Finite Element Analysis with Iterated Multiscale Analysis for Mechanical Parameters of Composite Materials with Multiscale Random Grains

Abstract

Based on the iterated statistically multiscale analysis (SMSA), we present the convergence of the equivalent mechanical parameters (effective moduli), obtain the error result, and prove the symmetric, positive and definite property of the equivalent mechanical parameters tensor computed by the finite element method. The numerical results show the proved results and illustrate that the SMSA-FE algorithm is a rational method for predicting the equivalent mechanical parameters of the composite material with multiscale random grains. In conclusion, we discuss the future work for the inhomogeneous composite material with multiscale random grains.

1. Introduction

Predicting the mechanical parameters of a composite material with the multiscale random grains is a very difficult problem because there are too many random grains in the composite material and the range of the scale of the grains is very large in the material field shown in Figure 1. Many studies on predicting physical and mechanical properties of composite materials with random grains have been done: the law of mixtures [1], the Hashin-Shtrikman bounds [2], the self-consistent method [3], the Eshelbys equivalent inclusion method [4] and the Mori-Tanaka method [5], microanalysis method [6], and so forth. These methods effectively promoted the development of composite materials, but they simplified the microstructure of real materials in order to reduce the computational complexity. The composite materials with large numbers of grains can be divided into two classes according to the basic configuration: the composite materials with periodic configurations, such as braided composites, and the composite materials with random distribution, such as concrete, foamed plastics. Some physical methods and mathematical methods [7–13] have been proposed to solve these problems. However, most of these techniques and methods are based on empirical, semiempirical models or based on the homogenization methods in the periodical structure. Due to the difference of basic configuration, it is difficult for them to handle the composite material with large numbers of multiscale random grains. Hence, in order to evaluate the physical and mechanical performance of the composite material with random grains, it is necessary to make use of the different advanced numerical methods.

(a)

(b)

(c)

Figure 1 

(a) Ω with multiscale grains, (b) equivalent matrix with random grains, (c) 𝜀2 cell.

In the recent decades, for the problems with the stationary random distribution, Jikov et al. [14] developed the homogenization method and proved the existence of the homogenization coefficients and the homogenization solution, however, not provided with the numerical techniques to carry out the methods for the stationary random distribution. In addition, their method only deals with the point randomly distributed, not with random grains.

For the porous medias with the random distribution, Hou and Wu [15] developed the the multiscale finite element base function method to compute these problems of the porous medias; this method is much valid to the problems with random coefficients and problems with continuous scale. As for the multiscale systems with stochastic effective, Vanden-Eijnden gave the specific step to carry the multiscale method out [16]; their methods are very effective to mainly solve the problem with time process. For the perforated domain with small holes, Wang et al. gave an effective macroscopic model for a stochastic microscopic system, and in theory, mainly proved that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution as the size of holes diminishes to zero [17]. However, the above methods are not for the equivalent performance of the composite material with large numbers of the multiscale random grains.

Duschlbauer et al. developed the homogenization method with the Mori-Tanaka scheme averaging microfields extracted for individual fibers and the finite element analysis to estimate the linear thermoelastic and thermophysical behavior of a short fiber reinforced composite material with planar random fiber arrangement [18]. Kari et al. developed a representative volume element (RVE) approach that was used to calculate effective material properties of randomly distributed short fibre composites and analyzed the properties for the volume of random short fibres [19]. Recently, Kalamkarov et al. gave an asymptotic homogenization model for the 3D grid-reinforced composite structures with the orthotropic reinforcements [20, 21], and Wang and Pan obtained the elastic property for the multiphase composites with the random microstructures [22, 23]. Their methods are the effective homogenous methods for the equivalent performance of the composites with the random grains. However, in fact, in the engineering fields, for the composite materials with a large number of multiscale random grains, such as concrete, asphalt mixture, rock mass, and foam plastics, owing to the random complexity in configuration and for that the grain scale range is very large from 10⁻¹ m to 10⁻⁶ m [24], shown in Figure 1, the above methods find it difficult to analyze the mechanical and physical performance. Hence, in order to deal with the composite materials with multiscale random grains, authors proposed a kind of statistically multiscale analysis (SMSA) method to predict the effective mechanical parameters of the composite materials with a large number of random multiscale grains [25–27]. In previous papers [25–27], we proposed an expression for predicting the equivalent mechanical parameters of a composite material with multiscale grains. This method cannot only show the macrocharacteristics and random configurations of a composite material, but also show the contribution of the small-scale grains. In addition, this method can greatly decrease the computing time for the required numerical result.

In these previous papers [25–27], we gave the multiscale formula to compute the composite material and proved that the expected displacement field is convergent to the equivalent displacement field based on the multiscale methods in the mathematical theory. However, we did not discuss the convergence, the error result, and the symmetric, positive definite property of the equivalent mechanical parameter tensor of the composite material with random grains by the SMSA-FE method. Therefore, in this paper, the convergence and the error result based on statistical multiscale analysis (SMSA) shall be presented, and the symmetric, positive definite property of the equivalent parameters tensor (the random parameters subjected to the uniform distribution) shall be proved.

The next section reviews a representation of a composite material with multiscale random grains, some results, and the SMSA-FE procedure [27]. Section 3 is devoted to proving the convergence of the equivalent mechanical parameters computed by the SMSA-FE algorithm. Section 4 obtains the error results of the SMSA-FE algorithm on iterated multiscale analysis. In Section 5, the symmetric, positive definite property of the equivalent mechanics parameters tensor computed by the SMSA-FE algorithm is proved. In Section 6, the numerical results are presented to demonstrate the validity, the convergence, and the proved results of the SMSA-FE algorithm. Finally, we discuss the future work for the inhomogeneous composites with multiscale random grains.

2. Iterated Multiscale Analysis Model and Algorithm

In the previous papers [27], the author had given the algorithm to compute the equivalent mechanical parameters in detail. In order to prove the finite element error and the convergence of the iterated multiscale computed model, we shall review the model and the algorithm in the brief.

2.1. Iterated Multiscale Analysis Model

For the brief, all of the grains are assumed as the ellipsoids. Set a domain Ω to represent a composite with multiscale random grains shown in Figure 1(a). Set Ω_𝑙 to be a set of cube cells of the size 𝜀^𝑙 shown in Figure 1(b). Based on [27], the iterated multiscale analysis model can be represented as follows. (1)Obtain the statistical data of the composites and specify the distributions 𝑃 of the ellipsoid’s parameters.(2)Set 𝑁 to denote the number of the 𝑙th scale ellipsoids in the cell 𝜀^𝑙𝑄^𝑠; we can describe the 𝑙th scale cube cell 𝜀^𝑙𝑄^𝑠 as follows: 𝜔^𝑠𝑙=𝑎^𝑠1,𝑏^𝑠1,𝑐^𝑠1,𝜃^𝑠𝑎_𝑥1𝑥21,𝜃^𝑠𝑎_𝑥11,𝜃^𝑠𝑏_𝑥1𝑥21,𝜃^𝑠𝑏_𝑥11,𝑥₁^0𝑠1,𝑥₂^0𝑠1,𝑥₃^0𝑠1…,𝜃^𝑠𝑏_𝑥1𝑁,𝑥₁^0𝑠𝑁,𝑥₂^0𝑠𝑁,𝑥₃^0𝑠𝑁,(2.1) where (𝑥₁⁰,𝑥₂⁰,𝑥₃⁰) is the center point, 𝑎 is the long axis, 𝑏 is the middle axis, 𝑐 is the short axis, and 𝜃_{𝑎𝑥1𝑥2}, 𝜃_𝑎𝑥1, 𝜃_{𝑏𝑥1𝑥2} and 𝜃_𝑏𝑥1,are the directions for the axis 𝑎 and 𝑏 of the ellipsoids, respectively. One sample 𝜔^𝑠𝑙 is shown in Figure 1(c).(3)Set the domain Ω_𝑙 to be logically composed of 𝜀^𝑙-sized samples: Ω_𝑙=⋃_{(𝜔𝑠,𝑡∈𝑍)}𝜀^𝑙(𝑄^𝑠+𝑡) shown in Figure 1(b). It can be defined as 𝜔_𝑙={𝜔^𝑠𝑙,𝑥∈𝜀𝑄^𝑠𝑙⊂Ω_𝑙}. By the SMSA-FE algorithm [27], the equivalent mechanical parameters can be predicted. Thus, the equivalent material with 𝑙th scale random grains in Ω_𝑙 can be formed.(4)Set 𝑚 to be the scale number in the composites Ω with multiscale random grains. For 𝑙=𝑚−1,…,1, using the above representation from step (2) to step (3), recursively and successively, the multiscale random model of Ω can be described.

For example, the asphalt concrete [24] can be considered as the composites with multiscale random grains, respectively. Set 𝜀²=0.01m; and 𝜀¹=0.1m, their configuration can be shown in Figure 1.

2.2. SMSA Algorithm Based on Finite Element Method

In the previous section, we introduced the equivalent composites Ω_𝑙,(𝑙=𝑚,𝑚−1,…,1) with the 𝑙th scale grains. In this section, we shall review the mathematical theory that predicts the equivalent mechanical parameters of these composites with random grains by the statistical multiscale analysis (SMSA) [27].

For the domain Ω_𝑙,(𝑙=𝑚−1,𝑚−2,…,0) shown in Figure 1(b), their corresponding elasticity equation system and the essential boundary condition can be shown as follows: 𝜕𝜕𝑥_𝑗𝑎^{𝜀𝑖𝑗ℎ𝑘}𝑥𝜀,𝜔12𝜕𝑢^𝜀ℎ(𝑥,𝜔)𝜕𝑥_𝑘+𝜕𝑢^𝜀𝑘(𝑥,𝜔)𝜕𝑥_ℎ=𝑓^𝑖(𝑥),𝑥∈Ω_𝑙,𝑢^𝜀(𝑥,𝜔)=𝑢(𝑥),𝑥∈𝜕Ω,(2.2) where 𝑖,𝑗,ℎ,𝑘=1,2,…,𝑛, 𝜔=𝜔^𝑠 for 𝑥∈Ω₁, 𝜔^𝑠∈𝑃, 𝑃 is the probability space, Ω_𝑙=⋃_{𝑠∈𝑝,𝑡∈𝑍𝑛}𝜀^𝑙(𝑄^𝑠+𝑡) shown in Figure 1, 𝑢^𝜀ℎ(𝑥,𝜔) are the displacement field, 𝑓_𝑖(𝑥) are the loads, and 𝑢(𝑥) is the boundary displacement vector.

In the paper [25–27], we had given the SMSA method [25] and established the finite element method [27] to compute the equivalent mechanical parameters. If the FE space (𝑉^ℎ0(𝑄^𝑠))^𝑛 can be established and 𝜉=𝑥/𝜀^𝑙, the equivalent mechanical parameters can be computed.

Theorem 2.1. If a composite material with random grains is subjected to the probability distribution 𝑃, the equivalent mechanical parameters of a composite material can be approximated as 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}=∑^𝑀𝑠=1𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)𝑀,(2.3) where 𝜔^𝑠∈𝑃(𝑠=1,2,…,𝑀) and 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠) is computed as 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)=1||𝑄^𝑠||_𝑄^𝑠𝑎^{𝑖𝑗ℎ𝑘}(𝜉,𝜔^𝑠)+𝑎_{𝑖𝑝ℎ𝑞}𝜀_𝑝𝑞𝑁^ℎ0𝑘𝑗(𝜉,𝜔^𝑠)𝐝𝜉,(2.4)𝐍^ℎ0_𝛼(𝜉,𝜔^𝑠)=𝐍^ℎ0_𝛼1,𝐍^ℎ0_𝛼2,…,𝐍^ℎ0_𝛼𝑛=⎛⎜⎜⎜⎜⎝𝑁^ℎ0𝛼11𝑁⋮(𝜉,𝜔^𝑠)⋯𝑁^ℎ0_𝛼1𝑛⋮_{ℎ0𝛼𝑛1}(𝜉,𝜔^𝑠)(𝜉,𝜔^𝑠)⋯𝑁^ℎ0_𝛼𝑛𝑛(𝜉,𝜔^𝑠).⎞⎟⎟⎟⎟⎠.(2.5)𝐍^ℎ0_𝛼(𝜉,𝜔^𝑠) are the FE solutions of (2.6) on unit cell 𝑄^𝑠. 𝜕𝜕𝜉_𝑗𝑎_{𝑖𝑗ℎ𝑘}(𝜉,𝜔^𝑠)𝜀_ℎ𝑘𝑁_𝛼𝑚(𝜉,𝜔^𝑠)=−𝜕𝑎^{𝛼𝑖𝑙𝑚}(𝜉,𝜔^𝑠)𝜕𝜉_𝑙𝐍_𝛼,𝜉∈𝑄^𝑠,(𝜉,𝜔^𝑠)=0,𝜉∈𝜕𝑄^𝑠.(2.6)

Therefore, the equivalent mechanical parameters can be determined by the following SMSA-FE algorithm.

SMSA-FE Algorithm (1)Specify the scale number 𝑚 of random grains in the composites and set the iterative number 𝑟=𝑚.(2)Model 𝑟-scale random grains in 𝑄^𝑠𝑟 and generate meshes of the sample domain according to the algorithm in [27].(3)If 𝑟=𝑚, 𝑎_{𝑖𝑗ℎ𝑘}(𝑥/𝜀^𝑟,𝜔^𝑠) in 𝑄^𝑠𝑟 can be indicated as follows: 𝑎_{𝑖𝑗ℎ𝑘}𝑥𝜀^𝑟,𝜔^𝑠=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑎^{1𝑖𝑗ℎ𝑘}𝑎_{2𝑖𝑗ℎ𝑘},𝑥∈𝜀^𝑟𝑄^𝑠,⋯𝑎_{𝑙𝑖𝑗ℎ𝑘},𝑥∈𝜀^𝑟𝑄^𝑠,,𝑥∈𝜀^𝑟𝑄^𝑠,(2.7) where 𝑄^𝑠⋃𝑄^𝑠⋃⋯⋃𝑄^𝑠=𝑄^𝑠𝑟 and 𝑄^𝑠⋂𝑄^𝑠⋯⋂𝑄^𝑠=𝜙, 𝜀^𝑟𝑄^𝑠𝑟⊂Ωdenotes the domain of the basic configuration, 𝜀^𝑟𝑄^𝑠 denotes the domain of the matrix, 𝜀^𝑟𝑄^𝑠 denotes the domain of the random grains in 𝜀^𝑟𝑄^𝑠𝑟, and 𝜀^𝑎𝑄^𝑠 denotes the interface domain between the grains and the matrix. Go to step (5).(4)If 𝑟<𝑚, 𝑎_{𝑖𝑗ℎ𝑘}(𝑥/𝜀^𝑟,𝜔^𝑠) in 𝑄^𝑠𝑟 can be indicated as follows: 𝑎_{𝑖𝑗ℎ𝑘}𝑥𝜀^𝑟,𝜔^𝑠=⎧⎪⎨⎪⎩𝑎̂^{ℎ0𝑖𝑗ℎ𝑘}𝜀𝑥_(𝑟+1),𝜔^𝑠,𝑥∈𝜀^𝑟𝑎_{1𝑖𝑗ℎ𝑘}𝑄^𝑠,,𝑥∈𝜀^𝑟𝑄^𝑠,(2.8) where 𝑄^𝑠⋃𝑄^𝑠=𝑄^𝑠𝑟 and 𝑄^𝑠⋂𝑄^𝑠=𝜙, 𝜀^𝑟𝑄^𝑠𝑟⊂Ω denotes the domain of the basic configuration, 𝜀^𝑟𝑄^𝑠 denotes the domain of the equivalent matrix, and 𝜀^𝑟𝑄^𝑠 denotes the domain of random grains. Go to step (5).(5)Compute the FE approximation 𝑁^ℎ0_𝛼𝑚(𝜉^𝑟,𝜔^𝑠) according to (2.6), obtain the FE approximation of sample 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜀^𝑟,𝜔^𝑠) on the 𝑟th scale according to (2.4), and compute the FE approximation of the equivalent mechanical parameter tensor {𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜀^𝑟)} on the 𝑟th scale using (2.3).(6)Set 𝑎̂_{𝑖𝑗ℎ𝑘}(𝜀^𝑟) equal to 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜀^𝑟) on the 𝑟th scale and 𝑟=𝑟−1. If 𝑟>1, go to step (2). Otherwise, the equivalent mechanical parameter tensor 𝑎̂_{𝑖𝑗ℎ𝑘}(𝜀^𝑟)(𝜀) is the equivalent mechanical parameter tensor of the composite material with multiscale random grains in Ω.

3. Convergence of SMSA-FE Algorithm

Lemma 3.1. If 𝑁^ℎ0_𝛼𝑚(𝜉,𝜔) and 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) are the finite element approximations of the random variables 𝑁_𝛼𝑚(𝜉,𝜔) and 𝑎̂_{𝑖𝑗ℎ𝑘}(𝜔), respectively, then there exists a constant 𝑀₂>0 such that |𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)|<𝑀₂.

Proof. If |𝑎_{𝑖𝑗ℎ𝑘}(𝜉,𝜔^𝑠)|<𝑀 for any sample 𝜔^𝑠∈𝑃, (2.6) has one unique finite element solution 𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)∈(𝐻¹(𝑄^𝑠))^𝑛(𝑛=2,3) such that ‖‖𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)‖‖_{(𝐻1(𝑄𝑠))𝑛}<𝐶||𝑎_{𝑖𝑗ℎ𝑘}(𝜉,𝜔^s)||_{𝐿∞(𝑄𝑠)}<𝐶𝑀₁,(3.1) where 𝐶 and 𝑀₁ are constants and independent of 𝜉 and 𝜔^𝑠.
From (2.4) and (3.1), one can obtain |||𝑎̂_{ℎ0𝑖𝑗ℎ𝑘}=(|||||𝜔^𝑠)|||1||mes(𝑄^𝑠)||_(𝑄^𝑠₎⎛⎜⎜⎝𝑎_{𝑖𝑗ℎ𝑘}(𝜉,𝜔^𝑠)+𝑎_{𝑖𝑝ℎ𝑞}(𝜉,𝜔^𝑠)12⎛⎜⎜⎝𝜕𝑁^{ℎ0𝑘𝑝𝑗}(𝜉,𝜔^𝑠)𝜕𝜉_𝑞+𝜕𝑁^{ℎ0𝑘𝑞𝑗}(𝜉,𝜔^𝑠)𝜕𝜉_𝑝<1⎞⎟⎟⎠⎞⎟⎟⎠𝐝𝜉|||||||mes(𝑄^𝑠)||‖‖𝑎^{𝑖𝑗ℎ𝑘}≤1(𝜉,𝜔^𝑠)‖‖_{𝐿∞(𝑄𝑠)}+‖‖𝑎_{𝑖𝑝ℎ𝑞}(𝜉,𝜔^𝑠)‖‖_{𝐿∞(𝑄𝑠)}‖‖𝑁_𝑘𝑗(𝜉,𝜔^𝑠)‖‖_{(𝐻1(𝑄𝑠))}^𝑛||mes(𝑄^𝑠)||||𝑀¹+𝑀₁∗𝐶𝑀₁||||mes(𝑄^𝑠)||=𝐶𝑀₁+𝑀²¹=𝑀₂,(3.2) where 𝑀₂=𝐶𝑀₁+𝑀²¹ and 𝑀₁ is independent of both the random variables 𝜔^𝑠 and the local coordinate 𝜉. Therefore, for the random variable 𝜔 in Section 2, |𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)|<𝑀₂.

Lemma 3.2. If 𝜔 is a random variable and 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) are defined as above, then one unique expected value of the equivalent mechanical parameters tensor 𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) exists in the probability space 𝑃=(𝑃_𝑎)^𝑁×(𝑃_𝑏)^𝑁×(𝑃_𝑏)^𝑁×(𝑃_{𝜃𝑎𝑥1𝑥2})^𝑁×⋯×(𝑃_𝑥0)^𝑁×(𝑃_𝑦0)^𝑁×(𝑃_𝑧0)^𝑁.

Proof. From the definition of the long axis 𝑎, the middle axis 𝑏, the short axis 𝑐, the directions of the long axis and the middle axis, 𝜃_{𝑎𝑥1𝑥2}, 𝜃_𝑎𝑥1, and 𝜃_{𝑏𝑥1𝑥2}, 𝜃_𝑏𝑥1, and the coordinates of the central points of the ellipsoids (𝑥₁⁰,𝑥₂⁰,𝑥₃⁰), their probability density functions are denoted by 𝑓_𝑎(𝑥), 𝑓_𝑏(𝑥), 𝑓_𝑐(𝑥), 𝑓_{𝜃𝑎𝑥1𝑥2}(𝑥), 𝑓_{𝜃𝑎𝑥1}(𝑥), 𝑓_{𝜃𝑏𝑥1𝑥2}(𝑥), 𝑓_{𝜃𝑏𝑥1}(𝑥), 𝑓_𝑥10(𝑥), 𝑓_𝑥20(𝑥), and 𝑓_𝑥30(𝑥), respectively. The united random variable 𝜔=(𝑎₁,𝑏₁,𝑐₁,𝜃_{𝑎𝑥1𝑥21},𝜃_𝑎𝑥11,𝜃_{𝑏𝑥1𝑥21},𝜃_𝑏𝑥11,𝑥₁⁰₁,𝑥₂⁰₁,𝑥₃⁰₁…,𝜃_{𝑏𝑥1𝑁},𝑥₁⁰_𝑁,𝑥₂⁰_𝑁,𝑥₃⁰_𝑁) has the the united probability density function 𝑓_{𝑎1,𝑏1,𝑐1,…,𝑥0𝑁,𝑦0𝑁,𝑧0𝑁}(𝜔)=𝑓^𝑁𝑎(𝜔)⋅𝑓^𝑁𝑏(𝜔)⋅𝑓^𝑁𝑐(𝜔)⋯𝑓^𝑁𝑥₀(𝜔)⋅𝑓^𝑁𝑦₀(𝜔)⋅𝑓^𝑁𝑧₀(𝜔). From Lemma 3.1, one can obtain 𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)=<𝑀_𝑃2𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)𝐝𝑃=^+∞−∞⋯^+∞−∞𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)𝑓_{𝑎1,𝑏1,𝑐1,…,𝑥0𝑁,𝑦0𝑁,𝑧0𝑁}(𝜔)𝐝𝜔^+∞−∞<𝑀₂⋯^+∞−∞𝑓_{𝑎1,𝑏1,𝑐1,…,𝑥0𝑁,𝑦0𝑁,𝑧0𝑁}(𝜔)𝐝𝜔^+∞−∞<𝑀₂⋯^+∞−∞𝑓^𝑁𝑎(𝜔)𝐝𝜔⋯^+∞−∞⋯^+∞−∞𝑓^𝑁𝑧₀(𝜔)𝐝𝜔.(3.3) Therefore, there exists one unique expected value of the equivalent mechanical parameters tensor 𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)(𝑖,𝑗,ℎ,𝑘=1,2,…,𝑛) in the probability space 𝑃.

Lemma 3.3. If 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)(𝑖,𝑗,ℎ,𝑘=1,2,…,𝑛.𝑠=1,2,….)have the expected value 𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) in the probability space and 𝜔 is the random variable, one obtain ∑^𝑀𝑠=1𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)𝑀^𝑎⋅𝑒→𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)(𝑀⟶∞).(3.4)

Proof. Because {𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠),𝑠⩾1} are the independent and identical distribution random variables and 𝑆_𝑀=^𝑀_𝑠=1𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)(𝑀=1,2,…),(3.5) from Lemma 3.2, |𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)|<∞. Set 𝑎₁=𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔); from Kolmogorov’s classical strong law of large numbers, one obtains 𝑆_𝑀𝑀−𝑎^1𝑎⋅𝑒→0(𝑀⟶∞).(3.6) Therefore, we have 𝑆_𝑀/𝑀→^𝑎⋅𝑒𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)(𝑀→∞), that is, ∑^𝑀𝑠=1𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)𝑀^𝑎⋅𝑒→𝐸_𝜔𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)(𝑀⟶∞).(3.7)

Theorem 3.4. If 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔)(𝑟=𝑚,𝑚−1,…,1) are computed as the equivalent mechanical parameter tensor of the composite material with 𝑟-scale random grains by the SMSA-FE algorithm, then the expected values of the equivalent mechanical parameters tensor 𝐸_𝜔𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) exist in the probability space 𝑃=(𝑃^𝑟𝑎)^𝑁×(𝑃^𝑟𝑏)^𝑁×(𝑃^𝑟𝑐)^𝑁×⋯×(𝑃^𝑟𝑥₀)^𝑁×(𝑃^𝑟𝑦₀)^𝑁×(𝑃^𝑟𝑧₀)^𝑁.

Proof. Set 𝑟=𝑚; define 𝑎_{𝑖𝑗ℎ𝑘}𝑥𝜀^𝑟,𝜔^𝑠=⎧⎪⎨⎪⎩𝑎^{2𝑖𝑗ℎ𝑘}𝑎_{1𝑖𝑗ℎ𝑘},𝑥∈𝜀𝑄^𝑠,,𝑥∈𝜀𝑄^𝑠,(3.8) because both 𝑎^{1𝑖𝑗ℎ𝑘} and 𝑎^{2𝑖𝑗ℎ𝑘} are constants satisfying |max{𝑎^{1𝑖𝑗ℎ𝑘},𝑎^{2𝑖𝑗ℎ𝑘}}|<𝑀₁. It is easy to see that 𝑎_{𝑖𝑗ℎ𝑘}(𝑥/𝜀^𝑟,𝜔^𝑠)(𝑖,𝑗=1,2,3) are bounded and measurable random variables. Based on Lemmas 3.1, 3.2, and 3.3, there exist the expected values 𝐸_𝜔𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) of the equivalent mechanical parameter tensor 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔) of the material with only the 𝑟th random grains. Therefore, the equivalent mechanical parameter tensor of the composites with the 𝑟th random grains can exist as follows: ∑^𝑀𝑠=1𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)𝑀^𝑎⋅𝑒→𝐸_𝜔𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)(𝑀⟶∞).(3.9) Set 𝑟=𝑟−1 and 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝑥/𝜀^(𝑟+1),𝜔^𝑠)=𝐸_𝜔𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝜔_𝑟), the mechanical parameters of the equivalent matrix material and the grain material in the equivalent composite material with the 𝑟th random grains can be obtained.
That is, 𝑎_{𝑖𝑗ℎ𝑘}𝑥𝜀^𝑟,𝜔^𝑠=⎧⎪⎨⎪⎩𝑎̂^{𝑟,ℎ0𝑖𝑗ℎ𝑘}𝑎_{1𝑖𝑗ℎ𝑘}𝜀𝑥_(𝑟+1),𝜔^𝑠,𝑥∈𝜀𝑄^𝑠,,𝑥∈𝜀𝑄^𝑠,(3.10)|𝑎_{𝑖𝑗ℎ𝑘}(𝑥/𝜀^𝑟,𝜔^𝑠)|<𝑀₁. It is easy to see that 𝑎_{𝑖𝑗ℎ𝑘}(𝑥/𝜀^𝑟,𝜔^𝑠)(𝑖,𝑗=1,2,3) are bounded and measurable random variables. By the iterated loop proof for 𝑟 as above, based on Lemmas 3.2 and 3.3, the convergence of the equivalent mechanical parameter tensor 𝐸_𝜔𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘},(𝑟=𝑚,𝑚−1,…,1) can be obtained.

4. Error Analysis for Equivalent Mechanical Parameter Tensor Computed by SMSA-FE Algorithm

Based on the SMSA-FE algorithm, if the equivalent mechanical parameter tensor is computed, three kinds of errors are considered: the homogenization error, the random error based on Monte Carlo simulation method, and the finite element computation error. For the homogenization error, the composite materials with multiscale random grains are the special cases of the random coefficient problems whose convergence was proved in [14]. For the random error, we have obtained the convergence of the equivalent mechanical parameter tensor of the composite material with multiscale random grains as above. Therefore, in the following section, we will devote to analyzing the finite element error based on SMSA.

Firstly, we give the finite element error estimation of the statistical two-scale analysis. Then we give the error estimation of the SMSA-FE algorithm.

Lemma 4.1. If 𝑁_𝛼𝑚(𝜉,𝜔^𝑠),𝛼=1,2,…𝑛, is the variational solution of (2.6) and 𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠) is the corresponding finite element solution of (2.6) and 𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)∈(𝐻²(𝑄))^𝑛, then we have ‖𝑁‖𝑁_𝛼𝑚𝜉_𝛼𝑚,𝜔^{0𝑥0206𝑒𝑠}−𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻¹⁰_(𝑄))^𝑛≤𝐶ℎ₀‖𝑁_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻²_(𝑄))^𝑛,(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐿²_(𝑄))^𝑛≤𝐶ℎ₀₂‖𝑁_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻²_(𝑄))^𝑛,(4.1) where the constant 𝐶>0 is independent of the size ℎ₀ of mesh.

Proof. Since 𝐸^𝑠𝛼𝑚=𝐍_𝛼𝑚(𝜉,𝜔^𝑠)−𝐍^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)∈𝐻¹⁰(𝑄), based on Korn inequality and the interpolation theorem, we have that ‖𝐸^𝑠𝛼𝑚‖^2(𝐻_1(𝑄))𝑛=≤𝐶𝐶𝑎𝑎𝐸𝐍^𝑠𝛼𝑚_𝛼𝑚,𝐸^𝑠𝛼𝑚≤𝐶ℎ₀(𝜉,𝜔^𝑠)−𝐍^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠),𝐍_𝛼𝑚(𝜉,𝜔^𝑠)−𝛾_ℎ0𝐍^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)‖𝐍_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻²_(𝑄))^𝑛‖𝐸^𝑠𝛼𝑚‖_(𝐻¹_(𝑄))^𝑛.(4.2) That is, ‖𝑁_𝛼𝑚(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻¹_(𝑄))^𝑛≤𝐶ℎ₀‖𝑁_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻²_(𝑄))^𝑛.(4.3) Aubin-Nitsche lemma [28] yields ‖𝑁_𝛼𝑚(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐿²_(𝑄))^𝑛≤𝐶ℎ₀₂‖𝑁_𝛼𝑚(𝜉,𝜔^𝑠)‖_(𝐻²_(𝑄))^𝑛.(4.4)

Lemma 4.2. Let 𝑁_𝛼𝑝𝑚(𝜉,𝜔^𝑠),𝛼₁=1,2,…,𝑛, be the variational solution of the (2.6) and 𝑁^ℎ0_𝛼𝑝𝑚(𝜉,𝜔^𝑠) the finite element solution such that 𝑁_𝛼𝑝𝑚(𝜉,𝜔^𝑠)∈𝑊^2,∞(𝑄), and then one has ‖‖∇_𝜉𝑁^𝛼𝑝𝑚(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝛼𝑝𝑚(𝜉,𝜔^𝑠)‖‖≤𝐶ℎ₀‖‖𝑁_𝛼𝑝𝑚||lnℎ₀||‖∇^2𝜉𝑁_𝛼𝑝𝑚‖_𝐿^∞_(𝑄),(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝛼𝑝𝑚(𝜉,𝜔^𝑠)‖‖≤𝐶ℎ₀₂||lnℎ₀||‖∇^2𝜉𝑁_𝛼𝑝𝑚‖_𝐿^∞_(𝑄).(4.5)

In fact, based on the idea and the method in [28], it is easy to prove Lemma 4.2.

Lemma 4.3. Let 𝑎̂_{𝑖𝑗ℎ𝑘} be the equivalent mechanical parameter tensor matrix based on STSA and 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘} its finite element approximation as above, if there exists one constant 𝑁 such that forall𝜔^𝑠∈𝑃,‖𝑁_𝛼𝑝𝑚(𝜉,𝜔^𝑠)‖_𝐻²_(Q)≤𝑁. Then one has ‖‖𝑎̂_{𝑖𝑗ℎ𝑘}−𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}‖‖_𝐿∞≤𝐶ℎ₀||lnℎ₀||𝑁.(4.6)

Proof. From the above algorithm, the following equation is held. 𝑎̂_{𝑖𝑗ℎ𝑘}=𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}+𝑅_{𝑖𝑗ℎ𝑘},(4.7) where 𝑅_{𝑖𝑗ℎ𝑘} is defined by 𝑅_{𝑖𝑗ℎ𝑘}=∑^𝑀𝑠=1∫_𝑄𝑎_{𝑖𝑝ℎ𝑞}(𝜉,𝜔^𝑠)(1/2)ℋ/𝜕𝜉_𝑝+𝒜/𝜕𝜉_𝑞𝑀,(4.8) where ℋ denotes 𝜕(𝑁_𝑘𝑞𝑗(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝑘𝑞𝑗(𝜉,𝜔^𝑠)) and𝒜 denotes 𝜕(𝑁_𝑘𝑝𝑗(𝜉,𝜔^𝑠)−𝑁^ℎ0_𝑘𝑝𝑗(𝜉,𝜔^𝑠))
Since ‖𝑅‖_∞ denote the maximum norm of matrix (𝑅_{𝑖𝑗ℎ𝑘})_𝑛×𝑛, applying Lemma 4.2 to the above (4.8), we deduce ‖𝑅‖_∞=‖‖‖‖∑^𝑀𝑠=1∫_𝑄𝑎_{𝑖𝑝ℎ𝑞}(𝜉,𝜔^𝑠)(1/2)ℋ/𝜕𝜉_𝑝+𝒜/𝜕𝜉_𝑞𝑀‖‖‖‖_∞≤1𝑀^𝑀_𝑠=1‖‖‖_𝑄𝑎_{𝑖𝑝ℎ𝑞}(𝜉,𝜔^𝑠)12𝜕ℋ𝜉_𝑝+𝒜𝜕𝜉_𝑞‖‖‖_∞≤𝐶1𝑀ℎ⁰||lnℎ₀||^𝑀_𝑠=1‖‖∇^2𝜉𝑁_𝑘(𝜉,𝜔^𝑠)‖‖_𝐿∞,(4.9)
Since there exists one constant 𝑁 such that the function matrices ‖𝑁_𝛼(𝜉,𝜔^𝑠)‖≤𝑁, the above inequality (4.9) yields ‖𝑅‖_∞≤𝐶ℎ₀||lnℎ₀‖𝑁||.(4.10) Then inequality (4.6) follows from the above inequality (4.10).
Based on the SMSA-FE algorithm, the equivalent mechanical parameter tensor of the equivalent material with the 𝑚th random grains can be obtained. Therefore, the matrix material with (𝑚−1)th random grains can be considered as the equivalent matrix material. Using the loop proof by Lemmas 4.2 and 4.3, it is easy to obtain the following theorem on the equivalent mechanical parameter tensor of a composite material with 𝑚-scale random grains.

Theorem 4.4. Let 𝑎̂^{𝑟𝑖𝑗ℎ𝑘}(𝑟=𝑚,𝑚−1,…,1) be the equivalent mechanical parameters tensor of the composite material with r th scale random grains and 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}(𝑟=𝑚,𝑚−1,…,1) its finite element approximation; set the size of the last mesh of the finite element in 𝑄^𝑠 to be ℎ₀; based on the SMSA-FE algorithm, if there exists one constant 𝑁 such that for all, 𝜔^𝑠∈𝑃,‖𝑁_𝛼𝑝𝑚(𝜉_𝑟,𝜔^𝑠)‖_𝐻²_(𝑄)≤𝑁,(𝑟=𝑚,𝑚−1,…,1), then one has ‖‖𝑎̂^{𝑟𝑖𝑗ℎ𝑘}−𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}‖‖≤𝐶ℎ₀||lnℎ₀||𝑁,(4.11) where 𝐶 is one constant that is independent of ℎ₀ but dependent on the sizes of the other finite element mesh ℎ_0𝑟(𝑟=𝑚,𝑚−1,…,2) in the cell 𝑄_𝑟(𝑟=𝑚,𝑚−1,…,2) with rth random grains.

From Theorem 4.4, it is easy to see the error of the equivalent mechanical parameter tensor of the composite material with the biggest grains being the main error by the SMSA-FE algorithm. Hence, we only need to consider the error in the composite material with the biggest random grains.

5. Symmetry and Positive Definite Property for Equivalent Mechanical Parameter Tensor

From [7, 26], if the parameters of the ellipsoids are subjected to the uniform probability 𝑃, the equivalent mechanical parameter tensor 𝑎̂_{𝑖𝑗ℎ𝑘} shall satisfy the following conditions: 𝑎̂_{𝑖𝑗ℎ𝑘}=𝑎̂_{𝑗𝑖𝑘ℎ}=𝑎̂_{ℎ𝑗𝑖𝑘,}𝜇₁𝜂_𝑖𝑗𝜂_𝑖𝑗≤𝑎̂_{𝑖𝑗ℎ𝑘}𝜂_𝑖𝑗𝜂_ℎ𝑘≤𝜇₂𝜂_𝑖𝑗𝜂_𝑖𝑗,(5.1) where for any symmetry matrix 𝜂=(𝜂_𝑖𝑗)_𝑛×𝑛, 𝜇₁≥0, and 𝜇₂≥0.

Therefore, if the parameters of the ellipsoids are subjected to the uniform probability 𝑃, it is important to keep the symmetric, positive definite property of mechanical parameter tensor computed by the finite element method. So we shall give the following theorem to illustrate it.

Lemma 5.1. Let {𝑎̂_{𝑖𝑗ℎ𝑘}} be the equivalent mechanical parameter tensor based on STSA [26] and 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘} its finite element approximation; if there exists one constant 𝑁 such that for all 𝜔^𝑠∈𝑃,‖𝑁_𝛼𝑝𝑚(𝜉,𝜔^𝑠)‖_𝐻²_(𝑄)≤𝑁, the matrix 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘} satisfies the following property: 𝐾𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}=𝑎̂^ℎ0_{𝑗𝑖𝑘ℎ}=𝑎̂^ℎ0_{ℎ𝑗𝑖𝑘},(5.2)₁𝜂_𝑖𝑗𝜂_𝑖𝑗≤𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}𝜂_𝑖𝑗𝜂_ℎ𝑘≤𝐾₂𝜂_𝑖𝑗𝜂_𝑖𝑗(5.3) for any symmetric matrix 𝜂=(𝜂_𝑖ℎ)_𝑛×𝑛, where 𝐾₁,𝐾₂ are positive constants.

Proof. Taking into consideration the fact that 𝑎̂_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)=𝑎̂_{𝑗𝑖𝑘ℎ}(1^𝑠)=𝑎̂_{ℎ𝑗𝑖𝑘}(𝜔^𝑠), based on the concept of 𝑁^ℎ0_𝛼(𝜉,𝜔^𝑠), taking into account the idea in [7], we deduce that 𝑎̂^ℎ0_{𝑗𝑖𝑘ℎ}(𝜔^𝑠)=1||𝑄^𝑠||_𝑄^𝑠⎡⎢⎢⎣𝑎_{𝑗𝑖𝑘ℎ}(𝜉,𝜔^𝑠)+𝑎_{𝑗𝑝𝑘𝑞}(𝜉,𝜔^𝑠)12⎛⎜⎜⎝𝜕𝑁^{ℎ0ℎ𝑝𝑖}(𝜉,𝜔^𝑠)𝜉_𝑞+𝜕𝑁^{ℎ0ℎ𝑞𝑖}(𝜉,𝜔^𝑠)𝜉_𝑝=1⎞⎟⎟⎠⎤⎥⎥⎦𝐝𝜉||𝑄^𝑠||_𝑄^𝑠⎡⎢⎢⎣𝑎_{ℎ𝑗𝑖𝑘}(𝜉,𝜔^𝑠)+𝑎_{ℎ𝑞𝑖𝑝}(𝜉,𝜔^𝑠)12⎛⎜⎜⎝𝜕𝑁^{ℎ0𝑘𝑝𝑗}(𝜉,𝜔^𝑠)𝜉_𝑞+𝜕𝑁^{ℎ0𝑘𝑞𝑗}(𝜉,𝜔^𝑠)𝜉_𝑝=𝑎̂_{ℎ0𝑘𝑗𝑖ℎ}⎞⎟⎟⎠⎤⎥⎥⎦𝐝𝜉(𝜔^𝑠).(5.4) So we have proved 𝑎̂^ℎ0_{𝑗𝑖𝑘ℎ}(𝜔^𝑠)=𝑎̂^ℎ0_{ℎ𝑗𝑖𝑘}(𝜔^𝑠). Let us establish the first equality in (5.2), which is equivalent to prove (𝐴^ℎ0ℎ𝑘)^∗=𝐴^ℎ0𝑘ℎ, where 𝐴^∗ denotes the transpose of the matrix 𝐴 and 𝐴^ℎ0ℎ𝑘 is the matrix (𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘})_𝑛×𝑛.
From the integral identity for solution of problem (2.6), for any matrix 𝑀^ℎ0(𝜉)∈(𝑉^ℎ0(𝑄^𝑠))^𝑛×𝑛, we have −_𝑄𝑠𝜕(𝑀(𝜉)𝜕𝜉_𝑝𝐴^𝑝𝑞(𝜉,𝜔^𝑠)𝜕𝑁^∗𝑘(𝜉,𝜔^𝑠)𝜕𝜉_𝑞𝑑𝜉=_𝑄𝑠𝐴^𝑝𝑘(𝜉,𝜔^𝑠)(𝑀(𝜉))𝜕𝜉_𝑝𝑑𝜉.(5.5)
Based on the relations (𝐴^𝑝𝑞(𝜉))^∗=𝐴^𝑞𝑝(𝜉) and (𝐴𝐵)^∗=𝐵^∗𝐴^∗ for matrices 𝐴,𝐵, it is easy to obtain 𝑁_𝛼 from (5.5) that −_𝑄𝑠𝜕𝑁^ℎ0_𝑘(𝜉,𝜔^𝑠)𝜕𝜉_𝑝𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝜕𝑀^∗(𝜉)𝜕𝜉_𝑞𝑑𝜉=_𝑄𝑠𝐴^𝑘𝑞(𝜉,𝜔^𝑠)𝜕𝑀^∗(𝜉)𝜕𝜉_𝑞𝑑𝜉.(5.6) If we set 𝑀=𝑁^ℎ0⋆ℎ, the following equations are obtained. −_𝑄𝑠𝜕𝑁^ℎ0_𝑘(𝜉,𝜔^𝑠)𝜕𝜉_𝑝𝐴^𝑝𝑞(𝜉,𝜔^𝑠)𝜕𝑁^ℎ0ℎ(𝜉,𝜔^𝑠)𝜕𝜉_𝑞𝑑𝜉=_𝑄𝑠𝐴^𝑘𝑞(𝜉,𝜔^𝑠)𝜕𝑁^ℎ0ℎ(𝜉,𝜔^𝑠)𝜕𝜉_𝑞𝑑𝜉.(5.7) Taking into account the idea in [7], (5.7), and the concept of 𝑁^ℎ0∗ℎ(𝜉,𝜔^𝑠), the following equations can be obtained. _𝑄𝑠𝜕𝜕𝜉_𝑞𝑁^ℎ0∗𝑘+𝜉_𝑘𝐸𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝜕𝜕𝜉_𝑝𝑁^ℎ0ℎ=+𝜉_ℎ𝐸𝑑𝜉_𝑄𝑠𝜕𝑁𝜕𝜉^ℎ0∗𝑘_𝑞𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝜕𝑁^ℎ0ℎ𝜕𝜉_𝑝+𝛿_𝑞𝑘𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝜕𝑁^ℎ0ℎ𝜕𝜉_𝑝+𝜕𝑁^ℎ0∗k𝜕𝜉_𝑞+𝛿_𝑞𝑘𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝛿_𝑝ℎ𝐸=𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝛿_𝑝ℎ𝐸𝑑𝜉_𝑄𝑠𝜕𝑁𝜕𝜉^ℎ0∗𝑘_𝑞𝐴^𝑞𝑝(𝜉,𝜔^𝑠)𝜕𝑁^ℎ0ℎ𝜕𝜉_𝑝+𝐴^𝑘𝑝(𝜉,𝜔^𝑠)𝜕𝑁^ℎ0ℎ𝜕𝜉_𝑝+𝜕𝑁^ℎ0∗𝑘𝜕𝜉_𝑞=𝐴^𝑞ℎ(𝜉,𝜔^𝑠)+𝐴^𝑘ℎ(𝜉,𝜔^𝑠)𝑑𝜉_𝑄𝑠𝜕𝑁𝜕𝜉^ℎ0∗𝑘_𝑞𝐴^𝑞ℎ(𝜉,𝜔^𝑠)+𝐴^𝑘ℎ(𝜉,𝜔^𝑠)𝑑𝜉=𝐴^{ℎ0ℎ𝑘⋆}(𝜔^𝑠).(5.8) In the second equation of the above equations, the property of 𝛿_𝑖𝑗 is used; in the third equation, (5.7) is applied. It follows that the equivalent mechanical parameter matrices can be written in the following form by the relationship of 𝐴^𝑗𝑞⋆=𝐴^𝑞𝑗: 𝐴^ℎ0ℎ𝑘(𝜔^𝑠)=_{𝑄0𝑥03631}𝜕𝜕𝜉_𝑞𝑁^ℎ0∗ℎ+𝜉_ℎ𝐸𝐴^𝑝𝑞(𝜉,𝜔^𝑠)𝜕𝜕𝜉_𝑝𝑁^ℎ0𝑘𝐴+𝜉_𝑘𝐸𝑑𝜉,_ℎ0𝑘ℎ(𝜔^𝑠)=_𝑄𝑠𝜕𝜕𝜉_𝑞𝑁^ℎ0∗𝑘+𝜉_𝑘𝐸𝐴^𝑞𝑗(𝜉,𝜔^𝑠)𝜕𝜕𝜉_𝑗𝑁^ℎ0ℎ+𝜉_ℎ𝐸𝑑𝜉.(5.9) Thus 𝐴^ℎ0𝑘ℎ^∗(𝜔^𝑠)==_𝑄𝑠𝜕𝜕𝜉_𝑗𝑁^ℎ0∗ℎ+𝜉_ℎ𝐸𝐴^𝑗𝑞(𝜉,𝜔^𝑠)𝜕𝜕𝜉_𝑞𝑁^ℎ0𝑘+𝜉_𝑘𝐸𝑑𝜉=𝐴^ℎ0ℎ𝑘(𝜔^𝑠).(5.10)
Hence, we see that 𝐴^ℎ0ℎ𝑘(𝜔^𝑠)=(𝐴^ℎ0𝑘ℎ(𝜔^𝑠))^∗. Thus we obtain the following equations: 𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)=𝑎̂^ℎ0_{𝑗𝑖𝑘ℎ}(𝜔^𝑠)=𝑎̂^ℎ0_{ℎ𝑗𝑖𝑘}(𝜔^𝑠).(5.11) Making use of the relations (5.11), one has 1𝑀^𝑀_𝑠=1𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}(𝜔^𝑠)=1𝑀^𝑀_𝑠=1𝑎̂^ℎ0_{𝑗𝑖𝑘ℎ}(𝜔^𝑠)=1𝑀^𝑀_𝑠=1𝑎̂^ℎ0_{ℎ𝑗𝑖𝑘}(𝜔^𝑠).(5.12) That is (5.2) is proved.
In the sequel, we shall prove (5.3). From the inequality (4.6) and Theorem 4.4, there exists one ℎ₀≥0 that is small enough such that 𝐶ℎ₀||lnℎ₀||||𝑁_𝛼||_𝐿∞<𝜇¹2.(5.13) We have 𝜇₁2𝜂^𝑖𝑗𝜂_𝑖𝑗≤𝑎̂^ℎ0_{𝑖𝑗ℎ𝑘}𝜂_𝑖𝑗𝜂_ℎ𝑘≤𝜇₂+𝜇¹2𝜂^𝑖𝑗𝜂_𝑖𝑗.(5.14) Setting 𝐾₁=𝜇₁/2 and 𝐾₂=𝜇₂+𝜇₁/2yields the inequality (5.3).
By the iterated multiscale analysis and Lemma 5.1, the finite element approximation of the equivalent mechanical parameter tensor of the composite material with multiscale random grains satisfies the following property.

Theorem 5.2. Let {𝑎̂^{𝑟𝑖𝑗ℎ𝑘}} be the equivalent mechanics parameter tensor of the composite material with r-scale random grains based on the SMSA algorithm and 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘} their finite element approximation by (2.3). Set the size of the last mesh of the finite element in 𝑄^𝑠 to be ℎ₀; if there exists the constant 𝑁_𝛼1 such that for all, 𝜔^𝑠∈𝑃,‖𝑁^{𝑟𝛼𝑝𝑚}(𝜉,𝜔^𝑠)‖_𝐻2(𝑄)≤𝑁_𝛼𝑝𝑚, then 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘} satisfy the conditions 𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}=𝑎̂^𝑟,ℎ0_{𝑗𝑖𝑘ℎ}=𝑎̂^𝑟,ℎ0_{ℎ𝑗𝑖𝑘},𝐾₁𝜂_𝑖𝑗𝜂_𝑖𝑗≤𝑎̂^𝑟,ℎ0_{𝑖𝑗ℎ𝑘}𝜂_𝑖𝑗𝜂_ℎ𝑘≤𝐾₂𝜂_𝑖𝑗𝜂_𝑖𝑗,(5.15) where 𝜂=(𝜂_𝑖ℎ)_𝑛×𝑛is the symmetry matrix and 𝐾₁,𝐾₂ are the positive constants that are independent of ℎ₀ but dependent on the sizes of the other finite element meshes ℎ_0𝑟,(𝑟=𝑚,𝑚−1,…,2) in the 𝑟th scale cells 𝑄_𝑟(𝑟=𝑚,𝑚−1,…,2,1).

6. Numerical Experiment

To test the validity of the error result, the convergence, and the symmetric, positive definite property of the mechanical parameter tensor computed by the SMSA-FE algorithm, two numerical examples are given as follows.

The first example models a composite material. The grains are divided into two classes according to the sizes of their long axis shown in Table 1. We use one statistical window 𝜀¹=0.1 to predict the mechanical parameters of the equivalent matrix with small random grains. In each window, small grains occupy approximate 30% of the volume. Their long axis 𝑎, short axis 𝑏, and angle 𝜃 are subjected to the uniform distributions shown in Table 1. Using the different finite element sizes ℎ₀, based on the data of Table 2, we obtain the equivalent mechanical parameters tensor {𝑎̂_{𝑖𝑗ℎ𝑘}} that are given in Table 3.

Table 1 

Probability distributions of grains in composite material.

Table 2 

Mechanical parameters of the matrix and grains.

Table 3 

Equivalent mechanical parameters {𝑎̂_𝑖𝑗(𝜔^𝑠)} for different mesh sizes ℎ₀.

From Table 3, it is easy to see that convergence of the equivalent mechanical parameter tensor computed by the SMSA-FE algorithm exists. From Table 3, the symmetric, positive definite property of the equivalent mechanical parameter tensor and the convergence of the finite element errors with the different mesh sizes ℎ₀ are proved.

The second example is a concrete named as C30 with three-scale random rock grains whose sizes are from 0.3 mm to 19 mm. Its matrix is made up of the cement and the sand. Their sizes of the three-scale rock grains in the concrete are shown in Table 4. Their elastic parameters are shown in Table 5. If all grains are generated in a large statistical window of 500 mm, the number of grains is approximately 6360. In each window, small grains, middle grains, and the large grains occupy approximately 57.7% of the volume and are subjected to the uniform probability distribution in the range of their sizes. Therefore, we set three kinds of sizes of the statistical windows: 𝜀³=5mm, 𝜀²=50mm, and 𝜀¹=100mm to compute the equivalent mechanical parameter tensor with the different scale random grains.

Table 4 

Size of rock random grains and the number in a statistical window.

Table 5 

Elasticity mechanical parameters of the matrix, grains, and joint interface materials.

Numerical results for the mechanical parameter tensor of a composite with only small, middle, and large rock grains are listed in Tables 6, 7, and 8 by the SMSA-FE algorithm. Tables 6–8 also show that the equivalent parameter tensor computed by the SMSA-FE algorithm possess the symmetrical, positive, and definite properties. The expected values of Young's modulus and Poisson's ratio for the different number of samples with three-scale random grains are shown in Table 9 and in Figure 2. Table 9 and Figure 2 show that the equivalent Young's modulus and Poisson's ratio are convergent.

Table 6 

Equivalent mechanical parameters of a composites with small rock random grains by the SMSA-FE algorithm (kPa).

Table 7 

Equivalent mechanical parameter tensor for concrete with middle rock random grains by the SMSA-FE algorithm (kPa).

Table 8 

Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (kPa).

Table 9 

Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (GPa).

(a)

(b)

Figure 2 

Young's modulus and Poisson's ratio for 5, 10, 15,…, 30 samples with different scale random grains by the SMSA-FE procedure.

A comparison of the numerical results for Young's modulus and Poisson's ratio by the SMSA-FE procedure, by the mixed volume method, and by the experiment method in the lab is also shown in Table 10. Table 10 also shows that Young's modulus and Poisson's ratios produced by SMSA-FE procedure are very close to that by the experiment method. It proves the SMSA-FE algorithm to be feasible and valid for predicting the effective modulus of the composites with random grains.

Table 10 

Equivalent mechanical parameters for a concrete by the different methods (GPa).

7. Conclusion

In this paper, we proved that the equivalent mechanical parameter tensor for the composite materials with multiscale random grains is convergent and obtained the error result by the finite element analysis. At the same time, we also prove that the equivalent parameter tensor matrix should satisfy the symmetric, positive, and definite property.

Various test examples were solved by the SMSA-FE procedure. The numerical results show that the SMSA-FE procedure is feasible and valid and that these data satisfy the properties of the equivalent mechanical parameter tensor proved in the pervious sections.

The procedure can also be extended to other composite materials with random short fibers, random foams, random cavities, and so forth. Although we have given the specific steps, some theory results and numerical examples to carry out the SMSA-FE method and to illustrate the validity, the influence of shape, size, component, orientation, spatial distribution, and volume fraction of inclusions on inhomogeneous macromechanical properties, analyze the calculated results, and capture the information of microbehaviors to the macromechanical properties are also our important future work for the composite materials with random grains.

Acknowledgments

This work is supported by National Natural Science Foundation of China (NSFC) Grants (11072041), by State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology (GZ1005), by Hunan province Natural Science Foundation Grants of China (10JJ6065), by Scientific Research Starting Foundation for Returned Overseas Chinese Scholars, Ministry of Education, China (20091001), and by China Postdoctoral Science Foundation (20100480944).