- Email: [email protected]
Finite element computation for mechanics parameters of composite material with randomly distributed multi-scale grains
ARTICLE IN PRESS
Engineering Analysis with Boundary Elements 32 (2008) 290–298 www.elsevier.com/locate/enganabound
Finite element computation for mechanics parameters of composite material with randomly distributed multi-scale grains Youyun Lia,b,, Shuyao Longa, Junzhi Cuic a
College of Mechanics and Aerospace, Hunan University, Hunan 410082, China College of Maths, Changsha University of Science and Technology, Hunan 410076, China c Academy of Mathematics and System Sciences, CAS, Beijing 100080, China
b
Received 9 October 2005; accepted 26 October 2007 Available online 9 January 2008
Abstract This paper presents a finite element (FE) computation model for estimating the equivalent mechanics parameters of a composite material with randomly distributed multi-scale grains. The model is based on statistical multi-scale analysis (SMSA). First, some definitions of the probability space and statistical two-scale analysis (STSA) of the mechanics parameters for a composite material with randomly distributed grains are briefly reviewed. Next, a computer simulation algorithm for composite materials with random grains is presented. Then an FE procedure based on STSA is proposed and some properties of the equivalent mechanics parameters are outlined. Finally, the FE procedure for a composite material with randomly distributed grains is described and two numerical examples are demonstrated. The numerical results show that the FE procedure based on SMSA is a valid and feasible method for predicting the mechanics parameters of a composite material with randomly distributed grains. r 2007 Elsevier Ltd. All rights reserved. Keywords: FE; STSA; SMSA; Equivalent mechanics parameter; Expected homogenization mechanics parameter
1. Introduction In the field of composites, predicting the mechanics parameters of a composite material with randomly distributed multi-scale grains is a very difficult problem. Some physical methods [1,2] have been proposed for solving this problem in the last decade. These techniques are based on empirical or semi-empirical models and cannot handle composite material with large numbers of random grains. Many researchers have shown that multiscale analysis is a valid method for predicting the mechanics parameters of composite materials [3–8]. This method is able not only to show the macrocharacteristics and random configurations of a composite material, but also to greatly decrease the computing time required for the numerical result.
Corresponding author at: College of Mechanics and Aerospace, Hunan University, Hunan 410082, China. E-mail address: [email protected] (Y. Li).
0955-7997/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2007.10.009
In previous papers [3,4], we proposed an expression for predicting the equivalent mechanics parameters of a composite material with randomly distributed grains. However, no research into how to obtain a numerical solution to these differential equations and compute the equivalent mechanics parameters has been carried out. Therefore, in the present paper, a finite element (FE) computation procedure based on statistical multi-scale analysis (SMSA) is presented. The equivalent mechanics parameters of a composite material with randomly distributed grains are predicted using this procedure. The next section describes some definitions in the probability space and a representation of a composite material with randomly distributed multi-scale grains. Statistical two-scale analysis (STSA) of the mechanics parameters for the composite material is reviewed in Section 3. In Section 4, computer modeling for composite materials is presented. Section 5 is devoted to setting up the FE formulation and indicating some properties of the equivalent mechanics parameters. In Section 6, the FE procedure used to compute the equivalent mechanics
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
parameters is described. Numerical results are presented in Section 7 to demonstrate the validity and convergence of the FE procedure based on SMSA.
2. Representation of composite material with randomly distributed multi-scale grains 2.1. Representation of composite material with randomly distributed one-scale grains Suppose that a composite material comprises a matrix and grains (foam particles or cavities) of one scale. For simplicity and without losing generality, all of the grains are considered as ellipses. The size of each grain is denoted by l, which is the size of the long axis in the corresponding ellipse. The long axis a for each ellipse satisfies r1 ol a or2 for one-scale grains. In this paper, we suppose that the difference between the long axis a and the short axis c of an ellipse is not very large, which means that ac od, where d is a positive constant. From engineering surveys and statistical data, the configuration of a composite material with randomly distributed one-scale grains can be described as follows:
be set up as follows: (a) Specify the distribution density of grains or the total volume of grains, and the distribution model of the central points ðx10 ; x20 ; x30 Þ. For example, a sample with randomly uniformly distributed grains in a screen of size is shown in Fig. 1. (b) Specify the distribution model of the long axis a, the middle axis b and the short axis c, and the distribution model of the directions for the axes a and b, yax1 x2 , yax1 , and ybx1 x2 , ybx1 , respectively. The ellipse for each grain is defined by 10 random parameters, i.e., the long axis a, the middle axis b, the short axis c, the directions of the long axis and the middle axis, yax1 x2 , yax1 , and ybx1 x2 , ybx1 , and the coordinates of the center points of the ellipse, ðx10 ; x20 ; x30 Þ. Their probability density functions are denoted by f a ðxÞ, f b ðxÞ, f c ðxÞ, f yax x ðxÞ, f yax ðxÞ, 1 2
1
f ybx x ðxÞ, f ybx ðxÞ, f x 0 ðxÞ, f x 0 ðxÞ and f x 0 ðxÞ, respectively. 1 2
1
1
2
3
Based on the above random parameters defining the ellipse in cell Qs , let N denote the maximum number of ellipses located in this cell. Then we can define a sample os of the grain distribution in the statistical screen as follows: os ¼ ðas1 ; bs1 ; cs1 ; ysax x ; ysax ; ysbx 1 21
1. In a statistical domain O representing a composite material, there exists a lowest constant ðl5Þ. Domain O can be considered as a set of a large number of cells of size , , which are cubes. This means that L 51, where L is the size of O. Thus, O is composed of all the cells of size , as shown in Fig. 1. 2. The probability distribution for the grains in each cell is the same and the statistical domain has a periodically random distribution of grains. The configuration of a composite material of randomly distributed one-scale grains is described by the probability distribution within a statistical screen of size , where ‘screen’ means the sample shown in one window in this paper. The probability model P inside a statistical screen can
291
11
1 x2 1
; ysbx , 11
x10 s1 ; x20 s1 ; x30 s1 ; . . . ; ysbx ; x10 sN ; x20 sN ; x30 sN Þ, 1N
for sample os shown in Fig. 1. Therefore, the statistical domain O is logically composed of -sized cells subject to S the same probability distribution model P, O ¼ ðos ;t2ZÞ ðQs þ tÞ, part of which is shown in Fig. 1. For the whole domain O, we define o ¼ fos ; x 2 Qs Og. Thus, the mechanics coefficient tensor of the composite material can be periodically randomly expressed as faijhk ðx ; oÞg, and for any sample the mechanics parameters faijhk ðx; os Þg can be defined as follows: 8 1 > < aijhk if x 2 ei1 ði1 ¼ 1; . . . ; NÞ i1 ¼N aijhk ðx; os Þ ¼ s 2 > ei 1 ; : aijhk if x 2 Q i [ ¼1
Fig. 1. Part and one sample of the statistical domain O.
1
ARTICLE IN PRESS 292
Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
where Qs denotes the domain of a cell belonging to O, ei1 is i1 th ellipse in Qs , a1ijhk and a2ijhk are constants satisfying jmaxfa1ijhk ; a2ijhk gjoM 1 . It is easy to see that faijhk ðx; oÞg ði; j ¼ 1; 2; 3Þ are bounded and measurable random variables.
10
2.2. Representation of a composite material with randomly distributed multi-scale grains The configuration of a composite material with randomly distributed multi-scale grains, such as concrete, as shown in Fig. 3, can be represented as follows: 1. All grains are considered as ellipses of different scales or polyhedrons inside ellipses, and all ellipses are divided into several classes according to their size (long axis). If m represents the scale number, this means that there exist i ði ¼ 1; . . . ; mÞ satisfying 1 bi bm 40, and grains of size l i ðri1 pl i ori2 Þ are subject to a certain probability distribution within an i statistical screen. 2. On a macroscopic scale, a brick of the material considered is characterized by multi-scale grains. This means that: (a) in terms of class m grains, any brick m m considered comprises grains of size l m a ð0pl a o Þ and the matrix for a statistical screen of size m . (b) For class i grains, i ¼ m 1; . . . ; 1, under the relative i 1 scale of i1 , any brick considered comprises grains of size l ia ðri1 pl ia ori2 Þ and a new matrix, which are equivalent to all k-class ðk ¼ i þ 1; . . . ; mÞ grains and their matrix materials, recursively and successively. For example, consider concrete containing randomly distributed multi-scale grains. The concrete material can be considered as three-scale grains with long axis length of 0:0000120:00004 m, 0:000220:001 m and 0:00520:03 m, denoted by l 3 , l 2 and l 1 , respectively. These can be subjected to the different probability models shown in Fig. 2. We can choose 3 ¼ 0:0002 m, 2 ¼ 0:005 m and 1 ¼ 0:15 m. It is apparent that 1 42 43 . 3. Equivalent mechanics parameters for a composite material with randomly distributed grains based on STSA In this section, we review the mathematical theory that predicts the equivalent mechanics parameters of a composite material with random distribution by STSA [3,4]. From the elasticity theory of a random domain O and the partial differential theory [9], the differential equation and the essential boundary condition can be obtained as 8 > < q a ðx; oÞ1 quh ðx; oÞ þ quk ðx; oÞ ¼ f i ðxÞ; x 2 O; qxj ijhk 2 qxk qxh > : u ðx; oÞ ¼ u¯ ðxÞ; x 2 qO; (1) s
s
where i; j; h; k ¼ 1; 2; . . . ; n, o ¼ S o for x 2 Q, o 2 P, P is the probability space, O ¼ s2p;t2Zn ðQs þ tÞ shown
1 0 0
1
10 Fig. 2. (a) Grains in an 1 screen and (b) grains in an 2 screen.
in Fig. 1, uh ðx; oÞ is the displacement, f i ðxÞ is the load and u¯ ðxÞ is the boundary displacement vector. We can obtain the expected homogenization parameters (equivalent mechanics parameters) from the following theorem [3,4]. Theorem 1. If a reinforced material is subject to some probability space P and the mechanics parameters of the composite material satisfy Eq. (1), the equivalent mechanics parameters based on STSA can be expressed as PM ^ ijhk ðos Þ s¼1 a , (2) a^ ijhk ¼ lim M!1 M where os 2 P; ðs ¼ 1; 2; . . . ; MÞ, M is the maximum number of samples and a^ ijhk ðos Þ is defined as Z 1 a^ ijhk ðos Þ ¼ aijhk ðx; os Þ þ aijpq ðx; os Þ s jmesQ j Qs 1 qN hpk ðx; os Þ qN hqk ðx; os Þ þ dx, ð3Þ 2 qxq qxp and 0
N a1 11 ðx; os Þ B ... Na1 ðx; os Þ ¼ B @ N a1 n1 ðx; os Þ
1 N a1 1n ðx; os Þ C C, ... A N a1 nn ðx; os Þ
where Na1 ðx; os Þ are the solutions of the following partial differential equations and the boundary conditions on unit cube Qs . 8 s q qN a1 km ðx; os Þ > s 1 qN a1 hm ðx; o Þ > a ðx; o Þ þ ijhk > > 2 qxk qxh < qxj (4) qaa1 ijm ðx;os Þ ¼ ; x 2 Qs ; > qxj > > > : N ðx; os Þ ¼ 0; x 2 qQs : a1 m
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
composite material can be evaluated as Z 1 h0 s aijhk ðx; os Þ þ aijpq ðx; os Þ a^ ijhk ðo Þ ¼ jmesQs j Qs !# h0 h0 s s 1 qN hpk ðx; o Þ qN hqk ðx; o Þ þ dx. xq xp 2
4. Computer modeling of a composite material with randomly distributed grains To use the FE method to analyze a composite material of randomly distributed grains, it is necessary to divide subdomain Qs into an FE mesh. To this end, a suitable algorithm is briefly described. Algorithm 1. 1. Specify screen Qs for simulating the composite material and set the volume of reinforced material (VOL) in the composite material. 2. Input the probability distribution of random grains in screen Qs and generate random numbers subject to the probability distribution. 3. Obtain the random grains in Qs such that all the grains do not intersect with each other in Qs . 4. If the volume of grains in the screen reaches VOL, then go to step 5, otherwise go to step 2. 5. Partition domain Qs according to the algorithm [10]. 6. Output the FE mesh and the material data for domain Qs with random grains. Using Algorithm 1, a modeling example in two dimensions is shown in Fig. 4. Although only a triangular mesh is used in this paper, other FE meshes can also be used in the algorithm.
According to mesh data, the FE space ðV h0 ðQs ÞÞn can be established. In this section, we derive an FE formulation to compute the equivalent mechanics parameters by STSA step by step. 1. FE solutions N ha01 m of N a1 m ðx; os Þ. According to Eq. (4), the FE solutions N ha01 m can be derived from the variational principle in ðV h0 ðQs ÞÞn . Let 1 qV j qV i ij ðV Þ ¼ þ 2 qxi qxj
(5)
and then Z Z ij ðV Þaijhk ðx; os Þhk ðN ha01 m Þdx ¼ ij ðV Þaija1 m ðx; os Þ dx Qs
Qs
8V 2 ðV h ðQs ÞÞn ; ðs ¼ 1; 2; . . . ; MÞ.
ð6Þ
0 ðos Þ of a^ ijhk ðos Þ. According to 2. FE approximation a^ hijhk 0 Eq. (3), the FE approximation a^ hijhk ðos Þ of the homogenization mechanics parameters of a sample of
ð7Þ
3. The equivalent mechanics parameters a^ ijhk of a composite material with random grains are approximated as PM h0 ^ ijhk ðos Þ s¼1 a h0 a^ ijhk ¼ . (8) M From the above FE formulation and Theorem 1, it is obvious that the equivalent mechanics parameters can be computed. Corollary 1. If a composite material with random grains is subject to probability distribution P, the equivalent mechanics parameters of the composite material can be approximated as PM h 0 ^ ijhk ðos Þ s¼1 a h0 a^ ijhk ¼ , (9) M 0 where os 2 P; ðs ¼ 1; 2; . . . ; MÞ and a^ hijhk ðos Þ is computed as Z 1 0 a^ hijhk ðos Þ ¼ ðaijhk ðx; os Þ jmesQs j Qs þ aijpq pq ðN ha01 m ðx; os ÞÞÞ dx
5. Two-scale FE formulation for a composite material with random grains
293
ð10Þ
and Nha01 ðx; os Þ ¼ ðNha01 1 ; Nha01 2 ; ; Nha01 n Þ 1 0 h0 N a1 11 ðx; os Þ N ha01 1n ðx; os Þ C B C B .. .. C. B ¼B . . C A @ h0 h0 s s N a1 n1 ðx; o Þ N a1 nn ðx; o Þ
ð11Þ
Nha01 ðx; os Þ are the FE solutions of the following partial differential equations and boundary conditions on unit cell Qs . 8 s > < q ðaijhk ðx; os Þhk ðN a1 m ðx; os ÞÞÞ ¼qaa1 ijm ðx; o Þ; x 2 Qs ; qxj qxj > : Na m ðx; os Þ ¼ 0; x 2 qQs : 1 (12) We can prove that the equivalent mechanics parameter tensor fa^ ijhk g satisfies properties (13) and (14) based on the following hints: 1. Using the FE error estimate technique [11], the error between Nha01 ðx; os Þ and Na1 ðx; os Þ can be estimated. 2. Based on homogenization theory [4,5] and the G-convergence theory [5], property (13) can be obtained.
ARTICLE IN PRESS 294
Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
5. If r ¼ m, aijhk ðxr ; os Þ in 8 > a1ijhk ; > > > > x < a2 ; ijhk aijhk r ; os ¼ > > > > > : alijhk ;
3. Based on homogenization theory [4,5] and the G-convergence theory [5], and using the results in hint 0 1, the error between fa^ hijhk g and fa^ ijhk g can be estimated. Using the triangular inequality, property (14) can be proved. Because the detailed proof requires a great deal of mathematical knowledge, here we only present the properties of the equivalent mechanics parameter tensor, hints for proving the properties and two numerical examples to verify the properties. The proof of the property will be presented in future papers. Let fa^ ijhk g be the equivalent mechanics parameter 0 tensor based on STSA and let fa^ hijhk g be its FE approximah0 tion in Corollary 1, then fa^ ijhk g satisfies the following properties: 0 0 0 a^ hijhk ¼ a^ hhkij ¼ a^ hjikh ,
(13)
0 K^ 1 Zij Zij pa^ hijhk Zij Zhk pK^ 2 Zij Zij ,
(14)
6. FE procedure for a composite material with randomly distributed multi-scale grains by SMSA
Algorithm 2. 1. Specify the scale number m of grains in the composite material and set the iterative number r ¼ 1. 2. Based on the probability distribution derived from the engineering statistics, generate the random numbers subject to the probability distribution of the grain materials F r , r ¼ m; m 1; . . . 1 for the rth scale. 3. Model the composite material using r-scale random grains in domain Qsr and generate meshes of the sample domain according to Algorithm 1. 4. Determine the mechanics parameter tensor of the random grains, the matrix material and their interface.
s x 2 Q^ ; s
x 2 Q~ ;
(15)
bs ; x 2 Q
S bs sS sS sT sT where Q^ ¼ Qsr and Q^ Q~ Q Q~ T bs s Q ¼ f, Qr O denotes the domain of the basic s configuration , Q^ denotes the domain of the matrix s and Q~ denotes the domain of the random grain bs denotes the interface domain material in Qsr and Q between the grains and the matrix. Go to step 7. 6. If rom, aijhk ðxr ; os Þ in Qsr can be indicated as follows: x 8 s s 0 x < a^ hijhk ; o ; x 2 Q^ ; ðrþ1Þ aijhk r ; os ¼ (16) s : a1 ; x 2 Q~ ; ijhk sS s sT s where Q^ Q~ ¼ Qsr and Q^ Q~ ¼ f; Qsr O des notes the domain of the basic configuration , Q^ s denotes the domain of the matrix and Q~ denotes the domain of random grains in Qsr . Go to step 7. 7. Compute the FE approximation N ha01 m ðxr ; os Þ according to Eq. (6), then obtain the FE approximation of 0 sample a^ hijhk ðr ; os Þ on the rth scale according to Eq. (7) and compute the FE approximation of the equivalent 0 mechanics parameters fa^ hijhk ðr1 Þg on the rth scale using Eq. (8). 0 8. Set a^ ijhk ðr1 Þ equal to a^ hijhk ðr1 Þ on the rth scale and r ¼ r 1. If r41, go to step 2. Otherwise, the equivalent mechanics parameters a^ ijhk are the equivalent mechanics parameters of the composite material with multi-scale random grains in O.
where any symmetric matrix Z ¼ ðZih Þn n and K^ 1 ; K^ 2 are positive constants. As a result, the equivalent mechanics parameter tensor in this paper retains symmetric, positive definite properties.
The FE procedure is normally used to determine the 0 unknown parameters Nha01 ðx; os Þ, a^ hijhk ðos Þ and fa^ ijhk g in (12), (10) and (9). Although this process is standard, implementation for the FE formulation still needs to be considered. In the following, we demonstrate implementation of the FE formulation described in Section 5. Let h0 be the maximum mesh size, os ðs ¼ 1; 2; . . . ; MÞ be a set of samples and M be the number of the samples. The equivalent mechanics parameters can be determined by the following FE method based on SMSA.
Qsr can be indicated as follows:
7. Numerical examples To check whether the FE formulation is valid and rational, we developed software for the FE algorithm based on SMSA and constructed some numerical experiments to predict the equivalent mechanics parameters of a composite material with randomly distributed grains. The results obtained are described below. The first example models a composite material with twoscale grains randomly distributed in the two-dimension case. All the grains are divided into two classes according to the size of their long axis, as shown in Table 1, and the size of the statistical screens is 1 and 2 , respectively. In
Table 1 Random distribution of grains in the screens Small grains y a b
Large grains ½0; 2p ½0:003; 0:008 ½0:002; a
y a b
½0; 2p ½0:03; 0:08 ½0:02; a
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
295
each screen, the grains occupy approximately 30% of the volume. From the engineering data, we can assume that the long axis a, short axis b and angle y are subject to the uniform distributions shown in Table 1, and that the mechanics parameters of the matrix and the grains are as
shown in Table 2. One statistical screen with 43 grains for one sample can be easily generated, as shown in Fig. 3a. The FE meshes of Fig. 4a are depicted in Fig. 3b. We generated 50 samples for this example. The homogenization coefficient tensors fa^ ijhk g for some samples in the
Table 2 Mechanics parameters of the matrix and grains
Table 4 Equivalent mechanics parameters for composite material with grains subject to a uniform distribution 0 1 1:095 106 8:302 105 0 B C @ 8:302 105 1:099 105 0 A 5 0 0 1:295 10
Matrix 8:415 105 7:041 105 0
Grains 7:041 105 8:415 105 0
0 0 6:870 104
2:99 106 1:50 106 0
1:50 106 2:99 106 0
0 0 7:50 105
1
0
1 1
Fig. 3. (a) Grains in screen and (b) triangular meshes in an 1 screen.
30 25 20 15 10 5 0 0
40 5
10
15
20
25
30
35
40
0
20
Fig. 4. (a) Part of a sample containing all classes of grains and (b) a sample with only small grains.
Table 3 Homogenization coefficient tensor fa^ ijhk ðos Þg for some samples No.
a11
a12
a13
a22
a23
a33
1 10 20 30 40 50
1:119 106 1:069 106 1:093 106 1:079 106 1:097 106 1:094 106
8:395 105 8:350 105 8:395 105 8:205 105 8:310 105 8:358 105
2:91 103 1:74 103 2:24 103 1:38 103 3:05 103 0:14 103
1:119 103 1:083 103 1:102 103 1:083 103 1:097 103 1:090 103
0:47 103 1:05 103 2:82 103 0:09 103 1:64 103 2:99 103
1:339 106 1:266 106 1:320 106 1:234 106 1:315 106 1:308 106
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
296
Table 5 Size of rock grains and number within a statistical screen Class
Grain size range (mm)
Average size (mm)
Number of rock grains
Large grains Middle grains Small grains
40–80 20–40 5–20
60 30 12.5
34 207 3256
Table 6 Elasticity mechanics parameters of the matrix, grains and joint interface materials Class
Young’s modulus (GPa)
Poisson’s ratio
Volume (kg/m3 )
Rock grains Matrix Joint interface
55.5 26 25
0.16 0.22 0.18
2700 2100 210
The Young module E
27.75 27.7 27.65 27.6 27.55 27.5 0
5
10
15
20 25 30 35 sample numbers
40
45
50
Fig. 5. Young’s modulus for 10, 20, 30, 40 and 50 samples with random small-ellipse rock grains according to the SMSA FE procedure.
Table 7 Homogenization mechanics parameters for samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 147 404:7 B 815 961:5 B B B 812 725:1 B B 0:0000 B B @ 2030:8 1049:6 0
3 106 882:6 B 817 434:5 B B B 818 738:3 B B 0:0000 B B @ 515:6 543:0
815 961:5 812 725:1 3138 775:3 814 195:0
897:8 273:0
2030:8 1696:6
1049:6 192:3
814 195:0 0:0000
3137 595:8 0:0000
450:6 1 161 332:3
809:5 0:0000
1400:0 0:0000
1696:6 192:3
809:5 0:0000
0:0000 303:9
1 160 464:1 628:8
628:8 1 158 896:4
817 434:5
818 738:3
1 C C C C C C C C A
515:6
543:0
3 100 766:2 817 173:1 734:3 817 173:1 3 095 507:1 478:3
571:2 1719:6
277:5 106:5
0:0000 571:2
0:0000 1719:6
1 140 169:4 0:0000
0:0000 1 140 414:4
0:0000 1155:8
277:5
0:0000
807:2
1155:8
1 138 434:2
1275:4
statistical screen with large grains are shown in Table 3. The equivalent mechanics parameters (expected homogenization coefficients) are obtained by Algorithm 2. The final results are given in Table 4. Table 4 demonstrates that if the random parameters defining the grain ellipses are subject to uniform probability distributions, a composite material with randomly distributed grains shows isotropic mechanics performance and satisfies the symmetric, positive definitive properties in (13) and (14). In this numerical example, it should be noted that the original data are not all from physical experiments, and the results obtained cannot be compared with those from physical experiments. Table 4 only shows that the FE procedure based on SMSA is feasible for computation of the equivalent mechanics parameters for a composite material with a random grain distribution. However, in the next example, we present numerical results based on data from engineering practice. The second example is based on a concrete material with a large number of three-scale random rock grains in three dimensions, for which data were derived from an engineering project. The matrix, which is considered as a type of isotropic material, is made from sand and cement. The sizes of the three-scale rock grains in the concrete are shown in Table 5. Their elasticity coefficients are shown in Table 6. If all grains are generated in a large statistical screen of 500 mm, the number of grains is approximately 3500. Part of a sample containing all classes of the random grains is shown in Fig. 5 and the number of grains for three classes is listed in the last column of Table 5. Numerical results for the mechanics parameters for some samples with only small rock grains, which were generated using the FE procedure based on SMSA, are listed in Table 7. The expected values of the equivalent mechanics parameters evaluated using the FE procedure for 10, 30 and 50 samples with only small-ellipse grains are listed in Tables 8–10, respectively. The expected Young’s modulus and Poisson’s ratio for the equivalent mechanics parameters evaluated by Algorithm 2 for 10, 20, 30, 40 and 50 samples with only small-ellipse grains are shown in Figs. 5 and 6, respectively. Figs. 5 and 6 show that the equivalent mechanics parameters evaluated for the different samples using Algorithm 2 are convergent. Numerical results for the mechanics parameters computed using the FE procedure based on SMSA for 50
1 C C C C C C C C A
Table 8 Expected values of the mechanics parameters for 10 samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 119 188:8 B 820 319:4 B B B 819 923:7 B B 0:0000 B B @ 952:9 292:7
820 319:4
819 923:7
1002:6
952:9
292:7
3 112 201:4 819 734:6
819 734:6 3 109 392:8
726:7 77:0
771:5 5:9
892:9 1023:8
0:0000 771:5
0:0000 5:9
1 145 336:6 0:0000
0:0000 1 144 667:7
0:0000 808:1
892:9
0:0000
964:1
808:1
1 143 172:1
1 C C C C C C C C A
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298 Table 9 Expected values of the mechanics parameters for 30 samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 113 742:9 B 819 643:6 B B B 819 016:0 B B 0:0000 B B @ 141:7 308:6
819 643:6 819 016:0 3 106 156:0 819 206:6
880:4 646:5
141:7 641:9
308:6 191:1
819 206:6 0:0000
3 103780:0 0:0000
162:1 1 142 567:6
502:0 0:0000
178:7 0:0000
641:9 191:1
502:0 0:0000
0:0000 585:0
1 141 982:5 752:3 752:3 1 140 693:8
1 C C C C C C C C A
Table 10 Expected values of the mechanics parameters for 50 samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 114 007:0 B 819 970:9 B B B 819 622:9 B B 0:0000 B B @ 176:8 355:5
1
819 970:9
819 622:9
909:9
176:8
355:5
310 6579:8 819 470:0
819 470:0 3 104 792:6
555:4 203:9
505:4 422:8
215:5 186:5
0:0000 505:4
0:0000 422:8
1 142 642:4 0:0000
0:0000 0:0000 1 142 541:2 762:5
215:5
0:0000
730:5
762:5
C C C C C C C C A
1 140 862:9
0.21 0.208
The Posion V
0.206 0.204 0.202 0.2 0.198 0.196 0.194 0.192 0.19 0
5
10
15
20 25 30 35 sample numbers
40
45
50
Fig. 6. Poisson’s ratio for 10, 20, 30, 40 and 50 samples with random small-ellipse rock grains according to the SMSA FE procedure.
Table 11 Equivalent mechanics parameters for a composite material with random small grains using the SMSA FE procedure or ANSYS software (10 kPa) E ellipse ¼ 2762557:7 E ball ¼ 2847824 E average ¼ 3044468 E ANSYS ¼ 2820000
V ellipse ¼ 0:208846 V ball ¼ 0:199713 V average ¼ 0:194606 V ANSYS ¼ 0:198000
samples with only small-ellipse rock grains are listed in the first line of Table 11. We also used ANSYS software to compute Young’s modulus and Poisson’s ratio for a larger brick with a uniform distribution of random small-ellipse grains using three times the length of the examples and the average volume. A comparison of the numerical results
297
Table 12 Equivalent mechanics parameters for concrete with multi-scale rock grains randomly distributed according to the SMSA FE procedure (10 kPa) 3 238 870.7 789 174.8 789 124.4 0.0000 36.8 21.4
789 174.8 3 238 878.0 789 152.8 0.0000 17.7 0.23
789124.4 789 152.8 3 239 046.7 0.0000 86.6 0.0000
8.7 19.6 16.6 1 224 722.6 0.000 5.4
36.8 17.7 86.6 0.0000 1 224 728.9 33.2
21.4 0.23 14.0 0.0000 33.2 1 224 798.4
obtained using ellipse grains and ball grains for Young’s modulus and Poisson’s ratio is also shown in Table 11. From Tables 8–10 it is evident that the numerical results for the equivalent parameters computed using the SMSAbased FE procedure possess the symmetrical, positive and definitive properties (13) and (14). From Table 11, it follows that values for Young’s modulus and Poisson’s ratios produced by the SMSA FE procedure and by the ANSYS software for the domain are very close. This demonstrates that the FE procedure based on SMSA is valid in our example. Because composite material with both uniformly distributed random elliptical grains and random ball grains is isotropic and their equivalent mechanics parameters are very close, to reduce the computing time and to use uniform probability distribution model, three-scale ellipse grains can be substituted by three-scale ball grains. The equivalent mechanics parameters for the concrete defined previously were computed, and the numerical results are listed in Table 12. It should be noted that if the random parameters for elliptical grains are not subject to a uniform probability model, materials with random grains may not possess the isotropic property. To verify the model’s feasibility and accuracy, in future work, we will continue to use our research results in an engineering project on the Xiaowan dam being constructed in Yunnan province in southern China. Moreover, using mathematical theory, we hope to obtain an error estimation for the SMSA-based FE procedure to estimate the accuracy of the equivalent mechanics parameters for composite material with multi-scale random grains. 8. Conclusion In this paper we developed an FE computational procedure to evaluate the equivalent mechanics parameters for a composite material with random multi-scale grains. The FE procedure is based on SMSA and the assumption that the random grains are located within a statistical screen. An FE formulation was derived and a procedure for evaluating the equivalent mechanics parameters of composite material with random multi-scale grains was presented. Various test examples were solved using the FE procedure developed. The numerical results show that the SMSA-based FE procedure is feasible and valid. The procedure can also be extended to other composite
ARTICLE IN PRESS 298
Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
materials with random short fibers, random foams/cavities, etc. Acknowledgments This work was supported by the National Natural Science Foundation of China (10590353,10672055,70601003) and by a project supported by the Scientific Research Fund of Hunan Provincial Education Department (06B003) by a project supported by the Scientific Research Fund of Hunan Provincial Science Department (2007RS4007). References [1] Du SY, Wang B. Micromechanics for composite materials. Science Press; 1998. [2] Dierk R. Computational materials science. Wiley-VCH; 1998. [3] Li YY, Cui JZ. Two-scale analysis method predicting heat transfer performance of composite material with random grain distribution. Sci China Ser A Math 2004;47:100–10.
[4] Li YY, Cui JZ. The multi-scale computational method for the mechanics parameters of the materials with random distribution of multi-scale grains. Compos Sci Technol 2005;65:1447–58. [5] Oleinik OA, Shamaev AS, Yosifian GA. Mathematical problems in elasticity and homogenization. Amsterdam: North-Holland; 1992. [6] Jikov VV, Kozlov SM, Oleinik OA. Homogenization of differential operators and integral functions. Springer; 1994. [7] Cui JZ, Yang HY. A dual coupled method of boundary value problems of PDE with coefficients of small period. Int J Comput Math 1996;14:159–74. [8] Li YY. The Multi-scale Analysis for the Elastics Mechanics Performance of the Composite Material with Rectangle/Cuboid Periodicity. J Hunan City Univ (Nat Sci) 2005;4:30–4. [9] Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer; 1977. [10] Li YY, Cui JZ. The computer simulated method for the domain with large numbers of random ellipse grains/cavities and the improved automatic generation triangle mesh algorithm. Chin J Comput Mech 2004;21:290–5. [11] Ciarlet PG. The finite element method for elliptic problems. Berlin: Springer; 1978.
Engineering Analysis with Boundary Elements 32 (2008) 290–298 www.elsevier.com/locate/enganabound
Finite element computation for mechanics parameters of composite material with randomly distributed multi-scale grains Youyun Lia,b,, Shuyao Longa, Junzhi Cuic a
College of Mechanics and Aerospace, Hunan University, Hunan 410082, China College of Maths, Changsha University of Science and Technology, Hunan 410076, China c Academy of Mathematics and System Sciences, CAS, Beijing 100080, China
b
Received 9 October 2005; accepted 26 October 2007 Available online 9 January 2008
Abstract This paper presents a finite element (FE) computation model for estimating the equivalent mechanics parameters of a composite material with randomly distributed multi-scale grains. The model is based on statistical multi-scale analysis (SMSA). First, some definitions of the probability space and statistical two-scale analysis (STSA) of the mechanics parameters for a composite material with randomly distributed grains are briefly reviewed. Next, a computer simulation algorithm for composite materials with random grains is presented. Then an FE procedure based on STSA is proposed and some properties of the equivalent mechanics parameters are outlined. Finally, the FE procedure for a composite material with randomly distributed grains is described and two numerical examples are demonstrated. The numerical results show that the FE procedure based on SMSA is a valid and feasible method for predicting the mechanics parameters of a composite material with randomly distributed grains. r 2007 Elsevier Ltd. All rights reserved. Keywords: FE; STSA; SMSA; Equivalent mechanics parameter; Expected homogenization mechanics parameter
1. Introduction In the field of composites, predicting the mechanics parameters of a composite material with randomly distributed multi-scale grains is a very difficult problem. Some physical methods [1,2] have been proposed for solving this problem in the last decade. These techniques are based on empirical or semi-empirical models and cannot handle composite material with large numbers of random grains. Many researchers have shown that multiscale analysis is a valid method for predicting the mechanics parameters of composite materials [3–8]. This method is able not only to show the macrocharacteristics and random configurations of a composite material, but also to greatly decrease the computing time required for the numerical result.
Corresponding author at: College of Mechanics and Aerospace, Hunan University, Hunan 410082, China. E-mail address: [email protected] (Y. Li).
0955-7997/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2007.10.009
In previous papers [3,4], we proposed an expression for predicting the equivalent mechanics parameters of a composite material with randomly distributed grains. However, no research into how to obtain a numerical solution to these differential equations and compute the equivalent mechanics parameters has been carried out. Therefore, in the present paper, a finite element (FE) computation procedure based on statistical multi-scale analysis (SMSA) is presented. The equivalent mechanics parameters of a composite material with randomly distributed grains are predicted using this procedure. The next section describes some definitions in the probability space and a representation of a composite material with randomly distributed multi-scale grains. Statistical two-scale analysis (STSA) of the mechanics parameters for the composite material is reviewed in Section 3. In Section 4, computer modeling for composite materials is presented. Section 5 is devoted to setting up the FE formulation and indicating some properties of the equivalent mechanics parameters. In Section 6, the FE procedure used to compute the equivalent mechanics
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
parameters is described. Numerical results are presented in Section 7 to demonstrate the validity and convergence of the FE procedure based on SMSA.
2. Representation of composite material with randomly distributed multi-scale grains 2.1. Representation of composite material with randomly distributed one-scale grains Suppose that a composite material comprises a matrix and grains (foam particles or cavities) of one scale. For simplicity and without losing generality, all of the grains are considered as ellipses. The size of each grain is denoted by l, which is the size of the long axis in the corresponding ellipse. The long axis a for each ellipse satisfies r1 ol a or2 for one-scale grains. In this paper, we suppose that the difference between the long axis a and the short axis c of an ellipse is not very large, which means that ac od, where d is a positive constant. From engineering surveys and statistical data, the configuration of a composite material with randomly distributed one-scale grains can be described as follows:
be set up as follows: (a) Specify the distribution density of grains or the total volume of grains, and the distribution model of the central points ðx10 ; x20 ; x30 Þ. For example, a sample with randomly uniformly distributed grains in a screen of size is shown in Fig. 1. (b) Specify the distribution model of the long axis a, the middle axis b and the short axis c, and the distribution model of the directions for the axes a and b, yax1 x2 , yax1 , and ybx1 x2 , ybx1 , respectively. The ellipse for each grain is defined by 10 random parameters, i.e., the long axis a, the middle axis b, the short axis c, the directions of the long axis and the middle axis, yax1 x2 , yax1 , and ybx1 x2 , ybx1 , and the coordinates of the center points of the ellipse, ðx10 ; x20 ; x30 Þ. Their probability density functions are denoted by f a ðxÞ, f b ðxÞ, f c ðxÞ, f yax x ðxÞ, f yax ðxÞ, 1 2
1
f ybx x ðxÞ, f ybx ðxÞ, f x 0 ðxÞ, f x 0 ðxÞ and f x 0 ðxÞ, respectively. 1 2
1
1
2
3
Based on the above random parameters defining the ellipse in cell Qs , let N denote the maximum number of ellipses located in this cell. Then we can define a sample os of the grain distribution in the statistical screen as follows: os ¼ ðas1 ; bs1 ; cs1 ; ysax x ; ysax ; ysbx 1 21
1. In a statistical domain O representing a composite material, there exists a lowest constant ðl5Þ. Domain O can be considered as a set of a large number of cells of size , , which are cubes. This means that L 51, where L is the size of O. Thus, O is composed of all the cells of size , as shown in Fig. 1. 2. The probability distribution for the grains in each cell is the same and the statistical domain has a periodically random distribution of grains. The configuration of a composite material of randomly distributed one-scale grains is described by the probability distribution within a statistical screen of size , where ‘screen’ means the sample shown in one window in this paper. The probability model P inside a statistical screen can
291
11
1 x2 1
; ysbx , 11
x10 s1 ; x20 s1 ; x30 s1 ; . . . ; ysbx ; x10 sN ; x20 sN ; x30 sN Þ, 1N
for sample os shown in Fig. 1. Therefore, the statistical domain O is logically composed of -sized cells subject to S the same probability distribution model P, O ¼ ðos ;t2ZÞ ðQs þ tÞ, part of which is shown in Fig. 1. For the whole domain O, we define o ¼ fos ; x 2 Qs Og. Thus, the mechanics coefficient tensor of the composite material can be periodically randomly expressed as faijhk ðx ; oÞg, and for any sample the mechanics parameters faijhk ðx; os Þg can be defined as follows: 8 1 > < aijhk if x 2 ei1 ði1 ¼ 1; . . . ; NÞ i1 ¼N aijhk ðx; os Þ ¼ s 2 > ei 1 ; : aijhk if x 2 Q i [ ¼1
Fig. 1. Part and one sample of the statistical domain O.
1
ARTICLE IN PRESS 292
Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
where Qs denotes the domain of a cell belonging to O, ei1 is i1 th ellipse in Qs , a1ijhk and a2ijhk are constants satisfying jmaxfa1ijhk ; a2ijhk gjoM 1 . It is easy to see that faijhk ðx; oÞg ði; j ¼ 1; 2; 3Þ are bounded and measurable random variables.
10
2.2. Representation of a composite material with randomly distributed multi-scale grains The configuration of a composite material with randomly distributed multi-scale grains, such as concrete, as shown in Fig. 3, can be represented as follows: 1. All grains are considered as ellipses of different scales or polyhedrons inside ellipses, and all ellipses are divided into several classes according to their size (long axis). If m represents the scale number, this means that there exist i ði ¼ 1; . . . ; mÞ satisfying 1 bi bm 40, and grains of size l i ðri1 pl i ori2 Þ are subject to a certain probability distribution within an i statistical screen. 2. On a macroscopic scale, a brick of the material considered is characterized by multi-scale grains. This means that: (a) in terms of class m grains, any brick m m considered comprises grains of size l m a ð0pl a o Þ and the matrix for a statistical screen of size m . (b) For class i grains, i ¼ m 1; . . . ; 1, under the relative i 1 scale of i1 , any brick considered comprises grains of size l ia ðri1 pl ia ori2 Þ and a new matrix, which are equivalent to all k-class ðk ¼ i þ 1; . . . ; mÞ grains and their matrix materials, recursively and successively. For example, consider concrete containing randomly distributed multi-scale grains. The concrete material can be considered as three-scale grains with long axis length of 0:0000120:00004 m, 0:000220:001 m and 0:00520:03 m, denoted by l 3 , l 2 and l 1 , respectively. These can be subjected to the different probability models shown in Fig. 2. We can choose 3 ¼ 0:0002 m, 2 ¼ 0:005 m and 1 ¼ 0:15 m. It is apparent that 1 42 43 . 3. Equivalent mechanics parameters for a composite material with randomly distributed grains based on STSA In this section, we review the mathematical theory that predicts the equivalent mechanics parameters of a composite material with random distribution by STSA [3,4]. From the elasticity theory of a random domain O and the partial differential theory [9], the differential equation and the essential boundary condition can be obtained as 8 > < q a ðx; oÞ1 quh ðx; oÞ þ quk ðx; oÞ ¼ f i ðxÞ; x 2 O; qxj ijhk 2 qxk qxh > : u ðx; oÞ ¼ u¯ ðxÞ; x 2 qO; (1) s
s
where i; j; h; k ¼ 1; 2; . . . ; n, o ¼ S o for x 2 Q, o 2 P, P is the probability space, O ¼ s2p;t2Zn ðQs þ tÞ shown
1 0 0
1
10 Fig. 2. (a) Grains in an 1 screen and (b) grains in an 2 screen.
in Fig. 1, uh ðx; oÞ is the displacement, f i ðxÞ is the load and u¯ ðxÞ is the boundary displacement vector. We can obtain the expected homogenization parameters (equivalent mechanics parameters) from the following theorem [3,4]. Theorem 1. If a reinforced material is subject to some probability space P and the mechanics parameters of the composite material satisfy Eq. (1), the equivalent mechanics parameters based on STSA can be expressed as PM ^ ijhk ðos Þ s¼1 a , (2) a^ ijhk ¼ lim M!1 M where os 2 P; ðs ¼ 1; 2; . . . ; MÞ, M is the maximum number of samples and a^ ijhk ðos Þ is defined as Z 1 a^ ijhk ðos Þ ¼ aijhk ðx; os Þ þ aijpq ðx; os Þ s jmesQ j Qs 1 qN hpk ðx; os Þ qN hqk ðx; os Þ þ dx, ð3Þ 2 qxq qxp and 0
N a1 11 ðx; os Þ B ... Na1 ðx; os Þ ¼ B @ N a1 n1 ðx; os Þ
1 N a1 1n ðx; os Þ C C, ... A N a1 nn ðx; os Þ
where Na1 ðx; os Þ are the solutions of the following partial differential equations and the boundary conditions on unit cube Qs . 8 s q qN a1 km ðx; os Þ > s 1 qN a1 hm ðx; o Þ > a ðx; o Þ þ ijhk > > 2 qxk qxh < qxj (4) qaa1 ijm ðx;os Þ ¼ ; x 2 Qs ; > qxj > > > : N ðx; os Þ ¼ 0; x 2 qQs : a1 m
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
composite material can be evaluated as Z 1 h0 s aijhk ðx; os Þ þ aijpq ðx; os Þ a^ ijhk ðo Þ ¼ jmesQs j Qs !# h0 h0 s s 1 qN hpk ðx; o Þ qN hqk ðx; o Þ þ dx. xq xp 2
4. Computer modeling of a composite material with randomly distributed grains To use the FE method to analyze a composite material of randomly distributed grains, it is necessary to divide subdomain Qs into an FE mesh. To this end, a suitable algorithm is briefly described. Algorithm 1. 1. Specify screen Qs for simulating the composite material and set the volume of reinforced material (VOL) in the composite material. 2. Input the probability distribution of random grains in screen Qs and generate random numbers subject to the probability distribution. 3. Obtain the random grains in Qs such that all the grains do not intersect with each other in Qs . 4. If the volume of grains in the screen reaches VOL, then go to step 5, otherwise go to step 2. 5. Partition domain Qs according to the algorithm [10]. 6. Output the FE mesh and the material data for domain Qs with random grains. Using Algorithm 1, a modeling example in two dimensions is shown in Fig. 4. Although only a triangular mesh is used in this paper, other FE meshes can also be used in the algorithm.
According to mesh data, the FE space ðV h0 ðQs ÞÞn can be established. In this section, we derive an FE formulation to compute the equivalent mechanics parameters by STSA step by step. 1. FE solutions N ha01 m of N a1 m ðx; os Þ. According to Eq. (4), the FE solutions N ha01 m can be derived from the variational principle in ðV h0 ðQs ÞÞn . Let 1 qV j qV i ij ðV Þ ¼ þ 2 qxi qxj
(5)
and then Z Z ij ðV Þaijhk ðx; os Þhk ðN ha01 m Þdx ¼ ij ðV Þaija1 m ðx; os Þ dx Qs
Qs
8V 2 ðV h ðQs ÞÞn ; ðs ¼ 1; 2; . . . ; MÞ.
ð6Þ
0 ðos Þ of a^ ijhk ðos Þ. According to 2. FE approximation a^ hijhk 0 Eq. (3), the FE approximation a^ hijhk ðos Þ of the homogenization mechanics parameters of a sample of
ð7Þ
3. The equivalent mechanics parameters a^ ijhk of a composite material with random grains are approximated as PM h0 ^ ijhk ðos Þ s¼1 a h0 a^ ijhk ¼ . (8) M From the above FE formulation and Theorem 1, it is obvious that the equivalent mechanics parameters can be computed. Corollary 1. If a composite material with random grains is subject to probability distribution P, the equivalent mechanics parameters of the composite material can be approximated as PM h 0 ^ ijhk ðos Þ s¼1 a h0 a^ ijhk ¼ , (9) M 0 where os 2 P; ðs ¼ 1; 2; . . . ; MÞ and a^ hijhk ðos Þ is computed as Z 1 0 a^ hijhk ðos Þ ¼ ðaijhk ðx; os Þ jmesQs j Qs þ aijpq pq ðN ha01 m ðx; os ÞÞÞ dx
5. Two-scale FE formulation for a composite material with random grains
293
ð10Þ
and Nha01 ðx; os Þ ¼ ðNha01 1 ; Nha01 2 ; ; Nha01 n Þ 1 0 h0 N a1 11 ðx; os Þ N ha01 1n ðx; os Þ C B C B .. .. C. B ¼B . . C A @ h0 h0 s s N a1 n1 ðx; o Þ N a1 nn ðx; o Þ
ð11Þ
Nha01 ðx; os Þ are the FE solutions of the following partial differential equations and boundary conditions on unit cell Qs . 8 s > < q ðaijhk ðx; os Þhk ðN a1 m ðx; os ÞÞÞ ¼qaa1 ijm ðx; o Þ; x 2 Qs ; qxj qxj > : Na m ðx; os Þ ¼ 0; x 2 qQs : 1 (12) We can prove that the equivalent mechanics parameter tensor fa^ ijhk g satisfies properties (13) and (14) based on the following hints: 1. Using the FE error estimate technique [11], the error between Nha01 ðx; os Þ and Na1 ðx; os Þ can be estimated. 2. Based on homogenization theory [4,5] and the G-convergence theory [5], property (13) can be obtained.
ARTICLE IN PRESS 294
Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
5. If r ¼ m, aijhk ðxr ; os Þ in 8 > a1ijhk ; > > > > x < a2 ; ijhk aijhk r ; os ¼ > > > > > : alijhk ;
3. Based on homogenization theory [4,5] and the G-convergence theory [5], and using the results in hint 0 1, the error between fa^ hijhk g and fa^ ijhk g can be estimated. Using the triangular inequality, property (14) can be proved. Because the detailed proof requires a great deal of mathematical knowledge, here we only present the properties of the equivalent mechanics parameter tensor, hints for proving the properties and two numerical examples to verify the properties. The proof of the property will be presented in future papers. Let fa^ ijhk g be the equivalent mechanics parameter 0 tensor based on STSA and let fa^ hijhk g be its FE approximah0 tion in Corollary 1, then fa^ ijhk g satisfies the following properties: 0 0 0 a^ hijhk ¼ a^ hhkij ¼ a^ hjikh ,
(13)
0 K^ 1 Zij Zij pa^ hijhk Zij Zhk pK^ 2 Zij Zij ,
(14)
6. FE procedure for a composite material with randomly distributed multi-scale grains by SMSA
Algorithm 2. 1. Specify the scale number m of grains in the composite material and set the iterative number r ¼ 1. 2. Based on the probability distribution derived from the engineering statistics, generate the random numbers subject to the probability distribution of the grain materials F r , r ¼ m; m 1; . . . 1 for the rth scale. 3. Model the composite material using r-scale random grains in domain Qsr and generate meshes of the sample domain according to Algorithm 1. 4. Determine the mechanics parameter tensor of the random grains, the matrix material and their interface.
s x 2 Q^ ; s
x 2 Q~ ;
(15)
bs ; x 2 Q
S bs sS sS sT sT where Q^ ¼ Qsr and Q^ Q~ Q Q~ T bs s Q ¼ f, Qr O denotes the domain of the basic s configuration , Q^ denotes the domain of the matrix s and Q~ denotes the domain of the random grain bs denotes the interface domain material in Qsr and Q between the grains and the matrix. Go to step 7. 6. If rom, aijhk ðxr ; os Þ in Qsr can be indicated as follows: x 8 s s 0 x < a^ hijhk ; o ; x 2 Q^ ; ðrþ1Þ aijhk r ; os ¼ (16) s : a1 ; x 2 Q~ ; ijhk sS s sT s where Q^ Q~ ¼ Qsr and Q^ Q~ ¼ f; Qsr O des notes the domain of the basic configuration , Q^ s denotes the domain of the matrix and Q~ denotes the domain of random grains in Qsr . Go to step 7. 7. Compute the FE approximation N ha01 m ðxr ; os Þ according to Eq. (6), then obtain the FE approximation of 0 sample a^ hijhk ðr ; os Þ on the rth scale according to Eq. (7) and compute the FE approximation of the equivalent 0 mechanics parameters fa^ hijhk ðr1 Þg on the rth scale using Eq. (8). 0 8. Set a^ ijhk ðr1 Þ equal to a^ hijhk ðr1 Þ on the rth scale and r ¼ r 1. If r41, go to step 2. Otherwise, the equivalent mechanics parameters a^ ijhk are the equivalent mechanics parameters of the composite material with multi-scale random grains in O.
where any symmetric matrix Z ¼ ðZih Þn n and K^ 1 ; K^ 2 are positive constants. As a result, the equivalent mechanics parameter tensor in this paper retains symmetric, positive definite properties.
The FE procedure is normally used to determine the 0 unknown parameters Nha01 ðx; os Þ, a^ hijhk ðos Þ and fa^ ijhk g in (12), (10) and (9). Although this process is standard, implementation for the FE formulation still needs to be considered. In the following, we demonstrate implementation of the FE formulation described in Section 5. Let h0 be the maximum mesh size, os ðs ¼ 1; 2; . . . ; MÞ be a set of samples and M be the number of the samples. The equivalent mechanics parameters can be determined by the following FE method based on SMSA.
Qsr can be indicated as follows:
7. Numerical examples To check whether the FE formulation is valid and rational, we developed software for the FE algorithm based on SMSA and constructed some numerical experiments to predict the equivalent mechanics parameters of a composite material with randomly distributed grains. The results obtained are described below. The first example models a composite material with twoscale grains randomly distributed in the two-dimension case. All the grains are divided into two classes according to the size of their long axis, as shown in Table 1, and the size of the statistical screens is 1 and 2 , respectively. In
Table 1 Random distribution of grains in the screens Small grains y a b
Large grains ½0; 2p ½0:003; 0:008 ½0:002; a
y a b
½0; 2p ½0:03; 0:08 ½0:02; a
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
295
each screen, the grains occupy approximately 30% of the volume. From the engineering data, we can assume that the long axis a, short axis b and angle y are subject to the uniform distributions shown in Table 1, and that the mechanics parameters of the matrix and the grains are as
shown in Table 2. One statistical screen with 43 grains for one sample can be easily generated, as shown in Fig. 3a. The FE meshes of Fig. 4a are depicted in Fig. 3b. We generated 50 samples for this example. The homogenization coefficient tensors fa^ ijhk g for some samples in the
Table 2 Mechanics parameters of the matrix and grains
Table 4 Equivalent mechanics parameters for composite material with grains subject to a uniform distribution 0 1 1:095 106 8:302 105 0 B C @ 8:302 105 1:099 105 0 A 5 0 0 1:295 10
Matrix 8:415 105 7:041 105 0
Grains 7:041 105 8:415 105 0
0 0 6:870 104
2:99 106 1:50 106 0
1:50 106 2:99 106 0
0 0 7:50 105
1
0
1 1
Fig. 3. (a) Grains in screen and (b) triangular meshes in an 1 screen.
30 25 20 15 10 5 0 0
40 5
10
15
20
25
30
35
40
0
20
Fig. 4. (a) Part of a sample containing all classes of grains and (b) a sample with only small grains.
Table 3 Homogenization coefficient tensor fa^ ijhk ðos Þg for some samples No.
a11
a12
a13
a22
a23
a33
1 10 20 30 40 50
1:119 106 1:069 106 1:093 106 1:079 106 1:097 106 1:094 106
8:395 105 8:350 105 8:395 105 8:205 105 8:310 105 8:358 105
2:91 103 1:74 103 2:24 103 1:38 103 3:05 103 0:14 103
1:119 103 1:083 103 1:102 103 1:083 103 1:097 103 1:090 103
0:47 103 1:05 103 2:82 103 0:09 103 1:64 103 2:99 103
1:339 106 1:266 106 1:320 106 1:234 106 1:315 106 1:308 106
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
296
Table 5 Size of rock grains and number within a statistical screen Class
Grain size range (mm)
Average size (mm)
Number of rock grains
Large grains Middle grains Small grains
40–80 20–40 5–20
60 30 12.5
34 207 3256
Table 6 Elasticity mechanics parameters of the matrix, grains and joint interface materials Class
Young’s modulus (GPa)
Poisson’s ratio
Volume (kg/m3 )
Rock grains Matrix Joint interface
55.5 26 25
0.16 0.22 0.18
2700 2100 210
The Young module E
27.75 27.7 27.65 27.6 27.55 27.5 0
5
10
15
20 25 30 35 sample numbers
40
45
50
Fig. 5. Young’s modulus for 10, 20, 30, 40 and 50 samples with random small-ellipse rock grains according to the SMSA FE procedure.
Table 7 Homogenization mechanics parameters for samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 147 404:7 B 815 961:5 B B B 812 725:1 B B 0:0000 B B @ 2030:8 1049:6 0
3 106 882:6 B 817 434:5 B B B 818 738:3 B B 0:0000 B B @ 515:6 543:0
815 961:5 812 725:1 3138 775:3 814 195:0
897:8 273:0
2030:8 1696:6
1049:6 192:3
814 195:0 0:0000
3137 595:8 0:0000
450:6 1 161 332:3
809:5 0:0000
1400:0 0:0000
1696:6 192:3
809:5 0:0000
0:0000 303:9
1 160 464:1 628:8
628:8 1 158 896:4
817 434:5
818 738:3
1 C C C C C C C C A
515:6
543:0
3 100 766:2 817 173:1 734:3 817 173:1 3 095 507:1 478:3
571:2 1719:6
277:5 106:5
0:0000 571:2
0:0000 1719:6
1 140 169:4 0:0000
0:0000 1 140 414:4
0:0000 1155:8
277:5
0:0000
807:2
1155:8
1 138 434:2
1275:4
statistical screen with large grains are shown in Table 3. The equivalent mechanics parameters (expected homogenization coefficients) are obtained by Algorithm 2. The final results are given in Table 4. Table 4 demonstrates that if the random parameters defining the grain ellipses are subject to uniform probability distributions, a composite material with randomly distributed grains shows isotropic mechanics performance and satisfies the symmetric, positive definitive properties in (13) and (14). In this numerical example, it should be noted that the original data are not all from physical experiments, and the results obtained cannot be compared with those from physical experiments. Table 4 only shows that the FE procedure based on SMSA is feasible for computation of the equivalent mechanics parameters for a composite material with a random grain distribution. However, in the next example, we present numerical results based on data from engineering practice. The second example is based on a concrete material with a large number of three-scale random rock grains in three dimensions, for which data were derived from an engineering project. The matrix, which is considered as a type of isotropic material, is made from sand and cement. The sizes of the three-scale rock grains in the concrete are shown in Table 5. Their elasticity coefficients are shown in Table 6. If all grains are generated in a large statistical screen of 500 mm, the number of grains is approximately 3500. Part of a sample containing all classes of the random grains is shown in Fig. 5 and the number of grains for three classes is listed in the last column of Table 5. Numerical results for the mechanics parameters for some samples with only small rock grains, which were generated using the FE procedure based on SMSA, are listed in Table 7. The expected values of the equivalent mechanics parameters evaluated using the FE procedure for 10, 30 and 50 samples with only small-ellipse grains are listed in Tables 8–10, respectively. The expected Young’s modulus and Poisson’s ratio for the equivalent mechanics parameters evaluated by Algorithm 2 for 10, 20, 30, 40 and 50 samples with only small-ellipse grains are shown in Figs. 5 and 6, respectively. Figs. 5 and 6 show that the equivalent mechanics parameters evaluated for the different samples using Algorithm 2 are convergent. Numerical results for the mechanics parameters computed using the FE procedure based on SMSA for 50
1 C C C C C C C C A
Table 8 Expected values of the mechanics parameters for 10 samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 119 188:8 B 820 319:4 B B B 819 923:7 B B 0:0000 B B @ 952:9 292:7
820 319:4
819 923:7
1002:6
952:9
292:7
3 112 201:4 819 734:6
819 734:6 3 109 392:8
726:7 77:0
771:5 5:9
892:9 1023:8
0:0000 771:5
0:0000 5:9
1 145 336:6 0:0000
0:0000 1 144 667:7
0:0000 808:1
892:9
0:0000
964:1
808:1
1 143 172:1
1 C C C C C C C C A
ARTICLE IN PRESS Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298 Table 9 Expected values of the mechanics parameters for 30 samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 113 742:9 B 819 643:6 B B B 819 016:0 B B 0:0000 B B @ 141:7 308:6
819 643:6 819 016:0 3 106 156:0 819 206:6
880:4 646:5
141:7 641:9
308:6 191:1
819 206:6 0:0000
3 103780:0 0:0000
162:1 1 142 567:6
502:0 0:0000
178:7 0:0000
641:9 191:1
502:0 0:0000
0:0000 585:0
1 141 982:5 752:3 752:3 1 140 693:8
1 C C C C C C C C A
Table 10 Expected values of the mechanics parameters for 50 samples of a composite material with small rock grains using the SMSA FE procedure (10 kPa) 0
3 114 007:0 B 819 970:9 B B B 819 622:9 B B 0:0000 B B @ 176:8 355:5
1
819 970:9
819 622:9
909:9
176:8
355:5
310 6579:8 819 470:0
819 470:0 3 104 792:6
555:4 203:9
505:4 422:8
215:5 186:5
0:0000 505:4
0:0000 422:8
1 142 642:4 0:0000
0:0000 0:0000 1 142 541:2 762:5
215:5
0:0000
730:5
762:5
C C C C C C C C A
1 140 862:9
0.21 0.208
The Posion V
0.206 0.204 0.202 0.2 0.198 0.196 0.194 0.192 0.19 0
5
10
15
20 25 30 35 sample numbers
40
45
50
Fig. 6. Poisson’s ratio for 10, 20, 30, 40 and 50 samples with random small-ellipse rock grains according to the SMSA FE procedure.
Table 11 Equivalent mechanics parameters for a composite material with random small grains using the SMSA FE procedure or ANSYS software (10 kPa) E ellipse ¼ 2762557:7 E ball ¼ 2847824 E average ¼ 3044468 E ANSYS ¼ 2820000
V ellipse ¼ 0:208846 V ball ¼ 0:199713 V average ¼ 0:194606 V ANSYS ¼ 0:198000
samples with only small-ellipse rock grains are listed in the first line of Table 11. We also used ANSYS software to compute Young’s modulus and Poisson’s ratio for a larger brick with a uniform distribution of random small-ellipse grains using three times the length of the examples and the average volume. A comparison of the numerical results
297
Table 12 Equivalent mechanics parameters for concrete with multi-scale rock grains randomly distributed according to the SMSA FE procedure (10 kPa) 3 238 870.7 789 174.8 789 124.4 0.0000 36.8 21.4
789 174.8 3 238 878.0 789 152.8 0.0000 17.7 0.23
789124.4 789 152.8 3 239 046.7 0.0000 86.6 0.0000
8.7 19.6 16.6 1 224 722.6 0.000 5.4
36.8 17.7 86.6 0.0000 1 224 728.9 33.2
21.4 0.23 14.0 0.0000 33.2 1 224 798.4
obtained using ellipse grains and ball grains for Young’s modulus and Poisson’s ratio is also shown in Table 11. From Tables 8–10 it is evident that the numerical results for the equivalent parameters computed using the SMSAbased FE procedure possess the symmetrical, positive and definitive properties (13) and (14). From Table 11, it follows that values for Young’s modulus and Poisson’s ratios produced by the SMSA FE procedure and by the ANSYS software for the domain are very close. This demonstrates that the FE procedure based on SMSA is valid in our example. Because composite material with both uniformly distributed random elliptical grains and random ball grains is isotropic and their equivalent mechanics parameters are very close, to reduce the computing time and to use uniform probability distribution model, three-scale ellipse grains can be substituted by three-scale ball grains. The equivalent mechanics parameters for the concrete defined previously were computed, and the numerical results are listed in Table 12. It should be noted that if the random parameters for elliptical grains are not subject to a uniform probability model, materials with random grains may not possess the isotropic property. To verify the model’s feasibility and accuracy, in future work, we will continue to use our research results in an engineering project on the Xiaowan dam being constructed in Yunnan province in southern China. Moreover, using mathematical theory, we hope to obtain an error estimation for the SMSA-based FE procedure to estimate the accuracy of the equivalent mechanics parameters for composite material with multi-scale random grains. 8. Conclusion In this paper we developed an FE computational procedure to evaluate the equivalent mechanics parameters for a composite material with random multi-scale grains. The FE procedure is based on SMSA and the assumption that the random grains are located within a statistical screen. An FE formulation was derived and a procedure for evaluating the equivalent mechanics parameters of composite material with random multi-scale grains was presented. Various test examples were solved using the FE procedure developed. The numerical results show that the SMSA-based FE procedure is feasible and valid. The procedure can also be extended to other composite
ARTICLE IN PRESS 298
Y. Li et al. / Engineering Analysis with Boundary Elements 32 (2008) 290–298
materials with random short fibers, random foams/cavities, etc. Acknowledgments This work was supported by the National Natural Science Foundation of China (10590353,10672055,70601003) and by a project supported by the Scientific Research Fund of Hunan Provincial Education Department (06B003) by a project supported by the Scientific Research Fund of Hunan Provincial Science Department (2007RS4007). References [1] Du SY, Wang B. Micromechanics for composite materials. Science Press; 1998. [2] Dierk R. Computational materials science. Wiley-VCH; 1998. [3] Li YY, Cui JZ. Two-scale analysis method predicting heat transfer performance of composite material with random grain distribution. Sci China Ser A Math 2004;47:100–10.
[4] Li YY, Cui JZ. The multi-scale computational method for the mechanics parameters of the materials with random distribution of multi-scale grains. Compos Sci Technol 2005;65:1447–58. [5] Oleinik OA, Shamaev AS, Yosifian GA. Mathematical problems in elasticity and homogenization. Amsterdam: North-Holland; 1992. [6] Jikov VV, Kozlov SM, Oleinik OA. Homogenization of differential operators and integral functions. Springer; 1994. [7] Cui JZ, Yang HY. A dual coupled method of boundary value problems of PDE with coefficients of small period. Int J Comput Math 1996;14:159–74. [8] Li YY. The Multi-scale Analysis for the Elastics Mechanics Performance of the Composite Material with Rectangle/Cuboid Periodicity. J Hunan City Univ (Nat Sci) 2005;4:30–4. [9] Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer; 1977. [10] Li YY, Cui JZ. The computer simulated method for the domain with large numbers of random ellipse grains/cavities and the improved automatic generation triangle mesh algorithm. Chin J Comput Mech 2004;21:290–5. [11] Ciarlet PG. The finite element method for elliptic problems. Berlin: Springer; 1978.