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x3) of elements of the second order infinitesimal. Consider the random modulus of volumetric strain, K(X), the shear modulus, G(X), the random Young’s modulus, E(X), and the determined Poisson ratio, ν. Corner brackets denote the averaging of random variables. The means of distribution or modules are marked E, K, and G.Modules of elasticity satisfy the following known ratios:E=9KG3K+G,ν=3K-2G6K+2G,G=E2(1+ν),K=E3(1-2ν).Tensor of random elastic modules Θ(X) in point X of micro-heterogeneous medium can be expressed through a pair of modules K(X) and G(X) or E(X) and ν. Also included are the volumetric component V and the deviation component D of the fourth rank unit tensor V |
+ |
D |
= |
I. Variations of tensor of microstructure modules of elasticity 〈Θ∘(X)〉 = |
Θ(X) − 〈(X)〉 will depend on variations of the elastic modules E∘(X), K∘(X), and G∘(X). As a result, the two notations of the tensor (X) are obtained.Θ(X)=3K(X)V+2G(X)D,Θ∘(X)=3K∘(X)V+2G∘(X)DorΘ(X)=E(X)11-2νV+11+νD,Θ∘(X)=E∘(X)11-2νV+11+νD.Tensor elements I,V, and D are expressed through δij – the Kronecker delta:Tensor contraction over the two indexes is indicated as 〈〈··〉〉. Summarized by repetitive Greek indices: (V |
·· |
D)ijkm |
= |
VijαβDαβkm. Tensors V, D are orthonormal for the contraction over the two indexes.To convert to the matrix form, the pairs of indices will be replaces according to the following rule:11→1,22→2,33→3,23→4,31→5,12→6.Uijmn=Upq,eij=ep,ij→p,mn→q.Consider random functions, which can be used for the probability distribution of microstructure values K(X), G(X), and E(X). If composite properties are discretely distributed, the indicator function of the components will be used. Discrete distribution of properties is characteristic of the two-component and multi-component composites with determined properties but random arrangement of the components, such as matrix-filler. For powdered metals and other polycrystalline solids, the microstructure elastic modules are continuously distributed over a certain interval. In this case, three kinds of probability density function are considered: the uniform distribution, the quadratic distribution and the limited normal distribution. The formulas for centered random variables distributed over a symmetrical range are derived. The moments about mean are given next to the distribution formulas.Let composite consist of two components, and λ(X) is the indicator function of the first component. This function is equal to one if the first component is found at point X, and is equal to zero if the second component is at that point. Relative content of the first component is p |
= 〈λ(X)〉. The content of the second component is then q |
= 1 − |
p. The probability density function λ(X) is expressed through the Dirac delta function δ(x): fλ(x) = |
pδ(x |
− 1) + |
qδ(x |
− 0) To calculate the moments of random variables λ(X) use the known rule for the indicator functions: λ2 |
= |
λ3 |
= |
λ4 |
= |
⋯ |
= |
λ. The central moments will be as follows:〈(X)n+1〉=pq(qn-(-p)n)or〈(X)n〉=pqn+q(-p)n.From now on λ(X) will also be used as an indicator of material continuity. Thus λ(X) is equal to one, if there is no damage in the element X, and equal to zero if X is a void or contains shrinkage porosity. For the latter grains, let us introduce the damage function ω(X).Uniform distribution over an interval [−α, |
α]:fξ(x)=12α,ifx∈[-α,α]0,ifx∉[-α,α],〈(ξ∘)2m〉=12m+1α2m.Quadratic distribution over an interval [−α, |
α]:fξ(x)=34α2(α2-x2),ifx∈[-α,α]0,ifx∉[-α,α],〈(ξ∘)2m〉=3α2m+1(2m+1)(2m+3).Limited normal distribution on an interval [− 3α, 3α]:fξ(x)=Aα2πexp-x22α2,ifx∈[-3α,3α]0,ifx∉[-3α,3α]The change of the central moments 〈(ξ0)n〉 = |
ξ∘(n, |
α) at α |
= 0.3, and α |
= 0.35 is shown in (n |
= 2, 3, …). Both graphs decrease faster than the converging sequence n−1.1. This comparison will be useful when convergence of series defining the macro properties of components is considered.Macroscopic properties of a composite are the result of statistic averaging of its microstructure properties. Tensor of medium elasticity modules C |
= 〈Θ(X)〉 is added to the tensor of adjustments h, taking into account the interaction of microstructure elements To calculate the adjustments, the formulas derived in Here R |
= −aV |
− |
bD is the integral of the second derivative of Green–Somigliana tensor over a sphere with the center in the origin of coordinates. This is an integral of singular functions, and is calculated using the limit is the degree of contraction of tensors over two indexes. U2¨=U··U,U3¨=U2¨··U,… The parameters a and b depend on the means of distribution of elastic modules of microstructure. Depending on the given conditions, formulas for a and b can be written using a pair of K and G, or E and ν, or K and ν, or G and ν. Following are different versions of the formula:a=13K+4G,b=3K+6G5G(3K+4G).a=(1+ν)(1-2ν)3E(1-ν),b=2(1+ν)(0.8-ν)3E(1-ν)a=1+ν9K(1-ν),b=0.8-ν3G(1-ν)Taking into account orthonormality of tensors V and D with respect to contraction, the tensor of adjustments is transformed as follows:h=∑n=2∞3(-3a)n-1〈(K∘)n〉V+∑n=2∞2(-2b)n-1〈(G∘)n〉D.The result is the adjustments ΔK and ΔG to average modules K and G of volumetric and shear deformations, respectively. Knowing the adjustments, let us calculate macro modules of elasticity of the composite K∼, G∼.ΔK=∑n=2∞(-3a)n-1〈(K∘)n〉,ΔG=∑n=2∞(-2b)n-1〈(G∘)n〉If the probability density function is symmetrical, then the central odd moments are equal to zero. In this case, the adjustments ΔK and ΔG to modules of elasticity areΔK=-∑m=1∞(3a)2m-1〈(K∘)2m〉,ΔG=-∑m=1∞(2b)2m-1〈(G∘)2m〉.It is more convenient to represent the elements of the series through dimensionless values. These values are normalized moment functions 〈K∘(X)n〉/Kn, 〈G∘(X)n〉/Gn, and 〈E∘(X)n〉/En. Using Eq. ΔK=K∑n=2∞-1+ν3(1-ν)n-1K∘(X)nKn,ΔG=G∑n=2∞-2(0.8-ν)3(1-ν)n-1G∘(X)nGnΔK=E3(1-2ν)∑n=2∞-(1+ν)(1-2ν)(1-ν)n-1〈(E∘)n〉En,ΔG=E2(1+ν)∑n=2∞-2(1+ν)(1.6-2ν)3(1-ν)n-1〈(E∘)n〉En.Study of series convergence should be carried out separately for each type of probability distribution. One can see that most of the standard distributions over infinite range failed to meet the conditions of convergence. For example, if the Young’s modulus E(X) is defined by normal distribution with a variation coefficient k, thenk2=〈E∘(X)2〉/E2,〈E∘(X)2m〉=k2m(2n-1)!!E2m.ΔK=-E3(1-2ν)∑m=1∞(2m-1)!!(1+ν)(1-2ν)(1-ν)2m-1k2m,ΔG=-E2(1+ν)∑m=1∞(2m-1)!!4(1+ν)(0.8-ν)3(1-ν)2m-1k2m.These series diverge with any k |
∈ [0; 1] and ν |
∈ [0; 0.5]. In this case, the natural way to ensure convergence is to use random variables distributed over a small limited interval. This is quite acceptable for the real physical values, including variations of modules of elasticity. Distribution may well reflect random microstructure properties and, under certain conditions, ensure the convergence of the method.Calculate deformational properties of the major types of composites. Tensor (ΘX) can be refined depending on the properties and microstructure and on special features that should be considered in a particular problem. Here are the most common cases.Continuous distribution of random microstructure properties is common for many construction materials. For example, in powder metals deformation and strength properties are heavily dependent on the relative content of alloying elements. The technology of adding alloying elements to the composite leads to some fluctuation of their content between grains. Random content of alloying elements is distributed continuously in the allowable range. As a result, the properties of powder composite have the same distribution.Random turns of crystallographic axes in polycrystalline metals also lead to the dispersion of elastic properties of the microstructure For composites formed by two or more components, it is also useful to consider the actual range of the components properties. This can be achieved through the continuous probability distributions. It should be noted that the existing problem definitions and solutions are focused on discrete distribution laws, usually applied to two-component composites. The development of composite models with continuous distribution of random microstructure properties is a promising research direction.Calculate macro modules of elasticity K∼ and G∼ for distribution in Eqs. will be performed with the relevant moment distribution functions.ΔK=-K∑m=1∞1+ν3(1-ν)2m-112m+1α2m,ΔG=-G∑m=1∞2(4-5ν)15(1-ν)2m-112m+1β2m.To summarize these series apply the formulaΔK=-Kαx12∑m=1∞x12m+12m+1,x1=α(1+ν)3(1-ν),ΔG=-Gβx22∑m=1∞x22m+12m+1,x2=2β(4-5ν)15(1-ν).ΔK=-K9(1-ν)2α(1+ν)212ln3(1-ν)+α(1+ν)3(1-ν)-α(1+ν)-α(1+ν)3(1-ν),ΔG=-G9(1-ν)24β(4-5ν)212ln15(1-ν)+2β(4-5ν)15(1-ν)-2β(4-5ν)-2β(4-5ν)15(1-ν)ΔK=-K∑m=1∞1+ν3(1-ν)2m-13α2m+1(2m+1)(2m+3),ΔG=-G∑m=1∞2(4-5ν)15(1-ν)2m-13β2m+1(2m+1)(2m+3).g(x)=∑m=1∞x2m+1(2m+1)(2m+3)=141-1x2ln1+x1-x+12x-x2with|x|<1.ΔK=-K·gα(1+ν)3(1-ν)·27(1-ν)2(1+ν)2,ΔG=-G·g2β(0.8-ν)3(1-ν)·27(1-ν)24(0.8-ν)2.Moment functions of this distribution are difficult to represent in the exact format, so the approximate calculations for the sum of the series Approximate calculations for four primary moments:ΔK≈-E(1+ν)3(1-ν)k21+3(1+ν)2(1-2ν)2(1-ν)2k2,ΔG≈-E(1.6-2ν)3(1-ν)k21+4(1+ν)2(1.6-2ν)23(1-ν)2k2,where k is the coefficient of variation, 〈(E∘)2〉 = |
k2E2, 〈(E)3〉 = 0, 〈(E)4〉 = 3k4E4.Let λ(X) be the indicator function of the first component. The function is equal to one if there is the first component at point X, and is equal to zero, if point X does not contain the first component. Relative content of the first component is p |
= 〈λ(X)〉, and of the second component is q |
= 1 − |
p. K1, K2, G1, and G2 are modules of elasticity of the components. ThenK(X)=λ(X)K1+(1-λ(X))K2,K=pK1+qK2,K∘(X)=λ∘(X)(K1-K2),G(X)=λ(X)G1+(1-λ(X))G2,G=pG1+qG2,G∘(X)=λ∘(X)(G1-G2),Θ∘(X)=3λ∘(X)(K1-K2)V+2λ∘(X)(G1-G2)D.ΔK=-pq(K1-K2)∑n=1∞(pn-(-q)n)sn,ΔG=-pq(G1-G2)∑n=1∞(pn-(-q)n)tn.ΔK=-(K1-K2)pqs(1-ps)(1+qs),ΔG=-(G1-G2)pqt(1-pt)(1+qt).s=3(K1-K2)3(pK1+qK2)+4(pG1+qG2),t=6(G1-G2)(pK1+qK2+2pG1+2qG2)5(pG1+qG2)(3pK1+3qK2+4pG1+4qG2). it is sufficient that max{ps, |
qs, |
pt, |
qt} < 1.For porous composites, let us use a two-component model. Assume the elastic modules of one component are equal to zero: K2 |
= 0 and G2 |
= 0. Let K1 |
= |
K and G1 |
= |
G. The parameter q will represent in this case a relative content of pores or damage of microstructure. The parameter p |
= 1 − |
q describes continuity of material. Transforming formula ΔK=-Kpqs(1-ps)(1+qs),s=3K3K+4G,ΔG=-Gpqt(1-pt)(1+qt),t=6(K+2G)5(3K+4G).Using adjustments, the macro modules of elasticity K∼ and G∼ can be calculated.K∼=pK1-qs(p+qs)(1-s),G∼=pG1-qt(p+qt)(1-t).The same result can be expressed in terms of the Young modulus and the Poisson ratio.K∼=pE3(1-2ν)1-qs(p+qs)(1-s),s=1+ν3(1-ν),G∼=pE2(1+ν)1-qt(p+qt)(1-t),t=8-10ν15(1-ν).Knowing K∼ and G∼, the macro Young modulus E∼ and the macro Poisson ratio ν˜ can be refined as follows:During the material’s lifetime, its damage q increases and its deformational properties change. show the correlations of macro Young modulus E∼, volumetric deformation K∼, shear modulus G∼, and Poisson ratio ν˜ with porosity q. The starting Young modulus of undamaged solid material is set as one.The greater is the initial Poisson ratio, the faster the changes due to increasing porosity q are accumulated. Modulus of volumetric strain reacts faster then the shear modulus. The smaller is the Poisson ratio, the greater is the relative change of volume in the process of deformation. Therefore, the change of volume depends on the material porosity to a lesser degree. With critical damage approaching, the macro modules E∼, K∼, and G∼ become negative. This happens at 30–40% damage of material. At that stage the strain curve turns downward, and micro-fractures develop.Tear and shear are considered the two basic ways of polycrystalline metal fracture Consider random indicator functions of microstructure damage for tear, ω1(X) and for shear, ω2(X).Then q1 and q2 are the relative numbers of microstructure elements that failed in tear and shear, respectively. Failed grains alter deformational and strength properties of the composite. This model assumes that the grains that lost their resistance to shear, can still withstand the tear stress. Likewise, the grains that failed in tear, still work in shear. The relative number p of undamaged elements of microstructure is calculated by the formula p |
= 1 − |
q1 |
− |
q2 |
− |
q1q2. Ignoring the small value q1q2, then p |
= 1 − |
q1 |
− |
q2.If K and G are elastic modules of solid material, then for ω1 and ω2, arrive at macro modules of elasticity K∼ and G∼.K∼=(1-q1)K1-q1s(1-q1+q1s)(1-s),G∼=(1-q2)G1-q2t(1-q2+q2t)(1-t).s=3(1-q1)K3(1-q1)K+4(1-q2)G,t=2(3(1-q1)K+6(1-q2)G)5(3(1-q1)K+4(1-q2)G).Suppose that the microstructure includes randomly positioned spherical shrinkage porosities. Random indicator function of the degree of element damage ω(X) equals to one if element X includes a porosity. The function ω(X) is zero if element X maintained continuity of material. The degree of elements’ continuity also assumed random. Shrinkage porosity leads to the loss of load capacity and reduces K, the volume modulus and G, the shear modulus of solid material in the considered grain of microstructure. Let random multiplier ξ(X) describe this change of elastic modulus for the solid material on macro level. Therefore, for each grain the modules ξ(X)K and ξ(X)G can be obtained. Assume that the probability density function ξ(X) is known from the experiment and is continuous over a certain interval. Let also 〈ξ(X)〉 = 1 − |
z, where z |
∈ [0; 1]. z parameter is a relative decrease in average microstructure modules of elasticity caused by porosity.Random function ω(X) and ξ(X) are independent of each other. Given the degree of damage, we arrive at the formula for the random microstructure elastic modules K (X) and G(X).η(X)=λ(X)+ω(X)ξ(X),〈η〉=1-qz,η∘=zλ∘+ωξ∘.K(X)=Kη(X),G(X)=Gη(X),K∘(X)=Kη∘(X),G∘(X)=Gη∘(X).Thus, the random parameter η(X) takes into account location and shrinkage porosity of damaged microstructure elements. Series for calculation of the adjustments ΔK and ΔG will become as follows:ΔK=-K(1-qz)3(1-ν)1+ν∑n=2∞-1+ν3(1-ν)(1-qz)n〈η∘(X)n〉,ΔG=-G(1-qz)3(1-ν)2(0.8-ν)∑n=2∞-2(0.8-ν)3(1-ν)(1-qz)n〈η∘(X)n〉.Moments of high order 〈η∘(X)n〉 have quite complex structure, so finding the exact sum of series can be difficult. The approximate formulas can be derived using the first four moments about mean of distribution η(X).〈(η∘)2〉=η2=z2pq+q〈(ξ∘)2〉,〈(η∘)3〉=η3=z3pq(q-p)-3zpq〈(ξ∘)2〉+q〈(ξ∘)3〉,〈(η∘)4〉=η4=z4pq(1-3pq)+6z2p2q〈(ξ∘)2〉-4z2pq〈(ξ∘)3〉+q〈(ξ∘)4〉.It yields the macro modules of elasticity K∼,G∼ for the composite.K∼=K(1-qz)1-1+ν3(1-ν)(1-qz)2η2-1+ν3(1-ν)(1-qz)η3+1+ν3(1-ν)(1-qz)2η4,G∼=G(1-qz)1-1.6-2ν3(1-ν)(1-qz)2η2-1.6-2ν3(1-ν)(1-qz)η3+1.6-2ν3(1-ν)(1-qz)2η4. can be refined depending on the distribution type of parameter of ξ(X) and the values of moment functions 〈(ξ∘)2〉, 〈(ξ∘)3〉, and 〈(ξ∘)4〉. Consider a uniform, quadratic, and normal distribution ξ(X).For uniform distribution ξ∘(X) over the segment [−α, |
α]〈(ξ∘)2〉=α2/3,〈(ξ∘)3〉=0,〈(ξ∘)4〉=α4/5,〈(η∘)2〉=z2pq+qα2/3,〈(η∘)3〉=z3pq(q-p)-3zpqα2/3,〈(η∘)4〉=z4pq(1-3pq)+6z2p2qα2/3+qα4/5.For quadratic distribution ξ∘(X) over the segment [−α, |
α]〈(ξ∘)2〉=α3/5,〈(ξ∘)3〉=0,〈(ξ∘)4〉=α5/35,〈(η∘)2〉=z2pq+qα3/5,〈(η∘)3〉=z3pq(1-2p)-3zpqα3/5,〈(η∘)4〉=z4pq(1-3pq)+6z2p2qα3/5+3qα5/35.For normal distribution ξ∘(X) the moment functions can be represented through k-coefficient of variation of ξ(X).〈(ξ∘)2〉=k2(1-z)2,〈(ξ∘)3〉=0,〈(ξ∘)4〉=3k4(1-z)4,〈(η∘)2〉=z2pq+q(1-z)2k2,〈(η∘)3〉=z3pq(q-p)-3zpq(1-z)2k2,〈(η∘)4〉=z4pq(1-3pq)+6z2p2q(1-z)2k2+3q(1-z)4k4. show the calculation results for macro modulus of volumetric strain K∼ using the formulas for the normal distribution ξ(X). Different values of the relative number of damaged grains q, average degree of porosity z, and the coefficient of variation of porosity k are considered. Macro Poisson’s ratio of the material ν |
= 0.3. The value K∼=K(k,ν,q,z) with damage as shrinkage porosity changes at a much slower rate than with damage modeled as voids in . With growth of damage q and the porosity z, the macro modules of elasticity continue to decrease. Negative K∼ indicates the onset of development of macroscopic cracks in composite. The worse is the composite quality the faster macro crack develops. shows the correlation between K∼ and the coefficient of variation of shrinkage porosity k for three average values of the degree of porosity z |
= 0.1, z |
= 0.3, and z |
= 0.5. Here ν |
= 0.3 and q |
= 0.2. shows how module K∼ changes with increasing number of damaged grains q for composites with the degree of porosity z |
= 0.1, z |
= 0.3, and z |
= 0.5. Here ν |
= 0.3 and k |
= 0.4. With higher degree of porosity the module K∼ decreases more rapidly., the top two graphs have coefficient of variation of shrinkage porosity k |
= 0.4 and small degrees of grain damage q |
= 0.1 and q |
= 0.2, respectively. The two bottom graphs in decrease faster. Coefficient of variation of shrinkage porosity for them is k |
= 0.9, and the grain damage is increased to q |
= 0.5 and q |
= 0.6. The increase of degree of grain porosity z in composites with low dispersion of the microstructure properties may not lead to the development of macro-cracks.Let us check the solution for composites with shrinkage porosity K2+p·ΔK1+q3ΔK3K2+4G2⩽K∼⩽K1-q·ΔK1+p3ΔK3K1+4G1,G2+p·ΔG1+q6ΔG(K2+2G2)5G2(3K2+4G2)⩽G∼⩽G1-q·ΔG1+p3ΔG(K1+2G1)5G1(3K1+4G1).Here K1, G1 and K2, G2 are the average modules of elasticity of the first component and the second component, respectively. The relative contents of the components are p and q |
= 1 − |
p. ΔK |
= |
K1 |
− |
K2, ΔG |
= |
G2 |
− |
G1.Consider a composite with the Young’s modulus of solid material E |
= 1 and Poisson ratio ν |
= 0.31. Let q be the part of damaged grains containing shrinkage porosity. Average degree of porosity z |
= 0.5. Variation of porosity is evenly distributed over an interval [−0.2; 0.2]. Let’s assume the undamaged grains are the first component, and the damaged grains are the second component. Then K1 |
= 0.877, G1 |
= 0.382, K2 |
= 0.438, G2 |