Chapter 3:Adaptive Concurrent Multilevel Model for Multiscale Analysis of Composite Materials Including Damage Somnath Ghosh John B.Nordholt Professor,Department of Mechanical Engineering, The Ohio State University,Columbus,OH,USA 3.1 Introduction The past few decades have seen rapid developments in the science and technology of a variety of advanced heterogeneous materials like polymer, ceramic,or metal matrix composite,functionally graded materials,and porous materials,as well as various alloy systems.Many of these engineered materials are designed to possess optimal properties for different functions, e.g.,low weight,high strength,superior energy absorption and dissipation, high impact and penetration resistance,superior crashworthiness,better structural durability,etc.Tailoring their microstructures and properties to yield high structural efficiency has enabled these materials to provide ena- bling mission capabilities,which has been a key factor in their successful deployment in the aerospace,automotive,electronics,defense,and other industries. Reinforced composites are constituted of stiff and strong fibers,whiskers or particulates of,e.g.,glass,graphite,boron,or aluminum oxide,which are dispersed in primary phase matrix materials made of,e.g.,epoxy,steel, titanium,or aluminum.Micrographs of a silicon particulate reinforced aluminum alloy (DRA)and an epoxy matrix composite (PMC),consisting of graphite fibers,are shown in Fig.3.1.The presence of reinforcing phases generally enhances physical and mechanical properties like strength, thermal expansion coefficient,and wear resistance of the composite
Chapter 3: Adaptive Concurrent Multilevel Model for Multiscale Analysis of Composite Materials Including Damage Somnath Ghosh John B. Nordholt Professor, Department of Mechanical Engineering, The Ohio State University, Columbus, OH, USA 3.1 Introduction The past few decades have seen rapid developments in the science and technology of a variety of advanced heterogeneous materials like polymer, ceramic, or metal matrix composite, functionally graded materials, and porous materials, as well as various alloy systems. Many of these engineered materials are designed to possess optimal properties for different functions, e.g., low weight, high strength, superior energy absorption and dissipation, high impact and penetration resistance, superior crashworthiness, better structural durability, etc. Tailoring their microstructures and properties to yield high structural efficiency has enabled these materials to provide enabling mission capabilities, which has been a key factor in their successful deployment in the aerospace, automotive, electronics, defense, and other industries. Reinforced composites are constituted of stiff and strong fibers, whiskers or particulates of, e.g., glass, graphite, boron, or aluminum oxide, which are dispersed in primary phase matrix materials made of, e.g., epoxy, steel, titanium, or aluminum. Micrographs of a silicon particulate reinforced aluminum alloy (DRA) and an epoxy matrix composite (PMC), consisting of graphite fibers, are shown in Fig. 3.1. The presence of reinforcing phases generally enhances physical and mechanical properties like strength, thermal expansion coefficient, and wear resistance of the composite
84 S.Ghosh a)】 (b) 28KV X630 (c) Fig.3.1.Micrographs of(a)SiC particle-reinforced aluminum matrix composite showing particle cracking,(b)graphite-epoxy,fiber-reinforced polymer matrix composite,(c)fiber breakage in a polymer matrix composite
Fig. 3.1. Micrographs of (a) SiC particle-reinforced aluminum matrix composite showing particle cracking, (b) graphite-epoxy, fiber-reinforced polymer matrix composite, (c) fiber breakage in a polymer matrix composite 84 S. Ghosh
Chapter 3:Adaptive Concurrent Multilevel Model 85 Processing methods,like powder metallurgy or resin transfer molding, often contribute to nonuniformities in microstructural morphology,e.g.,in reinforcement spatial distribution,size or shape,or in the constituent mate- rial and interface properties.These nonuniformities can influence the degree of property enhancement.However,the presence of the nonuniform microstructural heterogeneities can have a strong adverse effect on their failure properties like fracture toughness,strain to failure,ductility,and fatigue resistance.Damage typically initiates at microstructural "weak spots"by inclusion(fiber or particle)fragmentation or decohesion at the inclusion-matrix interface.The cracks often bifurcate into the matrix and link up with other damage sites and cracks to evolve across larger scales and manifest as dominant cracks that cause structural failure.Structural failure of composite materials is thus inherently a multiple scale phenome- non.Microstructural damage mechanisms and structural failure properties are sensitive to the local variations in morphology,such as clustering, directionality,or connectivity and variations in reinforcement shape or size.Figure 3.la shows particle and matrix cracking in a SiC-reinforced DRA microstructure,and Fig.3.1c is the micrograph of a graphite-epoxy PMC showing failure by fiber breakage and matrix rupture.Experimental studies,e.g.,in [5,18],have established that particles in regions of cluster- ing or alignment have a greater propensity toward fracture. The need for robust design procedures for reliable and effective compo- site materials provides a compelling reason for the accelerated development of competent modeling methods that can account for the structure-material interaction and relate the microstructure to properties and failure character- istics.The models should accurately represent phenomena at different length scales and also optimize the computational efficiency through effec- tive multiscale domain decomposition. 3.2 Homogenization and Multiscale Models It is prudent to use the notion of multispatial scales in the analysis of com- posite materials and structures due to the inherent existence of various scales.Conventional methods of analysis have used effective properties obtained from homogenization of response at microscopic length scales.A number of analytical models have evolved within the framework of small deformation linear elasticity theory to predict homogenized macroscale constitutive response of heterogeneous materials,accounting for the char- acteristics of microstructural behavior.The underlying principle of these models is the Hill-Mandel condition of homogeneity [41],which states
Processing methods, like powder metallurgy or resin transfer molding, often contribute to nonuniformities in microstructural morphology, e.g., in reinforcement spatial distribution, size or shape, or in the constituent material and interface properties. These nonuniformities can influence the degree of property enhancement. However, the presence of the nonuniform microstructural heterogeneities can have a strong adverse effect on their failure properties like fracture toughness, strain to failure, ductility, and fatigue resistance. Damage typically initiates at microstructural “weak spots” by inclusion (fiber or particle) fragmentation or decohesion at the inclusion-matrix interface. The cracks often bifurcate into the matrix and link up with other damage sites and cracks to evolve across larger scales and manifest as dominant cracks that cause structural failure. Structural failure of composite materials is thus inherently a multiple scale phenomenon. Microstructural damage mechanisms and structural failure properties are sensitive to the local variations in morphology, such as clustering, directionality, or connectivity and variations in reinforcement shape or size. Figure 3.1a shows particle and matrix cracking in a SiC-reinforced DRA microstructure, and Fig. 3.1c is the micrograph of a graphite-epoxy PMC showing failure by fiber breakage and matrix rupture. Experimental studies, e.g., in [5, 18], have established that particles in regions of clustering or alignment have a greater propensity toward fracture. The need for robust design procedures for reliable and effective composite materials provides a compelling reason for the accelerated development of competent modeling methods that can account for the structure–material interaction and relate the microstructure to properties and failure characteristics. The models should accurately represent phenomena at different length scales and also optimize the computational efficiency through effective multiscale domain decomposition. 3.2 Homogenization and Multiscale Models It is prudent to use the notion of multispatial scales in the analysis of composite materials and structures due to the inherent existence of various scales. Conventional methods of analysis have used effective properties obtained from homogenization of response at microscopic length scales. A number of analytical models have evolved within the framework of small deformation linear elasticity theory to predict homogenized macroscale constitutive response of heterogeneous materials, accounting for the characteristics of microstructural behavior. The underlying principle of these models is the Hill–Mandel condition of homogeneity [41], which states Chapter 3: Adaptive Concurrent Multilevel Model 85
86 S.Ghosh that for large differences in microscopic and macroscopic length scales,the volume averaged strain energy is obtained as the product of the volume averaged stresses and strains in the representative volume element or RVE, 1.e, ∫no6dn=(o)=(oXe} (3.1) Here and are the general statically admissible stress field and kine- matically admissible strain field in the microstructure,respectively,and is a microstructural volume that is equal to or larger than the RVE.The repre- sentative volume element or RVE in(3.1)corresponds to a microstructural subregion that is representative of the entire microstructure in an average sense.For composites,it is assumed to contain a sufficient number of inclu- sions,which makes the effective moduli independent of assumed homoge- neous tractions or displacements on the RVE boundary.The Hill-Mandel condition introduces the notion of a homogeneous material that is energeti- cally equivalent to a heterogeneous material.Cogent reviews of various ho- mogenization models are presented in Mura [9,52].Based on the eigenstrain formulation,an equivalent inclusion method has been introduced by Eshelby [22]for stress and strain distributions in an infinite elastic medium contain- ing a homogeneous inclusion.Mori-Tanaka estimates,e.g.,in [8],consider nondilute dispersions where inclusion interaction is assumed to perturb the mean stress and strain field.Self-consistent schemes introduced by Hill [40]provide an alternative iterative methodology for obtaining mean field estimates of thermoelastic properties by placing each heterogeneity in an effective medium.Notable among the various estimates and bounds on the elastic properties are the variational approach using extremum principles by Hashin et al.[39]and Nemat-Nasser et al.[53],the probabilistic approach by Chen and Acrivos [14],the self-consistent model by Budiansky [11], the generalized self-consistent models by Christensen and Lo [16],etc. These predominantly analytical models,however,do not offer adequate resolution to capture the fluctuations in microstructural variables that have significant effects on properties.Also,arbitrary morphologies,material nonlinearities,or large property mismatches in constituent phases cannot be adequately treated. The use of computational micromechanical methods like the finite element method,boundary element method,spring lattice models,etc.has become increasingly popular for accurate prediction of stresses,strains, and other evolving variables in composite materials [9,10,83].Within the framework of computational multispatial scale analyses of heterogeneous materials,two classes of methods have emerged,depending on the nature of coupling between the scales.The first group,known as "hierarchical
that for large differences in microscopic and macroscopic length scales, the volume averaged strain energy is obtained as the product of the volume averaged stresses and strains in the representative volume element or RVE, i.e., * * ** * * d . σ ij ij ij ij ij ij ε Ω σε σ ε = = ∫Ω (3.1) Here * ij σ and * ij ε are the general statically admissible stress field and kinematically admissible strain field in the microstructure, respectively, and Ω is a microstructural volume that is equal to or larger than the RVE. The representative volume element or RVE in (3.1) corresponds to a microstructural subregion that is representative of the entire microstructure in an average sense. For composites, it is assumed to contain a sufficient number of inclusions, which makes the effective moduli independent of assumed homogeneous tractions or displacements on the RVE boundary. The Hill–Mandel condition introduces the notion of a homogeneous material that is energetically equivalent to a heterogeneous material. Cogent reviews of various homogenization models are presented in Mura [9, 52]. Based on the eigenstrain formulation, an equivalent inclusion method has been introduced by Eshelby [22] for stress and strain distributions in an infinite elastic medium containing a homogeneous inclusion. Mori–Tanaka estimates, e.g., in [8], consider nondilute dispersions where inclusion interaction is assumed to perturb the mean stress and strain field. Self-consistent schemes introduced by Hill [40] provide an alternative iterative methodology for obtaining mean field estimates of thermoelastic properties by placing each heterogeneity in an effective medium. Notable among the various estimates and bounds on the elastic properties are the variational approach using extremum principles by Hashin et al. [39] and Nemat-Nasser et al. [53], the probabilistic approach by Chen and Acrivos [14], the self-consistent model by Budiansky [11], the generalized self-consistent models by Christensen and Lo [16], etc. These predominantly analytical models, however, do not offer adequate resolution to capture the fluctuations in microstructural variables that have significant effects on properties. Also, arbitrary morphologies, material nonlinearities, or large property mismatches in constituent phases cannot be adequately treated. The use of computational micromechanical methods like the finite element method, boundary element method, spring lattice models, etc. has become increasingly popular for accurate prediction of stresses, strains, and other evolving variables in composite materials [9, 10, 83]. Within the framework of computational multispatial scale analyses of heterogeneous materials, two classes of methods have emerged, depending on the nature of coupling between the scales. The first group, known as “hierarchical 86 S. Ghosh
Chapter 3:Adaptive Concurrent Multilevel Model 87 models"[17,23,30,31,37,43,63,77,78]entails bottom-up coupling in which information is passed unidirectionally from lower to higher scales. usually in the form of effective material properties.A number of hierarchical models have incorporated the asymptotic homogenization theory developed by Benssousan [7],Sanchez-Palencia [68],and Lions [47]in conjunction with computational micromechanics models.Homogenization implicitly assumes uniformity of macroscopic field variables.Uncoupling of govern- ing equations at different scales is achieved through incorporation of periodicity boundary conditions on the microscopic representative volume elements or RVEs,implying periodic repetition of a local microstructural region.Consequently,the models are used to predict evolution of variables at the macroscopic scale using homogenized constitutive relations,as well as in the periodic microstructural RVE.The latter analysis can be con- ducted as a postprocessor to the macroscopic analysis with macroscopic strain as the input.Hierarchical multiscale computational analyses of rein- forced composites have been conducted by,e.g.,Fish et al.[23],Kikuchi et al.[37],Terada et al.[78],Tamma and Chung [17,77],and Ghosh et al. [30,31,43].Hierarchical models involving homogenization for damage in composites have also been developed by Ghosh et al.in [63,65]from the microstructural Voronoi cell FEM model,Lene et al.[21,44],Fish et al. [25],and Allen et al.[2,3,20],among others. While the "bottom-up"hierarchical models are efficient and can accu- rately predict macroscopic or averaged behavior,such as stiffness or strength,their predictive capabilities are limited with problems involving localization,failure,or instability.Macroscopic uniformity of response variables,like stresses or strains,is not a suitable assumption in regions of high gradients like free edges,interfaces,material discontinuities,or in regions of localized deformation and damage.On the other hand,RVE periodicity is unrealistic for nonuniform microstructures,e.g.,in the presence of clustering of heterogeneities or localized microscopic damage.Even with a uniform phase distribution in the microstructure,the evolution of localized stresses,strains,or damage path can violate periodicity condi- tions.Such shortcomings for composite material modeling have been dis- cussed for modeling heterogeneous materials by Pagano and Rybicki [58 67],Oden and Zohdi [55,84],Ghosh et al.[35,62,64],Fish et al.[24]. The solution of micromechanical problems in the vicinity of stress singu- larity was suggested in [58,67]in the context of composite laminates with free edges.These problems have been effectively tackled by the second class of models known as "concurrent"multiscale modeling methods [24, 29,35,36,51,55,56,58,61,62,64,67,71,79,82,84] Concurrent multiscale models differentiate between regions requiring dif- ferent resolutions to invoke two-way (bottom-up and top-down)coupling
models” [17, 23, 30, 31, 37, 43, 63, 77, 78] entails bottom-up coupling in which information is passed unidirectionally from lower to higher scales, usually in the form of effective material properties. A number of hierarchical models have incorporated the asymptotic homogenization theory developed by Benssousan [7], Sanchez-Palencia [68], and Lions [47] in conjunction with computational micromechanics models. Homogenization implicitly assumes uniformity of macroscopic field variables. Uncoupling of governing equations at different scales is achieved through incorporation of periodicity boundary conditions on the microscopic representative volume elements or RVEs, implying periodic repetition of a local microstructural region. Consequently, the models are used to predict evolution of variables at the macroscopic scale using homogenized constitutive relations, as well as in the periodic microstructural RVE. The latter analysis can be conducted as a postprocessor to the macroscopic analysis with macroscopic strain as the input. Hierarchical multiscale computational analyses of reinforced composites have been conducted by, e.g., Fish et al. [23], Kikuchi et al. [37], Terada et al. [78], Tamma and Chung [17, 77], and Ghosh et al. [30, 31, 43]. Hierarchical models involving homogenization for damage in composites have also been developed by Ghosh et al. in [63, 65] from the microstructural Voronoi cell FEM model, Lene et al. [21, 44], Fish et al. [25], and Allen et al. [2, 3, 20], among others. While the “bottom-up” hierarchical models are efficient and can accurately predict macroscopic or averaged behavior, such as stiffness or strength, their predictive capabilities are limited with problems involving localization, failure, or instability. Macroscopic uniformity of response variables, like stresses or strains, is not a suitable assumption in regions of high gradients like free edges, interfaces, material discontinuities, or in regions of localized deformation and damage. On the other hand, RVE periodicity is unrealistic for nonuniform microstructures, e.g., in the presence of clustering of heterogeneities or localized microscopic damage. Even with a uniform phase distribution in the microstructure, the evolution of localized stresses, strains, or damage path can violate periodicity conditions. Such shortcomings for composite material modeling have been discussed for modeling heterogeneous materials by Pagano and Rybicki [58, 67], Oden and Zohdi [55, 84], Ghosh et al. [35, 62, 64], Fish et al. [24]. The solution of micromechanical problems in the vicinity of stress singularity was suggested in [58, 67] in the context of composite laminates with free edges. These problems have been effectively tackled by the second class of models known as “concurrent” multiscale modeling methods [24, 29, 35, 36, 51, 55, 56, 58, 61, 62, 64, 67, 71, 79, 82, 84]. Concurrent multiscale models differentiate between regions requiring different resolutions to invoke two-way (bottom-up and top-down) coupling Chapter 3: Adaptive Concurrent Multilevel Model 87