《纺织复合材料》课程参考文献（Multiscale Modeling and Simulation of Composite Materials and Structures）Chapter 3 Adaptive Concurrent Multilevel Model for Multiscale Analysis of Composite Materials Including Damage

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 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￾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, 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 microstruc￾tural 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 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. (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, 48–50]. 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 vector￾valued 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 con￾sideration of multiple phases in Ω het with the location dependent elasticity tensor E x( ) , which is a bounded function in R9 9× 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 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 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 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 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 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 num￾ber 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 para￾spatial coordinates about their respective macroscopic values. The com￾posite 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 micro￾scopic tensor components. 92 S. Ghosh