Table Of ContentComputational Mesomechanics of
Composites
ComputationalMesomechanicsofComposites: Numerical analysis of the effect of microstructures of composites on their
strength and damage resistance L. Mishnaevsky Jr ©2007JohnWiley&Sons,Ltd. ISBN: 978-0-470-02764-6
Computational
Mesomechanics of
Composites
Numerical analysis of the effect of microstructures of
composites on their strength and damage resistance
LEONMISHNAEVSKYJR
RisøNationalLaboratory,TechnicalUniversityofDenmark,Roskilde,Denmark
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Mishnaevsky,L.(Leon)
Computationalmesomechanicsofcomposites:numericalanalysisoftheeffectofmicrostructuresofcomposites
ontheirstrengthanddamageresistance/LeonMishnaevsky,Jr.
p. cm.
ISBN978-0-470-02764-6(cloth)
1. Compositematerials—Mechanicalproperties—Mathematicalmodels. 2.Micromechanics—Mathematicalmodels.
3. Numericalanalysis. I. Title.
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Contents
About the Author xi
Preface xiii
Acknowledgements xvii
1 Composites 1
1.1 Classification and types of composites 1
1.2 Deformation, damage and fracture of composites: micromechanisms
and roles of phases 5
1.2.1 Particle and short fiber reinforced composites 5
1.2.2 Long fiber reinforced composites 8
1.2.3 Laminates 10
References 11
2 Mesoscale level in the mechanics of materials 13
2.1 On the definitions of scale levels: micro- and mesomechanics 13
2.2 Size effects 14
2.2.1 Brittle and quasi-brittle materials 14
2.2.2 Metals 15
2.2.3 Thin films 16
2.3 Biocomposites 17
2.3.1 Nacre 17
2.3.2 Sponge spicules 18
2.3.3 Bamboo 19
2.3.4 Teeth 20
2.3.5 Bones 21
2.4 On some concepts of the improvement of material properties 22
2.4.1 Gradient composite materials 23
2.4.2 The application of coatings 23
2.4.3 Layered metal matrix composites 23
2.4.4 Surface composites 24
2.4.5 Agglomerates of small scale inclusions and the ‘double
dispersion’ microstructures of steels 24
2.4.6 Inclusion networks 25
vi Contents
2.4.7 Interpenetrating phase composites (IPCs) 25
2.4.8 Hyperorganized structure control 26
2.4.9 Summary 27
2.5 Physical mesomechanics of materials 27
2.6 Topological and statistical description of microstructures of composites 29
References 32
3 Damage and failure of materials: concepts and methods of modeling 37
3.1 Fracture mechanics: basic concepts 37
3.1.1 Griffith theory of brittle fracture 38
3.1.2 Stress field in the vicinity of a crack 39
3.1.3 Stress intensity factor and energy release rate 39
3.1.4 J-integral and other models of plastic effects 40
3.2 Statistical theories of strength 42
3.2.1 Worst flaw and weakest link theories 42
3.2.2 Random processes and stochastic equations 44
3.2.3 Fiber bundle models and chains of fiber bundles 46
3.3 Damage mechanics 48
3.3.1 Models of elastic solids with many cracks 48
3.3.2 Phenomenological analysis of damage evolution (continuum
damage mechanics) 49
3.3.3 Micromechanical models of void growth in ductile materials 50
3.3.4 Thermodynamic damage models 51
3.3.5 Nonlocal and gradient enhanced damage models 53
3.4 Numerical modeling of damage and fracture 55
References 60
4 Microstructure–strength relationships of composites: concepts and
methods of analysis 65
4.1 Interaction between elements of microstructures: physical and
mechanical models 65
4.1.1 Theories of constrained plastic flow of ductile materials
reinforced by hard inclusions 66
4.1.2 Shear lag model and its applications 69
4.2 Multiscale modeling of materials and homogenization 71
4.2.1 Multiscale modeling 72
4.2.2 Homogenization 74
4.3 Analytical estimations and bounds of overall elastic properties of
composites 75
4.3.1 Rule-of-mixture and classical Voigt and Reuss approximations 76
4.3.2 Hashin–Shtrikman bounds 78
4.3.3 Dilute distribution model 80
4.3.4 Effective field method and Mori–Tanaka model 80
4.3.5 Composite sphere and composite cylinder assemblage 81
4.3.6 Self-consistent models and other effective medium methods 82
4.3.7 Method of cells and transformation field analysis 84
4.3.8 Incorporation of detailed microstructural information:
three-point approximation 86
Contents vii
4.3.9 Generalized continua: nonlocal and gradient-enhanced models 87
4.3.10 Nonlinear material behavior 88
4.4 Computational models of microstructures and strength of composites 89
4.4.1 Unit cell models of composites 90
4.4.2 How to incorporate real microstructures of materials into
numerical models 98
References 105
5 Computational experiments in the mechanics of materials:
concepts and tools 115
5.1 Concept of computational experiments in the mechanics of materials 115
5.2 Input data for the simulations: determination of material properties 116
5.2.1 Nanoindentation 117
5.2.2 In-situ experiments using a scanning electron microscope 117
5.2.3 Inverse analysis 118
5.3 Program codes for the automatic generation of 3D microstructural
models of materials 120
5.3.1 Program Meso3D for the automatic geometry-based
generation of 3D microstructural FE models of materials 120
5.3.2 Program Voxel2FEM for the automatic voxel-based
generation of 3D microstructural FE models of materials 123
References 127
6 Numerical mesomechanical experiments: analysis of the effect of
microstructures of materials on the deformation and damage resistance 129
6.1 Finite element models of composite microstructures 130
6.2 Material properties used in the simulations 130
6.3 Damage modeling in composites with the User Defined Fields 131
6.3.1 Damage mechanisms and failure conditions 131
6.3.2 Subroutine User Defined Field 132
6.3.3 Interface debonding 133
6.4 Stability and reproducibility of the simulations 134
6.5 Effect of the amount and volume content of particles on the
deformation and damage in the composite 137
6.6 Effect of particle clustering and the gradient distribution of particles 141
6.7 Effect of the variations of particle sizes on the damage evolution 145
6.8 Ranking of microstructures and the effect of gradient orientation 147
References 149
7 Graded particle reinforced composites: effect of the parameters of
graded microstructures on the deformation and damage 151
7.1 Damage evolution in graded composites and the effect of the degree
of gradient 153
7.2 ‘Bilayer’ model of a graded composite 159
7.3 Effect of the shape and orientation of elongated particles on the
strength and damage evolution: nongraded composites 161
7.4 Effect of the shape and orientation of elongated particles on the
strength and damage evolution: graded composites 165
viii Contents
7.5 Effect of statistical variations of local strengths of reinforcing
particles and the distribution of the particle sizes 168
7.6 Combined Reuss–Voigt model and its application to the estimation
of stiffness of graded materials 172
7.6.1 Estimation of the stiffness of materials with arbitrarily
complex microstructures: combined Reuss–Voigt model 172
7.6.2 Effect of the degree of gradient on the elastic properties of
the composite 172
7.6.3 Inclined interface: effect of the orientation of interfaces on
the elastic properties 176
References 177
8 Particle clustering in composites: effect of clustering on the mechanical
behavior and damage evolution 179
8.1 Finite element modeling of the effect of clustering of particles on the
damage evolution 179
8.1.1 Numerical models of clustered microstructures and
statistical characterization of the microstructures 181
8.1.2 Simulation of damage and particle failure in clustered
microstructures 183
8.2 Analytical modeling of the effect of particle clustering on the
damage resistance 187
8.2.1 Cell array model of a composite 187
8.2.2 Probabilistic analysis of damage accumulation in a cell 191
8.2.3 Effect of particle clustering on the fracture toughness 193
References 195
9 Interpenetrating phase composites: numerical simulations of
deformation and damage 197
9.1 Geometry-based and voxel array based 3D FE model generation:
comparison 200
9.2 Gradient interpenetrating phase composites 202
9.3 Isotropic interpenetrating phase composites 207
9.3.1 Effect of the contiguity of interpenetrating phases on the
strength of composites 207
9.3.2 Porous plasticity: open form porosity 210
References 212
10 Fiber reinforced composites: numerical analysis of damage initiation
and growth 215
10.1 Modeling of strength and damage of fiber reinforced composites: a
brief overview 215
10.1.1 Shear lag based models and load redistribution schemas 215
10.1.2 Fiber bundle model and its versions 219
10.1.3 Fracture mechanics based models and crack
bridging 220
10.1.4 Micromechanical models of damage and fracture 223
Contents ix
10.2 Mesomechanical simulations of damage initiation and evolution in
fiber reinforced composites 225
10.2.1 Unit cell model and damage analysis 226
10.2.2 Numerical simulations: effect of matrix cracks on the fiber
fracture 228
10.2.3 Numerical simulations: interface damage initiation and its
interaction with matrix cracks and fiber fractures 229
References 235
11 Contact damage and wear of composite tool materials: micro–macro
relationships 239
11.1 Micromechanicalmodelingofthecontactwearofcomposites:abrief
overview 240
11.2 Mesomechanical simulations of wear of grinding wheels 243
11.3 Micro–macro dynamical transitions for the contact wear of
composites: ‘black box modeling’ approach 245
11.3.1 Model of the tool wear based on the ‘black box modeling’
approach 245
11.3.2 Effect of the loading conditions on the tool wear 250
11.3.3 Experimental verification: approach and steady state
cutting regimes 252
11.4 Microscale scattering of the tool material properties and the
macroscopic efficiency of the tool 253
11.4.1 Statistical description of tool shapes 254
11.4.2 Effect of the scattering of the tool material properties on
the efficiency of the tool 257
11.4.3 Principle of the optimal tool design 259
References 261
12 Future fields: computational mesomechanics and nanomaterials 265
References 267
13 Conclusions 269
Index 271
About the Author
Leon Mishnaevsky Jr, born in 1964 in Kiev (Ukraine),
graduatedfromtheKievCivilEngineeringUniversityin
1987,receivedhisdoctoratefromtheUSSRAcademyof
Sciencesin1991andhisDr.-Habil.degreeinMechanics
fromtheDarmstadtUniversityofTechnology(Germany)
in 2005. From 1981 till 1994 he worked at the Institute
forSuperhardMaterials,UkrainianAcademyofSciences,
Kiev.AfteraoneyearresearchstayattheViennaUniver-
sity of Technology (Austria) in 1994–1995, he joined
the State Materials Testing Institute (MPA), Univer-
sity of Stuttgart as a Humboldt fellow and later as a
ResearchAssociate.AfterbeingawardedtheHeisenberg
Fellowship of the German Research Council (DFG) in
2003, he started his work at the Darmstadt University
of Technology, sharing his time between Darmstadt and Stuttgart. The author has held
appointmentsasaVisitingResearchProfessorattheRutgersUniversity(USA),Visiting
Scholar at The University of Tokyo, and Science University of Tokyo (Japan) and
Massachusetts Institute of Technology (USA), Invited Professor at the China University
of Mining and Technology (China) and Ecole Nationale Supérieure d’Arts et Métiers
(France). His honors include Fellowships of the Japan Society for the Promotion of
Science (JSPS) and Japan Science and Technology Agency (STA), Engineering Foun-
dation Fellowship, Humboldt Fellowship, as well as the Heisenberg Fellowship of the
DFG, mentioned above. After his Habilitation, he joined the Risø National Laboratory
in Denmark as a Senior Scientist. He has published a book Damage and Fracture in
Heterogeneous Materials and over 100 research papers in the areas of computational
mechanics of materials, micromechanics, fracture and damage mechanics, tribology and
mechanical engineering.
Preface
Thestrengthanddamageresistanceofpartsandcomponents,madefromcompositemate-
rials, determine the quality and reliability of machines and devices, used in many areas
of industry and in everyday life. The mechanical properties and strength of composites
depend on their microstructures, i.e. on the content, geometries, distribution and proper-
tiesofphasesandconstituentsinthecomposites.Thisdependencecanbeusedtoimprove
the reliability, strength and damage resistance of the materials. For example, drilling
tools, produced from hard alloys with graded microstructures, exhibit four to five times
higher service life than the tools made from the alloys with homogeneous microstruc-
tures (Lisovsky, 2001). The fracture toughness and lifetime of tool steels are increased
by 30% if large primary carbides in the steels are replaced by dense dispersion of small
carbides (‘double dispersion’ structure) (Berns et al., 1998). The fracture toughness of
metal matrix composites can be doubled, if the reinforcing elements are localized in
layers,whichalternatewithlayersofunreinforcedmetalmatrix(McLellandetal.,1999).
The fracture toughness of Al O ceramics can be increased 2.3 times by introducing
2 3
SiC nanoparticles in the ceramics (Tan and Yang, 1998). There are a lot of examples
ofunusualpropertiesofbiomaterials,whicharerelatedtotheirpeculiarmicrostructures.
Forinstance,nacre,motherofpearl,whichconsistsof95%CaCO ,hasaworkoffrac-
3
ture that is 3000 times more than that of the monolithic CaCO (Ramachamndra Rao,
3
2003). The high fracture toughness of nacre is determined, among other factors, by the
brick-mortar,layeredmicrostructureofnacre,andtheinterlockingofthemineralplatelets
(Sarikaya et al., 2002; Katti et al., 2005).
Thus,serviceproperties,strengthanddamageresistanceofcompositescanbeimproved
by changing their microstructures. With the development of material technologies
(e.g.powdermetallurgy,heattreatmenttechnology,nanotechnology,etc.),theproduction
of materials with required, pre-defined microstructures became technologically possible.
Therefore,theproblemofthedeterminationoftheoptimalmicrostructuresofcomposites
has acquired practical importance.
In order to investigate interrelationships between the microstructures and strength of
composites, a lot of experimental, theoretical and numerical investigations have been
carried out. In many cases, links between some averaged parameters of microstruc-
tures (e.g. grain size, volume content of phases and distance between inclusions) and
the overall properties or strength of composites were established experimentally or
theoretically. However, not only the volume content, and other averaged properties
