— 60 —

COMPUTATIONAL MECHANICS

ISCM2007, July 30-August 1, 2007, Beijing,China

©2007 Tsinghua University Press & Springer

Statistical Two-Scale Method for Strength Prediction of Composites

with Random Distribution and Its Applications

Junzhi Cui1

*, X. G. Yu1

, Fei Han, Yan Yu2

1

Academy of Mathematics and System Sciences, CAS, Beijing, 100080 China

2

School of Science, Northwestern Polytechnical University, Xi’an, 710072 China

Email: cjz@lsec.cc.ac.cn

Abstract A Statistical two-order and Two-Scale computational Method (STSM) based on two-scale

homogenization approach is developed and successfully applied to predicting the strength parameters of

random particle reinforced composites. Firstly, the probability distribution model of composites with

random distribution of a great number of particles in any ε − size statistic screen, as a ε − size cell, is

described. And then, the stochastic two-order and two-scale computational expressions for the strain tensor

in the structure, which is made from the composites with random distribution model of ε − size cell, are

formulated in detail. And the effective expected strength and the minimum strength for the composites with

random distribution are expressed, and the computational formulas of them and the algorithm procedure for

strength parameter prediction are shown. Finally, some numerical results of its application to the random

particle reinforced composites, the concrete with random distribution of a great number of particles in

anyε − size statistic screen, are demonstrated, and the comparisons with physical experimental data are

given. They show that STSM is validated and efficient for predicting the strength of random particle

reinforced composites.

Key words: Statistical two-scale computational method, strength prediction, composites with random

particle distribution, meso-scale cell

INTRODUCTION

According to the basic configuration, composite materials can be divided into two classes: the composite

materials with periodic configuration, such as periodically honeycomb materials and braided composites,

and the composite materials with random distribution, such as the particle reinforced metal matrix

composites (MMCs) and concrete. In recent years, the interest in particulate composites has revived since

the multi-phase mixtures often provide an advantageous blend of the properties of basic components. For

example, most polymers in homogenous form are glassy and brittle, the addition of rubber particles into a

polymer matrix can greatly improve the impact resistance [1, 2]. Likewise, the addition of rigid fillers (for

example, carbon black) into rubberlike elastomers can greatly improve the stiffness and strength of the

materials [3].

It has been proved that the shape, size and spatial distributions of particles significantly influence the

macroscopic mechanical properties of random particle reinforced composites, which means that the

meso-scale configuration has to be taken into account to evaluate the macroscopic mechanical properties.

Until now, lots of work has been done on predicting the mechanical properties of particulate composites.

Many approaches can be used to the calculation of macroscopic stiffness parameters, such as the law of

mixture [4,5], Hashin-Shtrikman upper and lower bounds method [6], self-consistent approach [7-9] and

Eshelby effective inclusion method [10] etc. However, in regard to the prediction for strength parameters,

— 61 —

there are rare theoretical techniques available. For some specific kind of random particle reinforced

composites, such as MMCs and concrete, a few methods [11-15] have been developed for predicting the

strength. However, most of them are based on the greatly simplification of real composite structures and can

not really reflect the characteristics of random particulate composites. There is still no theoretical method,

which can provide an effective support for designing the structure of random particle reinforced composites.

Therefore, the most effective technique in strength evaluation is still physical experiments [16-18].

For predicting the mechanical and physical properties of the structure made from the composite materials

with periodic configuration the Multi-Scale Analysis (MSA) method has been developed in [19-22]. For the

physics field problems of composite materials with the stationary random distribution, Jikov and Kozlov

[23] gave a proof of the existence of the macroscopic homogenization coefficients and the solution. And the

statistical multi-scale computational (SMSA) method for predicting the macroscopic stiffness parameters

and physics parameters of the composite materials with random distribution was proposed by Li and Cui [24,

25] and numerical results were given. In this paper, the SMSA method is extended, and applied to predicting

the strength of the structures of random particle reinforced composites.

The remainder of this paper is organized as follows: At first introduces a method to represent the composites

with random particle distribution, and the computer simulation algorithm of random particle reinforced

composites. The next section is devoted to the statistical two-scale computational method for the strength

prediction of the composites with random particles distribution. Then some numerical results are shown, and

compared with physical experimental data. Finally, the conclusions are given.

COMPUTER SIMULATION OF THE COMPOSITES WITH RANDOM PARTICLE DISTRIBUTION

1. Representation of the composites with random particle distribution Suppose that the investigated

composite materials are made from matrix and random particles. All the particles are considered as ellipsoids

or the polyhedrons inscribed inside the ellipsoids, which are randomly distributed in the matrix. In this paper

all of the ellipsoid particles are also considered as “same scale”, which means all of their long axes satisfy

1 2

r a r< < where r1 and r2 are the given upper and lower bounds. For this kind of composite materials, we

can represent them, see ref. [24-26], as follows:

1) There exists a least constant ε satisfying 0 Lε< � , where L denotes the macro scale of the investigated

structure Ω . Thus, the structure made from composite materials can be regarded as a set of cells with the

ε -size, as shown in Fig. 1.

2) In each cell, the probability distribution of the particles is identical. Then the investigated structure has

periodically random distribution of particles, and then can be represented by the probability distribution of

the particles inside a typical cell.

3) Each ellipsoid can be defined by 10 random parameters, including the shape, size, orientation and spatial

distribution of ellipsoid particles: a, b, c, θaxy, θax, θbxy, θbx, x0, y0, z0, where a, b and c denote length of three

axes; x0, y0 and z0 the coordinates of the center; θaxy, θax, θbxy and θbx the direction of the long axis and middle

axis. Their probability density functions are denoted by 0

( )x

f x , 0

( )y

f x , 0

( )z

f x , ( )a

f x , ( )b

f x , ( )c

f x ,

( )axy

f xθ

, ( )ax

f xθ

, ( )bxy

f xθ

, ( )bx

f xθ

, respectively.

4) Suppose that there are N ellipsoids inside a cell s

Qε , then we can define a sample of particles distribution

as follow:

01 01 01 1 1 1 1 1 1 1 0 0 0

( , , , , , , , , , , ..., , , , , , , , , , )s s s s s s s s s s s s s s s s s s s s s

axy ax bxy bx N N N N N N axyN axN bxyN bxNx y z a b c x y z a b cω θ θ θ θ θ θ θ θ= (1)

as shown in Fig. 1(b).

Base on above representation, the structure Ω logically consists of ε –size cells subjected to identical

probability distribution model P, and can be denoted as

( , )

( )s

s

t Z

Q tω

ε

∈

Ω = +U (2)

— 62 —

For Ω , define

{ }|s s

x Qω ω ε= ∈ ⊂ Ω (3)

Thus the elasticity tensor of the composite materials can be periodically expressed as ( ){ },

ijhk

x

a ω

ε

, and for

a sample s

ω , the material parameters ( ){ },

s

ijhk

x

a ω

ε

can be defined as follow:

(a) A part of investigated structure Ω (b) The particle distribution for a sample

Figure 1: Composite structure with periodically random distribution of particles

( )

1

2

,

1

,

,

s

ijhk i

sN

ijhk s

ijhk i

i

a x e Q

xa

a x Q e

ε

ω

εε

=

⎧ ∈ ⊂

⎪= ⎨

∈ −⎪

⎩

U (4)

where s

Qε denotes the domain of a cell belonging to Ω ; i

e is the i-th ellipsoid in s

Qε , 1

ijhka and

2

ijhka are

elasticity constants of particles and matrix, respectively.

2. Modeling of composites of random distribution with plenty of particles Since the first “numerical

concrete” model was developed by Wittmann et al [27, 28], lots of work have been done on the meso-scopic

models of composite materials with random distribution of particles, especially on concrete, to investigate

the influence of the meso-scale configuration on the macro-scale properties. In regard to the spatial

distribution of particles, several techniques have been developed, such as take-and-place method [28-31] and

divide-and-fill method [32]. They are all efficient to generate a meso-scopic model of random composites

with low particle volume fractions. For achieving high particle volume fraction, Mier et al [32] and Wriggers

et al [27] used alternative algorithms, which take much more time.

In this paper, the computer generation method proposed by Yu et al [26], which is not only an algorithm with

high computing efficiency, but also is able to generate a random particle samples with high particle volume

fraction and high stochastic behavior, is employed. We can simulate several kinds of composite structures

through adjusting ten random parameters of ellipsoids. For instance, when one of the three axes becomes

short enough comparing with two others, ellipsoids look like coins, then we can use them to simulate

structures with random distribution of cracks. When two of the three axes are both insignificant, the model

can be used to simulate the composite structures reinforced with short fibers.

The developed take-and-place method is used. The computer generation method can be divided into two

parts: particle generating part, filtering and location part. In particle generating part, plenty of ellipsoids

satisfied to specified distribution models in shape, size and orientation, are randomly generated, and put into

the cuboid container one next to one, when the cuboid is full filled, this step is over. In filtering and location

part, no particle is generated, only thing to be done is to filter the particles generated previously and make

them satisfy the given spatial distribution model, and the given particle volume fraction. For the detail of

computer generation method, please see [26].

— 63 —

STATISTICAL TWO-SCALE FORMULATION FOR THE STRENGTH COMPUTATION OF

THE COMPOSITES WITH RANDOM PARTICLE DISTRIBUTION

1. Statistical two-scale formulation for the composites with random distribution For the structures

made from composites with random distribution of particles, based on the representation previously, the

elasticity problem with mixed boundary conditions can be expressed as follows:

( )

( )( )

( )

( ) ( ) ( ) ( )

1

(1)

2

( , ) ( , )1

( , ) ( )

2

, ,

( , ) ( )

( , ) ( , )1

( , ) ( , ) ( )

2

h k

ijhk i

j k h

p p m m

ij j ij j p m

p m

p m

h k

i j jihk i

k h

u x u x

a x f x x

x x x

x

xx x

x x x

u x u x

x a x p x x

x x

ε ε

ε

ε ε

ε

ε ε

ε

ω ω

ω

σ ν σ ν

ω ω

ω

ω ω

σ ω ν ω

⎡ ⎤⎛ ⎞∂ ∂∂

+ = ∈Ω⎢ ⎥⎜ ⎟∂ ∂ ∂

⎝ ⎠⎣ ⎦

= ∈∂Ω ∂Ω

∈∂Ω ∂Ω=

= ∈Γ

⎛ ⎞∂ ∂

= + = ∈Γ⎜ ⎟∂ ∂

⎝ ⎠

u u

u u

I

I

( )1 2 1 2

,φ

⎧

⎪

⎪

⎪

⎪

⎪⎪

⎨

⎪

⎪

⎪

⎪

⎪

Γ Γ = Γ Γ = ∂Ω⎪⎩

I U

(5)

where ( , )ijhka x

ε

ω (i,j,h,k=1,…,n) are the elastic coefficients of the randomicity with ε-size statistic screen,

( , )x

ε

ωu is the solution of vector-valued displacement. ( )p

jν and

( )m

jν are the normal direction cosine of

,

p m

∂Ω ∂Ω , and p m

∂Ω ∂ΩI means the interface between particles and matrix.

If the interfaces between particles and matrix are considered continuous transition zones, ( , )ijhka x

ε

ω is a

continuous variable. If not, ( , )ijhka x

ε

ω is a piece-wise constant. However, we can always construct a smooth

operator : ( , ) ( , )ijhk ijhk

S a x a x

ε ε

δ

ω ω→ % , where ( , )ijhka x

ε

ω% is continuous and smooth enough, and

( , ) ( , )ijhk ijhka x a x

ε ε

ω ω δ− <% , which means ( , ) ( , )ijhk ijhka x a x

ε ε

ω ω→% when 0δ → . Thus, the elasticity

problem (5) with mixed boundary conditions changes into following

( )

1

2

1 2 1 2

( , ) ( , )1

( , ) ( )

2

( , ) ( )

( , ) ( , )1

( , ) ( , ) ( )

2

,

h k

ijhk i

j k h

h k

i j jihk i

k h

u x u x

a x f x x

x x x

x x x

u x u x

x a x p x x

x x

ε ε

ε

ε

ε ε

ε

ω ω

ω

ω

ω ω

σ ω ν ω

φ

⎧ ⎡ ⎤⎛ ⎞∂ ∂∂

+ = ∈Ω⎪ ⎢ ⎥⎜ ⎟∂ ∂ ∂

⎝ ⎠⎪ ⎣ ⎦

⎪= ∈Γ⎪

⎨

⎛ ⎞∂ ∂⎪= + = ∈Γ⎜ ⎟⎪

∂ ∂⎝ ⎠

⎪

⎪ Γ Γ = Γ Γ = ∂Ω⎩

u u

% %%

%

% %%

I U

(6)

( )xε

u% is infinitely close to the solution ( )xε

u of problem (5) as 0δ → . Following discussion on STSM is

for the problem (6).

Let s

x x

Qξ

ε ε

⎡ ⎤= − ∈

⎢ ⎥⎣ ⎦

denotes the local coordinates on 1-normalized cell. Then ( ) ( ), ,

ijhk ijhka x a

ε

ω ξ ω= and

( , ) ( , , )x x

ε

ω ξ ω=u u . In [25], by using constructive way following formulas on STSM solution of previous

problem were obtained: The solution of problem (6) can be expressed in the statistic two-scale formulation

as follows

( ) ( ) ( ) ( )2

1 1 2

0 2 0

0 2

3

1

( ) ( )

, , ,

( , , ) ,

x x

x x

x x x

x x

ε

α α α

α α α

ω ε ξ ω ε ξ ω

ε ξ ω

∂ ∂

= + +

∂ ∂ ∂

+ ∈Ω

1 1

u u

u u N N

P

(7)

— 64 —

where 0

( )xu is the homogenization solution and defined on global Ω , ( ),

α

ξ ω1

N and ( )2

,

α α

ξ ω1

N

( )1 2

, 1, ,nα α = L are n-order matrix-valued functions defined on 1-normalized Q , and they have following

forms

( )

( ) ( )

( ) ( )

1 1

1

1 1

11 1

1

, ,

,

, ,

n

n nn

N N

N N

α α

α

α α

ξ ω ξ ω

ξ ω

ξ ω ξ ω

⎛ ⎞

⎜ ⎟

= ⎜ ⎟

⎜ ⎟

⎝ ⎠

N

L

M L M

L

(8)

( )

( ) ( )

( ) ( )

1 2 1 2

1 2

1 2 1 2

11 1

1

, ,

,

, ,

n

n nn

N N

N N

α α α α

α α

α α α α

ξ ω ξ ω

ξ ω

ξ ω ξ ω

⎛ ⎞

⎜ ⎟

= ⎜ ⎟

⎜ ⎟

⎝ ⎠

N

L

M L M

L

(9)

And ( ),

α

ξ ω1

N , ( )2

,

α α

ξ ω1

N ( )1 2

, 1, ,nα α = L and 0

( )xu are determined in following ways:

1) For any sample s

ω , ( ) ( )1

, , 1, ,s

m

m nα

ξ ω α =

1

N L are the solutions of following problems

1 1 1

1

( , ) ( , ) ( , )1

( , )

2

( , ) 0

s s s

hm km ij ms s

ijhk

j k h j

s s

m

N N a

a Q

Q

α α α

α

ξ ω ξ ω ξ ω

ξ ω ξ

ξ ξ ξ ξ

ξ ω ξ

⎧ ⎡ ⎤⎛ ⎞∂ ∂ ∂∂⎪ + = − ∈⎢ ⎥⎜ ⎟⎪

⎜ ⎟∂ ∂ ∂ ∂⎨ ⎢ ⎥⎝ ⎠⎣ ⎦

⎪

= ∈∂⎪⎩N

(10)

2) From ( ),

s

mα

ξ ω1

N , the homogenization elasticity parameters { }ˆ ( )s

ijhka ω corresponding to the sample

s

ω

are calculated in following formula

( , ) ( , )1

ˆ ( ) ( , ) ( , )

2s

s s

hpk hqks s s

ijhk ijhk ijpqQ

q p

N N

a a a d

ξ ω ξ ω

ω ξ ω ξ ω ξ

ξ ξ

⎛ ⎞⎛ ⎞∂ ∂

= + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

∫ (11)

3) One can evaluate the expected homogenized coefficients { }ijhka

)

in following formula

1

ˆ ( )

,

M

s

ijhk

s

ijhk

a

a M

M

ω

=

= → +∞

∑)

(12)

4) For any sample s

ω , ( ) ( )2

1 2

, , , 1, ,s

m

m nα α

ξ ω α α =

1

N L are the solutions of following problems

( )

1 2 1 2

2 1

1

2 1 2

2 1

1 2

( , ) ( , )1

ˆ( , )

2

( , )

( , ) ( , )

( , ) ( , )

( , ) 0

s s

hm kms

ijhk i m

j k h

s

shms s

i m i hk

k

s s

ijh hm

j

s s

m

N N

a a

NQ

a a

a N

Q

α α α α

α α

α

α α α

α α

α α

ξ ω ξ ω

ξ ω

ξ ξ ξ

ξ ωξ

ξ ω ξ ω

ξ

ξ ω ξ ω

ξ

ξ ω ξ

⎧ ⎡ ⎤⎛ ⎞∂ ∂∂⎪ + =⎢ ⎥⎜ ⎟

⎜ ⎟∂ ∂ ∂⎪ ⎢ ⎥⎝ ⎠⎣ ⎦

⎪

∂⎪∈

⎪ − −

⎨ ∂

⎪

∂⎪

−⎪ ∂

⎪

= ∈∂⎪⎩N

(13)

5) 0

( )xu is the solution of the homogenization problem with the homogenized parameters { }ijhka

)

defined on

global Ω

— 65 —

( )

( )

0 0

0

1

0 0

2

1 2 1 2

( ) ( )1

( ),

2

( ) ( ),

( ) ( )1

,

2

,

h k

ijhk i

j j k h

h k

i j jihk i

k h

u x u x

a f x x

x x x x

x x x

u x u x

x a p x

x x

σ ν

φ

⎧ ⎡ ⎤⎛ ⎞∂ ∂∂ ∂

+ = ∈Ω⎪ ⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎪ ⎝ ⎠⎣ ⎦

⎪= ∈Γ⎪

⎨

⎛ ⎞∂ ∂⎪= + = ∈Γ⎜ ⎟⎪

∂ ∂⎝ ⎠

⎪

⎪ Γ Γ = Γ Γ = ∂Ω⎩

u u

)

)

I U

(14)

2. Strain expressions of the structures of random particulate composites based on STSM To make

strength analysis onto the composites with random distribution of particles, it’s necessary to know the strain

and stress field inside the structure Ω and employ strength criterions for both particles and matrix. The

STSM formulas of strain fields for three kinds of classical components and the strength criterions will be

given below. From the expressions of strains, the stresses can be evaluated through Hooke’s Law

( , ) , ,ij ijhk hk

x x

x aσ ω ω ε ω

ε ε

⎛ ⎞ ⎛ ⎞=

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

. (15)

According to the elasticity theory, from the fore three terms of STSM formula (7), the strains can be

evaluated approximately in following formulas:

0 02

1 0 1 0

1

2

1 0

1

( ) ( )1 1

, , ( ) , ( )

2 2

1

, ( )

2

l l lh k

hk hm k m km h m

l lk h

l lhm km

m

l lk h

u x u xx x x

N D u x N D u x

x x

N N x

D u x

ε ω ε ω ω

ε ε ε

ε ω

ξ ξ ε

+ +

= < >=

−

= < >=

⎛ ⎞∂ ∂ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥

∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠

⎡ ⎤∂ ∂ ⎛ ⎞+ +

⎢ ⎥ ⎜ ⎟∂ ∂ ⎝ ⎠⎣ ⎦

∑ ∑

∑ ∑

α α α α

α

α α

α

α

(16)

where ( )l

αααα ,,,

21

L= , ( )( )

1 2

0

0

l

l

ml

m

u x

D u x

x x x

α

α α α

∂

=

∂ ∂ ∂L

.

In the evaluation of material strength, three kinds of classical components, i.e. tension of a bar in the axial

direction, bending of a beam with rectangular cross section and twist of circular column with constant cross

section, are often considered. And if the materials of above three kinds of components have isotropic

properties, from solid mechanics, it is easy to obtain the exact solution of the displacements for the elasticity

problems corresponding to three kinds of components. Therefore, the computational formulas of the

displacements and strains inside these components made from random composites can be derived from

previous STSM formulas.

1) Tension of a bar in the axial direction: For the axial tension of a bar with rectangle cross section, shown in

Figure 2, the exact displacements of homogenization problem can be expressed as

0 13

1 1

11

0 23

2 2

22

0

3 3

33

u px

E

u px

E

p

u x

E

ν

ν

⎧

= −⎪

⎪

⎪

= −⎨

⎪

⎪

=⎪

⎩

(17)

where /p T A= , 11

E , 22

E , 33

E , 13

ν and 23

ν are the elasticity moduli of three axis directions and Poisson

ratio, respectively.

From (17) it follows that

— 66 —

( )( )

1 2

0

0

0, for 2

l

l

ml

m

u

D u l

x x x

α

α α α

∂

= = ≥

∂ ∂ ∂L

x

x . (18)

Thus, the displacement vector of the tension problem of the bar made from composites with symmetrical

basic configuration can be expressed as

1

1

0

0( )

( ) ( ) ( , )

x

ε

α

α

ε ω

ε

∂

= +

∂

u xx

u x u x N (19)

And then the two-scale formulas on the strains and stresses inside previous column can be written into

1 1

1

0 0

0

( ) ( )1

( )

2

( )1

2

h k

hk

k h

hm kmm

k h

u u

x x

N N u

x x x

α α

α

ε

ε

ε ε

⎛ ⎞∂ ∂

= +⎜ ⎟∂ ∂

⎝ ⎠

∂ ∂⎡ ⎤ ∂⎛ ⎞ ⎛ ⎞+ +

⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

x x

x

xx x

(20)

Figure 2: Axial tension of a bar with rectangle cross section

Substituting (17) into (20) and respecting the symmetry of1

( )α

ξN , one obtains the expression on each

component of the strain tensor inside any cell of the column

( )13 13 23

11 313 111 212

11 1 33 11 22

1

, ,p p N N N

E E E E

ν ν ν

ε ω ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= − + − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x (21)

( )13 23

12 323 121 222

1 33 11 22

13 23

313 111 212

2 33 11 22

1

, ,

2

1

,

2

p

N N N

E E E

p

N N N

E E E

ν ν

ε ω ω

ξ ε

ν ν

ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

x

(22)

( )13 23

13 333 131 232

1 33 11 22

13 23

313 111 212

3 33 11 22

1

, ,

2

1

,

2

p

N N N

E E E

p

N N N

E E E

ν ν

ε ω ω

ξ ε

ν ν

ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

x

(23)

( )23 13 23

22 323 121 222

22 2 33 11 22

1

, ,p p N N N

E E E E

ν ν ν

ε ω ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= − + − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x (24)

— 67 —

( )13 23

23 333 131 232

2 33 11 22

13 23

323 121 222

3 33 11 22

1

, ,

2

1

,

2

p

N N N

E E E

p

N N N

E E E

ν ν

ε ω ω

ξ ε

ν ν

ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

x

(25)

( )13 23

33 333 131 232

33 3 33 11 22

1

, ,

p

p N N N

E E E E

ν ν

ε ω ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= + − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x (26)

Furthermore, from above strains the stresses are evaluated anywhere inside every cell belonging to the bar.

Then based on the yield criterion of basic materials, such as matrix, reinforced particles and their interfaces,

the critical point of tensile bar of composites previously can be evaluated.

2) Pure bending of a beam with rectangle cross section: For the pure bending of a beam with rectangle cross

section, shown in Figure 3, which is made from the composites with random distribution of particles, let

3

0x = denotes fixed end, and at 3

x L= the bend moment round 2

x axis is imposed. From solid mechanics

the bending problem of the cantilever with orthogonal an-isotropic material coefficients has following

solution

2

2

2

0 2 2 213 23

1 3 1 2

33 11 22

0 23

2 1 2

22

0

3 1 3

33

1

2x

x

x

M

u x x x

I E E E

M

u x x

E I

M

u x x

E I

ν ν

ν

⎧ ⎛ ⎞

= − + −⎪ ⎜ ⎟

⎝ ⎠⎪

⎪⎪

= −⎨

⎪

⎪

⎪ =

⎪⎩

(27)

Figure 3: Pure bending of a beam with rectangle cross section

Where 2

3

12

x

bh

I = is the moment of inertia round 2

x .

It is easy to see that the displacements are two-order polynomial. So

( )( )

1 2

0

0

0, for 3

l

l

ml

m

u

D u l

x x x

α

α α α

∂

= = ≥

∂ ∂ ∂L

x

x . (28)

Thus the displacement vector of the bending problem of the cantilever made from composites can be

expressed as

( ) ( )

1 1 2

1 1 2

2

0 2

( ) ( ) , ,

x x x

ε

α α α

α α α

ε ω ε ω

ε ε

∂ ∂⎛ ⎞ ⎛ ⎞= + +

⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

0 0

u x u xx x

u x u x N N (29)

— 68 —

Thus, respecting the symmetry of 1

( , )α

ωN ξ and 1 2

( , )α α

ωN ξ , the components of strain tensor inside every

cell belonging to the cantilever are evaluated in following formulas:

( )

2

2

2

13 23

11 313 111 212

33 11 22

23 131

313 212 111

1 33 22 11

23 13

3113 2112 1111

1 33 22 11

13

1

, , , ,

1

,

1

,

x

x

x

M

N N N

I E E E

Mx

N N N

I E E E

M

N N N

I E E E

M

E

ν ν

ε ω ε ω ω ω

ε ε ε

ν ν

ω

ξ ε

ν νε

ω

ξ ε

ν

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

−

x x x

x

x

x

2

1

11 x

x

I

(30)

( )

2

2

2

23 13

12 323 222 121

33 22 11

23 131

323 222 121

1 33 22 11

23 13

3123 2122 1121

1 33 22 11

1

, , , ,

2

1

,

2

1

,

2

x

x

x

M

N N N

I E E E

Mx

N N N

I E E E

M

N N N

I E E E

Mx

ν ν

ε ω ε ω ω ω

ε ε ε

ν ν

ω

ξ ε

ν νε

ω

ξ ε

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

+

x x x

x

x

x

2

2

23 131

313 212 111

2 33 22 11

23 13

3113 2112 1111

2 33 22 11

1

,

2

1

,

2

x

x

N N N

I E E E

M

N N N

I E E E

ν ν

ω

ξ ε

ν νε

ω

ξ ε

⎛ ⎞∂ ⎛ ⎞− −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

(31)

( )

2

2

2

23 13

13 333 232 131

33 22 11

23 131

333 232 131

1 33 22 11

23 13

3133 2132 1131

1 33 22 11

1

, , , ,

2

1

,

2

1

,

2

x

x

x

M

N N N

I E E E

Mx

N N N

I E E E

M

N N N

I E E E

Mx

ν ν

ε ω ε ω ω ω

ε ε ε

ν ν

ω

ξ ε

ν νε

ω

ξ ε

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

+

x x x

x

x

x

2

2

23 131

313 212 111

3 33 22 11

23 13

3113 2112 1111

3 33 22 11

1

,

2

1

,

2

x

x

N N N

I E E E

M

N N N

I E E E

ν ν

ω

ξ ε

ν νε

ω

ξ ε

⎛ ⎞∂ ⎛ ⎞− −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

(32)

( )

2

2

2

23 131

22 323 222 121

2 33 22 11

23 13

3123 2122 1121

2 33 22 11

23 1

22

1

, ,

1

,

x

x

x

Mx

N N N

I E E E

M

N N N

I E E E

Mx

E I

ν ν

ε ω ω

ξ ε

ν νε

ω

ξ ε

ν

⎛ ⎞∂ ⎛ ⎞= − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

−

x

x

x

(33)

— 69 —

( )

2

2

2

2

23 131

23 333 232 131

2 33 22 11

23 13

3133 2132 1131

2 33 22 11

23 131

323 222 121

3 33 22 11

31

3 33

1

, ,

2

1

,

2

1

,

2

1

2

x

x

x

x

Mx

N N N

I E E E

M

N N N

I E E E

Mx

N N N

I E E E

M

N

I E

ν ν

ε ω ω

ξ ε

ν νε

ω

ξ ε

ν ν

ω

ξ ε

ε

ξ

⎛ ⎞∂ ⎛ ⎞= − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

∂

+

∂

x

x

x

x

23 13

23 2122 1121

22 11

,N N

E E

ν ν

ω

ε

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

x

(34)

( )

2

2

2

23 231

33 333 232 131

3 33 22 22

23 23

3133 2132 1131

3 33 22 22

1

33

1

, ,

1

,

x

x

x

Mx

N N N

I E E E

M

N N N

I E E E

Mx

E I

ν ν

ε ω ω

ξ ε

ν νε

ω

ξ ε

⎛ ⎞∂ ⎛ ⎞= − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

+

x

x

x

(35)

Using the stress-strain relation one can evaluate the stresses anywhere inside each cell belonging to the

cantilever.

From previous formulas it is easy to see that only 1

x component of macroscopic coordinate appear in the

strain expressions. It means that the strains do not depend on macroscopic coordinate 2

x and 3

x . And then

the maximum strain occur in the cells near the above or below surface of the cantilever, but it is uncertain

that the maximum strain occur on the above or below surface 1

/ 2x h= ± , since the strains and stresses

change very sharply inside each cell. According to maximum principal stress or/and principal strain one can

evaluate the elasticity strength limit of the beam bending of composite materials with any symmetric basic

configuration.

It is worthy of note, the composite materials must lead to macroscopically orthogonal an-isotropic material

coefficients. If not, (27) does not hold, herewith, formulas (30)-(35) are wrong.

3) Twist of circular shafts with constant cross section: The twist of circular shafts with constant cross

section, shown in Fig. 4, which is made from random particle composites, is shown in Fig. 4. Let r denotes

the radius of cross section, L the length of the column, x3

= 0 fixed end, and at x3

= L the twist moment is

imposed. If the shaft macroscopically has the orthogonal an-isotropic property, from elasticity mechanics the

displacement solution of the homogenization problem can be expressed as

Figure 4: Twist of circular shafts with constant cross section

— 70 —

0 2 3

1 4

13 23

0 1 3

2 4

13 23

0 1 2

3 4

13 23

1 1

1 1

1 1

Tx x

u

r G G

Tx x

u

r G G

Tx x

u

r G G

π

π

π

⎧ ⎛ ⎞

= − +⎪ ⎜ ⎟

⎝ ⎠⎪

⎪⎛ ⎞⎪

= +⎨ ⎜ ⎟

⎝ ⎠⎪

⎪⎛ ⎞

⎪ = − −⎜ ⎟⎪

⎝ ⎠⎩

(36)

where 13

G , 23

G denote the shear moduli in x1-x

3 plane and x

2-x

3 plane. It is easy to see that the displacements

are 2-order polynomial. And respecting the symmetry of 1

( , )α

ωN ξ and 1 2

( , )α α

ωN ξ the components of

strain tensor anywhere inside the shaft are expressed as follows

( )1 2

11 213 213 1134 4

23 1 23 13

2 2

, , ,

x xT T

N N N

r G r G G

ε

ε ω ω ω

π ε π ξ ε

⎛ ⎞∂⎛ ⎞ ⎛ ⎞= + −⎜ ⎟⎜ ⎟ ⎜ ⎟

∂⎝ ⎠ ⎝ ⎠⎝ ⎠

x x

x (37)

( )12 223 1134

23

1 2

223 1234

1 23 13

1 2

213 1134

2 23 13

, , ,

,

,

T

N N

r G

x xT

N N

r G G

x xT

N N

r G G

ε

ε ω ω ω

π ε ε

ω

π ξ ε

ω

π ξ ε

⎛ ⎞⎛ ⎞ ⎛ ⎞= −

⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x x

x

x

x

(38)

( )2

13 2334 4

23 13

1 2

233 1334

1 23 13

1 2

213 1134

3 23 13

, ,

,

,

TxT

N

r G r G

x xT

N N

r G G

x xT

N N

r G G

ε

ε ω ω

π ε π

ω

π ξ ε

ω

π ξ ε

⎛ ⎞= −

⎜ ⎟

⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

x

x

(39)

( )1 2

22 123 223 1234 4

13 2 23 23

2 2

, , ,

x xT T

N N N

r G r G G

ε

ε ω ω ω

π ε π ξ ε

⎛ ⎞∂⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟

∂⎝ ⎠ ⎝ ⎠⎝ ⎠

x x

x (40)

( )1

23 1334 4

23 13

1 2

233 1334

2 23 13

1 2

223 1234

3 23 13

, ,

,

,

Tx T

N

r G r G

x xT

N N

r G G

x xT

N N

r G G

ε

ε ω ω

π π ε

ω

π ξ ε

ω

π ξ ε

⎛ ⎞= −

⎜ ⎟

⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞+ −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x

x

x

(41)

( )1 2

33 233 1334

3 23 13

2

, ,

x xT

N N

r G G

ε ω ω

π ξ ε

⎛ ⎞∂ ⎛ ⎞= −⎜ ⎟⎜ ⎟

∂ ⎝ ⎠⎝ ⎠

x

x (42)

Using the stress-strain relation one can evaluate the stresses anywhere inside shaft. And then according to

maximum principal stress and principal strain one can determine the elasticity critical point of the twist shaft

of random particle composites.

Remark: The random particle composites must macroscopically leads to orthogonal an-isotropic

coefficients. If not, (36) does not hold, herewith, (37)-(42) are wrong.

— 71 —

3. Computation of the strength As the strain and stress anywhere inside the investigated structure are

obtained, the strength for the failure of the structure made from the composites can be discussed. Until now

there is no strength criterion specialized for composites. Here, we employ the classical strength criterions

developed for traditional homogenous materials instead.

It’s worthy to note that the employed strength criterion should be different for different kind of materials, such as

Von-Mises criterion should be employed for ductile materials, the maximum principal stain theory will be more

suitable while brittle rupture always happens and the investigated structure is subjecting to compressive load.

Here, we just present the formulation of maximum strain criterion, which is employed in concrete numerical

experiments below. The formulas of other strength criterions can be easily found in textbook of solid

mechanics or mechanics of materials.

The maximum principal stain theory assumes that failure occurs when the maximum principal strain in the

complex stress system equals to that at the yield point in the tensile test, i.e.

1

31 2

1 2 3

y

y

y

E E E E

ε ε

σσσ σ

ν ν

σ νσ νσ σ

=

− − =

− − =

(43)

where 1

σ , 2

σ and 3

σ are the three principal stresses under the three dimensional complex stress states, 1

ε

is the first principal stain and y

ε and y

σ are the strain and stress at the yield point in the simple tension test.

For a sample s

ω , all of strains and stresses inside any ε − cell belonging to the structure can be obtained

through the formulas presented previously. Then, the strength ( )s

S ω of the structure with random particle

distribution can be evaluated as the elasticity failure criterions for the sample s

ω . Thus to repeat previous

calculation so many times, from Kolmogorov strong law of the large number, it follows that the expected

strength ˆ

S can be evaluated by following formula:

1

( )

ˆ

M

s

s

S

S

M

ω

=

=

∑

. (44)

However, the expected strength ˆ

S can not totally represent the strength properties of the structure of random

particular composites. The yield of some cells may lead to the collapse of the whole structure. Therefore, the

minimal strength of the structure of random particular composites is sometime worthier than the expected

one for the design of random composite structures. The minimal strength can be defined as following

formula:

min

1, ,

min ( )s

s M

S S ω

=

=

L

(45)

FEM COMPUTATION FOR THE STRENGTH OF THE RANDOM PARTICLE COMPOSITES

BASED ON STSM

1. FE formulation based STSM From the formulation in previous section we can see that in the STSM

computation on the strength parameters of the composites with random particle distribution the major work

is to solve the problems (10) and (13) to obtain, ( ),

s

mα

ξ ω1

N and ( )2

,

s

mα α

ξ ω1

N ( )1

, 1, ,m nα = L for M

samples s

Pω ∈ , and then to evaluate { }ˆ ( )s

ijhka ω by formula (11) and { }

ijhka

)

by formula (12), and according

to the elasticity strength criterions of matrix and particles, respectively, to determine the expected effective

elasticity strength of the random particle composites and the minimal elasticity strength by (44) and (45).

From partial differential equation theory and finite element method to solve problem (10) to obtain

( ) ( )1

, , 1, ,s

m

m nα

ξ ω α =

1

N L is equivalent to solve following virtual work equation

— 72 —

( ) ( ) ( )

( ) ( ) ( )

1

1

1 s

0

, d

, d , , for

s

s

ij ijhk hk mQ

ss

ij ij mQ

a

a H Q P

α

α

ε ξ ω ε ξ

ε ξ ω ξ ω= −∀ ∈ ∈

∫

∫

v N

vv

(46)

where

( )

( ) ( ), ,1

2

s s

i j

ij

j i

v vξ ω ξ ω

ε

ξ ξ

⎛ ⎞∂ ∂

⎜ ⎟= +

⎜ ⎟∂ ∂⎝ ⎠

v . (47)

Similarly, ( ) ( )2

1 2

, , , 1, ,s

m

m nα α

ξ ω α α =

1

N L can be obtained by solving following virtual work equation

( ) ( ) ( )

( )

( )( )

1 2

2 1 2 1

1

2

2 1

1

0

, d

ˆ ( ) ( , ) d

( , )

( , ) d

( , ) ( , )d, , for

s

s

s

s

s

ij ijhk hk mQ

s s

i m i mQ

s

hms

i hk rQ

k

s r s

s

ij ijh hmQ

a

a a

N

a

a NH Q P

α α

α α α α

α

α

α α

ε ξ ω ε ξ

ω ξ ω ξ

ξ ω

ξ ω ξ

ξ

ε ξ ω ξ ω ξω

= − −

⎛ ⎞∂

+ ⎜ ⎟⎜ ⎟∂⎝ ⎠

−∀ ∈ ∈

∫

∫

∫

∫

v N

v

v

vv

. (48)

It is well known that by using FEM program the finite element solution ( ),

h s

mα

ξ ω1

N and ( )2

,

h s

mα α

ξ ω1

N of

( ),

s

mα

ξ ω1

N and ( )2

,

s

mα α

ξ ω1

N ( )1

, 1, ,m nα = L are easily evaluated. If it is necessary to obtain the

homogenization solution0

( )xu , one can also solve homogenization problem (14) by FE program.

2. Finite element model Based on the representation of composites with random distribution of particles in

section 0, as STSM and FEM are applied to calculate the strength parameters of random particle composites,

an essential work is, again and again, to generate the sample cell and partition it to form its finite element

model.

Different techniques have been developed for generating mesh of random composite structures. The

advancing front method was used by George [34] and Gheung et al. [35]. However, it’s rather a long and

tedious process to generate a finite element mesh using this method. To avoid such difficulties, some

researchers [36,37] employed the projection method. Unfortunately, this method has major drawbacks that

the shape of the particle/matrix boundaries can not be closely simulated.

In this paper, the projection method is developed. Firstly, we project a regular mesh onto the particle

distribution structure, then we move the existing node near the particle/matrix interfaces or insert a new node

on the interfaces to make the nodes match the shape of particle surfaces. It turns out to be validated, and it is

efficient.

Fig. 5 shows the finite element model of random particle distribution structure. There are 27 ellipsoid

particles in the model.

(a) The mesh of particles (b) The mesh of the entire model

Figure 5: FE model of a sample of random particle distribution cell

— 73 —

3. The algorithm procedure based on STSM and FEM Based on the representation of composites with

random distribution of particles in sections 2, the algorithm procedure of predicting the s mechanical

parameters of composites with random distribution of particles by statistical two scale method is following:

1) Step 1: Generate a distribution model P of particles based on the statistical characteristics of the random

particle distribution, and determine the material coefficients ( ){ },

s

ijhk

x

a ω

ε

on ( )Qε ε as follows:

( )s

, ( )

, , ,

, ( )

ijhks

ijhk

ijhk

a x Qx

a P

a x Q

ε ε

ω ω

εε ε

⎧ ∈⎪= ∈⎨

′ ∈⎪⎩

)

% (49)

where ( )Qε ε

)

is the domain of matrix and ( )Qε ε%

the domain of particles in ( )Qε ε and { }ijhka and { }

ijhka′

are the material coefficients of matrix and particles, respectively.

2) Step 2: Evaluate FE solution ( ) ( )1

, , 1, ,h s

m

m nα

ξ ω α =

1

N L of ( ),

s

mα

ξ ω1

N by solving problem (46) for

s

Pω ∈ . Then the sample homogenization coefficients { }ˆ ( )r s

ijhka ω can be calculated through formula (11). If

it’s necessary, evaluate FE solution ( )2

,

h s

mα α

ξ ω1

N of ( )2

,

s

mα α

ξ ω1

N ( )1

, 1, ,m nα = L for s

Pω ∈ by solving

problem (48).

3) Step 3: For s

Pω ∈ , 1,2,...,s M= , step 1 to 3 are repeated M times. Then M sample homogenization

coefficients { }ˆ ( )s

ijkha ω are obtained. The expected homogenization coefficients { }

ijkha

)

for the materials

with random distribution of particles can be evaluated as follows:

1

ˆ ( )

M

s

ijkh

s

ijkh

a

a

M

ω

=

=

∑)

(50)

If it is necessary, the homogenization solution 0

( )xu can be obtained by solving homogenization problem

(14) with the homogenization coefficients{ }ijkha

)

. For some typical structures/components. 0

( )xu can be

exactly obtained from solid mechanics.

4) Step 4: For the sample s

ω , evaluate the stain fields anywhere inside the investigated structure by

,

h s

m

x

α

ω

ε

⎛ ⎞

⎜ ⎟

⎝ ⎠1

N , 2

,

h s

m

x

α α

ω

ε

⎛ ⎞

⎜ ⎟

⎝ ⎠1

N ( )1

, 1, ,m nα = L , and 0

( )xu through formulas in section 3. The stresses can

be calculated through Hooke’s Law (15).

5) Step 5: Using corresponding strength criterions, along with the strength of matrix m

S and the strength of

particlesP

S , the critical load can be determined by using iteration procedure. After that, compute the strength

limit of the structure for s

ω , denoted by ( )s

S ω , according to the critical load and the homogenization

stiffness parameters { }ˆ ( )s

ijkha ω .

6) Step 6: For s

Pω ∈ , 1,2,...,s M= , step 6 to 7 are repeated. Then M sample strength ( )s

S ω are obtained.

The expected strength S

)

and the minimal strength min

S for the composites with random distribution of

particles can be evaluated as follows:

1

( )

M

s

s

S

S

M

ω

=

=

∑)

(51)

min

1, ,

min ( )s

s M

S S ω

=

=

L

(52)

— 74 —

If the difference of particle sizes inside random particle composites is very large and there are so plenty of

random particles in a small brick, like the concrete. We should divide all of random particles into several

classes according to their size, long axis of ellipsoid particles. For the concrete, the particles inside it can be

divided into rock grains, sand grain and cements. In this case the strength computation should start from the

class with smallest size of particles, i.e. r=N=3. As { }r

ijhka

)

, r

S

)

and min

r

S obtained, they are used as the

elastic coefficients and strength of new matrix in the next cycle with r=N-1, i.e. if it’s not the first cycle

( r N≠ ), the material coefficients of the matrix are the homogenization coefficients { }1r

ijhka

+)

and the strength

of the matrix is the strength 1r

S

+

)

/min

r

S , which are evaluated in former cycle with (r+1) class of particles.

As the last cycle 1r = is completed, the expected homogenization coefficients { }1

ijhka

)

and expected strength

1

S

)

and 1

min

S are obtained. And then 1

S

)

and 1

min

S are defined as the effective elastic coefficients and

expected/minimal strength of the investigated structure/component made from the composites with random

distribution of multi-scale particles.

NUMERICAL EXPERIMENTS

1. Influence of the number of samples As you have known, in physics experiments of material strength

there is a random characteristic that the experimental data for strength testing of the structure made from

random particle reinforced composites are always decentralized. The same happens to numerical simulation.

There also exists a noticeable difference between the numerical strengths of two different samples with the

same shape, size and spatial distribution of particles. That is the reason why it is necessary to take a certain

number of samples but not just one. This sub-section is devoted to show the influence of the number of

samples on the strength quantitatively.

Plenty of numerical experiments are made. Here we use a two-phase model with ellipsoidal particles and

matrix. The bond between particles and matrix is assumed to be perfect. Furthermore, we assume the size of

the ellipsoids (the lengths of the three axes) has a uniform distribution in the field [10mm, 30mm] and the

orientation (the angles between the axes of ellipsoids and coordinates) has a uniform distribution between 0

and π. The positions of the particles are also assumed to have a uniform distribution inside the cell with the

sizes 100×100×100(mm3

). The mechanical properties of the particles and matrix are shown in Table 1.. The

particle volume fraction is 30%.

Table 1: Mechanical properties of the particles and matrix

Matrix Particles

Em

(GPa)

vm

fm

(MPa)

Ep

(GPa)

vp

fp

(MPa)

30 0.18 45.1 68 0.16 295

Figure 6: The expected strength value with different number of samples (ellipsoidal particles)

— 75 —

Fig. 6 shows the numerical results. For each number of samples, 5 predicted values are given. Obviously, the

decentralization of data becomes weaker and weaker as the increase of the number of samples. In fact, for the

composite materials with random distribution of plenty of particles there are two factors to influence the

predicted results. One is the particle volume fraction. Another is the orientation distribution of the ellipsoidal

particles. In most cases, the particle volume fraction is easy to be controlled and can be made stable, while it

is very hard to realize the real randomicity of the orientation of the ellipsoidal particles. For instance, in one

case, the long axes of most ellipsoidal particles are along the direction of load, and in the other case, the short

axes of most ellipsoidal particles are along the direction of load. The predicted strengths in the both cases

must be different. To validate this, another set of numerical data is shown in Fig. 7. Here the particles are

assumed to be spheres with identical radius 20mm. All the other parameters are the same as that of the above

numerical experiment. From Fig. 7, it is easy to see that the convergence is higher than that with ellipsoidal

particles. Therefore, if the orientation of particles insignificantly influences the predicted result, such as

sphere particles made of isotropic materials, less samples will give a stable prediction, otherwise, plenty of

samples must be taken to avoid the decentralization of the numerical results while the strength of random

particle reinforced composites is predicted. The number of samples depends on the required precision.

Figure 7: The expected strength value with different number of samples (spherical particles)

2. Comparisons with experimental data In this section, the strength of the concrete is calculated using

STSM in this paper. In regard to normal strength concrete, the interfacial zone is the weakest part. Cracks are

always initialized in the interfacial zone and propagate along the surface of coarse aggregates, which means

the strength of interfacial zone has to be taken into account when the strength of normal concrete is

predicted. However, the failure mechanism of high strength concrete has a obvious change while the quality

of interfacial zone is greatly improved. The fracture surfaces pass through the coarse aggregates as well as

the cement paste. It has been demonstrated that the strength of high strength concrete is controlled by the

weakest component [38,39].

The experimental data from Yang et al. [39] is excerpted. In [39], the strength of high strength concrete made

of different artificial aggregates and matrix was tested in the laboratory. The matrix was cement-based with

two different water/(cement + silicafume) ratios (w/b=0.28 and 0.6). The aggregates were cast in the

spherical mold using cement pastes with three water/cement ratios (w/c=0.4, 0.5 and 0.6). The mechanical

properties, including the elastic constants and compressive strength of the matrix and aggregates at the age of

28 days, are shown in Table 2. The details of the cylindrical specimens (Ф150×300 mm) tested in [39] are

also shown in Table 2, where Va means the volume fraction of aggregates. Corresponding numerical

computations have been made. The unit cell is taken as a cube with size 200×200×200(mm3

). Aggregates are

simulated by spheres with radius 25mm. The number of samples is taken as 30.

Table 3 shows the numerical result comparing with measured values. In the table exp

c

f means the expected

compressive strength predicted by STSM, and min

c

f means the minimum strength among the 30 samples. It

is easy to see that there is a great agreement between STSM-predicted results and measured values.

— 76 —

Table 2: Mechanical properties of the aggregates and matrix [39]

Matrix Aggregates

Designation w/b

Em

(GPa)

vm

fm

(MPa)

w/c

Ea

(GPa)

va

fa

(MPa)

Va

(%)

341

342

343

0.4 18.06 0.24 53.24

10

20

30

351

352

353

0.5 15.43 0.20 40.09

10

20

30

361

362

363

0.28 24.81 0.22 66.36

0.6 13.96 0.20 29.72

10

20

30

641

642

643

0.4 18.06 0.24 53.24

10

20

30

651

652

653

0.5 15.43 0.20 40.09

10

20

30

661

662

663

0.6 15.71 0.20 34.22

0.6 13.96 0.20 29.72

10

20

30

Besides compressive strength, the bending and torsional strengths are also evaluated here. All the parameters

are the same as presented above. Only the number of samples is changed to 20. The results are shown in

Table 4, and no comparison is made since there are rare experimental data about bending and torsional

strength available. Here, bending strength means the maximum homogeneous normal stress of a beam made

of random particle reinforced composites under pure bending load till the maximum stress inside the beam

reaches the elastic limit. Torsional strength means the maximum homogeneous shear stress of a composite

bar with constant circular section shape under torsional load till the stress inside the bar reaches the elastic

limit. Table 4 shows that the bending strengths (inside compressive stress zone) are very close to

compressive strengths. It is because the state of the unit cell belongs to dangerous area (the upper/lower

surfaces of the beam) under bending load is similar to that under compressive load.

Table 3: Comparison between measured and predicted compressive strength

STSM-Predicted (MPa)

Designation

Measured [39]

mea

c

f (MPa) exp

c

f min

c

f

exp

%

mea

c c

mea

c

f f

f

−

341

342

343

61.57

60.30

58.14

59.075

57.675

56.293

56.764

55.704

54.173

-4.05

-4.35

-3.18

351

352

353

52.13

47.42

45.30

51.025

48.793

46.671

48.228

45.575

43.295

-2.12

2.90

3.03

361

362

363

41.70

38.63

35.94

40.005

37.713

36.158

35.149

34.892

33.765

-4.06

-2.37

0.61

641

642

643

33.18

33.35

33.98

31.931

31.692

31.877

31.299

31.122

31.340

-3.76

-4.97

-6.19

651

652

653

35.66

34.61

33.77

34.001

33.944

33.829

33.961

33.884

33.228

-4.65

-1.92

0.17

661

662

663

29.34

28.28

27.56

31.449

30.998

30.738

31.046

30.198

30.237

7.19

9.61

11.53

— 77 —

Table 4: Bending and torsional strength predicted by STSM

Bending strength (MPa)

Torsional strength

(MPa) Designation

Exp. Min Exp. Min

341

342

343

60.513

58.864

56.635

58.450

56.722

55.720

46.3192

44.7703

43.1664

45.5933

43.8468

42.5574

351

352

353

51.407

49.454

47.602

46.207

47.436

45.804

39.0217

37.2942

35.882

38.1409

35.7913

34.8644

361

362

363

40.198

38.020

36.899

35.36

34.184

34.422

30.3532

28.9381

27.8361

29.6113

27.8433

26.6142

641

642

643

32.196

31.820

31.822

31.755

31.128

31.336

25.1788

25.2462

25.1728

24.5555

24.5874

24.7821

651

652

653

34.032

33.965

33.839

33.986

33.864

33.110

26.1643

26.1008

26.0393

26.1444

25.9441

26.0026

661

662

663

31.454

31.152

30.741

30.965

30.489

30.243

24.1128

23.9023

23.6325

23.9184

23.6465

23.4348

It should be pointed out that here we mainly pay attention to the compressive strength as well as the strengths

of aggregates and matrix. With respect to the concrete, it is worthy to note that there is always great

difference between tensile and compressive strength for both aggregates and cement. So the tensile strength

of aggregates and matrix must be treated carefully and correctly when predicting the strength of concrete

under bending, torsional or tensile load.

NUMERICAL EXPERIMENTS

The strength of random particle-reinforced composites depends on both macro parameters, such as structural

parameters, load type and boundary condition, and meso-scale configuration like the shape, size and spatial

distribution of particles, the material coefficients of particles and matrix etc. In this paper, a statistical

two-scale method is developed to predict the strength of structures made of random particle-reinforced

composites. The formulas of stain field of three kinds of the conventional components, such as a bar under

tensile load, a cantilever under pure bending load and a column under torsional load, are given. And the

STSM algorithm procedure is described in detail.

To validate the statistical two-scale method presented in this paper abundant numerical experiments are

made. Comparison with physical experimental data indicates that STSM in thispaper is feasible and valid for

strength prediction of random particle-reinforced composite structures.

Because of the Limit of physical experimental data no comparison on bending and torsional strength is

made. However, STSM can be used theoretically to predict the strength of any structures of random particle

composites under any load.

Acknowledgements

This work is supported by the Special Funds for Major State Basic Research Project (2005CB321704) and

National Natural Science Foundation of China (10590353 and 90405016).

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