Constructional Viscoelastic Composite Materials: Theory and Application 9819917859, 9789819917853

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Pouria Hajikarimi Alireza Sadat Hosseini

Constructional Viscoelastic Composite Materials Theory and Application

Constructional Viscoelastic Composite Materials

Pouria Hajikarimi · Alireza Sadat Hosseini

Constructional Viscoelastic Composite Materials Theory and Application

Pouria Hajikarimi Department of Civil and Environmental Engineering Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran

Alireza Sadat Hosseini School of Civil Engineering College of Engineering University of Tehran Tehran, Iran

ISBN 978-981-99-1785-3 ISBN 978-981-99-1786-0 (eBook) https://doi.org/10.1007/978-981-99-1786-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fibrous Composites in Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Bituminous Composites in Construction . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 5 7

2 Viscoelasticity Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Relaxation Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Creep Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mechanical Models for Representing Viscoelastic Behavior . . . . . . 2.3.1 Maxwell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Kelvin-Voigt Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Burgers Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 The Generalized Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 General Format of Constitutive Equations . . . . . . . . . . . . . . . 2.4 Hereditary Approach for Representing Viscoelastic Behavior . . . . . 2.5 Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Interconversion of Constitutive Viscoelastic Functions . . . . . . . . . . . 2.6.1 Hopkins and Hamming Method . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 System of Linear Algebraic Equations . . . . . . . . . . . . . . . . . . 2.7 Time–Temperature Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Effect of Time, Temperature, and Loading Rate . . . . . . . . . . 2.7.2 Master Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 11 12 14 16 19 20 21 24 27 27 29 35 35 37 41

3 Nonlinear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Schapery Single-Integral Nonlinear Model . . . . . . . . . . . . . . . . . . . . . 3.2.1 A Brief Review on Linear Viscoelasticity . . . . . . . . . . . . . . . . 3.2.2 The Schapery Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Schapery Equation for a Two-Step Stress . . . . . . . . . . . . . . . .

43 43 43 44 46 48

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3.2.4 Schapery Equation for a Creep and Creep-Recovery Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Determination of Material Parameters from a Creep and Creep-Recovery Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 One-Dimensional Formulation . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 53 58 59 62

4 Experimental Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fibers and Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Single Fiber Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Ultrasonic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Matrix Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Experiments Under Static Loading . . . . . . . . . . . . . . . . . . . . . 4.2.2 Dynamic Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Experiments Under Static Loading . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 65 70 71 71 83 85 85 95

5 Analytical and Empirical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Rule of Mixture in Composite Materials with Continuous Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Analytical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Semi-Analytical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rule of Mixture in Composite Materials with Discontinuous Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Fibrous Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Granular Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A Review on Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Fem for Two-Dimensional Solids . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 FEM for Plates and Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 FEM for 3D Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Incremental Viscoelastic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Geometrically Homogenous Composite Modeling . . . . . . . . . . . . . . . 6.3.1 Geometry Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Geometrically Heterogeneous Composite Modeling . . . . . . . . . . . . . 6.4.1 Geometry Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Boundary Conditions and Interactions . . . . . . . . . . . . . . . . . . . 6.5 An Extension of FEM for Discontinuity Modeling . . . . . . . . . . . . . . 6.5.1 Fundamentals and Formulations . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 102 104 104 109 117 119 119 122 126 129 133 139 139 143 145 148 148 150 153 153 157 165

Chapter 1

Introduction

This book is dedicated to the theories and applications of viscoelastic composite materials which are used in construction. Theories of viscoelasticity are explained in simple words aiming to familiarize the reader with this topic in the most efficient way. Therefore, this book explains the application of complicated viscoelastic equations and principles on constructional viscoelastic composite materials considering time, temperature, and loading rate dependency of viscoelastic materials and heterogeneity of composite substances. Fibrous and bituminous composites consisting of continuous and discontinuous inclusions are investigated. The available analytical and empirical approaches for calculation of the mechanical and physical properties of constructional viscoelastic composites are presented. Some practical examples are provided to make the application of such materials easier to follow for similar problems. Consequently, most of the standard test methods which are commonly used to characterize the mechanical properties of fiber reinforced polymer (FRP) and bituminous materials are summarized as a handy tool for the researchers in this area. Finally, the use of finite element method (FEM) in solving practical numerical examples is explained. All M.Sc. and Ph.D. students of Civil Engineering who work in the area of constructional composite structures and viscoelasticity (experimental, analytical, and numerical work) can use this book as a reference to theoretical background, experimental investigations, and numerical modeling methods, especially those who work on modeling FRPs and bituminous composites such as asphalt mastic, asphalt mortar, and asphalt concrete. Several students and researchers who study and work in the field of composite materials characterization and modeling perform experimental tests which need to be analyzed to construct phenomenological models or develop numerical simulations with appropriate verifications. Also, engineers who work in research and development departments of industries need to interpret experimental results beyond standard specifications to improve their quality and develop new products, such as modified bitumen or strengthening and rehabilitation schemes. This book

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. Hajikarimi and A. Sadat Hosseini, Constructional Viscoelastic Composite Materials, https://doi.org/10.1007/978-981-99-1786-0_1

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offers a package for experimental investigations and using analytical and numerical modeling of commonly used constructional viscoelastic composite materials, including bituminous and fiber-reinforced composites.

1.1 Viscoelasticity The mechanics of continuous media involves calculating three types of quantities: stresses, strains, and displacements. These quantities are interconnected and are governed by three types of equations: equilibrium conditions, kinematic relations, and constitutive equations as described in Flügge (2013). Due to the complex and varied behavior of constructional materials, it is not feasible to precisely describe their mechanical behavior. Therefore, a common approach is to develop an idealized mechanical model that best replicates the desired material’s mechanical behavior. Sadd (2009) notes that one of the simplest material behaviors is elastic behavior, which is governed by Hook’s law and involves a one-to-one relationship between stress and strain. This law is commonly used to explain the mechanical behavior of solid constructional materials, such as concrete, steel, aggregates, fibers, and so on. When stress is applied to an ideal elastic material, it immediately deforms but returns to its original configuration once the stress is removed. However, some materials exhibit plastic flow, which can be characterized by three statements: 1. The material behaves elastically until it reaches its yield limit. 2. Further strain can occur without an increase in stress, and 3. The additional strain is permanent and cannot be recovered once the applied stress is removed. In describing the viscous behavior of a simple fluid, the Newton’s law relates shear stress to shear strain rate, with viscosity or coefficient of viscosity being the constant ratio between the two. Figure 1.1 illustrates shear stress versus shear strain rate for a Newtonian fluid, where the coefficient of viscosity is independent of the shear rate but may vary with pressure and temperature. However, Non-Newtonian fluids exhibit non-constant viscosity, which depends on the shear rate and can demonstrate shear-thinning or shear-thickening behavior, as depicted in Fig. 1.1 (Massey and Ward-Smith 1998). In contrast to elastic materials, a viscous material does not return to its original configuration after removing the load as it has lost its memory of the original state. When materials display both elastic and viscous characteristics under deformation or force, they are referred to as viscoelastic materials. This term was coined to indicate that the properties of these materials come from a combination of idealized elastic and viscous models. Viscoelasticity was first observed in the 19th and 20th centuries by physicists such as Maxwell (1867), Kelvin (1875), and Boltzmann through their research on the creep and recovery of metals, glasses, and rubbers. The foundation of viscoelasticity can be traced back to polymer rheology (Brinson and Brinson 2008; Hajikarimi and Moghadas Nejad 2021) when synthetic polymers

1.1 Viscoelasticity

3

Fig. 1.1 Shear stress versus shear strain rate for Newtonian and Non-Newtonian’s fluids

were designed, produced, and utilized in various applications in the late 20th century. However, viscoelastic materials are not exclusive to polymers, as several engineering and constructional materials, such as bituminous materials (Di Benedetto et al. 2004; Kim 2009) and resins used to fabricate fiber-reinforced materials (Zhang and Wang 2011; Ascione et al. 2011), exhibit viscoelastic behavior. There is a common belief that viscoelasticity has little practical application in design methods. However, neglecting the viscoelastic characteristics of constructional materials like fiber-reinforced composites and bituminous materials can lead to significant overestimation or underestimation in the process of designing required parameters. The theory of viscoelasticity is based on the concept of “fading memory” for materials subjected to time-dependent stress or strain (Morrison 2001). The viscoelastic behavior of a specific material may be linear or nonlinear. When the excited strain (or stress) is directly proportional to the response stress (or strain), the material’s behavior is linear. For nonlinear behavior, the response strain (or stress) is a function of the applied stress (or strain). Viscoelastic materials are the result of the movement of atoms or molecules within an amorphous substance, and their behavior is influenced by several factors such as time or frequency of loading, temperature, and loading rate (Phan-Thien and Rossikhin 2004). Furthermore, their response to deformation or loading may also be influenced by the material’s load or deformation history. In the field of viscoelasticity, there are two commonly observed time-dependent behaviors: creep and stress relaxation, both of which are well-documented in the technical literature (Findley and Davis 2013). Designers may be able to overcome some of the challenges associated with using viscoelastic materials in load-bearing conditions if they have access to relevant laboratory data on the materials’ behavior under short-term and long-term stress or strain. The analysis of creep and stress relaxation phenomena is also critical for assessing the fundamental viscoelastic properties of constructional viscoelastic materials.

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1.2 Fibrous Composites in Construction Generally, fibrous composites, which are known as fiber reinforced polymer (FRP) materials, are composed of fibers (continuous or discontinuous) which are responsible for load bearing, and a viscoelastic matrix which holds the fibers together, helps to transfer loads and protects the fibers from environmental and physical damages. The use of FRP materials can be categorized into two groups. The first is new construction, in which FRP is used as one of the basic components of a structural element. An example of this application is FRP rebars, which are used as the reinforcement in sustainable concrete structural elements or FRP pipes. The other group consists of FRP materials which are used for strengthening and retrofit such as FRP sheets and fabrics. In practice, the behavior of FRP materials is assumed linear elastic up to the failure. This is not an incorrect assumption but a good estimation of their behavior due to the fact that the mechanical properties of an FRP are mostly inherited from the fibers which show negligible viscoelastic behavior and are almost linear elastic. However, when long-term properties of FRPs are the subject (Arao et al. 2012), taking into account the time-dependency of polymer matrix is of great importance. Moreover, since matrix part of the FRP is viscoelastic, the loading frequency and temperature are other contributing parameters in the mechanical behavior of the FRP materials. These can affect the design when long-term behavior (due to creep) and other effects in case of elevated temperature of loading frequency is of designers’ interest. In this book, only linear viscoelastic properties of matrix are considered and nonlinear viscoelastic behavior is not within the scope. It is shown that the nonlinear models, at least in the case of creep, do not add much to the precision of the results when long-term (over 1 year) is investigated (Wang et al. 2022). Despite the dependency of matrix to the mentioned parameters, FRPs are yet an interesting alternative to the conventional construction materials. The most important advantage of these materials over the traditional materials is their corrosion resistance feature. This supremacy provides the designer with a durable alternative for the design in corrosive and harsh environments such as ocean front and coastal areas. In such an environment, conventional materials are prone to corrosion and deterioration which poses a threat to safety and financial resources. For instance, FRP profiles such as tubes, boxes, and other forms, which are produced by pultrusion process, can be used in waterlines and also load-bearing members such as columns, piles, and beams. Although some techniques such as using stainless steel or impenetrable concrete are available and used, they are comparatively costly. Therefore, considering the structural load-bearing efficacy of FRP construction materials, they are an attractive competitor in reducing the capital and/or maintenance expenditures of construction. The other considerable advantage of FRPs is their relatively light weight for the strength they provide. FRP weighs about one-third of steel for an equal volume. Considering the replacement of steel rebars in a reinforced concrete structure (Alkhrdaji et al. 2006) can remarkably reduce the structure’s weight. The tailorability and flexibility of these materials allow the designer to meet the construction requirements proposed by the codes and standards.

1.3 Bituminous Composites in Construction

5

It is however important to have a look at the effects of harsh environment on FRP materials too (Fang et al. 2019). The elevated temperature can be a case when FRP profiles are used for new construction or FRP sheets and fabrics are used for wrapping an existing structure in very warm regions or accidents such as fire. These conditions have been shown to considerably affect the mechanical properties of FRP profiles (Turvey and Wang 2007) and wraps due to the increase in the tensile and bond strengths. The other effective parameter is humidity and immersion of alkaline or de-ionized water in marine environments. Being in contact with de-ionized water is shown to result in a range of reversible and irreversible changes in the mechanical properties of Pultruded E-Glass/Vinyl ester Composites (Cerniauskas et al. 2020) such as plasticization, hydrolysis, fiber-matrix debonding, matrix microcracking, and microcrack coalescence. Moreover, it is seen that de-ionized water has a negligible effect on tensile properties of GFRP bars while alkaline solution can drop the strength. Also, the stress level on FRPs can be another effective parameter on changing the size of micro-cracks in the FRP materials and thus the level of penetration of the ions (Nkurunziza et al. 2005). All in all, the use of FRP materials has some considerable advantages over the traditional competitors but there are indeed drawbacks to their use. Part of this relates to the less-developed knowledge about their material performance compared with the basic construction materials. The research in this area is rapidly growing and codes and standards are becoming more specialized for the use of FRP as a sustainable construction material.

1.3 Bituminous Composites in Construction Bitumen is a petroleum-based substance that can range in consistency from a viscous liquid to a glassy solid (Hunter et al. 2015). It can be obtained as a byproduct of petroleum distillation or found in natural deposits. Bitumen contains carbon and hydrogen compounds with small amounts of sulfur, nitrogen, and oxygen (Wess 2004). The use of bitumen is very old, dating back to its use as a water stop between brick walls of a reservoir at Mohenjo-Daro (about the 3rd millennium B.C.) in Pakistan. It was widely used for paving roads and sealing waterworks in the Middle East, important applications even today. Petroleum asphalt is produced in all consistencies from light road oils to heavy, high-viscosity industrial types. Asphalt mixture, as a widespread constructional material for paving applications, consists of three main components: bitumen, aggregates, and air voids (Huang 1993). Asphalt mixture is a temperature-sensitive constructional material; its mechanical properties and operational performance (laying and compaction) will also intensely alter with temperature differences (Yildirim et al. 2000). Regarding viscoelastic properties of bitumen, asphalt mixture behaves as a viscoelastic material which means that its behavior depends on time, temperature, and loading rate (Hajikarimi and Moghadas Nejad 2021). Figure 1.2 simply demonstrates dependency of bitumen

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Fig. 1.2 Flow behavior of bitumen depending on time and temperature

behavior as a viscoelastic material to temperature to flow. The flow behavior of a bitumen could be the same for one hour at 60 °C or 10 h at 25 °C. The U.S. has about 3,600 asphalt production sites and produced about 420 million tons of asphalt mixture in 2019. In 2020, the total production of hot and warm mix asphalt was 208,3 million tonnes for EU-27. It shows that asphalt production industry is a huge industry all over the world to construct new highways, airports, and rehabilitate and maintain the existing ones. The asphalt mixture is a composite material composed of 3–5% air voids, 9– 15% bitumen, and 80–88% aggregates by total volume. Although some researchers have treated the asphalt mixture as a homogeneous material by making simplifying assumptions that may not accurately reflect its structure and behavior, others have taken a multi-scale approach to investigate the behavior of the mixture’s components at different length scales (Fakhari Tehrani et al. 2013; Allen et al. 2017; Hajikarimi et al. 2019). In their studies, they examined the asphalt mixture at various length scales, including bitumen, asphalt mastic, and asphalt mortar. All of these length scales are bituminous composites that have a viscoelastic matrix containing elastic inclusions (aggregates). The multi-scale evaluation of asphalt mixtures is a technique whereby material behaviors are considered as a consequence of mechanisms that are active at different length scales within the asphalt composite. Evaluations in this field of study involve identifying the functional mechanisms at the specific length scales where they are most active. This framework may suggest further insights into fundamental behaviors of asphalt mixture as well as provide information on the link between constituent material behaviors and bulk mixture characteristics. Developing new technologies for paving roads like using different types of additives (Yildirim 2007) for producing high-performance asphalt mixture, warm mix asphalt (Chowdhury and Button 2008), cold mix asphalt (Hanson et al. 2012), and the necessity of recycling for preserving the environment (Al-Qadi et al. 2007) enforces all researchers, designers, and who should make critical decisions to change their point of view about asphalt mixture behavior, durability, and requirements. Therefore, it is needed to better understand the mechanical behavior of these complex constructional materials for both experimental and numerical applications.

References

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References Al-Qadi IL, Elseifi M, Carpenter SH (2007) Reclaimed asphalt pavement—a literature review. Fed Highw Adm. http://hdl.handle.net/2142/46007 Alkhrdaji T, Fyfe ER, Korff J et al (2006) Guide for the design and construction of structural concrete reinforced with FRP bars Allen DH, Little DN, Soares RF, Berthelot C (2017) Multi-scale computational model for design of flexible pavement–part III: two-way coupled multi-scaling. Int J Pavement Eng 18:335–348 Arao Y, Yukie O, Koyanagi J et al (2012) Simple method for obtaining viscoelastic parameters of polymeric materials by incorporating physical-aging effects. Mech Time-Dependent Mater 16:169–180. https://doi.org/10.1007/s11043-011-9143-z Ascione L, Berardi VP, D’Aponte A (2011) A viscoelastic constitutive law For FRP materials. Int J Comput Methods Eng Sci Mech 12:225–232. https://doi.org/10.1080/15502281003660211 Brinson HF, Brinson LC (2008) Polymer engineering science and viscoelasticity: an introduction. Springer Cerniauskas G, Tetta Z, Bournas DA, Bisby LA (2020) Concrete confinement with TRM versus FRP jackets at elevated temperatures. Mater Struct 53:58. https://doi.org/10.1617/s11527-02001492-x Chowdhury A, Button JW (2008) A review of warm mix asphalt Di Benedetto H, Olard F, Sauzéat C, Delaporte B (2004) Linear viscoelastic behaviour of bituminous materials: from binders to mixes. Road Mater Pavement Des 5:163–202 Fakhari Tehrani F, Quignon J, Allou F et al (2013) Two-dimensional/three-dimensional biphasic modelling of the dynamic modulus of bituminous materials. Eur J Environ Civ Eng 17:430–443. https://doi.org/10.1080/19648189.2013.786243 Fang H, Bai Y, Liu W et al (2019) Connections and structural applications of fibre reinforced polymer composites for civil infrastructure in aggressive environments. Compos Part B Eng 164:129–143. https://doi.org/10.1016/j.compositesb.2018.11.047 Findley WN, Davis FA (2013) Creep and relaxation of nonlinear viscoelastic materials. Courier Corporation Flügge W (2013) Viscoelasticity. Springer Science & Business Media Hajikarimi P, Fakhari Tehrani F, Moghadas Nejad F et al (2019) Mechanical behavior of polymermodified bituminous mastics. I: experimental approach. J Mater Civ Eng 31:4018337 Hajikarimi P, Moghadas Nejad F (2021) Applications of viscoelasticity: bituminous materials characterization and modeling. Elsevier Hanson CS, Noland RB, Cavale KR (2012) Life-Cycle Greenhouse Gas Emissions of Materials Used in Road Construction. Transp Res Rec 2287:174–181. https://doi.org/10.3141/2287-21 Huang YH (1993) Pavement analysis and design. Prentice hall Englewood Cliffs, NJ Hunter RN, Self A, Read J, Hobson E (2015) The shell bitumen handbook. ICE Publishing London, UK Kelvin W (1875) On the theory of viscoelastic fluids. Math Phys Pap 3:27–84 Kim RY (2009) Modeling of asphalt concrete. Mc-Graw Hill Massey BS, Ward-Smith J (1998) Mechanics of fluids. Crc Press Maxwell JC (1867) IV. On the dynamical theory of gases. Philos Trans R Soc London 49–88 Morrison FA (2001) Understanding rheology. Oxford University Press, USA Nkurunziza G, Benmokrane B, Debaiky AS, Masmoudi R (2005) Effect of sustained load and environment on long-term tensile properties of glass fiber-reinforced polymer reinforcing bars. ACI Struct J 102:615–621 Phan-Thien N, Rossikhin YA (2004) Understanding viscoelasticity: basics of rheology. Appl Mech Rev 57:B4–B4 Sadd MH (2009) Elasticity: theory, applications, and numerics. Academic Press Turvey GJ, Wang P (2007) Failure of pultruded GRP single-bolt tension joints under hot–wet conditions. Compos Struct 77:514–520. https://doi.org/10.1016/j.compstruct.2005.08.024

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Wang S, Stratford T, Reynolds TPS (2022) A comparison of the influence of nonlinear and linear creep on the behaviour of FRP-bonded metallic beams at warm temperatures. Compos Struct 281:115117. https://doi.org/10.1016/j.compstruct.2021.115117 Wess J (2004) Asphalt (bitumen). World Health Organization Yildirim Y (2007) Polymer modified asphalt binders. Constr Build Mater 21:66–72. https://doi.org/ 10.1016/j.conbuildmat.2005.07.007 Yildirim Y, Solaimanian M, Kennedy TW (2000) Mixing and compaction temperatures for hot mix asphalt concrete. University of Texas at Austin, Center for Transportation Research Zhang C, Wang J (2011) Viscoelastic analysis of FRP strengthened reinforced concrete beams. Compos Struct 93:3200–3208. https://doi.org/10.1016/j.compstruct.2011.06.006

Chapter 2

Viscoelasticity Theoretical Background

This chapter briefly provides an appropriate and sufficient knowledge for those who want to face serious problems related to constructional viscoelastic materials. The information presented in this chapter is condensed and it is evident that for going into more depth, it is required to refer to textbooks written particularly in the field of viscoelasticity theorem. In this chapter, the main background on viscoelasticity theorem is presented, including introducing constitutive viscoelastic functions: relaxation modulus and creep compliance, mechanical models for representing viscoelastic behavior: the Maxwell model, the Kelvin-Voigt model, the Burgers model, the generalized Maxwell model, and the generalized Kelvin-Voigt model, and the hereditary approach for representing rheological and mechanical behavior of viscoelastic materials. The correspondence principle is also introduced for finding solution of viscoelastic problems having correspondent analytical solutions for elastic materials. Interconversion of viscoelastic materials is another topic presented in this chapter for determining the desired viscoelastic constitutive function having another function. The last topic presented in this chapter is time–temperature superposition principle to construct master curves of viscoelastic functions having experimental data.

2.1 Relaxation Modulus The relaxation modulus, E(t), of a viscoelastic material can be calculated by measuring the resulting stress while applying a constant strain for a specific duration of time, using Eq. (2.1) (Ottosen and Ristinmaa 2005): E(t) =

σ (t) ε0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. Hajikarimi and A. Sadat Hosseini, Constructional Viscoelastic Composite Materials, https://doi.org/10.1007/978-981-99-1786-0_2

(2.1)

9

10

2 Viscoelasticity Theoretical Background

in which ε0 is the constant applied strain and σ (t) is the measured stress varied with time. Figure 2.1 shows the schematic excited strain and response stress of a relaxation test in which t 1 is the required time for achieving the required strain in the step-strain experiment. In the literature, two distinct relaxation behaviors for viscoelastic materials are commonly described: thermoplastic and thermoset. An ideal thermoplastic polymer will exhibit a stress that tends to zero, while a thermoset polymer will reach a plateau, known as residual stress. Figure 2.2 depicts the relaxation modulus of an epoxy resin at various test temperatures, as well as its master curve at a reference temperature of 25 °C. As can be seen in this figure, increasing time and temperature resulted in decreasing relaxation modulus, which shows the dependency of relaxation modulus as a constitutive viscoelastic function to time and temperature.

Fig. 2.1 The schematic diagram for excited stain and response stress of a relaxation test

Fig. 2.2 Relaxation modulus for epoxy resin versus time (Theocaris 1962)

2.3 Mechanical Models for Representing Viscoelastic Behavior

11

Fig. 2.3 The schematic diagram for excited stress and response strain of a creep test

2.2 Creep Compliance By applying constant stress during a specific time and measuring the resulted strain, the creep compliance, D(t), for a viscoelastic material can be calculated using the following Eq. (2.2) (Ottosen and Ristinmaa 2005): D(t) =

ε(t) σ0

(2.2)

in which ε(t) is the measured strain and σ 0 is the constant step stress. A creep test schematic is depicted in Fig. 2.3, showcasing the excited stress and response strain. The time required to reach the desired stress level during the step stress experiment is represented by t 1 . A simple test setup for determining creep compliance for a viscoelastic material is a three-point bending setup in which mid-span deflection is measured versus time, and then using Eq. (2.3), creep compliance can be determined (Hajikarimi et al. 2018): P L3 1 = D(t) 48I δ(t)

(2.3)

where D(t) is the creep compliance, P is the applied constant load at the mid-span of the beam, L is the span length, I is the moment of inertia of the beam section, and δ(t) is the mid-span deflection of the beam.

2.3 Mechanical Models for Representing Viscoelastic Behavior Viscoelastic materials’ mechanical behavior is represented by combining two fundamental elements: the spring and the dashpot, as illustrated in Fig. 2.4. The spring

12

2 Viscoelasticity Theoretical Background

=

=

(a)

(b)

̇

Fig. 2.4 Basic elements of a spring and b dashpot for representing viscoelastic behavior

represents ideal elastic behavior (with an elastic modulus, E), while the dashpot represents ideal viscous behavior (with a coefficient of viscosity, η). These elements can be arranged in series or parallel to create various mechanical models, including the Kelvin-Voigt model, the Maxwell model, the Burgers model, and generalized models. This section introduces essential mechanical models that describe viscoelastic behavior, which are valuable for comprehending the stress-strain relationship for various types of viscoelastic materials.

2.3.1 Maxwell Model To create the Maxwell model (Maxwell 1867), an elastic spring and a viscous dashpot are linked in series, as depicted in Fig. 2.5. The total stress and strain for the Maxwell model are expressed in Eqs. (2.4.1, 2.4.2): d/dt

ε = εe + εv ⇒ ε˙ = ε˙ e + ε˙ v

(2.4.1)

σ = σe = σv

(2.4.2)

Using the stress–strain relationship for spring and dashpot elements (see Fig. 2.4), it is possible to re-write Eq. (2.4.1) for the total strain of the Maxwell model as follows: ε˙ =

σ˙ e σ + E η

(2.5)

The elastic modulus of the spring is represented by E, and the viscosity coefficient of the dashpot is represented by η in the Maxwell model. By conducting a relaxation test under an applied strain of ε0 and a creep test under the applied stress of σ 0 , it Fig. 2.5 The Maxwell model

2.3 Mechanical Models for Representing Viscoelastic Behavior

13

is possible to establish the relaxation modulus and creep compliance as constitutive functions for the Maxwell model. For a relaxation test (ε = ε0 ), the first derivative of the strain function will be zero. Then, Eq. (2.5) can be written and integrated as follows:   σ 1 dσ σ E σ˙ + =0⇒ + = 0 ⇒ σ (t) = C exp − t E η E dt η η

(2.6)

It is required to consider the initial condition of the Maxwell model to determine the constant C in Eq. (2.6). At t = 0, the dashpot behaves like a rigid element without deformation. Therefore, the spring carries the total stress equal to σ = Eε0 and having t = 0, C = Eε0 . Substituting the stress function, σ (t), from Eq. (2.6) into Eq. (2.1) resulted in the following equation for the relaxation modulus of the Maxwell model:     C exp − Eη t C=Eε σ (t) E E(t) = = ⇒ 0 E(t) = E exp − t ε0 ε0 η

(2.7)

The relaxation time, which is the duration required for the stress to decrease and reach 36.8% of the initial value, is determined by the ratio of η to E. The initial value corresponds to the condition in which σ 0 = Eε0 . For a creep test (σ = σ 0 ), Eq. (2.5) is re-organized as follows: 1 dσ σ dσ σ dε = + ⇒ dε = + dt dt E dt η E η

(2.8)

and then by integrating both sides of Eq. (2.8), the following equation can be derived: t 

t dε = 0

0

 dσ σ + dt E η



σ (t)=σ0

⇒ ε(t) =

 t 1 + σ0 E η

(2.9)

Substituting the strain function, ε(t), from Eq. (2.9) into Eq. (2.2) resulted in the following equation for the creep compliance of the Maxwell model: ε(t) = D(t) = σ0



t 1 + E η

 (2.10)

The creep compliance of the Maxwell model is directly proportional to time, as depicted in Fig. 2.6. Under creep loading and subsequent unloading, the spring’s elastic behavior causes an instantaneous elastic strain, followed by incremental viscous strain from the dashpot. While the elastic strain recovers immediately upon unloading, the viscous strain does not recover. Therefore, the Maxwell model is limited to demonstrating only the elastic recovery of viscoelastic materials and is unable to account for the recovery phenomenon.

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2 Viscoelasticity Theoretical Background

Fig. 2.6 Maxwell model behavior under creep loading and after unloading

2.3.2 Kelvin-Voigt Model The Kelvin-Voigt model (Voigt 1892) is proposed by connecting an elastic spring and a viscous dashpot in parallel, as shown in Fig. 2.7. Equations (2.11.1, 2.11.2) indicated the total stress and strain of the Kelvin-Voigt model: ε = εe = εv

(2.11.1)

σ = σe + σv

(2.11.2)

Having the relationship between stress and strain for spring and dashpot elements (see Fig. 2.4), Eq. (2.4.2) can be re-written for the total stress of the Kelvin-Voigt model as indicated in Eq. (2.12): σ = Eε e + η˙εv

ε=εe =εv

⇒

σ = Eε + η˙ε

(2.12)

The Kelvin-Voigt model employs the elastic modulus of the spring, represented by E, and the viscosity coefficient of the dashpot, represented by η. The approach for Fig. 2.7 The Kelvin-Voigt model

2.3 Mechanical Models for Representing Viscoelastic Behavior

15

determining the relaxation modulus and creep compliance for this model is comparable to that used for the Maxwell model. To establish the constitutive functions, a relaxation test is conducted at an applied strain of ε0 , while a creep test is performed at an applied stress of σ 0 . For a creep test (σ = σ 0 ), Eq. (2.12) as a first-order differential equation can be solved as follows:   E σ0 E σ0 = ε + ε˙ ⇒ ε(t) = + Cex p − t (2.13) η η E η To determine the C coefficient appeared in Eq. (2.13), the initial condition of the Kelvin-Voigt model is considered in which at t = 0, when the load is suddenly applied, no deformation (ε = 0) is produced in the Kelvin-Voigt model as the dashpot behaves as a rigid element. Substituting this initial value in the Eq. (2.13), the C coefficient will be C = −σ 0 /E and the total deformation of the Kelvin-Voigt model is as follows:    E σ0 1 − exp − t ε(t) = E η

(2.14)

Substituting the strain function, ε(t), from Eq. (2.14) into Eq. (2.2) resulted in the following equation for the creep compliance of the Kelvin-Voigt model: D(t) =

   E 1 1 − exp − t E η

(2.15)

Figure 2.8, the Kelvin-Voigt model’s response strain under creep loading and after unloading is shown. It can be observed that the strain increased exponentially over time and gradually recovered after unloading. Due to the presence of the dashpot, the strain tends to zero for an infinite time. Unlike the Maxwell model, the Kelvin-Voigt model can fully recover all the strain produced in viscoelastic materials, which may not be appropriate for simulating permanent deformation induced phenomena. The ratio of viscosity coefficient of the dashpot to the stiffness of linear spring (ρ = η/E) is called retardation time which is the needed time for the strain to reach 63.2% of the total retarded strain (Huang 2004). When performing a relaxation test at a constant strain (ε = ε0 ), the elastic spring undergoes an immediate stress of σ = Eε0 upon application of the sudden strain, while the dashpot acts like a rigid element under step-strain loads. In order to deform the Kelvin-Voigt model and achieve a steady-state strain of ε0, an infinitely large stress is required, after which the stress will attain a magnitude of σ = Eε0 . The response stress for a Kelvin-Voigt model during a relaxation test, along with the applied strain, is shown in Fig. 2.9.

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2 Viscoelasticity Theoretical Background

Fig. 2.8 Kelvin-Voigt model behavior under creep loading and after unloading

Fig. 2.9 Kelvin-Voigt model behavior under a constant strain

2.3.3 The Burgers Model In earlier sections (Sects. 2.3.1 and 2.3.2), it was noted that both the Maxwell and Kelvin-Voigt models have limitations in accurately representing the stress-strain behavior of viscoelastic materials. While the Maxwell model can only illustrate

2.3 Mechanical Models for Representing Viscoelastic Behavior

17

Fig. 2.10 The Burgers model

elastic recovery, the Kelvin-Voigt model exhibits full recovery without capturing permanent deformation. To address these shortcomings, the Burgers model (1935) was introduced by combining the Maxwell model and Kelvin-Voigt model in series, as shown in Fig. 2.10. The total stress and strain of the Burgers model is described in Eqs. (2.16.1, 2.16.2) considering that these two models are connected in series. σBurgers = σMaxwell = σKelvin-Voigt

(2.16.1)

εBurgers = εMaxwell + εKelvin-Voigt

(2.16.2)

Considering a creep test (σ = σ 0 ), the creep compliance of the Burgers model can be determined using Eqs. (2.10) and (2.15) for the Maxwell model and the Kelvin-Voigt model, respectively. εMaxwell (t) + εKelvin (t) σ0    EK 1 t 1 1 − exp − = + + t EM ηM EK ηK

D(t) =

(2.17)

This graph reveals that the model can capture both instantaneous elastic recovery and permanent deformation due to the individual dashpots of the Maxwell model. Upon unloading, the model exhibits an instantaneous recovery of elastic strain (εE ), a delayed elastic strain recovery (εDE ) that takes a specific time to recover, and a

18

2 Viscoelasticity Theoretical Background

Fig. 2.11 The stress excitation and response strain of the Burgers model under creep loading and after unloading

viscous strain (εV ) that results in permanent deformation of the viscoelastic material and cannot recover (Fig. 2.11). The total stress of the Burgers model can be derived using Eqs. (2.16.2) and (2.12) as follows: σ = E K (ε − ε M ) + η K (˙ε − ε˙ M )

(2.18)

By taking the derivative of Eq. (2.18) with respect to time and using Eq. (2.5) to represent the Maxwell model, we can obtain a second-order differential equation that serves as a fundamental equation for describing the connection between stress and strain in a viscoelastic substance. This equation applies specifically to the Burgers model.   EK ηK d 2σ d 2ε dε ηK E K dσ + (2.19) = 1 + + σ = η + EK K E M dt 2 ηM E M dt ηM dt 2 dt The energy per volume, W, can be integrated for each spring and dashpot element in the Burgers model with the following general relation,  W =

σ dε

(2.20)

which is according to the definition of mechanical work as force times path (Johansson and Isacsson 1998). For the stored energy, Eq. (2.21), and the dissipated

2.3 Mechanical Models for Representing Viscoelastic Behavior

19

energy, Eq. (2.22), of the Burgers model, the following equations are derived:      EM 1 1 EK 1 − 2 exp − = + t + exp −2 t EM 2E K ηM ηK     EK t 1 2 1 − exp −2 Wdissipated = σ0 + t ηM 2E M ηK 

Wstored

σ02

(2.21) (2.22)

2.3.4 The Generalized Models To improve the accuracy of predicting the relaxation modulus or creep compliance of viscoelastic materials over a wider range of time or frequency, the concept of generalized models was introduced (Brinson and Brinson 2008). The generalized Maxwell model is created by connecting multiple Maxwell models in parallel, as depicted in Fig. 2.12. Similarly, the generalized Kelvin-Voigt model is formed by connecting multiple Kelvin-Voigt models in series, as shown in Fig. 2.13. Considering the possibility of superposing stresses of n parallel elements (see Eqs. 2.11.1, 2.11.2), and the definition of relaxation modulus presented in Eq. (2.1), the following equation is derived for the relaxation modulus, E(t), of a generalized Maxwell model under a step-strain load (ε = ε0 ): n E(t) =

i=1

σi (t)

ε0

Fig. 2.12 The generalized Maxwell model

= E∞ +

n

i=1

  Ei E i exp − t ηi

(2.23)

20

2 Viscoelasticity Theoretical Background

Fig. 2.13 The generalized Kelvin-Voigt model

in which E ∞ is equilibrium modulus, E i and ηi are the stiffness and viscosity coefficient of ith spring and dashpot, respectively. The creep compliance, D(t), of the generalized Kelvin-Voigt model under a creep loading (σ = σ 0 ) can be determined thanks to the opportunity of superposing strains of n elements connected in series (see Eqs. 2.4.1, 2.4.2) and the definition of creep compliance introduced in Eq. (2.2), as follows: m D(t) =

j=1 εi (t)

σ0

= Dg +

m

j=1

   Ej D j 1 − exp − t ηj

(2.24)

where Dg is glassy compliance, Dj is compliance of each spring (1/E j ), E j and ηj are the stiffness and viscosity coefficient of jth spring and dashpot, respectively. The Prony series equations mentioned earlier for the relaxation modulus of a generalized Maxwell model and the creep compliance of a generalized Kelvin-Voigt model are crucial in accurately predicting the mechanical behavior of viscoelastic materials. However, the challenge lies in selecting the minimum number of elements needed to achieve this accuracy. This can be determined through research in technical literature or statistical investigation (Hajikarimi et al. 2018). Once the appropriate number of elements is identified, the coefficients in Eqs. (2.23) and (2.24) can be calculated using a nonlinear regression method and an appropriate solver or commercial toolboxes. Figure 2.14 shows the effect of number of elements on improving the accuracy of the Prony series to describe the mechanical behavior of a viscoelastic material. In this figure, the performance of a fractional model entitled FDM-4 (Xu et al. 2019) is compared with the generalized Maxwell model having different number of elements.

2.3.5 General Format of Constitutive Equations The previous sections primarily focused on formulating functions for relaxation modulus and creep compliance for various configurations of springs and dashpots

2.4 Hereditary Approach for Representing Viscoelastic Behavior

21

Fig. 2.14 The effect of the number of elements on the generalized Kelvin-Voigt model (Hajikarimi et al. 2022)

under specific loading situations. Along with constitutive viscoelastic functions that explain the mechanical properties of viscoelastic materials, the stress-strain correlation is also crucial. The general format of the stress–strain relation of viscoelastic materials is (Findley et al. 1976): n

i=1

d jε di σ = bj j dt i dt j=1 m

ai

(2.25)

where ai and bj are constant material parameters and d i /dt i is derivative operation. Equation (2.19) shows a general equation for explaining the stress–strain relation of the Burgers model. Both m and n should be integer numbers.

2.4 Hereditary Approach for Representing Viscoelastic Behavior The hereditary approach is introduced in this section to establish the stress– strain relationship of linear viscoelastic materials using Boltzmann’s superposition principle. The arbitrary uniaxial strain function shown in Fig. 2.15 is considered as input excitation. By discretizing the whole time of strain application into n small time steps, assuming that the applied strain is a constant amount of ∆ε in each time step

22

2 Viscoelasticity Theoretical Background

Fig. 2.15 Boltzmann’s superposition for an arbitrary strain excitation and the correspondant stress response

will be satisfactory. Having the relaxation modulus of a viscoelastic material, E(t), the corresponding stress, ∆σ (t), at time ψ can be determined by using Eq. (2.26): ∆σ (t) = E(t − ψ)∆ε

(2.26)

The corresponding stresses over the whole strain history (n time steps) can be determined as: σ (t) =

n n



∆εi ∆ψ i = E(t − ψi )∆ψ i [∆εi E(t − ψi )] ∆ψ ∆ψ i i i=0 i=0

(2.27)

and by considering an infinitesimal strain alteration dε applied at time ψ, Eq. (2.27) can be written in a differential form as follows: t σ (t) =

E(t − ψ) −∞

dε(ψ) dψ dψ

(2.28)

which is the constitutive relationship between response time-dependent stress and applied time-dependent strain for a smooth strain history of ε = ε(ψ). Assuming an arbitrary uniaxial stress function is used as an input excitation function, it is possible to discretize the entire duration of stress application into n time steps, as illustrated in Fig. 2.16. To do so, the time steps must be small enough to allow for the assumption of a constant applied stress, ∆σ, during each time interval to be valid. Using the creep compliance, D(t), of a viscoelastic material, the corresponding strain, ∆ε(t), at a given time, ψ, can be calculated using Eq. (2.29): ∆ε(t) = D(t − ψ)∆σ

(2.29)

2.4 Hereditary Approach for Representing Viscoelastic Behavior

23

Fig. 2.16 Boltzmann’s superposition for an arbitrary stress excitation and the correspondant strain response

The corresponding strain over the whole stress history (n time steps) can be determined as: ε(t) =

n n



∆σ i ∆ψ i = D(t − ψi )∆ψ i [∆σ i D(t − ψi )] ∆ψ ∆ψ i i i=0 i=0

(2.30)

and by considering an infinitesimal stress alteration dσ applied at time ψ, Eq. (2.30) can be written in a differential form as: t ε(t) =

D(t − ψ) −∞

dσ (ψ) dψ dψ

(2.31)

which similar to Eq. (2.28), is the constitutive relationship between time-dependent response strain and applied time-dependent stress for a smooth stress history of σ = σ (ψ). If the strain history or the stress history exhibits some jumps, the constitutive equations can be re-written as follows: t σ (t) =

t E(t − ψ)dε(ψ) = E(t)ε0 +

−∞

0

t

t

ε(t) =

D(t − ψ)dσ (ψ) = D(t)σ0 + −∞

0

E(t − ψ)dε(ψ)

(2.32)

D(t − ψ)dσ (ψ)

(2.33)

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2 Viscoelasticity Theoretical Background

2.5 Correspondence Principle The general format of the stress–strain relation of viscoelastic materials is represented in Eq. (2.25) as a linear differential equation. Therefore, the superposition principle can be simply represented by using the following equations for linear viscoelastic materials and problems containing such substances: If σ1 (t) ⇒ ε1 (t) Then kσ1 (t) ⇒ kε1 (t)  σ (t) ⇒ ε1 (t) If 1 Then σ1 (t) + σ2 (t) ⇒ ε1 (t)+ ε2 (t) σ2 (t) ⇒ ε2 (t)

(2.34)

Equation (2.25) can be re-written in a new format as: Fσ = Qε

(2.35)

in which F and Q are operators that can be defined as: F=

n

ai

di dt i

(2.36)

bj

dj dt j

(2.37)

i=0

Q=

m

j=1

Using the Laplace transform (Boyce et al. 2017), it is possible to convert a differential equation to an algebraic equation. Applying the Laplace transform on Eq. (2.25) and considering the following formula for the Laplace transform of a function’s derivation,

L f ' (t) = s f (s) − f (0)

(2.38)

the following equation can be derived: n

ai s i σ (s) =

m

b j s j ε(s)

(2.39)

j=1

i=0

where s is the Laplace variable. Applying Eq. (2.38) on Eqs. (2.36) and (2.37) results in the following equations: F(s) =

n

i=0

ai s i

(2.40)

2.5 Correspondence Principle

25

Q(s) =

m

bjs j

(2.41)

j=1

Inserting Eqs. (2.40) and (2.41) into Eq. (2.39) results in the following equation: F(s)σ (s) = Q(s)ε(s)

(2.42)

which can be re-organized to reach the following equation, which is similar to Hook’s law in the elastic domain (Timoshenko and Young 1962): σ (s) =

Q(s) F(s)

ε(s)

(2.43)

Using Eq. (2.43), stress and strain in the Laplace transformed domain are connected via the following equations: ∗

σ (s) = E (s)ε(s) ∗

ε(s) = D (s)σ (s)

(2.44) (2.45)

which are known as correspondence principle (CP) or Alfrey’s Correspondence Principle equations, and are commonly used to convert viscoelastic problems from the time domain into elastic ones in the Laplace transform domain. Alfrey (1944) first introduced this concept, followed and generalized by Read (1950) and Lee (1955). Equations (2.44) and (2.45) represent a linear elastic form between stress and strain in the Laplace transformed domain in which E¯ * (s) is similar to Young’s modulus. Applying the Laplace transform to the stress–strain relationship developed based on the hereditary approach in Sect. 2.4, and recalling the Laplace transform of a convolution integral, ⎧ t ⎫ ⎧ t ⎫ ⎨ ⎬ ⎨ ⎬ L f (t − ξ )g(ξ )dξ = L f (ξ )g(t − ξ )dξ = L{ f (t)}L{g(t)} (2.46) ⎩ ⎭ ⎩ ⎭ 0

0

it is possible to derive the following equation based on Eq. (2.28): ⎧ t ⎫   ⎨ dε(t) dε(ψ) ⎬ dψ = L{E(t)}L L{σ (t)} = L E(t − ψ) ⎩ ⎭ dψ dt 0

= L{E(t)}[sL{ε(t)} − ε(0)] = sL{E(t)}L{ε(t)} where ε(0) = 0. Therefore, Eq. (2.47) can be represented as follows:

(2.47)

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2 Viscoelasticity Theoretical Background

σ (s) = s E(s)ε(s)

(2.48a)

and comparing Eqs. (2.48a) and (2.44) results in the following equality: ∗

E (s) = s E(s)

(2.48b)

In addition, based on Eq. (2.31), the following equation can be derived in a similar procedure: ⎧ t ⎫   ⎨ dσ (t) dσ (ψ) ⎬ dψ = L{D(t)}L L{ε(t)} = L D(t − ψ) ⎩ ⎭ dψ dt 0

= L{D(t)}[sL{σ (t)} − σ (0)] = sL{D(t)}L{σ (t)}

(2.49)

where σ (0) = 0. Thus, Eq. (2.49) can be represented as: ε(s) = s D(s)σ (s)

(2.50)

and again, by comparing Eqs. (2.50) and (2.45), the following equality can be drawn: ∗

D (s) = s D(s)

(2.51)

By substituting Eq. (2.50) for ε(s) into Eqs. (2.48a, 2.48b), the following equation is derived: E(s)D(s) =

1 s2

(2.52)

Furthermore, considering both Eqs. (2.48a, 2.48b) and (2.51), the following equation can also be written: ∗

E (s) =

1 ∗

D (s)

(2.53)

Equation (2.52) can be transformed to the time domain by implementing the convolution theorem as: t E(ψ)D(t − ψ)dψ = t

(2.54)

0

which indicates that the creep compliance is not the same as the inverse of the relaxation modulus, meaning that E(t) /= 1/D(t) (Flugge 2013). However, despite this difference, these two constitutive functions are directly related in the Laplace transformed domain, as mentioned in Eq. (2.53).

2.6 Interconversion of Constitutive Viscoelastic Functions

27

Here, a step-by-step procedure is introduced implementing the CP to find a solution for a viscoelastic problem in the time domain using the correspondent elastic solution. This procedure is as follows (Hajikarimi and Nejad 2021): (1) Determine an analytical solution for a linear elastic boundary value problem that has the same geometry, loading, and boundary conditions as the desired linear viscoelastic problem. (2) Replace all variables, such as applied loads, stresses, strains, and displacements, in the analytical elastic solution with their Laplace transforms. (3) Replace all elastic constants, such as Young’s modulus and Poisson’s ratio, with s times the transform of the time-dependent moduli. (4) Utilize the inverse Laplace transform to obtain the viscoelastic solution in the time domain.

2.6 Interconversion of Constitutive Viscoelastic Functions In Sect. 2.3, the mechanical models that employ spring and dashpot elements must be capable of accurately producing both relaxation modulus and creep compliance, which can be based on analytical solutions or experimental data. When characterizing linearly behaving viscoelastic materials, the primary difficulty is accurately measuring relaxation modulus and creep compliance by devising and conducting experiments. Depending on the nature of the viscoelastic material, the desired temperature, and loading rate, measuring one of these constitutive functions may be impossible, complex, costly, or imprecise. For instance, monitoring a constant strain to determine relaxation modulus is more difficult than regulating constant stress. Therefore, stress-control experiments are more common compared to strain-control experiments. Equations (2.52) and (2.53), can be utilized for interconversion. Many studies have focused on developing a straightforward and efficient method to perform such an interconversion to determine a desired viscoelastic function based on another known function. This section introduces and describes two well-known methods: the Hopkins and Hamming method and the system of linear algebraic equations based on the Prony series.

2.6.1 Hopkins and Hamming Method Hopkins and Hamming (1957) proposed a straightforward technique for converting between constitutive viscoelastic functions. This method involves dividing a specific time domain into n time steps and making certain simplifying assumptions to obtain the desired function. Referring to Eq. (2.54), it is possible to re-write this equation using the convolution theorem as follows (Ottosen and Ristinmaa 2005):

28

2 Viscoelasticity Theoretical Background

t E(t − ψ)D(ψ)dψ = t

(2.55)

0

where ψ is an auxiliary integration parameter. As mentioned, E(t) /= 1/D(t) and it is impossible to directly determine the relaxation modulus using the creep compliance or the creep compliance based on the relaxation modulus. A simple mathematical procedure was proposed by Hopkins and Hamming (1957) to solve Eqs. (2.54) and (2.55) to determine the relaxation modulus or the creep compliance. Assume that creep compliance data are available (D(t) is measured), and it is required to determine the relaxation modulus (E(t) is unknown). Equation (2.54) should be discretized into n time steps to solve it as follows: n 

ti

tn =

E(ψ)D(tn − ψ)dψ

(2.56)

i=1 t

i−1

The integral can be calculated using the trapezoidal rule for each time step: tn =

n  D(t − t ) + D(t − t )  

 n i n i−1 tn − ti−1 E ti− 21 2 i=1

(2.57)

in which ∆t = [t i − t i−1 ] is the time step, and each of the following equations can be considered for E(t i-1/2 ) regarding the viscoelastic material behavior:   E(ti ) + E(ti−1 ) E ti− 21 = 2     ti + ti−1 1 E ti− 2 = E 2

(2.58) (2.59)

Figure 2.17 shows an illustration for both Eqs. (2.58) and (2.59). It is evident that selecting one of these equations to proceed with solving the problem is directly dependent on the experimental data and fitted curve. The following equation can be rapidly derived for the first time step by considering n = 1 and t 0 = 0 in Eq. (2.57):   D(0) + D(t )   2 1 t1 = E t 21 t1 ⇒ E t 21 = 2 D(0) + D(t1 )

(2.60)

Also, by removing the nth term of Eq. (2.57) from the summation and keeping it as an individual term, the following equation can be derived:

2.6 Interconversion of Constitutive Viscoelastic Functions

29

Fig. 2.17 Different approximation of relaxation modulus, E(t), in each time step in the Hopkins and Hamming’s method

n−1   D(t − t ) + D(t − t ) 

 n i n i−1 tn − ti−1 E ti− 21 2 i=1  D(0) + D(t − t )    n n−1 tn − tn−1 + E tn− 21 2

tn =

(2.61)

By re-organizing Eq. (2.61), E(t n−1/2 ) can be derived using a recursive function as follows:  E t

n− 21

⎡

 = 



2



D(0) + D tn − tn−1 ∆t n

⎣tn −

n−1

i=1

 E t

i− 21



   ⎤ D tn − ti + D tn − ti−1 ⎦∆t i 2

(2.62)

in which ∆t i = t i − t i−1 . A similar procedure can be applied when the relaxation modulus data are available and creep compliance is desired (Hajikarimi and Nejad 2021).

2.6.2 System of Linear Algebraic Equations As discussed in Sect. 2.6.1, the method proposed by Hopkins and Hamming (1957) is a general technique that can be applied to a dataset representing the relaxation modulus or creep compliance, or a mathematical function obtained through curve fitting or mechanical modeling to describe constitutive functions. Using the Prony series (see Sect. 2.3.4) which represents the generalized Maxwell model or KelvinVoigt model, Park and Schapery (1999) developed a numerical method of interconversion between linear viscoelastic material functions. The following integral can be derived to relate the creep compliance, D(t), and the relaxation modulus, E(t), of linear viscoelastic materials: t E(t − ψ) 0

d D(ψ) dψ = 1 dψ

(2.63)

30

2 Viscoelasticity Theoretical Background

Recalling Eq. (2.24), the creep compliance of the generalized Kelvin-Voigt model is as follows:    n

t t D(t) = Dg + + D j 1 − exp − η0 ρj j=1

(2.64)

in which Dg , η0 , Dj, and ρ j are the glassy compliance, zero-shear viscosity, retardation strength, and retardation times, respectively (see Fig. 2.13). Having a data set for creep compliance of a viscoelastic material, it will be possible to determine all the parameters mentioned above based on the principle of multiple nonlinear regression. The term of t/η0 is quite small and neglectable for solid-like viscoelastic materials. Therefore, Eq. (2.64) can be written as: D(t) = Dg +

n

j=1

   t D j 1 − exp − ρj

(2.65)

The relaxation modulus can be determined using the creep compliance data implementing the following equation: E(t) = E e +

m

i=1

  t E i exp − τi

(2.66)

in which E e and E i are the equilibrium modulus and the relaxation strength, respectively. In order to solve Eq. (2.63), both viscoelastic solids and liquids can be considered based on the following characteristics:

2.6.2.1

Viscoelastic Solids : E e > 0 and η0 → ∞

(2.67a)

Viscoelastic Liquids : E e = 0 and η0 is finite

(2.67b)

Viscoelastic Solids

In order to use Prony series for interconverting constitutive viscoelastic functions, Eq. (2.66) and Eq. (2.65), which is related to viscoelastic solids, are substituted in Eq. (2.63): t  Ee + 0

m

i=1



t −ψ E i exp − τi

⎡ ⎤    n

D ψ j ⎣ Dg δ(ψ) + ⎦dψ = 1 exp − ρ ρ j j j=1 (2.68)

2.6 Interconversion of Constitutive Viscoelastic Functions

31

and by re-ordering summation and integration, the following equation can be written: 

m

Dg E e +

i=1

+

n m

i=1 j=1

    t n

Dj t −ψ ρ dψ E i exp − exp − + Ee τi ρj ρj j=1 0

 t       Ei D j ψ t ψ − dψ = 1 exp − exp − ρj τi ρj τi

(2.69)

0

where δ(.) represents the Dirac delta function. There are two simple integrals in Eq. (2.69) that can be determined as follows: t 0

     ρ dψ = ρ j 1 − exp −t/ρ j exp − ρj

(2.70)

and, t 0

           τi ρ j ψ ψ ψ ψ exp − − dψ = 1 − exp − − ρj τi τi − ρ j ρj τi (2.71)

for τ i /= ρ j , and, t 0

     ψ ψ − dψ = t exp − ρj τi

(2.72)

for τ i = ρ j . Having Eqs. (2.71) and (2.72), Eq. (2.69) can be re-organized as follows: Ei

m

⎡

   

   n t τi t ⎣ Dg exp − t + D j exp − 1 − exp − τi τi τi − ρ j ρj

i=1

⎡

= 1 − E e ⎣ Dg +

j=1

n

j=1





t D j 1 − exp − ρj

 −

t τi

⎤   ⎦

⎤ ⎦

(2.73)

for τ i /= ρ j , and, Ei

m

⎤ ⎡   

   n n

Dj t ⎦ t ⎣ Dg exp − t + = 1 − E e ⎣ Dg + t exp − D j 1 − exp − τi ρj τi ρj ⎡

i=1

j=1

⎤ ⎦

(2.74)

j=1

for τ i = ρ j . If the creep compliance, D(t), is known and the relaxation modulus, E(t), is the target function, it is needed to use the initial-value and final-value problems of the Laplace transform to determine E e (Churchill 1972):

32

2 Viscoelasticity Theoretical Background

1

Ee ∼ = lim E(t) = lim E(s) = s→∞

t→0

lim D(s)

=

1 1 n = (2.75) lim D(t) Dg + j=1 D j

t→0

s→∞

Similar to the Hopkins and Hamming’s method, it is possible to determine the relaxation modulus based on the creep compliance and vice versa. If the relaxation modulus, E(t), is known and the creep compliance, D(t), is desired, Eq. (2.69) can be arranged using Eqs. (2.70) and (2.71) as: 

  

     m t t t τi + exp − Ei Dj E e 1 − exp − τi τi − ρ j τi ρi j=1 i=1      m

t = 1 − Dg E e + E i 1 − exp − (2.76) τi i=1 n

for τ i /= ρ j , and, Dj

n

⎤ ⎤ ⎡ 

      m m

Ei t ⎦ t ⎦ ⎣ ⎣ E e 1 − exp − t + = 1 − Dg E e + t exp − E i exp − τi ρj τi τi ⎡

i=1

j=1

(2.77)

i=1

for τ i = ρ j . To determine Dg , a similar procedure which is used to determine E e and presented in Eq. (2.75), the following procedure can be applied: Dg ∼ = lim D(t) = lim D(s) = t→0

s→∞

1 lim E(s)

s→∞

=

1 1 n = lim E(t) E e + i=1 Ei

(2.78)

t→0

If we have the desired time constants and constant parameters for either the creep compliance, {Dg , Dj , ρ j (j = 1, 2, 3, …, n), and η0 }, or for the relaxation modulus, {E e , E i , and τ i (i = 1, 2, 3, …, m)}, we can calculate other unknown parameters for the relaxation modulus or the creep compliance by solving the following system of linear algebraic equations: [A]{D} = {B}

(2.79)

for the case E(t) is known, and the creep compliance, D(t), is desired. In Eq. (2.79), {D} is the array of desired constant parameters of the Prony series that describes the creep compliance function, including D1 , D2 , …, Dn . Also, the matrix [A] and the vector {B} are as follows: Ak j

  

     m tk tk tk τi + exp − − exp − Ei = E e 1 − exp − ρj τi − ρ j τi ρj i=1 (2.80)

for τ i /= ρ j , and,

2.6 Interconversion of Constitutive Viscoelastic Functions

 Ak j = E e



tk 1 − exp − ρj

33

 +

m

Ei

i=1

tk ρj

(2.81)

for τ i = ρ j .

Bk = 1 −

Ee +

  tk E exp − i i=1 τj m E e + i=1 Ei m

(2.82)

in which t k represents a discrete time corresponding to the time that appeared in Eq. (2.63). Also, for the case, D(t) is known and the relaxation modulus, E(t), is desired, the following system of a linear algebraic equation is used: [A]{E} = {B}

(2.83)

in which, 

      n tk tk τi tk + exp − − exp − Dj Aki = Dg exp − τi τi − ρ j τi ρi j=1

(2.84)

for τ i /= ρ j , and, 

 n tk tk + Aki = Dg exp − Dj τi ρ j j=1

(2.85)

for τ i = ρ j .

Bk = 1 −

Dg +

!  " D j 1 − exp − ρtkj Dg + nj=1 D j

n

j=1

(2.86)

where t k again represents a discrete time corresponding to the time that appeared in Eq. (2.63).

2.6.2.2

Viscoelastic Liquids

Similar to the procedure introduced in Sect. 2.6.2.1 for viscoelastic solids, it is possible to develop equations required for viscoelastic liquids, with consideration of some assumptions (Eq. 2.67b) specific to this type of material. By substituting the Prony series form of relaxation modulus and creep compliance into Eq. (2.63), it can be written as:

34

2 Viscoelasticity Theoretical Background

 t 

m i=1

0

⎡ ⎤     n

D 1 ψ t −ψ j ⎣ Dg δ(ψ) + ⎦dψ = 1 + exp − E i exp − τi η0 ρ ρ j j j=1 (2.87)

and by re-ordering summation and integration, the following equation can be written:

Dg

m

i=1

+

    t n 1

t (t − ψ) + dψ E i exp − E i exp − τi η0 i=1 τi

n m

i=1 j=1

0

 t       Ei D j ψ t ψ − dψ = 1 exp − ex p − ρj τi ρj τi

(2.88)

0

in which δ(.) represents the Dirac delta function. There are two simple integrals in Eq. (2.69) that can be determined as follows: There are two simple integrals in Eq. (2.69) that can be determined as follows: t 0

    (t − ψ) dψ = τi 1 − exp(−t/τi ) exp − τi

(2.89)

and, t 0

           τi ρ j ψ ψ ψ ψ − dψ = 1 − exp − − exp − ρj τi τi − ρ j ρj τi (2.90)

for τ i /= ρ j , and, t 0

     ψ ψ − dψ = t exp − ρj τi

(2.91)

for τ i = ρ j . By inserting Eqs. (2.90) or (2.91) and Eq. (2.89) into Eq. (2.88), the following equation can be derived:  m   n

E i τi   t  t exp − − exp − Dj τ − ρj τi ρj j=1 i=1 i      m m

1

t t − E i τi 1 − exp − E i exp − = 1 − Dg τi η0 i=1 τi i=1

(2.92)

2.7 Time–Temperature Superposition

35

for τ i /= ρ j , and,  m     n m

Ei

t t Dj t exp − E i exp − =1− D ρj τi τi i=1 j=1 i=1    m t 1

− E i τi 1 − exp − η0 i=1 τi

g

(2.93)

for τ i = ρ j . Equation (2.78) can be re-written by considering the main assumptions of viscoelastic liquids (E e = 0): 1 Dg ∼ = m i=1

Ei

(2.94)

and η0 can also be determined as: η0 =

m

τi E i

(2.95)

i=1

2.7 Time–Temperature Superposition 2.7.1 Effect of Time, Temperature, and Loading Rate Viscoelastic materials exhibit time-dependent, temperature-dependent, and loading rate-dependent behavior. Some constructional materials, such as bitumen and resin, are known to exhibit viscoelastic behavior. Figure 2.18 shows the complex shear modulus, G*, for an epoxy resin at three temperatures of 70, 90, and 100 °C measured by performing a frequency sweep test on a disk of resin with 8 mm diameter and 2 mm thickness using a dynamic shear rheometer (DSR). It can be observed that the complex shear modulus increases with increasing frequency and decreases with increasing temperature. Another example is shown in Fig. 2.19, which illustrates the flexural creep stiffness, S(t), of a bituminous composite containing different volume filling ratios (0, 10, 15, 20, 25, 30, 35, and 40%) of siliceous filler particulates at a specific time at different sub-zero temperatures. The flexural creep stiffness was measured based on ASTM D6648 (Hajikarimi et al. 2022). As can be seen in this figure, S(t) increases with decreasing temperature and increasing volume filling ratio.

36

2 Viscoelasticity Theoretical Background

Fig. 2.18 Complex shear modulus versus applied angular frequency for epoxy resin at different temperatures

Fig. 2.19 Flexural creep stiffness at a specific time, t = 60 s, for a bituminous composite containing different volume filling ratios of siliceous filler at sub-zero temperatures

2.7 Time–Temperature Superposition

37

Fig. 2.20 Relaxation modulus of an original bitumen at five test temperatures

2.7.2 Master Curve A master curve is a useful tool for visualizing the behavior of a viscoelastic material across a wide range of frequencies or times at a specific temperature, known as the “reference temperature.” To construct a master curve, the time–temperature superposition principle (TTSP) should be used. An example is presented in this section to better describe the concept of master curve and the method to construct it. Figure 2.20 shows the relaxation modulus for an original bitumen at five temperatures of −6, −12, −18, −24, and −30 °C measured by performing a standard three-point bending test using a bending beam rheometer in a limited time interval, i.e., 240 s. In Fig. 2.21, it can be observed that by taking a reference temperature of −18 °C, the curves corresponding to temperatures above −18 °C were shifted towards the right, whereas those corresponding to temperatures below −18 °C were shifted towards the left. These shifts were made to create a continuous curve, known as the “master curve”, which represents the relaxation modulus of the original bitumen at −18 °C for a wide range of time, compared to the 240 s duration of the main test performed at five different test temperatures. When a master curve is formed through the horizontal shifting of curves to the left and right, the viscoelastic material is classified as a thermo-rheologically simple material (Brinson and Brinson 2008). Some notable facts can be found by considering the procedure of master curve construction and the master curve constructed in Fig. 2.21, as follows: (1) The selection of the reference temperature (−18 °C) is critical as it should be chosen from the set of all test temperatures at which the desired viscoelastic

38

2 Viscoelasticity Theoretical Background

Fig. 2.21 Constructing master curve of relaxation modulus of an original bitumen at the reference temperature of −18 °C, a the horizontally shifting process, and b the final master curve

2.7 Time–Temperature Superposition

39

function is measured. The master curve constructed represents only the rheological behavior of the original bitumen at the reference temperature, and it is not possible to obtain its rheological properties directly for other temperatures. (2) Even though the relaxation modulus was measured through a bending experiment for 240 s at each test temperature, the master curve describes the relaxation modulus in a wider time range from 0.002 to 20,000 s, indicating a broad range of applicability. This unique feature of the master curve can be utilized to overcome experimental limitations and testing constraints, such as the duration and cost of performing tests that require a longer duration (e.g., 20,000s or 5.5 hours). (3) The higher temperature is associated with the right-hand side of the master curve, indicating a longer time, while the lower temperature is linked with the left-hand side of the master curve, representing a shorter time compared to the reference temperature. (4) By horizontally shifting the constructed master curve to the left or right, a master curve can be generated for a lower or higher reference temperature. The TimeTemperature Superposition Principle (TTSP) can be employed to adjust the time range for the master curves at various temperatures, trading-off temperature for time (or frequency). The TTSP method allows the determination of viscoelastic functions such as relaxation modulus as a function of time over several decades in time (or frequency) by performing rheological or mechanical experiments at different temperatures within a limited time range with a short duration. Determining the horizontal shifting of a master curve for other desired temperatures in the absence of experimental data is a significant concern. To address this, two established shift functions have been introduced: the Williams-Landel-Ferry (WLF) function (Williams et al. 1955), and the Arrhenius Activation Energy equation. These functions provide a means to determine whether the curve should be shifted to the right or left in order to achieve the desired master curve. To calculate the shift factor of each temperature, Willimas et al. (1955) introduced the following equation: log(aT ) =

−c1 (T − Tref ) c2 + (T − Tref )

(2.96)

where aT is the shift factor at temperature T, T ref is the reference temperature, c1 and c2 are constant coefficients calculated by fitting Eq. (2.96) on shift factors for test temperatures. Regarding the main variable of the problem (time or frequency), the following definition can be used for the shift factor: aT =

t tR

(2.97)

in which t is the real time of the experiment and t R is reduced time in the master curve, and:

40

2 Viscoelasticity Theoretical Background

Fig. 2.22 Shift factors for the master curve shown in Fig. 2.21

aT =

ωR ω

(2.98)

where ωR is the reduced frequency in the master curve, and ω is the real frequency of the experiment. The shift factors used to construct the master curve shown in Fig. 2.21 is illustrated in Fig. 2.22. Having a shift function like WLF, it is possible to determine the desired viscoelastic function at any temperature within the test temperature range. The WLF equation is only valid for temperatures greater than glass transition temperature, T g since for temperatures lower than T g , it is not possible to consider a viscoelastic material as a super cooled liquid (Brinson and Brinson 2008). Ferry (1980) pointed out that the slope of the shift factor function should be discontinuous at the T g as the thermal expansion coefficient suffers a discontinuity at the T g . The Arrhenius activation energy equation (Menzinger and Wolfgang 1969) shown in Eq. (5.6) was used to develop a shift function for temperatures below the T g .   Ea τ (T ) = A exp − RT

(2.99)

where τ is the relaxation time, T is the absolute temperature, E a is the activation energy, and R is the gas constant. Applying the ln operation on both sides of Eq. (2.99) resulted in: ln[τ (T )] = ln(A) −

Ea RT

(2.100)

Then, taking the ratio at an arbitrary temperature and the T g will give after converting to base ten logarithm as: 

τ (T ) log(aT ) = log   τ Tg

  1 1 Ea − =− 2.303R T Tg

(2.101)

References

41

References Alfrey T (1944) Non-homogeneous stresses in visco-elastic media. Q Appl Math 2(2):113–119 Boyce WE, DiPrima RC, Meade DB (2017) Elementary differential equations. John Wiley & Sons Brinson HF, Brinson LC (2008) Polymer engineering science and viscoelasticity: an introduction. Springer Burgers JM (1935) First and second report on viscosity and plasticity. Acad Sci Amsterdam, Amsterdam, NL [reference not Consult 368] Ferry JD (1980) Viscoelastic properties of polymers. Wiley Findley WN, Lai JS, Onaran K (1976) Creep and relaxation of nonlinear materials. North-Holland, Amsterdam Flügge W (2013) Viscoelasticity. Springer Science & Business Media Hajikarimi P, Nejad FM (2021) Applications of viscoelasticity: Bituminous materials characterization and modeling. Elsevier Hajikarimi P, Nejad FM, Aghdam MM (2018) Implementing general power law to interconvert linear viscoelastic functions of modified asphalt binders. J Transp Eng Part B Pavements 144. https://doi.org/10.1061/JPEODX.0000038 Hajikarimi P, Ehsani M, EL Haloui Y, et al (2022) Fractional viscoelastic modeling of modified asphalt mastics using response surface method. Constr Build Mater 317:125958. https://doi.org/ 10.1016/j.conbuildmat.2021.125958 Hopkins IL, Hamming RW (1957) On creep and relaxation. J Appl Phys 28:906–909. https://doi. org/10.1063/1.1722885 Huang YH (2004) Pavement analysis and design. Upper Saddle River, NJ: Pearson Prentice Hall Lee EH (1955) Stress analysis in visco-elastic bodies. Q Appl Math 13(2):183–190 Maxwell JC (1867) IV. On the dynamical theory of gases. Philos Trans R Soc London 49–88 Menzinger M, Wolfgang R (1969) The meaning and use of the arrhenius activation energy. Angew Chemie Int Ed English 8:438–444. https://doi.org/10.1002/anie.196904381 Ottosen NS, Ristinmaa M (2005) The mechanics of constitutive modeling. Elsevier Park SW, Schapery RA (1999) Methods of interconversion between linear viscoelastic material functions. Part I—a numerical method based on Prony series. Int J Solids Struct 36:1653–1675. https://doi.org/10.1016/S0020-7683(98)00055-9 Read WT (1950) Stress analysis for compressible viscoelastic materials. J Appl Phys 21(7):671–674. https://doi.org/10.1063/1.1699729 Theocaris PS (1962) Viscoelastic properties of epoxy resins derived from creep and relaxation tests at different temperatures. Rheol Acta 2:92–96 Voigt W (1892) Ueber innere Reibung fester Körper, insbesondere der Metalle. Annalen Der Physik 283(12):671–693. https://doi.org/10.1002/andp.18922831210 Williams ML, Landel RF, Ferry JD (1955) The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J Am Chem Soc 77:3701–3707 Xu Y, Shan L, Tian S (2019) Fractional derivative viscoelastic response model for asphalt binders. J Mater Civ Eng 31(6):04019089. https://doi.org/10.106/(ASCE)MT.1943-5533.0002716

Chapter 3

Nonlinear Viscoelasticity

3.1 Introduction There are two main categories for mechanical behavior of viscoelastic materials regarding their stress–strain relationship, including linear and nonlinear viscoelasticity. The response stress (or response strain) is proportional to the applied strain (or stress) for a specific time and temperature in linear viscoelastic materials. Figure 3.1 simply represents the main difference between linear and nonlinear behavior of a viscoelastic material. However, there is no direct relationship between variation in the applied stress (or strain) and the corresponding strain (or stress) variation in nonlinear viscoelastic materials (Ottosen and Ristinmaa 2005). In this chapter, a brief review on nonlinear viscoelasticity is presented for explaining the stress–strain relationship based on Schapery’s robust framework. After that, the method for determination of parameters of a nonlinear viscoelastic material is introduced. Finally, the numerical framework to simulate a nonlinear viscoelastic material is developed.

3.2 Schapery Single-Integral Nonlinear Model The Schapery single-integral approach (Schapery 1964, 1969), which originated from the irreversible thermodynamic procedures devised by Biot (1958), has become the most commonly utilized technique for characterizing the nonlinear time-dependent behavior of polymers. This section aims to introduce the method as a means of representing viscoelastic materials data and to provide a basic understanding of how to determine the necessary material parameters from experimental data. It should be noted that the goal of developing a relatively simple and easy-to-use single integral approach is not only to simplify the calculation of required material parameters, but also to provide a technique that can be used with greater ease and

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. Hajikarimi and A. Sadat Hosseini, Constructional Viscoelastic Composite Materials, https://doi.org/10.1007/978-981-99-1786-0_3

43

44

3 Nonlinear Viscoelasticity

Fig. 3.1 Difference between a linear and b nonlinear viscoelastic behavior

confidence in solving nonlinear boundary value problems to find stress, strain, and displacement distributions for engineering design purposes.

3.2.1 A Brief Review on Linear Viscoelasticity In Chap. 2, it was shown that the hereditary convolution integrals can be used to represent linear viscoelastic materials, t ε(t) =

D(t − ξ ) −∞

dσ (ξ ) dξ dξ

(3.1)

dε(ξ ) dξ dξ

(3.2)

t σ (t) =

E(t − ξ ) −∞

in which all former history of loading is included in the lower limit. Without previous history prior to time zero, the lower limit was t = 0− and the equations are, t ε(t) =

D(t − ξ ) 0−

dσ (ξ ) dξ dξ

(3.3)

3.2 Schapery Single-Integral Nonlinear Model

45

t σ (t) =

E(t − ξ ) 0−

dε(ξ ) dξ dξ

(3.4)

Also, Eqs. (3.3) and (3.4) can be written as, t ε(t) = σ0 D(t)H (t) +

D(t − ξ )

dσ (ξ ) dξ dξ

(3.5)

E(t − ξ )

dε(ξ ) dξ dξ

(3.6)

0+

t σ (t) = σ0 E(t)H (t) + 0+

When a step input is applied at time t = 0, it is possible to obtain an alternate form by decomposing the creep compliance, D(t), into two distinct components, namely instantaneous and transient terms, as follows: ˜ D(t) = D0 + D(t)

(3.7)

or by decomposing the relaxation modulus into equilibrium and transient terms as, ˜ E(t) = E ∞ + E(t)

(3.8)

For a three-parameter solid, the creep compliance may be written as an example,  D(t) =

   t 1 1 ˜ 1 − exp − = D0 + D(t) + E0 E1 τ

(3.9)

where D0 =

   t 1 1 ˜ 1 − exp − and D(t) = E0 E1 τ

(3.10)

Similarly, for a three-parameter solid, the relaxation modulus is,  E(t) = q0 +

   t q1 − q0 exp − p1 p1

(3.11)

where, E∞

E0 E1 ˜ and E(t) = = q0 = E0 + E1



 −t q1 − q 0 e / p1 p1

(3.12)

46

3 Nonlinear Viscoelasticity

By utilizing the decomposed representations of creep compliance or relaxation modulus, it becomes possible to reformulate the linear viscoelastic constitutive equations in the following manner. t ε(t) = D0 σ (t)H (t) + 0−

t σ (t) = E 0 σ (t)H (t) + 0−

˜ − ξ ) dσ (ξ ) dξ D(t dξ

(3.13)

˜ − ξ ) dε(ξ ) dξ E(t dξ

(3.14)

which are used as the base forms for the Schapery nonlinear model.

3.2.2 The Schapery Equation Schapery (1964, 1966) formulated a single-integral representation to illustrate the strains arising from a stress input that varies with time, utilizing the principles of irreversible thermodynamics (or energy) to describe the state of a viscoelastic material subjected to external loads. t ε(t.σ ) = g0 D0 σ (t)H (t) + g1 0−

  d[g2 .σ (ξ )H (ξ )] ' dσ (ξ ) ˜ dξ D(Ψ − Ψ ) dξ dξ (3.15)

in which g0 , g1 , g2 , and aσ are stress-dependent material parameters. The parameter aσ serves as a shift factor that alters the time scale in a similar fashion to the temperaturedependent shift factor, aT , which modifies the time scale for temperature-related effects. The shifted time scale that is dependent on stress can be expressed as follows, t Ψ(t.σ ) = 0

dt aσ (t)

(3.16)

dξ aσ (ξ )

(3.17)

and '

ξ

Ψ (ξ.σ ) = 0

3.2 Schapery Single-Integral Nonlinear Model

47

The Schapery method given by Eqs. (3.15), (3.16), and (3.17) is a mathematical definition of a time-stress-superposition-principle or TSSP which is similar to the time–temperature-superposition-principle (TTSP) introduced in Sect. 2.7.2. Having a variable strain input, a similar equation for stress was also developed by Schapery as follows, t σ (t.ε) = h ∞ E ∞ σ (t)H (t) + h 1 0−

  d[h 2 .ε(ξ )H (ξ )] ' ˜ dξ E(Ψ −Ψ ) dξ t

Ψ(t.ε) = 0

'

ξ

Ψ (ξ.ε) = 0

(3.18)

dt aε (t)

(3.19)

dξ aε (ξ )

(3.20)

It should be noted that if the nonlinear parameters are uniformly equal to one, the Boltzmann superposition integral for linear viscoelasticity can be retrieved from Eq. (3.15). g0 = g1 = g2 = aσ = 1

(3.21)

Additionally, If all parameters with the exception of aσ are equivalent to unity, Knauss’s free volume model (Knauss and Emri 1981) can be obtained in which, g0 = g1 = g2 = 1

(3.22a)

aσ / = 1

(3.22b)

aσ ∼ FreeVol.(f)

(3.23c)

f = f 0 + αΔT + βΔσ + γ ΔC

(3.24d)

in which, α denotes the coefficient of thermal expansion, β represents a parameter that links stress with the quantity of free volume, and γ denotes the association between moisture concentration and free volume.

48

3 Nonlinear Viscoelasticity

3.2.3 Schapery Equation for a Two-Step Stress To apply the Schapery equation, it is crucial to identify the material parameters. This section focuses on the derivation of the Schapery equation for a basic two-step load. It is assumed that a general two-step stress distribution is applied in such a way that, σ (t) = σa H (t) − (σ b − σa )H (t − t a )  σ (t) =

σa .0 ≤ t ≤ ta σb .t > ta

(3.25a)

(3.25b)

Figure 3.2 also illustrates the graphical form of the two-step stress. Since stress remains constant in each of the two time regions (before and after t a ), the nonlinear parameters are also constant in each of those regions. Denoting gi and aσ for t ≤ t a as gi a and aσ a respectively; and similarly for t > t a as gi b and aσ b , the Schapery’s Eq. (3.15) can be rewritten by adding the corresponding superscripts. The superscript for the aσ and the g1 coefficients indicate the time interval for evaluating the term, while those for g0 and g2 coefficients correspond to the stress value that each coefficient modifies. Thus, the modified Eq. (3.15) can be expressed as follows,

ε(t) = g0a σa D0 H (t) + g0b σb − g0a σa D0 H (t − ta ) t d ' ˜ − Ψ ) (g2a σa H (ξ ) + (g2b σb − g2a σa )H (ξ − ta ))dξ (3.26) + g1 D(Ψ dξ 0−

It is possible to discretize the integral into two separate integrals by dividing it at a point before and after t a . This approach involves using the derivative of the step function as the dirac delta function. Fig. 3.2 Two-step creep load

3.2 Schapery Single-Integral Nonlinear Model

49

ε(t) = g0a σa D0 H (t) + g0b σb − g0a σa D0 H (t − ta ) −

ta + g1

' ˜ D(Ψ − Ψ )(g2a σa δ(ξ ) + (g2b σb − g2a σa )δ(ξ − ta ))dξ

0−

t +

g1b

' ˜ D(Ψ − Ψ )(g2a σa δ(ξ ) + (g2b σb − g2a σa )δ(ξ − ta ))dξ

(3.27)

ta−

Regarding the first integral, the exponent assigned to the g1 term is undetermined, as it varies based on the time period being evaluated. Any terms denoted with a zero are considered inconsequential and do not contribute to the integral in which they are present. Therefore, the resulting expression is simplified by ignoring such terms,  ε(t) = g0a σa D0 H (t) + g0b σb − g0a σa D0 H (t − ta ) −

ta + g1

'

˜ D(Ψ − Ψ )(g2a σa δ(ξ ))dξ + g1b

0−

t

'

˜ D(Ψ − Ψ )(g2b σb − g2a σa )δ(ξ − ta ))dξ

ta−

(3.28) For 0 ≤ t ≤ t a , the expression simplifies to, t ε(t) =

g0a D0 σa H (t)

+

g1a

' ˜ D(Ψ − Ψ )(g2a σa δ(ξ ))dξ

(3.29)

0−

in which the integrand must be determined at ξ = 0. Consequently, the effective times may be calculated as t Ψ= 0− '

t

Ψ = 0−

dt t = a aσa aσ

(3.30)

'

dt ξ = a =0 a aσ aσ

(3.31)

With these considerations, Eq. 3.29 becomes,    t σa H (t)0 ≤ t ≤ ta ε(t) = g0a D0 + g1a g2a D˜ aσa

(3.32)

50

3 Nonlinear Viscoelasticity

For the interval, t > t a , the strain is presented by, −

ta

˜ ε(t) = g1b D0 σb H (t − ta ) + g1b D(Ψ − Ψ ' )(g1a σa δ(ξ ))dξ

  0−

  t ˜ − Ψ ' )(g2b σb − g2a σa )δ(ξ − ta ))dξ + g1b D(Ψ

ta−



(3.33)



In Eq. (3.33) represents the impact of a stress step input, whereby σ b = σ a + (σ b – σ a ), occuring at t = t a . This term accounts for the effect of the σ a stress step input that takes place at t = 0. The subsequent term represents the temporary portion of the σ a , stress step input that extends beyond t = t a . The third term represents the temporary portion of the σ b – σ a , stress step input that occurs at t = t a . The first integral must be determined at τ = 0 and the second at τ = t a to calculate the effective times. For the first integral, t Ψ= 0−

dt = aσ Ψ=

ta 0−

dt + aσa

t ta

dt ' andΨ = b aσ

ξ 0−

dξ aσ

ta t − ta ' + and Ψ = 0 aσa aσb

(3.34)

(3.35)

For the second integral, t Ψ= ta

Ψ=

dt ' and Ψ = b aσ

τ τa

dξ aσb

t − ta ' andΨ = 0 aσb

(3.36)

(3.37)

Consequently, the final equation for creep strain for t > t a is, ε(t) =

g0b D0 σb

+

g1b g2a D˜



    ta t − ta t − ta b b a ˜ + σa + g1 (g2 σb − g2 σa ) D aσa aσb aσb (3.38)

3.2 Schapery Single-Integral Nonlinear Model

51

3.2.4 Schapery Equation for a Creep and Creep-Recovery Test Schapery proposed a procedure for calculating the stress dependent parameters g0 , g1 , g2 , aσ by using a creep and creep-recovery test (see Fig. 3.3). It is a special case of the two-step stress input in which σa = σ0 and σb = 0 and therefore σ (t) = σ0 H (t) − σ0 H (t − t 1 )  σ (t) =

σ0 , 0 ≤ t ≤ t1 0, t > t1

(3.39)

and the parameters related to σ b are, g0b = g1b = g2b = gσb = 1

(3.40)

∼

It is necessary to specify D (t) to calculate the parameters and for that Schapery proposed using a power law given by, D(t) = D0 + D1 t n

(3.41)

∼

where D0 , D1 , and n are constants and D (t) is given by, ˜ D(t) = D1 t n

(3.42)

Based on Eq. (3.32), the creep strain within the time interval of 0 to t1 becomes after substituting the conditions provided in Eqs. (3.39) to (3.42),  n   t1 σ0 H (t) 0 ≤ t ≤ t1 ε(t) = g0 D 0 + g1 g 2 D1 aσ

Fig. 3.3 Two-step creep load and creep-recovery load

(3.43)

52

3 Nonlinear Viscoelasticity

The recovery strain for t > t 1 is determined from Eq. (3.38) after substituting the same conditions given in Eq. (3.37) and Eqs. (3.39) to (3.42) is, 

 εr (t) = D1

t1 + t − t1 aσ

n

 − D1 (t − t1 ) g2 σ 0 n

(3.44)

To accurately describe the nonlinear uniaxial creep and creep-recovery behavior of a viscoelastic material, seven material properties must be determined. These include the constants D0 , D1 , and n as well as the stress dependent parameters g0 , g1 , g2 , and aσ which are defined in Eqs. (3.43) and (3.44). If only creep behavior is being analyzed, five parameters are required, with g1 and g2 being combined with D1 . However, to differentiate between g1 and g2 , the recovery strain must also be considered. Additionally, the value of n is sensitive to the length of the creep or creep recovery test, and it is recommended to use recovery data to determine its value as proposed by (Brinson and Brinson 2008), Lou and Schapery (1971). Determining the values of these parameters requires considering the strain jumps at the initial load and unloading, which arise logically from the theory. However, when dealing with experimental data, it is important to note that ε(t = 0+ ) and ε(t = t 1 + ) are not well-defined quantities, as creep or recovery stresses (or strains) cannot be instantaneously applied without causing dynamic effects. Furthermore, if the jump stresses are instantaneous at t = 0+ and at t = t 1 , the theory predicts that the instantaneous jump strains will not be equal in magnitude. For instance, Eq. (3.44) can be re-written to reflect this,  εr (t) = g2 D 1

  n  n aσ aσ (t − t1 ) − (t − t1 )n σ0 t1 t1  n  t1  εr (t) = g2 D 1 [1 + aσ λ]n − (aσ λ)n σ0 aσ t1 aσ

n 



1+

(3.45) (3.46)

where λ=

t − t1 t1

(3.47)

The creep and creep-recovery data are as shown in Fig. 3.4. The instantaneous creep recovery, Δε(t 1 ) is as follows,



Δε(t1 ) = ε t1+ − ε t1−

(3.48)

Based on Eqs. (3.43) and (3.44),  n  

t1 σ0 ε t1− = g0 D0 + g1 g2 D 1 aσ

(3.49)

3.2 Schapery Single-Integral Nonlinear Model

53

Fig. 3.4 Creep and creep-recovery: applied stress and strain response



εr t1+ = g2 D 1



t1 aσ

n σ0

(3.50)

or



Δε(t1 ) = ε t1+ − ε t1− = −g0 D0 σ0 + g2 D 1



t1 aσ

n (1 − g1 )σ 0

(3.51)

It is necessary to mention that the jump in strain when the creep load is applied at t = 0 is, ε(t = 0) = g0 D0 σ0

(3.52)

Through a comparison of Eqs. (3.51) and (3.52), it becomes evident that the difference in strain jump discontinuity at times t = 0 and t = t 1 is not equal, despite an equal magnitude of stress change. This discrepancy is linked to the nonlinearity of the material, which is reliant on its load history. The parameter g1 can be identified as the origin of this variation, as in the case of a linear material g1 = 1 and Eq. (3.51) condenses to the negative of Eq. (3.52) as is anticipated for a linear viscoelastic material.

3.2.5 Determination of Material Parameters from a Creep and Creep-Recovery Test To determine the seven material parameters to represent nonlinear viscoelastic behavior implementing the Schapery model, the technique most often found in the technical literature is numerical fitting of experimental data both in the linear and nonlinear range at various stress levels (Peretz and Weitsman 1982; Rochefort and Brinson 1983; Tuttle and Brinson 1985). In this section, the approach originally introduced by Schapery and his coworkers (1969; Lou and Schapery 1971) is presented,

54

3 Nonlinear Viscoelasticity

which provides an appropriate insight into the meaning and source of the nonlinear parameters. In order to calculate the seven material parameters using this original approach, it is required to perform creep and creep-recovery experiments at several stress levels.

3.2.5.1

Determination of D0 , D1, and n

Before the nonlinear parameters can be accurately determined, it is necessary to establish the linear material constants from experimental data within the linear stress range. In the case of linear viscoelastic response, if the transient strain is significantly greater than the initial step input, the strain versus time on a log-log plot follows a straight line at extended times, with the slope of the line representing the power law exponent, n. Furthermore, the initial strain jump, ε0 , can be expressed as g0 D0 σ 0 , and since g0 equals 1 in the linear range, this provides a means to determine D0 . However, these methods of determining n and ε0 are imprecise, owing to the inability to apply a truly instantaneous stress jump as previously noted. Consequently, in general, ε0 must be treated as a fitting parameter that necessitates determination in addition to the seven other parameters. The following section presents and discusses this approach. From Eq. (3.43), the transient creep strain at t = t 1 shown in Fig. 3.4 is,  εT (t1 ) = g1 g2 D1

t1 aσ

n σ0

(3.53)

The creep and creep-recovery strain, Eqs. (3.49) and (3.50), can now be determined as,

ε t1− = [g0 D0 + εT (t1 )]σ0 εr (t) =

εT (t1 ) {[1 + aσ λ]n − (aσ λ)n } g1

(3.54) (3.55)

in which, λ=

t − t1 t1

(3.56)

Equation (3.55) can be implemented to determine n and εT (t 1 ) for a creep and creep-recovery stress in the linear range by recalling that all nonlinear terms are unity, i.e., g0 = g1 = g2 = aσ = 1, and thus   εr (t) = (1 + λ)n − λn εr (t1 )

(3.57)

3.2 Schapery Single-Integral Nonlinear Model

55

In Eq. (3.57), the left-hand side numerator represents the experimental data, while the right-hand side represents a mathematical representation of the data used to determine n and εT (t 1 ). The right-hand side can be displayed as a parametric family of curves with respect to n as depicted in Fig. 3.5 by the solid lines. The left-hand side numerator corresponds to creep-recovery data for a stress level within the linear range and is depicted by square symbols in Fig. 3.5. The denominator indicates the amount, εT (t 1 ), that the linear recovery data must be shifted downward on a log scale to fit the curve with the appropriate exponent, which is equal to the transient creep strain at t1 for the same stress level in the linear range. The diamond symbol shows that the recovery strain, when shifted downward by the correct amount, corresponds to the exponent n = 0.15. As a result, the power law exponent and the transient creep strain, εT (t 1 ), for the specific stress level utilized in the linear range, can be determined. Subsequently, by solving the Eq. (3.57) using data collected at two different values of λ, both n and εT (t 1 ) can be calculated. Ideally, the values of n and εT (t1 ) should be consistent across various values of λ, but minor variations may arise due to experimental inaccuracies. To account for this, an average of the values obtained from different points should be taken. Using the determined values of n and εT (t1 ),

Fig. 3.5 Procedure for finding the parameter n (Brinson and Brinson 2008)

56

3 Nonlinear Viscoelasticity

D1 can be computed from Eq. (3.53) when the nonlinear parameters are set to unity. However, the original methodology deviates slightly from this approach. The equation for creep strain, Eq. (3.43), for a linear viscoelastic material can be written as,   ε(t) = D0 + D1 t n σ0 = ε0 + ε1 t n

(3.58)

in which ε0 represents the initial strain and ε1 represents the transient strain coefficient. By selecting two different values of transient strain at two distinct times for a stress level within the linear range, two equations can be established to determine ε0 and ε1 . Subsequently, the coefficients D0 and D1 can be computed accordingly. It is important to note that the chosen strains should exceed five times the duration required to apply the initial stress (Lou and Schapery 1971). Once ε0 and ε1 are known for a stress level within the linear range, it is possible to determine εT (t 1 ) for the same stress level, and the calculated value should correspond to the value obtained by shifting the linear data in Fig. 3.5. However, a slight discrepancy may arise due to experimental error.

3.2.5.2

Determination of the Quantities g0 and

g1 g2 aσn

Implementing the creep strain presented in Eq. (3.43),  n   t σ0 ε(t) = g0 D 0 + g1 g2 D 1 aσ

(3.59)

and two values of measured strain for two time values will allow determining of g0 and ga1 gn 2 at each nonlinear stress level. This also resulted in the initial strain, ε0 , ε1, σ and the transient strain, εT (t 1 ), for each stress level in the nonlinear range.

3.2.5.3

Determination of g1 , g2 , and aσ

Consider the recovery data depicted in Fig. 3.6 and the recovery strain determined in Eq. (3.55) written as,   εr (t) = (1 + aσ λ)n − (aσ λ)n εT (t1 )/g1

(3.60)

By applying the power law exponent n = 0.15, the data for all stress levels can be shifted to align with the linear viscoelastic data depicted in Fig. 3.5. This can be achieved by displacing each curve downwards by a certain amount and to the left by a certain amount aσ , as shown in ε Tg(t 1 ) Fig. 3.5, resulting in the formation of a 1

3.2 Schapery Single-Integral Nonlinear Model

57

Fig. 3.6 Creep-recovery data and shifting process to form a master curve

master curve. This process is analogous to the creation of time-temperature master curves as discussed in Chap. 2, and is known as the analytical basis for the timestress-superposition-principle (TSSP). Previously calculated transient strain, εT (t 1 ), for each stress level in the nonlinear range can be used to calculate g1 which in conjunction with aσ provides a complete set of known parameters to determine g2 . As a result, all other quantities in the expression ε1 = gg1 g1 2 D1 σ0 are known. All parameters are now known that are required to predict the response of a nonlinear viscoelastic material using the Schapery technique. Obtaining reliable experimental data from creep and creep-recovery tests for high performance materials such as fiber reinforced polymer matrix composites and thin film adhesives is challenging. Strain gauges are commonly used to measure strains in the test specimens. However, the reinforcement of strain gauges is estimated to be around 2% for glass-epoxy specimens, which is considered negligible. Nevertheless, for very thin specimens, the error caused by strain gauge heating effects cannot be ignored, but it can be minimized by limiting the amount of current used or pulsing the current to the gauge. Prior to conducting creep and creep-recovery tests, it is advantageous to mechanically condition specimens made of continuous or chopped glass fibers composites and film type adhesives. This is because the accumulation of deformation arising from flaws created during processing can result in an initial strain and transient strain, which cannot be identified without mechanical conditioning prior

58

3 Nonlinear Viscoelasticity

to a creep test. For instance, Lou and Schapery (1971) used cyclic constant strain rate tests of about 50% of ultimate to condition each test specimen. A pertinent issue arises from the discrepancy in the duration of short-term laboratory tests and the actual long-term service life of structures: "What is the level of trustworthiness associated with using predictive equations that were derived from short-term tests to design structures intended to last for significantly longer periods of time?" This question is relevant since the laboratory testing protocols are typically designed to run for less than one hour of creep and less than two hours of recovery, which contrasts with the extended periods of time that materials are expected to withstand in real-world applications, spanning from days, to months, or even years. Tuttle (Tuttle and Brinson 1985) conducted short-term tests on 90º and 10º unidirectional graphite epoxy specimens with a creep time of 480 min and a recovery time of 120 min to address the reliability of predictive equations derived from shortterm tests. The data obtained from these tests for the seven required parameters was used in the Schapery model along with laminate theory analysis to predict the longterm creep response in the matrix-dominated direction for a symmetric composite laminate. However, when the results were compared to independent long-term creep tests on the appropriate laminate, it was observed that the analysis underestimated the response at 105 min by approximately 8%. Hence, for relatively longer periods, such as several years, the error would be even greater. A sensitivity analysis revealed that the power law exponent was the primary contributor to the error. This could be due to the power law’s inability to provide the best fit to general viscoelastic creep compliance data, as previously observed. The Schapery model’s parameters can be determined using the power law due to its limited number of parameters, making it well-suited for semi-graphical methods. Nonetheless, the use of numerical methods mentioned previously can lead to superior outcomes, particularly when employing a Prony series expansion for the creep compliance, as shown by (Tuttle et al. 1995). It is essential to consider that the nonlinear characterization method proposed by Schapery is highly suitable for materials that do not display any permanent deformation after the stress is eliminated, making it an appropriate choice for cross-linked or thermosetting polymers rather than linear polymers or thermoplastics. Nonetheless, in case there is any residual permanent deformation present, this approach can still be employed by compensating for it through the subtraction of its contribution from the creep and creep recovery response. Nonetheless, the residual deformation ought to be limited in magnitude, and an effective procedure must be employed to distribute it uniformly over the entire time scale.

3.3 Numerical Formulation To use the Schapery nonlinear theory in numerical problems, the uniaxial form of the theory given in Eq. (3.15) needs to be incrementally formulated. In the technical literature, various numerical approaches are implemented to solve the nonlinear

3.3 Numerical Formulation

59

viscoelastic integral equations (Lai and Bakker 1996; Poon and Ahmad 1999; Eyad and Niranjanan 2002; Haj-Ali and Muliana 2004). The formulation given by Lai and Bakker (1996) is based on the assumption that the nonlinear stress-based parameters are constant over the time increment. A recent formulation (Haj-Ali and Muliana 2004; Huang et al. 2007) incorporates the possible variation of the nonlinear stressbased parameters over the time increment. It has been reported that the latter approach allows larger time steps to be considered in performing simulations. To utilize the latest numerical method, it is necessary to have the knowledge of nonlinear parameters g1 and g2 to be known. The method presented by Lai and Bakker (1996), which assumes constant nonlinear stress-based parameters over each time increment, requires only the product of nonlinear parameters g1 and g2 to be known. Huang et al. (2007) state that this method can face convergence issues when using large time increments. However, it is expected to produce comparable results with smaller time increments.

3.3.1 One-Dimensional Formulation Equation (3.15) involves a convolution integral, where the creep compliance of the material needs to be mathematically represented for numerical calculations. However, using the Modified Huet-Sayegh (MHS) model in the integral formulation can be complicated. Additionally, when performing computations with small time steps, as required in nonlinear simulations, using the MHS model can result in a significant computational memory requirement. In contrast, linear springdashpot models are computationally efficient and are commonly used in nonlinear formulations, as noted by (Henriksen 1984). Following the selected numerical approach, the transient component of the creep compliance ΔD(ψ) in the Schapery theory can be given by: ΔD(ψ) =

N Σ n=1

   −ψ Dn 1 − exp τn

(3.61)

The parameters Dn and τ n are acquired from experimental data. It is necessary to mention that these values correspond to the various Kelvin-Voigt parameters in the generalized Burgers’ model (see Chap. 2). Similarly, the nonlinear parameter determination work presented in Sect. 3.2.5 also utilized the generalized Burgers’ model. The shear strain ε(t) for an applied stress, σ, can be obtained using Eq. 6.1 as: t ε(t) = g0 D0 σ + g1

ΔJ[ψ(t) − ψ(ξ )] 0

d(g2 σ) dξ dξ

60

3 Nonlinear Viscoelasticity

ΔD(ψ) =

N Σ n=1

   −ψ Dn 1 − exp ξn

(3.62)

Substituting the exponential form of the transient creep compliance into the integral form in Eq. (3.62), the strain can be given as: ε(t) = g0 D0 σ +

N Σ

   −ψ(t) − ψ(ξ ) Dn 1 − exp ξn

t g1

n=1

0

(3.63)

which can be re-written as: ε(t) = ε (t) + 0

N Σ

εn (t)

(3.64)

n=1

where:  ε (t) = σ g0 D0 + g1 g2 0

N Σ

 Dn

n=1

   −ψ(t) − ψ(ξ ) d(g2 σ) dξ Dn exp ξn dξ

t εn (t) = −g1 0

(3.65)

For an infinitesimal time step increment, the integral in Eq. (3.65) can be written as: t−Δt 

ε (t) = −g1 n

0

t − g1 t−Δt

   −ψ(t) − ψ(ξ ) d(g2 σ) dξ Jn exp ξn dξ

   −ψ(t) − ψ(ξ ) d(g2 σ) dξ Dn exp ξn dξ

Since,  exp

−ψ(t) − ψ(τ ) ξn



 = exp

   −Δψ −(ψ(t) − Δψ − ψ(ξ )) exp ξn ξn

the first integral in Eq. (3.66) can be re-written as:

(3.66)

3.3 Numerical Formulation t−Δt 

g1 0

61

   −ψ(t) − ψ(ξ ) d(g2 σ) dξ Dn exp ξn dξ

 t−Δt     −(ψ(t) − Δψ − ψ(ξ )) d(g2 σ) −Δψ = g1 exp Dn ex p dξ ξn ξn dξ 0   −Δψ n ε (t − Δt) = −exp (3.67) ξn 

If for an infinitesimal time increment Δt the nonlinear functions; g1 , g2, and aσ are assumed constant and σ is assumed to vary linearly, the second integral in Eq. (3.66) can be evaluated as: 



 −ψ(t) − ψ(ξ ) d(g2 σ) dξ g1 Dn exp ξn dξ t−Δt    −Δψ Δσξn 1 − exp = Dn g1 g2 Δψ ξn t

(3.68)

where Δψ = Δψ − Δψ(t) − Δψ(t-Δ t). Substituting Eqs. (3.67) and (3.68) into Eq. (3.66) one obtains:     −Δψ Δσξn −Δψ n ε (t − Δt) − Dn g1 g2 1 − exp ε (t) = exp ξn Δψ ξn 

n

(3.69)

From Eq. 6.13 the incremental shear strain can be obtained as:  Δε(t) = Δε (t) + 0

N Σ

 Δε (t) n

(3.70)

n=1

Using the expressions given in Eqs. (3.69) and (3.65) into Eq. (3.70) the incremental formulation for the strain becomes:   N Σ ∗ n Δε(t) = D Δσ + αn ε (t − Δt) (3.71) n=1

where D* and αn are given by: ∗

D = g0 D0 + g1 g2

N Σ n=1

     ξn −Δψ −1 exp Dn 1 + Δψ ξn

(3.72)

62

3 Nonlinear Viscoelasticity

 αn = exp

−Δψ ξn

 −1

(3.73)

The numerical formulations presented by Lai and Bakker (1996) enable the calculation of incremental strain for a given incremental stress, as shown in Eq. (3.71). In practical numerical simulations, the viscoelastic strain component should be updated at the end of each time increment using Eq. (3.69) εn (t). In finite element (FE) environments that are strain-controlled, such as ABAQUS, the incremental strain is inputted into the numerical routine, and the corresponding stress is computed. However, this also means that the current stress and nonlinear parameters cannot be directly determined at the start of the time step, as they are mutually dependent. Thus, an iterative procedure is necessary, where trial stress values are used to compute the correct stress.

References Biot MA (1958) Linear thermodynamics and the mechanics of solids. Cornell Aeronautical Lab., Inc., Buffalo Brinson HF, Brinson LC (2008) Polymer engineering science and viscoelasticity: an introduction. Springer Eyad M, Niranjanan S (2002) Microstructural finite-element analysis of influence of localized strain distribution on asphalt mix properties. J Eng Mech 128:1105–1114. https://doi.org/10. 1061/(ASCE)0733-9399(2002)128:10(1106) Haj-Ali RM, Muliana AH (2004) Numerical finite element formulation of the Schapery nonlinear viscoelastic material model. Int J Numer Methods Eng 59:25–45. https://doi.org/10.1002/ nme.861 Henriksen M (1984) Nonlinear viscoelastic stress analysis—a finite element approach. Comput Struct 18:133–139 Huang C-W, Masad E, Muliana AH, Bahia H (2007) Nonlinearly viscoelastic analysis of asphalt mixes subjected to shear loading. Mech Time-Dependent Mater 11:91–110. https://doi.org/10. 1007/s11043-007-9034-5 Knauss WG, Emri IJ (1981) Non-linear viscoelasticity based on free volume consideration. In: Computational methods in nonlinear structural and solid mechanics. Elsevier, pp 123–128 Lai J, Bakker A (1996) 3-D schapery representation for non-linear viscoelasticity and finite element implementation. Comput Mech 18:182–191. https://doi.org/10.1007/BF00369936 Lou YC, Schapery RA (1971) Viscoelastic characterization of a nonlinear fiber-reinforced plastic. J Compos Mater 5:208–234. https://doi.org/10.1177/002199837100500206 Ottosen NS, Ristinmaa M (2005) The mechanics of constitutive modeling. Elsevier Peretz D, Weitsman Y (1982) Nonlinear viscoelastic characterization of FM-73 adhesive. J Rheol (n Y N y) 26:245–261. https://doi.org/10.1122/1.549666 Poon H, Ahmad MF (1999) A finite element constitutive update scheme for anisotropic, viscoelastic solids exhibiting non-linearity of the Schapery type. Int J Numer Methods Eng 46:2027–2041. https://doi.org/10.1002/(SICI)1097-0207(19991230)46:123.0.CO;2-5 Rochefort MA, Brinson HF (1983) Nonlinear viscoelastic characterization of structural adhesives. Virginia Polytechnic Inst and State Univ Blacksburg Dept of Engineering … Schapery RA (1964) Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic media. J Appl Phys 35:1451–1465. https://doi.org/10.1063/1. 1713649

References

63

Schapery RA (1966) A theory of non-linear thermoviscoelasticity based on irreversible thermodynamics (stress-strain temperature equations and energy equation with heat conduction for isotropic nonlinear viscoelastic materials, using irreversible thermodynamics) Schapery RA (1969) On the characterization of nonlinear viscoelastic materials. Polym Eng Sci 9:295–310. https://doi.org/10.1002/pen.760090410 Tuttle ME, Pasricha A, Emery AF (1995) The nonlinear viscoelastic-viscoplastic behavior of IM7/5260 composites subjected to cyclic loading. J Compos Mater 29:2025–2046. https://doi. org/10.1177/002199839502901505 Tuttle ME, Brinson HF (1985) Accelerated viscoelastic characterization of T3OO/5208 graphiteepoxy laminates. Virginia Polytechnic Inst and State Univ Blacksburg Dept of Engineering

Chapter 4

Experimental Characterization

To study the mechanical behavior of composite structures, a measurement of the mechanical properties of composite materials is needed which can be acquired by conducting experiments on standard specimens. Composite structures may be constructed using elastic and viscoelastic components, and to better understand their mechanical behavior for constructional and modeling purposes, the mechanical properties of composites and their components have to be measured using standard test methods. This chapter is dedicated to introducing the fundamental experiments for characterizing the elastic and viscoelastic properties of fibrous and bituminous composites. Among the common international and national test standards for the test methods, emphasis was put on ASTM, and explanations about the instrumentation and methodology are presented. Furthermore, the most important outputs of each test are presented and discussed.

4.1 Fibers and Inclusions The elastic behavior of the fibers and other inclusions such as fillers of bituminous composites has a determining role in load bearing capacity of composite structural members. The most common test methods for determining the mechanical properties are discussed in this section.

4.1.1 Single Fiber Test To find the mechanical properties of the unidirectional and woven fibers, especially the tensile strength and elasticity modulus, the mechanical test on a single fiber can be a good indicator.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. Hajikarimi and A. Sadat Hosseini, Constructional Viscoelastic Composite Materials, https://doi.org/10.1007/978-981-99-1786-0_4

65

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4 Experimental Characterization

Fig. 4.1 The general scheme of single fiber tensile testing machine

The determination of the breaking force and corresponding tenacity, initial modulus, chord modulus, tangent modulus, tensile stress at specified elongation, and breaking toughness can be achieved using ASTM D3822 (2020) test method. Breaking toughness is directly proportional to the area beneath the force-elongation curve from the origin to the breaking force, and in textile strands, it is measured as work (joules) per unit linear density. Figure 4.1 provides an illustration of the single fiber testing machine setup. A force-extension curve is obtained in this test method using test specimens having gage lengths of 10 mm (0.4 in.) or greater to calculate the mentioned properties of the fiber. After the test, the tension testing machine directly gives the breaking force. Therefore, the breaking tenacity is calculated using: ϒ=

F DL

(4.1)

wherein, U is the breaking tenacity in mN/tex (tex number is the measured grams (g) of 1,000 metres (m) length textile), F is the breaking force in centinewton (cN), and DL is the linear density in tex. To determine the initial modulus of fibers, the maximum slope of the stress-strain curve needs to be located. A tangent line is then drawn from the tangent point for this slope to the proportional elastic limit, passing through the zero-stress axis (strain axis). The stress and corresponding strain are measured using the following method: Ji =

S εp

(4.2)

wherein, J i is the initial modulus (cN/tex), and S and εp are the determined stress on the drawn tangent line (cN/tex) and the corresponding strain, respectively.

4.1 Fibers and Inclusions

67

The chord modulus between two specific elongations is determined by labeling the stress values at zero and n percent elongation. In order to find the chord modulus between two specific elongations, two stress values corresponding to the 0% elongation and n percent elongation are labeled. The chord modulus can be calculated by: Jch =

S εp

(4.3)

In this equation, J ch is the chord modulus between the specified elongations (cN/tex), and S and εp are the stress on the line passing from the stresses on the stress–strain curve and the strains corresponding to the specified elongations, respectively. From the information presented in Fig. 4.2, it is possible to calculate the chord modulus by dividing the stress at any point along line AA' (or its extension) by the corresponding strain point, which is measured from point A, the zero strain axis. If a different stress point, such as point B, is chosen instead of A, the corresponding line intersection with the zero strain axis (B'' ) needs to be subtracted from the stress value. The breaking toughness of the fiber can be calculated by measuring the area under the stress–strain curve. Note that, the area which precedes the initial modulus line corresponds to the work to remove the slack caused by the clamp error (see Fig. 4.3). The breaking toughness can be calculated by: Fig. 4.2 The methodology of finding chord modulus from the stress–strain data

Fig. 4.3 Removing the slack caused by the clamp error

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4 Experimental Characterization

Fig. 4.4 A schematic comparison of the stress–strain relationship for different fiber types

Tu =

A c F f s Rs Wc C s D L L e

(4.4)

Ir F f s Rs Ic D L L e

(4.5)

Tu =

where, T u is the breaking toughness (J/g), Ac is the area under the stress–strain curve (mm2 ), F fs is the range of full-scale force (cN), Rs is the rate of speed testing, W c is the width of recording chart (mm), C s is the recording chart speed (mm/min), DL is the linear density (dtex), L e is the effective length of specimen (mm), and I r and I c are the integrator reading and integrator constant (per minute) as determined by the manufacturer. A comparative diagram which illustrates the stress–strain relationship for different types of fibers is presented in Fig. 4.4. It is also possible to measure the tensile strength of fiber strap instead of a single fiber as Soric et al. (2008) measured for glass fibers. Recently, a new test method was introduced by Jenket et al. (2019) to measure tensile strength of a single fiber. Compressive strength and elastic moduli of intact rock core specimens can be measured using ASTM D7012 (2017) under varying states of stress and temperature. A rock core specimen with a specified diameter and length and the flat ends should be placed in a testing machine and subjected to applied pressure. Preferable specimen diameter to length ratios are between 1:2 and 1:2.5. The testing machine may equip with a chamber to provide a specific test temperature by heating or cooling the test specimen before the start of the test. The axial load applied on the specimen is then increased and recorded continuously. Regarding the desired output, deformation may be measured or not until the peak load and failure. Figure 4.5 shows a test setup in which a servo control Dartec-9600 with load and displacement measurement accuracy of 10 N and 0.001 mm is used to measure compressive strength of rock core. The uniaxial compressive strength, σ u , of the test specimen can be calculated as:

4.1 Fibers and Inclusions

69

Fig. 4.5 Determination of aggregate stiffness using Dartec-9600 device

σu =

P A

(4.6)

where P is failure load (kN), and A is cross-sectional area (mm2 ). The axial strain, εu , can also be calculated using the following equation: εu =

∆L L

(4.7)

in which ∆L is the change in measured axial gauge length (mm), and L is the original undeformed axial gauge length (mm). The lateral strain, εl , shall be calculated as: εl =

∆D D

(4.8)

in which ∆D is the change in diameter (an increase in diameter is positive and a decrease is negative) (mm), and D is the original undeformed diameter (mm). Figure 4.6 depicts the stress–strain curve for the axial and lateral directions. There are three methods illustrated in Fig. 4.7 to determine Young’s modulus from the axial stress–strain curve. Also, the Poisson’s ratio can be calculated using the following equations: υ=−

E slope of axial curve =− slope of lateral curve slope of lateral curve

(4.9)

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4 Experimental Characterization

Fig. 4.6 The axial stress–strain curve

Fig. 4.7 Various methods to determine the Young’s modulus from axial stress–strain curve: a Tangent modulus measured at a fixed percentage of ultimate strength, b Average modulus of linear portion of axial stress–strain curve, c Secant modulus measured up to a fixed percentage of the ultimate strength

4.1.2 Ultrasonic Test Section 4.1.2 describes a destructive method, but an alternative non-destructive approach involves the use of ultrasonic technology to determine both Young’s modulus and Poisson’s ratio of a rock core sample. Figure 4.8 illustrates the steps involved in the test procedure of the ultrasonic technique to measure the velocity of different waves according to ASTM C597 (2016). Based on the ultrasonic results, Eqs. (4.10) and (4.11) should be used to calculate the stiffness and Poisson’s ratio of aggregate, respectively:   3V p 2 − 4Vs 2  E = ρVs  2 V p − Vs 2   2 V p − 2Vs 2  ν=  2 2 V p − Vs 2 2

(4.10)

(4.11)

4.2 Matrix Component

71

Fig. 4.8 Determination of Young’s modulus and Poisson’s ratio using an ultrasonic test

In the given equation, the elastic modulus of aggregate is represented by E, the Poisson’s ratio of aggregate is represented by ν, and V p and V s represent the compression and shear wave velocity, respectively.

4.2 Matrix Component The elastic and viscoelastic properties of the matrix component in an FRP material can be found using the experiments discussed in this section.

4.2.1 Experiments Under Static Loading 4.2.1.1

Tensile

To determine the tensile properties of unreinforced and reinforced plastics, ASTM D638 (2014) is described. The alternative standard which covers the same subject matter, but differs in technical content is ISO 527-1 (2019). ASTM D638 is a test method suitable for evaluating the tensile properties of sheet, plate, and molded plastics with thicknesses up to 14 mm (0.55 in.), while materials with thicknesses greater than 14 mm (0.55 in.) require machining to reduce the thickness. However, for thin sheeting and films with thicknesses less than 1.0 mm (0.04 in.), it is recommended to use ASTM D882 (2018). The latter test method provides an option for determining Poisson’s ratio at room temperature. Standard “dogbone” shaped specimen of equal to or less than 14 mm in thickness is used in this standard. Figure 4.9 illustrates the shape and dimensions of the specimens. In this figure, W and L are the width and length of the narrow section, respectively, and WO and LO are the overall width and length of the specimen, respectively. G is

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4 Experimental Characterization

Fig. 4.9 The standard dogbone sample dimensions for tensile test on unreinforced and reinforced plastics (ASTM D638): a Sample types I, II, III, and V, b Sample type IV

the gauge length, D is the distance between grips, R is the fillet radius, and RO is the outer radius. There are five different geometries of these specimens which are used depending on the behavior of the material. The reader is referred to part 6 of ASTM D638 for further information about the specimen geometries. After cutting the material into “dogbone” sample, the test is followed by loading the sample into a pair of tensile grips. The speed of testing has to be determined based on the type of specimen and the required strain rate. If the speed is not specified, the lowest speed recommended by the standard should be used. The load-extension curve of the specimen is recorded up to the end of the test which corresponds to the sample rupture. The outputs of this test are the tensile strength, elongation at yield, elongation at break, nominal strain at break, modulus of elasticity, secant modulus, and Poisson’s ratio which requires transverse extensometer. The maximum load that the specimen can bear is divided by its average original cross-sectional area in the gage length segment to give the tensile strength. Elongation of the specimen is the change in gage length with respect to the original gage length of the specimen. So, the reading value of extension at yield and break gives the elongation at yield and break, respectively. The change in grip separation at the point of rupture yields the nominal strain at break. The elasticity modulus can be found by dividing the difference in stress (from two arbitrary points on the initial linear portion of the load-elongation curve) by the corresponding difference in strain. The secant modulus is found by dividing the nominal stress by the corresponding designated strain. Instances of different stress–strain relationships and the failure points are provided in Fig. 4.10.

4.2 Matrix Component

73

Fig. 4.10 Yield and break point in different stress–strain relationships: a, e tensile strength at break, b tensile strength at yield, c break point, d yield point

Fig. 4.11 Calculating the Poisson’s ratio from the load-strain curve for different stress values

To find the Poisson’s ratio, ν, the axial strain, εa , read from the axial extensometer, and the transverse strain, εt , indicated by the transverse extensometers, should be plotted against the load, P, applied to the specimen (Fig. 4.11). The Poisson’s ratio can be calculated using: ν=

4.2.1.2

εt εa

(4.12)

Compression

This section conforms to ASTM D695 (2015), which outlines the procedure for determining the mechanical properties of both reinforced and unreinforced rigid plastics, including high-modulus composites, under compressive loading at low and uniform rates of straining or loading. To obtain stress-strain data, the speed of testing should be limited to 1.3 mm/min. However, this speed can be increased to 5–6 mm/min after the yield point. Two examples of the suitable compression tools are presented in Fig. 4.12.

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4 Experimental Characterization

Fig. 4.12 Schematic of the compression test tool

Standard shape should be employed for the test specimens. As indicated in the standard, the standard form of a right cylinder or prism whose length is twice its principal width or diameter. Ideal dimensions for cylinder specimens are 12.7 mm in diameter and 25.4 mm in length (also valid for tubes), or 12.7 by 12.7 by 25.4 mm prism. When it is desired to find elastic modulus and offset yield-stress the slenderness of the specimen should be in the range from 11:1 to 16:1. In this case, the preferred dimensions for cylinder specimens are 12.7 mm in diameter and 50.8 mm in length (also valid for bars), and for prism are 12.7 by 12.7 by 50.8 mm. For thin materials with the thickness of under 3.2 mm, a special specimen with its specific dimensions given in Fig. 4.13 shall be used. The compressive strength can be calculated by dividing the maximum compressive load, carried by the specimen, by the minimum original cross-sectional area of the specimen: Sc,u =

Fc,u A

Fig. 4.13 The specimen for the compression test for materials less than 3.2 mm thick

(4.13)

4.2 Matrix Component

75

Similarly, the compressive yield strength can be calculated by dividing the value of load carried at the yield point to the minimum original cross-sectional area of the specimen: Sc,y =

Fc,y A

(4.14)

The elasticity modulus is the slope of the line drawn tangent to the linear part of the stress–strain curve of the material, resulting from the test.

4.2.1.3

Flexural

In this section, two ASTM standards (ASTM D790 2017; ASTM D6272 2017) are briefly explained. The test setup of ASTM D790 (Fig. 4.14) consists of a bar (having a rectangular cross section) on two supports and is loaded by a loading nose midway between the two supports. The supports and loading nose have cylindrical surfaces to avoid premature failures due to stress concentration. The loading is applied to the specimen and continued until failure (rupture) on the outer surface is seen or the maximum strain in the specimen reaches 5 percent. This test method is not applicable for strains beyond 5%. To determine the flexural strength, chord or secant modulus or the tangent modulus of elasticity, and the total work of the specimen, load–deflection curves are plotted. The deflection of the test specimen is calculated using: D=

r L2 6d

(4.15)

wherein, D is the mid-span deflection (mm), r is the maximum strain in the outer surface of the test specimen, L is the span length between supports, and d is the depth of the beam. The maximum stress in the outer surface of a test specimen constructed from homogeneous elastic material and loaded at the midpoint occurs at the midpoint.

Fig. 4.14 The test setup of ASTM D790 and the allowable ranges of dimensions

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4 Experimental Characterization

The maximum flexural stress (σ fM ) sustained by the test specimen is the flexural strength. The flexural stress can be calculated using the following equations. For span-to-depth ratios less than 16:1: σf =

3P L 2bd 2

(4.16)

For span-to-depth ratios greater than 16:1:      2 d D D 3P L −4 σf = 1+6 2 2bd L L L

(4.17)

wherein, σ is the stress in the outer fibers at midpoint (MPa), P is the loading value (N), L is the length between supports (m), b is the specimen width (m), d is the specimen depth (m), and D is the deflection of the centerline of the specimen at the middle of the support span (mm). The flexural stress at break (σ fB ) is the stress experienced by the specimen at its breaking point or when the deflection reaches 5%. A set of typical flexural stress versus strain curves is illustrated in Fig. 4.15. The flexural strain can be calculated by: εf =

6Dd L2

(4.18)

where, εf is the strain in the outer surface if the beam, and the other parameters are the same as for Eq. (4.15). The modulus of elasticity, which is also called the tangent modulus of elasticity is the ratio of stress (within the elastic limit) to its corresponding strain. To find this property, a straight line is drawn tangent to the initial linear portion of the load– deflection curve. Reading the slope of this straight line, m (mm/mm), the flexural elastic modulus, E B (MPa) can be calculated using: Fig. 4.15 A set of typical flexural stress versus strain curves and examples of maximum flexural stress (σ fM ) and flexural stress at break (σ fB )

4.2 Matrix Component

77

EB =

L 3m 4bd 3

(4.19)

L, b, and d are the same as for Eq. (4.16). Note that, shear deflections in highly anisotropic composites can significantly reduce the modulus of elasticity. Therefore, it is recommended to use a span-to-depth ratio of 60 to 1. ASTM D6272 recommends the four-point bending test method which can be used for those materials that do not fail within the strain limits (i.e., 5%) imposed by test method of ASTM D790. The location of the maximum bending moment and maximum axial fiber stress are different under four-point and three-point bending moments. In three-point bending scheme, the maximum bending moment is located at the midpoint under the load, while in the four-point bending scheme the maximum bending moment is distributed uniformly between two loads. The test setup consists of a bar (having a rectangular cross section) on two supports and is loaded by two loading nose between the two supports as per the configurations presented in Fig. 4.16. The beam deflection in this test setup can be calculated using Eqs. (4.20) and (4.21) for a load span of one-third and one-half of the overall span, respectively. D = 0.21

Fig. 4.16 Loading diagrams of the four-point bending test setup: a One-half of support span, b One-third of the support span

r L2 d

(4.20)

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4 Experimental Characterization

D = 0.23

r L2 d

(4.21)

wherein, r, L, and d are the same as for Eq. (4.15) and D is the mid-span deflection (mm). The flexural stress can be calculated using the following equations for spanto-depth ratios less than 16:1. For the load span equal to one-third of the overall span: σ =

PL bd 2

(4.22)

For the load span equal to one-half of the overall span: σ =

3P L 4bd 2

(4.23)

and for span-to-depth ratios greater than 16:1 the following equations can be used: For the load span equal to one-third of the overall span:  D2 Dd PL σ = 2 1 + 4.70 2 − 7.04 2 L L bd

(4.24)

And for the load span equal to one-half of the overall span:  Dd 3P L 1 − 10.91 2 σ = L 4bd 2

(4.25)

wherein, σ is the stress in the outer fibers along the load span (MPa), P is the loading value (N), L is the length between supports (m), b is the specimen width (m), d is the specimen depth (m), and D is the deflection of the centerline of the specimen at the middle of the support span (mm). The maximum strain which occurs at the mid-span can be calculated using Eqs. (4.26) and (4.27) for the load span equal to one-third and one-half of the overall span, respectively. r = 4.70

Dd L2

(4.26)

r = 4.36

Dd L2

(4.27)

wherein, r, L, d, and D are the same as for Eq. (4.20). The modulus of elasticity, which is also called the tangent modulus of elasticity is the ratio of stress (within the elastic limit) to its corresponding strain. It can be calculated by reading the slope of the straight line tangent to the initial straight portion of the stress–strain curve, m (mm/mm), for a load span of one-third the support span

4.2 Matrix Component

79

using Eq. (4.28) and for a load span of one-half of the support span using Eq. (4.29), as follows: E B = 0.21

L 3m bd 3

(4.28)

E B = 0.17

L 3m bd 3

(4.29)

wherein, E B , L, m, b, and d are the same as for Eq. (4.19). A similar concept was used in well-known standard test methods introduced to determine flexural creep and creep rupture of plastics (ASTM D2990 2017), and flexural creep stiffness of asphalt binder (ASTM-D6648-08 2001) under a specified creep loading. In these standards, a three-point bending beam test setup is used to determine the applied stress and the response strain by measuring the mid-span deflection. The test specimen used in ASTM D2990 shall be in the form of a right prism or cylinder having a cross section of 12.7 by 12.7 mm or 12.7 mm in diameter. The test specimen used in ASTM D6648 shall be in the form of a right prism with a dimension of 127 × 12.7 × 6.35 mm. The applied load on the specimen must be maintained within ±1% of the desired load. The mid-span deflection of the specimen shall be measured using a dial gauge or a cathetometer. It is evident that regarding the temperature-dependent behavior of viscoelastic materials, it is necessary to control the specimen’s temperature during the test using a chamber. Figure 4.17 shows a bending beam rheometer used in ASTM D6648 for applying a concentrated constant load of 980 ± 50 mN at the mid-span of beam fabricated with asphalt binder at sub-zero temperatures. The maximum mid-span deflection for an elastic material in three-point bending setup can be calculated by using elementary bending theory (Timoshenko and Young 1962):

Fig. 4.17 Bending beam rheometer for measuring the flexural creep stiffness of asphalt binder

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4 Experimental Characterization

Fig. 4.18 Mid-span deflection of crumb rubber modified asphalt binders under creep loading at different test temperatures

δ=

P L3 48E I

(4.30)

where δ is mid-span deflection of beam (mm), P is concentrated constant load (N), L is length of beam span (mm), E is elastic modulus, I = bh3 /12 is moment of inertia, b is width, and h is depth of beam. Figure 4.18 shows mid-span deflection of crumb rubber modified asphalt binders (containing different extent of crumb rubber) under creep loading at different test temperatures, including −12, −18, and −24 °C (Hajikarimi et al. 2021). As can be seen in this figure, increasing the time and test temperature resulted in increasing mid-span deflection. For a viscoelastic material, using the correspondence principle (Alfrey 1944), already introduced in Sect. 2.5, it is possible to derive Eq. (4.31) in a time domain using the step-by-step procedure. Firstly, the applied load and deflection are substituted with their Laplace transforms. Then, the elastic modulus of E is substituted with ∗ s E(s), i.e., E (s). Thus, the following equation is derived in the Laplace transformed domain (Schiff 2013): δ(s) =

P(s)L 3 48s E(s)I

(4.31)

Since L and I are just dependent to beam geometry, no transformation is required for these two parameters. Recalling that the mid-span concentrated load is constant, its Laplace transform is as follows: P(s) =

P0 s

(4.32)

4.2 Matrix Component

81

Then, Eq. (4.32) can be inserted into Eq. (4.33): δ(s) =

P0 L 3   48 s 2 E(s) I

(4.33)

and regarding to Eq. (2.52), it is possible to re-write Eq. (4.33) as: δ(s) =

P0 L 3 D(s) 48I

(4.34)

It is evident that Eq. (4.34) can be transformed to time domain by applying inverse Laplace transform. By re-arranging this equation, the following formula can be developed for the creep compliance function: D(t) =

48δ(t)I P0 L 3

(4.35)

It can be seen that having the mid-span deflection for a bending beam, it is possible to calculate the creep compliance as a constitutive viscoelastic function. Then, it will be possible to determine the relaxation modulus using the inter-conversion method described in Sect. 2.6. In ASTM D6648, the inverse of creep compliance is named flexural creep stiffness, S(t), which can be determined as follows: S(t) =

P0 L 3 1 = D(t) 48δ(t)I

(4.36)

The maximum flexural stress of the specimen is: σmax =

3P L 2bd 2

(4.37)

and the maximum strain in the outer face at the mid-span is as follows: εmax = 6δmax

4.2.1.4

d L2

(4.38)

Relaxation

ASTM E328 (2021) outlines four different types of stresses—uniform tension, uniform compression, bending, and torsion—that can be applied during stress relaxation tests for materials and structures. The type of stress applied during the relaxation test should be representative of the stress experienced by the component during reallife performance. Figure 2.1 shows the stress application for all four types of stress.

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4 Experimental Characterization

During the test, an increasing force is applied to the specimen until the specified initial strain is reached. The specimen constraint is kept constant throughout the test. Using the initial force (moment or torque), specimen geometry, and the appropriate elastic constants, the initial stress can be calculated using simple elastic theory. The remaining stress can be determined from the force (moment or torque) measured under constrained conditions, either continuously, periodically, or through elastic spring back at the end of the test period. This section provides details on the test specimen, test setup, calculations, and presentation of results for each of the four types of stress. Uniform Tension The size and shape of specimens are shown in Fig. 4.19 in accordance with ASTM E8/E8M (2022). The cross section of the specimen should be uniform all through the length of the reduced section and to simplify controlling the limiting strain, it is desirable that the gauge length be longer than those specified in ASTM E8/E8M. Since mechanical behavior of viscoelastic materials directly depends on the test temperature, in the case that the test temperature is different from ambient, it is required to expose the specimen to the test temperature for an adequate period of time or use a chamber to provide this test temperature. Figure 4.20 shows the setup required for conducting the test. It is important that the testing machine used should have a precision of 1% through its working range. The testing machine should also have the capability to automatically and continuously adjust the force to maintain a constant constraint, thus ensuring that the strain on the specimen does not exceed ±0.000025 mm/mm. The remaining stress, relaxed stress, or applied force can be plotted versus time or log time. Also, the difference between the initial stress and the remaining stress divided by the initial stress is entitled “Fraction Initial Stress Relaxed” and can be plotted versus time. Uniform Compression The size and shape of specimens should be selected according to ASTM E9 (2009), a solid circular cylinder with an L/D (length/diameter ratio) of 8 to 10. Figure 4.21 shows the test setup in which the axial misalignment should be minimized using

Fig. 4.19 Test specimen for applying uniform tension in the relaxation test

4.2 Matrix Component

83

Fig. 4.20 Tension stress relaxation test

an alignment device. The extensometer and thermocouples are attached to the specimens. For the test temperature which is different from ambient, it is required to keep the specimen within a chamber to reach desired test temperature. The force application rate should not exceed 690 MPa/min. The total strain constant should be maintained in a specified limit in which the combined stress resulting from differential thermal expansion between the test specimen and the constraint and other variations should not exceed ±0.000025 mm/mm. The testing machine should have similar specification as mentioned for uniform tension. In addition, it is so important to apply the force axially. Similar results mentioned for uniform tension can be measured and reported for uniform compression.

4.2.2 Dynamic Mechanical Analysis Harmonic loading is a commonly used excitation for characterizing the linear or nonlinear behavior of viscoelastic materials. This type of loading is applied using a dynamic mechanical analyzer (DMA) setup, as illustrated in Fig. 4.22. This setup involves subjecting the sample to a sinusoidal wave of stress or strain, usually at frequencies between 0.01 Hz and 100 Hz, depending on the material being tested and the specific measurement being taken. The deformation (strain) and corresponding stress response are simultaneously recorded, and the complex modulus of the material is determined by dividing the stress by the strain. This modulus encompasses both the storage modulus, which reflects the elastic response, and the loss modulus, which indicates the energy dissipated due to viscous flow and converted to heat. γ (t) = γ0 sin(ωt)

(4.39)

84

4 Experimental Characterization

Fig. 4.21 Test setup of uniform compression test

in which γ 0 is excitation amplitude of shear strain and ω is excitation frequency which shows loading rate. It is expected that the response to shear stress has the following form: τ (t) = τ0 sin(ωt − ϕ)

(4.40)

where ϕ is time lag between application of strain and stress response (see Fig. 4.23). This parameter is called phase angle in the viscoelastic literature. It is evident that having ϕ = 0°, we have ideal elastic behavior and having ϕ = 90°, we have ideal viscous behavior.

4.3 Composites

85

Fig. 4.22 Harmonic loading using the dynamic shear rheometer

Fig. 4.23 Stress and strain versus time under a harmonic loading

4.3 Composites 4.3.1 Experiments Under Static Loading 4.3.1.1

Tensile

The tensile mechanical properties of the laminate composites can be found using in-plane tensile testing according to ASTM D3039 (2008), EN 2561 (1995), and ISO 527-4/5 (2021, 2009). Among these test procedures, ASTM D3039 is described in this section. Note that, for fiber reinforced polymer matrix composite bars ASTM D7205 (2006) can be used.

86

4 Experimental Characterization

Fig. 4.24 The general scheme of flat strip tensile testing machine

The test specimen in ASTM D3039 is a thin flat strip of material having a constant rectangular cross section. This specimen is subject to static tensile loading using a mechanical testing machine (Fig. 4.24). Note that, the laminate is balanced and symmetric with respect to the test direction. The maximum force carried by the specimen before failure determines the ultimate tensile strength. The corresponding strain to this stress is the ultimate tensile strain. In addition, the tensile chord modulus of elasticity, Poisson’s ratio, and transition strain can be obtained from this test. The minimum number of strain gauges is three and they should be of similar type. Two of the gauges are installed on the front face while one is on the back face of the coupon. It is recommended to test five specimens per test condition at least. Figure 4.25 shows the two types of test coupons. There are ordinary test specimens without tabs and coupons with tabs. Tabs are not a requirement if premature failure at the ends can be prevented by correct gripping within the allowable tolerances. The standard dimensions of the tensile test coupon is presented in Table 4.1. These dimensions are only for recommendation and they can be changed as long as the requirements of the standard is met. For instance, the minimum length of the coupon is equal to the summation of gripping, 2 times width, and the gage length. Note that the direction of fibers along the specimen length indicates the 0° orientation. Among the outputs of the test, the tensile strength and stress are the most important parameters that can be calculated using: Ftu = σi =

Pmax A

(4.41)

Pi A

(4.42)

4.3 Composites

87

Fig. 4.25 Tensile test specimen for fiber-reinforced polymers: a coupon without tabs, b coupons with tabs

Table 4.1 The standard recommended dimensions of the tensile test specimen for fiber-reinforced polymers based on ASTM D3039 Fibers orientation

Lo

W

Ts

Lt

Tt

θt

Unidirectional 0°

250

15

1.0

56

1.5

7 or 90

Unidirectional 90°

175

25

2.0

25

1.5

90

Balanced and symmetric

250

25

2.5

EC

EC

EC

Randomly distributed discontinuous fibers

250

25

2.5

EC

EC

EC

EC Emery cloth

where, F tu is the ultimate tensile strength (MPa), Pmax is the maximum tensile load (N), and σ i is the tensile stress at any data point which is calculated using the force at the data point (N) divided by the average cross-sectional area at three places in the gage section (mm2 ). The tensile stress at a given data point (see Fig. 4.26 as an example) can be used together with the corresponding tensile strain to find the tensile modulus. The tensile strain from an indicated displacement at a specific data point is calculated using:

εi =

δi Lg

(4.43)

88

4 Experimental Characterization

Fig. 4.26 Examples of stress–strain response in tensile test

where, εi is the tensile strain at any data point, δ i is the displacement measured by the extensometer at the data point (mm), and L g is the extensometer gauge length (mm). To determine the modulus of elasticity, it is recommended to average the strains from all the strain gauges installed on the specimen. This can minimize the potential effects of bending on the results. To find the tensile chord modulus of elasticity, the appropriate chord modulus strain range should be selected based on the recommendations of the test method and substituted in the following equation: E chord =

∆σ ∆ε

(4.44)

wherein, E chord is the tensile chord elastic modulus (MPa), ∆σ is the range of stress between two corresponding strains (The range of strain is ∆ε and normally is 0.002). To calculate the Poisson’s ratio, appropriate longitudinal strain ranges can be selected from the standard’s recommendations and use the following equation: ν=−

∆εt ∆εl

(4.45)

where, ν is the Poisson’s ratio, and ∆εt and ∆εl are the range of lateral and longitudinal strains, respectively.

4.3.1.2

Compression

There are a variety of standard test methods under compression loading, such as ASTM D3410 (2016), ASTM D6641 (2016), and ISO 14126 (1999). Compression testing can also be performed between hydraulic grips as long as they are well-aligned and have high lateral stiffness. This section will focus on ASTM D3410, although it should be noted that ASTM D6641 also offers the combined loading compression (CLC) test, which will not be covered here due to space limitations.

4.3 Composites

89

According to D 3410, the specimen which is in the form of a flat strip of material having a constant rectangular cross section is inserted into the test fixture and loaded in compression by a shear force acting along the grips. A schematic illustration of compressive test machine is presented in Fig. 4.27. Figure 4.28 shows the two types of test coupons including without tabs and with tabs. The maximum allowable variation in width and thickness of the specimen is ±1 and ±2%, respectively. The variation of thickness at the tabbed ends should be limited to ±1% of the thickness. Recommended dimensions for the specimens of Fig. 4.27 are presented in Table 4.2. The specimens of compressive test are very sensitive to poor material fabrication practices. Although there is no specific procedure to avoid misalignment of the fibers, a recommended procedure is proposed by ASTM D3410 which should be used. By extracting the stress–strain data from the compressive test, the ultimate compressive stress and strain (measured by the strain or displacement transducers), Fig. 4.27 A schematic presentation of the fixture of compression test according to ASTM D3410

90

4 Experimental Characterization

Fig. 4.28 Compression test specimen for fiber-reinforced polymers: a coupon without tabs, b coupons with tabs

Table 4.2 The standard recommended dimensions of the compressive test specimen for fiberreinforced polymers based on ASTM D3410 Lo

W

Lg

Lt

Tt

Unidirectional 0°

140–155

10

10–25

65

1.5

Unidirectional 90°

140–155

25

10–25

65

1.5

Specially orthotropic

140–155

25

10–25

65

1.5

the compressive modulus of elasticity, Poisson’s ratio in compression, and transition strain can be derived. The compressive strength of the composite material is calculated using: Fcu =

Pmax A

(4.46)

where, F cu is the compressive strength, Pmax is the maximum compressive force before failure, and A is the cross-sectional area of the specimen. The compressive chord modulus of elasticity can be obtained from the stress– strain data. For this purpose, the stress values corresponding to the strain values of 0.001 and 0.003 are read and if significant change in the slope of the stress–strain

4.3 Composites

91

curve is not observed in the mentioned strain range, the chord modulus of elasticity can be calculated using: E chord =

∆σ ∆ε

(4.47)

wherein, ∆σ and ∆ε are the stress and strain ranges, respectively. To calculate the compressive Poisson’s ratio, the transverse strain and the longitudinal strain ranges from a single transverse strain gage and a single longitudinal gage mounted in an adjacent location on the same side of the specimen should be read. The Poisson’s ratio is calculated by: νc =

∆εt ∆εl

(4.48)

where, ∆εt and ∆εl are the difference in transverse and longitudinal compressive strains, respectively.

4.3.1.3

Shear

There are three common test methods for determining the shear strength of the composite materials. The inter-laminar shear strength test, also known as short beam shear test (ASTM D2344 2022; EN 2563 1997; ISO 14130 1997), is a simple test which is conducted using the three-point bending setup. A shear failure occurs due to the ratio of the specimen thickness to its span. An apparent strength value rather than a true shear strength is obtained by this test, which is the main shortcoming of this setup. The other available test setup (ASTM D3518 2018; ISO 14129 1997) to find the in-plane shear modulus and strength of fiber-reinforced composites is the common tensile test but on composite with ±45-degree fiber orientation. The axial and transverse strains are measured and the shear strain is obtained. The main drawback of this method is that the stress in the specimen is a combination of shear and axial stress. A more liable alternative to the mentioned setups for the determination of shear properties is the V-notched specimens (ASTM D7078 2005; ASTM D5379 2019), which can produce a pure shear state in the region of the notch. Here in this section, ASTM D5379 is briefly discussed. The test specimen is a rectangular flat strip with symmetrical v-notches located at the both sides in the middle (Fig. 4.29). The shear response of the material can be measured by using two strain gauges oriented at ±45° to the loading axis while the specimen is under vertical displacements induced by the testing machine fixture in the opposite directions. This loading mechanism results in the shear and bending moment diagrams of Fig. 4.30. The speed of testing shall be set in a way to achieve a nearly constant strain rate in the gage section. At the end of the test, the load–deflection curve is extracted and the shear stress at each required data point can be calculated using:

92

4 Experimental Characterization

Fig. 4.29 The v-notched specimen dimensions for shear test (T: the thickness as required)

Fig. 4.30 The v-notched specimen: a Force diagram, b Shear diagram, c Moment diagram

τi =

Pi A

(4.49)

The ultimate strength of the specimen can be calculated by: Fu =

Pu A

(4.50)

4.3 Composites

4.3.1.4

93

Flexural

This section presents a summary of three-point bending and four-point bending test setups according to ASTM D7264 (2007) to determine the flexural stiffness and strength properties of polymer matrix composites. This test method should be used for continuous fiber-reinforced polymer matrix composites and differs from other flexure methods such as ASTM D790 and D6272 that described before. In this test method, the standard span-to-thickness ratio is 32:1, however, the 16:1 ratio is used by Test Methods D790 and D6272. The technical equipment used in this test method is very similar to those used in ASTM D790 and D6272. The testing machine operates at a constant rate of 1.0 mm/min, and shall be essentially free of inertia lag. The loading noses and supports may have rotatable or fixed arrangements, and shall have hall have cylindrical contact surfaces of radius 3.00 mm. Other equipment such as Micrometers and Conditioning Chamber shall be in accordance with the test instructions. The configurations of the three-point and four-point loading setups are illustrated in Fig. 4.31. The standard span-to-thickness ratio in this setup is 32:1, and the standard thickness and width of the specimen shall be 4 and 13 mm, respectively. The results of this test are considered valid if failure occurs on either one of the outer surfaces of the specimen, without a preceding inter-laminar shear failure or a crushing failure around a support or the loading nose. The maximum flexural stress at the outer surface of the specimen depending on the test setup can be calculated using Eqs. (4.51) and (4.52). For the three-point bending setup: Fig. 4.31 The test setups for flexural testing: a Three-point bending setup with loading nose at the middle and fixed supports under the specimen, b Four-point bending setup with two rolling loading noses and fixed supports

94

4 Experimental Characterization

σ =

3P L 2bh 2

(4.51)

wherein, σ is the stress at the outer surface at mid-span (MPa), P is the applied force (N), L is the span length between supports (mm), and b and h are the width (mm) and thickness (mm) of the specimen, respectively. For the four-point bending setup: σ =

3P L 4bh 2

(4.52)

wherein, σ is the stress at the outer surface at mid-span region (MPa), and P, L, b, and h are the same as for Eq. (4.51). The maximum stress at the outer surface corresponding to the peak applied load to the specimen prior to failure is the flexural strength of the specimen. The maximum strain, which occurs at mid-span, corresponding to the three-point and four-point bending setups is calculated using Eqs. (4.53) and (4.54), respectively: 6δh L2

(4.53)

4.36δh L2

(4.54)

ε= ε=

ε is the maximum strain at the outer surface (mm/mm), d is the mid-span deflection (mm), L is the length of span between supports (mm), and h is the thickness of the specimen (mm). To find the flexural chord modulus of elasticity, it is recommended to divide the ratio of stress range corresponding to the strain range of 0.002 with a start point of 0.001 and an end point 0.003 (or the closest available data points). E chord = f

∆σ ∆ε

(4.55)

E chord is the flexural chord modulus of elasticity (MPa), ∆σ is the difference in f flexural stress (MPa) corresponding to the selected strain range of 0.002. The flexural secant modulus of elasticity is same as the flexural chord modulus in which the initial strain point is zero. It can be found for the three-point and four-point bending setups using Eqs. (4.56) and (4.57), respectively: L 3m 4bh 3

(4.56)

0.17L 3 m bh 3

(4.57)

E secant = f E secant = f

References

95

wherein, E secant is the flexural secant modulus of elasticity (MPa), L is the support f span (mm), b and h are the width and thickness of the specimen (mm), and m is the slope of the secant of the load–deflection curve.

References Alfrey T (1944) Non-homogeneous stresses in visco-elastic media. Q Appl Math 2:113–119 ASTM D 790 (2017) Standard test methods for flexural properties of unreinforced and reinforced plastics and electrical insulating materials ASTM D 3039 (2008) Standard test method for tensile properties of polymer matrix composite materials ASTM D 5379 (2019) Standard test method for shear properties of composite materials by the v-notched beam method ASTM D 6272 (2017) Standard test method for flexural properties of unreinforced and reinforced plastics and electrical insulating materials by four-point bending ASTM D 7264 (2007) Standard test method for flexural properties of polymer matrix composite materials ASTM C 597 (2016) Standard test method for pulse velocity through concrete ASTM D 638 (2014) Standard test method for tensile properties of plastics ASTM D 695 (2015) Standard test method for compressive properties of rigid plastics ASTM D 882 (2018) Standard test method for tensile properties of thin plastic sheeting ASTM D 2344 (2022) Standard test method for short-beam strength of polymer matrix composite materials and their laminates ASTM D 2990 (2017) Standard test methods for tensile, compressive, and flexural creep and creep-rupture of plastics ASTM D 3410 (2016) Standard test method for compressive properties of polymer matrix composite materials with unsupported gage section by shear loading ASTM D 3518 (2018) Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a ±45° laminate ASTM D 3822 (2020) Standard test method for tensile properties of single textile fibers ASTM D 6641 (2016) Standard test method for compressive properties of polymer matrix composite materials with unsupported gage section by shear loading ASTM D 7012 (2017) Standard test methods for compressive strength and elastic moduli of intact rock core specimens under varying states of stress and temperatures ASTM D 7078 (2005) Standard test method for shear properties of composite materials by v-notched rail shear method ASTM D 7205 (2006) Standard test method for tensile properties of fiber reinforced polymer matrix composite bars ASTM E 328 (2021) Standard test methods for stress relaxation for materials and structures ASTM E8/E8M (2022) Standard test methods for tension testing of metallic materials ASTM E9 (2009) Standard test methods of compression testing of metallic materials at room temperature ASTM-D6648-08 (2001) Standard test method for determining the flexural creep stiffness of asphalt binder using the bending beam rheometer (BBR). In: American society for testing and materials west conshohocken, PA B.E. 2561 (1995) Carbon fibre reinforced plastics. Undirectional laminates. Tensile test parallel to the fibre direction EN 2563 (1997) Carbon fibre reinforced plastics—unidirectional laminates: determination of apparent interlaminar shear strength

96

4 Experimental Characterization

Hajikarimi P, Sadat Hosseini A, Fini EH (2021) A heterogeneous micromechanical model for bituminous composites containing rigid and flexible particulates. Constr Build Mater 275:122102. https://doi.org/10.1016/j.conbuildmat.2020.122102 I. 527-5 (2009) Plastics—determination of tensile properties—part 5: test conditions for unidirectional fibre-reinforced plastic composites ISO 527-1 (2019) Plastics—determination of tensile properties—part 1: general principles ISO 527-4 (2021) Plastics—determination of tensile properties—part 4: test conditions for isotropic and orthotropic fibre-reinforced plastic composites ISO 14126 (1999) Fibre-reinforced plastic composites—determination of compressive properties in the in-plane direction ISO 14129 (1997) Fibre-reinforced plastic composites—determination of the in-plane shear stress/shear strain response, including the in-plane shear modulus and strength, by the plus or minus 45 degree tension test method ISO 14130 (1997) Fibre-reinforced plastic composites—determination of apparent interlaminar shear strength by short-beam method Jenket DR, Jenket DR, Engelbrecht-Wiggans AE, Forster AL, Al-Sheikhly M (2019) A new method for tensile testing UHMMPE single fibers at high temperatures and strain-rates. US Department of Commerce, National Institute of Standards and Technology Schiff JL (2013) The Laplace transform: theory and applications. Springer Science & Business Media Soric Z, Galic J, Rukavina T (2008) Determination of tensile strength of glass fiber straps. Mater Struct 41:879–890. https://doi.org/10.1617/s11527-007-9291-4 Timoshenko S, Young DH (1962) Elements of strength of materials. Van Nostrand Princeton

Chapter 5

Analytical and Empirical Formulation

5.1 Rule of Mixture in Composite Materials with Continuous Inclusions 5.1.1 Analytical Equations A unidirectional fiber reinforced polymer (FRP) lamina as illustrated in Fig. 5.1 consists of fibers along the main direction of the lamina (1-axis), and matrix all around the fibers. The 2-axis is perpendicular to 1-axis in the lamina plane and 3axis is normal to this plane. The 1–2–3 system is the local coordinate system of an anisotropic material which is required to be defined when mechanical properties of a lamina are formulated. All the formulations presented here are in the lamina local coordinate system unless it is stated that the global coordinate system (x–y–z system) is considered. The rule of mixture in composites is defined as a weighted average of the matrix and inclusions properties. To this aim, two definitions, volume fraction and weight fraction, are required: ϑf = wf =

Vf Vm , ϑm = V V

(5.1)

Wf Wm , wm = W W

(5.2)

wherein, ϑ f and ϑ m are the fibers and matrix volume fraction, respectively, which are defined based on the assumption that the overall volume of composite, V, consists only of fibers and matrix and the volume of void and other impurities are negligible. The same definition and assumptions are considered for fibers weight fraction, wf , and matrix weight fraction, wm , respectively, and W is the overall weight of the composite. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. Hajikarimi and A. Sadat Hosseini, Constructional Viscoelastic Composite Materials, https://doi.org/10.1007/978-981-99-1786-0_5

97

98

5 Analytical and Empirical Formulation

Fig. 5.1 A scheme of a unidirectional lamina and its local coordinate system

Fig. 5.2 Simplification of the composite microstructure

The other assumption is a perfect bond between fibers and matrix with no slip between these two materials. Therefore, the overall composite can be approximated as illustrated in Fig. 5.2, and material properties can be calculated based on the direction of the loading on the lamina. By assuming a linear elastic stress–strain behavior for the matrix in relatively low temperatures and high loading frequencies, the material properties and engineering constants of a fiber-reinforced polymer lamina could be approximately driven using the following equations: ρ = ρ f ϑ f + ρm ϑm

(5.3)

E 11 = E f ϑ f + E m ϑm

(5.4)

ϑf 1 ϑm = + E 22 Ef Em

(5.5)

ϑf 1 ϑm = + G 12 G f,12 Gm

(5.6)

5.1 Rule of Mixture in Composite Materials with Continuous Inclusions

99

ϑf 1 ϑm = + G 23 G f,23 Gm

(5.7)

ν12 = ν f ϑ f + νm ϑm

(5.8)

ν21 = ν12

E 22 E 11

(5.9)

in which ρ is mass density, E 11 and E 22 are longitudinal and transverse elastic modulus, G12 is in-plane shear modulus, ν12 and ν21 are the major and minor Poisson’s ratios, and ϑ f and ϑ m are volume fractions of fibers and matrix in the composite lamina, respectively. Gf is defined as follows: Ef ) Gf = ( 1 + νf

(5.10)

One application of the above set of equations is finding the approximate required weight of resin to achieve a specific fibers volume fraction in the wet layup process. The dry weight of fabrics is generally provided by the manufacturer in gr/m2 , however, one can find or double-check it by weighing one square meter of the fiber, Wf . The density of fibers, ρf , and matrix, ρm , are known. Therefore, the fiber volume fraction or the fiber content of the resulting layup, assuming Ap as the ratio of entrapped air bobble in the product can be calculated as follows: V = V f + Vm + V A and V A = A p V → ϑ f + ϑm = 1 − A p ∫

(5.11)

( ) 1 − Ap Wf ρf ϑf ρm Wf = ρf V f → = × Wm = Wf −1 Wm ρm 1 − Ap − ϑ f ρf ϑf Wm = ρm V m (5.12)

By using Eq. (5.12), for a unidirectional fabric with dry weight of 800 gr/m2 , ρ f = 2.6 t/m3 , and ρ m , = 1.3 t/m3 , 320 gr resin per each used square meter of fabric is required to achieve a fiber volume fraction of 55%, considering 1% void in the composite (Ap = 0.01). The composite longitudinal tensile strength along 1-axis is calculated using the following relationship: ( ) Em SL T = S f T ϑ f + ϑm Ef

(5.13)

where S fT represents the fiber tensile strength which can be determined by performing the single fiber tensile test (see Chap. 4). In Eq. (5.13), it is assumed that all fibers in the

100

5 Analytical and Empirical Formulation

composite fail simultaneously, and the failure strain of fibers is the governing strain for both fibers and matrix. This is a deterministic approach toward the failure mechanism in a bundle of fibers. There is also a probabilistic approach which describes the fraction of failed fibers using Weibull distribution as follows: [

(

ε R(ε) = ex p −L ε0

)λ ] (5.14)

in which L is equal length of fibers subjected to a strain ε, ε0, and λ are constants. The SfT is represented as (Tsai and Hahn 2018):

SfT

) ) ( ∫∞ ( 1 dR − λ1 dε = ε0 L Γ 1 + = ε − dε λ

(5.15)

0

where Γ(.) denotes Gamma function. The composite transverse tensile strength is calculated using the following formula: ) ( [ ( )] σm,23 avg 1 ) 1+ϑf (5.16) ST T = ( S η23 − 1 σm,23 max mT in which SmT is matrix tensile strength, and η23 is stress partitioning parameter. In the literature, the ratio of (σ m,23 )max /(σ m,23 )avg is introduced as stress concentration factor. Having a ductile behavior for matrix, this factor becomes closer to unity. Considering the viscoelastic behavior of matrix, the following methodology can be used to predict the mechanical properties of FRPs, assuming that there is a perfect bond between matrix and fibers, and the matrix shows linear viscoelastic behavior while fibers behave as a linear time-independent elastic material. The following equilibrium is the starting point for derivation of material properties of FRP unidirectional laminate subjected to a creep loading along the fibers’ direction: σ0 = σm (t)ϑm + σ f (t)ϑ f

(5.17)

wherein, σ 0 is the normal stress in composite which is calculated using rule of mixture on time-dependent stresses in matrix and fibers. Considering the first basic assumption of perfect bond between matrix and fibers, the normal strain in the matrix and fibers should be equal: ε f (t) = εm (t)

(5.18)

5.1 Rule of Mixture in Composite Materials with Continuous Inclusions

101

Assuming linear viscoelastic behavior for matrix and fibers, the following constitutive equations can be used: ∫t εm (t) = σm (t0 )Dm (t0 ) +

dσm (t) Dm (ξ )dξ dξ

(5.19)

dσ f (t) D f (ξ )dξ dξ

(5.20)

t0

∫t ε f (t) = σ f (t0 )D f (t0 ) + t0

wherein Dm (t) and Df (t) are creep compliance of the matrix and fibers, respectively, t is time, t 0 is the starting time of creep loading, and ξ is the integration variable of time. Substituting Eqs. (5.19) and (5.20) into Eq. (5.18) yields: ∫t σm (t0 )Dm (t0 ) +

dσm (t) Dm (ξ )dξ = σ f (t0 )D f (t0 ) + dξ

∫t

dσ f (t) D f (ξ )dξ dξ

t0

t0

(5.21) and then, by using Eq. (5.17), σ f (t) can be re-written and substituted in Eq. (5.21): ( ) σ0 vm D f (t0 ) + σm (t0 ) Dm (t0 ) + D f (t0 ) Vf vf t ] ∫ [ dσ f (t) dσm (t) + Dm (ξ ) − D f (ξ ) dξ = 0 dξ dξ

−

(5.22)

t0

By defining the following parameter: ∆

D (t) =

ϑm D f (t) + Dm (t) ϑf

(5.23)

and considering the convolution integral property (see Chap. 2), Eq. (5.22) can be re-written as follows: σ0 D f (t0 ) + σm (t) = ϑ f D (t0 )

∫t

∆

σm (ξ ) d D (ξ ) dξ D (t) dξ ∆

∆

t0

(5.24)

Applying the Laplace transformation on both sides of the Eq. (5.24), the following equation can be derived:

102

5 Analytical and Empirical Formulation

⎤ ⎡ t ] ∫ σ0 D f (t0 ) 1 d D (ξ ) dξ ⎦ L[σm (t)] = L + L⎣ σm (ξ ) ϑ f D (t0 ) D (t) dξ [

∆

∆

(5.25)

∆

t0

Assuming t 0 = 0 and σ m (t) = 0 for t < t 0 , the convolution theorem reveals the following equation: ⎡ L⎣

∫t

⎤ [ ] 1 d D (ξ ) ⎦ 1 d D (t) 1 d D (s) σm (ξ ) dξ = L[σm (ξ )]L = σm (s) D (t) dξ D (t) dt D (s) ds ∆

∆

∆

t0

∆

∆

∆

(5.26) in which σ m (s) and D(s) represent the Laplace transforms of the σ m (t) and D(t), respectively. Therefore, Eq. (5.25) can be written as: ] [ D f (t0 ) 1 d D (s) σ0 + σm (s) L σm (s) = ϑf D (s) ds D (t0 ) ∆

∆

∆

(5.27)

It is obvious that σ m (s) can be found by using the inverse Laplace transform as follows: σm (t) = L−1 [σm (s)]

(5.28)

Various mechanical models, such as the Burgers model, generalized Maxwell model, or generalized Kelving-Voigt model, can be used to represent the creep compliance function of viscoelastic materials, as explained in Chap. 2. Considering the Burgers model as a simple and efficient mechanical model for representing both creep and recovery behavior of a viscoelastic material, the creep compliance function can be represented as: D(t) =

[ ( )] E2 t 1 1 1 − exp −t + + E1 η1 E2 η2

(5.29)

In Eq. (5.23), the creep compliance functions of matrix and fibers, Dm (t) and Df (t) can be represented by using Eq. (5.25).

5.1.2 Semi-Analytical Equations Experimental observations have shown that the simple rule of mixture cannot properly predict the mechanical behavior of fiber-reinforced matrix lamina in transverse direction and shear (Jones 1968). However, using a modification on the basic Eqs. (5.5) to (5.7) implementing stress partitioning parameter, η, can enhance the

5.1 Rule of Mixture in Composite Materials with Continuous Inclusions

103

prediction’s accuracy. The following equations can be derived using the mentioned parameter for the transverse elastic and shear moduli (see Fig. 5.1): 1 1 = E 22 ϑ f + η22 ϑm 1 1 = G 12 ϑ f + η12 ϑm 1 1 = G 23 ϑ f + η23 ϑm

( ( (

ϑf ϑm + η22 Ef Em

)

ϑf ϑm + η12 Gf Gm ϑf ϑm + η23 Gf Gm

(5.30) ) (5.31) ) (5.32)

where, η22 and η12 can be approximated to 0.5. Meanwhile, based on some advanced experiments (Hermans 1967; Hashin 1972), the shear stress partitioning parameter can be written as follows: ) ( Gm ≈ 0.5 (5.33) η12 = 0.5 × 1 + Gf For η23 , the following equation is used: η23

) ( 3 − 4ϑm 1 Gm ≈ 3 − 4ϑm + = 4(1 − ϑm ) G f,23 4(1 − ϑm )

(5.34)

The transverse plane strain bulk modulus, k 2 , is: 1 1 = K ϑ f + ηk ϑm

(

ϑf ϑm + ηk Kf Km

) (5.35)

wherein K f and K m are bulk moduli of fiber and matrix, respectively. The ηk is calculated using: ) ( 1 Gm 1 ≈ 1+ ηk = 2(1 − ϑm ) K f,23 2(1 − ϑm )

(5.36)

If fiber is isotropic, the subscript 23 can be eliminated. Thus, Eqs. (5.34) and (5.36) can be simplified considering Gm /Gf > Gm , or in other words, Gf /Gm >> 1 or even 10. Therefore, knowing that the 0 < υ f < 1 and 0 < υ m < 1, following equalities can be approximately considered without significant error: ) ( ) Gf −1 ≈ Gm (( ) ) ( ) Gf Gf −8 ≈ Gm Gm (( ) ) ( ) Gf Gf +4 ≈ Gm Gm ((

Gf Gm

)

(5.91a) (5.91b) (5.91c)

5.2 Rule of Mixture in Composite Materials with Discontinuous Inclusions

115

((

) ) ) ( ) ( ) ) Gf ( Gf ( ν f − 2νm + 2ν f − νm ≈ ν f − 2νm Gm Gm ) ( ) ) (( Gf Gf (8 − 10νm ) + (7 − 5νm ) ≈ (8 − 10νm ) Gm Gm

(5.91d) (5.91e)

Implementing Eqs. (41a)–(41e), Eqs. (40a)–(40c) can be simplified as follows: ) ) Gf ( 7 + 5ν f (7 − 10νm ) Gm ) ( ) Gf ( 7 + 5ν f η2 = Gm ) ( Gf η3 = (8 − 10νm ) Gm (

η1 =

(5.92a) (5.92b) (5.92c)

and Eqs. (39a)–(39c) are simplified accordingly as: )( ) ) ) ( Gf ( Gf 7 + 5ν f × −4(1 − 5νm )(7 − 10νm )φ f (10/3) A= Gm Gm ) ( −50 7 − 12νm + 8νm 2 φ f (7/3) + 252φ f (5/3) ) ( ) −50 7 − 12νm + 8νm 2 φ f + 8(4 − 5νm )(7 − 10νm ) (5.93a) ((

)( ) ) ( ) Gf ( Gf 7 + 5ν f × − 4(1 − 5νm )(7 − 10νm )φ f (10/3) B= Gm Gm ) ( + 100 7 − 12νm + 8νm 2 φ f (7/3) − 504φ f (5/3) ) (5.93b) + 150(3 − νm )νm φ f + 6(4 − 5νm )(15νm − 7) ((

)( ) ) ( ) Gf ( Gf 7 + 5ν f × 4(5νm − 7)(7 − 10νm )φ f (10/3) Gm Gm ) ( − 50 7 − 12νm + 8νm 2 φ f (7/3) + 252φ f (5/3) ) ( ) (5.93c) + 25 νm 2 − 7 φ f − (7 + 5νm )(8 − 10νm ) ((

C=

5.2.2.6

Differential Scheme Approach

To increase the accuracy of the predicted results, especially for composites having higher volume fractions of inclusions (dense inclusions), some researchers (McLaughlin 1977; Norris 1985) proposed the differential scheme (DS) approach.

116

5 Analytical and Empirical Formulation

In the DS approach, inclusions are added gradually in steps and the (C com * ) can be calculated by using the incremental formulation as: (

) ( )) ( ) Δφ void (i + 1) ( ( )) ( ) ) ( ) Δφ inc (i + 1) (( C inc − C mat ∗ (i) : Ainc (i) + − C mat ∗ (i) : Avoid (i) C com ∗ (i + 1) = C com ∗ (i) + φmat (i ) φmat (i )

(5.94)

in which (C com * (i + 1)) and (C com * (i)) are the effective stiffness tensor of the composite in step i + 1 and step i, respectively. (Ainc (i)) and (Avoid (i)) are (Ainc ) and (Avoid ) in step i which can be determined using the Dilute model, the Mori– Tanaka model, the self-consistent model, the Lielens’ model, or the GSCS model, depending on the characteristics of composite’s components and the volume fraction of inclusions. Also, Δφ inc (i + 1) and Δφ void (i + 1) are the increments of φ inc and φ void in step i + 1, respectively.

5.2.2.7

Empirical Models

For some specific composites, researchers have proposed some empirical models based on experimental observations. Buttlar et al. (1999) suggested the following exponential expression for asphalt mastic using nonlinear regression analysis: Gc = aexp(bG m ) Gm

(5.95)

a = 25.083φ f 3 − 10.154φ f 2 + 4.876φ f + 0.831

(5.96)

in which,

and ) ) ) ) ( ( ( ( b = −1.28 × 10−3 φ f 3 − 3.37 × 10−4 φ f 2 + 3.08 × 10−4 φ f + 6.53 × 10−6 (5.97) Faheem and Bahia (2011) also developed an empirical model to calculate the relative complex shear modulus of an asphalt mastic. They proposed the following expression: G

∗

{

r

(φ − φc ) = G 1 + a1 (φ − φc ) + G 2 (a2 − a1 )ln 1 + exp G2

wherein, φ

[

filler volume fraction (%)

]} (5.98)

References

117

φ c critical filler concentration = 83.2 × Rigden voids percentage % + 4.79 × Methylene blue value (MBV) a1 Initial stiffening rate = 0.212–0.373 RV% a2 Terminal stiffening rate = 2.97–5.06 × Rigden voids percentage % + 0.383 × MBV–2.69 CaO% G1 1.5[(Binder G*)/(6 × Binder Asphaltene Content)] G2 log[(10 × D10 × D50 )/(φ × D90 )]

References Buttlar WG, Bozkurt D, Al-Khateeb GG, Waldhoff AS (1999) Understanding asphalt mastic behavior through micromechanics. Transp Res Rec 1681:157–169 Callister WD, Rethwisch DG (2007) Materials science and engineering: an introduction. John Wiley & Sons New York Christensen RM, Lo KH (1979) Solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solids 27:315–330. https://doi.org/10.1016/0022-5096(79)900 32-2 Christensen RM, Lo KH (1986) Erratum: solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solids 34:639 Cox HL (1952) The elasticity and strength of paper and other fibrous materials. Br J Appl Phys 3:72 Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc London Ser A Math Phys Sci 241:376–396 Hashin Z (1972) Theory of fiber reinforced materials. NASA Hermans JJ (1967) Elastic properties of fiber reinforced materials when fibers are aligned. K Ned Akad Van Weteschappen-Proceedings Ser B-Physical Sci 70:1-+ Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 13:213–222 Jones B (1968) Analytical design procedures for the strength and elastic propertiesof multilayer fibrous composites. In: 9th structural dynamics and materials conference. American Institute of Aeronautics and Astronautics Landel RF, Nielsen LE (1993) Mechanical properties of polymers and composites. CRC Press Lee L-H (1969) Strength-composition relationships of random short glass fiber-thermoplastics composites. Polym Eng Sci 9:213–224. https://doi.org/10.1002/pen.760090310 Lielens G, Pirotte P, Couniot A et al (1998) Prediction of thermo-mechanical properties for compression moulded composites. Compos Part A Appl Sci Manuf 29:63–70 McLaughlin R (1977) A study of the differential scheme for composite materials. Int J Eng Sci 15:237–244 Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574. https://doi.org/10.1016/0001-6160(73)90064-3 Ning P (1996) The elastic constants of randomly oriented fiber composites: a new approach to prediction. Sci Eng Compos Mater 5:63–72. https://doi.org/10.1515/SECM.1996.5.2.63 Norris AN (1985) A differential scheme for the effective moduli of composites. Mech Mater 4:1–16 Starink MJ, Syngellakis S (1999) Shear lag models for discontinuous composites: fibre end stresses and weak interface layers. Mater Sci Eng A 270:270–277 Tsai SW, Hahn HT (2018) Introduction to composite materials. Routledge

Chapter 6

Numerical Simulation

This chapter starts with the necessary theoretical background on Finite Element Modeling, such as introducing different types of the commonly used elements, the stiffness matrix, mass matrix, and so forth. This chapter then introduces the numerical modeling procedure of composite materials with linear and nonlinear viscoelastic matrices by using incremental viscoelastic relations developed in Chap. 2. Consequently, two numerical modeling approaches, geometrically homogenous and geometrically heterogeneous composite modeling, are discussed, and some practical examples are provided. An extension of the finite element method for using in discontinuum medium is also presented in this chapter for using eXtended Finite Element Method (XFEM) to simulate inclusions and crack initiation and propagation.

6.1 A Review on Finite Element Method To solve structural problems using the finite element method (FEM), the geometry domain is initially discretized into small pieces that are called elements. Subsequently, to have the desired results for the global domain which consists of elements, simplified element displacement equations are assembled to form the overall FEM equation and then solved. One simple method to produce the FEM formulation of structural mechanics is Hamilton’s principle. This powerful principle can be used to derive equations for discretized dynamic system generally and can be derived for static cases especially. For more information in this regard, readers are referred to the study by Vance and Sitchin (1972) and Cavin and Dusto (1977). The FEM procedure is very well described by Liu and Quek (2013). Here, each step is briefly explained for practical users, and in the following sub-sections, FEM for two- and three-dimensional shells and solids is explained. The first step in the FEM procedure is to discretize the domain. For this purpose, the geometry is divided into n elements, and these elements construct the mesh

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. Hajikarimi and A. Sadat Hosseini, Constructional Viscoelastic Composite Materials, https://doi.org/10.1007/978-981-99-1786-0_6

119

120

6 Numerical Simulation

Fig. 6.1 Global and local coordinate systems

configuration of the domain. Numbers are attributed to the elements and their corresponding nodes in the whole domain. The larger the number of elements, means the finer the mesh size, the higher the accuracy and run time. Therefore, the size of the mesh can be determined based on the required accuracy and indeed the size of the structure. The second step is the interpolation of the displacements. Polynomial interpolation is used on the displacements at the nodes of each element to find the element displacement. This is called interpolation of nodal displacements to the element displacements (Eq. 6.1) which needs a coordinate system for the FEM formulations. Usually, the required coordinate system is a local coordinate system for each element about the global coordinate system (Fig. 6.1).

Uh (x.y.z) =

nd ⎲

Ni (x.y.z)di = N(x.y.z)de

(6.1)

i=1

wherein, nd is the number of nodes in an element, di is the displacement of the ith node considering the number of degrees of freedom (DoF) at each node, and Ni is the matrix of shape functions for each node which is normally defined based on the shapes of the displacement variations in the displacement method. In the third step, the shape functions are constructed. In a standard procedure, we consider an element with nd nodes that requires nd shape functions. The shape functions in a matrix form, N(x), can be written as follows: Ni (x) = pT (x)Pi−1

(6.2)

wherein, Pi −1 is the ith column of matrix P−1 , assuming that we can have the inverse of matrix P. P is the moment matrix consisting of nd rows of PT (x i ), and PT (x) is the basic function of monomials in the space coordinates x. Since basis functions are linearly independent, the shape functions are also linearly independent. The other property of the shape functions is that they are unit at their home node and zero at the

6.1 A Review on Finite Element Method

121

remote node. Moreover, based on the reproduction feature of the shape functions, it is proved that the sum of shape functions at the nodes of an element is equal to one (Eq. 6.3) and they have linear field reproduction property (Eq. 6.4). For more information, readers are referred to Liu and Quek (2013). n ⎲

Ni (x) = 1

(6.3)

Ni (x)xi = x

(6.4)

i=1 n ⎲ i=1

In the fourth step, the finite element equations are formulated in the local coordinate system. Substituting Eq. (6.1) into the strain energy equation over the global domain (Eq. 6.5) and considering the strain–displacement relationships, the strain energy in elements can be written in the form of Eq. (6.6). 1 ∏= 2

∫

1 ε σdV = 2

V

1 ∏= 2

∫

1 ε T cεd V = 2

Ve

∫

∫ ε T cεd V

T

(6.5)

V

⎞ ⎛ ∫ 1 de T BT cBde d V = de T ⎝ BT cBd V ⎠de 2

Ve

(6.6)

Ve

wherein, subscript e represents the element, and B is the strain matrix. The integral term inside the parentheses is called the element stiffness matrix, ke , and is symmetrical. Moreover, by substituting Eq. 3.6 in Eq. (6.7) (the kinetic energy), the kinetic energy in the element domain can be found in Eq. (6.8). T=

1 2

∫

ρ U˙ T U˙ d V

(6.7)

V

1 T= 2

∫ V

1 ρ U˙ T U˙ d V = 2

∫ Ve

⎞ ⎛ ∫ 1 ρ d˙ e NT Nd˙ e d V = d˙ eT ⎝ ρNT Nd V ⎠d˙ e 2 T

(6.8)

Ve

wherein, the integral term inside the parentheses is called the element mass matrix, me , and is symmetrical. The amount of energy expended by external forces within a specific element or subset of elements, or the work done by the external forces in the element domain (Eq. 6.10) can be found by substituting Eq. (6.1) into the external force equation (Eq. 6.9). Wherein, the surface integration is considered only for elements on the force boundary of the domain.

122

6 Numerical Simulation

∫

∫

Wf =

U T fb d V + V

∫ Wf =

T

∫

de T N fb d V + Ve

Se

U T fs d S f

(6.9)

Sf

⎞ ⎞ ⎛ ⎛ ∫ ∫ de T N fs d S = de T ⎝ NT fb d V ⎠ + de T ⎝ NT f S d V ⎠ T

Ve

Se

(6.10) In Eq. (6.10), the integral terms inside the parentheses are the body forces and surface forces applied on elements, respectively. The summation of these terms forms the nodal force vector, fe . By considering Hamilton’s principle and further mathematical procedure (Liu and Jerry 2002) the FEM equation for an element can be written as: ke de + me d¨ e = fe

(6.11)

wherein, ke , me, and fe are the element stiffness matrix, mass matrix, and force vector, respectively, and de and d¨ e are the element displacement and acceleration vectors, respectively. Since Eq. (6.11) is formulated based on the local coordinate system of a single element, the coordination of the element has to be transformed in the sixth step to the global coordinate system for assembly purposes. This can be done by applying the transformation matrix to the displacements and forces. After transformation, in the seventh step, the finite element equations of all elements can be assembled to add up the contributions from all elements connected to a node. Consequently, the displacement constraints are imposed in the eighth step by removing the columns and rows that are corresponding to the constrained nodal displacements. Finally, at the ninth step, the displacements at the nodes can be calculated by solving the finite element equation in the global coordinate system.

6.1.1 Fem for Two-Dimensional Solids A review of the finite element modeling formulation for two-dimensional (2D) solids subjected to external loads is presented in this section. Although in reality there is no 2D solid structure, 2D analysis of the three-dimensional problems (plane stress and plane strain problems) is an effective and economical simulation method. Plane stress conditions mostly are applied to structures that have a relatively small thickness and normal stresses are negligible. However, in plane strain condition, the thickness of the structure is relatively large and uniform stress is applied to the thickness.

6.1 A Review on Finite Element Method

123

Fig. 6.2 The linear triangular element

The element that is used in this analysis methodology is either a plane stress or a plane strain element which can be triangular, rectangular, or quadrilateral in shape depending on the shape function order. In plane stress problems, the elements have the thickness of the actual model for computing the stiffness matrix and stresses. However, plane strain elements are assumed to have a thickness of unity. The first and simplest type of element developed for 2D solids was the linear triangular element (Fig. 6.2). Although this element is less accurate with respect to linear quadrilateral elements, it has higher adaptivity to be used in complex geometries. The linear triangular element in Fig. 6.2 has two degrees of freedom at each node, u and v, as the displacement field variables. The general formulation of the linear triangular element is: ⎡ N ode I N ode 2 [ h ] ⎢   u (x.y) ⎢ N 0 N2 0 =⎢ 1 v h (x.y) ⎣ 0 N1 0 N2

 Uh (x. y)

 N

⎧ ⎪ u ⎪ ⎤ ⎪ 1 N ode 3 ⎪ ⎪  ⎪ ⎪ v1 ⎥⎨ N3 0 ⎥ u 2 ⎥ v2 0 N3 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ u ⎪ 3 ⎩

⎪ v3



⎫ } ⎪ ⎪ ⎪ N ode 1 ⎪ ⎪ ⎪ ⎪ ⎬ } N ode 2 ⎪ ⎪ } ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ N ode 3

(6.12)

de

wherein, N is the matrix of shape functions, d e is the nodal displacement vector, and Uh is the displacement averaged for thickness, h. The standard procedure for finding the shape functions for any type of element starts with an assumption for the displacements using polynomial basis functions. These functions have unknown constants which can be found by using the known displacements at the nodes of the element. Afterward, the strain matrix should be derived for computing the element’s stiffness matrix. For 2D solids, the stress components and corresponding strains are as follows: ] [ σT = σx x σ yy σx y

(6.13a)

] [ εT = εx x ε yy εx y

(6.13b)

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6 Numerical Simulation

Once the matrices of shape function and strain are obtained, the strain and displacement can be expressed in terms of the element’s nodal displacement, and hence the stiffness matrix, mass matrix, and nodal force vector can be found. For higher order triangular elements such as the quadratic triangular element with six nodes and cubic triangular element with nine nodes (Fig. 6.3) the shape function can be found using an area coordinate system (Argyris et al. 1968). The second type of element for 2D solids is the linear rectangular element. This element is more accurate than the triangular element and it is preferred to be used for problems where the geometry is not too complex. Unlike the triangular element, the strain matrix of the rectangular element is not constant which provides more realistic results. The shape functions can form easier than the triangular element due to regularity in the element shape. Figure 6.4 shows the rectangular element with four nodes and two degrees of freedom per node.

Fig. 6.3 a The quadratic triangular element, and b the cubic triangular element

6.1 A Review on Finite Element Method

125

Fig. 6.4 The rectangular element and the coordinate system, a Rectangular element in physical system, and b square element in natural coordinate system

The displacement vector, Uh , can be written as:

⎡ N ode I [ h ] ⎢  u (x.y) ⎢N 0 =⎢ 1 h v (x.y) ⎣ 0 N1

 Uh (x. y)

⎧ ⎫ ⎪ ⎪ u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v1 ⎪ ⎪ ⎪ ⎤ ⎪ ⎪ N ode 2 N ode 3 N ode 4 ⎪ ⎪ u    ⎪ 2⎪ ⎪ ⎪ ⎪ ⎥⎨ ⎪ ⎬ N2 0 N3 0 N4 0 ⎥ v2 ⎥ u3 ⎪ 0 N2 0 N3 0 N4 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪v ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 

⎪ ⎪ ⎪ u ⎪ ⎪ 4 ⎪ N ⎩ ⎪ ⎭ v4

} N ode 1 } N ode 2 }

(6.14) N ode 3

} N ode 4 

de

wherein, N is the shape function of the element, and d e is the nodal displacement vector. The element matrices such as mass matrix, stiffness matrix, and force vector can be found when the matrices of shape functions and strains are found. Two examples of the higher order rectangular elements are Lagrange and Serendipity type elements as illustrated in Figs. 6.5 and 6.6. For more information regarding the shape functions the reader is referred to Zienkiewicz and Taylor (2005), and Liu and Quek (2013). The third type of 2D element is the linear quadrilateral element which can address the shortcoming of rectangular elements in discretizing non-rectangular domains. Fig. 6.5 The higher order rectangular Lagrange elements

126

6 Numerical Simulation

Fig. 6.6 The higher order rectangular Serendipity elements

Fig. 6.7 Coordinates mapping between coordinate systems

The quadrilateral element is more practical than the other two, however, the irregular shape makes the integration of the mass and stiffness matrices difficult. To cope with this problem, the quadrilateral element is mapped into the natural coordinates system and the resulting square element can be dealt with the same as that was done for rectangular elements. The quadrilateral element in Fig. 6.7 with four nodes and two degrees of freedom per node. A local natural coordinate system (ξ η, 0) is defined whose origin is at the center of the mapped element. This local coordinate system is used for constructing the shape functions.

6.1.2 FEM for Plates and Shells The finite element matrices of plates and shells need more computational effort since they involve more degrees of freedom. Herein, at first, the plate elements are reviewed and then the shell element whose finite element matrices are obtained by superimposing the plate elements and 2D solids matrices.

6.1 A Review on Finite Element Method

127

Geometry wise, a plate structure is like two-dimensional plane stress, but it only carries transversal out-of-plane loads which produce a bending moment in the plate. The schematic representation of the plane structure is its middle plane laying on a surface like x–y, and the deformations in plane are presented by deflection and rotation of the normal of the mid-plane. It means that the plane elements of a plane have the function of beam elements but in two dimensions. They are used for analyzing the bending moments and shear forces in the plane. One of the theories that was developed for plane structures is the Reissner–Mindlin plate theory which works for thick plates. For more information, the reader is referred to Bathe and Dvorkin (1985). In this theory, the plate element has a constant thickness of h. If the thickness of the structure varies, the structure can be divided into smaller pieces assuming constant thickness for each or it might be more convenient to develop the formulation for varying thicknesses (Kohn and Vogelius 1984). The plate in Fig. 6.8 is properly divided into rectangular elements each of which has four nodes and four straight edges. Each node of an element has three degrees of freedom, deflection, and the rotation about x- and y-axis. Therefore, the rectangular plate element has twelve degrees of freedom. Since in Reissner–Mindlin plate theory, the straight fibers along thickness (perpendicular to mid-plane) remain straight after deformation. Hence, the in-plane displacement components are a function of the rotations with respect to the x and y axes. By using Hamilton’s principle, the in-plane and out-of-plane stresses and strains form the strain energy of the element to form the constitutive equations. Here, for the sake of brevity, the formulations are not provided. For more information, readers are referred to Liu and Quek (2013). The deflection of the plate and the rotations around the x and y axes are proved to be independent and hence they have independent shape functions. The matrix form of the displacement vector is like the two-dimensional solid except for the type and number of the degrees of freedom: Fig. 6.8 2D domain of a plate meshed by rectangular elements

128

6 Numerical Simulation

⎧ ⎫ w1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θx1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ y1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w 2 ⎪ ⎪ ⎤ ⎡ N ode 1 ⎪ ⎪ N ode 2 N ode 3 N ode 4 ⎪ ⎪ ⎪







 ⎪ ⎪ θx2 ⎪ ⎡ ⎤ ⎪ ⎪ ⎥ ⎢ ⎪ ⎪ w ⎢ N1 0 0 N2 0 0 N3 0 0 N4 0 0 ⎥ ⎪ ⎬ ⎨θ ⎪ y2 ⎢ ⎥ ⎢ ⎥ ⎣ θx ⎦ = ⎢ 0 N1 0 0 N2 0 0 N3 0 0 N4 0 ⎥ ⎥ ⎪ w3 ⎪ ⎢ ⎪ ⎪ ⎣ 0 0 N1 0 0 N2 0 0 N3 0 0 N4 ⎦ ⎪ θy ⎪ ⎪ ⎪



⎪ ⎪ ⎪ θx3 ⎪

 ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ Uh ⎪ ⎪ ⎪ θ y3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w 4⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ x4 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ θ y4

⎫ ⎬ ⎭ ⎫ ⎬

N ode 1

⎭ ⎫ ⎬

N ode 2

⎭ ⎫ ⎬

N ode 3

⎭

N ode 4



de

(6.15) wherein, N is the shape function of the plate element, and de is the nodal displacement vector consisting of one deflection and two rotational components. Having the shape function and nodal variables, the element matrices can be formulated. Practically, it is assumed that the shell elements are flat, and a curved shell structure is constructed by changing the orientation of the shell elements in space. Flat shell elements are used in commercially available software packages, and it is required by the user to refine the mesh when the curvature of the shell structure is very large. By using the formulation of the plate element and the two-dimensional solid element, the shell element which has both out-of-plane and in-plane degrees of freedom can be formulated. A similar procedure such as that is used for formulating the frame element from the beam and truss element can be used. Thus, such as a frame element, there are six degrees of freedom per node for a shell element, three of which represent the displacements while the rest are rotational deformations (Fig. 6.9). The stiffness and mass matrices for a rectangular shell element in the local axis are generated using the superposition of the two-dimensional solid element matrices (membrane effects) and plate element matrices (bending effects). Therefore, the stiffness and mass matrices for a rectangular shell element are 24 × 24. To transform the elements from the local axis to the global coordinate system, a transformation matrix consisting of direction cosines is used (Liu and Quek 2013). In the superposition procedure, it is assumed that the membrane and the bending effects are uncoupled. However, these two effects are coupled globally. Piecewise fiat elements can be used to discretize the curved shell structure with a strong curvature.

6.1 A Review on Finite Element Method

129

Fig. 6.9 The middle plane of a rectangular shell element

6.1.3 FEM for 3D Solids The three-dimensional solid element is the general form of the solid element considering the displacements in x, y, and z-axis (three degrees of freedom per node). A three-dimensional solid element has three normal and three shear stress components. Although this element is the most general type of element, it is always preferred to simplify the structure into a one-dimensional or two-dimensional structure and use corresponding elements within acceptable tolerances. Because using the most general form of the elements demands more computational resources, is time consuming, and is not economical. The formulation of three-dimensional solid elements is basically an extension of the two-dimensional solid elements. In the rest of this section, the solid elements are introduced. Figure 6.10 illustrates a tetrahedron element having four nodes, each of which has three degrees of freedom. The displacement vector of the tetrahedron element, U h , is as general the multiplication of the shape function matrix, N, by the nodal displacement vector, de :

130

6 Numerical Simulation

Fig. 6.10 A tetrahedron element

⎧ ⎫ u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v ⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u2 ⎪ ⎪ ⎪ ⎤ ⎡ N ode 1 ⎪ ⎪ N ode 2 N ode 3 N ode 4 ⎪ ⎪     ⎪ ⎪ ⎪ v2 ⎪ ⎪ ⎪ ⎢ N 0 0 N 0 0 N 0 0 N 0 0 ⎥⎪ ⎪ ⎥⎪ ⎢ 1 ⎬ ⎨w ⎪ 2 3 4 2 ⎥ ⎢ h U (x.y.z) = ⎢ 0 N1 0 0 N2 0 0 N3 0 0 N4 0 ⎥ ⎥⎪ ⎢ ⎪ ⎪ u3 ⎪ ⎣ 0 0 N1 0 0 N2 0 0 N3 0 0 N4 ⎦ ⎪ ⎪ ⎪ ⎪



⎪ ⎪ ⎪ v3 ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪w ⎪ ⎪ 3⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u 4⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v4 ⎪ ⎪ ⎭ ⎩ ⎪ w4

⎫ ⎬ ⎭ ⎫ ⎬

N ode 1

⎭ ⎫ ⎬

N ode 2

⎭ ⎫ ⎬

N ode 3

⎭

N ode 4



de

(6.16) The shape functions of this element are found using volume coordinates, and extension of the area coordinates for two-dimensional solids. By using this method, the shape function of node i in a tetrahedron element can be written as:

6.1 A Review on Finite Element Method

⎛

131

⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎞ x j yj z j 1 yj z j yj 1 z j yj z j 1 ⎝det⎣ xk yk z k ⎦ − det⎣ 1 yk z k ⎦x − det⎣ yk 1 z k ⎦ y − det⎣ yk z k 1 ⎦z ⎠ xl yl zl 1 yl zl yl 1 zl yl zl 1 Ni = ⎤ ⎡ 1 xi yi z i ⎢ 1 x j yj z j ⎥ ⎥ det⎢ ⎣ 1 xk yk z k ⎦ 1 xl yl zl (6.17) in which the subscript i varies from 1 to 4, and j, k, and l are determined by a cyclic permutation in the order of i, j, k, l. The shape function in Eq. (6.17) is a linear function of x, y, and z, and therefore, the tetrahedron element is a linear function. The strain matrix can be found from the shape function matrix, and subsequently, the stiffness matrix and mass matrix can be obtained. There exist higher order tetrahedron elements with ten and twenty nodes (Fig. 6.11). Unlike the 4-node tetrahedron element that was linear, the 10-node and 20-node elements are quadratic and cubic, respectively. Therefore, complete polynomials of up to second and third order are needed, respectively, to develop these higher order elements. The second element type that is introduced here is the hexahedron element with eight nodes, each of which has three degrees of freedom, and six sides as shown in Fig. 6.12. Therefore, in total, a hexahedron element has 24 degrees of freedom. Using the natural coordinate system, shown in Fig. 6.12, the coordinate mapping is conducted like the procedure done for quadrilateral elements. The shape function for each node is a linear function of natural coordinates (ξ, η, ζ ): Ni =

1 (1 + ξ ξi )(1 + ηηi )(1 + ζ ζi ) 8

(6.18)

Having the shape function for each node, the matrix of shape function, N, can be obtained. Consequently, the displacement vector, U, is: Fig. 6.11 Higher order 3D tetrahedron elements, a 10-node tetrahedron element, b 20-node tetrahedron element

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6 Numerical Simulation

Fig. 6.12 An eight-nodal hexahedron element

[ U = N1 N2 N3

N4 N5 N6

⎧ de1 ⎪ ⎪ ⎪ ⎪ ⎪ d e2 ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ e3 ] ⎨ de4 N7 N8 ⎪ de5 ⎪ ⎪ ⎪ ⎪ de6 ⎪ ⎪ ⎪ ⎪ de7 ⎪ ⎪ ⎩ de8



⎫ node 1 ⎪ ⎪ ⎪ ⎪ ⎪ node 2 ⎪ ⎪ ⎪ ⎪ node 3 ⎪ ⎪ ⎬ node 4 ⎪ node 5 ⎪ ⎪ ⎪ ⎪ node 6 ⎪ ⎪ ⎪ ⎪ node 7 ⎪ ⎪ ⎭ node 8

(6.19)

de

in which: ⎧ ⎫ ⎤ Ni 0 0 ⎨ u1 ⎬ Ni = ⎣ 0 Ni 0 ⎦. dei = v1 (i = 1, 2 . . . 8) ⎩ ⎭ w1 0 0 Ni ⎡

wherein, de is the nodal displacement vector. The element matrices can then be found from the strain matrix. The higher order three-dimensional elements are Lagrange type element, which is not widely used because of the interior nodes, and Serendipity type element, without internal nodes. The latter has eight corners and can have one and two nodes on each edge. The 20-nodal quadratic and 32-nodal cubic elements are shown in Fig. 6.13.

6.2 Incremental Viscoelastic Formulation

133

Fig. 6.13 High order 3D serendipity elements, a 20-node quadratic element, b 32-node cubic element

6.2 Incremental Viscoelastic Formulation To solve viscoelastic problems using the Finite Element Method (FEM), an incremental formulation is necessary. This involves breaking down the entire time period into a series of discrete time steps in order to calculate the stresses and strains. Viscoelastic materials can be characterized using both integer-order and fractional-order constitutive equations, both of which can be used in this formulation. The present section outlines an incremental formulation for solving viscoelastic problems using integer-order viscoelastic constitutive equations. For those interested in fractional-order constitutive viscoelastic equations, refer to Hajikarimi and Moghadas Nejad (2021) for more information. In the uniaxial condition, the relationship between stress and strain can be expressed using Boltzmann’s superposition integration, which can be described as follows: ∫t σ (t) = E(t)ε(t = 0) +

E(t − ξ )

dε(ξ ) dξ dξ

(6.20)

0

where E(t) is relaxation modulus, σ (t) is stress, and ε(t) is strain. The stress–strain relation for multi-axial condition can be represented as: ∫t {σ (t)} = E(t)[C]{ε(t = 0)} +

{ } dε(ξ ) dξ E(t − ξ )[C] dξ

(6.21)

0

wherein, the stress and strain tensors are represented by {σ } and {ε}, respectively, while the elastic matrix [C] depends on Poisson’s ratio. In a two-dimensional scenario, the elastic matrix can be determined for plane stress and plane strain

134

6 Numerical Simulation

conditions using the following equations, respectively: ⎡

⎤ 1ν 0 1 ⎣ [C] = ν1 0 ⎦ 1 − ν2 0 0 1−ν 2 ⎤ ⎡ ν 0 1 1−ν 1−ν ⎥ ⎢ ν 1 0 ⎦ [C] = ⎣ (1 + ν)(1 − 2ν) 1−ν 1−2ν 0 0 2(1−ν)

(6.22)

(6.23)

where υ denotes Poisson’s ratio. It is important to note that assuming Poisson’s ratio to be a constant parameter may not always be acceptable for all viscoelastic problems. Typically, Poisson’s ratio is a time-dependent parameter in viscoelastic problems and can only be treated as constant if its variation with time is negligible. In the incremental formulation presented below, if the variation of Poisson’s ratio with time needs to be considered, then the elastic matrices (Eqs. (6.22) and (6.23)) must be updated in every time step. To solve a viscoelastic problem numerically, the constitutive variables (stress and strain) are known at time t = t L , and it is required to determine them after passing a time step of ∆t L+1 = t L+1 − t L under loading or unloading conditions by considering material properties variation. To derive an incremental formulation for a viscoelastic problem, it is needed to find relation of {∆σ L+1 } and {∆εL+1 } which are incremental stress and incremental strain, respectively. To this end, Eq. (6.21) can be implemented to determine the stress tensor for two successive time steps at t L and t L+1 as follows: ∫tL {σ L } = E(t L )[C]{ε(t = 0)} +

{

} dε(ξ ) dξ E(t L − ξ )[C] dξ

(6.24)

0

{

∫tL+1 {σ L+1 } = E(t L+1 )[C]{ε(t = 0)} +

E(t L+1 − ξ )[C]

} dε(ξ ) dξ dξ

(6.25)

0

Substituting t L+1 = ∆t L+1 + t L into Eq. (6.25) resulted in: {

t L +∆t ∫ L+1

{σ L+1 } = E(t L+1 )[C]{ε(t = 0)} +

E((t L + ∆t L+1 ) − ξ )[C]

} dε(ξ ) dξ dξ

0

(6.26) Then by subtracting Eq. (6.24) from Eq. (6.25), {∆σ L+1 } = {σ L+1 } − {σ L } can be determined as:

6.2 Incremental Viscoelastic Formulation

135

{∆σ L+1 } = E(t L+1 )[C]{ε(t = 0)} − E(t L )[C]{ε(t = 0)}



I

+

t L +∆t L+1

∫ 0

{

E((t L + ∆t L+1 ) − ξ )[C]  {

} dε(ξ ) dξ dξ

II

} dε(ξ ) − ∫ E(t L − ξ )[C] dξ dξ 0



tL

(6.27)

III

Equation (6.27) is separated into three sub-equations of (I), (II), and (III) to facilitate solving this equation. The first part is straight forward and can be determined having relaxation modulus function and the initial strain. By dividing the integration domain into two parts, the second part, (II), is simplified as: {

} dε(ξ ) E((t L + ∆t L+1 ) − ξ )[C] dξ dξ

t L +∆t ∫ L+1

0

{

∫tL E((t L + ∆t L+1 ) − ξ )[C]

=

} dε(ξ ) dξ dξ

0 ∆t ∫ L+1

+

{

} dε(ξ ) E((t L + ∆t L+1 ) − ξ )[C] dξ dξ

(6.28)

tL

To proceed with solving Eq. (6.28), dε/dt is assumed as {∆εL+1 /∆t L+1 } for each time step of ∆t L+1 . This assumption should be carefully checked by considering mechanical behavior of the desired material under boundary and loading conditions of the problem. Therefore, the second term in the right-hand side of Eq. (6.27) is integrated as: } dε(ξ ) E((t L + ∆t L+1 ) − ξ )[C] dξ dξ {

t L +∆t ∫ L+1

tL

= [C]

{∆ε L+1 } ∆t L+1

t L +∆t ∫ L+1

∫

E((t L + ∆t L+1 ) − ξ )dξ

(6.29)

tL

Using the Prony series for representing the relaxation modulus of a viscoelastic material (see Eq. (2.23)), the remained integral in Eq. (6.29) can be calculated by using the properties of exponential function as follows:

136

6 Numerical Simulation t L +∆t ∫ L+1

E((t L + ∆t L+1 ) − ξ )dξ tL

t L +∆t ∫ L+1(

E∞ +

=

) E m ex p(ξ − (t L + ∆t L+1 ))/τm dξ

m=1

tL

= E∞ξ +

M ⎲

M ⎲ m=1

|tL +∆t L+1 | | E m τm ex p(ξ − (t L + ∆t L+1 ))/τm | | tL

)] [ ( ∆t L+1 = E ∞ ∆t L+1 + E m τm 1 − ex p − τm m=1 M ⎲

(6.30)

By substituting the calculated integral into Eq. (6.29), the following equation is derived: {

t L +∆t ∫ L+1

E((t L + ∆t L+1 ) − ξ )[C]

} dε(ξ ) ˜ dξ = E[C]{∆ε L+1 } dξ

(6.31)

tL

in which, E˜ = E ∞ +

[ ( )] M ⎲ ∆t L+1 E m τm 1 − exp − ∆t L+1 τm m=1

(6.32)

(II) in Eq. (6.27) is identical to that of the expression (III), and therefore, it can be added to the latter directly. This addition leads to a new integral expression, which can be written as follows: ∫tL

[

{ } ] dε(ξ ) dξ E((t L + ∆t L+1 ) − ξ ) − E(t L − ξ ) [C] dξ

(6.33)

0

and to calculate this integral, the following equation can be written considering the Prony series for representing the relaxation modulus function: E((t L + ∆t L+1 ) − ξ ) − E(t L − ξ ) =

[ ( ) ] ∆t L+1 −1 E m τm exp(ξ − t L ) exp − τm m=1 M ⎲

(6.34) Then, by substituting Eq. (6.34) into Eq. (6.33), it would be:

6.2 Incremental Viscoelastic Formulation

∫tL

[

137

{ } ] dε(ξ ) dξ E((t L + ∆t L+1 ) − ξ ) − E(t L − ξ ) [C] dξ

0

=

( ) ] M [ ⎲ { } ∆t L+1 exp − − 1 αmL τm m=1

(6.35)

in which, (

∫tL

{ L} αm =

E m exp 0

) { } dε(ξ ) ξ − tL dξ [C] τm dξ

(6.36)

Again, the integration range of Eq. (6.36) can be divided into two parts by considering the simple relation of ∆t L = t L – t L−1 and it will be possible to write Eq. (6.36) as follows: {

αmL

}

∫tL−1 = 0

) { } dε(ξ ) ξ − tL dξ E m exp [C] τm dξ (

(

∫tL +

E m exp

t L−1

) { } dε(ξ ) ξ − tL dξ [C] τm dξ

(6.37)

The second term in the right-hand side of Eq. (6.37) can be approximately calculated by considering a constant amount of ∆ε/∆t L for dε/dt within the time step of ∆t L : ∫tL t L−1

) { [ ( )] } dε(ξ ) ∆t L E m τm ξ − tL 1 − exp − dξ = E m exp [C] [C]{∆ε L } τm dξ ∆t L τm (

(6.38) Recalling Eq. (6.36), it is possible to write the following equation: {

αmL−1

}

(

∫tL−1 =

E m exp 0

) { } dε(ξ ) ξ − t L−1 dξ [C] τm dξ

(6.39)

Therefore, the first term in the right-hand side of Eq. (6.37) can be determined considering the relation of ∆t L = t L – t L−1 as follows:

138

6 Numerical Simulation

∫tL−1 0

) { ) } ( dε(ξ ) ∆t L { L−1 } ξ − tL αm dξ = exp − E m exp [C] τm dξ τm (

(6.40)

Finally, {α m L } is calculated by using Eq. (6.40): {

αmL

}

) [ ( )] ( ∆t L ∆t L { L−1 } E m τm αm 1 − exp − = exp − + [C]{∆ε L } τm ∆t L τm

(6.41)

The summation of Eq. (6.34) and the expression (I) of Eq. (6.27) is called residual stress and can be determined using the following equation: ( ) ] M [ ⎲ { } [ ] ∆t L+1 {σr es } = E(t L+1 ) − E(t L ) [C]{ε(t = 0)} + exp − − 1 αmL τm m=1 (6.42) Based on Eq. (6.36), {α m 0 } = 0. Having all expressions of I, II, and III in Eq. (6.27), {∆σ L+1 } is accordingly determined as follows: ˜ {∆σ L+1 } = E[C]{∆ε L+1 } + {σr es }

(6.43)

Regarding Eq. (6.43), it becomes apparent that the application of strain during each time step causes the emergence of residual stress, which, in turn, affects the subsequent time step. The incremental formulation presented in this section can be employed to derive the discretized force-displacement equation for both integer-order and fractionalorder viscoelastic models. This can be achieved by utilizing Eq. (6.1) and implementing the virtual work principle in its incremental form for an element denoted as e. The incremental virtual work principle for an element e, can be expressed as follows: ∫ ∫ ∫ T T (6.44) δ{∆ε} {∆σ }dΩ = δ{∆u} {∆b}dΩ + δ{∆u}T {∆p}dΓ Ωe

Ωe

Γe

wherein, δ is the first-order derivation operator, {}T is the matrix transpose operator, {∆u}, {∆p}, and {∆b} are the incremental displacement, the prescribed traction, and the body force, respectively. Based on Eq. (6.1), the incremental form of this interpolation would be: {∆U } = N{∆d}

(6.45)

On the other hand, the following equation can be written by assuming small deformation and via the strain–displacement relation:

6.3 Geometrically Homogenous Composite Modeling

139

{∆ε} = B{∆d}

(6.46)

The residual force, F res , should be calculated from the residual stress, σ res . According to the above-mentioned general equations for a finite element framework, Zhang and Li (2009) developed the following equations for a linear viscoelastic domain: [K]{∆d} = {∆F} + {Fr es }

(6.47)

wherein, ∫ [K] = ∫ {∆F} =

˜ T [C]BdΩ EB

Ω

∫

N T {∆b L+1 }dΩ + Ωe

∫ {Fr es } = −

{ } N T ∆p L+1 dΓ

(6.48)

(6.49)

Γe

BT {σr es }dΩ

(6.50)

Ωe

where K is the equivalent stiffness matrix, {∆F} is the equivalent incremental load in the interval ∆t L+1 , and {F res } is the residual force due to the residual stress. Figure 6.14 represents a flowchart proposed by Zhang and Li (2009) to solve viscoelastic problems by implementing a finite element framework.

6.3 Geometrically Homogenous Composite Modeling Among the different numerical methods available to solve the problems in this area, FEM is selected due to its simplicity and applicability. Since numerical modeling of composite materials is the target, two main methods, geometrically homogenous composite method and geometrically heterogeneous composite method are described. The former is discussed in this section and the next section is dedicated to the latter. Here, Abaqus software package is selected for practical examples due to its generality and interesting graphical user interface.

6.3.1 Geometry Construction Generating the geometry of a homogeneous media has no complexity in terms of struggling with the distribution of inclusions such as fibers or particles. The only bottle neck in this regard could be constructing a complex geometry due to the

140

6 Numerical Simulation

Fig. 6.14 Flowchart for solving viscoelastic problems by using finite element method (FEM)

inherent complexity of the overall shape of the geometry. A lot of time and effort have to be spent for generating the mesh geometry of complex objects in terms of having sharp edges or a number of curved surfaces. In most cases of modeling an experimental specimen, geometry is not the issue, however, if complexity exists, one can use different meshing methods to tackle it. For instance, modelers may prefer using first-order tetrahedral elements for discretizing complex geometries. As discussed earlier, this type of element can be attributed to almost every geometry but the element itself is considered a poor element. Therefore, the proper mesh size that is found in mesh sensitivity analysis is relatively smaller than when other element types are used. This approach may seem convenient but when it comes to the computation costs and the required calculating power of the machine, it is no longer a practical approach. Instead, one can consider using the second-order tetrahedral elements to enhance the precision of the elements and increase the mesh size. However, this type of element has its own drawbacks. One of which is that it is hard to converge for some problems in which node-to-surface contact is introduced. It is generally preferred to use hexahedral elements. Although this element type seems not applicable to many complex geometries at first, with spending some time on partitioning the geometry to smaller segments, it can be used. Figure 6.15 shows an example of using tetrahedral

6.3 Geometrically Homogenous Composite Modeling

141

Fig. 6.15 Mesh generation for an arbitrary complex geometry: a tetrahedral elements, b hexahedral elements

versus hexahedral elements to discretize the snowman geometry. This geometry as a whole is a complex one and hexahedral elements cannot be assigned to it. However, by considering some partition surfaces, the geometry is divided into a few simpler geometries on which hexahedral elements are applicable. In practice, most experimental specimens that are used in mechanical tests have simple prismatic geometries or simplified in finite element simulations. We have seen that it is possible to discretize such geometries using hexahedral mesh elements. Figure 6.16 illustrates an example finite element model of an Aluminum sheet with a hole, strengthened with an FRP sheet. The Aluminum sheet is constructed using solid elements and the FRP sheet is modeled as a homogeneous geometry but having anisotropic material properties. Assigning the material properties to the geometrically homogeneous geometries is discussed in the next section. The FRP sheet is modeled using solid elements here but there is no limit to use shell elements instead and it depends on the problem conditions and the outputs we would like to extract from the model. In the older versions of ABAQUS software package, the only available method for damage propagation of the composites was Hashin criteria (Hashin 1980) for two-dimensional elements. In the case of threedimensional elements, writing a subroutine of the three-dimensional Hashin damage criteria (Hashin 1980) was required. However, in the newer versions, LaRC05

Fig. 6.16 An aluminum sheet with a hole, strengthened with a CFRP sheet

142

6 Numerical Simulation

damage criteria (Pinho et al. 2012) is added to the software which has made it possible for the user to consider damage of the three-dimensional elements. As can be seen in Fig. 6.17, half of the geometry is modeled because it is symmetric about the surface passes perpendicularly from the longitudinal axis of the geometry. This technique makes it possible to run the model with a smaller number of elements and less computation time. However, there are limitations for using this technique such as the loading and displacement that is applied to the model, the type of elements that are used for different parts in a geometry and so forth (Sadat Hosseini et al. 2023).

Fig. 6.17 Meshed geometry of model with two-dimensional solid elements

6.3 Geometrically Homogenous Composite Modeling

143

As discussed in the review of the finite element method, an effective and economical simulation method for three-dimensional problems is to simplify them as plane stress or plane strain models. The reason for this simplification is to decrease the computational costs even with powerful modern computers through using 2D planar elements instead of 3D solid elements. When the in-plane thickness of the model is relatively small, the model can be simplified into a representative plane stress model, and no stresses through thickness in the out-of-plane direction is considered. Therefore, for a three-dimensional problem in which the in-plane stresses are constant through thickness and there is no through-thickness stress, the problem can be simplified into a two-dimensional plane stress problem. In this method, the two-dimensional geometry is discretized into 2D solid mesh elements considering a constant value of thickness for the geometry. As we know, in case of variable thickness of the specimen, the model geometry can be divided into segments of constant, but different with each other, thicknesses. The notched-beam specimen is modeled using plane stress elements. The area of interest in this model was around the notch and through-thickness stresses are not the subject of study. The plane strain method is used when through-thickness stresses are developed but strains are zero and only in-plane strains are considered in the analysis. This method is mostly used for structures with a relatively large out-of-plane thickness. An example of this method of simulation is presented in Fig. 6.17. The out-of-plane thickness and the height of the notched beam are both equal to 100 mm.

6.3.2 Material Characterization The first step in characterizing the material properties is to define the mechanical properties of the material according to its type. Isotropic materials, within the linear elastic range, have two independent elastic constants, the Elastic Modulus and the Poisson’s Ratio. For particulate composites which are composed of particles dispersed in a matrix, such as bituminous composites, the matrix and inclusion are both isotropic materials but with different behaviors. For fibrous composites which are composed of orderly oriented chopped or continuous fibers, again each of the ingredients are isotropic. However, the resulting composite lamina is no longer an isotropic material but an anisotropic one. Anisotropic materials have different material properties along the local axis of the material. In other words, their material properties are direction dependent. Therefore, when an anisotropic composite is defined as a homogenous media, this simplification must be compensated for by defining the material properties of the composite correctly in the direction of the fibers and the transverse direction. Also, the material orientation has to be defined in the model so that the engineering material properties can be assigned to the geometry. However, for composites with randomly distributed inclusions, isotropic assumption is not incorrect, but it is always required to verify the results with an experimental and/or micromechanical model to find the accuracy of the predictions.

144

6 Numerical Simulation

In the case of a geometry which is composed of FRP layers, a solid or a conventional shell composite layup can be defined and the material properties, layer thickness, fibers orientation, and the number of integration points in each ply of the layup can be defined. Figure 6.18 shows the section properties and the layup modeled for an FRP plate which is defined using three-dimensional shell elements. As an instance, a practical use of modeling the FRP material as a homogeneous geometry and defining the material properties and layup is for the applications where FRP is used as strengthening material (Sadat Hosseini et al. 2020, 2021). The FRP layup is modeled as a shell covering the solid steel substrate. As explained before and showed in Fig. 6.19, the composition of unidirectional glass fiber of E-glass type and Vinyl ester resin is modeled as a homogeneous geometry but the material properties and the layup configuration is defined for the mentioned geometry. An example work for modeling the distributed fibers of arbitrary orientation in matrix medium is proposed by Pajerski (2010) as Biphasic Non-Linear Transversely Isotropic, Transversely Homogeneous (NLTITH) continuum model. The Mises distribution is used as the probability distribution function which is transversely

Fig. 6.18 A three-dimensional shell part and the ply stack orientation definition

Fig. 6.19 Transversely isotropic probability density functions describing fiber orientation

6.3 Geometrically Homogenous Composite Modeling

145

isotropic in the desired direction. / (− b exp[b(cos(2Θ) + 1)] 1 →) (√ ) ψ M = ρ(Θ) = π 2π er f i 2b

(6.51)

wherein, erfi is the imaginary error function defined as erfi(x) = −ierf (ix), and b is the concentration parameter quantifying the orientation of the fibers. In this approach, when b is tending toward −∞ the fibers are oriented isotopically in the transverse plane, while b = 0 randomly oriented fibers, and when b is tending toward +∞ the fibers are aligned in the polar direction (Fig. 6.19). To evaluate material properties of viscoelastic materials, it is possible to measure complex shear modulus (G*) and phase angle (δ) as two main experimental outputs at each frequency. A frequency sweep test can be fulfilled in strain-control mode with a specified strain amplitude, within a frequency range at different test temperatures. The Prony series is a well-known method to express viscoelastic properties of a viscoelastic material. The shear relaxation modulus of the generalized Maxwell model can be explained by following equation so called Prony series (Brinson and Brinson 2008): G(t) = G ∞ +

n ⎲ i=1

) ( t G i exp − τi

(6.52)

in which, G∞ and Gi are the equilibrium modulus and the shear relaxation strength, respectively and τ i is relaxation time. Also, n is number of the generalized Maxwell model element as it is shown in Fig. 2.13. Table 6.1 presents the generalized Maxwell model components for an original asphalt binder at a reference temperature of 22 °C. As can be seen in Table 6.1, 25 elements are used to construct the generalized Maxwell model. Hajikarimi et al. (2018) showed that using F-test help researchers to find the optimum number of elements that should be considered to construct the generalized Maxwell model to ensure about the modeling accuracy.

6.3.3 Boundary Conditions Model boundary conditions are defined based on the observed conditions of the supports and loadings on the structure or the specimen. Most of times some modifications are considered for boundary conditions subject to that the model is still an acceptable representative of the real condition. This judgment requires experience in modeling, being familiar with the boundary conditions modeling techniques and a good understanding of the problem. Some example models are introduced here to show how boundary conditions are defined for actual test setups.

146

6 Numerical Simulation

Table 6.1 Parameters of the Prony series for the original bitumen at a reference temperature of 22 °C i

Gi (MPa)

τ i (s)

i

Gi (MPa)

τ i (s)

1

1.061E−01

3.079E−14

14

5.097E+01

1.381E−04

2

1.414E−01

1.702E−13

15

2.349E+01

7.653E−04

3

5.494E−01

9.408E−13

16

8.418E+00

4.235E−03

4

2.618E−01

5.200E−12

17

2.577E+00

2.341E−02

5

2.475E−14

2.875E−11

18

5.903E−01

1.294E−01

6

5.460E+00

1.589E−10

19

1.128E−01

7.151E−01

7

1.191E+01

8.784E−10

20

2.505E−02

3.953E+00

8

2.405E+01

4.856E−09

21

1.730E−03

2.185E+01

9

5.123E+01

2.684E−08

22

4.927E−04

1.208E+02

10

1.449E+02

1.484E−07

23

2.242E−05

6.677E+02

11

2.632E+02

8.202E−07

24

6.174E−06

3.691E+03

12

2.705E+02

4.467E−06

25

4.765E−07

2.040E+04

13

1.391E+02

2.507E−05

∞

1.437E−03

–

For a viscoelastic bituminous material such as original bitumen, modified bitumen, and bituminous composites containing fine particles, the mechanical properties can be measured using two well-known test setups, bending beam rheometer (BBR) and dynamic shear rheometer (DSR). The flexural creep stiffness, S(t), and creep rate, m-value, as two fundamental criteria for evaluating the performance of bituminous materials at low temperatures (ASTM-D6648-08 2001) are measured using BBR setup. The two constitutive properties of viscoelastic materials, complex shear modulus (G*) and phase angle (δ), which depend on time/frequency, temperature, and loading rate (ASTM D7175 2005) are determined using DSR setup. The BBR test setup simply consists of a constant point load which is applied to the center of a horizontally supported specimen beam. The flexural test on an asphalt binder and so forth can be simulated by a three-dimensional or a two-dimensional beam that lies on two supports. Supports are cylindrical in shape and the beam can roll over them. A two-dimensional model is preferred here because through-thickness stresses are not the subject. Figure 6.20 illustrates a homogeneous finite element model geometry for a bituminous beam under three-point bending creep test.

Fig. 6.20 Two-dimensional model of a beam in BBR test setup

6.3 Geometrically Homogenous Composite Modeling

147

The finite element model of the DSR setup (Fig. 6.21) was simulated using a simplified method. The specimen is modeled as a cylindrical part, and an analytical rigid ring is modeled on the top of it and fixed in its position by defining displacement and rotational constraint at the central reference point. Only the rotational degree of freedom around the z-axis is unconstraint and the rotational displacement (according to the test frequency) is applied to the rigid ring. Since the ring is fixed to the upper surface of the bituminous specimen, the rotational displacement is transferred to the disk. The bottom surface of the disk specimen is fixed, therefore, the rotational displacement on the top produces shear strains and stresses in the model. Due to the simplification that is made in homogenous geometry approach, there are some limitations in the case of the results that can be extracted from the model. Since the inclusions are not modeled here, local stresses that are produced due to the difference between the stiffness of the inclusions and matrix cannot be observed. Also, the interaction between matrix and inclusions cannot be modeled. Moreover, different material properties and behaviors of the matrix and inclusions cannot be applied, and consequently, the results are highly dependent on the approximation that is made for the overall material properties or the layup definition for FRPs. For instance, in the simulation of a bituminous composite containing randomly distributed particles, wherein bitumen is considered to have a viscoelastic behavior while the inclusions are linear elastic, results will defer based on the assumption of linear elastic or viscoelastic for the mixture. Although it is preferred to model this problem by using viscoelastic material properties to capture the time–temperature dependencies, the verification of the results is highly affected by the simplifications. In some cases, wherein the material properties of the matrix and inclusions are approximately the same this method is not only precise but also is less time consuming. An example Fig. 6.21 Numerical finite element model for DSR setup

148

6 Numerical Simulation

of this condition is modeling of the fiber reinforced polymer (FRP) composites in relatively low and moderate temperatures in which resin has a linear elastic behavior within an acceptable approximation.

6.4 Geometrically Heterogeneous Composite Modeling 6.4.1 Geometry Construction When the microstructure of the composite material is of interest of the researchers, a more complex geometry consisting of the matrix and inclusions is generated. Although the heterogeneous model is more time consuming and requires more computational capabilities, there are some advantages to this method and sometimes due to the focus of the problem, it is required to have a detailed model. By constructing the geometry of the different constituents of a composite geometry, specific mechanical properties can be attributed to each part. Also, it is possible to define contact properties between the matrix and inclusions. This helps to study the effect of contact stiffness and strength in some composites. Local outputs such as stresses and strains in vicinity of the contact can be extracted and studied from the model. Since the composite geometry consists of distinct parts, defining the material properties in this method is more straight forward. For instance, for FRP materials, the material properties of fiber and resin can be distinctively defined. This provides the possibility of considering the viscoelastic behavior of the resin material (Theocaris 1962). In the case of a bituminous composite containing filler, viscoelastic material properties can be assigned to the bitumen, while fillers have elastic properties. In the case of composite FRP materials containing resin matrix and continuous fibers, micromechanical numerical models can be constructed using the experimental characterization of the microconstituents. In this approach, matrix and fibers are modeled distinctively and very local effects can be studied. For instance, fatigue strength of CFRP composites was studied by Cai et al. (2008) through a micromechanical model, considering the viscoelastic behavior of matrix resin under arbitrary frequency, load ratio, and temperature. Naya et al. (2017b) studied the effect of different environmental conditions on the longitudinal compression properties of unidirectional FRP plies through a realistic micromechanical model. Gagani et al. (2017) proposed a micromechanical model in which matrix and fiber bundles are modeled separately. They investigated the anisotropic fluid diffusion of the composite. Three-dimensional micromechanical model was proposed by Sharifpour and Montesano (2022) for [0/90/0] laminates to study the local deformations and failures considering the effects of thermal residual stress, ply constraints, and fiber volume fraction. These example studies show how flexibly researchers can investigate various local effects and parameters through modeling the heterogeneous geometry of the FRP materials. Indeed a verified and correct micromechanical model can be used for extensive parametric studies which are very costly as experimental specimens.

6.4 Geometrically Heterogeneous Composite Modeling

149

There are different methods introduced by researchers to generate the heterogeneous microstructure of composites containing chopped fibers. The representative volume element (RVE) for composite materials was used by a number of researchers. A very first definition is proposed by Hill (1963) as follows: a sample that (a) is structurally entirely typical of the whole mixture on average, and (b) contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the surface values of traction and displacement, so long as these values are macroscopically uniform. That is, they fluctuate about a mean with a wavelength small compared with the dimensions of the sample, and the effects of such fluctuations become insignificant within a few wavelengths of the surface. The contributions of this surface layer to any average can be made negligible by taking the sample large enough.

Gusev (2001) and Duschlbauer et al. (2006) used a Monte Carlo procedure on RVE. The former presented a finite element-based approach to predict the behavior of composites containing anisotropic, arbitrarily shaped, and oriented phases. The latter investigated the behavior of a short fiber reinforced composite material by introducing a finite element-based multi-fiber unit cells and using the analytical method of Mori–Tanaka (Mori and Tanaka 1973; Benveniste 1987) which is a mean field approach. Pan and Pelegri (2011) generated an RVE based on microscopic observations and analyzed the mechanical properties of random chopped fiber composites. Damage and failure of the composite material are investigated in the finite element model through a progressive damage model. Using micromechanical model, it was possible to capture matrix cracking and fiber breakage by using local stress and strain fields. Also, the interfacial debonding was modeled using traction–separation cohesive law. The interfacial constraints are discussed further in the next section. A random sequential adsorption (RSA) algorithm was proposed by Rintoul and Torquato (1997) to model composite cubic unit cells containing up to 50% particle volume fraction. In this method, the coordination of the center of each spherical particle is randomly generated and the radius of the spheres is calculated based on the volume fraction of the particles. There are two main conditions for this random particle generation. Firstly, a minimum distance between the ith particle and all the previously generated (1 to i − 1) particles must be considered. Meanwhile, spheres have to keep enough distance from the cubic container boundaries to avoid distorted mesh. Segurado and Llorca (2002) found that the two conditions provided in the RSA model are not capable of accommodating more than a certain number of particles in the container (about 30% volume fraction). This threshold is called the jamming limit over which no more particles can be added to the cell. They modified the method to reach 50% volume fraction. In their modified method, the new position of the particle center is found using a compression factor (below 1) multiplied by the current position. Then, the two conditions of RSA model are checked and if satisfied, the volume fraction is increased. This process is conducted up to the desired volume fraction of the inclusions. In the case of any overlapping of the particles, the particle is moved to a randomly generated direction.

150

6 Numerical Simulation

Trias et al. (2006) reviewed the methodologies proposed for random generation of RVE size until that time and argued that the studies had failed to consider both mechanical and statistical criteria in depth. To determine the finite size of the statistical representative volume element (SRVE), first they defined a dimensionless variable as the ratio of the side length of SRVE and the fiber radius, which was between 4 and 100. Consequently, the finite element models are constructed and solved for imposed displacement and force. The dependency of the contributing parameters such as Hill condition, fiber content, stress and strain fields statistics and probability density functions, and so forth on the dimensionless parameter was studied. As a result, the minimum valid SRVE size was determined by the criterion which is satisfied for a larger value of the dimensionless parameter. It is known that periodic spatial arrangement of inclusions inside the container cannot correctly predict the damage mechanism due to the nonperiodic nature of the damage (Swaminathan et al. 2005). Vaughan and McCarthy (2010) conducted digital image analysis on the transverse cross section of CFRP unidirectional laminate composites and statistical functions were generated to characterize the microstructure of CFRP laminates. By comparing the second-order intensity functions and the complete spatial random (CSR) pattern it was found that the CSR pattern is uncapable of characterizing the distribution of fibers in CFRP microstructure. Because experiments showed some regularity for the fibers having relatively shorter distances. Therefore, they felt it is needed to develop an algorithm to generate high volume fractions of fiber having the same geometry features as the experimental microstructure. In another approach, random generation of different particle shapes inside the matrix media has been conducted based on the Delaunay triangulation method (Joliff et al. 2007; Hajikarimi et al. 2020, 2021). Figure 6.22 illustrates a sample heterogeneous model geometry which is constructed using Delaunay triangulation method and can be used for simulating the bituminous composites in DSR setup.

6.4.2 Boundary Conditions and Interactions Defining the boundary conditions follows similar rules and methods as described for the geometrically homogeneous materials. However, it is very important to consider any local boundary conditions that may affect the simulation. For instance, applying the point load needs some consideration since if it is applied on each of the matrix or inclusions local stresses can be affected. In ABAQUS-Standard, the inclusion/matrix interaction can be modeled in different ways. One approach is to completely neglect the interaction and consider a tie constraint between matrix and inclusions. In this way some major aspects of the interaction such as sliding of fibers in the resin medium, interfacial zone around the fillers of the bituminous composites, and so forth cannot be characterized. A conceptual definition is introduced (Hajikarimi et al. 2022) as effective volume fraction ratio (EVFR) which can quantify the stiffening mechanism in the bituminous composites

6.4 Geometrically Heterogeneous Composite Modeling

151

Fig. 6.22 Constructing the geometry of an example heterogeneous material: a sequences of model generation by deducting the inclusions geometry from the cylindrical container and adding them to the resulting geometry to make the composite, b discretizing the composite geometry using tetrahedral elements

and stand for the interfacial effects in the calculations. By assuming an equilibrium spherical shape for the filler materials, EVFR is defined as follows: EVFR =

R1 R2

(6.53)

In this formula, R2 is the radius of the equivalent sphere from biphasic numerical simulation on the experimental condition. While R1 is the equivalent radius calculated from the volumetric relationships: ρbitumen = ρfiller =

mb vb

mf vf

(6.54) (6.55)

152

6 Numerical Simulation

M = mb + m f

(6.56)

V = vb + v f

(6.57)

%mass =

mf mf = M mb + m f

(6.58)

wherein, m and v are the representatives of mass and volume, and subscripts b and f stand for bitumen and filler, respectively. M and V are the mass and volume of the composite. When it comes to considering the mechanism of load transfer between matrix and inclusions and the corresponding parameters, the inclusion/matrix interfacial properties must be defined to study the stress–strain behavior of composites. To this aim, the cohesive interaction is used wherein the traction–separation law governs. This type of interaction starts with a linear elastic response between the interfacial displacement and the developed stresses (Fig. 6.23). The slope of this line indicates the penalty stiffness of the interfacial zone which is normally a large number of 105 (can vary depending on the problem conditions) order in the numerical models which ensures the continuity of interfacial displacement with no damage. The higher the K value, the more difficult convergence of the numerical model. The onset of damage is defined by quadratic relationship between the normal and shear strengths: (

〈σn 〉 σn0

)2

( +

τT τT0

)2

( +

τL τ L0

)2 =1

(6.59)

Fig. 6.23 a Schematic of the uniaxial tension–compression response of the epoxy matrix based on damage-plasticity model for quasi-brittle materials, b schematic the shear response of the damagefriction model for fiber/matrix interfaces (Naya et al. 2017a)

6.5 An Extension of FEM for Discontinuity Modeling

153

wherein, σ n is the normal stress, and τ T and τ L are the transversal and longitudinal shear strengths. denotes that only positive values of normal stress have to be taken into account for separation. When the above criterion is satisfied and the damage is initiated, the interface strength decreases to zero. The Benzeggagh–Kenane (Benzeggagh and Kenane 1996) damage propagation criterion (also known as BK law) is used by researchers (Naya et al. 2017a) as a good indicator of the fracture energy dependence on the fracture modes: (

G C = G I C + (G I I C

GII − GIC) GI + GII

)m (6.60)

where, GIC , GIIC, and GTC are mode I, mode II, and total critical energy release rate. GI and GII are resulting from the cantilever beam mode I and the end notch flexure mode II tests. m is an interaction material dependent parameter which comes from fitting the BK law on the experimental results.

6.5 An Extension of FEM for Discontinuity Modeling The finite element method (FEM) is an appropriate technique for solving problems in a continuous medium. However, appearing discontinuities enforce serious difficulties, especially for re-meshing the domain for simulating crack growth and debonding in material interfaces. Ted Belytschko and collaborators (Moës et al. 1999) developed the extended finite element method (XFEM) in 1999 to help alleviate inadequacies of the FEM and used it to model the propagation of different discontinuities, including strong (cracks) and weak (material interfaces). The idea behind XFEM is to retain the most benefits of mesh-free methods while lessening their negative sides. The main advantage of XFEM compared to FEM is that in this method the mesh is not dependent on the internal boundaries such as cracks interfaces or inclusion boundaries. Thus, no mesh updating is needed to solve the problems by using the XFEM (Moës et al. 1999). Accordingly, the XFEM is widely used to simulate crack propagation problems and fracture analysis of composites (Mohammadi 2008).

6.5.1 Fundamentals and Formulations The extended finite element method (XFEM) is a numerical method based on the generalized finite element method (GFEM) and the partition of unity method (PUM). By enriching the solution space for solutions to differential equations with discontinuous functions, the XFEM extends the FEM approach.

154

6.5.1.1

6 Numerical Simulation

Partition of Unity Method

The PUM has been implemented in different computational fields (Melenk and Babuška 1996). A partition of unity is defined as a set of n functions f i (x) within a domain ΩPU such that n ⎲

f i (x) = 1

(6.61)

i=1

It can be shown that by choosing of any arbitrary function ψ(x), the following equation is satisfied: n ⎲

f i (x)ψ(x) = ψ(x)

(6.62)

i=1

The set of isoparametric finite element shape functions, N j , also satisfy the partition of unity condition as follows: m ⎲

N j (x) = 1

(6.63)

j=1

in which m is the number of nodes for each finite element. The concept of partition of unity offers a mathematical framework for developing an enriched solution. Equation (6.62) represents the definition of completeness based on the order of the polynomial ψ(x) = p(x), which must be expressed exactly by approximating functions f i (x). From a theoretical point of view, enrichment is the principal of increasing the completeness order that can be achieved. From computational point of view, the enrichment is achieving higher accuracy of the approximation using the information acquired from the analytical solution. Thus, the selection of enriched function directly depends on the prior solution of the problem. As previously mentioned in this chapter, the classical approximation of a field variable d within a finite element method, in terms of the n basis functions p, is as follows: d = pT a =

n ⎲

pi ai

(6.64)

i=1

where unknowns ai are calculated from the approximation at nodal points. The basis function can be defined for various completeness orders for one- and two-dimensional problems as presented in Table 6.2.

6.5 An Extension of FEM for Discontinuity Modeling Table 6.2 Basis functions for various types of problems and different order of completeness

6.5.1.2

155

Type of problem

1st order

One-dimensional problem

pT

Two-dimensional problem

pT = {1, x, y}

= {1, x}

2nd order pT = {1, x, x 2 } pT = {1, x, y, x 2 , xy, y2 }

Enrichment

There are two basic methods for enriching an approximation, including the enrichment of the basis vector (intrinsic enrichment), and the enrichment of the approximation (extrinsic enrichment). The main concept of the intrinsic enrichment is to increase the approximation accuracy of Eq. (6.64) by including new terms obtained from the analytical solution to satisfy a specific condition of reproducing a complex field (Fries and Belytschko 2006). Considering the classical crack problem, it is well-known that the asymptotic near tip displacement field can be presented as: /

( ) r θ θ K I cos (κ − cosθ ) + K I I sin (κ + cosθ + 2) 2π 2 2 / ( ) r 1 θ θ K I sin (κ − cosθ ) + K I I cos (κ + cosθ − 2) uy = μ 2π 2 2

ux =

1 μ

(6.65)

(6.66)

in which μ is the shear modulus, κ is determined based on plane stress or plane strain condition using the Poisson’s ratio, r and θ are shown in Fig. 6.24, K I and K II are the stress intensity factors (SIF) of the mode I and II, respectively. The asymptotic near crack tip displacement field (Eqs. (6.65) and (6.66)) can be represented by the following basis function p(x) in the polar coordinate system:

Fig. 6.24 Polar coordinates at crack tip

156

6 Numerical Simulation

[ pT (x) = [P1 .P2 .P3 .P4 ] =

√ θ √ θ √ θ √ θ r sin . r cos . r sinθ sin . r sinθ cos 2 2 2 2

] (6.67)

For the total solution, the basis function must include the constant and linear terms: ] [ √ θ √ θ √ θ √ θ T (6.68) p (x) = 1.x.y. r sin . r cos . r sinθ sin . r sinθ cos 2 2 2 2 As can be seen in Eq. (6.68), the parameters obtained from the analytical solution in conjunction with the FEM ordinary basis functions increase the accuracy of solution. For the extrinsic enrichment, to increase the order of completeness, extrinsic basis can be used as follows: d h (x) =

m ⎲

N j (x)d j +

j =1

n ⎲

f i (x) p(x)a i

(6.69)

i=1

in which f i (x) are set of the partition of unity functions defined over the support domain, and ai are additional unknowns or degree of freedom related to the enriched solution.

6.5.1.3

Extended Finite Element Method

In this section, the eXtended finite element method (XFEM) is demonstrated as an application of this method for simulating crack propagation within a viscoelastic medium based on fracture mechanics. The approximation in element e of a cracked medium is implemented using the framework of XFEM (Mohammadi 2008): u(x) =

⎲ P∈D O

N P (x)u P +

⎲

N Q (x)HQ a Q +

Q∈D S

⎲

[ N R (x)

R∈DT

4 ⎲

] ΦmR (x)C Rm

m=1

(6.70) in which DO , DS, and DT indicated ordinary nodes, enriched nodes related with crack surfaces, and enriched nodes related with the crack tip, respectively as it is illustrated in Fig. 6.25. aQ and C R m are virtual enriched degree of freedoms. H(x) is Heaviside function used to reflect the displacement discontinuity as: { H(x) =

1 for x above the crack −1 for x below the crack

(6.71)

The major function in Eq. (6.70) is Φ R m (m = 1–4) which is an enrichment function to reflect the local property at the crack tip. Actually, by using this function the mesh

6.5 An Extension of FEM for Discontinuity Modeling

157

Fig. 6.25 Enriched nodes in a cracked domain

can be independent of the internal discontinuities. Implementing the viscoelastic asymptotic displacement field by using correspondence principle, this function can be derived as Eq. (6.67) in which Pi are Φ R i . Referring to Sect. 6.2, it is possible to use the incremental viscoelastic formulation for solving the force–displacement equation.

6.5.2 Applications In this section, two examples are provided to demonstrate the practical application of the incremental viscoelastic formulation within the XFEM. The first example considers an infinite viscoelastic plate containing a circular hole under uniaxial tension, while the second example examines the behavior of an edge crack within an infinite viscoelastic plate.

6.5.2.1

Infinite Viscoelastic Plate with a Circular Hole Under Uniaxial Tension

The problem under consideration is a viscoelastic quadrangle plate with dimensions (L = T = 2), and a circular void with a radius of a = 0.4 in its center, as shown in Fig. 6.26. This plate is a part of an infinite viscoelastic plate subject to a constant creep load of σ ∞ (creep load) in the x-direction on its boundaries. The displacement on the boundaries can be determined by using the analytical solution for creep loading, which is obtained using the correspondence principle introduced in Chap. 2. As

158

6 Numerical Simulation

illustrated in Fig. 6.27, a three-parameter standard viscoelastic model is used to describe viscoelastic behavior of this plate in which E ∞ and E 1 = 1, and η1 = 10.0. The Poisson’s ratio is also supposed to be a constant value of 0.3. The exact displacement field for an infinite viscoelastic plate can be formulated using the correspondence principle (as discussed in Chap. 2) as follows: [ ] σ∞ (1 + υ)ϑ(t)a r a a3 ux = (1 + κ)cosθ + 2 [(1 − κ)sinθ + cos3θ ] − 2 3 cos3θ 4 a r r (6.72) ] [ σ∞ (1 + υ)ϑ(t)a r a a3 uy = (κ − 3)sinθ + 2 [(1 − κ)sinθ + sin3θ ] − 2 3 sin3θ 4 a r r (6.73) where a is the radius of circular void, the polar coordinates r and θ are used in this problem as plane strain condition is being considered, where κ = 3 − 4υ. To describe the viscoelastic behavior of the plate, a three-parameter standard viscoelastic model Fig. 6.26 An infinite viscoelastic plate with a circular void subjected to uniaxial tension

Fig. 6.27 Three-parameters standard viscoelastic model

6.5 An Extension of FEM for Discontinuity Modeling

159

is used and the creep compliance is calculated accordingly: ) ( t 1 E1 exp − D(t) = − E∞ E ∞ (E ∞ + E 1 ) τ

(6.74)

) ( in which τ = η1 E1∞ + E11 . The exact displacements are determined based on analytical solution and presented in Eqs. (6.72) and (6.73) are applied on boundaries having creep load of σ ∞ = 0.25. The problem is solved for a loading time of 200 s and time step of ∆t = 5 s with a reasonable mesh size (907 elements) as illustrated in Fig. 6.28. The displacement of Point A(located at r = 0.5, θ = 0.0) (see Fig. 6.26). The analytical solution for ux and uy are ux = 0.139035(1 – 0.5exp(−t/10)) and uy = 0. To determine the values of ux and uy at Point A, it is needed to use finite element approximation. Figure 6.7 depicts a 4-node element in x–y coordinate system and its corresponding mapped form in ξ-η coordinate system. The values of x T = (x, y) can be determined by interpolating the nodal values of x T as follows: x=

4 ⎲ j=1

where N j is shape function matrix: Fig. 6.28 Meshed viscoelastic plate with a circular void included

Nj x j

(6.75)

160

6 Numerical Simulation

[ Nj =

Nj 0 0 Nj

] (6.76)

It is well-known that in a finite isoperimetric element, the displacement field of uT = (ux , uy ) can be calculated by interpolating nodal displacements as follows: u=

4 ⎲

Njuj

(6.77)

j=1

x–y coordinate system using nodal displacements, it is necessary to determine the corresponding point in the ξ-η coordinate system. For structured meshes, this process is straightforward, but for the present unstructured mesh, an iterative procedure must be employed. Figure 6.29 shows displacement of Point A during loading time of 200 s. The vertical displacements at Point A are zero due to pure tension loading and symmetry condition.

Fig. 6.29 Results of displacement versus time for 200 s loading time for a viscoelastic plate with circular void

6.5 An Extension of FEM for Discontinuity Modeling

161

Fig. 6.30 Edge crack within a finite viscoelastic plate

6.5.2.2

The First and Second Opening Mode for an Edge Crack with In-plane Behavior

The problem involves a finite viscoelastic plate with an edge crack that is loaded by an analytical displacement field on its boundaries. The plate is separated from an infinite viscoelastic plate and the goal is to use the XFEM to solve this problem. The specific geometry of the plate and the analytical displacement field applied to its boundaries are shown in Fig. 6.30. The viscoelastic behavior of the plate is described by a three-parameter standard viscoelastic model (Fig. 6.27) with E ∞ = 1.0, E 1 = 1.0, and η1 = 10.0, and a constant Poisson’s ratio of 0.3 is assumed. The plate is subjected to a loading time of 200 s with a time step of ∆t = 0.5 s. The displacement field around the plate with an edge crack is determined for both Mode I (plane crack opening) and Mode II (in-plane shear) of fracture, with K I (t) = 0.5 and K II (t) = 0.25. The analytical solution for two-dimensional linear viscoelastic fracture mechanics is used to derive the displacement field around the crack tip. The displacement field is presented as follows: / θ r cos (κ − cosθ ) u x (r.θ.t) = (1 + ν)D(t)K I (t) (6.78) 2π 2 / θ r sin (κ − cosθ ) u y (r.θ .t) = (1 + ν) D(t)K I (t) (6.79) 2π 2 for mode I of fracture, and: / r θ u x (r.θ.t) = (1 + ν)D(t)K I I (t) sin (κ + cosθ + 2) 2π 2

(6.80)

162

6 Numerical Simulation

/ u y (r.θ.t) = (1 + ν)D(t)K I I (t)

r θ cos (κ + cosθ − 2) 2π 2

(6.81)

for mode II of fracture, in which D(t) is presented in Eq. (6.74) and κ = 3 − 4υ due to the plane strain condition of this problem. A structured 39 × 39 mesh shown in Fig. 6.31 is considered and the displacement fields determined based on analytical solution of mode I and mode II are applied on its boundaries. The deformed shape of mesh after applying the load for t = 200 s is shown in Fig. 6.32 for both mode I and mode II of fracture. As can be seen in Fig. 6.31, the finite elements used in this problem are divided into three categories: (1) ordinary elements, (2) split elements, and (3) tip elements. To incorporate the analytical solution near the crack tip into the numerical simulation, additional degrees of freedom are imposed for split and tip elements, as illustrated in Fig. 6.31. The split elements are modeled using the Heaviside function to represent the discontinuity, with a multiplication sign (×) to denote enriched nodes. The tip element is enriched using the function described in Eq. (6.67) denoted by a square symbol (⏿), and has eight degrees of freedom for enriched nodes. In order to verify the XFEM solution, two points A and B are considered, located at the same distance from the crack tip but on opposite sides of the crack plane, A(r = 0.5, θ = π ) and B(r = 0.5, θ = −π ). By using the symmetry of the problem, it

Fig. 6.31 Structured mesh to model an edge crack discontinuity within a viscoelastic finite plate

6.5 An Extension of FEM for Discontinuity Modeling

163

Fig. 6.32 Deformed shape due to opening the edge crack of a finite viscoelastic plate

is possible to determine the crack opening displacement (COD) in mode I and the crack sliding displacement (CSD) in mode II, based on the displacements at points A and B, which can be calculated using the analytical solution presented earlier. The COD is given by the difference between the y-displacements at A and B, while the CSD is given by the difference between the x-displacements at A and B. COD = u y.I (r.π.t) − u y.I (r. − π.t)

(6.82)

CSD = u x.I (r.π.t) − u x.I (r. − π.t)

(6.83)

164

6 Numerical Simulation

The XFEM approximation for COD and CSD is as follows: uh (x) =

n ⎲ j=1

N j (x)u j +

m ⎲

Nk (x)[H (ξ ) − H (ξk )]ak

(6.84)

k=1

where H(x) = 1 if x is above the crack and H(x) = −1 if x is below the crack. The j index is for ordinary degrees of freedom and k is for enriched ones. In Figs. 6.33 and 6.34, the crack opening displacement (COD) and crack sliding displacement (CSD) are presented for both analytical and numerical solutions. These figures demonstrate the capability of XFEM to solve viscoelastic problems that contain discontinuity.

Fig. 6.33 Analytical and numerical results of crack opening displacement (COD) for mode I of fracture for an edge crack within an infinite viscoelastic plate

References

165

Fig. 6.34 Analytical and numerical results of crack sliding displacement (CSD) for mode II of fracture for an edge crack within an infinite viscoelastic plate

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