2 Elasticity Problems Numerical experiments devoted to multi-component and multiscale media modelling are still one of the most important part of modern computational mechanics and engineering [98,161,272,312].The main idea of this chapter in this context is to present a general approach to numerical analysis of elastostatic problems in 1D and 2D heterogeneous media [105,274,300,317]and the homogenisation method of periodic linear elastic engineering composite structures exhibiting randomness in material parameters [32,83,356,372,375].As is shown below,the effective elasticity tensor components of such structures are obtained as the closed-form equations in the deterministic approach,which enables a relatively easy extension to the stochastic analysis by the application of the second order perturbation second central probabilistic moment analysis.On the other hand, the Monte Carlo simulation approach is employed to solve the cell problem.As is known from numerous books and articles in this area,the main difficulty in homogenisation is the lack of one general model valid for various composite structures;different nature homogenised constitutive relations are derived for beams,plates,shells etc.and even for the same type of engineering structure different effective relations are fulfilled for composites with constituents of different types(with ceramic,metal or polymer matrices and so forth).That is why numerous theoretical and numerical homogenisation models of composites are developed and applied in engineering practice. All the theories in this field can be arbitrarily divided,considering especially the method and form of the final results,into two essentially different groups.The first one contains all the methods resulting in closed form equations characterizing upper and lower bounds [108,138,156,285,339]or giving direct approximations of the effective material tensors [122,123,248].In alternative,so-called cell problems are solved to calculate,on the basis of averaged stresses or strains,the final global characteristics of the composite in elastic range [11,214,304,383],for thermoelastic analysis [117],for composites with elasto-plastic [50,57,58,146,332]or visco-elasto-plastic components [366],in the case of fractured or porous structures [38,361]or damaged interfaces [224,252,358].The very recently even multiscale methods [236,340]and models have been worked out to include the atomistic scale effects in global composite characteristics [67,145].The results obtained for the first models are relatively easy and fast in computation.However, usually these approximations are not so precise as the methods based on the cell problem solutions.In this context,the decisive role of symbolic computations and the relevant computational tools (MAPLE,MATHEMATICA,MATLAB,for instance)should be underlined [64,111,268].By using the MAPLE program and any closed form equations for effective characteristics of composites as well as thanks to the stochastic second order perturbation technique (in practice of any finite order),the probabilistic moments of these characteristics can be derived and computed.The great value of such a computational technique lies in its usefulness
2 Elasticity Problems Numerical experiments devoted to multi-component and multiscale media modelling are still one of the most important part of modern computational mechanics and engineering [98,161,272,312]. The main idea of this chapter in this context is to present a general approach to numerical analysis of elastostatic problems in 1D and 2D heterogeneous media [105,274,300,317] and the homogenisation method of periodic linear elastic engineering composite structures exhibiting randomness in material parameters [32,83,356,372,375]. As is shown below, the effective elasticity tensor components of such structures are obtained as the closed-form equations in the deterministic approach, which enables a relatively easy extension to the stochastic analysis by the application of the second order perturbation second central probabilistic moment analysis. On the other hand, the Monte Carlo simulation approach is employed to solve the cell problem. As is known from numerous books and articles in this area, the main difficulty in homogenisation is the lack of one general model valid for various composite structures; different nature homogenised constitutive relations are derived for beams, plates, shells etc. and even for the same type of engineering structure different effective relations are fulfilled for composites with constituents of different types (with ceramic, metal or polymer matrices and so forth). That is why numerous theoretical and numerical homogenisation models of composites are developed and applied in engineering practice. All the theories in this field can be arbitrarily divided, considering especially the method and form of the final results, into two essentially different groups. The first one contains all the methods resulting in closed form equations characterizing upper and lower bounds [108,138,156,285,339] or giving direct approximations of the effective material tensors [122,123,248]. In alternative, so-called cell problems are solved to calculate, on the basis of averaged stresses or strains, the final global characteristics of the composite in elastic range [11,214,304,383], for thermoelastic analysis [117], for composites with elasto-plastic [50,57,58,146,332] or visco-elasto-plastic components [366], in the case of fractured or porous structures [38,361] or damaged interfaces [224,252,358]. The very recently even multiscale methods [236,340] and models have been worked out to include the atomistic scale effects in global composite characteristics [67,145]. The results obtained for the first models are relatively easy and fast in computation. However, usually these approximations are not so precise as the methods based on the cell problem solutions. In this context, the decisive role of symbolic computations and the relevant computational tools (MAPLE, MATHEMATICA, MATLAB, for instance) should be underlined [64,111,268]. By using the MAPLE program and any closed form equations for effective characteristics of composites as well as thanks to the stochastic second order perturbation technique (in practice of any finite order), the probabilistic moments of these characteristics can be derived and computed. The great value of such a computational technique lies in its usefulness
Elasticity problems 31 in stochastic sensitivity studies.The closed form probabilistic moments of the homogenised tensor make it possible to derive explicitly the sensitivity gradients with respect to the expected values and standard deviations of the original material properties of a composite. Probabilistic methods in homogenisation [116,120,141,146,259,287,378]obey (a)algebraic derivation of the effective properties,(b)Monte-Carlo simulation of the effective tensor,(c)Voronoi-tesselations of the RVE together with the relevant FEM studies,(d)the moving-window technique.The alternative stochastic second order approach to the cell problem solution,where the SFEM analysis should be applied to calculate the effective characteristics,is displayed below.Various effective elastic characteristics models proposed in the literature are extended below using the stochastic perturbation technique and verified numerically with respect to probabilistic material parameters of the composite components.The entire homogenisation methodology is illustrated with computational examples of the two-component heterogeneous bar,fibre-reinforced and layered unidirectional composites as well as the heterogeneous plate.Thanks to these experiments,the general computational algorithm for stochastic homogenisation is proposed by some necessary modifications with comparison to the relevant theoretical approach. Finally,it is observed that having analytical expressions for the effective Young modulus and their probabilistic moments,the model presented can be extended to the deterministic and stochastic structural sensitivity analysis for elastostatics or elastodynamics of the periodic composite bar structures.It can be done assuming ideal bonds between different homogeneous parts of the composites or even considering stochastic interface defects between them and introducing the interphase model due to the derivations carried out or another related microstructural phenomena both in linear an nonlinear range.In the same time, starting from the deterministic description of the homogenised structure,the effective behaviour related to any external excitations described by the stochastic processes can be obtained. 2.1 Composite Model.Interface Defects Concept The main object of the considerations is the random periodic composite structure Y with the Representative Volume Element (RVE)denoted by Let us assume that contain perfectly bonded,coherent and disjoint subsets being homogeneous(for classical fibre-reinforced composites there are two components, for instance)and let us assume that the scale between corresponding geometrical diameters of and the whole Y is described by some small parameter g>0;this parameter indexes all the tensors rewritten for the geometrical scale connected with Further,it should be mentioned that random periodic composites are considered;it is assumed that for an additional c belonging to a suitable probability space there exists such a homothety that transforms into the entire
Elasticity problems 31 in stochastic sensitivity studies. The closed form probabilistic moments of the homogenised tensor make it possible to derive explicitly the sensitivity gradients with respect to the expected values and standard deviations of the original material properties of a composite. Probabilistic methods in homogenisation [116,120,141,146,259,287,378] obey (a) algebraic derivation of the effective properties, (b) Monte-Carlo simulation of the effective tensor, (c) Voronoi-tesselations of the RVE together with the relevant FEM studies, (d) the moving-window technique. The alternative stochastic second order approach to the cell problem solution, where the SFEM analysis should be applied to calculate the effective characteristics, is displayed below. Various effective elastic characteristics models proposed in the literature are extended below using the stochastic perturbation technique and verified numerically with respect to probabilistic material parameters of the composite components. The entire homogenisation methodology is illustrated with computational examples of the two-component heterogeneous bar, fibre-reinforced and layered unidirectional composites as well as the heterogeneous plate. Thanks to these experiments, the general computational algorithm for stochastic homogenisation is proposed by some necessary modifications with comparison to the relevant theoretical approach. Finally, it is observed that having analytical expressions for the effective Young modulus and their probabilistic moments, the model presented can be extended to the deterministic and stochastic structural sensitivity analysis for elastostatics or elastodynamics of the periodic composite bar structures. It can be done assuming ideal bonds between different homogeneous parts of the composites or even considering stochastic interface defects between them and introducing the interphase model due to the derivations carried out or another related microstructural phenomena both in linear an nonlinear range. In the same time, starting from the deterministic description of the homogenised structure, the effective behaviour related to any external excitations described by the stochastic processes can be obtained. 2.1 Composite Model. Interface Defects Concept The main object of the considerations is the random periodic composite structure Y with the Representative Volume Element (RVE) denoted by Ω. Let us assume that Ω contain perfectly bonded, coherent and disjoint subsets being homogeneous (for classical fibre-reinforced composites there are two components, for instance) and let us assume that the scale between corresponding geometrical diameters of Ω and the whole Y is described by some small parameter ε>0; this parameter indexes all the tensors rewritten for the geometrical scale connected with Ω. Further, it should be mentioned that random periodic composites are considered; it is assumed that for an additional ω belonging to a suitable probability space there exists such a homothety that transforms Ω into the entire
32 Computational Mechanics of Composite Materials composite Y.In the random case this homothety is defined for all probabilistic moments of input random variables or fields considered.Next,let us introduce two different coordinate systems:y=(yy2,y3)at the microscale of the composite and x=()at the macroscale.Then,any periodic state function F defined on Ycan be expressed as F(x)=F (2.1) This definition allows a description of the macro functions (connected with the macroscale of a composite)in terms of micro functions and vice versa.Therefore, the elasticity coefficients(being homogenised)can be defined as Ciu(x)=Cu(y) (2.2) Random fields under consideration are defined in terms of geometrical as well as material properties of the composite.However the periodic microstructure as well as its macrogeometry is deterministic.Randomising different composite properties,the set of all possible realisations of a particular introduced random field have to be admissible from the physical and geometrical point of view,which is partially explained by the below relations.Let each subset contain linear- elastic and transversely isotropic materials where Young moduli and Poisson coefficients are truncated Gaussian random variables with the first two probabilistic moments specified.There holds 0<e(ro)k∞ (2.3) Ele(co)= e1;x∈2 (2.4) e;xED2 cm(ak( 2.5) -1<v(co)< (2.6 E(co明=：xe (2.7) V2;x∈22 c(c(c (2.8) Moreover,it is assumed that there are no spatial correlations between Young moduli and Poisson coefficients and the effect of Gaussian variables cutting-off in the context of (2.3)and (2.6)does not influence the relevant probabilistic moments.This assumption will be verified computationally in the numerical
32 Computational Mechanics of Composite Materials composite Y. In the random case this homothety is defined for all probabilistic moments of input random variables or fields considered. Next, let us introduce two different coordinate systems: ( ) 1 2 3 y = y , y , y at the microscale of the composite and ( ) 1 2 3 x = x , x , x at the macroscale. Then, any periodic state function F defined on Y can be expressed as ( ) ( ) y x F x F ⎟ = F ⎠ ⎞ ⎜ ⎝ ⎛ = ε ε (2.1) This definition allows a description of the macro functions (connected with the macroscale of a composite) in terms of micro functions and vice versa. Therefore, the elasticity coefficients (being homogenised) can be defined as () () x y Cijkl = Cijkl ε (2.2) Random fields under consideration are defined in terms of geometrical as well as material properties of the composite. However the periodic microstructure as well as its macrogeometry is deterministic. Randomising different composite properties, the set of all possible realisations of a particular introduced random field have to be admissible from the physical and geometrical point of view, which is partially explained by the below relations. Let each subset Ωa contain linearelastic and transversely isotropic materials where Young moduli and Poisson coefficients are truncated Gaussian random variables with the first two probabilistic moments specified. There holds 0 < e( ) x;ω < ∞ (2.3) [ ] ( ) ⎩ ⎨ ⎧ ∈Ω ∈Ω = 2 2 1 1 ; ; ; e x e x E e x ω (2.4) ( ) ( )( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 2 1 0 0 ; ; ; Vare Vare Cov e x e x i ω j ω ; i, j = 1, 2 (2.5) ( ) 2 1 −1 <ν x;ω < (2.6) [ ] ( ) ⎩ ⎨ ⎧ ∈Ω ∈Ω = 2 2 1 1 ; ; ; x x E x ν ν ν ω (2.7) ( ) ( )( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 2 1 0 0 ; ; ; ν ν ν ω ν ω Var Var Cov x x i j ; i, j = 1, 2 (2.8) Moreover, it is assumed that there are no spatial correlations between Young moduli and Poisson coefficients and the effect of Gaussian variables cutting-off in the context of (2.3) and (2.6) does not influence the relevant probabilistic moments. This assumption will be verified computationally in the numerical
Elasticity problems 33 experiments;a discussion on the cross-property correlations has been done in [315].Further,the random elasticity tensor for each component material can be defined as v(x;@) Cw)8uvv) ij,k,l=1,2 (2.9) +6.d,+dirxlh0ieortcoj Considering all the assumptions posed above,the random periodicity of Y can be understood as the existence of such a translation which,applied to enables to cover the entire Y(as a consequence,the probabilistic moments of e(x;0)and v(x;0)are periodic too).Thus,let us adopt Y as a random composite if relevant properties of the RVE are Gaussian random variables with specified first two probabilistic moments;these variables are bounded to probability spaces admissible from mechanical and physical point of view. Let us note that the probabilistic description of the elasticity simplifies significantly if the Poisson coefficient is assumed to be a deterministic function so that v(x)=v,for a=1,2,....n;xe (2.10) Finally,the random elasticity tensor field C(x is represented as follows: Cw(xω) (2.11) =co6a+a西*低5，+.l30同 v(x) Because of the linear relation between the elasticity tensor components and the Young modulus these components have the truncated Gaussian distribution and can thus be derived uniquely from their first two moments as ElCi (x:)]=Aik(a(x)-Elea(x:)] for ij.k,l=1,2,a=1,2,...n;xE (2.12) and Var(C(x:))=Ajkt(a)(x)Ajl(a (x)Var(ea(x:)) for ij,k,1=1,2,a=1,2,xe2。, (2.13) with no sum over repeating indices at the right hand side
Elasticity problems 33 experiments; a discussion on the cross-property correlations has been done in [315]. Further, the random elasticity tensor for each component material can be defined as ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω ν ω δ δ δ δ ω ν ω ν ω ν ω ω δ δ ; 2 1 ; 1 ; 1 ; 1 2 ; ( ; ) ; e x x e x x x x C x ik jl il jk ijkl ij kl + + + + − = ; i,j,k,l = 1,2 (2.9) Considering all the assumptions posed above, the random periodicity of Y can be understood as the existence of such a translation which, applied to Ω, enables to cover the entire Y (as a consequence, the probabilistic moments of e(x;ω) and ν(x;ω) are periodic too). Thus, let us adopt Y as a random composite if relevant properties of the RVE are Gaussian random variables with specified first two probabilistic moments; these variables are bounded to probability spaces admissible from mechanical and physical point of view. Let us note that the probabilistic description of the elasticity simplifies significantly if the Poisson coefficient is assumed to be a deterministic function so that a ν ( , for x) =ν a=1,2,...,n; a x∈Ω (2.10) Finally, the random elasticity tensor field ) Cijkl (x;ω is represented as follows: ( )( ) ( ) ( ) ⎩ ⎨ ⎧ ⎭ ⎬ ⎫ + + + + − = 2 1 ( ) 1 1 ( ) 1 2 ( ) ( ) ( ; ) ( ; ) x x x x e x C x ij kl ik jl il jk ijkl ν δ δ δ δ ν ν ν ω δ δ ω (2.11) Because of the linear relation between the elasticity tensor components and the Young modulus these components have the truncated Gaussian distribution and can thus be derived uniquely from their first two moments as [ ( ; )] ( ) [ ] ( ; ) E Cijkl x ω = Aijkl(a) x ⋅ E ea x ω for i,j,k,l=1,2, a=1,2,...,n; a x∈Ω (2.12) and ( ( ; )) ( ) ( ) ( ) ( ; ) Var Cijkl x ω = Aijkl(a) x Aijkl(a) x Var ea x ω for i,j,k,l=1,2, a=1,2,...,n; a x∈Ω , with no sum over repeating indices at the right hand side. (2.13)
34 Computational Mechanics of Composite Materials There holds 4国=66a+v是-2网*6.2+65：l20+同 v(x) (2.14) ij,k,l=1,2 General methodology leading to the final results of the effective elasticity tensor is to rewrite either strain energy (or complementary energy,for instance)or equilibrium equations for a homogeneous medium and the heterogeneous one. Next,the effective parameters are derived by equating corresponding expressions for the homogeneous and for the real structure.This common methodology is applied below,particular mathematical considerations are included in the next sections and only the final result useful in further general stochastic analysis is shown.The expected values for the effective elasticity tensor in the most general case can be obtained by the second order perturbation based extension as [162,208] ECg)小-门Cog)+AbCy)+5山△C")PRb)d (2.15) Using classical probability theory definitions and theorems it is obtained that jpx(6(y))db=1.jAbpx(b(y))db=0 (2.16 ∫△b'△bPRb(y)db=Comb',b)51≤r,s≤R (2.17) Therefore EC(y)]=C°y)+C“Cow6',b） (2.18) Further,the covariance matrix CovC C)of the effective elasticity tensor is calculated using its integral definition CovC':Cg） =jc-ccg-c）g6.b,), (2.19) whereas inserting the second order perturbation expansion it is found that
34 Computational Mechanics of Composite Materials There holds ( )( ) ( )2( ) 1 ( ) 1 1 ( ) 1 2 ( ) ( ) ( ) x x x x A x ijkl ij kl ik jl il jk ν δ δ δ δ ν ν ν δ δ + + + + − = i,j,k,l=1,2 (2.14) General methodology leading to the final results of the effective elasticity tensor is to rewrite either strain energy (or complementary energy, for instance) or equilibrium equations for a homogeneous medium and the heterogeneous one. Next, the effective parameters are derived by equating corresponding expressions for the homogeneous and for the real structure. This common methodology is applied below, particular mathematical considerations are included in the next sections and only the final result useful in further general stochastic analysis is shown. The expected values for the effective elasticity tensor in the most general case can be obtained by the second order perturbation based extension as [162,208] E[ ] C y ( ) C y b C y b b C pR ( ) b db eff rs ijkl eff r r s ijkl eff r ijkl eff ijkl ∫ +∞ −∞ = + ∆ + ∆ ∆ ( ), 2 ( ) ( )0 ( ), 1 ( ) ( ) ( ) (2.15) Using classical probability theory definitions and theorems it is obtained that ( ) ∫ +∞ −∞ pR b(y) db =1, ( ) ∫ +∞ −∞ ∆bpR b(y) db = 0 (2.16) ( ) ( ) r s R r s ∫ ∆b ∆b p b( ) db = Cov b ,b +∞ −∞ y ; 1 ≤ r,s ≤ R (2.17) Therefore [ ] ( ) eff rs r s ijkl eff ijkl eff E Cijkl ( ) C ( ) C Cov b ,b ( ), 2 ( ) ( )0 1 y = y + (2.18) Further, the covariance matrix ( ) ( ) ( ) ; eff pqmn eff Cov Cijkl C of the effective elasticity tensor is calculated using its integral definition ( ) ( )( ) ( ) ∫ +∞ −∞ = − − i j i j eff pqmn eff pqmn eff ijkl eff ijkl eff pqmn eff ijkl C C C C g b b db db Cov C C , ; ( ) ( )0 ( ) ( )0 ( ) ( ) (2.19) whereas inserting the second order perturbation expansion it is found that