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)