Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach

By Jacob Aboudi, Steven M. Arnold and Brett A. Bednarcyk

With composites under increasing use in industry to replace traditional materials in components and structures, the modeling of composite performance, damage and failure has never been more important.

Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach brings together comprehensive background information on the multiscale nature of the composite, constituent material behaviour, damage models and key techniques for multiscale modelling, as well as presenting the findings and methods, developed over a lifetime’s research, of three leading experts in the field.

The unified approach presented in the book for conducting multiscale analysis and design of conventional and smart composite materials is also applicable for structures with complete linear and nonlinear material behavior, with numerous applications provided to illustrate use.

Modeling composite behaviour is a key challenge in research and industry; when done efficiently and reliably it can save money, decrease time to market with new innovations and prevent component failure. This book provides the tools and knowledge from leading micromechanics research, allowing researchers and senior engineers within academia and industry with to improve results and streamline development workflows.

• Brings together for the first time the findings of a lifetime’s research in micromechanics by recognized leaders in the field
• Provides a comprehensive overview of all micromechanics formulations in use today and a unified approach that works for the multiscale analysis and design of multi-phased composite materials, considering both small strain and large strain formulations
• Combines otherwise disparate theory, code and techniques in a step-by-step manner for efficient and reliable modeling of composites

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 31, 2012
• ISBN: 9780123977595

Jacob Aboudi

Jacob Aboudi is a Professor Emeritus at the School of Mechanical Engineering, Tel Aviv University, Israel. He was formerly Head of the University’s Department of Solid Mechanics, Materials and Structures, and Dean of their Faculty of Engineering. He has held visiting appointments at the University of Strathclyde,?Northwestern?University, Virginia Tech, and the University of Virginia, and has over 45 years of research experience. He has written over 300 journal articles and 2 prior books.

Book preview

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Acknowledgments

Acronyms

Chapter 1. Introduction

1.1 Fundamentals of Composite Materials and Structures

1.2 Modeling of Composites

1.3 Description of the Book Layout

1.4 Suggestions on How to Use the Book

Chapter 2. Constituent Material Modeling

2.1 Reversible Models

2.2 Irreversible Deformation Models

2.3 Damage/Life Models

2.4 Concluding Remarks

Chapter 3. Fundamentals of the Mechanics of Multiphase Materials

3.1 Introduction of Scales and Homogenization/Localization

3.2 Macromechanics versus Micromechanics

3.3 Representative Volume Elements (RVEs) and Repeating Unit Cells (RUCs)

3.4 Volume Averaging

3.5 Homogeneous Boundary Conditions

3.6 Average Strain Theorem

3.7 Average Stress Theorem

3.8 Determination of Effective Properties

3.9 Mechanics of Composite Materials

3.10 Comparison of Various Micromechanics Methods for Continuous Reinforcement

3.11 Levin’s Theorem: Extraction of Effective CTE from Mechanical Effective Properties

3.12 The Self-Consistent Scheme (SCS) and Mori-Tanaka (MT) Method for Inelastic Composites

3.13 Concluding Remarks

Chapter 4. The Method of Cells Micromechanics

4.1 The MOC for Continuously Fiber-Reinforced Materials (Doubly Periodic)

4.2 The Method of Cells for Discontinuously Fiber-Reinforced Composites (Triply Periodic)

4.3 Applications: Unidirectional Continuously Reinforced Composites

4.4 Applications: Discontinuously Reinforced (Short-Fiber) Composites

4.5 Applications: Randomly Reinforced Materials

4.6 Concluding Remarks

Chapter 5. The Generalized Method of Cells Micromechanics

5.1 GMC for Discontinuous Reinforced Composites (Triple Periodicity)

5.2 Specialization of GMC to Continuously Reinforced Composites (Double Periodicity)

5.3 Applications

5.4 Concluding Remarks

Chapter 6. The High-Fidelity Generalized Method of Cells Micromechanics

6.1 Three-Dimensional (Triply Periodic) High-Fidelity Generalized Method of Cells with Imperfect Bonding Between the Phases

6.2 Specialization to Double Periodicity (for Continuous Fibers, Anisotropic Constituents, and Imperfect Bonding)

6.3 Reformulation of the Two-Dimensional (Doubly Periodic) HFGMC with Debonding and Inelasticity Effects

6.4 Contrast Between HFGMC and Finite Element Analysis (FEA)

6.5 Isoparametric Subcell Generalization

6.6 Doubly Periodic HFGMC Applications

6.7 Triply Periodic Applications

6.8 Concluding Remarks

Chapter 7. Multiscale Modeling of Composites

7.1 Introduction

7.2 Multiscale Analysis Using Lamination Theory

7.3 HyperMAC

7.4 Multiscale Generalized Method of Cells (MSGMC)

7.5 FEAMAC

7.6 Concluding Remarks

Chapter 8. Fully Coupled Thermomicromechanical Analysis of Multiphase Composites

8.1 Introduction

8.2 Classical Thermomicromechanical Analysis

8.3 Fully Coupled Thermomicromechanical Analysis

8.4 Applications

8.5 Concluding Remarks

Chapter 9. Finite Strain Micromechanical Modeling of Multiphase Composites

9.1 Introduction

9.2 Finite Strain Generalized Method of Cells (FSGMC)

9.3 Applications Utilizing FSGMC

9.4 Finite Strain High-Fidelity Generalized Method of Cells (FSHFGMC) for Thermoelastic Composites

9.5 Applications Utilizing FSHFGMC

9.6 Concluding Remarks

Chapter 10. Micromechanical Analysis of Smart Composite Materials

10.1 Introduction

10.2 Electro-Magneto-Thermo-Elastic Composites

10.3 Hysteresis Behavior of Ferroelectric Fiber Composites

10.4 The Response of Electrostrictive Composites

10.5 Analysis of Magnetostrictive Composites

10.6 Nonlinear Electro-Magneto-Thermo-Elastic Composites

10.7 Shape Memory Alloy Fiber Composites

10.8 Shape Memory Alloy Fiber Composites Undergoing Large Deformations

10.9 Applications

10.10 Concluding Remarks

Chapter 11. Higher-Order Theory for Functionally Graded Materials

11.1 Background and Motivation

11.2 Generalized Three-Directional HOTFGM

11.3 Specialization of the Higher-Order Theory

11.4 Higher-Order Theory for Cylindrical Functionally Graded Materials (HOTCFGM)

11.5 HOTFGM Applications

11.6 HOTCFGM Applications

11.7 Concluding Remarks

Chapter 12. Wave Propagation in Multiphase and Porous Materials

12.1 Full Three-Dimensional Theory

12.2 Specialization to Two-Dimensional Theory for Thermoelastic Materials

12.3 The Inclusion of Inelastic Effects

12.4 Two-Dimensional Wave Propagation with Full Thermoelastic Coupling

12.5 Applications

12.6 Concluding Remarks

Chapter 13. Micromechanics Software

13.1 Accessing the Software

13.2 Method of Cells Source Code

13.3 MAC/GMC 4.0

13.4 Concluding Remarks

Color Plate

References

Index

Copyright

Butterworth-Heinemann is an imprint of Elsevier

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First edition 2013

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12 13 14 15 16 10 9 8 7 6 5 4 3 2 1

Dedication

This book is dedicated to our families, with all of our love

To my wife Ilana, who let me have peace of mind, which enabled me to do what I like to do most.

Jacob Aboudi

To my wife Debbie, and our children Graham, Leah, and Julianne.

Steven M. Arnold

To my wife Jennifer, and our daughter Adia.

Brett A. Bednarcyk

Preface

This book provides a detailed treatment of a unified family of micromechanics theories for multiphase materials developed by the authors over the past 30 years. These theories are applicable to composites with both periodic and nonperiodic (bounded) microstructures. A unique and important feature of these theories is their ability to provide not only the global effective composite properties, but also the varying local field distributions within the constituent materials. This capability enables the modeling of localized nonlinear phenomena such as damage and inelasticity, which are critical to the prediction of composite failure and life. In addition, because these theories can produce a macroscopic, nonlinear, anisotropic constitutive relation for the multiphase material, they are ideal for incorporation within multiscale analyses. Any higher scale method or model can therefore call these theories as an effective constitutive equation to obtain the local nonlinear response and to recover the local fields at any point within the composite structure. The resulting micro-macro-structural analysis capability is quite unique and is facilitated by the inherent computational efficiency of these micromechanics theories. Further, the nonperiodic versions of the micromechanics theories explicitly link the macro and micro scales, thus enabling concurrent analysis of problems where no repeating unit cell exists.

An additional unique feature of the unified micromechanics approach described herein is its ability to be readily extended to handle many technologically relevant aspects of advanced composite materials. These include composites (1) undergoing finite deformations, (2) subjected to dynamic impact conditions, (3) composed of smart (electro-magneto-thermo-elastic, electrostrictive, and shape memory alloy) constituents, and (4) exhibiting full (two-way) thermomechanical coupling. Thus, the authors believe that this book fills a void as most other books on composites emphasize the macromechanics approach and provide little treatment of nonlinearity in general and the above topics in particular.

The three of us wrote this book over the past several years, predominantly while the first author visited NASA Glenn Research Center in Cleveland, OH each year. We have attempted to highlight key lessons learned in developing and applying these theories over the past two decades. Consequently, we hope that this unified multiscale approach will help provide materials scientists, researchers, engineers, and structural designers with a better understanding of composite mechanics at all scales, and thereby contribute to composites reaching their full potential. More related materials to this book could be found at the companion website: http://booksite.elsevier.com/9780123970350/. The password is Solutions.

Jacob Aboudi

Steve Arnold

Brett Bednarcyk

August 2012

Acknowledgments

Without the hard work of many dedicated students and colleagues, the book you are reading would not exist. We extend a tremendous ‘Thank You’ to everyone who contributed to and helped us complete this book. At times, it certainly seemed like it would never reach this point.

Thank you to our NASA editor extraordinaire, Laura Becker, whose hard work, long hours, and attention to detail improved the book immensely, to Lorie Passe for laying out the book format, turning long-forgotten papers into live documents, and re-typing innumerable equations, and to Nancy Mieczkowski for her excellent work on the figures. Thank you to Caroline Rist for overseeing their work and to Robert Earp (GRC legal counsel) for making sure we are not exposed.

Thank you to Jennifer Bednarcyk for sending years of weekly update e-mails to Jacob, forcing Steve and Brett to stay on track, and to Shirley Arnold for hosting ‘NASA South’ on Fridays at her home in Akron.

Thank you to past and present students, Daniel T. Butler, Patrick Dunn, Yuval Freed, John W. Hutchins, Saiganesh K. Iyer, K.C. Lui, Albert M. Moncada, Len E. Necefer, Moshe Paley, Evan J. Pineda, Trent M. Ricks, Scott E. Stapleton, Benjamin T. Switala, and Edward Urquhart, whose labor led to many of the fruits in the book.

Thank you to our colleagues from academia, Professors Leslie Banks-Sills, Sol R. Bodner, Hugh A. Bruck, Aditi Chattopadhyay, Thomas E. Lacy, Cliff J. Lissenden, Rivka Gilat, Rami Haj-Ali, Carl T. Herakovich, Cornelius O. Horgan, Marek-Jerzy Pindera, Vipul Ranatunga, David Robinson, Samit Roy, Michael Ryvkin, Atef F. Saleeb, Rani W. Sullivan, Moshe Tur, and Anthony M. Waas, for working with us and providing us with access to your students.

Thank you to our colleagues working with us at NASA, and through other collaborative agreements,

Cheryl Bowman, Mike Castelli, Craig S. Collier, J. Rod Ellis, Robert K. Goldberg, Dale A. Hopkins, Serge Kruch, Bradley A. Lerch, Subodh K. Mital, Dieter H. Pahr, Sharon Priscak, Doron Shalev, Roy M. Sullivan, Daniel Trowbridge, Todd O. Williams, Thomas E. Wilt, and Phillip W. Yarrington.

Special thanks to NASA for sponsoring the development of the theory and computational tools associated with this book. Specific thanks go to the Integrated Vehicle Health Management Project and System-wide Safety Assurance Technologies within the Aviation Safety Program and to Ajay Misra, Structures and Materials Division Chief, and Leslie Greenbauer-Seng, Deputy Chief, for their encouragement and support of this effort.

I, Jacob Aboudi, would like to thank Professor Isaac Elishakoff, who kept urging me to write a book on the micromechanical analysis of composites.

Finally, I, Steve Arnold, would like to thank my Lord and Savior, Jesus Christ, for providing me with the numerous opportunities throughout my career that have made this endeavor possible and most of all for bringing two such excellent gentlemen (Jacob and Brett) into my life, with whom I am proud to have been associated for so many years. I specifically want to thank Jacob for his mentorship and Brett for his attention to detail, but most of all for their friendship. They have made this specific journey, although intense at times, a very special and memorable time in my life, and I consider myself blessed to have been able to make it with them.

Acronyms

5HS    5-harness satin

AR    aspect ratio

BK    Benzeggagh-Kenane

BP    Bodner-Partom

CCA    concentric cylinder assemblage

CCI    constant compliant interface (model)

CDM    Continuum Damage Mechanics

CFRP    carbon fiber-reinforced polymers

CMC    ceramic matrix composite

CTE    coefficient of thermal expansion

CVI    chemical vapor infiltration

DCB    double cantilever beam

DS    differential scheme

EAM    element array model

ECI    evolving compliant interface (model)

ER    electrorheological

FCTM    fully coupled thermomicromechanical

FE    finite element

FEA    finite element analysis

FGM    functionally graded material

FI    flexible interface (model)

FS    facesheet

FSGMC    Finite Strain Generalized Method of Cells

FSHFGMC    Finite Strain High-Fidelity Generalized Method of Cells

GMC    Generalized Method of Cells

GMC-3D    triply periodic Generalized Method of Cells

gps    generalized plane strain

GSCS    generalized self-consistent scheme

GVIPS    generalized viscoplasticity with potential structure (model)

HFGMC    High-Fidelity Generalized Method of Cells

HOTCFGM    Higher-Order Theory for Cylindrical Functionally Graded Materials

HOTFGM    Higher-Order Theory for Functionally Graded Materials

HOTFGM-1D    one-directional Higher-Order Theory for Functionally Graded Materials

HOTFGM-2D    two-directional Higher-Order Theory for Functionally Graded Materials

H-S    Hashin-Shtrikman

ICME    integrated computational materials engineering

MAC/GMC    Micromechanics Analysis Code with Generalized Method of Cells (software)

MCCM    multiple concentric cylinder model

MD    molecular dynamics

MI    melt infiltrated

MKM    modified Kabelka model

MM    multiscale modeling

MMC    metal matrix composite

MMCDM    Mixed Mode Continuum Damage Mechanics

MMPM    modified mosaic parallel model

MOC    Method of Cells

MOC-TI    Method of Cells, transversely isotropic

MR    magnetorheological

MSGMC    Multiscale Generalized Method of Cells

MT    Mori-Tanaka (theory)

NDE    nondestructive evaluation

NI    Needleman Interface (model)

NLCDR    NonLinear Cumulative Damage Rule

ONERA    Office Nationale d’Études et de Recherches Aérospatiales

PLS    proportional limit stress

PMC    polymer matrix composite

PMN    lead magnesium niobate

ps    plane strain

PVDF    polyvinylidene fluoride

PZT    lead zirconium titanate

QLV    quasilinear viscoelasticity

RCS    representative cross-section

ROM    rule of mixtures

RUC    repeating unit cell

RVE    representative volume element

SAM    slice array model

SCDR    surface of constant dissipation rate

SCIP    surface of constant inelastic power

SCISR    surface of constant inelastic strain rate

SCS    self-consistent scheme

SIF    statistical interfacial failure (model)

SMA    shape memory alloy

SOM    strength of materials

SPL    sound pressure level

TBC    thermal barrier coating

TE    thermoelastic

TGVIPS    transversely isotropic GVIPS

TMC    thermomechanical coupling

TMC    titanium matrix composite

TRIP    transformation-induced plasticity

TRL    Technology Readiness Level

TVE    thermoviscoelastic

UTS    ultimate tensile strength

VCCT    virtual crack closure technique

VCM    variable constraint model

VE    viscoelastic

VFD    vanishing fiber diameter

WWFE    World-Wide Failure Exercise

Chapter 1

Introduction

Chapter Outline

1.1 Fundamentals of Composite Materials and Structures

1.2 Modeling of Composites

1.3 Description of the Book Layout

1.4 Suggestions on How to Use the Book

Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach is the culmination of nearly 30 years of work by the first author and his co-workers on the development, implementation, and application of micromechanics theories for composites. The intent of the book is to place these theories in context, provide their theoretical underpinnings in a clear and concise manner, and illustrate their utility for the design and analysis of advanced composites, particularly in the nonlinear regime. The power of these theories becomes particularly clear with their application in multiscale modeling of composites. Because they provide an effective anisotropic constitutive equation for composite materials, these theories can be used to represent the macroscopic (global) nonlinear, inelastic, viscoelastic, or finite strain behavior at a point in a composite structure that is being analyzed using a higher scale model such as finite element analysis. In this context, nonlinearity in the composite constituent materials due to inelasticity and/or damage will affect the composite behavior, and this change will impact the higher scale structural response. Thus, the physics of damage and deformation in composites can be captured at a more fundamental scale by conducting multiscale analyses. However, for multiscale problems to remain tractable, the micromechanics methods must be very efficient—and efficiency is a hallmark of the micromechanics theories presented herein as they are closed form or semi-closed form.

Throughout this book, a basic knowledge of solid mechanics is assumed. Consequently, there is no chapter on the basics of solid and structural mechanics (e.g., introducing the concepts of stress and strain). Rather, Chapter 2 presents the constitutive models associated with deformation and damage that will be used throughout the book to describe the behavior of the constituent materials of composites. For the advanced topics covered in Chapters 8 to 12, it is further assumed that the reader has a general knowledge of each topic.

This introductory chapter provides some fundamental information about composites and then focuses on introducing modeling of composites, particularly micromechanics and multiscale modeling. There are many excellent texts, however, that go into much greater detail regarding the how and the why of composite materials and structures. Rather than repeat this information, the reader is referred to Jones (1975), Christensen (1979), Carlsson and Gillespie (1990), Herakovich (1998), Hyer (1998), Zweben and Kelly (2000), Miracle and Donaldson (2001), and Barbero (2011). It should also be noted that this book follows the foundation laid by the Aboudi (1991) book on micromechanics, which summarizes a great deal of his early work on the subject.

1.1 Fundamentals of Composite Materials and Structures

In the fields of Structural Engineering and Materials Science and Engineering, the difference between a structure and a material comes down to the presence of a boundary. A material is the substance of which a body is composed. The material itself has no boundaries, but rather may be thought of as what is present at a point in the body. Scientists and engineers have developed ways to represent materials through properties that describe how the material behaves at a point in a body, such as Young’s modulus, thermal conductivity, density, yield stress, Poisson’s ratio, and coefficient of thermal expansion. The body itself, on the other hand, is a structure. It has boundaries and its behavior is dependent on the conditions at these boundaries. For example, given a steel beam, the beam itself is a structure, while the material is steel. This distinction between materials and structures is natural and extremely convenient for structural engineers and materials scientists. Imagine attempting to combine properties of materials and structures in the case of the aforementioned beam. A beam’s bending characteristic is dictated by its flexural rigidity (the Young’s modulus E times the cross-section moment of inertia I, or EI, and not just its Young’s modulus). If this were not separated into a material property (E) and a structural property (I) but rather kept as a combined property, one would need to look up a value for every combination of beam shape and material.

The above discussion implies that a material is a continuum, meaning it is continuous and completely fills the region of space it occupies. The material can thus be modeled using continuum mechanics, which considers the material to be amorphous and does not explicitly account for any internal details within the material, such as the presence of inclusions, grain orientation, or molecular arrangement. To account for such internal details, some additional theory beyond standard continuum mechanics is needed.

In its broadest context, a composite is anything comprised of two or more entities. A composite structure would then be any body made up of two or more parts or two or more materials. Likewise, a composite material is a material composed of two or more materials with a recognizable interface between them. Because it is a material, it has no external boundaries; once an external boundary is introduced, it becomes a structure composed of composite materials, which is a particular type of composite structure. Clearly, however, a composite material does have distinct internal boundaries. If these internal boundaries are ignored, continuum mechanics can be used to model composite materials as pseudo-homogeneous anisotropic materials with directionally dependent ‘effective,’ ‘homogenized,’ or ‘smeared’ material properties. Micromechanics, on the other hand, attempts to account for the internal boundaries within a composite material and capture the effects of the composite’s internal arrangement. In micromechanics, the individual materials (typically referred to as constituents or phases) that make up a composite are each treated as continua via continuum mechanics, with their individual representative properties and arrangement dictating the overall behavior of the composite material.

In many cases, especially with composite materials used in structural engineering, the geometric arrangement of one phase is continuous and serves to hold the other constituent(s) together. This constituent is referred to as the matrix material. Whereas the other constituent(s), often referred to as inclusion(s) and/or reinforcement(s), are materials that can be either continuous or discontinuous and are held together by the matrix. There may also be interface materials, or interphases, present between the matrix and inclusion. A fundamental descriptor of composites that should always be indicated when denoting a given system (since it strongly influences the effective behavior) is the volume fraction of phases present. Typically, only the reinforcement phase is indicated unless multiple phases are present since the sum of all phases must equal 100%; for example, in a two-phase fiber-reinforced composite vf is the volume fraction of fibers and vm = 1 − vf is that of the matrix. Composites are typically classified at two distinct levels. The first level of designation is usually made with respect to the matrix constituent. This divides composites into three main categories: polymer matrix composites (PMCs), metal matrix composites (MMCs), and ceramic matrix composites (CMCs). The second level of classification refers to the form of the reinforcement: discontinuous (particulate or whisker), continuous fiber, or woven (textile) (braided or knitted fiber architectures are included in this classification). In the case of woven and braided composites, the weave or braiding pattern (e.g., plane weave, triaxially braided) is also often indicated. Examples of some of these types of composites are shown in Figure 1.1. Note that particulate composites are typically isotropic whereas most other composite forms have some level of anisotropy (e.g., a unidirectional continuous fiber composite is usually transversely isotropic).

Figure 1.1 Composite systems. (a) Reinforcement types. (b) Laminate and woven constructions.

Composites, particularly PMCs, are often manufactured as an assembly of thin layers joined together to form a laminate (see Figure 1.1(b)). Each layer is referred to as a lamina or ply. By orienting the reinforcement direction of each ply, the properties and behavior of the resulting laminate can be controlled. A quasi-isotropic laminate can be formed by balancing the orientations of the plies such that the extensional stiffness of the laminate is constant in all in-plane directions. Quasi-isotropic laminates have thus been very popular as—under in-plane elastic extension—they behave like isotropic metals, with which most engineers are familiar. However, this has also led to engineers attempting to simply replace metals with quasi-isotropic laminates in structures that were designed based on isotropic metallic properties. This is the origin of the expression ‘black aluminum,’ which refers to a black quasi-isotropic carbon/epoxy laminate, whose in-plane effective elastic properties are often very close to those of aerospace aluminum alloys. The tremendous pitfall of this approach, which has in many ways slowed the realization of the full potential of composites, is that quasi-isotropic carbon/epoxy laminates are not even close to isotropic in terms of their out-of-plane behavior. They are highly prone to delamination and interlaminar failure, failure modes which do not afflict isotropic metals. Care must therefore be taken to minimize out-of-plane loads and quantify out-of-plane margins of safety when designing structures with this type of composite laminate. The ‘black aluminum’ design approach, while simple, is typically very inefficient.

A key distinction among PMCs, MMCs, and CMCs is their maximum service temperature. As shown in Figure 1.2, most PMCs are limited to an operating temperature under 450 °F. Metal matrix composites extend this range to approximately 1200 °F, depending upon the capability of the chosen matrix, and typical CMCs can remain functional to over 2000 °F. Obviously, the temperature limitations are dependent on the limitations of the composite constituent materials. Indeed, the groups of small ovals in Figure 1.2 representing CMCs with maximum service temperatures of approximately 1100 °F are tungsten carbide ceramic matrix materials with particulate metallic cobalt inclusions. Thus, the lower operating temperature is due to the metallic reinforcement; most CMCs are composed of ceramic matrices and ceramic reinforcements.

Figure 1.2 Ashby diagrams (produced using CES Selector 2012 (Granta Design Limited, 2012)) for various PMCs, MMCs, and CMCs. (a) Young’s modulus versus maximum service temperature. (b) Specific strength (tensile strength divided by density) versus maximum service temperature.

The vertical axes in Figure 1.2 represent the composite’s (a) effective Young’s modulus and (b) specific strength (strength divided by density). The wide spread in properties, especially in the case of PMCs, is indicative of the anisotropy present in continuously reinforced composites. For the carbon/epoxy composite labeled in Figure 1.2, there is a factor of 20 between the Young’s moduli in the longitudinal direction (along the continuous carbon fibers, 0°) and the transverse direction (perpendicular to the fiber direction, 90°). For the specific strength, the corresponding factor is close to 40. The composite labeled ‘HS C/epoxy, QI lam’ represents the effective in-plane properties of a quasi-isotropic laminate composed of the previously discussed high strength carbon/epoxy material. As would be expected, this laminate’s effective properties are intermediate to those of its plies in each direction. It is also noteworthy that this quasi-isotropic laminate, which is actually a structure with external boundaries, is compared here to unidirectional composite materials. Such a laminate would only behave like a material if appropriate extensional in-plane boundary conditions were applied. If it were subjected to bending it would behave like an anisotropic plate, and its properties would be dependent on its thickness and ply stacking sequence, which are structural rather than material properties.

Without differences in properties between the constituents, a composite would cease to be a composite. That is, it is the difference in properties between the constituents that makes a composite behave differently than a monolithic material and enables its tailoring for specific purposes. PMCs are characterized by a large property mismatch between the constituents. As shown in Figure 1.3, the extremely low stiffness of the polymer matrix (epoxy shown here) results in a stiffness mismatch in the order of 80:1 in the case of carbon fibers in the longitudinal direction, and 20:1 in the case of glass fibers. Carbon fibers are transversely isotropic, so the mismatch in the transverse direction for carbon fiber-reinforced polymers (CFRP) is typically much lower. Alternatively, MMCs typically have constituent stiffness mismatches of 3:1 to 4:1. The longitudinal stiffness mismatch between the fiber and matrix in CMCs is much lower, close to 1:1 in the case of SiC/SiC composites and approximately 1:1.6 in the case of C/SiC. In the transverse direction, however, there is a large mismatch in stiffness in the case of C/SiC (in the order of 1:17). In SiC/SiC, the SiC fibers are coated with a very compliant material, such as BN, to present a barrier to matrix crack growth. This then sets up a large mismatch in stiffness for SiC/SiC in the transverse direction (in the order of 1:18) as well. C/SiC composites also typically have a compliant pyrolytic carbon interface material that serves the same purpose. Thus it is clear that CFRPs and CMCs represent nearly converse cases wherein CFRPs have high property mismatch in the longitudinal direction and lower property mismatch in the transverse direction and CMCs have low property mismatch in the longitudinal direction and high property mismatch in the transverse direction. Glass fiber-reinforced polymers have reasonably large property mismatch in both directions; typical MMCs (like SiC/Ti) have intermediate property mismatch in both directions. The extent of property mismatch is a key feature of composites that impacts the efficacy of composite models. For a model to be applicable to all types of composites, it must properly handle widely varying degrees of property mismatch in each direction.

Figure 1.3 Comparison of the Young’s moduli of typical PMC, MMC, and CMC constituent materials, with (L) indicating the longitudinal direction and (T) indicating the transverse direction in the case of transversely isotropic carbon fibers. The other constituents plotted are typically considered to be isotropic.

The development of high-performance composite materials started in the 1940s with the introduction of glass fiber-reinforced polymer matrix composites. It has continued to grow with the introduction of additional polymeric, metallic, and ceramic composite systems to become a major force in the world materials market. Composites have penetrated such key industries as aerospace, automotive, building and construction, sports and leisure, and most recently, wind energy. JEC (2011) estimates the worldwide market for composites in 2011 to be $90 billion (USD) with a total production mass of 7.9 million metric tonnes. This global composite market can be broken down into market sectors (see Table 1.1), wherein both value and volume percentages for each sector have been quantified by TenCate (2010). The volume share of the United States is 35% (value 36%), the European Union (EU) 22% (value 33%), and Asia 43% (value 31%). Furthermore, in the first decade of the twenty-first century the composite market as a whole saw an annual growth of approximately 4% to 5% per year in value and 3% in volume, with emerging countries seeing approximately twice as much growth per year in value compared to developed nations. Growth is expected to continue at an average rate of 6% per year in value for the next five years with a 5% shift from North America and Europe to Asian markets (JEC, 2011).

Table 1.1 Value and Volume of Composites Produced within Various Market Sectors

Arguably the most aggressive industry utilizing composites is aerospace, as illustrated in Figure 1.4, because of their attractive (weight-saving) properties, which translate directly to cost savings. The most recently publicized commercial aircraft is the Boeing 787 with over 50% by weight of its materials being composites, as shown in Figure 1.5. Composite usage in some military aircraft is shown in Figure 1.6. Both Figures 1.5 and 1.6 illustrate the significant increase in the use of composites over the past half century as manufacturing, joining, and analysis methods have improved and performance demands have increased. Prior to the mid-1990s, composites were mainly limited to use in secondary structures (i.e., those that do not cause immediate danger upon failure). However, with the development of the Airbus A380, Boeing 787, and Lockheed Martin F-22 and F-35, composites are now being extensively utilized in primary structures such as wings and fuselage components as well.

Figure 1.4 Example of the evolution of aerospace composite applications over time, with time increasing from left to right.

(Raju, 2011)

Figure 1.5 Composite material usage by weight in commercial aircraft.

(Harris et al., 2001)

Figure 1.6 Composite use in military aircraft.

(Harris et al., 2001)

Usage of composites (primarily PMCs) in spacecraft is lagging behind that in aircraft, mainly because of unique environmental and loadings requirements, and the fact that spacecraft structures are typically designed for a single launch and are certified to perform their entire mission without interim inspections or repairs. Probably the best known use of composites in spacecraft is the 60-ft-long payload doors of the space shuttle; however, other launch vehicles using carbon fiber-reinforced composites are the Delta IV, Atlas V, EELV, and Pegasus. NASA and its industrial partners are actively pursuing the development and use of composites in large structures (e.g., composite crew module, space launch systems, antennas and solar arrays, and propellant tanks, to name a few) for future space missions.

1.2 Modeling of Composites

In this text, the phrase ‘modeling of composites’ is intended to refer to simulating or analyzing the behavior of a fully consolidated composite material or structure. Process modeling, the simulation of the manufacturing and forming of composite materials and parts, is not addressed. In this context, there are two basic approaches to modeling composites: the macromechanical approach and the micromechanical approach. The macromechanical approach involves constructing models strictly at the macro/global scale (see Figure 1.7), wherein the composite is viewed as an anisotropic material, and the details of the underlying arrangement of the constituent materials are ignored. In the linear elastic regime, this approach is straightforward; it involves only determining (usually experimentally) the anisotropic elastic properties of the composite material. These can then be entered into a structural analysis code, such as finite element analysis (FEA), to determine structural performance. In fact, this is the current standard design procedure for composite structures. In addition to the elastic properties, statistically meaningful design allowable stresses are determined for the composite material through extensive testing. Then, the part is designed such that the stresses never exceed these allowables, with a sufficient margin of safety. Of course, while computationally efficient and straightforward, this approach is heavily reliant on costly experimental data (both coupon and structural), which must be generated for each variation of the composite (e.g., change in fiber volume fraction).

Figure 1.7 Illustration of the relevant levels of scale for multiscale composite analysis.

In the nonlinear regime (e.g., high temperature applications), and when trying to predict damage and failure, the macromechanical approach becomes somewhat problematic, as anisotropic constitutive, damage, and failure models must be constructed to account for the widely varying behavior and failure mechanisms of the composite in the various directions and characterized through extensive composite testing. A benefit is that the intrinsic, history-dependent, interactive effects of the composite constituents are embedded in the experimental results, and their in-situ behavior is automatically captured. However, such models are hampered by the fact that, physically, the deformation, damage, and failure occur in the actual constituent materials of the composite, not within an idealized effective anisotropic material. Thus, while many have attempted to model the nonlinear behavior of composites with macromechanics, the approach will always be highly phenomenological since, by definition, it does not consider what is happening in each constituent at the appropriate physical scale.

In contrast, the micromechanical approach to modeling composites explicitly considers the constituent materials and how they are arranged to form the composite. The goal of micromechanics is to predict the effective behavior of a heterogeneous material based on the behavior of the constituent materials and their geometric arrangement. By determining a composite’s effective behavior via micromechanics, it can then be treated as a material in higher scale analyses (similar to the macromechanical approach). For example, the effective material properties of the composite determined via micromechanics can be used in a laminate analysis to represent the ply materials, or in FEA of a composite structure to represent the materials in different regions. One benefit of micromechanics is that composite properties can be determined, in any direction, for any fiber volume fraction or reinforcement architecture, even if the composite has never been manufactured. It can therefore assist in designing the composite materials themselves as well as the structures comprised of them. In terms of nonlinearity associated with inelasticity, damage, and failure, micromechanics allows the physics of these mechanisms to be captured at the constituent scale, where they are actually occurring (provided the micromechanics theory is capable of solving for the local fields in the constituents). There are many (molecular dynamicists, for example) who argue that even this scale is too high to properly account for these mechanisms. It is clear, however, that micromechanics allows the physics to be captured at a more fundamental scale compared to macromechanics. Further, if the interface between the composite constituents contributes significantly to the overall composite behavior, it can be addressed through micromechanics, wherein information as to the state of the interface is available.

Multiscale modeling of composites refers to simulating their behavior through multiple time and/or length scales. Although the nomenclature in the literature varies, typically a multiscale modeling analysis will follow length scales shown in Figure 1.7 for continuum-based modeling. These scales, progressing from left to right in Figure 1.7, are the microscale (constituent level; fiber, matrix, interface), the mesoscale (tow), the macroscale (woven repeating unit cell (RUC)), and the global/structural scale. Traditionally, one traverses (transcends (moves right) or descends (moves left)) these scales via homogenization and localization techniques, respectively (Figures 1.7 and 1.8); a homogenization technique provides the properties or response of a ‘structure’ (higher level) given the properties or response of the structure’s ‘constituents’ (lower scale). Conversely, localization techniques provide the response of the constituents given the response of the structure. Figure 1.8 illustrates the interaction of homogenization and localization techniques in that, during a multiscale analysis, a particular stage in the analysis procedure can function on both levels simultaneously. For example, during the process of homogenizing the stages represented by X and Y to obtain properties for the stage represented by V, X and Y should be viewed as the constituent level, while V is on the structure level. However, during the process of homogenizing V and W to obtain properties for U, V is now on the constituent level (as is W). Obviously, the ability to homogenize and localize accurately requires a sophisticated theory that relates the geometric and material characteristics of structure and constituents.

Figure 1.8 Multilevel tree diagram relating constituents and structures.

Multiscale modeling methods can be classified as hierarchical, synergistic, or concurrent (Sullivan and Arnold, 2010). Hierarchical, or sequential, methods are typically strategies that systematically pass information in a bottom-up (or top-down) approach from a finer (coarser) scale to the next coarser (finer) scale as either boundary conditions or effective properties. The hierarchical approach involves strictly one-way coupling of the scales, either bottom-up (homogenization) or top-down (localization), but not both. Concurrent methods are fully coupled such that the scales are interwoven in a parallel fashion for simultaneous computation. Essentially, all scales are handled at once in both time and space. Synergistic methods represent an intermediate approach wherein data is passed between the scales like the hierarchical approach, but with a two-way information flow. These methods typically handle field quantities spatially sequentially and temporally concurrently, or spatially concurrently and temporally sequentially.

Figure 1.9 illustrates these three major overarching approaches, wherein within each approach there is typically a range of methods from fully analytical (e.g., rule of mixtures (ROM) and Mori-Tanaka (MT)) to fully numerical (e.g., finite element analysis (FEA) and molecular dynamics (MD)), or from semi-analytical (e.g., Generalized Method of Cells (GMC), High-Fidelity Generalized Method of Cells (HFGMC), and Higher Order Theory for Functionally Graded Materials (HOTFGM)) to fully numerical. All of these methods are discussed within this book. A key point of Figure 1.9, and the premise behind the development of all of the theories based on the Method of Cells (MOC) presented herein, is the balance between fidelity and efficiency that must be met when conducting multiscale modeling of composites. The ultimate goal is the highest efficiency and the highest fidelity possible (upper right corner in the figure). Hierarchical multiscale models provide the highest efficiency, but the lowest fidelity. An example of this approach would be using a micromechanics model to determine a composite’s effective anisotropic elastic properties and then entering these properties into a finite element (FE) code to analyze a composite part. The execution of the FE model would be as efficient as possible in this case. Conversely, concurrent multiscale models provide the highest fidelity, but lowest efficiency. An example would be modeling the aforementioned composite part down to the level of every individual fiber using finite elements. While this would be very computationally intensive, it would provide the highest fidelity representation of the local stress fields in the fiber and matrix throughout the part. The synergistic multiscale approach is intermediate. It would involve a micromechanics model for the composite material (e.g., GMC) embedded in the FE structural analysis of the composite part. The goal of the synergistic approach is to realize an optimum balance between local and global stress and strain field accuracy and the computational demands to determine these fields. This is the multiscale modeling approach taken in this book, and as demonstrated in Chapter 7, it provides an excellent balance of fidelity and efficiency when the MOC-based theories are utilized.

Figure 1.9 Schematic illustrating the balance of multiscale model fidelity and efficiency for hierarchical, synergistic, and concurrent multiscale composite modeling approaches.

Also shown in Figure 1.9 within each multiscale modeling approach is a gradation intended to further characterize the types of models used within each approach. For example, in the synergistic multiscale approach, any type of micromechanics model, ranging from simple fully analytical rule of mixtures equations to a numerical FE-based approach, could be used to provide the global FE model with the composite effective behavior. This too would have fidelity and efficiency consequences. Similarly, a hierarchical approach could use a range of approaches to determine effective composite properties for use in an FE model of a part. Concurrent multiscale methods typically need to be fully numerical in order to capture all geometric details of the problem at all scales simultaneously. An exception is the semi-analytical HOTFGM presented in Chapter 11.

The multiscale modeling examples described above represent the simplest case, that of linear elastic constituent materials. In such a case, on the global structural scale hierarchical, synergistic, and concurrent models should all provide the same answer (assuming the models used are sufficiently realistic). They should also match the macromechanical approach where test data would be used for the composite material elastic properties. In such a situation, all that is gained from the more complex and higher fidelity hierarchical and concurrent methods would be the ability to discern the stress and strains locally in the constituent materials. This is because, in the linear elastic regime, all information affecting the higher scale is captured through the homogenized (effective) elastic stiffness tensor of the composite. In the nonlinear regime, the situation is vastly different. If any constituents (including any interfaces) experience damage, inelasticity, or nonlinear elasticity, which all depend on the local stress or strain state in the composite, the effective composite response becomes dependent on the local stress and strain history. The macromechanical approach would then require some sort of evolving anisotropic phenomenological model to try to capture the effects of the constituent nonlinearity on the global composite response. The hierarchical multiscale approach breaks down as it includes only one-way uncoupled information flow, and thus there is no systematic way to enable local nonlinearity and pass the effects to the global scale. This leaves synergistic and concurrent multiscale methods as the only legitimate options for capturing local path-dependent nonlinearity based on the local physics and percolating the effects to the higher scale. With infinite time and infinite computational resources, the concurrent approach would be preferable. Recall the example where in some composite structure a three-dimensional FE model is constructed to the scale where every individual fiber is meshed. For real structures (e.g., an aircraft wing), the number of degrees of freedom in such a model would be astronomical. For small composite parts in the linearly elastic regime, such a concurrent model is currently feasible. However, in the nonlinear regime, wherein a loading history is applied with a small time step, such concurrent models are typically intractable. Synergistic models, as mentioned previously, offer a balance of fidelity and efficiency. They discern the local constituent fields, enable nonlinearity to be captured at the local scale, and pass this information on to the higher scales at each increment in time. As computational power continues to increase, the feasibility of concurrent models will increase, but synergistic models will still always be much faster.

As a final note on material nonlinearity, it must be remembered that the in-situ nonlinear behavior of the composite constituents will always be multiaxial and non-proportional. If a monolithic material is loaded uniaxially, the internal stress field is constant and uniaxial. In a composite, even if it is loaded uniaxially, the mismatch in constituent properties in the various directions sets up multiaxial stress fields in each constituent. When material nonlinearity is present in the constituents, as the composite is loaded monotonically, stress redistribution occurs internally among the constituents—and even within the constituents—because higher stress regions of the composite behave differently to lower stress regions. The result is local non-proportionality as every point in every constituent affects the response of every other such point. Local unloading can often occur even as the global monotonic loading on the composite continues. For these reasons, care must be taken when implementing a nonlinear material model into a micromechanics model to represent the constituents. Many constitutive and damage models are only validated for monotonic uniaxial loading. One might think that such a model is satisfactory for use in a composite micromechanics model for uniaxial loading on the composite. However, because of the in-situ multiaxiality and non-proportionality, this is not necessarily the case. Nonlinear constitutive and damage models should therefore be validated in multiaxial loading situations and under an array of varying loading and unloading scenarios.

1.3 Description of the Book Layout

The remainder of the book begins with a chapter on constitutive models. Chapter 2 presents the linear and nonlinear models that are used throughout the book to represent the material behavior of the individual composite constituents. Constitutive models handling time-dependent and time-independent reversible and irreversible deformation are presented. Damage models for continuum damage, cracks, debonding, fatigue, and static failure are also presented.

Chapter 3 deals with the fundamentals of micromechanics, presenting the theoretical underpinnings of the subject in general, as well as details of many of the classical micromechanics methods. This is followed by three chapters that present each of the major micromechanics theory contributions of the first author and co-workers. Each of these chapters is divided into theory and application sections. Chapter 4 lays out the MOC theory for continuous (doubly periodic) and discontinuous (triply periodic) composites. Applications include the calculation of effective properties, effective thermal properties, yield surface prediction, weak interfacial bonding, and the effective viscoplastic behavior of composite materials. The GMC, which is obtained through geometric generalization of the MOC to an arbitrary number of subvolumes, is presented in Chapter 5. Here the theoretical development begins with the most general case, the discontinuously reinforced (triply periodic) version of the GMC theory. The continuously reinforced (doubly periodic) version of GMC is given as a specialization. A version of the theory that has been reformulated to maximize computational efficiency is also provided. The effective properties and the nonlinear deformation, damage, and failure behavior of continuous and discontinuous composites in the longitudinal and transverse directions are examined as applications, as are effective yield surfaces and the behavior of woven composites. Chapter 6 develops the HFGMC micromechanics theory. This relies on a higher-order displacement field to provide higher-fidelity local field predictions in the composite constituents. Versions for discontinuous (triply periodic) and continuous (doubly periodic) composites, a reformulation for computation speed, and an isoparametric (non-orthogonal) formulation are presented, along with a number of applications.

Chapter 7 is dedicated to the multiscale analysis of composites, wherein the micromechanics theories developed in the previous three chapters are used within higher-scale analyses to represent composite materials. Multiscale lamination theory uses GMC and HFGMC to model the response of the composite plies within composite laminates. HyperMAC uses GMC to represent the plies in composite stiffened and sandwich panels that are analyzed and sized using the HyperSizer commercial structural sizing software (Collier Research Corp., 2012). Multiscale GMC (MSGMC) enables the constituents of a composite being analyzed with GMC to themselves be composites, which are analyzed using GMC. GMC is thus called recursively an arbitrary number of times to consider an arbitrary number of scales in a multiscale analysis. Finally, FEAMAC is the implementation of GMC within FEA to represent composite materials in arbitrary structures. The many application examples in the chapter focus on nonlinearity due to inelasticity and damage as well as how these microscale effects percolate to the higher scales.

In the remaining chapters, individual advanced topics are discussed separately. Chapters 8, 9, and 10 use GMC and HFGMC to examine effects beyond the standard infinitesimal thermal and mechanical strain behavior typically predicted using micromechanics. Chapter 8 deals with two-way thermomechanical coupling, wherein now not only does temperature change induce deformation, but also material deformation induces a temperature change within the material. Chapter 9 presents finite strain versions of GMC and HFGMC, including an array of hyperelastic, viscoelastic, and viscoplastic constitutive theories that are used to represent the constituents in composite materials. Chapter 10 examines smart composites: those composed of so-called smart constituent materials. Piezoelectric, piezomagnetic, ferroelectric, electrostrictive, magnetostrictive, nonlinear electro-magneto-thermal-elastic, and shape memory alloy phases are considered, and the GMC and HFGMC theories are presented along with an enhanced lamination theory enabling analysis of these advanced morphing composite systems.

Chapter 11 presents a micromechanics model for composites with external boundaries and thus represents a composite structure rather than a composite material. The formulation is similar to HFGMC, but instead of periodicity conditions imposed on composite RUCs, the theory imposes arbitrary boundary conditions on a composite body. The theory is known as the Higher-Order Theory for Functionally Graded Materials (HOTFGM), as it was originally applied to functionally graded materials in which periodicity is not present and no representative volume element can truly be identified. Versions of the theory for one, two, and three directions of microstructural grading in Cartesian coordinates are presented, along with cylindrical coordinate versions. Applications include free-edge stress analysis, delamination, smart materials, and several thermal stress examples.

Chapter 12 examines wave propagation in composites via an extension of HOTFGM to consider the full dynamic equations of motions. A full triply periodic (discontinuous reinforced) version of the theory is presented, along with specialization to doubly periodic (continuously reinforced). Inelasticity and two-way thermomechanical coupling are also introduced. Applications focus on modeling the acoustic behavior of composite plates and dynamic cracking of composites.

Chapter 13, the final chapter in the book, describes the micromechanics software that is available to readers of the book through the Elsevier website. Source code for the MOC micromechanics theory is provided, along with executable code for the Micromechanics Analysis Code with Generalized Method of Cells (MAC/GMC) 4.0 software package (Bednarcyk and Arnold, 2002a). MAC/GMC 4.0 includes the GMC and HFGMC micromechanics theories along with multiscale lamination theory. MAC/GMC 4.0 also contains a library of inelastic constitutive models, damage models, and failure models. Many of the application examples presented in the book were generated using the MAC/GMC 4.0 code.

1.4 Suggestions on How to Use the Book

For those desiring an overview of the fundamental tenets of micromechanics and their application to composite materials, Chapter 3 will be of interest. Researchers, professionals, and students desiring a complete understanding of the family of micromechanics theories based on the MOC should focus on Chapters 4, 5, and 6: Chapter 4 addresses the original MOC, Chapter 5 presents the generalization of that theory, and Chapter 6 provides the more recent, and more accurate, HFGMC theory. Consequently, a person who is new to the field of composite micromechanics should be able to gain an appreciation of the field and the basics of the approach recommended herein by reading Chapters 3 to 5. Chapter 7 then demonstrates through example the suitability of these micromechanics theories for implementation into multiscale analyses. Chapters 9 to 12 are relatively independent, and can be used by readers interested in these particular advanced topics. The software provided with the book (Chapter 13) may also be useful to those who wish to actually use the methods presented to perform composite analyses. For researchers, in particular, the provided MOC source code can be adapted and used for any purpose without restriction.

The text is also useful as a reference for advanced undergraduate and graduate courses on composite mechanics in which the topic of micromechanics is addressed. It is recommended that such courses focus on Chapters 3 to 5, so that students can gain an understanding of classical micromechanics theories, as well as the MOC and GMC. Use can also be made of the MOC and MAC/GMC 4.0 software provided as described in Chapter 13 to solve problems or contribute to course-related projects. The advanced topics in Chapters 8 to 12 are recommended for graduate students and professional researchers.

Chapter 2

Constituent Material Modeling

Chapter Outline

2.1 Reversible Models

2.1.1 Elasticity

2.1.1.1 Isotropic Hooke’s Law

2.1.1.2 Transversely Isotropic in Global Coordinates

2.1.1.3 Transversely Isotropic with Arbitrary Plane of Isotropy

2.1.1.4 Orthotropic Elastic

2.1.1.5 Anisotropic Elastic

2.1.2 Ramberg-Osgood Nonlinear Elastic Constitutive Equations

2.1.3 Viscoelasticity

2.1.3.1 Linear Viscoelasticity

2.1.3.2 Schapery Single-Integral Nonlinear Viscoelasticity

2.2 Irreversible Deformation Models

2.2.1 Incremental Plasticity

2.2.2 Power-Law Creep

2.2.3 Viscoplasticity

2.2.3.1 Original Bodner-Partom Model

2.2.3.2 A Modified Bodner-Partom Model

2.2.3.3 Generalized Viscoplasticity with Potential Structure (GVIPS)

2.3 Damage/Life Models

2.3.1 Continuum-Based Damage

2.3.1.1 Subvolume Elimination Method

2.3.1.2 Triaxial Stress-Driven Damage Evolution

2.3.1.3 Mixed-Mode Continuum Damage Mechanics (MMCDM) Model

2.3.1.4 Curtin-Stochastic Fiber Breakage Model

2.3.1.5 Combined Plasticity-Damage Model

2.3.1.6 Multimechanism, Viscoelastoplastic with Coupled Damage (GVIPS)

2.3.1.7 Cyclic Fatigue Damage Analysis (ADEAL)

2.3.1.8 Creep Damage

2.3.1.9 Creep-Fatigue Interaction

2.3.2 Interface Models

2.3.2.1 Flexible Interface Model

2.3.2.2 Constant Compliant Interface (CCI) Model

2.3.2.3 Evolving Compliant Interface (ECI) Model

2.4 Concluding Remarks

The solution of a solid mechanics problem involves the establishment of a statically admissible field (one that satisfies equilibrium internally along with traction boundary conditions), a kinematically admissible field (one that satisfies strain-displacement relations and displacement boundary conditions) and the satisfaction of material constitutive laws. Constitutive theory concerns the mathematical modeling of the physical response (output) of a material to a given stimulus (input), where the input can be a generalized force or displacement. The importance of accurate constitutive relationships is illustrated in Figure 2.1, as they form the primary link between stress (σij) and strain (εij) components at any point within a body. Note that in Figure 2.1 F is force and u is displacement. The appropriate relations may be simple (as in the case of linear isotropic elasticity) or extremely complex (as in the case of anisotropic viscoplasticity), depending upon the material comprising the body and the conditions to which the body is subjected (e.g., temperature, loading, environment). Constitutive relations for a particular material are traditionally established experimentally, and they may involve both physically (directly) measurable quantities (e.g., strain, temperature, time) as well as internal parameters that are not directly measurable, often referred to as internal state variables. Note that constitutive models are not limited to deformation models (e.g., Hooke’s Law) but can, and often do, include continuum damage models as well (e.g., Lemaitre and Chaboche, 1990).

Figure 2.1 Key aspects of solid mechanics problem.

Three types of experimentation are necessary to support the rational formulation of constitutive theories:

1) Exploratory tests, which illuminate the salient material response (e.g., time dependence and/or time independence, sensitivity to hydrostatic stress field, and material symmetry and/or anisotropy), identify fundamental deformation and damage mechanisms, and guide the mathematical structure of the model;

2) Characterization tests, which provide the required database for determining the material specific functional forms and associated parameters so as to represent a particular material over a given range of conditions;

3) Validation tests, often structural (multiaxial) in nature, which provide the prototypical response data, enabling validation of a constitutive model through comparison of structural response with predictions based on the model. Results from validation tests ideally provide feedback for subsequent developmental and/or refinement efforts.

The observed behavior of real materials in response to thermal and mechanical stimuli can vary greatly depending upon the magnitude and multiaxiality of loading and the magnitude of the homologous temperature (TH  =  T/Tm, where Tm represents the melting temperature of the material). For example, at room temperature, material response is typically time-independent and either reversible (linear elastic) or irreversible (inelastic) depending upon whether or not the ‘yield stress’ of the material has been exceeded. Alternatively, when TH is, say, greater than or equal to 0.25, time-dependent behavior (both reversible and irreversible) is commonly observed. Figure 2.2 illustrates schematically (i) strain-rate sensitivity, (ii) creep, (iii) relaxation, (iv) thermal recovery, (v) dynamic recovery, and (vi) creep-plasticity interaction, all of which are examples of time-dependent behavior. Other complex time- and path-dependent behavior such as cyclic ratcheting, creep-fatigue interaction, and thermal mechanical fatigue are also often times observed depending upon the magnitude and type of loading being applied (e.g., thermal or mechanical); see Dowling (1999).

Figure 2.2 Schematics showing representative hereditary material behavior at elevated temperature.

A prerequisite for meaningful assessment of component (or composite material) durability and life, and consequently design of structural components (or composite materials), is the ability to estimate the stresses and strains occurring within a loaded structure (or composite). Because constitutive material models provide the required mathematical link between stress and strain, the selection and characterization of an appropriate constitutive model is required before a material can be used in design. Thus, a wide range of constitutive models with varying levels of idealization have been proposed and utilized, each with its own shortcomings and/or limitations.

The most well-known and widely used constitutive model is Hooke’s Law:

(2.1)

where σij and εij are the stress and strain components, respectively, and Cijkl are the elastic stiffness tensor components. This equation describes time-independent, isothermal, linear (proportional), reversible material behavior. Extension into the thermal and irreversible regimes has been accomplished by assuming an additive decomposition of the total strain tensor, εij,

(2.2)

or

(2.3)

, a reversible thermal strain. Substitution of Eq. (2.3) into Eq. (2.1) yields a stress-strain relation (known as the Generalized Hooke’s Law) that incorporates both reversible and irreversible strains:

(2.4)

The thermal strain is generally taken to be linear with temperature change. Thus, full determination of Eq. . Numerous models describing the evolution of the inelastic strain have been proposed in the literature (e.g., Skrzypek and Hetnarski, 1993; Lemaitre, 2001; Yip, 2005). A critical issue when modeling composite materials using micromechanics is the need for multiaxial constitutive models, as the in-situ stress state (that is, the stress state within the constituent phases of the composite) is always multiaxial and the history is usually non-proportional. Most inelastic models struggle under these circumstances and are not typically validated under multiaxial loading scenarios. For example, someone unfamiliar with composites might develop a constitutive model based on uniaxial tension tests on a monolithic material, then assume the model is applicable to the matrix phase in a composite when the composite is subjected to uniaxial tension. The in-situ stress state of the matrix is, of course, multiaxial; thus, the constitutive model may not be applicable. Consequently, before conducting micromechanics analyses, one must be careful when selecting a given constitutive model to represent a specific constituent material. Clearly, in the linear thermoelastic case, much of this issue can be ignored.

Figures 2.3 and 2.4 illustrate how micromechanics can not only provide insight into the actual behavior of a composite and its constituents, but also how the local (i.e., in-situ) behavior may not even be of the same character as that described by the macro loading condition. Figure 2.3 shows the simulated response of a unidirectional titanium matrix composite (TMC) system when subjected to a constant global load as a function of time (i.e., creep) in the longitudinal and transverse directions. The longitudinal creep response appears to be a pure primary creep tending toward a zero steady-state creep rate. In contrast, in the transverse direction, a primary creep zone is present, followed by pronounced steady-state (or constant) creep rate zone. Therefore, it might be claimed that the TMC exhibits creep behavior in both the longitudinal and transverse directions. Yet the fiber behaves linear elastically; therefore, why should the composite creep at all in the fiber direction? This question can be answered by using micromechanics to analyze the composite response. In Figure 2.4, one can see that, although the applied composite stress is constant, in the longitudinal case the fiber stress increases while that of the matrix decreases with time. Therefore, although the longitudinal composite response appears to ‘creep,’ in actuality the matrix constituent mainly relaxes while the fiber merely elongates elastically as it is subjected to the additional stress shed by the matrix. In the transverse direction, the stress in both the fiber and the matrix remain nearly constant, resulting in matrix creep that is obvious in the composite creep response (Figure 2.3). The point of this illustration is to demonstrate that, irrespective of the character of globally applied loading, the in-situ constituent behavior can be of a completely different nature. This strongly argues for caution in selecting constituent constitutive models. The models will be subjected not only to highly multiaxial in-situ loading, but also to a full spectrum of local loading scenarios (tensile, compressive, relaxation, creep, cyclic), thus demanding highly robust, well validated, constituent constitutive models.