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
Elasticity problems 35 =j了C0+,Cr+号山，Ab,Ca-CO） (2.20) 00 C0+如，Cy+Ab,ab,Ca-C}g6,b,)abab After all algebraic transformations and neglecting terms of order higher than second,there holds pqmn (2.21) JAb,CAb,Cb)db,db;=CCov(b.b.) Then,starting from two-moment characterization of the effective elasticity tensor and the corresponding homogenisation models presented in (2.15)-(2.21),the stochastic second order probabilistic moment analysis of a particular engineering composites can be carried out.In the general case,these equations lead to a rather complicated description of probabilistic moments for the effective elasticity tensor particular components. In the theory of elasticity the continuum is usually uniquely represented by its geometry and elastic properties;most often a character of these features is considered as deterministic.It has been numerically proved for the fibre composites that the influence of the elastic properties randomness on the deterministically represented geometry can be significant.The most general model of the linear elastic medium,and intuitively the nearest to the real material,is based on the assumption that both its geometry and elasticity are random fields or stochastic processes.The phenomenon of random,both interface [5,27,131,200, 225,242]and volumetric [74,316,342,353,388],non-homogeneities occur mainly in the composite materials.While the interface defects(technological inaccuracies, matrix cracks,reinforcement breaks or debonding)are important considering the fracturing of such composites,the volume heterogeneities generally follow the discrete nature of many media.The existing models of stochastic media(based on various kinds of geometrical tesselations)do not make it possible to analyse such problems and that is why a new formulation is proposed. The main idea of the proposed model is a transformation of the stochastic medium into some deterministic media with random material parameters,more useful in the numerical analysis.Such a transformation is possible provided the probabilistic characteristics of geometric dimensions and total number of defects occuring at the interfaces are given,assuming that these random fields are Gaussian with non-negative or restricted values only.All non-homogeneities introduced are divided into two groups:the stochastic interface defects (SID), which have non-zero intersections with the interface boundaries,and the volumetric stochastic defects (VSD)having no common part with any interface or external composite boundary.Further,the interphases are deterministically
Elasticity problems 35 {( ) ( )} ( ) i j i j eff mnpq eff rs r s mnpq eff r r mnpq eff mnpq eff ijkl eff rs r s ijkl eff r r ijkl eff ijkl C b C b b C C g b b db db C b C b b C C , ( ), ( )0 2 ( )0 ( ), 1 ( ), ( )0 2 ( )0 ( ), 1 + ∆ + ∆ ∆ − = ∫ + ∆ + ∆ ∆ − +∞ −∞ (2.20) After all algebraic transformations and neglecting terms of order higher than second, there holds ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ( eff ) pqmn ( eff ) ijkl Cov C ;C ( ) ( ) r s eff s mnpq eff r i j i j ijkl eff s s mnpq eff r r ijkl b C b C g b ,b db db C C Cov b ,b ( ), ( ), ( ), ( ), = ∫ ∆ ∆ = +∞ −∞ . (2.21) Then, starting from two-moment characterization of the effective elasticity tensor and the corresponding homogenisation models presented in (2.15) - (2.21), the stochastic second order probabilistic moment analysis of a particular engineering composites can be carried out. In the general case, these equations lead to a rather complicated description of probabilistic moments for the effective elasticity tensor particular components. In the theory of elasticity the continuum is usually uniquely represented by its geometry and elastic properties; most often a character of these features is considered as deterministic. It has been numerically proved for the fibre composites that the influence of the elastic properties randomness on the deterministically represented geometry can be significant. The most general model of the linear elastic medium, and intuitively the nearest to the real material, is based on the assumption that both its geometry and elasticity are random fields or stochastic processes. The phenomenon of random, both interface [5,27,131,200, 225,242] and volumetric [74,316,342,353,388], non-homogeneities occur mainly in the composite materials. While the interface defects (technological inaccuracies, matrix cracks, reinforcement breaks or debonding) are important considering the fracturing of such composites, the volume heterogeneities generally follow the discrete nature of many media. The existing models of stochastic media (based on various kinds of geometrical tesselations) do not make it possible to analyse such problems and that is why a new formulation is proposed. The main idea of the proposed model is a transformation of the stochastic medium into some deterministic media with random material parameters, more useful in the numerical analysis. Such a transformation is possible provided the probabilistic characteristics of geometric dimensions and total number of defects occuring at the interfaces are given, assuming that these random fields are Gaussian with non-negative or restricted values only. All non-homogeneities introduced are divided into two groups: the stochastic interface defects (SID), which have non-zero intersections with the interface boundaries, and the volumetric stochastic defects (VSD) having no common part with any interface or external composite boundary. Further, the interphases are deterministically
36 Computational Mechanics of Composite Materials constructed around all interface boundaries using probabilistic bounds of geometric dimensions of the SID considered.Finally,the stochastic geometry is replaced by random elastic characteristics of composite constituents thanks to a probabilistic modification of the spatial averaging method (PAM).Let us note that the formulation proposed for including the SID in the interphase region has its origin in micro-mechanical approach to the contact problems rather than in the existing interface defects models. Having so defined the composite with deterministic geometry and stochastic material properties,the stochastic boundary-value problem can be numerically solved using either the Monte Carlo simulation method,which is based on computational iterations over input random fields,or the SFEM based on second- order perturbation theory or based on spectral decomposition.The perturbation- based method has found its application to modeling of fibre-reinforced composites and,in view of its computational time savings,should be preferred. Finally,let us consider the material discontinuities located randomly on the boundaries between composite constituents(interfaces)as it is shown in Figs.2.1 and 2.2. Figure 2.1.Interface defects geometrical sample 2 Bubble Figure 2.2.A single interface defect geometric idealization Numerical model for such nonhomogeneities is based on the assumption that [193,194:
36 Computational Mechanics of Composite Materials constructed around all interface boundaries using probabilistic bounds of geometric dimensions of the SID considered. Finally, the stochastic geometry is replaced by random elastic characteristics of composite constituents thanks to a probabilistic modification of the spatial averaging method (PAM). Let us note that the formulation proposed for including the SID in the interphase region has its origin in micro-mechanical approach to the contact problems rather than in the existing interface defects models. Having so defined the composite with deterministic geometry and stochastic material properties, the stochastic boundary-value problem can be numerically solved using either the Monte Carlo simulation method, which is based on computational iterations over input random fields, or the SFEM based on secondorder perturbation theory or based on spectral decomposition. The perturbationbased method has found its application to modeling of fibre-reinforced composites and, in view of its computational time savings, should be preferred. Finally, let us consider the material discontinuities located randomly on the boundaries between composite constituents (interfaces) as it is shown in Figs. 2.1 and 2.2. Ωa-1 Ωa Figure 2.1. Interface defects geometrical sample Ω a-1 Ω a r b Bubble Figure 2.2. A single interface defect geometric idealization Numerical model for such nonhomogeneities is based on the assumption that [193,194]:
Elasticity problems 37 (1)there is a finite number of material defects on all composite interfaces;the total number of defects considered is assumed as a random parameter(with nonnegative values only)defined by its first two probabilistic moments; (2)interface defects are approximated by the semi-circles (bubbles)lying with their diameters on the interfaces;the radii of the bubbles are assumed to be the next random parameter of the problem defined by the expected value and the variance; (3)geometric dimensions of every defect belonging to any are small in comparison with the minimal distance between the I2 and I boundaries for a=3,...,n or with geometric dimensions; (4)all elastic characteristics specified above are assumed equal to 0 if x D.,for =1,2n. It should be underlined that the model introduced approximates the real defects rather precisely.In further investigations the semi-circle shape of the defects should be replaced with semi-elliptical [353]and their physical model should obey nucleation and growth phenomena [345,346]preserving a random character. However to build up the numerical procedure,the bubbles should be appropriately averaged over the interphases,which they belong to.Probabilistic averaging method is proposed in the next section to carry out this smearing. Let us consider the stochastic material non-homogeneities contained in some c.The set of the defects considered D.can be divided into three subsets D2,D and D,where D contains all the defects having a non-zero intersection with the boundary ID having zero intersection with T and T and D having a non-zero intersection with IFurther,all the defects belonging to subsets D and D are called the stochastic interface defects (SID)and those belonging to D the volumetric stochastic defects(VSD).Let us consider such and where =that with probability equal to 1,there holds DD and D(cf.Figure 2.3). D0— 0时1 a0) 丁e》 Figure 2.3.Interphase schematic representation
Elasticity problems 37 (1) there is a finite number of material defects on all composite interfaces; the total number of defects considered is assumed as a random parameter (with nonnegative values only) defined by its first two probabilistic moments; (2) interface defects are approximated by the semi-circles (bubbles) lying with their diameters on the interfaces; the radii of the bubbles are assumed to be the next random parameter of the problem defined by the expected value and the variance; (3) geometric dimensions of every defect belonging to any Ωa are small in comparison with the minimal distance between the Γ(a−2,a−1) and Γ(a−1,a) boundaries for a=3,...,n or with Ω1 geometric dimensions; (4) all elastic characteristics specified above are assumed equal to 0 if Da x ∈ , for a=1,2,...,n. It should be underlined that the model introduced approximates the real defects rather precisely. In further investigations the semi-circle shape of the defects should be replaced with semi-elliptical [353] and their physical model should obey nucleation and growth phenomena [345,346] preserving a random character. However to build up the numerical procedure, the bubbles should be appropriately averaged over the interphases, which they belong to. Probabilistic averaging method is proposed in the next section to carry out this smearing. Let us consider the stochastic material non-homogeneities contained in some Ωa ⊂ Ω . The set of the defects considered Da can be divided into three subsets Da ′ , Da ′′ and Da ′′′ , where Da ′ contains all the defects having a non-zero intersection with the boundary Γ(a −1,a) , Da ′′ having zero intersection with Γ(a −1,a) and Γ(a,a+1) , and Da ′′′ having a non-zero intersection with Γ(a,a+1) . Further, all the defects belonging to subsets Da ′ and Da ′′′ are called the stochastic interface defects (SID) and those belonging to Da ′′ the volumetric stochastic defects (VSD). Let us consider such Ωa ′ , Ωa ′′ and Ωa ′′′ , where a a a Ωa Ω = Ω′ ∪ Ω′′ ∪ ′′′ , that with probability equal to 1, there holds Da Ωa ′ ⊂ ′ , Da Ωa ′′ ⊂ ′′ and Da Ωa ′′′⊂ ′′′ (cf. Figure 2.3). Figure 2.3. Interphase schematic representation
38 Computational Mechanics of Composite Materials The subsets can be geometrically constructed using probabilistic moments of the defect parameters(their geometric dimensions and total number). To provide such a construction let us introduce random fields A(x@)and A(x;@)as upper bounds on the norms of normal vectors defined on the boundaries T and I and the boundaries of the SID belonging to D, and D respectively.Next,let us consider the upper bounds of probabilistic distributions of A(x)and A()given as follows: △=EA(x:o)]+3 Var(A(xo】 (2.22) △=EA(xo)]+3VarA(xo)】 (2.23) Thus,can be expressed in the following form: 2a=P(x)e2a:d(P,ra-la)≤△a} (2.24) 2a=P()e2a:d(P,「a.a+l)≤△g (2.25) where i=1,2 and d(P,T)denotes the distance from a point P to the contour T.Let us note that can be obtained as 2°=2-2'U2 (2.26) Deterministic spatial averaging of properties Y on continuous and disjoint subsets of is employed to formulate the probabilistic averaging method. The averaged property Y(characterizing the region is given by the following equation [65,129]: y. y(@)=a=1 (2.27) ；x∈2 whereis the two-dimensional Lesbegue measure of Deterministic averaging can be transformed to the probabilistic case only if is defined deterministically,and Y and are uncorrelated random fields.The expected value of probabilistically averaged (@)on can be derived as
38 Computational Mechanics of Composite Materials The subsets a a Ωa Ω′ , can be geometrically constructed using probabilistic Ω′′, ′′′ moments of the defect parameters (their geometric dimensions and total number). To provide such a construction let us introduce random fields ) ∆′ a (x;ω and (x;ω) ∆a ′′′ as upper bounds on the norms of normal vectors defined on the boundaries Γ(a −1,a) and Γ(a,a+1) and the boundaries of the SID belonging to Da ′ , and Da ′′′ , respectively. Next, let us consider the upper bounds of probabilistic distributions of ) ∆′ a (x;ω and ) ∆′ a ′′(x;ω given as follows: E[ ] (x;ω) 3 Var( ) (x;ω) a a ∆a ∆′ = ∆′ + ′ (2.22) E[ ] (x;ω) 3 Var( ) (x;ω) a a ∆a ∆′′′ = ∆′′′ + ′′′ (2.23) Thus, a Ωa Ω′ , can be expressed in the following form: ′′′ Ω′ a = {P(xi)∈Ωa : d(P,Γ(a−1,a) ) ≤ ∆′ a } (2.24) Ω′ a ′′ = {P(xi)∈Ωa : d(P,Γ(a,a+1) ) ≤ ∆′ a ′′} (2.25) where i=1,2 and ) d(P,Γ denotes the distance from a point P to the contour Γ . Let us note that Ωa ′′ can be obtained as a a a a Ω′′ = Ω − Ω′ ∪ Ω′′′ (2.26) Deterministic spatial averaging of properties Ya on continuous and disjoint subsets Ωa of Ω is employed to formulate the probabilistic averaging method. The averaged property (av) Y characterizing the region Ω is given by the following equation [65,129]: Ω Ω = ∑ = n a a a av Y Y ( ) 1 ; x∈Ω (2.27) where Ω is the two-dimensional Lesbegue measure of Ω . Deterministic averaging can be transformed to the probabilistic case only if Ω is defined deterministically, and Ya and Ωa are uncorrelated random fields. The expected value of probabilistically averaged ) ( ( ) ω pav Y on Ω can be derived as
Elasticity problems 39 tm@小2top.ol (2.28) and,similarly,the variance as tmo时守 var化.(o)Var(o0 (2.29) Using the above equations Young moduli are probabilistically averaged over all regions and their subsets.Finally,a primary stochastic geometry of the considered composite is replaced by the new deterministic one.In this way, the n-component composite having m interfaces with stochastic interface defects on both sides of each interface and with volume non-homogeneities can be transformed to a n+m-component structure with deterministic geometry and probabilistically defined material parameters.More detailed equations of the PAM can be derived for given stochastic parameters of interface defects (if these defects can be approximated by specific shapes -circles,hexagons or their halves for instance). Let us suppose that there is a finite element number of discontinuities in the matrix region located on the fibre-matrix interface.These discontinuities are approximated by bubbles-semicircles placed with their diameters on the interface, see Figure 2.4.The random distribution of the assumed defects is uniquely defined by the expected values and variances of the total number and radius of the bubbles; it is shown below,there is a sufficient number of parameters to obtain a complete characterization of semicircles averaged elastic constants. Using (2.28)and (2.29)one can determine the expected value and the variance of the effective Young modulus e,the terms included in the covariance matrix of this modulus and also the Poisson ratio.It yields for the expected value ase小女 (2.30)
Elasticity problems 39 [ ] [ ] ( ) [ ] ( ) 1 ( ) 1 ( ) ω ω a ω n a a pav E Y E Y E Ω Ω = ∑ = (2.28) and, similarly, the variance as ( ) ( ) ( ) ( ) ( ) 1 ( ) 1 2 ( ) ω ω a ω n a a pav Var Y Var Y Var Ω Ω = ∑ = (2.29) Using the above equations Young moduli are probabilistically averaged over all Ωa regions and their a a Ωa Ω′ , subsets. Finally, a primary stochastic geometry Ω′′, ′′′ of the considered composite is replaced by the new deterministic one. In this way, the n-component composite having m interfaces with stochastic interface defects on both sides of each interface and with volume non-homogeneities can be transformed to a n+m-component structure with deterministic geometry and probabilistically defined material parameters. More detailed equations of the PAM can be derived for given stochastic parameters of interface defects (if these defects can be approximated by specific shapes - circles, hexagons or their halves for instance). Let us suppose that there is a finite element number of discontinuities in the matrix region located on the fibre-matrix interface. These discontinuities are approximated by bubbles – semicircles placed with their diameters on the interface, see Figure 2.4. The random distribution of the assumed defects is uniquely defined by the expected values and variances of the total number and radius of the bubbles; it is shown below, there is a sufficient number of parameters to obtain a complete characterization of semicircles averaged elastic constants. Using (2.28) and (2.29) one can determine the expected value and the variance of the effective Young modulus k e , the terms included in the covariance matrix of this modulus and also the Poisson ratio. It yields for the expected value [ ] [ ]⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⋅ − ⋅ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⋅ − = Ω Ω Ω b b c E S S e E e S S S E e E c c c 2 2 2 1 [ ] 1 2 2 2 (2.30)
