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Mathematical Problems in Engineering
Volume 2011, Article ID 585624, 19 pages
http://dx.doi.org/10.1155/2011/585624
Research Article

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

1School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Hunan 410004, China
2State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
3Division of Mathematical Sciences, School of Physical and Mathematical Science (SPMS), Nanyang Technological University, 21 Nanyang Link, Singapore 637371

Received 11 August 2010; Revised 22 December 2010; Accepted 17 March 2011

Academic Editor: Moran Wang

Copyright © 2011 Youyun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [713] 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.

fig1
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 [2527]. In previous papers [2527], 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 [2527], 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 Ω𝑙 to be a set of cube cells of the size 𝜀𝑙 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 𝜀𝑙𝑄𝑠; we can describe the 𝑙th scale cube cell 𝜀𝑙𝑄𝑠 as follows: 𝜔𝑠𝑙=𝑎𝑠1,𝑏𝑠1,𝑐𝑠1,𝜃𝑠𝑎𝑥1𝑥21,𝜃𝑠𝑎𝑥11,𝜃𝑠𝑏𝑥1𝑥21,𝜃𝑠𝑏𝑥11,𝑥10𝑠1,𝑥20𝑠1,𝑥30𝑠1,𝜃𝑠𝑏𝑥1𝑁,𝑥10𝑠𝑁,𝑥20𝑠𝑁,𝑥30𝑠𝑁,(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 𝜃𝑎𝑥1𝑥2, 𝜃𝑎𝑥1, 𝜃𝑏𝑥1𝑥2 and 𝜃𝑏𝑥1,are the directions for the axis 𝑎 and 𝑏 of the ellipsoids, respectively. One sample 𝜔𝑠𝑙 is shown in Figure 1(c).(3)Set the domain Ω𝑙 to be logically composed of 𝜀𝑙-sized samples: Ω𝑙=(𝜔𝑠,𝑡𝑍)𝜀𝑙(𝑄𝑠+𝑡) shown in Figure 1(b). It can be defined as 𝜔𝑙={𝜔𝑠𝑙,𝑥𝜀𝑄𝑠𝑙Ω𝑙}. By the SMSA-FE algorithm [27], the equivalent mechanical parameters can be predicted. Thus, the equivalent material with 𝑙th scale random grains in Ω𝑙 can be formed.(4)Set 𝑚 to be the scale number in the composites Ω with multiscale random grains. For 𝑙=𝑚1,,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 Ω𝑙,(𝑙=𝑚,𝑚1,,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 Ω𝑙,(𝑙=𝑚1,𝑚2,,0) shown in Figure 1(b), their corresponding elasticity equation system and the essential boundary condition can be shown as follows: 𝜕𝜕𝑥𝑗𝑎𝜀𝑖𝑗𝑘𝑥𝜀1,𝜔2𝜕𝑢𝜀(𝑥,𝜔)𝜕𝑥𝑘+𝜕𝑢𝜀𝑘(𝑥,𝜔)𝜕𝑥=𝑓𝑖(𝑥),𝑥Ω𝑙,𝑢𝜀(𝑥,𝜔)=𝑢(𝑥),𝑥𝜕Ω,(2.2) where 𝑖,𝑗,,𝑘=1,2,,𝑛, 𝜔=𝜔𝑠 for 𝑥Ω1, 𝜔𝑠𝑃, 𝑃 is the probability space, Ω𝑙=𝑠𝑝,𝑡𝑍𝑛𝜀𝑙(𝑄𝑠+𝑡) shown in Figure 1, 𝑢𝜀(𝑥,𝜔) are the displacement field, 𝑓𝑖(𝑥) are the loads, and 𝑢(𝑥) is the boundary displacement vector.

In the paper [2527], we had given the SMSA method [25] and established the finite element method [27] to compute the equivalent mechanical parameters. If the FE space (𝑉0(𝑄𝑠))𝑛 can be established and 𝜉=𝑥/𝜀𝑙, 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 ̂𝑎0𝑖𝑗𝑘=𝑀𝑠=1̂𝑎0𝑖𝑗𝑘(𝜔𝑠)𝑀,(2.3) where 𝜔𝑠𝑃(𝑠=1,2,,𝑀) and ̂𝑎0𝑖𝑗𝑘(𝜔𝑠) is computed as ̂𝑎0𝑖𝑗𝑘(𝜔𝑠)=1||𝑄𝑠||𝑄𝑠𝑎𝑖𝑗𝑘(𝜉,𝜔𝑠)+𝑎𝑖𝑝𝑞𝜀𝑝𝑞𝑁0𝑘𝑗(𝜉,𝜔𝑠)𝐝𝜉,(2.4)𝐍0𝛼(𝜉,𝜔𝑠𝐍)=0𝛼1,𝐍0𝛼2,,𝐍0𝛼𝑛=𝑁0𝛼11(𝜉,𝜔𝑠)𝑁0𝛼1𝑛(𝜉,𝜔𝑠)𝑁0𝛼𝑛1(𝜉,𝜔𝑠)𝑁0𝛼𝑛𝑛(𝜉,𝜔𝑠)..(2.5)𝐍0𝛼(𝜉,𝜔𝑠) are the FE solutions of (2.6) on unit cell 𝑄𝑠. 𝜕𝜕𝜉𝑗𝑎𝑖𝑗𝑘(𝜉,𝜔𝑠)𝜀𝑘𝑁𝛼𝑚(𝜉,𝜔𝑠)=𝜕𝑎𝛼𝑖𝑙𝑚(𝜉,𝜔𝑠)𝜕𝜉𝑙,𝜉𝑄𝑠,𝐍𝛼(𝜉,𝜔𝑠)=0,𝜉𝜕𝑄𝑠.(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 𝑄𝑠𝑟 and generate meshes of the sample domain according to the algorithm in [27].(3)If 𝑟=𝑚, 𝑎𝑖𝑗𝑘(𝑥/𝜀𝑟,𝜔𝑠) in 𝑄𝑠𝑟 can be indicated as follows: 𝑎𝑖𝑗𝑘𝑥𝜀𝑟,𝜔𝑠=𝑎1𝑖𝑗𝑘,𝑥𝜀𝑟𝑄𝑠,𝑎2𝑖𝑗𝑘,𝑥𝜀𝑟𝑄𝑠,𝑎𝑙𝑖𝑗𝑘,𝑥𝜀𝑟𝑄𝑠,(2.7) where 𝑄𝑠𝑄𝑠𝑄𝑠=𝑄𝑠𝑟 and 𝑄𝑠𝑄𝑠𝑄𝑠=𝜙, 𝜀𝑟𝑄𝑠𝑟Ωdenotes the domain of the basic configuration, 𝜀𝑟𝑄𝑠 denotes the domain of the matrix, 𝜀𝑟𝑄𝑠 denotes the domain of the random grains in 𝜀𝑟𝑄𝑠𝑟, and 𝜀𝑎𝑄𝑠 denotes the interface domain between the grains and the matrix. Go to step (5).(4)If 𝑟<𝑚, 𝑎𝑖𝑗𝑘(𝑥/𝜀𝑟,𝜔𝑠) in 𝑄𝑠𝑟 can be indicated as follows: 𝑎𝑖𝑗𝑘𝑥𝜀𝑟,𝜔𝑠=̂𝑎0𝑖𝑗𝑘𝑥𝜀(𝑟+1),𝜔𝑠,𝑥𝜀𝑟𝑄𝑠,𝑎1𝑖𝑗𝑘,𝑥𝜀𝑟𝑄𝑠,(2.8) where 𝑄𝑠𝑄𝑠=𝑄𝑠𝑟 and 𝑄𝑠𝑄𝑠=𝜙, 𝜀𝑟𝑄𝑠𝑟Ω denotes the domain of the basic configuration, 𝜀𝑟𝑄𝑠 denotes the domain of the equivalent matrix, and 𝜀𝑟𝑄𝑠 denotes the domain of random grains. Go to step (5).(5)Compute the FE approximation 𝑁0𝛼𝑚(𝜉𝑟,𝜔𝑠) according to (2.6), obtain the FE approximation of sample ̂𝑎0𝑖𝑗𝑘(𝜀𝑟,𝜔𝑠) on the 𝑟th scale according to (2.4), and compute the FE approximation of the equivalent mechanical parameter tensor {̂𝑎0𝑖𝑗𝑘(𝜀𝑟)} on the 𝑟th scale using (2.3).(6)Set ̂𝑎𝑖𝑗𝑘(𝜀𝑟) equal to ̂𝑎0𝑖𝑗𝑘(𝜀𝑟) on the 𝑟th scale and 𝑟=𝑟1. If 𝑟>1, go to step (2). Otherwise, the equivalent mechanical parameter tensor ̂𝑎𝑖𝑗𝑘(𝜀𝑟)(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 𝑁0𝛼𝑚(𝜉,𝜔) and ̂𝑎0𝑖𝑗𝑘(𝜔) are the finite element approximations of the random variables 𝑁𝛼𝑚(𝜉,𝜔) and ̂𝑎𝑖𝑗𝑘(𝜔), respectively, then there exists a constant 𝑀2>0 such that |̂𝑎0𝑖𝑗𝑘(𝜔)|<𝑀2.

Proof. If |𝑎𝑖𝑗𝑘(𝜉,𝜔𝑠)|<𝑀 for any sample 𝜔𝑠𝑃, (2.6) has one unique finite element solution 𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐻1(𝑄𝑠))𝑛(𝑛=2,3) such that 𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐻1(𝑄𝑠))𝑛||𝑎<𝐶𝑖𝑗𝑘(𝜉,𝜔s)||𝐿(𝑄𝑠)<𝐶𝑀1,(3.1) where 𝐶 and 𝑀1 are constants and independent of 𝜉 and 𝜔𝑠.
From (2.4) and (3.1), one can obtain |||̂𝑎0𝑖𝑗𝑘(𝜔𝑠)|||=|||||1||mes(𝑄𝑠)||(𝑄𝑠)𝑎𝑖𝑗𝑘(𝜉,𝜔𝑠)+𝑎𝑖𝑝𝑞(𝜉,𝜔𝑠)12𝜕𝑁0𝑘𝑝𝑗(𝜉,𝜔𝑠)𝜕𝜉𝑞+𝜕𝑁0𝑘𝑞𝑗(𝜉,𝜔𝑠)𝜕𝜉𝑝|||||<1𝐝𝜉||mes(𝑄𝑠)||𝑎𝑖𝑗𝑘(𝜉,𝜔𝑠)𝐿(𝑄𝑠)+𝑎𝑖𝑝𝑞(𝜉,𝜔𝑠)𝐿(𝑄𝑠)𝑁𝑘𝑗(𝜉,𝜔𝑠)(𝐻1(𝑄𝑠))𝑛1||mes(𝑄𝑠)||||𝑀1+𝑀1𝐶𝑀1||||mes(𝑄𝑠)||=𝐶𝑀1+𝑀21=𝑀2,(3.2) where 𝑀2=𝐶𝑀1+𝑀21 and 𝑀1 is independent of both the random variables 𝜔𝑠 and the local coordinate 𝜉. Therefore, for the random variable 𝜔 in Section 2, |̂𝑎0𝑖𝑗𝑘(𝜔)|<𝑀2.

Lemma 3.2. If 𝜔 is a random variable and ̂𝑎0𝑖𝑗𝑘(𝜔) are defined as above, then one unique expected value of the equivalent mechanical parameters tensor 𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔) exists in the probability space 𝑃=(𝑃𝑎)𝑁×(𝑃𝑏)𝑁×(𝑃𝑏)𝑁×(𝑃𝜃𝑎𝑥1𝑥2)𝑁××(𝑃𝑥0)𝑁×(𝑃𝑦0)𝑁×(𝑃𝑧0)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, 𝜃𝑎𝑥1𝑥2, 𝜃𝑎𝑥1, and 𝜃𝑏𝑥1𝑥2, 𝜃𝑏𝑥1, and the coordinates of the central points of the ellipsoids (𝑥10,𝑥20,𝑥30), their probability density functions are denoted by 𝑓𝑎(𝑥), 𝑓𝑏(𝑥), 𝑓𝑐(𝑥), 𝑓𝜃𝑎𝑥1𝑥2(𝑥), 𝑓𝜃𝑎𝑥1(𝑥), 𝑓𝜃𝑏𝑥1𝑥2(𝑥), 𝑓𝜃𝑏𝑥1(𝑥), 𝑓𝑥10(𝑥), 𝑓𝑥20(𝑥), and 𝑓𝑥30(𝑥), respectively. The united random variable 𝜔=(𝑎1,𝑏1,𝑐1,𝜃𝑎𝑥1𝑥21,𝜃𝑎𝑥11,𝜃𝑏𝑥1𝑥21,𝜃𝑏𝑥11,𝑥101,𝑥201,𝑥301,𝜃𝑏𝑥1𝑁,𝑥10𝑁,𝑥20𝑁,𝑥30𝑁) has the the united probability density function 𝑓𝑎1,𝑏1,𝑐1,,𝑥0𝑁,𝑦0𝑁,𝑧0𝑁(𝜔)=𝑓𝑁𝑎(𝜔)𝑓𝑁𝑏(𝜔)𝑓𝑁𝑐(𝜔)𝑓𝑁𝑥0(𝜔)𝑓𝑁𝑦0(𝜔)𝑓𝑁𝑧0(𝜔). From Lemma 3.1, one can obtain 𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔)=𝑃̂𝑎0𝑖𝑗𝑘(𝜔)𝐝𝑃=++̂𝑎0𝑖𝑗𝑘(𝜔)𝑓𝑎1,𝑏1,𝑐1,,𝑥0𝑁,𝑦0𝑁,𝑧0𝑁(𝜔)𝐝𝜔<𝑀2++𝑓𝑎1,𝑏1,𝑐1,,𝑥0𝑁,𝑦0𝑁,𝑧0𝑁(𝜔)𝐝𝜔<𝑀2++𝑓𝑁𝑎(𝜔)𝐝𝜔++𝑓𝑁𝑧0(𝜔)𝐝𝜔<𝑀2.(3.3) Therefore, there exists one unique expected value of the equivalent mechanical parameters tensor 𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔)(𝑖,𝑗,,𝑘=1,2,,𝑛) in the probability space 𝑃.

Lemma 3.3. If ̂𝑎0𝑖𝑗𝑘(𝜔𝑠)(𝑖,𝑗,,𝑘=1,2,,𝑛.𝑠=1,2,.)have the expected value 𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔) in the probability space and 𝜔 is the random variable, one obtain 𝑀𝑠=1̂𝑎0𝑖𝑗𝑘(𝜔𝑠)𝑀𝑎𝑒𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔𝑠)(𝑀).(3.4)

Proof. Because {̂𝑎0𝑖𝑗𝑘(𝜔𝑠),𝑠1} are the independent and identical distribution random variables and 𝑆𝑀=𝑀𝑠=1̂𝑎0𝑖𝑗𝑘(𝜔𝑠)(𝑀=1,2,),(3.5) from Lemma 3.2, |𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔𝑠)|<. Set 𝑎1=𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔); from Kolmogorov’s classical strong law of large numbers, one obtains 𝑆𝑀𝑀𝑎1𝑎𝑒0(𝑀).(3.6) Therefore, we have 𝑆𝑀/𝑀𝑎𝑒𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔𝑠)(𝑀), that is, 𝑀𝑠=1̂𝑎0𝑖𝑗𝑘(𝜔𝑠)𝑀𝑎𝑒𝐸𝜔̂𝑎0𝑖𝑗𝑘(𝜔)(𝑀).(3.7)

Theorem 3.4. If ̂𝑎𝑟,0𝑖𝑗𝑘(𝜔)(𝑟=𝑚,𝑚1,,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 𝐸𝜔̂𝑎𝑟,0𝑖𝑗𝑘(𝜔) exist in the probability space 𝑃=(𝑃𝑟𝑎)𝑁×(𝑃𝑟𝑏)𝑁×(𝑃𝑟𝑐)𝑁××(𝑃𝑟𝑥0)𝑁×(𝑃𝑟𝑦0)𝑁×(𝑃𝑟𝑧0)𝑁.

Proof. Set 𝑟=𝑚; define 𝑎𝑖𝑗𝑘𝑥𝜀𝑟,𝜔𝑠=𝑎2𝑖𝑗𝑘𝑄,𝑥𝜀𝑠,𝑎1𝑖𝑗𝑘𝑄,𝑥𝜀𝑠,(3.8) because both 𝑎1𝑖𝑗𝑘 and 𝑎2𝑖𝑗𝑘 are constants satisfying |max{𝑎1𝑖𝑗𝑘,𝑎2𝑖𝑗𝑘}|<𝑀1. It is easy to see that 𝑎𝑖𝑗𝑘(𝑥/𝜀𝑟,𝜔𝑠)(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 𝐸𝜔̂𝑎𝑟,0𝑖𝑗𝑘(𝜔) of the equivalent mechanical parameter tensor ̂𝑎𝑟,0𝑖𝑗𝑘(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=1̂𝑎𝑟,0𝑖𝑗𝑘(𝜔𝑠)𝑀𝑎𝑒𝐸𝜔̂𝑎𝑟,0𝑖𝑗𝑘(𝜔𝑠)(𝑀).(3.9) Set 𝑟=𝑟1 and ̂𝑎𝑟,0𝑖𝑗𝑘(𝑥/𝜀(𝑟+1),𝜔𝑠)=𝐸𝜔̂𝑎𝑟,0𝑖𝑗𝑘(𝜔𝑟), 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, 𝑎𝑖𝑗𝑘𝑥𝜀𝑟,𝜔𝑠=̂𝑎𝑟,0𝑖𝑗𝑘𝑥𝜀(𝑟+1),𝜔𝑠𝑄,𝑥𝜀𝑠,𝑎1𝑖𝑗𝑘𝑄,𝑥𝜀𝑠,(3.10)|𝑎𝑖𝑗𝑘(𝑥/𝜀𝑟,𝜔𝑠)|<𝑀1. It is easy to see that 𝑎𝑖𝑗𝑘(𝑥/𝜀𝑟,𝜔𝑠)(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 𝐸𝜔̂𝑎𝑟,0𝑖𝑗𝑘,(𝑟=𝑚,𝑚1,,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 𝑁𝛼𝑚(𝜉,𝜔𝑠),𝛼=1,2,𝑛, is the variational solution of (2.6) and 𝑁0𝛼𝑚(𝜉,𝜔𝑠) is the corresponding finite element solution of (2.6) and 𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐻2(𝑄))𝑛, then we have 𝑁𝛼𝑚𝜉,𝜔0𝑥0206𝑒𝑠𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐻10(𝑄))𝑛𝐶0𝑁𝛼𝑚(𝜉,𝜔𝑠)(𝐻2(𝑄))𝑛,𝑁𝛼𝑚(𝜉,𝜔𝑠)𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐿2(𝑄))𝑛𝐶02𝑁𝛼𝑚(𝜉,𝜔𝑠)(𝐻2(𝑄))𝑛,(4.1) where the constant 𝐶>0 is independent of the size 0 of mesh.

Proof. Since 𝐸𝑠𝛼𝑚=𝐍𝛼𝑚(𝜉,𝜔𝑠)𝐍0𝛼𝑚(𝜉,𝜔𝑠)𝐻10(𝑄), based on Korn inequality and the interpolation theorem, we have that 𝐸𝑠𝛼𝑚2(𝐻1(𝑄))𝑛𝐸𝐶𝑎𝑠𝛼𝑚,𝐸𝑠𝛼𝑚𝐍=𝐶𝑎𝛼𝑚(𝜉,𝜔𝑠)𝐍0𝛼𝑚(𝜉,𝜔𝑠),𝐍𝛼𝑚(𝜉,𝜔𝑠)𝛾0𝐍0𝛼𝑚(𝜉,𝜔𝑠)𝐶0𝐍𝛼𝑚(𝜉,𝜔𝑠)(𝐻2(𝑄))𝑛𝐸𝑠𝛼𝑚(𝐻1(𝑄))𝑛.(4.2) That is, 𝑁𝛼𝑚(𝜉,𝜔𝑠)𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐻1(𝑄))𝑛𝐶0𝑁𝛼𝑚(𝜉,𝜔𝑠)(𝐻2(𝑄))𝑛.(4.3) Aubin-Nitsche lemma [28] yields 𝑁𝛼𝑚(𝜉,𝜔𝑠)𝑁0𝛼𝑚(𝜉,𝜔𝑠)(𝐿2(𝑄))𝑛𝐶02𝑁𝛼𝑚(𝜉,𝜔𝑠)(𝐻2(𝑄))𝑛.(4.4)

Lemma 4.2. Let 𝑁𝛼𝑝𝑚(𝜉,𝜔𝑠),𝛼1=1,2,,𝑛, be the variational solution of the (2.6) and 𝑁0𝛼𝑝𝑚(𝜉,𝜔𝑠) the finite element solution such that 𝑁𝛼𝑝𝑚(𝜉,𝜔𝑠)𝑊2,(𝑄), and then one has 𝜉𝑁𝛼𝑝𝑚(𝜉,𝜔𝑠)𝑁0𝛼𝑝𝑚(𝜉,𝜔𝑠)𝐶0||ln0||2𝜉𝑁𝛼𝑝𝑚𝐿(𝑄),𝑁𝛼𝑝𝑚(𝜉,𝜔𝑠)𝑁0𝛼𝑝𝑚(𝜉,𝜔𝑠)𝐶02||ln0||2𝜉𝑁𝛼𝑝𝑚𝐿(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 ̂𝑎𝑖𝑗𝑘 be the equivalent mechanical parameter tensor matrix based on STSA and ̂𝑎0𝑖𝑗𝑘 its finite element approximation as above, if there exists one constant 𝑁 such that forall𝜔𝑠𝑃,𝑁𝛼𝑝𝑚(𝜉,𝜔𝑠)𝐻2(Q)𝑁. Then one has ̂𝑎𝑖𝑗𝑘̂𝑎0𝑖𝑗𝑘𝐿𝐶0||ln0||𝑁.(4.6)

Proof. From the above algorithm, the following equation is held. ̂𝑎𝑖𝑗𝑘=̂𝑎0𝑖𝑗𝑘+𝑅𝑖𝑗𝑘,(4.7) where 𝑅𝑖𝑗𝑘 is defined by 𝑅𝑖𝑗𝑘=𝑀𝑠=1𝑄𝑎𝑖𝑝𝑞(𝜉,𝜔𝑠)(1/2)/𝜕𝜉𝑝+𝒜/𝜕𝜉𝑞𝑀,(4.8) where denotes 𝜕(𝑁𝑘𝑞𝑗(𝜉,𝜔𝑠)𝑁0𝑘𝑞𝑗(𝜉,𝜔𝑠)) and𝒜 denotes 𝜕(𝑁𝑘𝑝𝑗(𝜉,𝜔𝑠)𝑁0𝑘𝑝𝑗(𝜉,𝜔𝑠))
Since 𝑅 denote the maximum norm of matrix (𝑅𝑖𝑗𝑘)𝑛×𝑛, applying Lemma 4.2 to the above (4.8), we deduce 𝑅=𝑀𝑠=1𝑄𝑎𝑖𝑝𝑞(𝜉,𝜔𝑠)(1/2)/𝜕𝜉𝑝+𝒜/𝜕𝜉𝑞𝑀1𝑀𝑀𝑠=1𝑄𝑎𝑖𝑝𝑞(𝜉,𝜔𝑠)12𝜕𝜉𝑝+𝒜𝜕𝜉𝑞1𝐶𝑀0||ln0||𝑀𝑠=12𝜉𝑁𝑘(𝜉,𝜔𝑠)𝐿,(4.9)
Since there exists one constant 𝑁 such that the function matrices 𝑁𝛼(𝜉,𝜔𝑠𝑁), the above inequality (4.9) yields 𝑅𝐶0||ln0𝑁||.(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 (𝑚1)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 ̂𝑎𝑟𝑖𝑗𝑘(𝑟=𝑚,𝑚1,,1) be the equivalent mechanical parameters tensor of the composite material with r th scale random grains and ̂𝑎𝑟,0𝑖𝑗𝑘(𝑟=𝑚,𝑚1,,1) its finite element approximation; set the size of the last mesh of the finite element in 𝑄𝑠 to be 0; based on the SMSA-FE algorithm, if there exists one constant 𝑁 such that for all, 𝜔𝑠𝑃,𝑁𝛼𝑝𝑚(𝜉𝑟,𝜔𝑠)𝐻2(𝑄)𝑁,(𝑟=𝑚,𝑚1,,1), then one has ̂𝑎𝑟𝑖𝑗𝑘̂𝑎𝑟,0𝑖𝑗𝑘𝐶0||ln0||𝑁,(4.11) where 𝐶 is one constant that is independent of 0 but dependent on the sizes of the other finite element mesh 0𝑟(𝑟=𝑚,𝑚1,,2) in the cell 𝑄𝑟(𝑟=𝑚,𝑚1,,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 ̂𝑎𝑖𝑗𝑘 shall satisfy the following conditions: ̂𝑎𝑖𝑗𝑘=̂𝑎𝑗𝑖𝑘=̂𝑎𝑗𝑖𝑘,𝜇1𝜂𝑖𝑗𝜂𝑖𝑗̂𝑎𝑖𝑗𝑘𝜂𝑖𝑗𝜂𝑘𝜇2𝜂𝑖𝑗𝜂𝑖𝑗,(5.1) where for any symmetry matrix 𝜂=(𝜂𝑖𝑗)𝑛×𝑛, 𝜇10, and 𝜇20.

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 {̂𝑎𝑖𝑗𝑘} be the equivalent mechanical parameter tensor based on STSA [26] and ̂𝑎0𝑖𝑗𝑘 its finite element approximation; if there exists one constant 𝑁 such that for all 𝜔𝑠𝑃,𝑁𝛼𝑝𝑚(𝜉,𝜔𝑠)𝐻2(𝑄)𝑁, the matrix ̂𝑎0𝑖𝑗𝑘 satisfies the following property: ̂𝑎0𝑖𝑗𝑘=̂𝑎0𝑗𝑖𝑘=̂𝑎0𝑗𝑖𝑘𝐾,(5.2)1𝜂𝑖𝑗𝜂𝑖𝑗̂𝑎0𝑖𝑗𝑘𝜂𝑖𝑗𝜂𝑘𝐾2𝜂𝑖𝑗𝜂𝑖𝑗(5.3) for any symmetric matrix 𝜂=(𝜂𝑖)𝑛×𝑛, where 𝐾1,𝐾2 are positive constants.

Proof. Taking into consideration the fact that ̂𝑎𝑖𝑗𝑘(𝜔𝑠)=̂𝑎𝑗𝑖𝑘(1𝑠)=̂𝑎𝑗𝑖𝑘(𝜔𝑠), based on the concept of 𝑁0𝛼(𝜉,𝜔𝑠), taking into account the idea in [7], we deduce that ̂𝑎0𝑗𝑖𝑘(𝜔𝑠1)=||𝑄𝑠||𝑄𝑠𝑎𝑗𝑖𝑘(𝜉,𝜔𝑠)+𝑎𝑗𝑝𝑘𝑞(𝜉,𝜔𝑠)12𝜕𝑁0𝑝𝑖(𝜉,𝜔𝑠)𝜉𝑞+𝜕𝑁0𝑞𝑖(𝜉,𝜔𝑠)𝜉𝑝=1𝐝𝜉||𝑄𝑠||𝑄𝑠𝑎𝑗𝑖𝑘(𝜉,𝜔𝑠)+𝑎𝑞𝑖𝑝(𝜉,𝜔𝑠)12𝜕𝑁0𝑘𝑝𝑗(𝜉,𝜔𝑠)𝜉𝑞+𝜕𝑁0𝑘𝑞𝑗(𝜉,𝜔𝑠)𝜉𝑝𝐝𝜉=̂𝑎0𝑘𝑗𝑖(𝜔𝑠).(5.4) So we have proved ̂𝑎0𝑗𝑖𝑘(𝜔𝑠)=̂𝑎0𝑗𝑖𝑘(𝜔𝑠). Let us establish the first equality in (5.2), which is equivalent to prove (𝐴0𝑘)=𝐴0𝑘, where 𝐴 denotes the transpose of the matrix 𝐴 and 𝐴0𝑘 is the matrix (̂𝑎0𝑖𝑗𝑘)𝑛×𝑛.
From the integral identity for solution of problem (2.6), for any matrix 𝑀0(𝜉)(𝑉0(𝑄𝑠))𝑛×𝑛, we have 𝑄𝑠𝜕(𝑀(𝜉)𝜕𝜉𝑝𝐴𝑝𝑞(𝜉,𝜔𝑠)𝜕𝑁𝑘(𝜉,𝜔𝑠)𝜕𝜉𝑞𝑑𝜉=𝑄𝑠𝐴𝑝𝑘(𝜉,𝜔𝑠)(𝑀(𝜉))𝜕𝜉𝑝𝑑𝜉.(5.5)
Based on the relations (𝐴𝑝𝑞(𝜉))=𝐴𝑞𝑝(𝜉) and (𝐴𝐵)=𝐵𝐴 for matrices 𝐴,𝐵, it is easy to obtain 𝑁𝛼 from (5.5) that 𝑄𝑠𝜕𝑁0𝑘(𝜉,𝜔𝑠)𝜕𝜉𝑝𝐴𝑞𝑝(𝜉,𝜔𝑠)𝜕𝑀(𝜉)𝜕𝜉𝑞𝑑𝜉=𝑄𝑠𝐴𝑘𝑞(𝜉,𝜔𝑠)𝜕𝑀(𝜉)𝜕𝜉𝑞𝑑𝜉.(5.6) If we set 𝑀=𝑁0, the following equations are obtained. 𝑄𝑠𝜕𝑁0𝑘(𝜉,𝜔𝑠)𝜕𝜉𝑝𝐴𝑝𝑞(𝜉,𝜔𝑠)𝜕𝑁0(𝜉,𝜔𝑠)𝜕𝜉𝑞𝑑𝜉=𝑄𝑠𝐴𝑘𝑞(𝜉,𝜔𝑠)𝜕𝑁0(𝜉,𝜔𝑠)𝜕𝜉𝑞𝑑𝜉.(5.7) Taking into account the idea in [7], (5.7), and the concept of 𝑁0(𝜉,𝜔𝑠), the following equations can be obtained. 𝑄𝑠𝜕𝜕𝜉𝑞𝑁0𝑘+𝜉𝑘𝐸𝐴𝑞𝑝(𝜉,𝜔𝑠)𝜕𝜕𝜉𝑝𝑁0+𝜉𝐸=𝑑𝜉𝑄𝑠𝜕𝑁0𝑘𝜕𝜉𝑞𝐴𝑞𝑝(𝜉,𝜔𝑠)𝜕𝑁0𝜕𝜉𝑝+𝛿𝑞𝑘𝐴𝑞𝑝(𝜉,𝜔𝑠)𝜕𝑁0𝜕𝜉𝑝+𝜕𝑁0k𝜕𝜉𝑞𝐴𝑞𝑝(𝜉,𝜔𝑠)𝛿𝑝𝐸+𝛿𝑞𝑘𝐴𝑞𝑝(𝜉,𝜔𝑠)𝛿𝑝𝐸=𝑑𝜉𝑄𝑠𝜕𝑁0𝑘𝜕𝜉𝑞𝐴𝑞𝑝(𝜉,𝜔𝑠)𝜕𝑁0𝜕𝜉𝑝+𝐴𝑘𝑝(𝜉,𝜔𝑠)𝜕𝑁0𝜕𝜉𝑝+𝜕𝑁0𝑘𝜕𝜉𝑞𝐴𝑞(𝜉,𝜔𝑠)+𝐴𝑘(𝜉,𝜔𝑠)=𝑑𝜉𝑄𝑠𝜕𝑁0𝑘𝜕𝜉𝑞𝐴𝑞(𝜉,𝜔𝑠)+𝐴𝑘(𝜉,𝜔𝑠=𝐴)𝑑𝜉0𝑘(𝜔𝑠).(5.8) In the second equation of the above equations, the property of 𝛿𝑖𝑗 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 𝐴𝑗𝑞=𝐴𝑞𝑗: 𝐴0𝑘(𝜔𝑠)=𝑄0𝑥03631𝜕𝜕𝜉𝑞𝑁0+𝜉𝐸𝐴𝑝𝑞(𝜉,𝜔𝑠)𝜕𝜕𝜉𝑝𝑁0𝑘+𝜉𝑘𝐸𝐴𝑑𝜉,0𝑘(𝜔𝑠)=𝑄𝑠𝜕𝜕𝜉𝑞𝑁0𝑘+𝜉𝑘𝐸𝐴𝑞𝑗(𝜉,𝜔𝑠)𝜕𝜕𝜉𝑗𝑁0+𝜉𝐸𝑑𝜉.(5.9) Thus 𝐴0𝑘(𝜔𝑠)==𝑄𝑠𝜕𝜕𝜉𝑗𝑁0+𝜉𝐸𝐴𝑗𝑞(𝜉,𝜔𝑠)𝜕𝜕𝜉𝑞𝑁0𝑘+𝜉𝑘𝐸𝐴𝑑𝜉=0𝑘(𝜔𝑠).(5.10)
Hence, we see that 𝐴0𝑘(𝜔𝑠𝐴)=(0𝑘(𝜔𝑠)). Thus we obtain the following equations: ̂𝑎0𝑖𝑗𝑘(𝜔𝑠)=̂𝑎0𝑗𝑖𝑘(𝜔𝑠)=̂𝑎0𝑗𝑖𝑘(𝜔𝑠).(5.11) Making use of the relations (5.11), one has 1𝑀𝑀𝑠=1̂𝑎0𝑖𝑗𝑘(𝜔𝑠1)=𝑀𝑀𝑠=1̂𝑎0𝑗𝑖𝑘(𝜔𝑠1)=𝑀𝑀𝑠=1̂𝑎0𝑗𝑖𝑘(𝜔𝑠).(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 00 that is small enough such that 𝐶0||ln0||||𝑁𝛼||𝐿<𝜇12.(5.13) We have 𝜇12𝜂𝑖𝑗𝜂𝑖𝑗̂𝑎0𝑖𝑗𝑘𝜂𝑖𝑗𝜂𝑘𝜇2+𝜇12𝜂𝑖𝑗𝜂𝑖𝑗.(5.14) Setting 𝐾1=𝜇1/2 and 𝐾2=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 {̂𝑎𝑟𝑖𝑗𝑘} be the equivalent mechanics parameter tensor of the composite material with r-scale random grains based on the SMSA algorithm and ̂𝑎𝑟,0𝑖𝑗𝑘 their finite element approximation by (2.3). Set the size of the last mesh of the finite element in 𝑄𝑠 to be 0; if there exists the constant 𝑁𝛼1 such that for all, 𝜔𝑠𝑃,𝑁𝑟𝛼𝑝𝑚(𝜉,𝜔𝑠)𝐻2(𝑄)𝑁𝛼𝑝𝑚, then ̂𝑎𝑟,0𝑖𝑗𝑘 satisfy the conditions ̂𝑎𝑟,0𝑖𝑗𝑘=̂𝑎𝑟,0𝑗𝑖𝑘=̂𝑎𝑟,0𝑗𝑖𝑘,𝐾1𝜂𝑖𝑗𝜂𝑖𝑗̂𝑎𝑟,0𝑖𝑗𝑘𝜂𝑖𝑗𝜂𝑘𝐾2𝜂𝑖𝑗𝜂𝑖𝑗,(5.15) where 𝜂=(𝜂𝑖)𝑛×𝑛is the symmetry matrix and 𝐾1,𝐾2 are the positive constants that are independent of 0 but dependent on the sizes of the other finite element meshes 0𝑟,(𝑟=𝑚,𝑚1,,2) in the 𝑟th scale cells 𝑄𝑟(𝑟=𝑚,𝑚1,,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 0, based on the data of Table 2, we obtain the equivalent mechanical parameters tensor {̂𝑎𝑖𝑗𝑘} that are given in Table 3.

tab1
Table 1: Probability distributions of grains in composite material.
tab2
Table 2: Mechanical parameters of the matrix and grains.
tab3
Table 3: Equivalent mechanical parameters {̂𝑎𝑖𝑗(𝜔𝑠)} for different mesh sizes 0.

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 0 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.

tab4
Table 4: Size of rock random grains and the number in a statistical window.
tab5
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 68 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.

tab6
Table 6: Equivalent mechanical parameters of a composites with small rock random grains by the SMSA-FE algorithm (kPa).
tab7
Table 7: Equivalent mechanical parameter tensor for concrete with middle rock random grains by the SMSA-FE algorithm (kPa).
tab8
Table 8: Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (kPa).
tab9
Table 9: Equivalent mechanical parameter tensor for concrete with large rock random grains by the SMSA-FE algorithm (GPa).
fig2
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.

tab10
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).

References

  1. W. Voigt, Lehrbuch der Kristallphysik, Teubner, Leipzig, Germany, 1928.
  2. Z. Hashin and S. Shtrikman, “A variational approach to the theory of the elastic behaviour of multiphase materials,” Journal of the Mechanics and Physics of Solids, vol. 11, no. 2, pp. 127–140, 1963. View at Google Scholar · View at Scopus
  3. B. Budiansky, “On the elastic moduli of some heterogeneous materials,” Journal of the Mechanics and Physics of Solids, vol. 13, no. 4, pp. 223–227, 1965. View at Google Scholar · View at Scopus
  4. R. J. Farris, “Prediction of the viscosity of multimodal suspensions from unimodal viscosity data,” Journal of Rheology, vol. 12, no. 2, pp. 281–301, 1968. View at Google Scholar
  5. J. Zuiker and G. Dvorak, “The effective properties of functionally graded composites-I. Extension of the mori-tanaka method to linearly varying fields,” Composites Engineering, vol. 4, no. 1, pp. 19–35, 1994. View at Google Scholar · View at Scopus
  6. R. Dierk, Computational Materials Science, Wiley-VCH, 1998.
  7. O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, The Netherlands, 1992.
  8. J. Z. Gui and H. Y. Yang, “A dual coupled method for boundary value problems of pde with coefficients of small period,” Journal of Computational Mathematics, vol. 14, no. 2, pp. 159–174, 1996. View at Google Scholar · View at Scopus
  9. Y. M. Poon, W. L. Luk, and F. G. Shin, “Statistical spherical cell model for the elastic properties of particulate-filled composite materials,” Journal of Materials Science, vol. 37, no. 23, pp. 5095–5099, 2002. View at Publisher · View at Google Scholar · View at Scopus
  10. C. P. Tsui, C. Y. Tang, and T. C. Lee, “Finite element analysis of polymer composites filled by interphase coated particles,” Journal of Materials Processing Technology, vol. 117, no. 1-2, pp. 105–110, 2001. View at Publisher · View at Google Scholar · View at Scopus
  11. Y. P. Feng and J. Z. Cui, “Multi-scale FE computation for the structure of composite materials with small periodic configuration under condition of thermo-elasticity,” Acta Mechanical Sinica, vol. 20, no. 1, pp. 54–63, 2004. View at Google Scholar
  12. W. M. He and J. Z. Cui, “A pointwise estimate on the 1-order approximation of Gεx0,” IMA Journal of Applied Mathematics, vol. 70, no. 2, pp. 241–269, 2005. View at Publisher · View at Google Scholar · View at Scopus
  13. L. Q. Cao, “Multiscale asymptotic method of optimal control on the boundary for heat equations of composite materials,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 1103–1118, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer, Berlin, Germany, 1994.
  15. T. Y. Hou and X. H. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media,” Journal of Computational Physics, vol. 134, no. 1, pp. 169–189, 1997. View at Publisher · View at Google Scholar · View at Scopus
  16. E. Vanden-Eijnden, “Numerical techniques for multi-scale dynamical systems with stochastic effects,” Communication Mathemtical Science, vol. 1, no. 2, pp. 385–391, 2003. View at Google Scholar
  17. W. Wang, D. Cao, and J. Duan, “Effective macroscopic dynamics of stochastic partial differential equations in perforated domains,” SIAM Journal on Mathematical Analysis, vol. 38, no. 5, pp. 1508–1527, 2006. View at Publisher · View at Google Scholar · View at Scopus
  18. D. Duschlbauer, H. J. Böhm, and H. E. Pettermann, “Computational simulation of composites reinforced by planar random fibers: homogenization and localization by unit cell and mean field approaches,” Journal of Composite Materials, vol. 40, no. 24, pp. 2217–2234, 2006. View at Publisher · View at Google Scholar · View at Scopus
  19. S. Kari, H. Berger, and U. Gabbert, “Numerical evaluation of effective material properties of randomly distributed short cylindrical fibre composites,” Computational Materials Science, vol. 39, no. 1, pp. 198–204, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. A. L. Kalamkarov, E. M. Hassan, A. V. Georgiades, and M. A. Savi, “Asymptotic homogenization model for 3D grid-reinforced composite structures with generally orthotropic reinforcements,” Composite Structures, vol. 89, no. 2, pp. 186–196, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. A. Kalamkarov, I. Andrianov, and V. Danishevskyy, “Asymptotic homogenization of composite materials and structures,” Applied Mechanics Reviews, vol. 62, no. 3, Article ID 030802, 20 pages, 2010. View at Google Scholar
  22. M. Wang and N. Pan, “Predictions of effective physical properties of complex multiphase materials,” Materials Science and Engineering R, vol. 63, no. 1, pp. 1–30, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Wang and N. Pan, “Elastic property of multiphase composites with random microstructures,” Journal of Computational Physics, vol. 228, no. 16, pp. 5978–5988, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. P. W. Hao and D. L. Zhang, “Performance evaluation of polythene modified asphalt,” in Asia-Pacific Reginal Meeting, Taipei, Taiwan, May 1996.
  25. Y. Li and J. Gui, “Two-scale analysis method for predicting heat transfer performance of composite materials with random grain distribution,” Science in China, Series A, vol. 47, pp. 101–110, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. Y. Li and J. Z. Cui, “The multi-scale computational method for the mechanics parameters of the materials with random distribution of multi-scale grains,” Composites Science and Technology, vol. 65, no. 9, pp. 1447–1458, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. Li, S. Long, and J. Cui, “Finite element computation for mechanics parameters of composite material with randomly distributed multi-scale grains,” Engineering Analysis with Boundary Elements, vol. 32, no. 4, pp. 290–298, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Springer, Berlin, Germany, 1978.