Micromechanics of Composite Materials (Aboudi J , Arnold S M , Bednarcyk B A )

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Micromechanics of Composite
Materials
A Generalized Multiscale Analysis Approach
Jacob Aboudi
Tel Aviv University
Steven M. Arnold
NASA Glenn Research Center
Brett A. Bednarcyk
NASA Glenn Research Center
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK
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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 NASAGlenn 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
xvii
xviii Preface
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
http://booksite.elsevier.com/9780123970350/
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.
xix
xx Acknowledgments
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
xxi
xxii Acronyms
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 2
1.2 Modeling of Composites 9
1.3 Description of the Book Layout 15
1.4 Suggestions on How to Use the Book 17
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 efficientdand
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.
Micromechanics of Composite Materials. http://dx.doi.org/10.1016/B978-0-12-397035-0.00001-X
Copyright � 2013 Elsevier Inc. All rights reserved. 1
http://dx.doi.org/10.1016/B978-0-12-397035-0.00001-X
2 Chapter 1
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
Introduction 3
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 ¼ 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).
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
Long fibers
(monofilaments)
Short fibers
(whiskers)
Particles
Woven and braided reinforcement patterns
Laminate
(a)
(b)
Figure 1.1:
Composite systems. (a) Reinforcement types. (b) Laminate and woven constructions.
4 Chapter 1
in-plane directions. Quasi-isotropic laminates have thus been very popular asdunder in-plane
elastic extensiondthey 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
Yo
un
g’
s 
m
od
ul
us
, M
si
100
10
1
200
HS C/epoxy [0°]
HS C/epoxy
[90°]
HS C/epoxy
QI lam
500
Maximum service temperature, °F
1000 2000
Te
ns
ile
 s
tre
ng
th
/d
en
si
ty
 {(
ks
i-i
n.
3 )
/lb
} 5
2
1
0.5
0.2
0.1
0.05
C/C (vf = 0.40)
C/C (vf = 0.50)
C/C (vf = 0.50)
C/C (vf = 0.40)
SiC/Ti (vf = 0.40) [90°]
SiC/SiC
(vf = 0.35–0.45)
QI lam
SiC/SiC
(vf = 0.35–0.45)
QI lam
SiC/Ti (vf = 0.40) [0°]
HS C/epoxy [0°]
HS C/epoxy [90°]
HS C/epoxy
QI lam
B/Mg (vf = 0.70) [90°]
B/Mg (vf = 0.70) [0°]
SiC/Ti (vf = 0.40) [90°]
SiC/Ti (vf = 0.40) [0°]
B/Mg (vf = 0.70) [90°]
B/Mg (vf = 0.70) [0°]
500
Maximum service temperature, °F
1000 2000
(a)
(b)
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.
Introduction 5
temperature is due to the metallic reinforcement; most CMCs are composed of ceramic
matrices and ceramic reinforcements.
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,
6 Chapter 1
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
400
350
300
250
200
150
100
50
0
Carbon
fiber
(L)
Carbon
fiber
(T)
Glass
fiber
SiC
fiber
Epoxy
matrix
Ti
matrix
Yo
un
g’
s 
m
od
ul
us
, G
Pa
MMCPMC
Carbon
fiber
(L)
Carbon
fiber
(T)
SiC
fiber
BN
fiber
coating
SiC
matrix
PMC
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.
Introduction 7
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.
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
Table 1.1: Value and Volume of Composites Produced within
Various Market Sectors
Market sector Value (%) Volume (%)
Aerospace 21 4
Building and construction 18 27
Consumer goods 10 9
Electrical and electronics 10 16
Marine 8 6
Pipe and tank 3 7
Transportation 24 28
Wind energy 6 3
8 Chapter 1
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).
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.
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
Graphite/epoxy
Graphite/polyimide
Graphite/bismalimide
B/epoxy
Material systems
Fuselage
Wings
Elevators
Stabilizers
Rudders
Ailerons
Doors
Rotors
Tank structure
Structural components
B727
elevator
DC10
rudder and
vertical
stabilizer
Beech
starship
Shuttle
cargo 
bay doors
B2
F-18
F-14
V-22
F-22
A310
A380
B777
B787
Composite
crew module
Honda jet
composite
fuselage
Ares V
Ares I
L 1011 aileron
B737 horizonal stabilizer
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)
C
om
po
si
te
 s
tru
ct
ur
al
 w
ei
gh
t, 
pe
rc
en
t
50
45
40
35
30
25
20
15
10
5
0
1960 1970 1980
Year
1990 2000 2010
F-16
F-18
F-22
F-35
AV-8B
F-15F-14
Figure 1.6:
Composite use in military aircraft. (Harris et al., 2001)
Introduction 9
10 Chapter 1
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
Structure
PlyTowFiber/inclusion
Interphase
Matrix
Mesoscale MacroscaleMicroscale
Localization
Homogenization
Localization
Homogenization
Laminate
Woven/braided
RUC
Homogenized
material
element
Figure 1.7:
Illustration of the relevant levels of scale for multiscale composite analysis.
Introduction 11
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).
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.
12 Chapter 1
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 Vand 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.
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
Level
X
Exploratory or
characterization
Testing
ValidationY
W
U
V
Figure 1.8:
Multilevel tree diagram relating constituents and structures.
Introduction 13
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
HOTFGM,
FEA, MD
M
od
el
 e
ffi
ci
en
cy
Model fidelity
Concurrent
multiscale
Analytical
Semi-analytical
Numerical
Synergistic
multiscale
Goal
Hierarchical (one-way)
multiscale
FEA
GMCMT
ROMMD
HFGMC
Figure 1.9:
Schematic illustrating the balance of multiscale model fidelity and efficiency for hierarchical,
synergistic, and concurrent multiscale composite modeling approaches.
14 Chapter 1
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.
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
Introduction 15
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 constituentsdand even within the
constituentsdbecause 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.
16 Chapter 1
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
Introduction 17
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,
18 Chapter 1
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 andMAC/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 25
2.1.1 Elasticity 26
2.1.1.1 Isotropic Hooke’s Law 26
2.1.1.2 Transversely Isotropic in Global Coordinates 28
2.1.1.3 Transversely Isotropic with Arbitrary Plane of Isotropy 29
2.1.1.4 Orthotropic Elastic 30
2.1.1.5 Anisotropic Elastic 32
2.1.2 Ramberg-Osgood Nonlinear Elastic Constitutive Equations 33
2.1.3 Viscoelasticity 34
2.1.3.1 Linear Viscoelasticity 34
2.1.3.2 Schapery Single-Integral Nonlinear Viscoelasticity 43
2.2 Irreversible Deformation Models 46
2.2.1 Incremental Plasticity 47
2.2.2 Power-Law Creep 49
2.2.3 Viscoplasticity 50
2.2.3.1 Original Bodner-Partom Model 51
2.2.3.2 A Modified Bodner-Partom Model 53
2.2.3.3 Generalized Viscoplasticity with Potential Structure (GVIPS) 53
2.3 Damage/Life Models 58
2.3.1 Continuum-Based Damage 59
2.3.1.1 Subvolume Elimination Method 61
2.3.1.2 Triaxial Stress-Driven Damage Evolution 62
2.3.1.3 Mixed-Mode Continuum Damage Mechanics (MMCDM) Model 64
2.3.1.4 Curtin-Stochastic Fiber Breakage Model 70
2.3.1.5 Combined Plasticity-Damage Model 72
2.3.1.6 Multimechanism, Viscoelastoplastic with Coupled Damage (GVIPS) 74
2.3.1.7 Cyclic Fatigue Damage Analysis (ADEAL) 76
2.3.1.8 Creep Damage 78
2.3.1.9 Creep-Fatigue Interaction 80
2.3.2 Interface Models 80
2.3.2.1 Flexible Interface Model 80
2.3.2.2 Constant Compliant Interface (CCI) Model 81
2.3.2.3 Evolving Compliant Interface (ECI) Model 83
2.4 Concluding Remarks 85
Micromechanics of Composite Materials. http://dx.doi.org/10.1016/B978-0-12-397035-0.00002-1
Copyright � 2013 Elsevier Inc. All rights reserved. 19
http://dx.doi.org/10.1016/B978-0-12-397035-0.00002-1
20 Chapter 2
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 (sij) 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).
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;
F
Equilibrium Compatibility
Constitutive
σ
u
ε
Figure 2.1:
Key aspects of solid mechanics problem.
Constituent Material Modeling 21
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 ¼ 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).
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:
sij ¼ Cijklεkl (2.1)
where sij 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,
εij ¼ εeij þ εIij þ εthij (2.2)
Strain-rate sensitivity
Small strain
recovery
Large “state”
recovery
t
Thermal recovery Dynamic recovery
Creep-plasticity interactions
Creep
Sometimes
·ε2 > ε1·
·ε1 > ε2·
Creep
σ
σ
σ
ε·ε1
·ε2
σ
σ
σ
ε
ε
t
1
1
t
Indications
of hereditary
behavior
2
2
t
ε
Small reversed
inelastic strain
σ
ε
ε
t
Relaxation
ε
t
σ
t
Figure 2.2:
Schematics showing representative hereditary material behavior at elevated temperature.
22 Chapter 2
or
ε
e
ij ¼ εij � εIij � εthij (2.3)
into three components: εeij, a reversible elastic or viscoelastic strain; ε
I
ij, an irreversible
inelastic or viscoplastic strain; and εthij , a reversible thermal strain. Substitution of Eq. (2.3)
Constituent Material Modeling 23
into Eq. (2.1) yields a stress-strain relation (known as the Generalized Hooke’s Law) that
incorporates both reversible and irreversible strains:
sij ¼ Cijkl
�
εkl � εIkl � εthkl
�
(2.4)
The thermal strain is generally taken to be linear with temperature change. Thus, full
determination of Eq. (2.4) requires only a model for εIkl. 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
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 1000 2000 3000 4000 5000
C
re
ep
 s
tra
in
, p
er
ce
nt
Time, s
Longitudinal
Transverse
Figure 2.3:
Simulated creep response of a titanium matrix composite loaded in the longitudinal and transverse
directions at 650 �C. Note that the applied load in the longitudinal direction is 145 ksi, and in the
transverse direction the applied load is 35 ksi.
0
50
250
300
0 100 200 300 500 600 700400 800
St
re
ss
, k
si
Time, s
Longitudinal matrix
Longitudinal fiber
Transverse fiber
Transverse matrix
Figure 2.4:
In-situ average constituent stress response as a function of time in the longitudinal and transverse
directions during a simulated creep test. Note that the applied load in the longitudinal direction is
145 ksi (constant), while in the transverse direction the applied load is 35 ksi (constant).
24 Chapter 2
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.
The impact of this can best be demonstrated through example using the popular, yet simple,
uniaxial Norton-Bailey power law (see Section 2.2.2). Here both creep
_εI ¼ Asn (2.5)
Constituent Material Modeling 25
)
)
)
and relaxation
_s ¼ �EAsn (2.6
behavior can be represented quite easily using the same model parameters A and n. However,
the two material coefficients A and n are obtained from completely different sets of data
(steady-state creep data or relaxation data) and are typically vastly different based on the
source. In the case of steady-state creep test data, the coefficients are obtained from
a logarithmic plot of creep strain rate versus stress:
logð_εIÞ ¼ logðAÞ þ nlogðsÞ (2.7
and in the case of relaxation test data (a stress versus time plot):
ð�nþ 1ÞlogðsÞ ¼ logððn� 1ÞEAÞ þ logðtÞ (2.8
The time scales associated with these types of tests are orders of magnitude apart. For
example, in the case of the TIMETAL 21S titanium alloy at 565 �C, the parameters A and n
obtained from relaxation tests are 4.24�10�16 (1/hr) and 4.67, whereas those obtained from
creep data are 1.13�10�16 (1/hr) and 10, respectively. Consequently, even if one were to
utilize micromechanics to predict the composite response, if one then chose to represent the
inelastic response of the matrix with a Norton-Bailey (power-law) creep model, either the
longitudinal or transverse behavior would be poorly predicted depending upon which set of
parameters were used (because in the longitudinal direction the matrix response is driven by
relaxation, and in the transverse direction the response is driven by creep).
Clearly, much more can be said regarding deformation and damage modeling as evidenced by
the large volume of literature on this topic. In this chapter, however, the discussion will be
limited to a brief description of the equations of the various reversible and irreversible
constitutive models used in the examples presented throughout the book. The corresponding
required material parameters are given later in each chapter when the specific example is
discussed. Also, in this chapter, the scope is limited to small-strain constitutive models,
because large-strain constitutive models (an advanced topic) will be discussed as required in
Chapter 9 and the more exotic material models associated with smart materials are discussed
in Chapter 10. Lastly, details regarding numerical
implementation are provided in specific
cases when the numerical integration of the model is not straightforward.
2.1 Reversible Models
All solid materials possess a domain in the stress-strain space in which the relationship
between stress and strain is fully reversible (i.e., when loaded the material deforms and when
completely unloaded all deformation is recovered). Within this domain the relationship
26 Chapter 2
between stress and strain may be linear (i.e., stress and strain are proportional) or nonlinear
(i.e., non-proportional) and either time-independent (wherein the behavior is termed elastic or
nonlinear elastic) or time-dependent (wherein the behavior is termed linear viscoelastic or
nonlinear viscoelastic). The extent of this domain is dependent upon the current homologous
temperature of the material as well as the prior load history during processing or service. As
stated before, the topic of elasticity (either time-independent or time-dependent) is
fundamental to mechanics and is well documented in the literature (e.g., Chen and Saleeb,
1982; or Lemaitre and Chaboche, 1990). Consequently, in this section we will limit the
discussion to a presentation of the essential forms of the stress-strain relations for isotropic,
transversely isotropic, orthotropic, and full anisotropic material behavior. Only in the case of
viscoelasticity was it felt that additional details needed to be provided in order to ensure that
the reader would have sufficient information to implement the model and solve problems
numerically.
2.1.1 Elasticity
2.1.1.1 Isotropic Hooke’s Law
Hooke’s Law for a thermoelastic material is given by:
sij ¼ Cijklðεkl � aijDTÞ (2.9)
where sij and εij are the stress and strain components, respectively, Cijkl are the elastic
stiffness tensor components, aij are the CTEs, DT is the temperature change from a reference
temperature, and the thermal strain is given by εthij ¼ aijDT. For isotropic materials:
Cijkl ¼ ldijdkl þ mðdikdjl þ dildjkÞ (2.10)
where l and m are the two Lamé constants that characterize the material elastic response and
dij is the Kronecker delta. The material engineering constants, Young’s modulus, E, and
Poisson’s ratio, n, are related to the Lamé constants by:
E ¼ mð3lþ 2mÞ
lþ m (2.11)
n ¼ l
2ðlþ mÞ (2.12)
The relationships among many of the common material parameters for isotropic materials are
given in Table 2.1.
Table 2.1: Relationships among Common Elastic Properties for Isotropic Materials
Young’s modulus Shear modulus Poisson’s ratio Bulk modulus Lamé constant
E G (or m) n K l
E, G E G E � 2G
2G
GE
9G� 3E
GðE � 2GÞ
3G� E
E, n E
E
2ð1þ nÞ n
E
3ð1� 2nÞ
nE
ð1þ nÞð1� 2nÞ
E, K E
3KE
9K � E
3K � E
6K
K
Kð9K � 3EÞ
9K � E
G, n 2Gð1þ nÞ G n 2Gð1þ nÞ
3ð1� 2nÞ
2Gn
1� 2n
G, K
9GK
3K þ G G
3K � 2G
2ð3K þ GÞ K K �
2G
3
G, l
Gð3lþ 2GÞ
lþ G G
l
2ðlþ GÞ lþ
2G
3
l
n, K 3Kð1� 2nÞ 3Kð1� 2nÞ
2ð1þ nÞ n K
3Kn
1þ n
K, l
9KðK � lÞ
3K � l
3ðK � lÞ
2
l
3K � l K l
Constituent Material Modeling 27
Only a single CTE is needed for an isotropic material:
aij ¼ a for i ¼ j
aij ¼ 0 for i 6¼ j (2.13)
In matrix form, isotropic Hooke’s Law can be written as:
2
666664
s11
s22
s33
s23
s13
s12
3
777775 ¼
2
6666666666666664
C11 C12 C12 0 0 0
C12 C11 C12 0 0 0
C12 C12 C11 0 0 0
0 0 0
ðC11 � C12Þ
2
0 0
0 0 0 0
ðC11 � C12Þ
2
0
0 0 0 0 0
ðC11 � C12Þ
2
3
77777777777777775
�
0
BBBBB@
2
666664
ε11
ε22
ε33
g23
g13
g12
3
777775�
2
666664
aDT
aDT
aDT
0
0
0
3
777775
1
CCCCCA
(2.14)
28 Chapter 2
where the stiffness components
C11 ¼ Eð1� nÞð1þ nÞð1� 2nÞ (2.15)
C12 ¼ Enð1þ nÞð1� 2nÞ (2.16)
are written in terms of the two independent material engineering constants, E and n. Note that
gij, i s j, are the engineering shear strain components and are related to the tensorial shear
strain components by gij¼ 2εij , is j. The normal and shear behavior of isotropic materials is
uncoupled, that is, imposing normal stress or strain results in no shear stresses or strains, and
vice versa.
2.1.1.2 Transversely Isotropic in Global Coordinates
Hooke’s Law for a transversely isotropic material, with an x2ex3 plane of isotropy, is
given by:
2
6666664
s11
s22
s33
s23
s13
s12
3
7777775
¼
2
6666666666664
C11 C12 C12 0 0 0
C12 C22 C23 0 0 0
C12 C23 C22 0 0 0
0 0 0
ðC22 � C23Þ
2
0 0
0 0 0 0 C66 0
0 0 0 0 0 C66
3
7777777777775
0
BBBBBB@
2
6666664
ε11
ε22
ε33
g23
g13
g12
3
7777775
�
2
6666664
aLDT
aTDT
aTDT
0
0
0
3
7777775
1
CCCCCCA
(2.17)
where the stiffness components can be expressed in terms of five independent constants:
EA; ET ; nA; nT ; GA (2.18)
where the subscripts A and T refer to axial and transverse properties, respectively, andG is the
shear modulus. The stiffness components are:
C11 ¼ EA þ 4cnA2 (2.19)
C12 ¼ 2cnA (2.20)
Constituent Material Modeling 29
0:5ET
C22 ¼ cþ ð1þ nTÞ (2.21)
0:5EA
C23 ¼ c� ð1þ nTÞ (2.22)
C66 ¼ GA (2.23)
with
c ¼ 0:25EA�
0:5ð1� nTÞ
�
EA
ET
�
� nA2
� (2.24)
Transversely isotropic materials retain uncoupled normal and shear behavior.
2.1.1.3 Transversely Isotropic with Arbitrary Plane of Isotropy
It is possible to express the fourth-order stiffness tensor, Cijkl, for a transversely isotropic
material for which the axis of symmetry is oriented in the direction defined by the unit vector
n ¼ (n1, n2, n3), as follows:
Cijkl ¼ ldijdkl þ mðdikdjl þ dildjkÞ þ 2ðdijnknl þ dklninjÞ
þ 4ðdiknjnl þ djkninl þ dilnjnk þ djlninkÞ þ zninjnknl
(2.25)
where l, m, 2, 4, and z are five independent constants that characterize the material. If, for
instance, n ¼ (1, 0, 0), these five constants are related to the components of the stiffness
matrix in Eq. (2.17) as follows:
l ¼ C23 (2.26)
m ¼ 1
2
ðC22 � C33Þ (2.27)
2 ¼ C12 � C33 (2.28)
4 ¼ C66 � 1
2
ðC22 � C33Þ (2.29)
30 Chapter 2
x ¼ C11 þ C22 � 2C12 � 4C66 (2.30)
with the Cij terms related to the engineering constants as given in Eqs. (2.19) to (2.24). Note
that, in the global coordinate system, the material will in general be anisotropic (see Section
2.1.1.5), with coupled normal-shear behavior.
The CTEs are given by:
aij ¼ ðaL � aTÞninj þ dijaT (2.31)
2.1.1.4 Orthotropic Elastic
Orthotropic thermoelastic materials retain uncoupled normal and shear behavior and are
characterized by nine independent elastic constants. The elastic stiffness tensor is given by
the following expression:
Cijkl ¼
X3
n¼1
fmnðdindjndkl þ dijdkndlnÞ þ lndindjndkndln
þ ynðdindjkdln þ djndikdln þ dindjldkn þ djndildknÞg
(2.32)
where
m1 ¼
1
2
ðC12 þ C13 � C23Þ (2.33)
m2 ¼
1
2
ðC12 þ C23 � C13Þ (2.34)
m3 ¼
1
2
ðC13 þ C23 � C12Þ (2.35)
y1 ¼ 1
2
ðC55 þ C66 � C44Þ (2.36)
1
y2 ¼
2
ðC44 þ C66 � C55Þ (2.37)
1
y3 ¼
2
ðC44 þ C55 � C66Þ (2.38)
Constituent Material Modeling 31
l1 ¼ C11 þ C23 þ 2C44 � ðC12 þ C13 þ 2C55 þ 2C66Þ (2.39)
l2 ¼ C22 þ C13 þ 2C55 � ðC12 þ C23 þ 2C44 þ 2C66Þ (2.40)
l3 ¼ C33 þ C12 þ 2C66 � ðC13 þ C23 þ 2C44 þ 2C55Þ (2.41)
and the components of the stiffness matrix are related to the engineering constants by:
C11 ¼ 1� n23n32
E2E3B
(2.42)
C12 ¼ n12 þ n23n13
E1E3B
(2.43)
C13 ¼ n13 þ n12n23
E1E2B
(2.44)
C22 ¼ 1� n13n31
E1E3B
(2.45)
C23 ¼ n23 þ n21n13
E1E2B
(2.46)
C33 ¼ 1� n12n21
E1E2B
(2.47)
C44 ¼ G23 (2.48)
C55 ¼ G13 (2.49)
C66 ¼ G12 (2.50)
32 Chapter 2
and
B ¼ 1� n12n21 � n23n32 � n31n13 � 2n21n32n13
E1E2E3
(2.51)
where E1, E2, and E3 are the Young’s moduli in the three orthogonal directions, nij are the
Poisson’s ratios, and Gij are the shear moduli. Note that:
n32 ¼ E3n23
E2
(2.52)
n31 ¼ E3n13
E1
(2.53)
n21 ¼ E2n12
E1
(2.54)
Thus, there are only nine independent elastic engineering constants for orthotropic materials.
In matrix form, the orthotropic Hooke’s Law is given by:
2
6666664
s11
s22
s33
s23
s13
s12
3
7777775
¼
2
6666664
C11 C12 C13 0 0 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
3
7777775
0
BBBBBB@
2
6666664
ε11
ε22
ε33
g23
g13
g12
3
7777775
�
2
6666664
a1DT
a2DT
a3DT
0
0
0
3
7777775
1
CCCCCCA
(2.55)
where ai are the three