model to the experimental data, the visco-hyperelastic parameters are identified as listed below (Table 8.3). Ding et al. [41] have used acidic and basic groups with concentration of C0AH 5 2400 mM and C0BOH 5 600 mM. pH-sensitive hydrogels swell when the pH reaches to the dissociation constant of their ionic groups. Ding et al. [41] have utilized acrylic acid (AAc) and 1-vinylimidazole (VI) for ionic groups. The dissociation constant of AAc and VI groups are 4.3 and 7.0, respectively. However, the hydrogel swells much higher or lower than the dissociation constant at pH 5 2 and pH 5 11 [41]. Because the critical pH to destroy the hydrogen bonds has no correlation to the pKa of conjugate bases or acids. Using the free swelling results, one may find the swelling parameter χ 5 9:5819 3 1026 =Pa. The experimental and fitted results of free swelling are illustrated in Fig. 8.16. To examine the accuracy of the proposed model, the self-recovery test of the hydrogel is simulated using the calibrated parameters, and the model results are compared with those of the experiments reported in Ref. [41]. In this test, the hydrogel is loaded with a maximum strain of 300% and then is unloaded to the

Figure 8.15 Experimental and numerical tensile stress-stretch curve of the tough pHsensitive hydrogel at various loading rates [42].

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Table 8.3 Visco-hyperelastic parameters obtained by fitting [42]. Parameter

Value

Parameter

Value

C10 ðPaÞ C20 ðPaÞ C30 ðPaÞ β 1 ðBÞ τ 1 ðsÞ β 2 ðBÞ τ 2 ðsÞ

63941.0630 216.8179 3.0692 3.0378e-4 15.7422 1.2520 14.2042

β 3 ðBÞ τ 3 ðsÞ β 4 ðBÞ τ 4 ðsÞ β 5 ðBÞ τ 5 ðsÞ β 6 ðBÞ τ 6 ðsÞ

0.1522 68.1954 0.9950 8.3135 22.1777 0.1142 0.3743 0.6613

Figure 8.16 Free swelling test: swelling ratio of the tough pH-sensitive hydrogel at various pH values [42].

stress-free state. The hydrogel then rests for a specified time, after which the loading-unloading process is repeated. The results depicted in Fig. 8.17 indicate that the simulation results are in good agreement with the experimental ones. The self-recovery ability of the hydrogel is arising from the dynamic nature of the hydrogen bonds. After the unloading process, a notable residual strain about 50% remains, which is due to the visco-hyperelastic deformation of the hydrogel. However, the residual strain disappears after about 2 min, and the following loading
unloading curve reaches the first cycle which shows the fast recovery of the hydrogen bonds. The hysteresis ratio is calculated as the area ratio of the second loop to the first one. The magnitude of the hysteresis ratio and the residual strain depend on the waiting time, as shown in Fig. 8.18. Another important issue about tough pH-sensitive hydrogels is that their mechanical strength diminishes as they highly swell. Ding et al. [41] also reported this

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Figure 8.17 Stress
stretch cyclic tensile loading
unloading curves of tough pH-sensitive hydrogel after different waiting times [42].

Figure 8.18 The residual strains and hysteresis ratios corresponding to the cyclic loading
unloading curves [42].

issue. They have performed uniaxial tension test on swollen hydrogels. The results of simulation are compared to the experimental data reported by Ding et al. [41] in Fig. 8.19, to verify our model. To simulate the uniaxial tension test of the swollen hydrogel, the swollen state of the hydrogel is set as the initial configuration, then the tension loading with the specified rate is applied. The presented swelling theory is employed in a nonlinear finite element framework to model a tough pH-sensitive hydrogel-based microvalve, which was first introduced by [39]. As shown in Fig. 8.20, we model the microvalve as a ring with

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

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Figure 8.19 Stress
stretch tensile curve of the pH-sensitive tough hydrogel at two different pH values [42].

Figure 8.20 Radial stress contour (left side) and tangential stress contour (right side) of a microvalve made of tough pH-sensitive hydrogel [42].

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inner and outer radii of 60 and 150 μm, respectively. Plane-strain condition is assumed for the problem. When pH 5 7, the channel is open, and the hydrogel is in its free stress state. The hydrogel gradually swells to push against the channel wall and closes the channel as the pH value of the external solution changes to 14. It is clear in Fig. 8.20 that it approximately takes 1000 s for the hydrogel to touch the channel wall. By reducing the size of the valve, this time can be lowered.

8.5

Summary and conclusion

In first part of this chapter, the equilibrium behavior of pH/temperature sensitive PNIPAM hydrogel was studied. The free and constrained swelling of the pH/temperature sensitive hydrogel was examined and some parameter studies were performed. One of the main advantageous of the presented model was its continuity especially against the temperature variations which makes it appropriate for implementing in inhomogeneous swelling problems whether analytically or numerically. Thereafter, the inhomogeneous swelling of a spherical shell on a hard core was analytically studied and the results were presented for various thicknesses of the shell. Then, the inhomogeneous swelling of a spherical shell on a hard core, with boundary conditions identical to the analytical one, was solved and the validity of the numerical results was successfully confirmed. Finally, the numerical simulation of the complicated problem of contact between the hydrogel shell and a channel with rigid walls due to the pH or temperature variations was performed with reasonable results. In the second part of this chapter, a coupled chemo-mechanical transient large deformation swelling theory for the pH-sensitive hydrogels was developed. The flux of the ionic species is generally due to diffusion, convection, and electrical migration. In the hydrogel swelling process the convection term in Nernst
Planck equation is usually negligible. In this theory, the electrical term was neglected due to the Debye length and because the hydrogels of this study are of the macro size and larger. After determining the concentration of hydrogen ion by the Nernst
Planck equation, the concentrations of other ions were efficiently computed employing the Donnan equilibrium. The main reason which causes the hydrogel as a hyperelastic material to swell is the osmotic pressure, which can be computed using the concentration of ions. The theory was implemented in a finite element framework. The experimental equilibrium results were used to find the hyperelastic material parameters of the hydrogel. The examination of the swelling and deswelling kinetics of the hydrogel showed that the deswelling process is approximately ten times faster than the swelling process. It was observed that the free swelling process is not homogeneous, and as a square hydrogel freely swells the corners swell faster because of the more considerable contact with the external solution in comparison with the other parts of the hydrogel. Finally, a pH-sensitive microfluidic valve was simulated to examine its radial and tangential stresses during operation. In third part of this chapter, a transient visco-hyperelastic model was developed for swelling, rate-dependent, and self-healing behavior of tough pH-sensitive

Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels

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hydrogels. The model consists of coupled electrochemical and mechanical fields. The main features of this model are similar to the previous transient model with the main difference that here the osmotic pressure is assumed to be visco-hyperelastic driven. The osmotic pressure was computed using the concentration of ions. The mechanical part is based on the Yeoh strain energy function and the Generalized Maxwell model to account for the visco-hyperelastic behavior of the hydrogel. Comparing the simulation results with the experimental ones, it was demonstrated that the proposed models do an excellent job in predicting the swelling response of soft and tough pH-sensitive hydrogels. In summary, accurate swelling theories for pH-sensitive hydrogels were presented which are useful for optimum utilization of these materials in various applications.

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Structural analysis of different smart hydrogel microvalves: the effect of fluid
structure interaction modeling

9

Chapter outline 9.1 Introduction 313 9.2 Different constitutive models’ description 9.2.1 9.2.2 9.2.3 9.2.4

314

Swelling theory of thermal-sensitive hydrogels 314 A stationary swelling theory for pH-sensitive hydrogels 316 A transient theory for pH-sensitive hydrogels 320 Coupled fields in fluid
structure interaction modeling 320

9.3 Results and discussion

320

9.3.1 Results for fluid
structure interaction analysis of temperature-sensitive hydrogel valves 320 9.3.2 Results for stationary response of pH-sensitive hydrogels 329 9.3.3 Transient results of the pH-sensitive hydrogel valve 340

9.4 Summary and conclusions References 346

9.1

345

Introduction

A polymeric gel is a network of covalently cross-linked polymer chains, aqueous and capable of absorbing a huge amount of water. The aggregate of the gel and the absorbed water is called hydrogel. In some hydrogels, called “smart hydrogels,” the absorbed water may be influenced by environmental stimuli, such as temperature [1], pH [2], ionic concentration, light [3,4], electric field, ultrasound [5,6], and mechanical loads. Due to their large reversible deformations, smart hydrogels are good candidates to be employed as microvalves [7,8], microactuators [9
11], and microsensors, which are utilized in applications, such as microfluidics, soft robotics, drug delivery [12], and responsive surfaces. The high compatibility of hydrogels with aqueous environment, its inherent sensitivity to the variety of stimuli, high biocompatibility, low price and their rapid prototyping are some of the main advantages reported for smart hydrogels. The widespread use emphasizes the need of a mathematical formulation capable of modeling the mechanical behavior of various types of hydrogels. The history of constitutive modeling of large deformation behavior of hydrogels dates back to 1943 when Flory [13] proposed that Helmholtz free energy of hydrogels can be decomposed into network stretch and mixing free energies. This theory examines only the equilibrium condition and accounts only for the initial and final states. Computational Modeling of Intelligent Soft Matter. DOI: https://doi.org/10.1016/B978-0-443-19420-7.00006-9 © 2023 Elsevier Inc. All rights reserved.

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Hong et al. [14] introduced a transient continuum theory for neutral hydrogels, which involves the water diffusion influence on large deformation behavior of hydrogels and is capable of modeling time-dependent problems. The theory is extended to thermal-sensitive [15] and pH-sensitive hydrogels [16]. In 2009 polymeric gel in equilibrium condition was modeled in a finite element framework and the model was implemented in ABAQUS software by developing a user-defined subroutine [17]. Later the model was expanded to different types of responsive hydrogels, involving various stimulus variations [2,18] and was generalized to time-dependent problems, considering the effect of solvent diffusion [19]. Some of these models exhibit snap-through instability and multiple solution near the phasetransition temperature, imposing some restrictions in finite element implementations. By modifying the mixing free energy, Mazaheri et al. [1] suggested a more stable model for PNIPAAM gel in equilibrium with an external solvent. Hydrogels are often employed in contact with fluid solvent. Since hydrogels are soft and highly deformable, in some applications, such as microfluidics, the effect of fluid pressure distribution on the polymeric gel deformation is significant and it should be appropriately accounted for in calculations. Moreover, due to nonlinearity in both contact phenomenon and the interaction between fluid and hydrogel, it is inaccurate to relate the contact pressure and maximum admissible inlet pressure qualitatively and simultaneous computations should be conducted in fluid and structural domain: the fluid
structure interaction (FSI) approach [20]. Two major techniques have been proposed [12]: (1) monolithic and (2) partitioned. In the first approach, the fluid and structural equations are combined and a single system of equations is achieved, weak form of which is employed for FEM simulations. In contrast, the second approach treats fluid and structural dynamic equations as two separate sets of equations discretized and solved in two separate mathematical frameworks and the interfacial parameters are exchanged between two solutions explicitly. The partitioned approach is subdivided into two different procedures [20]: (1) one-way and (2) two-way. While, the first procedure transfers only the fluid pressure to the structural solver and neglects the influence of structural displacement on the fluid domain; in the second procedure, effect of solid displacement on fluid domain and fluid pressure on solid domain are both involved.

9.2

Different constitutive models’ description

We first briefly describe necessary constitutive models for equilibrium response of both temperature- and pH-sensitive hydrogel. Then, a transient theory for pHsensitive hydrogels is discussed to be used in the following FSI simulations.

9.2.1 Swelling theory of thermal-sensitive hydrogels Similar to the hyperelastic solids, an energy-based approach is used to model the swelling behavior of a thermal-sensitive gel, which is in equilibrium with external solvent after the fluid diffusion [21]. In a variational approach, considering a

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specified temperature, the change in Helmholtz free energy density is expressed as a relation between deformation gradient F, first Piola
Kirchhoff stress tensor P, chemical potential μ, and the number of fluid molecules absorbed by the dry network per its unit volume Cs : δW 5 P:δF 1 μδCs

(9.1)

Due to the small stress in the gel and large deformations, both the polymers and solvent molecules are taken to be incompressible. By assuming the diffusion of the molecules as the only cause of the networks volume change, the molecular incompressibility can be expressed as J 5 1 1 ϑs Cs

(9.2)

where J 5 det F, and ϑs is the volume of a solvent molecule. The Helmholtz free energy can be written in an additive form as proposed by Flory [13]: W 5 Wmixing 1 Wstretch

(9.3)

where Wstretch is arising from stretching the network. Employing the neo-Hookean model, one may define: Wstretch 5

1 NKTðI1 2 3 2 2logðJ ÞÞ 2

(9.4)

In addition, Wmixing is the free energy density of mixing the polymer and the solvent. Considering the Flory
Huggins model, it is written as:     KT 1 1 ϑC χϑC ϑCln Wmix ðC Þ 5 1 ϑ ϑC 1 1 ϑC

(9.5)

where ϑ is the volume of a polymer molecule, and χ as a dimensionless parameter measures the pairwise interaction between species, which is calculated by fitting experimental results, and defined as χ 5 χ0 1

χ1 J

(9.6)

Parameters in Eq. (9.6) are χ0 5 A0 1 B0 T and χ1 5 A1 1 B1 T, in which A0, B0, A1, and B1 are material parameters. Substituting Eqs. (9.2) and (9.5) is recast as:     KT J 21 χ ðJ 2 1Þ ln Wmix ðJ Þ 5 1 ϑ J J

(9.7)

Replacing the logarithmic term in Eq. (9.5) by its polynomial expansion, Mazaheri et al. [1] proposed a modified energy model which solves the

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discontinuity and instabilities of the original model. Thus this model is a more suitable case for FEM frameworks and is defined as: Wmix ðJ Þ 5

  KT 1 1 1 χ ðJ 2 1Þ 2 2 2 2 3 1 ϑ J 2J 3J J

(9.8)

In the equilibrium condition, the chemical potential is assumed to be zero. Considering Eq. (9.2), the state equation for Cauchy stress σ s is written as σs 5

1 @W F J @F

(9.9)

where superscript s shows that the stress is computed within the solid region. In the equilibrium state, the hydrogel imbibes an amount of water and swells due to its environment temperature, which therefore makes this free-swollen condition as its new reference state. In the special case of free swelling, the deformation gradient is F 5 λ0I, where I is the identity matrix and λ0 is the free swelling stretch. As a result, at a specific temperature, the energy equation can be simplified to:    1 3 3 1 11 Nυλ0 1 B0 T 1 A0 2 Nυ 2 3Nυlnðλ0 Þ 2 1 λ90 1 λ60 W5 9 2 2 2 λ0   1 1 2 B1 T 2 A 1 1 1 ðB1 T 1 A1 2 A0 2 B0 T Þλ60 1 6 3

(9.10)

In this case, free energy turns into a polynomial of λ0, coefficients of which depend on the temperature and material parameters. Additionally, since the deformation is not restricted by any constraint, the Cauchy stress is equal to zero: 05

      1 1 1 9 2 B1 T 2 A1 1 B0 T 1 A0 λ60 1 6 B1 T 1 A1 2 λ30 2 3 3Nυλ11 0 2 3Nυλ0 1 3 12 2 6 λ0

(9.11) Solving Eq. (9.11) for λ0, the equilibrium state of free swelling can be determined in each temperature. Due to its simplicity, the free swelling equation is often used to illustrate the approximate behavior of hydrogels. To rewrite the free energy concerning the reference state, we divide the free energy by initial swelling ratio λ30 . Therefore the I1 and J formula should be multiplied by λ20 and λ30 , respectively. To simulate the swelling behavior of the valve, the prescribed model is implemented as a UHYPER subroutine in ABAQUS.

9.2.2 A stationary swelling theory for pH-sensitive hydrogels An energy based approach, developed by Marcombe et al. [2], was used to model the swelling behavior of the pH-sensitive hydrogel in which, following a common

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approach among the researchers, an additive decomposition of the Helmholtz free energy density is assumed. The free energy density for the pH-sensitive hydrogels is then taken to be in form of W 5 WðF; Cs ; C1 ; CH1 ; C2 Þ, where F is the deformation gradient tensor and, Cs ; C1 ; CH1 , and C2 are the concentrations of solvent molecules, counter-ions, hydrogen ions and co-ions in the reference configuration, respectively. Assuming incompressibility of the solvent molecules and polymer network individually, the deformation gradient of the network is related to the solvent absorption by det F 5 1 1 νCs where ν is the solvent volume [2]. Considering the incompressibility assumption, and applying Legendre transformation, now the free energy can be rewritten as W 5 WðF; C1 ; CH1 ; C2 Þ which is a function of F and concentration of mobile ions [2]. For pH-sensitive gels, the free energy density includes different contributions as below: WðF; C1 ; CH1 ; C2 Þ 5 Wnet ðFÞ 1 Wsol ðFÞ 1 Wion ðF; C1 ; CH1 ; C2 Þ 1 Wdis ðF; C1 ; CH1 ; C2 Þ

(9.12) in which Wnet ðFÞ, Wsol ðFÞ, Wion ðF; C1 ; CH1 ; C2 Þ, and Wdis ðF; C1 ; CH1 ; C2 Þ are contributions of energies due to the elastic deformation, mixing the solvent and network, mixing ions and the solvent, and dissociation of the acidic groups, respectively [2]. The neo-Hookean model is adopted for the elastic deformation as: Wnet ðFÞ 5

1 NkT ½I1 2 3 2 2logðJÞ 2

(9.13)

where N; k; T; and I1 are the number of chains divided by the volume of dry network, Boltzmann constant, the absolute temperature and the first invariant of C, respectively. The Flory-Huggins [13] model has been taken for the mixing energy of the solvent and polymer:     kT 1 χ ðJ 2 1Þlog 1 2 Wsol ðFÞ 5 2 ν J J

(9.14)

which includes both the entropic and enthalpic changes due to mixing between the solvent and polymer. The dimensionless parameter χ indicates enthalpy change of the mixing. Contribution of the mobile ions to the free energy is due to just the entropy of the mixing as their concentration is assumed to be low. The expression for the ions’ energy is: " Wion 5 kT CH1 log

CH1 cref H1 J

! 2 1 1 C1 log

!  # C2 2 1 1 C 2 1 log 2 ref ref J c2 c1 J C1

(9.15)

ref ref where cref H1 , c1 , and c2 are reference values of the concentration of the hydrogen ions, the counter-ions, and the co-ions, respectively. The last part of the free energy

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density includes both the enthalpy and the entropy of the acidic dissociation which is taken to be:   Wdis 5 kT CA2 log

   CA2 CAH 1 CAH log 1 γCA2 CA2 1 CAH CA2 1 CAH

(9.16)

where CA2 and CAH are the nominal concentration of the fixed charges and of the associated acidic groups, respectively. γ stands for theenthalpy  rise caused by dissociation of an acidic group which is equal to γ 5 2 kTln

NA Ka ref c1

where NA 5 6:023 3 1023 is

the Avogadro number and Ka is the constant of acidic dissociation. Assuming chemical and mechanical constrains, the Cauchy stress in the indicial format is calculated as: σij 5

 NkT  Fik Fjk 2 δij 2 ðΠsol 1 Πion Þδij J

(9.17)

in which Πion 5 kT ðcH1 1 c1 1 c2 2 c H1 2 c 1 2 c 2 Þ Πsol 5 2

    kT 1 1 χ log 1 2 1 1 2 ν J J J

(9.18) (9.19)

where Πsol and Πion are the osmotic pressures caused by the mixing of the network and solvent and the osmotic pressure due to the unbalanced distribution of ions inside and outside of the gel network and δij is the Kronecker delta. cα and c α are representatives of the α species concentration in the current configuration inside the gel and in the external solution, respectively. The equilibrium equation of ions concentration inside the gel network with the external solution, known as Donnan equations, states that the ion concentration inside the gel is related to the concentration of the ions outside of it, through the following relations: c1 =c 1 5 cH1 =c H1

(9.20)

c2 =c 2 5 ðcH1 =c H1 Þ21

(9.21)

The electro-neutrality of the external solution yields that c 2 5 c 1 1 c H1 . In light of Donnan Eqs. (9.20) and (9.21) along with electro-neutrality assumption, the concentration of co-ions and counter-ions inside the gel can be written as a function of the hydrogen ions and the counter-ions concentrations in the external solution. The equilibrium equation in terms of acidic dissociation is [2]: 

cH1 ðcH1 1 c1 1 c2 Þ  5 NA Ka f =ν J 21 2 ðcH1 1 c1 1 c2 Þ

(9.22)

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where f is the number of acidic groups attached to polymer chains divided by the total number of monomers that existed on the chain and ν is the volume of one monomer. The volume per solvent is assumed to be equal to volume per monomer ν 5 ν s . Using the Donnan equations and relation (9.22), a third-order polynomial equation for cH1 is found as:  11

     c1 3 c1 2 NA Ka f cH1 2 NA Ka cH1 ðc1 1 cH1 Þ 5 0 cH1 1 NA Ka 1 1 cH1 2 cH1 ðc1 1 cH1 Þ 1 νJ cH1 cH1

(9.23) The above equations are the foundation for developing analytical solutions for a variety of problems. An analytical solution for the cylindrical microvalve jacket is presented in the follow sections based on the fundamental equations. While material (Lagrangian) description is used to calculate solid mechanics equations, the fluid dynamics problem is solved using spatial (Eulerian) description. Therefore in FSI problems, in which fluid and solid media are both involved, a combination of these approaches are employed: an arbitrary Lagrangian
Eulerian (ALE) description [22]. In an ALE coordinate system, the mesh movement is incorporated into fluid dynamics equations explicitly. For an incompressible viscous fluid, the continuity and momentum equations in an ALE coordinate system are formulated as [12] Δ  u50 ρ

@u 1 ρðu 2 u0 ÞΔu 5 Δ  σ f 1 ρg @t

(9.24) (9.25)

where ρ, u, u0 , g and σ f denote fluid density, fluid velocity, grid velocity, gravitational acceleration and stress tensor, respectively. The stress tensor is defined as σ f 5 2 pI 1 2ϕe

(9.26)

where p and ϕ are pressure and dynamic viscosity, respectively and e stands for the strain rate tensor defined as e 5 (Δ  u 1 Δ  uT)/2. In the interface of fluid and solid, the no-slip condition is imposed by the following equations: u 5 x_ s

(9.27)

n  σf 5 n  σs

(9.28)

where x_ s and n are displacement rate in the solid region and normal vector of the interface, respectively and σs is the Cauchy stress calculated within the solid region (Eq. (9.9)). In FSI simulations, in each step, the governing equations of the solid and fluid domain are solved iteratively, until the two interface conditions [Eqs. (9.27) and (9.28)] are solved by an acceptable degree of accuracy.

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9.2.3 A transient theory for pH-sensitive hydrogels Since a transient theory involves different governing fields, each one should be explained in separately. Details of electrochemical field and mechanical field are reported in Section 8.2. The theory developments for this model are given in [23,24]. To account for the Fluid Flow Field, we assume an incompressible singlephase fluid flow for the channel. The equations of motion are the continuity equation: r  ðρf uÞ 5 0

(9.29)

and the momentum equation: ρf

h  i @u 1 ρf ðu  rÞu 5 r  2 pI 1 μf ru 1 ðruÞT 1 G @t

(9.30)

in which, ρf is the fluid mass density, u and p respectively stand for the fluid velocity vector and fluid pressure in the channel, μf refers to the fluid viscosity, and G is the external body force.

9.2.4 Coupled fields in fluid
structure interaction modeling In FSI modeling of the pH-sensitive hydrogel-based microvalve, three coupled fields (i.e., electrochemical, mechanical, and fluid flow fields) exist. The ionic species transport to the hydrogel through diffusion, electrical migration, and convection, which leads to the pH change of the hydrogel. As the pH of the hydrogel reaches the dissociation constant of the acidic groups’ pendant to the polymeric network of the hydrogel, the acidic groups release hydrogen ion. The difference of the concentration of the ionic species inside and outside the hydrogel constitutes an osmotic pressure, which in turn causes the hydrogel to swell. As the hydrogel swells, it closes the channel gradually. The fluid pressure on the hydrogel interface also causes the hydrogel to deform. Therefore the fluid flow equations are coupled with the swelling kinetics of the hydrogel. In this section, we present the governing equations of the related fields in the transient FSI modeling of the pH-sensitive hydrogel-based microvalve.

9.3

Results and discussion

9.3.1 Results for fluid
structure interaction analysis of temperature-sensitive hydrogel valves Using a two-way FSI method, the influence of various effective parameters in microvalve design is studied. ANSYS FLUENT is employed to carry out the CFD calculations and ABAQUS is utilized to perform the solid mechanics analysis.

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At first, the initial state of the flow sorter is modeled in both CFD software and FEA package. Quadratic elements using hybrid formulation (CPE4H) are employed to model the hydrogel jacket in FEA package. The influence of the rigid core is modeled by fixing the inner boundary of the jacket. Hydrogel swelling is considered a twodimensional (2D) plane-strain problem in steady-state condition. Putting to use the free energy defined in Eq. (9.10), a UHYPER subroutine is implemented in ABAQUS defining the mechanical behavior of hydrogel valves. Employing the penalty method, the contact between the hydrogel jacket and the channel walls, are modeled as a rigid component. It is noted that the temperature filed is applied in FEM model as an input. A T-channel is modeled as a 2D geometry in CFD software. A laminar viscous incompressible model is utilized to simulate the water flowing in the channel. A dynamic mesh is employed in the vicinity of microvalves to update hydrogel jackets configuration in each step. The inlet flow is defined as a constant pressure, While the outlets are considered with 0-G pressure. Discretized form of the governing equations in the fluid region are solved using the finite volume method. FEM and CFD software are coupled using MPCCI, a multiphysics software providing an interface for direct coupling of different simulation codes. After modeling the fluid and solid sketches separately, the outer jacket surface is defined as an interchanging surface between the solid and fluid domains. In each step, the coupling software receives the boundary pressure from the fluid domain and imposes it as a boundary condition on the solid domain. Then, the solid problem is solved and nodal displacement of the exchanging surface, arising as a result of large deformation, is transferred from the solid domain to fluid domain and the outer jacket boundary is redefined in the fluid domain. The transformation of the nodal displacement and boundary pressure is performed iteratively, until the variation in one of the exchanging parameters in two consecutive steps becomes less than a desirable tolerance. This process depicted as a flow-chart in Fig. 9.1.

Figure 9.1 Flow chart of the fluid
structure interaction simulation procedure [20].

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9.3.1.1 Hydrogels characteristics The microvalves utilized in the flow-sorter branches are made up of two different polymeric gels. Gels show reverse behaviors when they are exposed to temperature variation. The first gel, poly(N-isopropylacrylamide) (PNIPAAm), is a negative temperature-sensitive hydrogel shrinking upon heating, while the second one, comprised of poly(acrylic acid) (PAA) and polyacrylamide (PAAm), undergoes positive volume change (swells) under heating. The material parameters for the two utilized polymeric gels are listed in Table 9.1 [26,27]. Substituting material parameters from Table 9.1 in Eq. (9.11) and solving the equation for initial temperatures T 5 307K and T 5 293K, respectively, the initial free swelling stretch (λ0 ) can be calculated for both of utilized polymeric gels as λ0 5 2.34. By these assumptions, two valves swelling in the channel with temperature change, are simulated in ABAQUS. Table 9.2 illustrates two valves reverse stretch Table 9.1 The material parameters of two polymeric gels [25]. Material GEL 1 GEL 2

PNIPAAm PAA and PAAm

Nυ

A0

A1

B0 (1/K)

B1 (1/K)

0.002 0.002

2 12.947 14.0308

17.92 2 16.22

0.04496 2 0.04496

2 0.0569 0.0569

Table 9.2 Left outlet fluid flow rate of 2.5-, 5-, and 10-kPa inlet pressures [25]. Left outlet fluid flow rate of 2.5-kPa inlet pressure Outlet-left flow rate (mL/min) 2.5 kPa—FSI 2.5 kPa—non-FSI Difference %

Open
close state

Close
close state

Close
open state

0.0741 0.0600 23

1.3715e 2 04 1.1449e 2 04 20

7.1887e 2 05 6.7937e 2 05 5

Open
close state

Close
close state

Close
open state

0.1716 0.1176 46

3.6791e 2 04 2.2896e 2 04 60

1.4841e 2 04 1.3590e 2 04 8.8

Left outlet fluid flow rate of 5-kPa inlet pressure Left outlet flow rate (mL/min) 5 kPa—FSI 5 kPa—non-FSI Difference %

Left outlet fluid flow rate of 10-kPa inlet pressure Left outlet flow rate (mL/min) 10 kPa—FSI 10 kPa—non-FSI Difference %

Open
close state

Close
close state

Close
open state

0.3652 0.2270 60

0.0116 4.5789e 2 04 Breakage

3.3445e 2 04 2.7180e 2 04 23

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(λÞ versus the temperature. Simulation is initialized from point A, where both channels are sealed, which here is called the close
close state. At first, the temperature is raised to point B, that is, the close
open state. Then, the temperature is returned to point A and after that reduces to state C, which is called the open
close state. As mentioned, the simulation is conducted on a T-junction, branches of which are gated with two hydrogel valves at the entrances. Both valves are cylindrical pillars, on which the hydrogel jacket is coated. Each pillar has an outer radius of ro 5 63 μm and inner radius of ri 5 25 μm. The inlet and outlets path width are 200 μm and the inlet path length is assumed to be 500 μm. Fig. 9.3 represents the T-junctions configuration. The performance starts from state A according Fig. 9.2, at T 5 300K, in which both valves are swollen and both channels are sealed (close
close state). By heating to T 5 304K (Fig. 9.2B), the right branch is closed and the flow passes only through the left channel, which results in an open
close state. Cooling to T 5 300K, the close
close state takes place again (Fig. 9.2A). And finally, at T 5 296K, Fig. 9.2C results in an opposite direction flow through the right channel (the close
open state). The von Mises stress contour within the left valve without considering the fluid effects for three swelling states, are illustrated in Fig. 9.4. Valve swells in a symmetric way where the maximum stress of 50 kPa occurs in the inner radius of the valve, when the lowest temperature. The fluid within the channel is assumed to be water. To take in to account the FSI, the valves are studied at inlet pressure of 2.5, 5, and 10 kPa, while the outlet is taken to be in zero-gage pressure. Fig. 9.5 represents the contour of fluid pressure and von Mises stress for the inlet pressure of 10 kPa in different states. Side of the

Figure 9.2 Two hydrogels stretch versus the temperature change [25].

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Figure 9.3 T-junction flow sorter with one valve in each entrance [25].

Figure 9.4 von Mises stress contour in one of two valves without considering fluid
structure interaction in swollen states (A) open
close, (B) close
close, and (C) close
open [25].

valves, which confronts the incoming flow, undergoes lower stresses because this stress is neutralized by a compressive force applied by the fluid and therefore valves deform in asymmetric patterns. The deformed shape of valves due to the fluid pressure, results in a smaller effective blocking area. Consequently, the fluid flow rate is higher in FSI computation than the non-FSI solution, computed by a surface integration over the velocity profile. The inlet fluid flow rate is depicted in Fig. 9.6A for fluid pressure of 2.5, 5, and 10 kPa. As explained in Fig. 9.2, valves are assumed to be in the state of A, B, A, and C. As expected, increasing the incoming fluids pressure signifies the fluid

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Figure 9.5 Fluid pressure distribution and von Mises stress within two valves in open
close state (A) left valve and (B) right valve; close
close state (C) left valve and (D) right valve; and close
open state (E) left valve and (F) right valve [25].

(A)

(B)

Figure 9.6 ( A) The inlet fluid flow rate and (B) right valves contact length for different inlet fluid pressures [25].

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Table 9.3 Left outlet closing temperature in the close
close state [25]. Left outlet closure temperature

Non-FSI

FSI

T %

2.5 kPa 5 kPa 10 kPa

302.40 302.08 301.82

301.96 301.12 Not closed

5.5 12


flow rate. In addition, at higher fluid pressures, the difference between the predicted flow rate of FSI and non-FSI simulations is more noticeable. For 10-kPa inlet pressure, this difference has a significant influence on the functionality of the valves. In the close
close state that both valves seal branches in the non-FSI simulation and the inlet flow rate is zero, critical deformed shape of valves makes the paths stay rather open and subsequently the inlet flow rate remains nonzero. Table 9.2 presents the left branches outlet fluid flow rate of both FSI and non-FSI simulations. The breakage of valves due to the fluid pressure in the middle state for 10-kPa inlet pressure is visible in these results. Because of symmetry of the fluid flow in two branches, we just represented the results for the left outlet. Another interesting parameter in FSI study is the valves closing temperature. This is the temperature, in which the paths flow rate tends to zero. The closing temperatures of three fluid pressures for the close
close state are presented in Table 9.3. In higher fluid pressures, the closing temperature for the left valve decreases because it should swell more to stop the fluid flow. In addition, the closing temperature in FSI computation is smaller than the non-FSI ones, because of the impact of the fluid forces applied  2Tnon2FSI  on the valves. To illustrate the difference we define T  5 TFSI 3 100%. For 304 2 296 inlet pressure of 10 kPa, simultaneous effect of fluids flow on two valves results in valves failure as mentioned before. Another parameter, which plays a key role in microfluidic channel design, is the contact length between the valve and channel walls. It is obvious that this contact length determines the functionality of the hydrogel as a valve to seal the path. For 2.5-kPa inlet fluid pressure, the effect of FSI on the contact length is almost negligible. While for the fluid pressure of 10 kPa, this effect leads to the valves’ breakage. Fig. 9.6B shows the contact length during the swelling for the right valve.

9.3.1.2 Multimicrovalves in a channel The long response time of hydrogels is a drawback in their usage. Reducing the size of the hydrogel reduces the diffusion time and enhances its response time. For this purpose, there are two approaches available, first to use hydrogels as a jacket around rigid posts as applied in this section. The second is to use multiple valves with smaller radius in each path, while valves pattern is effective in their functionality. Three different patterns proposed in [20] are discussed here. Each one consists of three equal cylindrical valves with different configurations. The First is called inline (IL) pattern in which all valves are aligned on a straight line (Fig. 9.7A). In

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327

Figure 9.7 Multivalves’ pattern configurations: (A) upstream, (B) inline, and (C) downstream [25].

the second one called downstream (DS) pattern, valves make a triangle where its tip is facing the inflow (Fig. 9.7B), and the third one called upstream (US) pattern makes a triangle as well, while its tip is against the incoming flow (Fig. 9.7C). To preserve comparability, the outer radius of the valves in each pattern are determined in a way that the flow rate in the initial state is equal for all three patterns for the inlet pressure of 5 kPa as the average working pressure. Valves thicknesses in all patterns are constant as well. In IL pattern, valves inner and outer radii are taken to be ri 5 14 μm and ro 5 24 μm, while in other two patterns, the inner and outer radii are ri 5 18 μm and ro 5 28 μm, respectively. The valves’ functionality in each branch is the same as the single valve problem discussed in latter section. The von Mises stress within the valve and the fluid pressure distribution for an inlet pressure of 10 kPa for three patterns in open
close and close
open states are illustrated in Fig. 9.8. The influence of FSI on the inlet flow rate for each pattern are presented in Table 9.4. The Outer radius of IL pattern is smaller than others, but its flow rate is lower which reveals the impact of the valves pattern. The flow stream has sharper angles to pass in IL configuration, which means it is less consistent with the fluid stream. Although reducing the size enhances their response time, it decreases their mechanical properties. The significant difference between the flow rate of the FSI and non-FSI simulation of IL pattern can be explained by both the valves configuration effect and the reduced mechanical properties, which enhances the deformability of valves. The Maximum flow rate is computed for DS pattern, while there is minimum deviation between FSI and non-FSI simulations. One may conclude that DS and US configurations are more compatible with the fluid stream from a hydrodynamic viewpoint. In the presence of the fluid pressure, valves have to swell more to overcome the fluid forces and close the passages. As a result, in FSI simulations, the closing temperature of the left valve is less than in the non-FSI results, while it is higher than the non-FSI results for the right one. Table 9.5 represents the left outlet’s closing temperature for various patterns. One may observe that under the same conditions, IL pattern responds faster to the temperature changes. Thus in microfluidic applications where sharper response is needed, the IL pattern seems to be a more promising candidate. Other interesting point is that the difference between the FSI and non-FSI results for the closing temperature of IL pattern is the smallest (Table 9.5).

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Figure 9.8 von Mises stress and pressure distribution in inline pattern (A) open
close state and (B) close
open state; in upstream pattern (C) open
close state and (D) close
open state; and in downstream pattern (E) open
close state and (F) close
open state [25]. Table 9.4 Inlet flow rate in the open
close state for different valves patterns [25]. inlet flow rate (mL/min)

IL pattern

US pattern

DS pattern

FSI Non-FSI Difference %

0.2049 0.0860 58

0.2327 0.1470 37

0.2681 0.2072 22

US and DS patterns close the passage in a lower temperature due to their configurations, while in that closing temperatures, the FSI effects on the valves’ deformation is more significant. Thereupon, the effect of FSI simulation compared to IL pattern is more noticeable.

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Table 9.5 Left outlet closure temperature for various valve patterns [25]. Closure temperature (K)

IL pattern

US pattern

DS pattern

Non-FSI FSI T %

302.88 301.36 19

302 300.24 22

302 300.4 20

9.3.2 Results for stationary response of pH-sensitive hydrogels 9.3.2.1 Fluid
structure interaction procedure Similar to previous section, two-way FSI simulation is carried out for investigating effective parameters in microvalve design. The microvalves are considered to be normally open at low pH. Initial state of the jackets is modeled both in FEM and CFD software, separately. The hydrogel jacket is meshed in FEM software with plane-strain quadratic elements using hybrid formulation (CPE4H). In order to reach acceptable accuracy an adequately small mesh size is used. UHYPER subroutine defining the hydrogel behavior is passed in ABAQUS to define the hydrogel jacket behavior. Walls of the channel are modeled using rigid elements in FEM model to account for the contacts occurring between the hydrogel and the channel walls in the closed state. The penalty method is used for modeling the contacts. The surfaces are considered to be frictionless, as there are approximately no tangential relative movements between contacting surfaces. The pH value of external solution and its salt concentration are used as input parameters in FEM model. On the other side, assuming a 2D channel, the geometry is modeled in CFD software. The hydrogel jacket is considered as a wall inside the channel with the exact size of the jacket in FEM. Since the jacket geometry is changing during FSI analysis, the vicinity of the microvalve is meshed with triangular elements to be able to use dynamic mesh. The microvalve inlet is considered to have constant pressure and the outlet is with the 0-G pressure. The laminar incompressible viscous model is used to simulate the water inside the channel. The results are converged employing an optimum mesh size. Coupling of FEM and CFD software are the same as the previous section. The iterative process between CFD and FEM solver continues until the difference between pressure distribution or jacket surface displacement in two successive iterations become lower than a prespecified value. When this state is achieved, the FSI simulation is converged. Using this approach, the behavior of the microvalve is studied due to pH changes as well as fluid flow effects. First, analytical solution which is useful in conceptual design and FEM validation is presented and compared with numerical ones. Next, FSI simulation of the one-jacket as well as three- jackets is examined in detail. Water with viscosity of μ 5 0:001003kg=m s and density of ρ 5 998:2kg=m3 is considered in simulations. The material parameters of the pHsensitive hydrogel are presented in Table 9.6.

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Table 9.6 The material parameters of pH-sensitive hydrogel [2]. Parameter description

Symbol

Value

The number of chains divided by the volume of dry network The volume of one monomer The Flory-Huggins interaction parameter The constant of acidic dissociation The absolute temperature in unit of energy The number of acidic groups attached to polymer chains divided by the total number of monomers that existed on the chain The concentration of counter-ions in external solution

N ν χ Ka kT f

1025 =m3 10228 m3 0:1 1024:3 4 3 10221 J 0:05

c1

0:001M

9.3.2.2 Analytical solution The swelling behavior of one cylindrical hydrogel jacket coated on a rigid pillar with ri 5 180μm and ro 5 360μm is studied here, using both 2D plane-strain FEM model and an analytical approach. Based on the model described earlier, an analytical solution for this special case, in the polar coordinate, is developed. Implementing relations converting Cauchy stress into nominal stress Pθ 5 σθ λr λz and Pr 5 σr λθ λz (where λr ; λθ ; and λz are stretches in radial, hoop, and z direction, respectively), the nominal stresses are found: Pθ 5 NkTðλθ 2 λ21 θ Þ 2 λr λz ðΠsol 1 Πion Þ

(9.31)

Pr 5 NkTðλr 2 λ21 r Þ 2 λθ λz ðΠsol 1 Πion Þ

(9.32)

The stretch in z direction is λz 5 1. Hydrogels rarely exist in a dry state and often contain some water. As a result, one may assume the reference state as a free swelling process form the dry state with free-stress conditions [7]. Taking the equilibrium equation in the reference state at pH 5 2, the stretches are multiplied by the reference principal stretch with respect to the dry state (λ0 ). The whole energies, and consequently, the stresses must be divided by λ30 to account for the reference configuration. The initial free stretch is calculated to be λ0 5 3:39. Considering the symmetry in θ direction, the stretches are: λr 5

d r; dR

λθ 5

r R

(9.33)

where r and R denote current and initial states of the material, respectively. The equilibrium equation in polar coordinate is: dPr ðPr 2 Pθ Þ 50 1 R dR

(9.34)

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Putting the stresses into equilibrium equation and replacing λr ; λθ , we arrive at a nonlinear ordinary differential equation (ODE). The boundary conditions for this ODE are: λr ðri ; θÞ 5 1;

Pr ðro ; θÞ 5 0

(9.35)

that is unity radial stretch in the inner radius and stress-free surface in radial direction at the outer radius of the jacket. The known input variables in equations are c H1 and c 1 . The concentration of the hydrogen ions in the external solution is related to pH value by c H1 5 NA 102pH . Applying numerical techniques, such as finite difference method, the answer to the swelling problem is identified via solving the nonlinear ODE, with boundary condition, coupled with the algebraic Eq. (9.23). The final ODE equation is:  Nνr 3 Rλ80  1

2 Nν

   4 dr 5 d2 r dΠion 4 9 dr 1 Nν 2 r 3 R2 λ80 2 r Rλ0 2 Nνr 4 λ80 1 Nνr2 R2 λ60 2 Nνr 2 R2 λ50 dR dR dR dR

d 2 r 2 3 5 dΠion 3 2 6 r R λ0 2 Nνr 3 Rλ60 1 Nνr 3 Rλ50 2 NνrR3 λ30
2rR3 χλ30 1 rR3 λ30 r R λ0 1 dR dR2



dr dR

3

  2 d2 r 3 2 6 dr 3 2 2 3 2 2 2 2 3 4 1 Nν 2 r R λ0 1 Nνr R λ0 1 2r R χλ0 2 r R λ0 1 2R χ dR dR  1

 d2 r 2 3 3 d2 r 2 3 3 d2 r 2 3 3 dr d2 r 3 2 Nν 2 r R λ0
2 2 r R χλ0 1 r R λ0
2rR χ 1 2 2 rR4 χ 5 0 2 dR dR dR dR dR

(9.36) Increasing pH to 8, the outer radius reaches ro 5 566μm as the hydrogel swells. The change of the outer radius due to pH changes for both the analytical and 2D FEM models are depicted in Fig. 9.9A.

Figure 9.9 ( A) Change of the outer radius of hydrogel jacket valve due to pH variations of the external solution for both analytical and 2D FEM models. (B) The radial and hoop stress distribution inside the hydrogel jacket at pH 5 8 [20].

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The radial and hoop stress distributions inside the hydrogel jacket at pH 5 8 are presented in Fig. 9.9B. Since the hydrogel swells approximately freely in the vicinity of the outer boundary, the hoop stress is smaller there. The constraints applied on the inner radius prevent the hydrogel from radial movement, consequently, high stresses are found in this area. The good agreement between the FEM results and the analytical solution guarantees the FEM models performance to be used in next section for FSI simulations purposes.

9.3.2.3 One hydrogel jacket microvalve fluid
structure interaction In this section, a single jacket microvalve is studied employing two-way FSI simulation. The microvalve inner and outer radii are ri 5 180μm and of ro 5 360μm, respectively. The channel width is assumed to be 1mm, so that the contact between the jacket and the channel walls be possible in swollen state. In both FEM and CFD simulations, the jackets are perfectly cylindrical at their initial state. The microvalves performance is studied at three inlet pressures of 1, 2, and 4 kPa, while the valve outlet is to the atmosphere with 0-G pressure. To follow the microvalve status during the actuation, the pH value is increased step by step from pH 5 2 to 8. To demonstrate the FSI impacts more clearly, the contours of the fluid pressure in the channel and the radial stress field inside the hydrogel are plotted altogether for the inlet pressure of 4 kPa at pH 5 2, 5, and 8 in Fig. 9.10. Half of the microvalve is shown for the symmetry reasons and the velocity vectors are included as well. At pH 5 2, without considering the fluid stream, the hydrogel is in the stressfree state with a cylindrical shape. Imposing the fluid flow forces on the FEM model, the jacket is deformed into an asymmetric shape with respect to the y-axis and as a result, a stress field with a maximum compressive radial stress of 25 kPa is produced inside the hydrogel (Fig. 9.10A). The stress field inside the jacket follows the external fluid pressure distribution. The fluid velocity is higher at the gaps between the jacket and the channel walls. Following the famous Bernoulli equation in viscous flows, the pressure drops at locations with higher velocity. Therefore a negative fluid pressure exists within the gaps. The suction force of the negative pressure on the hydrogel surface induces tensile radial stresses in the vicinity of the gaps (Fig. 9.10A). It is necessary to state that the negative pressure values in all figures are obtained due to the fact that the ambient pressure is not taken into account. There is no vacuum in the outlet of the valves. In higher pH values (Fig. 9.10B and C), the tensile radial stresses induced during the swelling (Fig. 9.10B) are neutralized by the compressive stresses exerted by the fluid force at the locations facing the microvalve inlet (Fig. 9.10B and C). At pH 5 8, the microvalve is closed and the fluid stream stopped (Fig. 9.10C). The asymmetric stress distribution inside the jacket can be only detected in FSI simulation. An important parameter in microvalve operation is its flow rate for various operational states, such as different pHs. The fluid flow rate is extracted from the integration of the velocity profile at the inlet of the microvalve at different pHs. Fig. 9.11A depicts the flow rate of the microvalve in range of pH 5 2
7 for the

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333

Figure 9.10 The fluid
structure interaction results including fluid pressure contour inside the channel, velocity vectors, and induced radial stress inside the hydrogel jacket for the inlet pressure of 4 kPa at (A) pH 5 2, (B) pH 5 5, and (C) pH 5 8 [20].

inlet pressures of 1, 2, and 4 kPa. The flow rate is higher for the higher inlet pressure as expected. In addition, the microvalve closing pH value is affected by the inlet pressure. Table 9.7 lists the pH values in which the microvalve is closed at a specified pressure. For higher inlet pressures, the jacket should swell more to be able to resist the fluid stream. Therefore the microvalve is closed at higher pH values as the inlet pressure increases. The variation in the closing pH value might affect the correct operation of the microvalve in different applications and should be accurately predicted using FSI simulations. To further discuss on the flow rate trends, the deformation of the jackets outer surface in different states is presented in Fig. 9.11B. The hydrogel deformation is

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Figure 9.11 (A) The flow rate drop versus the pH for inlet pressure of P 5 1, 2, and 4 kPa. (B) Deformation of the outer surface of the hydrogel jacket at inlet pressures of 1, 2, and 4 kPa at pH 5 2, 5, and 8 [20]. Table 9.7 The closing pH of the microvalve for different inlet pressures [20]. Inlet pressure

1 kPa

2 kPa

4 kPa

Closing pH value

5.30

5.40

5.55

directly related to the inlet pressure magnitude and its swelling due to external solution pH. The more it swells, the difference between the jacket shapes in different inlet pressures are more noticeable. In shrunk state, the microvalve is fully open and its flow resistance is minimum. Therefore higher pressures only exist in the small areas near to the front of the microvalve and the pressure drops sharply over the surface of the jacket in the flow direction (Fig. 9.12). Moreover, the hydrogel is stiffer at lower pH values. Consequently, the difference between the shapes of the jacket in different inlet pressures in shrunk state is small. As illustrated in Fig. 9.12, the pressure distribution over the jacket is altered as the hydrogel swells. At pH 5 5, the average pressure applied by the fluid to the areas of the jacket surface facing the inlet is higher than the zones facing the outlet (negative normalized x in Fig. 9.12). Further deformation of the jacket in this state is the result of reduced stiffness and higher fluid pressure. At pH 5 8, the microvalve is fully closed and the pressure is equal to the inlet pressure at the front side and relative zero at the other side (Fig. 9.12). In the fully closed state, the contact length has been considered as an important factor defining the safety factor of the microvalves [7,12]. As shown in Fig. 9.11B, increasing the inlet pressure reduces the contact length and subsequently the reliability of the microvalve. The microvalve leaks if the contact length diminishes to zero. Thus the FSI simulation becomes more crucial at higher inlet pressures. A sudden drop in flow rate at pH 5 2 is also noticeable in Fig. 9.11A. The reference state flow rates are computed in CFD without considering the jacket deformation (non-FSI). Using FSI simulation, the flow rate drops as a result of the alteration in the pressure distribution due to jacket deformation. The microvalves flow rate at pH 5 2, at different inlet pressures, for the FSI and non-FSI CFD

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Figure 9.12 Pressure distribution over the surface of the jacket versus the normalized distance from the microvalve center at inlet pressures of 4 kPa and pH 5 2, 5, and 8. Normalized distance is the distance from the center of the pillar divided by the maximum distance of the jackets outer surface [20]. Table 9.8 The reference state (pH 5 2) flow rates for FSI and non-FSI simulations at different inlet pressures for pH 5 2. Inlet pressure

1 kPa 2 kPa 4 kPa

Flow rate (mL/min) [20] FSI

Non-FSI

Difference

3.65 5.55 8.39

3.58 5.30 7.37

1.96% 4.72% 13.84%

simulations are given in Table 9.8, revealing that the variation in the flow rate due to the jacket deformation is more significant at higher inlet pressures and should be considered through the FSI simulations.

9.3.2.4 Three jackets’ microvalve fluid
structure interaction Since diffusion is the main governing phenomenon in hydrogel materials, the response time of the smaller pieces of hydrogels is lower. Thus employing multiple

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Figure 9.13 Geometry of the three jacket microvalve patterns: (A) inline, (B) upstream, and (C) downstream [20].

smaller hydrogel jackets are in priority in hydrogel microvalve design. One interesting idea is to use three hydrogel jackets instead of one. In this way, the response time decreases while the functionality of the microvalve is maintained. Since the patterns of the jackets can influence the performance of the microvalve, we also examine this geometric property in this section. As depicted in Fig. 9.13, three patterns are studied: IL pattern in which the jackets are placed on a straight line with equal distance, US in which the jackets are placed on the corners of an Isosceles triangle with its tip facing the inflow, and the DS pattern in which the jackets are placed on the corners of a isosceles triangle which its tip is against the inflow. For the IL configuration, each hydrogel jackets inner and outer radii were chosen to be 60 and 120 μm, respectively. The flow rate has been calculated for the microvalve inlet pressure of 1.5 kPa considering rigid jackets. To make the microvalves comparable, the outer radius of jackets in other patterns has been computed so that the flow rate remained the same at the reference state (pH 5 2). The thickness of the hydrogel jackets is also assumed to be constant to make the patterns more comparable. The US and DS patterns have bigger outer radius, as they possess a more hydrodynamic shape (shapes with less drag coefficient) than that of the IL pattern. It means that if the jackets with the same size of those in IL pattern, are used in US and DS patterns, the microvalve flow rate would be higher at the same inlet pressure. All patterns with full geometrical details are illustrated in Fig. 9.13. The external solution pH is changed from 2 to 8 during FSI simulations. For better illustration of the closing process, the pressure distribution and von Mises stress contours are plotted on the deformed shape of the jackets for each pattern at pH 5 2, 5, and 8 in Fig. 9.14. At pH 5 2, the microvalve is in its stress-free state. However, the fluid stream imposes a pressure on the jackets perimeter that induces a stress field inside the hydrogel jackets. The maximum stresses occur at inner

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Figure 9.14 Stress field inside the jackets along with pressure distribution and velocity vectors in fluid flow at different pH 5 2, 5, and 8 for three studied patterns [20].

radius of the jackets in all patterns. The disturbance of the fluid passing the gaps between the jackets at the open state (pH 5 2) are less at US and DS patterns than the IL pattern. The reason is that the US and the DS microvalves conform to the streamlines and are near to the ideal hydrodynamic shape than the IL microvalve.

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Table 9.9 Maximum stress inside the jackets for different patterns at pH 5 2, 5, and 8 [20]. Pattern

IL US DS

Maximum stress (kPa) pH 5 2

pH 5 5

pH 5 8

1.67 1.43 1.44

13.77 13.20 13.46

27.38 26.49 26.59

That means, pressure break is more smoothly in the US and DS patterns as depicted in Fig. 9.14. The tensile stresses existing at the swollen jacket in the area that facing the microvalve inlet are neutralized by the compressive stresses imposed by the external fluid pressure (Fig. 9.14 for all microvalves at pH 5 5 and 8). In IL microvalve, the jackets size is small compared to those of DS and US microvalves, while, approximately the same fluid pressure is applied to the jackets for all patterns. The maximum stress of microvalves in different states is given in Table 9.9. Based on the FSI simulation, it is concluded that the IL microvalve experiences a higher maximum stress among others. Excessive stresses at the interface of the jacket and the pillar increase the possibility of the rupture of the jacket or its separation from the stand, and subsequently the microvalve failure. The bigger size of pillars in the US and DS patterns compared to the IL pattern increases the strength of these microvalves. Therefore US and DS patterns are recommended for their higher durability and strength. For better comparison of the flow rate drop inside the microvalve, the flow rates in the range of pH 5 2
6 are shown for each microvalve in Fig. 9.15. The initial flow rates in the reference state are equal for all patterns as it is a fixed parameter in the design procedure. However, increasing the pH value, the flow rate becomes different for each of the patterns. In IL configuration, the flow rate drops sharply compared to other patterns. In the lower pH values, the flow rate of the DS microvalve is higher than that of the US microvalve, but increasing the pH higher than 4, the DS microvalves flow rate becomes less than that of the US microvalve. The gaps in the DS microvalve are smaller than the US microvalve in pH values higher than 4 (pH 5 5 in Fig. 9.15), since the jackets deformations are different at each microvalve due to the dissimilarity of their pressure distribution. The closing pH value is also different for the microvalves (Table 9.9). The jackets in the US microvalve need to swell more in higher pH to be able to close the microvalve, since its resistance to the fluid stream is low (see Fig. 9.14). Thus for applications that sharp changes in the flow rate are required, the IL microvalve is recommended. However, whereas the microvalves should work in high-cycle processes during their life-time, the DS and US microvalves are more desired candidates, as they have higher strength and lower stress despite having a bit smoother pressure drop. Considering the jackets deformation due to fluid pressure results in lower flow rates at the open state (pH 5 2), in comparison with non-FSI computations. The initial flow rates of the FSI and non-FSI jackets for all microvalves are given in

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Figure 9.15 The flow rate drop for three patterns with increasing pH values at inlet pressure of 1.5 kPa [20]. Table 9.10 The closing pH values and FSI and non-FSI flow rates for each pattern from simulations [20].

Closing pH value Flow rate (non-FSI) (mL/min) Flow rate (FSI) (mL/min) FSI and non-FSI flow rate difference (%)

IL

US

DS

5.25 4.43 4.30 2.93

5.40 4.41 4.32 2.04

5.35 4.41 4.40 0.23

Table 9.10. The DS microvalve has the minimum flow rate change due to FSI simulation and the IL has the highest one. This trend is related to the deformation of the jackets in response to the fluid stream. As observed from Fig. 9.14, at pH 5 2, the area subjected to the excessive stresses inside the jackets in the IL microvalve is bigger than that of the US, and in the US is larger than the DS microvalve. The maximum stress inside the IL jackets is also higher than the others. Consequently, the deformation of the DS microvalve is lower than the others which reduces the flow drop of the DS microvalve at the initial state. In the DS microvalve, considering the FSI simulation has minor impacts on the flow rate. Consequently, the design of DS microvalve without performing FSI simulations (with less computational cost) produces minor errors to the problem in comparison with the two other microvalves. In the closed state, the pressure applied to the jackets is approximately equal to the inlet pressure, as there is no fluid stream inside the microvalve. If we increase

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Figure 9.16 The stress and velocity contours for each microvalve in their leakage pressures (A) inline, (B) upstream, and (C) downstream [20].

the inlet pressure, more force will be transferred to the jackets. Consequently, the gap size between the jackets grows as the hydrogel deforms, leading to the fluid permeation into the gaps between the jackets. There is a point in which the jackets are unable to withstand the pressure and fluid leakage occurs, that means the microvalve leaks. The leakage pressure is near to 11 kPa for all microvalves except the DS pattern, which was near to 10.5 kPa. Fig. 9.16 plots the stress and the velocity contours for each microvalve in their leakage pressures. As shown Fig. 9.16, at the leakage inlet pressure, the fluid seep through the contact areas between jackets while deforming them tremendously. The unstable nature of the leakage phenomenon violates the symmetry of the microvalves at this stage. High stresses are observed especially near to the interface of the jackets and the stands, making this area vulnerable to the rupture. The leakage pressure of the microvalves, which is a critical parameter in microvalve design, cannot be predicted without carrying out FSI simulation.

9.3.3 Transient results of the pH-sensitive hydrogel valve 9.3.3.1 FEM implementation procedure Using the described multiphysics theory in Section 8.2, we aim to examine the time-dependent behavior of a microvalve subjected to pH change and fluid flow effects. The schematic representation of the pH-sensitive hydrogel-based microvalve implanted in the microchannel is depicted in Fig. 9.17. The cross-section of the channel is a rectangle. Therefore we treat the problem as a 2D model. The fluid flows from the left side to the right side, and the microvalve is planted in the center of the channel. We assume the microvalve as a hollow cylinder and fix the inner radius, because in practice a stand is used at the center of the hydrogel to fix the microvalve [8]. The governing equations of the FSI problem of the microvalve are implemented in the multiphysics finite-element package, COMSOL. For the electrochemical

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341

Figure 9.17 Schematic representation of the pH-sensitive hydrogel-based microvalve implanted in the microchannel [28].

field, the “Transport of Diluted Species (TDS)” physics is used which employs the Nernst
Planck equation [29,30] to model the ionic flux. We take advantages of the chemical reaction option of the TDS physics to account for the chemical part of model. The pH of the outside fluid is applied as a concentration boundary condition for the hydrogel in the TDS physics. The “solid mechanics” physics coupled with the “laminar flow” physics is also employed to model the FSI problem. The hyperelastic material model is used and the osmotic pressure is applied as an external stress field, which causes the hydrogel to swell. The osmotic pressure, in turn, is the consequence of the difference of the concentration inside and outside of the hydrogel. Therefore the three used physics are fully coupled together. In COMSOL, there are two approaches for solving nonlinear coupled multiphysics problems: (1) the fully coupled and (2) the segregated methods. The fully coupled approach constitutes a unified large system of equations, which solves all fields within a single iteration including all multiphysics effects at once. However, the segregated approach will not solve all of the fields at once. Instead, it subdivides the problem up into some segregated steps. Then, within a single iteration, the segregated steps are solved sequentially. In this section, the fully coupled and the segregated method for transient and stationary problems are used, respectively. The transient behavior of the microvalve is studied employing a fully coupled FSI simulation. The inner and outer radii of the microvalve are ri 5 10μm and ro 5 25μm, respectively. The material and geometrical parameters of the model are listed in Table 9.11. For the sake of possible contact between the hydrogel and the channel wall at the swollen state, it is assumed that the height of the channel to be 85μm. The initial pH of the hydrogel is 2, and then, the boundary of the hydrogel is subjected to a pH of 12 for 40 s. The normal inflow velocity of 1 mm/s and the 0-G pressure conditions to the inlet and outlet of the channel are applied, respectively. To clearly illustrate the significance of the FSI in simulating soft microvalve, we compare the transient FSI and non-FSI swelling of the microvalve in various times in Fig. 9.18. The upper-half of each plot in Fig. 9.18 shows the FSI results, including the induced von Mises stress contour inside the hydrogel and the fluid pressure contour alongside with the fluid velocity vectors inside the channel. Moreover, the

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Table 9.11 The material and geometrical parameters of the FSI study of the pH-sensitive hydrogel-based microvalve [28] Parameter description

Symbol

Value

Hydrogel mass density Fluid mass density Fluid dynamic viscosity

ρs ρf

1000kg=m3 1200kg=m3 0:001Pa s

H S Rg DH Cfix csalt Ka μ κ

85μm 100μm 25μm 9:3 3 1029 2300mM 200mM 0:01mM 40kPa 272kPa

Channel height Channel width Hydrogel radius Diffusion coefficient of hydrogen ion Concentration of fixed acidic groups Concentration of salt Dissociation constant Hydrogel macroscopic shear modulus Hydrogel bulk modulus

μf ρf

m2 =s

lower-half of each plot in Fig. 9.18 shows the von Mises stress contour of the microvalve during the non-FSI simulation. As time goes on, the hydrogen ions migrate into the hydrogel, and the pH of the hydrogel increases. As the pH approaches the dissociation constant, the fixed acidic groups start to dissociate gradually, which in turn develops an osmotic pressure inside the hydrogel. As a consequence, the hydrogel swells by passing time and gently closes the channel. As shown in Fig. 9.18, at time 5 0.01 s the channel is open and the von Mises stress over the hydrogel is approximately zero except at the outer radius boundary. It is evident that the gradient of the von Mises stress is very high because the hydrogen ions are migrating from this boundary and cause a high osmotic pressure. Due to the Bernoulli’s law in incompressible viscous flows, as the velocity drops the pressure rises, which can induce deformation in the hydrogelbased microvalve. Therefore the FSI and non-FSI results of the microvalve are almost the same until the channel is open, and the fluid pressure is below 2 kPa. When the fluid pressure reaches 10 kPa at t 5 20 s, it produces an asymmetric deformation in hydrogel about the vertical axis. Also, the von Mises stress, which was mainly due to osmotic pressure and was axisymmetric before increasing of the fluid pressure, changes asymmetrically as the fluid pressure grows. The differences between FSI and non-FSI results including the distribution of the von Mises stress, the deformation, and the length and location of the contact between microvalve and the channel wall, clearly demonstrate the importance of the FSI simulation of the pH-sensitive hydrogel-based microvalves. The fluid pressure contour on the right and left side of the microvalve demonstrates that the microvalve has closed the channel perfectly. However, still some leakage is visible, which is due to the offset of 1μm between microvalve and the channel wall. This offset has been set to avoid mesh discontinuity in the fluid medium. To show that the microvalve appropriately closes the channel, the flow

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Figure 9.18 The time-dependent fluid
structure interaction (i.e., upper-half of each plot) and non-fluid
structure interaction (i.e., lower-half of each plot) results including the induced von Mises stress contour inside the hydrogel and the fluid pressure alongside with the fluid velocity vectors inside the channel in various times during the swelling [28].

rate of the channel is plotted during the swelling in Fig. 9.19A. It is revealed that the flow rate reduces ten times in the first 5 s of the swelling. Also, it is shown that the residual leakage due to the offset is negligible. The fluctuation of the flow rate after 20 s is due to the contact chattering which occurs between the microvalve and the channel, which is a purely numerical issue and should be neglected. The most highlighted novelty of is approach over other works in the literature [20] and simulations in previous sections, is its time dependency. The responsetime in microfluidic devices is a vital factor, which cannot be computed in

344

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-2

5 4.5 4 10

-3

3.5 3 2.5 10

-4

0

5

10

15

20

25

30

35

40

2 0

5

10

15

(A)

20

25

30

35

40

(B)

Figure 9.19 (A) The flow rate of the channel during the swelling time. (B) The average pH of the hydrogel during the transient simulation [28]. 35

1.5 1.4

30

1.3 25 1.2 1.1

20

1

15

0.9 10 0.8 5

0.7 0.6 0

5

10

15

20

25

(A)

30

35

40

45

50

0 -25

-20

-15

-10

-5

0

5

10

15

20

25

(B)

Figure 9.20 (A) The radial logarithmic strain on the horizontal radial line of the microvalve for transient and stationary simulations. (B) The contact pressure between microvalve and the channel wall at different times during the swelling [28].

stationary simulations, so time-dependent simulation of the microvalve is a must. The average pH of the hydrogel during the transient simulation is shown in Fig. 9.19B. Although, the pH of the outside fluid is 12, at time 40 s the average pH inside the hydrogel is 5. This occurs because some of the hydrogen ions reversibly bound to the polymer fixed charge, and this phenomenon slows down the swelling kinetics. Actually, at t 5 40 s, the hydrogel is not at its stationary state, but it perfectly closes the channel and this state is of high interest. At the closing state, the pH has a nonuniform distribution over the hydrogel; however, in the stationary simulation, the pH of the hydrogel is assumed to be constant inside the hydrogel [20]. Therefore the closing and stationary states of the hydrogel are different. We compare the radial logarithmic strain on the horizontal radial line of the microvalve for transient and stationary simulations in Fig. 9.20A. In the stationary simulation, we have set a uniform distribution of pH 5 12 for the hydrogel. The results show a huge deviation between the transient closing and stationary states. It should be noted that the transient simulation will reach the stationary state as time goes on. However, the status of the microvalve at the closing state is the key parameter. The contact pressure between the microvalve and the channel wall are plotted for both transient and stationary simulations in Fig. 9.20B, the horizontal axis stands for the spatial x coordinate with the microvalve center at x 5 0. This

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12 11 10 9 8 7 6 5 4 3 0

5

10

15

20

25

30

35

40

45

Figure 9.21 The distribution of pH on the horizontal radial path of the microvalve for transient and stationary simulations [28].

figure implicitly manifests the contact length (i.e., length on the horizontal axis with nonzero pressure). It can be deduced from Fig. 9.20B that the rate of growth of the contact pressure drops as time goes on. This is arising from the nonlinear swelling kinetics of the hydrogel, which is fast at first and then continues slowly. The fluid pressure is responsible for the nonuniform distribution of the contact pressure over the spatial coordinate. Again, this figure demonstrates that the stationary and the transient closing states are completely different. Not only the contact pressure at the stationary state is almost three times that of the closing state, but also the contact lengths of these two states are completely dissimilar. The most important parameter, which controls the swelling kinetics of a pHsensitive hydrogel is clearly the pH; because this parameter determines the magnitude of the dissociation of the acidic group and consequently the osmotic pressure. The pH of the microvalve over the horizontal radial path is plotted in Fig. 9.21 at various times during the transient simulation. In the stationary state, the pH is uniform over the whole geometry of the hydrogel. This figure depicts that the pH is 12 at the outer boundary as it is forced to be. Then, the pH decreases rapidly, which is due to the consumption of the hydrogen ion before the hydrogen ions reversibly bound to the polymer fixed charge. If the hydrogen ions migrate continuously into the hydrogel after a while, there will be no fixed charge, and the hydrogel will be saturated.

9.4

Summary and conclusions

Hydrogels are suitable candidates for sensing and actuating applications arising from their inherent ability to respond to the environmental stimulus. Employing

346

Computational Modeling of Intelligent Soft Matter

them as microvalves is one of these applications in microfluidic systems. In this chapter, we explored the FSI stationary swelling behavior of temperature-sensitive, and pH-sensitive hydrogel-based microvalves. Then, we dealt with the FSI transient behavior of a pH-sensitive hydrogel-based microvalve. Since in most previous studies, the fluids effect is neglected in designing of the hydrogel valves, in this study, we examined the effect of performing the FSI in a T-junction flow sorter with one valve in each branch’s gate. The results disclosed that the FSI has a noticeable effect on valves design parameters, such as valve stresses, closing temperature, and fluid flow rate. Besides, at a higher inlet flow pressure, the impact of FSI was more significant, as well. We found that the swelling rate is fast at first and then the hydrogel keeps on swelling very slowly, because the hydrogen ions can reversibly bond to the polymer fixed charge as they diffuse into the hydrogel and this slows down the saturation of hydrogen ions. Therefore the FSI simulation was found to be a powerful and essential tool for better understanding of the real behavior and accurate design of the microfluidic devices made of soft materials.

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Planck equations, Analytica Chimica Acta 1158 (2021).

Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively. A Aerospace application, 3
4 Agriculture, 9 Analytical solution, 293
298, 330
332 Applications of shape memory polymers aerospace application, 3
4 application in different fields, 3t biomedical applications, 2
3 3D laser printing, 3 suturing success, 3 thermomechanical response of system, 2 vascular and surgery applications, 2 elastic strain, 2 Arbitrary thermodynamic processes, 132 Arrhenius equation, 145
146 Arruda-Boyce model, 36
37 Automotive application, 4 B Baghani et al.’s model, 47
48, 49f Bending test, 104 Biomedical applications, 2
3 3D laser printing, 3 suturing success, 3 thermomechanical response of system, 2 vascular and surgery applications, 2 Biosensors, 9 Boltzmann constant, 275 Boltzmann superposition theory, 174
175 C Cauchy
Green deformation tensor, 271
272 Cauchy stress, 316 CitNetExplorer’s analytic software, 58 Clausius
Duhem inequality, 47
48, 132 Close-close state, 322
323

Cohesive zone model, 166
168 Cohesive zone parameters, 165 Coiled carbon nanotube-reinforced shape memory polymer nanocomposites addition of nano and microparticles, 173 based on thermo-visco elasticity, 174
178 Boltzmann superposition theory, 174
175 isothermal process, 177 material parameters of SMP and nanoparticles, 176t Maxwell-Wiechert, 174
175 Monte-Carlo technique, 178f neo-Hookean strain energy function, 175 representative volume element construction, 177
178 shear modulus equation, 175 time-temperature superposition principle (TTSP), 175 total deformation gradient tensor, 174 coiled carbon nanotube particle, 173f finite element model, 178
182 coiled carbon nanotube, 181f definition of boundary condition, 179f determination of representative volume element size, 181
182 force recovery versus time for four sizes of representative volume element, 182f initial and boundary conditions applied to the RVE, 180t thermomechanical cycle, 179f geometrical parameters and volume fraction, 173 helix-shaped graphite nanotubes, 173 numerical homogenization of, 173
187

350

Coiled carbon nanotube-reinforced shape memory polymer nanocomposites (Continued) numerical results and discussion, 182
187 effect of heating rates and prestrain, 185
187 effect of prestrain and heating rate, 186f Monte-Carlo algorithm, 187 orientation of coiled carbon nanotube, 185, 185f pitch of coiled carbon nanotube, 183
185, 184f spring length of coiled carbon nanotube, 183, 184f thermomechanical properties of SMPC, 183f, 187 thermo-visco-hyperelastic constitutive equation, 187 volume fraction of coiled carbon nanotube on SMPC, 182
183 SMP/CCNT nanocomposite, 174 Consistent tangent matrix, 137
138 Constitutive equations, for SMP, 161
163 first order rule of mixture, 163 heating process, 163 small strain constitutive model, 162 stored strain, 162
163 stress-strain-temperature, 161f typical thermomechanical cycle, 161
162 Constitutive modeling of pH-sensitive hydrogels anionic, cationic, and ampholytic pHsensitive hydrogels in various pH values, 247f applications of, 249 classification of hydrogels, 246, 247f experiments, 260
262 hydrogel, 246f introduction, 245
248 ionic hydrogels, 247
248 numerical implementation, 258
260 polymeric network of hydrogels, 245 predominant keywords, 248f swelling/deswelling phenomena, 250
255 swelling theories, 248, 255
258 Constitutive models for thermoresponsive shape memory polymers chemical structure of, 20
21

Index

classification of temperature-dependent polymers, 18
20 cooling process, 16 cyclic behavior view, 16 deformation and cooling the SMP, 17 empirical testing and mathematical modeling, 16 modeling thermoresponsive, 29
57 researches on, 18 shape memory polymers and their classification, 20
28 SMP programming, 16 statistical analysis of available, 58
59 stress-free-strain-recovery and fixedstrain-stress-recovery processes, 17f stress-strain-temperature, 17f structure and applicability of SMPs, 18 thermally activated, 16 thermomechanical cycle, 17f Control of microfluidic flow, 249 Coupled fields in fluid-structure interaction modeling, 320 D Darcy’s law, 253 Debye length, 253, 281
282, 282f Different constitutive models’ description Cauchy stress, 316 coupled fields in fluid-structure interaction modeling, 320 Flory-Huggins model, 315 gel and large deformations, 315 Helmholtz free energy, 315 neo-Hookean model, 315 Piola-Kirchhoff stress tensor, 314
315 stationary swelling theory for pH-sensitive hydrogels, 316
319 ALE coordinate system, 319 chemical and mechanical constrains, 318 Donnan equations, 318 equilibrium equation in terms of acidic dissociation, 318
319 ions’ energy, 317
318 neo-Hookean model, 317
318 swelling theory of thermal-sensitive hydrogels, 314
316 transient theory for pH-sensitive hydrogels, 320

Index

electrochemical field and mechanical field, 320 Differential scanning calorimetry, 80
81, 81f, 108
109 Divergence theorem, 283 Donnan equations, 277
278, 318 Downstream (DS) pattern, 326
327 Drug delivery, 6
7, 249 Dynamic mechanical analysis (DMA) test, 303 Dynamic-mechanical thermal analysis, 109
114 E Elastic memory composite hinge, 4 Electrical field, 252
253 Equilibrium and transient swelling of soft and tough pH-sensitive hydrogels acidic dissociation, 273
274 chemical potential of the co-ion, 276
277 constitutive laws for network, 274 Donnan equations, 277
278 equilibrium thermodynamically consistent theory, 271
278 external solvent molecules and ions, 271
272 free swelling of a square hydrogel, 301f test, 305f ill-posed logarithmic term, 275
276 Lagrange multiplier and its elimination, 274
275 mathematical manipulation, 277 numerical solution procedure, 286
289 pH/temperature sensitive hydrogel, 272f references state and its boundary, 272
273 results and discussion, 289
308 analytical solution, 293
298 chemical parameters, 299t constrained swelling, 302
303 crosslinking density, 290 current element radius, 294 cyclic loading-unloading curves, 306f deformation gradient tensor in uniaxial test, 303 dynamic mechanical analysis (DMA) test, 303 equilibrium swelling, 289
293

351

experimental and numerical tensile stress-stretch curve, 304f experimental data, 290
291 experiment versus numerical simulation, 300f FEM and analytical results, 299f free energy expressions, 294 free swelling, 301
302 inhomogeneous deformations, 293 inhomogeneous swelling of a pH/ temperature sensitive hydrogel shell, 296f inhomogeneous swelling of a spherical shell, 295f material parameters, 289t, 294
295 material properties, 299 microfluidic pH-sensitive hydrogelbases microvalve, 302f neutral PNIPAM hydrogel, 289
290, 291f normalized outer radius, 294
295 numerical implementation, 296
298 presented model, 291
293, 292f radial stress contour (left side) and tangential stress contour, 307f spherical pH/temperature sensitive hydrogel, 297f stress-stretch cyclic tensile loadingunloading curves, 306f stress-stretch tensile curve, 307f test setup, 298f theoretical and experimental stress data, 303
304 transient swelling response of pHsensitive hydrogels, 298
303 user-defined subroutine in ABAQUS, 289 visco-hyperelastic constitutive model for tough pH-sensitive hydrogels, 303
308 visco-hyperelastic parameters obtained by fitting, 305t swelling constraint, 274
275 transient electro-chemo-mechanical swelling theory, 279
286 Equilibrium thermodynamically consistent theory, 271
278 acidic dissociation, 273
274 Boltzmann constant, 275

352

Equilibrium thermodynamically consistent theory (Continued) Cauchy-Green deformation tensor, 271
272 co-ion and counter-ion concentrations, 278 constitutive laws for network, 274 external solvent molecules and ions, 271
272 free energy change of the system, 273 free energy density and constitutive laws, 276 Lagrange multiplier, 274
275 mathematical manipulation, 277 network and the external solution, 273 pH/temperature sensitive hydrogel, 272f PNIPAM hydrogel, 275 Empirical testing and mathematical modeling, 16 Euler-Bernoulli beam theory (EBBT), 188, 192
193 Euler time discretization scheme, 288 Experimental data, 290
291 Experiments of pH-sensitive hydrogels four cylindrical, 260
261 pH-sensitive cylindrical hydrogel, 261f Poisson’s ratio, 260 in a rigid glass channel, 261f swelling and deswelling experiment, 260 swelling response of, 260 transient swelling and deswelling behavior, 261
262, 262f Young’s modulus, 260 Experiments on shape memory polymers glass transition temperature, 77 investigation on structural design of, 90
104 methods of, 78
89 stent as an application, 104
121 F Finite difference method, 192
193 Finite element analysis, 229
233 determination of elements size, 231
233 final 3D periodic cell, 230
231 mesh dependency analysis for the auxetic structure, 233f representative volume element structures, 232f

Index

sample unit cell, 232f structured mesh, 233f thermomechanical programming of, 232t Finite element model, 178
182 coiled carbon nanotube, 181f definition of boundary condition, 179f determination of representative volume element size, 181
182 force recovery versus time for four sizes of representative volume element, 182f initial and boundary conditions applied to the RVE, 180t thermomechanical cycle, 179f Fixed-strain-stress-recovery process, 16 Flory-Huggins model, 315 Fluid-structure interaction modeling results flow chart of the fluid-structure interaction simulation procedure, 321f fluid pressure distribution, 325f fluid-structure interaction analysis of temperature-sensitive hydrogel valves, 320
328 hydrogels characteristics, 322
326 inlet flow rate in the open-close state for different valves patterns, 328t inlet fluid flow rate, 325f left outlet closing temperature in the close-close state, 326t left outlet closure temperature for various valve patterns, 329t multimicrovalves in a channel, 326
328 multivalves’ pattern configurations, 327f outlet fluid flow rate, 322t results for stationary response of pHsensitive hydrogels, 329
340 analytical solution, 330
332 closing pH of the microvalve for different inlet pressures, 334t closing pH values and FSI and non-FSI flow rates, 339t equilibrium equation in polar coordinate, 329 flow rate drop for three patterns, 339f flow rate drop versus the pH for inlet pressure, 334f fluid-structure interaction, 329, 333f FSI and non-FSI simulations, 335t

Index

geometry of the three jacket microvalve patterns, 336f material parameters of pH-sensitive hydrogel, 328t maximum stress inside the jackets, 338t one hydrogel jacket microvalve fluidstructure interaction, 332
335 outer radius of hydrogel jacket valve, 331f pressure distribution and velocity vectors, 337f radial and hoop stress distribution, 331f stress and velocity contours, 340f three jackets’ microvalve fluid-structure interaction, 335
340 right valves contact length, 325f T-junction flow sorter with one valve in each entrance, 324f transient results of the pH-sensitive hydrogel valve, 340
345 average pH of the hydrogel during the transient simulation, 344f contact pressure between microvalve and the channel wall, 344f FEM implementation procedure, 340
345 flow rate of the channel during the swelling time, 344f "laminar flow" physics, 341 material and geometrical parameters of the FSI study, 342t microvalve for transient and stationary simulations, 345f pH-sensitive hydrogel-based microvalve implanted in the microchannel, 341f radial logarithmic strain, 344f "solid mechanics" physics, 341 time-dependent fluid-structure interaction, 343f two hydrogels stretch versus the temperature change, 323f von Mises stress and pressure distribution in inline pattern, 328f von Mises stress contour in one of two valves, 324f zero-gage pressure, 323
324 Force recovery, 96

353

G Graphene nanoplatelets (GNPs), 160
161, 165t H Helmholtz free energy, 315 density function, 132 of hydrogels, 313
314 Homogenization of shape memory polymer nanocomposites constitutive equations, 161
163 first order rule of mixture, 163 heating process, 163 small strain constitutive model, 162 stored strain, 162
163 stress-strain-temperature, 161f typical thermomechanical cycle, 161
162 3D modeling and numerical considerations, 163
166 bilinear traction-separation law, 165 cohesive zone model parameters, 166t cohesive zone parameters, 165 computational costs and modeling complexities, 163 3D cubic RVE, 164 elastic constants of graphene nanoplatelet, 165t FEM model, 164 material parameters of SMP modeling, 165t size dependency checking simulations, 164f SMP matrix, 164 graphene nanoplatelets (GNPs), 160
161 modeling and, 160
173 numerical results, 166
173 cohesive zone model, 166
168 dimension and geometry of inclusions, 166 elastic modulus, 172f elastic property and recovery stress of SMP nanocomposite, 172t FEM simulation, 170 fixed-strain stress recovery cycles, 171f graphene nanoplatelets/shape memory polymer nanocomposite system, 168f imperfect GNP/SMP interface, 171
173

354

Homogenization of shape memory polymer nanocomposites (Continued) reconstructed RVE, 170 representative volume element size, 167f SMP nanocomposite, 170 representative volume elements (RVEs), 160
161 SMP matrix, 160
161 ultrasonic and static tests, 160
161 Hughes-Winget algorithm, 138
139 Hydrogels and agriculture, 9 and biosensors, 9 and cancer therapy, 8 and contact lens products, 8
9 and hygiene products, 9
10 and water treatment, 8 for wound dressing, 7
8 I Inhomogeneous deformations, 293 Inline (IL) pattern, 326
327 Intelligent soft matters applications of shape memory polymers, 2
5 automotive application, 4 breathable and waterproof intelligent textile, 5f calibration methods, 1
2 design and analysis of intelligent soft matters, 10
12 4D-printed scaffolds fabricated, 5f free recovery measurement, 5f other applications, 4
5 self-deployment origami 4D-printed honeycomb sandwich structure, 5f smart hydrogel applications, 5
10 versatile gripper, 5f Isothermal process, 177 K Kovacs-Aklonis-Hutchinson-Ramos (KAHR) model, 34
35 L Lagrangian multiplier, 144, 151 Linear viscoelastic model, 32f, 160

Index

M Macroscopic phase transition approach consistent tangent matrix, 137
138 constitutive models, 129 equivalent representative volume element, 129f extension of the model to the time dependent regime, 133
135 convex free-energy density function, 134
135 fixed-strain stress recovery, 133 Helmholtz free-energy density functions, 134
135 phenomenological or macromodeling approach, 133 proposed constitutive model, 134f shape memory polymer and hard segments, 133f SMP segment, 133 global mechanical response, 128 Hughes-Winget algorithm, 138
139 material model predictions, 141
143 fixed-strain stress recovery, 141 multiaxial loading path, 148 shape memory effect, 141f SMP-based syntactic foam sample, 141 stress-free strain recovery, 143f material parameters identification, 139
140 adopted for experiments, 140t characteristic temperatures, 139 curve fitting method, 139
140 elastic moduli and viscosity coefficients, 140 free-stress strain recovery process, 140f thermal strain, 139 volume fractions, 139 numerical solution of the constitutive model, 135
137 fourth-order positive definite tensors, 136 time interval of interest, 136 representative volume element (RVE), 129
130 shape memory effect, 128
129 strain storage and recovery, 130
131 convex free-energy density function, 131 cooling process, 130

Index

heating process, 130
131 Helmholtz free-energy density functions, 131 proposed constitutive model, 131f strain storage, 130 stress-free strain recovery, 128
129 stress-strain-temperature, 129f thermodynamic considerations, 132
133 arbitrary thermodynamic processes, 132 Clausius-Duhem inequality, 132 Helmholtz free energy density function, 132 thermomechanical cycle, 128
129 MathMod software, 213 Maxwell thermoviscoelastic model, 37
38 Maxwell-Wiechert model, 40 Mechanical metamaterials, 210 Melting temperature of PCL crystals, 81 Methods of shape memory polymer production additive manufacturing, 79 characterization in combined torsiontension loading, 79
89 angle recovery process, 85f, 86, 88f angle recovery versus temperature of polyurethane/poly, 89f biomedical applications, 84
85 combined tension-torsion shape recovery process, 87f differential scanning calorimetry, 80
81, 81f dynamic mechanical thermal analysis, 81
83, 83f heating rate on RST and RFT of PU/ PCL, 87
89 low-stress recovery, 83
84 materials, 80 normalized angle recovery versus temperature of polyurethane/poly, 88f polyurethane/poly solution, 81f recovery of prestretch versus temperature, 89f sample preparation, 80 shape memory behavior, 83
89 shape recovery versus temperature, 86f stress recovery response, 84f, 85f tensile test and maximum stress recovery, 84f thermo-viscoelastic behavior, 82f

355

melt mixing, 78 solution mixing, 78
79 Microfluidic valves, 7 Modeling thermoresponsive shape memory polymers constitutive models in engineering applications, 29
57 conventional thermally activated shape memory polymers, 30
54 Arruda-Boyce model, 36
37 Baghani et al.’s model, 49f based RVE, 45f constitutive models, 40
54 creep and creep recovery of the proposed model, 31f crystalline SMPs, 53 3D constitutive model, 51
52 3D finite deformation two-phase constitutive model, 50
51 3D finite strain rheological model, 39
40 3D finite strain thermoviscohyperelastic model, 36
37 3D model for application, 48 first constitutive model for, 31 first nonlinear 3D model, 34 frozen and crystalline volume fraction functions, 52t governing equation, 32
33 Helmholtz potential approach, 33
34 high-temperature tensile or compression test, 37
38 hyperelastic part of model, 38 Kovacs-Aklonis-Hutchinson-Ramos (KAHR) model, 34
35 linear viscoelastic model, 32f Maxwell thermoviscoelastic model, 37
38 Maxwell-Wiechert model, 40 model SMP nanocomposites, 36 Mooney-Rivlin model, 47 nonlinear three-element model, 51 3-parameter rheological model, 30f 4-parameter rheological model, 31f phase-transition-based phenomenological constitutive model, 44
46 Polyisoprene-based amorphous, 54 quite comprehensive 3D model, 34
35

356

Modeling thermoresponsive shape memory polymers (Continued) rheological models based on viscoelastic modeling approach, 41t rheological scheme, 34f rubbery phase, 50 semicrystalline PE-based, 53 SMP-based matrix, 51
52 strain 3D mechanical model, 49 strain recovery prediction of model, 46 stress recovery, 32
33 styrene-based thermoset resin, 39 and syntactic foam-based, 39 temperature- and rate-dependent constitutive model, 50 testing and calibration, 35 thermo-mechanical coupling theory, 35
36 under thermoviscoelastic approach, 30
40 three-element linear model, 33 three-element rheological model, 33 three nonequilibrium branches, 47
48 time-dependent effects, 48 Tool
Narayanaswamy
Moynihan model, 34
35 UMAT subroutine, 46
47 Veriflex-based, 46
47 viscoelastic models, 44 Voigt model, 47 Williams
Landel
Ferry (WLF), 36
37 polymer behavior, 30 thermally activated, 55
57 multibranch equilibrium and nonequilibrium elements, 55 presented models on multi-SMP, 56t series and parallel phases arrangement, 56 UMAT subroutine, 55 two-way thermally activated, 54
55 Molecular structure of shape memory polymers chemical structure of, 20
21 amorphous and crystalline states, 20 in amorphous SMPs, 21 entropic conformational motions, 21 glass transition temperature, 20 thermoplastic and thermoset, 21

Index

classification of, 21
28 acrylate polymer network test, 22 conventional, 22 Diglycidyl ether bisphenol A epoxy monomer, 27
28 experimental data, 25f free-standing two-way shape memory behavior, 25 in literature and its characteristics, 23t multilayer epoxy L- and H-based, 29f multishape memory polymers, 26
28, 28t temperature memory effect, 26
27 thermomechanical cycles, 28f two-way behavior, 26, 26f two-way property, 25 two-way shape memory polymers, 22
26, 26t and their classification, 20
28 Monte-Carlo technique, 178, 178f, 187 Mooney-Rivlin model, 47 N Neo-Hookean model, 315, 317
318 Neo-Hookean strain energy function, 175 Nernst-Planck equation, 256, 279
280 Net cooling history, 189 Nonlinear three-element model, 51 Numerical implementation, 258
260, 296
298 finite element method, 260 finite element simulation of hydrogel swelling, 258
260 user-defined heat transfer, 258
260 Numerical modeling in design and analysis of intelligent soft matters, 10
12 constitutive modeling, calibration, and simulation of SMPs, 10 constitutive models of pH-sensitive hydrogels, 11 equilibrium and transient swelling, 11 experimental aspects of SMPs’ response, 10 fluid-structure interaction (FSI), 11
12 metamaterials, 11 nanocomposites and corrugated structures, 10
11 thermomechanical response of SMPs, 10

Index

Numerical solution procedure development of time and space discretization, 287
288 Euler time discretization scheme, 288 development of weak form, 287 electro-chemo-mechanical swelling theory, 286
287 Lagrangian formulation, 286
287 residuals and tangent moduli, 288
289 displacement and hydrogen ion concentration, 288
289 element-level set of equations, 288 O Open-close state, 322
323 P Partial stress recovery regime, 198 pH-sensitive hydrogels, applications of control of microfluidic flow, 249 drug delivery, 249 hydrogel bi-layered structure, 249f soft actuators, 249 Phase-transition-based phenomenological constitutive model, 44
46 Piola-Kirchhoff stress tensor, 190, 285, 314
315 Pitch of coiled carbon nanotube, 183
185, 184f Poisson’s ratio, 222f, 227
228, 260 of FE simulation and topology optimization for PPR and NPR, 235t Polyisoprene-based amorphous, 54 Polymer functionalization, 16 R Recovered force, 99 Representative volume elements (RVEs), 130
131, 133, 160
161 Residuals and tangent moduli, 288
289 displacement and hydrogen ion concentration, 288
289 element-level set of equations, 288 S Shape memory effect, 114
116, 128
129, 141f and force generation, 198 Shape memory polymer composites

357

analytical and numerical simulations, 160 carbon and alumina nanoparticles, 160 characteristics of, 159
160 modeling and homogenization, 160
173 numerical homogenization of coiled carbon nanotube-reinforced, 173
187 temperature sensitive material, 159 thermomechanical behavior of, 187
205 Shape memory polymer constitutive model, 144
155 Shape memory polymer metamaterials based on triply periodic minimal surfaces, 209
225 finite element analysis of, 229
233 determination of elements size, 231
233 final 3D periodic cell, 230
231 mesh dependency analysis for the auxetic structure, 233f representative volume element structures, 232f sample unit cell, 232f structured mesh, 233f thermomechanical programming of, 232t materials and methods, 212
216 acrylate network composition, 215 behavior of, 216f 3D model of primitive, 214f generation and modeling of microstructures, 212
213 geometry generation, 213
216 MathMod software, 213 parameters used for generating TPMSbased structures, 214t shape recovery and force recovery processes, 215
216 TPMS-based structures, 212 using thermovisco-hyperplastic model, 213
216 mechanical metamaterials, 210 nature-based TPMS diamond and primitive topologies, 210
211 nature-inspired unit cells, 211f optics and electromagnetism fields, 210 results and discussion, 217
225, 234
238 bone defect, 225

358

Shape memory polymer metamaterials (Continued) force recovery, 219
220, 221f loading type on negative Poisson’s ratio structure, 235
237, 237f mechanical properties, 220
223 nondimensional elastic modulus, 222f Poisson’s ratio, 222f Poisson’s ratio of FE simulation and topology optimization for PPR and NPR, 235t positive Poisson’s ratio and negative Poisson’s ratio structures, 234
235, 234f potential application areas, 223
225 prestrain on negative Poisson’s ratio structure, 236f prestrain on the response of negative Poisson’s ratio structure, 235 self-fitting shape memory polymer, 224f shape fixity, 219, 220f shape recovery, 218
219, 218f temperature rate on negative Poisson’s ratio structure, 237
238, 238f thermomechanical cycle, 217f SMP-based liquid resins, 211 SMP-based metamaterials, 211
212 strut-based open-cell lattice structures, 210 TPMS structures, 210 Shape memory polymer nanocomposites modeling and homogenization of, 160
173 Shape memory polymer path, 144
155 applying the desired deformation, 146 constraint removal, 147 shape and force recovery, 147
148 temperature history in an SME path, 146f temporal shape fixation, 146
147 Shape memory polymers axial deformation and rotation angle, 155 axial strain and rotation angle, 155f force recovery and moment recovery curves, 155f force recovery response, 154
155 macroscopic phase transition approach, 128
143 major conventional methods for, 128

Index

material parameters adopted for experiments, 152t, 153t mesoscale method and molecular dynamics, 127 shape and force recovery (reheating step), 147
148 SME for the torsion-extension, 154t through thermo-viscoelastic approach, 144
155 torsion-extension problem, 151 torsion M and extension N, 154f Shape memory polymer stent as an application fabrication method, 104
105 materials, 105
107 blending and forming, 105
107 hot press process, 106
107 melt-mixing method, 106
107, 106f polyurethane as hard segment, 105f solution-mixing method, 106
107 Young’s modulus, 105 stent fabrication, 107
116 angle recovery results, 114
115, 115f cylindrical molding, 107 differential scanning calorimetry, 108
109 dimensional and physical specifications of fabricated stents, 107t DMTA testing, 111
114 dynamic-mechanical thermal analysis, 109
114 fabricated stent after removal from the mold, 108f fabrication process, 108f force/stress recovery, 114 loss modulus of polyurethane/poly, 111f melt-mixing method, 112f pasting process, 107 scanning calorimetry result of polyurethane, 108f, 109f shape memory effect, 114
116 solution-mixing method, 111
114 storage and loss modulus of polyurethane, 113f storage modulus, 110f, 113f storage modulus, loss modulus and tan delta, 109
110 stress recovery of polyurethane/poly, 116f

Index

stress recovery test, 114, 114f stent radial compression, 116
121 force-displacement graph, 116
118 force recovery and RST, 120
121 force versus normalized displacement, 118f loading process, 116
118 radial compression test, 119f radial force recovery for tubular stent versus temperature, 121f radial stiffness, 118f stent force recovery, 120
121, 120f stent radial compression, 117f Shape recovery, 96, 217 Shear modulus equation, 175 Smart auxetic 3D metastructures computational material distribution method, 226 design and manufacturing of a deployable antenna, 225
226 geometrical modeling of an representative volume element, 227
228 history-dependent materials, 227
228 initial guess of topology to generate unit cells, 230f optimality criteria algorithm (OCA), 227 Poisson’s ratios, 227
228 solid isotropic material with penalization methodology, 229f unit cells, 227
228, 231f Young’s modulus of void phase, 227 numerical investigation of, 225
238 programming path of an SMP material, 226 stiffness and low recovery stress, 225
226 stress-strain-temperature, 227f Smart hydrogel applications breathable and waterproof intelligent textile, 6f 4D-Printed honeycomb sandwich structure, 6f drug delivery, 6
7 free recovery measurements, 6f hydrogel and cancer therapy, 8 hydrogels and agriculture, 9

359

and biosensors, 9 and contact lens products, 8
9 and hygiene products, 9
10 and water treatment, 8 for wound dressing, 7
8 microfluidic valves, 7 tissue engineering, 6 Smart hydrogels, 313 SMP. See Shape memory polymers Soft actuators, 249 Spring length of coiled carbon nanotube, 183, 184f Stationary response of pH-sensitive hydrogels, 329
340 analytical solution, 330
332 closing pH of the microvalve for different inlet pressures, 334t closing pH values and FSI and non-FSI flow rates, 339t equilibrium equation in polar coordinate, 329 flow rate drop for three patterns, 339f flow rate drop versus the pH for inlet pressure, 334f fluid-structure interaction, 329, 333f FSI and non-FSI simulations, 335t geometry of the three jacket microvalve patterns, 336f material parameters of pH-sensitive hydrogel, 328t maximum stress inside the jackets, 338t one hydrogel jacket microvalve fluidstructure interaction, 332
335 outer radius of hydrogel jacket valve, 331f pressure distribution and velocity vectors, 337f radial and hoop stress distribution, 331f stress and velocity contours, 340f three jackets’ microvalve fluid-structure interaction, 335
340 Stationary swelling theory for pH-sensitive hydrogels, 316
319 ALE coordinate system, 319 chemical and mechanical constrains, 318 Donnan equations, 318 equilibrium equation in terms of acidic dissociation, 318
319 ions’ energy, 317
318 neo-Hookean model, 317
318

360

Statistical analysis of available shape memory polymer models amorphous and semicrystalline, 59 citation network and connections of core publication, 58f CitNetExplorer’s analytic software, 58 constitutive equations. P.T. and T.V. mean phase transition, 60f micromolecular types and strain range, 59 Stored strain, 162 Structural analysis of different smart hydrogel microvalves different constitutive models’ description, 314
320 Helmholtz free energy of hydrogels, 313
314 results and discussion, 320
345 soft and highly deformable, 314 Structural design of shape memory polymers characterization of printed material, 94 force recovery in bending and tensile tests, 96f and preforce values, 103t testing, 99
104 investigation on, 90
104 method of sample production, 92
93 additive manufacturing, 92
93 3D printer settings, 93t effect of 3D printing, 93 by FDM method, 93f material, 92 nonlinear equations and definite constraints, 90 shape recovery in bending and tensile tests, 96f process for beam III, 99f testing, 96
99 structural (geometrical) design, 90
92 2D schematic of beams I, II, and III geometrical cells, 91f nonlinear equations, 90
91 smart polymers, 90 system of nonlinear equations, 92 thermomechanical shape memory tests, 94
104 bending and tensile, 95, 98, 98t, 101
103 bending and uniaxial tensile testing, 95
96

Index

bending test, 104 cooling and heating stages, 103
104 cycle of the three-point bending test, 100f effect in water, 95
96 force-temperature, 99
100 principal strain, 98
99 programming cycle to shape recovery process, 97f shape recovery versus the temperature, 98f tensile test, 104 three-point bending thermomechanical test, 102f uniaxial tensile test, 101f uniaxial tensile thermomechanical test, 102f Swelling/deswelling phenomena, 250
255 anionic and cationic pH-sensitive hydrogels, 252 chemical reaction, 250
252 conservation of mass, 250 continuum theory, 253
254 Darcy’s law, 253 Debye length, 253 electrical field, 252
253 external solution and the osmotic pressure, 254 fluid flow, 253 free-energy, 255 ionic groups and fixed charges, 251
252 ion transfer, 252 mechanical field, 253
255 pH-sensitive hydrogel, 251f Poisson’s equation, 253 thermodynamically-consistent theories, 255 thermodynamics laws, 254 Swelling theories, 248, 255
258 monophasic models, 256
258 monophasic swelling theories for pHsensitive hydrogels, 257t multiphasic models, 258 swelling theories for pH-sensitive hydrogels, 259t theory, 258 Nernst-Planck equation, 256 osmotic pressure, 256 thermodynamically consistent monophasic theories, 256

Index

T Tangent matrix, 137 Temperature-dependent polymers amorphous and crystalline, 18
19 classification of, 18
20 effect on thermoset and thermoplastic polymers, 19
20 shear modulus, 19f thermoset and thermoplastic polymers, 18
19, 19f Textile application, 4 Thermomechanical behavior of shape memory polymer beams active and frozen phases, 189 bending of a reinforced, 192
196 developing formulation of, 192 different corrugated patterns, 193f EBBT, 192
193 finite difference method, 192
193 heating process in present theory, 195
196 symmetric and asymmetric sections, 194f carbon fiber fabric reinforcement, 188 constitutive model for SMPs, 188
189 corrugated sections with equal height, 204t with equal width, 204t equal SMP content and height, 200t content and width, 200t Euler-Bernoulli beam theory (EBBT), 188 numerical results and discussion, 196
205 curvature-temperature diagrams, 199
200 fixed end curvature in a cantilever beam, 197f force generated due to mechanical constraint in heating process, 203f geometry and the loading conditions of the investigated composite beam, 196f mechanical properties of composite beam constituents, 197t output results and the contours of the deformation of the composites beams, 201f

361

shape memory effect and force generation, 198 SMP-based actuators, 179f thermomechanical cycle, 196
197, 199f three-dimensional interaction study, 204
205 on phase transition concept, 189
192 Piola-Kirchhoff stress tensor, 190 reinforced by corrugated polymeric sections, 187
205 SMP-based actuators, 188 thermoelastic constitutive equation, 189 Thermomechanical shape memory tests, 94
104 bending and tensile, 95, 98, 98t, 101
103 bending and uniaxial tensile testing, 95
96 bending test, 104 cooling and heating stages, 103
104 cycle of the three-point bending test, 100f effect in water, 95
96 force-temperature, 99
100 principal strain, 98
99 programming cycle to shape recovery process, 97f shape recovery versus the temperature, 98f tensile test, 104 three-point bending thermomechanical test, 102f uniaxial tensile test, 101f uniaxial tensile thermomechanical test, 102f Thermo-viscoelastic approach elastic and viscous properties, 144 material parameters identification, 149
150 constitutive equations, 149
150 temperature effect, 149
150 shape memory polymer constitutive model, 144
155 in as shape memory polymer path, 144
155 applying the desired deformation, 146 constraint removal, 147 shape and force recovery, 147
148

362

Thermo-viscoelastic approach (Continued) temperature history in an SME path, 146f temporal shape fixation, 146
147 strain-dependent part of the stress, 144 hook’s law, 144 Lagrangian multiplier, 144 temperature-dependent modification of the stress, 145 Arrhenius equation, 145
146 total stress, 145
146 time-dependent part of the stress, 144
145 storage modulus, 145 storage modulus and Prony series, 144
145 time-discretization scheme for constitutive equations, 148
149 generic time interval, 148 TTSP equation, 149 unloading and shape recovery stage, 149 torsion-extension of SMP, 150
155 incompressibility constraint, 150 Lagrange multiplier, 151 nonzero equilibrium equation, 151 Thermovisco-hyperplastic model, 213
216 3D constitutive model, 51
52 3D finite deformation two-phase constitutive model, 50
51 3D finite strain rheological model, 39
40 3D finite strain thermo-viscohyperelastic model, 36
37 3D model for application, 48 3D modeling and numerical considerations, 163
166 bilinear traction-separation law, 165 cohesive zone model parameters, 166t cohesive zone parameters, 165 computational costs and modeling complexities, 163 3D cubic RVE, 164 elastic constants of graphene nanoplatelet, 165t FEM model, 164 material parameters of SMP modeling, 165t

Index

size dependency checking simulations, 164f SMP matrix, 164 Time-temperature superposition principle (TTSP), 175 Tissue engineering, 6 Tool
Narayanaswamy
Moynihan model, 34
35 Total deformation gradient tensor, 174 Transient electro-chemo-mechanical swelling theory, 279
286 chemical field, 279
281 dissociation constant, 281 Nernst-Planck equation, 279
280 polymer network, 280 polymer network of pH-sensitive hydrogel, 280
281 continuity of ions, 282
283 divergence theorem, 283 electro-neutrality condition, 283 electrostatic field, 281
282 Debye length, 281
282, 282f Poisson’s equation, 281 vacuum and hydrogel, 281 initial and boundary conditions, 286 large deformation theory, 279 concentration of species, 279 continuum theory, 279 pH-sensitive hydrogel, 280f mechanical field, 283
286 elastic part of energy, 284 energy sources, 283
284 first and second laws of thermodynamics, 284 Piola-Kirchhoff stress, 285 second Piola-Kirchhoff stress, 285
286 thermodynamics laws, 284 total free energy of the hydrogel, 283
284 osmotic pressure, 279 pH-sensitive hydrogels, 279 Transient swelling response of pH-sensitive hydrogels, 298
303 Triply periodic minimal surfaces and auxetic structures numerical investigation of smart auxetic 3D metastructures, 225
238 shape memory polymer metamaterials based on, 209
225

Index

V Visco-hyperelastic constitutive model for tough pH-sensitive hydrogels, 303
308 von Mises stress and pressure distribution in inline pattern, 328f contour in one of two valves, 324f

363

W Water treatment, 8 Williams
Landel
Ferry (WLF), 36
37 Y Young’s modulus, 105, 260 of void phase, 227