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
88 S.Ghosh of scales in the computational domain.These models provide effective means for analyzing heterogeneous materials and structures involving high solution gradients.Substructuring allows for macroscopic analysis using homogenized material properties in some parts of the domain while zoom- ing in at selected regions for detailed micromechanical modeling.Macro- scopic analysis,using bottom-up homogenization in regions of relatively benign deformation,enhances the efficiency of the computational analysis due to the reduced order models with limited information on the micro- structural morphology.The top-down localization process,on the other hand,incorporates cascading down to the microstructure in critical regions of localized damage or instability.These regions need explicit representa- tion of the local microstructure,and micromechanical analysis is con- ducted for accurately predicting localization or damage path.Microscopic computations involving complex microstructures are often intensive and computationally prohibitive.Selective microstructural analysis in the con- current setting makes the overall computational analysis feasible,provided the“zoom-in”regions are kept to a minimum. A variety of alternative methods have been explored for adaptive con- current multiscale analysis in [51,55,56,84,79,821.Concurrent multi- scale analysis using adaptive multilevel modeling with the microstructural Voronoi cell FEM model has been conducted by Ghosh et al.[29,35,36, 61,62,64]for modeling composites with free edges or with evolving damage resulting in dominant cracks.Guided by physical and mathematical considerations,the introduction of adaptive multiple scale modeling is a desirable feature for optimal selection of regions requiring different resolu- tions to minimize discretization and modeling errors.Ghosh and coworkers have also developed adaptive multilevel analysis using the microstructural Voronoi cell FEM model for modeling elastic-plastic composites with par- ticle cracking and porosities in [35]and for elastic composites with debond- ing at the fiber-matrix interface in [29,361.This chapter is devoted to a discussion of adaptive concurrent multiple scale models developed by the author for composites with and without damage. 3.3 Multilevel Computational Model for Concurrent Multiscale Analysis of Composites Without Damage A framework of an adaptive multilevel model is presented for macroscale to microscale analysis of composite materials in the absence of microstruc- tural damage.The model consists of three levels of hierarchy,as shown in Fig.3.2.These are:
of scales in the computational domain. These models provide effective means for analyzing heterogeneous materials and structures involving high solution gradients. Substructuring allows for macroscopic analysis using homogenized material properties in some parts of the domain while zooming in at selected regions for detailed micromechanical modeling. Macroscopic analysis, using bottom-up homogenization in regions of relatively benign deformation, enhances the efficiency of the computational analysis due to the reduced order models with limited information on the microstructural morphology. The top-down localization process, on the other hand, incorporates cascading down to the microstructure in critical regions of localized damage or instability. These regions need explicit representation of the local microstructure, and micromechanical analysis is conducted for accurately predicting localization or damage path. Microscopic computations involving complex microstructures are often intensive and computationally prohibitive. Selective microstructural analysis in the concurrent setting makes the overall computational analysis feasible, provided the “zoom-in” regions are kept to a minimum. A variety of alternative methods have been explored for adaptive conVoronoi cell FEM model has been conducted by Ghosh et al. [29, 35, 36, 61, 62, 64] for modeling composites with free edges or with evolving damage resulting in dominant cracks. Guided by physical and mathematical considerations, the introduction of adaptive multiple scale modeling is a desirable feature for optimal selection of regions requiring different resolutions to minimize discretization and modeling errors. Ghosh and coworkers have also developed adaptive multilevel analysis using the microstructural Voronoi cell FEM model for modeling elastic–plastic composites with particle cracking and porosities in [35] and for elastic composites with debonding at the fiber–matrix interface in [29, 36]. This chapter is devoted to a discussion of adaptive concurrent multiple scale models developed by the author for composites with and without damage. 3.3 Multilevel Computational Model for Concurrent Multiscale Analysis of Composites Without Damage A framework of an adaptive multilevel model is presented for macroscale to microscale analysis of composite materials in the absence of microstructural damage. The model consists of three levels of hierarchy, as shown in Fig. 3.2. These are: scale analysis using adaptive multilevel modeling with the microstructural current multiscale analysis in [51, 55, 56, 84, 79, 82]. Concurrent multi- 88 S. Ghosh
Chapter 3:Adaptive Concurrent Multilevel Model 89 (1)Level-0 macroscopic computational domain of Fig.3.2b using material properties that are obtained by homogenizing the material response in the microstructural RVE of Fig.3.2a. (2)Level-I computational domain of macroscopic analysis that is fol- lowed by a postprocessing operation of microscopic RVE analysis. This level,shown in Fig.3.2c,is used to decipher whether RVE-based homogenization is justified in this region. (3)Level-2 computational domain of pure microscopic analysis,where the assumption of the microscopic RVE for homogenization is not valid. (4)Intermediate transition layer sandwiched between the macroscopic (level-0/level-1)and microscopic (level-2)computational domains. LEVEL 0 图 LEVFL I LEVEL LEVEL2 LEVEL 2 。g ☒ LEVEL 1 (a) (b) (c) Fig.3.2.An adaptive two-way coupled multiscale analysis model:(a)RVE for constructing continuum models for level-0 analysis,(b)a level-0 model with adap- tive zoom-in,(c)zoomed-in level-1,level-2 and transition layers Physically motivated error indicators are developed for transitioning from macroscopic to microscopic analysis and tested against mathematically rigorous error bounds.All microstructural computations of arbitrary het- erogeneous domains are conducted using the adaptive Voronoi cell finite element model [26,34,48-501. 3.3.1 Hierarchy of Domains for Heterogeneous Materials Consider a heterogeneous domain composed of multiple phases of linear elastic materials,which occupies an open bounded domain, with a Lipschitz boundary he=U,T=.I and r, corresponds to displacement and traction boundaries,respectively.The
(1) Level-0 macroscopic computational domain of Fig. 3.2b using material properties that are obtained by homogenizing the material response in the microstructural RVE of Fig. 3.2a. (2) Level-1 computational domain of macroscopic analysis that is followed by a postprocessing operation of microscopic RVE analysis. This level, shown in Fig. 3.2c, is used to decipher whether RVE-based homogenization is justified in this region. (3) Level-2 computational domain of pure microscopic analysis, where the assumption of the microscopic RVE for homogenization is not valid. (4) Intermediate transition layer sandwiched between the macroscopic (level-0/level-1) and microscopic (level-2) computational domains. (a) (b) (c) Fig. 3.2. An adaptive two-way coupled multiscale analysis model: (a) RVE for constructing continuum models for level-0 analysis, (b) a level-0 model with adaptive zoom-in, (c) zoomed-in level-1, level-2 and transition layers Physically motivated error indicators are developed for transitioning from macroscopic to microscopic analysis and tested against mathematically rigorous error bounds. All microstructural computations of arbitrary heterogeneous domains are conducted using the adaptive Voronoi cell finite element model [26, 34, 48–50]. 3.3.1 Hierarchy of Domains for Heterogeneous Materials Consider a heterogeneous domain composed of multiple phases of linear elastic materials, which occupies an open bounded domain 3 Ω het⊂ R , with a Lipschitz boundary het ∂Ω = = Γ ΓΓ Γ u tu t ∪ ∩ , . ∅ Γu and Γt corresponds to displacement and traction boundaries, respectively. The Chapter 3: Adaptive Concurrent Multilevel Model 89
90 S.Ghosh body forces feL()and surface tractions teL()are vector- valued functions.The multilevel computational model for this domain uses problem descriptions for two types of domains. Micromechanics problem for the heterogeneous domain The micromechanics problem for the entire domain includes explicit con- sideration of multiple phases in with the location dependent elasticity tensor E(x),which is a bounded function in x that satisfies conven- tional conditions of ellipticity (positive strain energy for admissible strain fields)and symmetry.The displacement field u for the actual problem can be obtained as the solution to the conventional statement of principle of virtual work,expressed as Findu,ulr=u, such that ∫nv:E:ud2=ovd2+∫，t.vdr VveV(2ah (3.2) where V()is a space of admissible functions defined as V(2)={v:veH'(2vlr=0} (3.3) For heterogeneous materials with a distribution of different phases, such as fibers,particles,or voids,the constituent material properties E(x) may vary considerably with spatial position.Consequently,conventional finite element models are likely to incorporate inordinately large meshes for accuracy,which results in expensive computations.A regularized ver- sion of the actual problem,using homogenization methods can be of sig- nificant value in reducing the computing efforts through reduced order models. Regularized problem in a homogenized domain m A regularized solution u to the actual problem can be obtained by using a homogenized linear elasticity tensor C(x)in solving the boundary value problem,which is characterized by the principle of the virtual work: Findu,uHlr,=ū
body forces 2 het f ∈ L ( ) Ω and surface tractions 2 het t ∈ L ( ) Ω are vectorvalued functions. The multilevel computational model for this domain uses problem descriptions for two types of domains. Micromechanics problem for the heterogeneous domain Ω het The micromechanics problem for the entire domain includes explicit consideration of multiple phases in Ω het with the location dependent elasticity tensor E x( ) , which is a bounded function in R9 9× that satisfies conventional conditions of ellipticity (positive strain energy for admissible strain fields) and symmetry. The displacement field u for the actual problem can be obtained as the solution to the conventional statement of principle of virtual work, expressed as Find , | , Γu u u =u such that het het het : : d d d ( ), Ω ΩΓ t ∫ ∫∫ ∇ ∇ = ⋅ + ⋅ Γ ∀∈ vE u fv tv v V ΩΩ Ω (3.2) where V( ) Ω is a space of admissible functions defined as ( ) { : ( ); | 0}. Ω Ω =∈ = Γ u 1 V vv H v (3.3) For heterogeneous materials with a distribution of different phases, such as fibers, particles, or voids, the constituent material properties E x( ) may vary considerably with spatial position. Consequently, conventional finite element models are likely to incorporate inordinately large meshes for accuracy, which results in expensive computations. A regularized version of the actual problem, using homogenization methods can be of significant value in reducing the computing efforts through reduced order models. Regularized problem in a homogenized domain Ω hom A regularized solution H u to the actual problem can be obtained by using a homogenized linear elasticity tensor ( ) H C x in solving the boundary value problem, which is characterized by the principle of the virtual work: Find , | H H Γ =u uu u 90 S. Ghosh
Chapter 3:Adaptive Concurrent Multilevel Model 91 such that oVv:c":Vu"do-Jofvdo+tvdr Wev() (3.4) The homogenized elasticity tensor is assumed to satisfy symmetry and ellipticity conditions,and it is required to produce an admissible stress fieldo(=C:Vu)satisfying the traction boundary condition:no= t(x)VxEl,.Determination of statistically homogeneous material para- meters requires an isolated representative volume element or RVE Y(x)c3 over which averaging can be performed.The resulting field variables like stresses and strains are also statistically homogeneous in the RVE and may be obtained from volumetric averaging as g-owa,e”=(ynar.d业 (3.5) In classical methods of estimating homogenized elastic moduli C(x), the RVE is subjected to prescribed surface displacements or tractions, which in turn produce uniform stresses or strains in a homogenous me- dium.Various micromechanical theories have been proposed to predict the overall constitutive response by solving RVE-level boundary value prob- lems,followed by volumetric averaging [9,52].The scale of the RVE Y(x)is typically very small in comparison with the dimension L of the structure.The asymptotic homogenization theory,proposed in [7,47,68], is also effective in multiscale modeling of physical systems that contain multiple length scales.This method is based on asymptotic expansion of the solution fields,e.g.,displacement and stress fields,in the microscopic spatial coordinates about their respective macroscopic values.The com- posite microstructure in the RVE is assumed to be locally Y-periodic. Correspondingly,any variable fs in the RVE is also assumed to be Y-periodic,i.e.,f(x,y)=f(x,y+kY).Here y=x/g corresponds to the microscopic coordinates in Y(x).Here,<1 is a small positive num- ber representing the ratio of microscopic to macroscopic length scales,and k is a 3 x 3 array of integers.Superscript denotes the association with both length scales (x,y).In homogenization theory,the displacement field is asymptotically expanded aboutx with respect to the parameter as
such that hom hom hom : : d d d ( ). H H Ω Ω Γ ∫ ∫∫ ∇ ∇ = ⋅ + ⋅ ∀∈ Ω ΩΓ Ω t v u fv tv v V C (3.4) The homogenized elasticity tensor is assumed to satisfy symmetry and ellipticity conditions, and it is required to produce an admissible stress field ( :) HHH σ = ∇ C u satisfying the traction boundary condition: H n⋅ = σ t 3 over which averaging can be performed. The resulting field variables like stresses and strains are also statistically homogeneous in the RVE and may be obtained from volumetric averaging as 1 1 ( )d , ( )d , | | d . | | | | H H Y YY Y Y Y Y Y Y σ σε ε == = ∫ ∫∫ y y (3.5) In classical methods of estimating homogenized elastic moduli ( ) H C x , the RVE is subjected to prescribed surface displacements or tractions, which in turn produce uniform stresses or strains in a homogenous medium. Various micromechanical theories have been proposed to predict the overall constitutive response by solving RVE-level boundary value problems, followed by volumetric averaging [9, 52]. The scale of the RVE Y( ) x is typically very small in comparison with the dimension L of the structure. The asymptotic homogenization theory, proposed in [7, 47, 68], is also effective in multiscale modeling of physical systems that contain multiple length scales. This method is based on asymptotic expansion of the solution fields, e.g., displacement and stress fields, in the microscopic Correspondingly, any variable ε f in the RVE is also assumed to be Y-periodic, i.e., (,) (, + ) ε ε f x y f = x y kY . Here y = x/ ε corresponds to the microscopic coordinates in Y( ) x . Here, ε 1 is a small positive number representing the ratio of microscopic to macroscopic length scales, and k is a 3 × 3 array of integers. Superscript ε denotes the association with both length scales (,) x y . In homogenization theory, the displacement as meters requires an isolated representative volume element or RVE Y(x)⊂ R , t( ) x x ∀ ∈Γ . Determination of statistically homogeneous material paraspatial coordinates about their respective macroscopic values. The composite microstructure in the RVE is assumed to be locally Y-periodic. field is asymptotically expanded about x with respect to the parameter ε Chapter 3: Adaptive Concurrent Multilevel Model 91
92 S.Ghosh 4(x)=4(x,y)+e4(x,y)+e24(x,y)+… (3.6) Since the stress tensor is obtained from the spatial derivative of u(x)as (..+a.)+cj(.a(. (3.7) where (3.8) ox,oy, By applying periodicity conditions on the RVE boundary,i.e., 0it is possible to decouple the govemingionnto a set of microscopic and macroscopic problems,respectively.These are: Microscopic equations a6州=0 (Equilibrium), dy (y)= (Constitutive) (3.9) The superscripts k and in(3.9)correspond to the components of the macroscopic strain that cause the microscopic stress components The subscripts i,j,etc.in this equation on the other hand correspond to micro- scopic tensor components. Macroscopic equations a远，(r) +f=0 (Equilibrium), Ox swc人+答r]-ca.C) The interscale transfer operators in these relations are defined as
0 12 2 () (,) (,) (,) . ii i i uu u u ε x xy xy xy =+ + + ε ε " (3.6) Since the stress tensor is obtained from the spatial derivative of ( ) i uε x as 0 1 22 3 (,) (,) (,) (,) (,) , ij ij ij ij ij ε σ σ σ εσ ε σ ε = ++ + + 1 xy xy xy xy xy " (3.7) where 0 0 1 1 2 0 1 2 ,,. k k k k k ij ijkl ij ijkl ij ijkl t t t t t u uu uu CC C y x y x y ε ε ε σσ σ ∂ ∂∂ ∂∂ ⎛⎞ ⎛⎞ = =+ =+ ⎜⎟ ⎜⎟ ∂ ∂∂ ∂∂ ⎝⎠ ⎝⎠ (3.8) By applying periodicity conditions on the RVE boundary, i.e., d 0 ij j Y σ n Y ∂ ∂ = ∫ , it is possible to decouple the governing equations into a set of microscopic and macroscopic problems, respectively. These are: Microscopic equations ˆ ( ) 0 (Equilibrium), ˆ ( ) kl ij j kl kl p ij ijpm kp lm m y C y ε σ χ σ δ ∂ = ∂ ⎡ ⎤ ∂ = + ⎢ ⎥ ⎢ ⎥ ∂ ⎣ ⎦ y y (3.9) Macroscopic equations 0 ( ) 0 1 ( ) d | | (Equilibrium), ij i j mn k m H ij ijkl km lm ijmn mn Y l n f x u C Y Y yx ε χ δ δ ∂Σ + = ∂ ∂ ∂ Σ = + ∂ ∂ ⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ∫ x x x (3.10) The interscale transfer operators in these relations are defined as δ (Constitutive). = C e ( ) (Constitutive). The superscripts k and l in (3.9) correspond to the components of the macroscopic strain that cause the microscopic stress components ˆ kl σ ij . The subscripts i, j, etc. in this equation on the other hand correspond to microscopic tensor components. 92 S. Ghosh