The Two-Scale Methods for Mechanics Parameter Computation
of Composite Materials with Periodic Distribution
J. Z. Cui and X. G. Yu
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, China,
E-mail:cjz@lsec.cc.ac.cn
Abstract: In this paper, the Two-Scale computation Method (TSM) is presented for the mechanics
parameters of composite materials with small periodic configuration, including stiffness
parameters and strength parameters. First the major formulation of the Two-Scale Asymptotic
expressions (TSA) for composite materials with periodic configuration is briefly given. And then
the two-scale computation (TSM) formulation of the strains and stresses in conventionally
strength experimental components, such as the tension of the column with square cross section, the
bending of the cantilever with rectangular cross section and the twist of the column with circle cross
section, which are made from the composite materials with the same basic cells, is developed by
means of the fundamental solutions, and the procedure of TSM computation is discussed. Finally
the numerical results for both stiffness and strength parameter computation are demonstrated in three
kinds of typical components. And they have been compared with the experimental those, both TSM and
experimental results are very close. They show that the TSM method given in this paper is feasible and
valid for both stiffness and strength parameter prediction of the composite materials with periodic
configuration.
Key words: Two-scale method, stiffness parameters, strength parameters, composite materials
with small periodic configuration.
1.
Introduction
With the rapid advance of composite materials, they have been widely used to a variety of
industrial fields besides high science and technology products. Therefore in order to develop new
composite materials it is necessary to predict the physical and mechanical properties of the
composite materials at first. In stiffness parameter prediction a variety of meso-scopic and
microscopic methods in mechanics and engineering [1-13] was proposed by mechanical experts
and engineers.
The composite materials can be divided into two classes according to the basic configuration:
the composite materials with periodic configuration, such as periodically honeycomb materials
and braided composites, and the composite materials with random distribution, such as concrete,
foamed plastics. Due to the difference of basic configuration it is necessary to make use of
different numerical methods to evaluate the physical and mechanical properties of different
materials. Based on the homogenization method proposed by J.L. Lions, O.A. Oleinik and others,
the Multi-Scale Analysis (MSA) method for the structure of composite materials with periodic
configuration was proposed in [8-10]. It can be applied to evaluating the macroscopic stiffness
parameter of the composite materials with periodic configuration. For the physics field problems
of composite materials with the stationary random distribution Jikov and Kozlov developed the
homogenization method, and proved the existence of macroscopic homogenization parameters [7],
however, they did not give the numerical method to compute the macroscopic homogenization
parameters. One of the authors in this paper proposed a kind of Statistic Multi-Scale Analysis
(SMSA) method for the macroscopic mechanics and physics parameters of the composite
materials of random distribution [11, 12]. Up to now the literature pertaining to the theoretic
prediction for the strength parameters of 3D braided composites is very limited [13-15]. In this
paper, we are concerned with the mechanics parameter computation of the composite materials
with periodic configuration, including the effectively stiffness parameter and strength parameters.
A kind of two-scale method for the strength parameter computation of the composite materials
with periodic configuration is developed.
The remainder of this paper is outlined as follows: In section 2 the main results on the
two-scale asymptotic analysis for the composite materials with small periodic configuration is
briefly expressed. Section 3 is devoted to the Two-Scale Analysis (TSA) formulation for the
strength parameters of the composite materials with periodicity. In section 4 the procedure based
on TSA is stated for both stiffness and strength parameter computation of the composite materials
with periodicity. In section 5 the numerical results for mechanical parameter computation of the
composite materials with small periodicity are shown. They show that the TSM method given in
this paper is feasible and valid for both stiffness and strength parameter computation of the
composite materials with periodic configuration.
2.
Two-Scale Approximate Formulation for Composite Materials with Periodicity
Suppose that the structure or component made from the composite materials with small
periodic configuration is denoted by  
c
i
, where c i denotes basic cell, and all of
iT
ci i  T  are of same configuration with size  . From solid mechanics the elasticity problem
for this kind of structure  can be expressed as follows
   
1  uh ( x) uk ( x)  
a
(
x
)

x

 ijhk

   fi ( x)
2  xk
xh  
 x j 
u ( x)  u( x)
x  1


1  uh ( x) uk ( x) 



(
u
)


a
(
x
)


  pi ( x) x   2
j ijhk

2  xk
xh 

 1  2  0, 1  2   
where
(2.1)

( x) i, j, h, k  1,, n are the elastic
 denotes the size of basic cell ci , aijhk
coefficients of
  periodicity with cells, u ( x) is the solution of vector-valued displacement,
and  j are the normal direction cosine of  2 .
From composite material science it is well known that

L
 1 , where L denotes the size

of the structure  . Since aijhk ( x) i, j, h, k  1,, n vary very sharply and periodically it is
very difficult to numerically solve the problem (2.1) as the detailed configuration of materials and
the structure together are considered in a macroscopic scale.
From material science it has been shown that the mechanical properties of previous structure
depends not only on the macroscopic conditions, such as the geometry of the structure, the
macroscopic constants of materials, the loading and constraints, but also on detailed configuration.
Therefore the detailed configuration of composite materials should be considered with the
macroscopic conditions of structure together to investigate the mechanical properties of structure.
Let
x

  , then aijhk
 x   aijhk    aijhk   , aijhk   is 1-periodic function. From

 
x

[8] u ( x) can be expressed into u ( x)  u( x,  ) , where x denotes the global coordinates of

the structure and global behaviors, and   x /  the local coordinates on normalized cell and

the effects of cell configuration. Then u ( x)  u( x,  ) can be expressed formally as follows

u  x   u 0  x     l
l 1

1 l 1 n
N1
l
 
l u0
, x 
x1 xl
(2.2)
0
where u ( x) is the homogenization solution and defined on global  , N12
1,2 ,
,l  1,
, n, l  1, 2,3,

l
 
are n-order matrix valued functions with 1-periodicity,
and they will be defined on 1-square Q normalized basic cell. They can be expressed as follows
N1
 N1

l    
N
 1
 l 11
 l n1
 
 
N1
N1
 l 1n
  

  N1

 l nn  


l 1
 
N1
l n
 
(2.3)
0
In concrete, all of N 1 2  l m   1 ,  2 , ,  l , m  1, , n, l  1,2,3,  and u ( x) are
determined as follows:
1)
For l  1 , N 1m   1 , m  1, , n  are the solutions of following problems
  
aij1m ( )
1  N1hm ( ) N1km ( )  

 Q

 aijhk ( ) 
  
2   k
 h  
 j
  j 

  Q
N1m ( )  0
2)
From N 1m   1 , m  1, , n  , the homogenization elasticity parameters
calculated as follows
(2.4)
â 
ijhk
are

1  N hpk ( ) N hqk ( )  
aˆijhk    aijhk ( )  aijpq ( ) 

 d
 
Q

2


q
p



3)
(2.5)
For l  2, N 1 2 m   1 ,  2 , m  1,  , n  are the solutions of following problems
  
1  N1 2 hm ( ) N1 2 km ( )  


 aijhk ( ) 
   aˆi 2 m1  ai 2 m1 ( )
2
 k
 h

  j 
  Q

N1hm ( ) 

ai 2 hk ( )

aijh 2 ( ) N1hm ( )





k
j

N  m ( )  0
  Q
 12

4)

(2.6)
u0 ( x) is the solution of the homogenization problem defined on global  with the
 
homogenized parameters â ijhk

1   uh0 ( x) uk0 ( x) 

 aˆijhk

  fi ( x), x  
2 x j  xk
xh 

 u 0 ( x)  u( x),
x  1


0
0
 x   aˆ 1  uh ( x)  uk ( x)   p , x  




j ijhk
i
2
 i
2  xk
xh 


 1  2  0, 1  2   
5)
(2.7)
N 1 2  l m   1 ,  2 ,,  l , m  1,, n, l  3, are the solutions of following
problems,
  
1  N1 2 ...l hm ( ) N1 2 ...l km ( )  


 aijhk ( ) 

2
 k
 h
  j 
 

 ail hl 1 ( ) N1 2 ...l 2 hm ( )

  Q

N1 2 ...l 1hm ( )

ail hk ( )
 k





aijhl ( ) N1 2 ...l 1hm ( )
 j


  Q
N1 2 ...l m ( )  0

6)
Since aij1m ( )  aijm1 ( ) , then N12
l m
(2.8)

  1,2 ,
,l , m  1,
, n, l  3,

satisfies symmetric relation
N12
7)
l m
   Nm
2
l1
 
(2.9)
In practical computation of engineering problem only the sum of fore M terms in (2.2) are
evaluated
M
u
(M )
 x   u0  x    
l 1
l

1 l 1 n
N1
l u0
, x 
l  
x1 xl
(2.10)
generally, M  1,2,3 . And then the strains and stresses are evaluated in the formulas
anywhere
1  uh ( x) uk ( x) 


2  xk
xh 
 hk ( x,  )  
1  uh0 ( x) uk0 ( x) 
 


2  xk
xh 
(2.11)
M
1
   l   N αhm ( ) Dαl k1um0 ( x)  N αkm ( ) Dαl h1um0 ( x) 
l 1
α l 2
M
   l 1
l 1
where
1  N αhm ( ) N αkm ( )  l 0


Dα um ( x)
 k
 h 
α  l 2 

  1, 2 ,,l  , Dl u m0 x  
 l u m0 x 
x1 x 2 x l
 ij ( x,  )  aijhk ( ) hk ( x,  )

 u 0 ( x) uk0 ( x) 
1
aijhk ( )  h


2
xh 
 xk
M
  l
l 1
M
1
aijhk ( )  N αhm ( ) Dαl k 1um0 ( x)  N αkm ( ) Dαl h1um0 ( x) 
2
 α  l

   l 1
l 1
(2.12)
 N ( ) N αkm ( )  l 0
1
aijhk ( )  αhm

Dα um ( x)
 h 
 α  l 2
  k

It has been proved that if the basic configuration of cells compounded into composite materials is
constructed into symmetrical-structure for xi  axis
i  1,
, n  , then the approximate
displacement (2.10), strains (2.11) and stresses (2.12) strongly converge to the true displacement,
strains and stresses of problem (2.1) as
 0.
Figure 1. The 2-D structures with periodic configuration
and their basic cell
3.
The Formulation for Strains and Stresses of Some Special Structures
It is well known that the strength parameters of one kind of materials are obtained by the
mechanics experiment of typical components made from those. By the maximum of elasticity
strains/stresses inside component the strength parameters are worked out. In this section the
formulas on the strains and stresses inside the structure of composite materials with periodic
configuration will be discussed by the means of simulating the mechanics experiments.
From (2.11) and (2.12) it follows that the distribution of the strains and stresses inside the
structure/component made from composite materials depend on both the macroscopic condition,
0
that is the homogenization displacement u ( x ) defined on global  , and the detailed
configuration
of
cells,
1 , 2 ,, l , m  1,, n, l  1,2,3
the
local
solutions
N  1 2  l m  
determined on normalized cell Qcell . In the mechanics
experiment three kinds of experimental components, such as tensile column, bend beam and twist
column, are often adopted. The formulas on the strains and stresses inside them are discussed
below, respectively.
3.1
Column tension
The tensile behavior of the column with rectangular cross section shown in Figure 2, which
3
x3
T
1
o
2
x2
x1
Figure 2. Column with square section
secton 矩形截面柱体的均匀拉伸
is made from composite materials with small periodicity, is investigated at first. Let A denotes
the area of cross section, L length of the column, x3  0 fixed end and x3  L loading end
with load T . From elasticity mechanics for the tension of the column with orthogonal-anisotropic
material coefficients there exists a true displacement as follows
 0
 13
px1
u1  
E11

 0
 23
px2
u2  
E
22

 0
p
x3
u3 
E33

(3.1)
where p  T / A , E11 , E22 , E33 ,  12 and  23 are the elasticity modulus of three axis
directions and Poisson ratio.
From (3.1) it follows that
Dl um0  x  
 l um0  x 
 0, for l  2 .
x1 x 2 xl
Thus the displacement vector of the tension problem of the column made from composite
materials with periodic configuration can be expressed as
0
x u1 ( x)
u ( x)  u ( x)   N1 ( )
.
 x1

0
(3.2)
And then the formulas on the strains and stresses inside previous column can be exactly written
into
1  uh0 ( x) uk0 ( x ) 


2  xk
xh 
 hk ( x)  
x
x 

N ( ) N1km ( )  0
1  1hm 
 um ( x)
 


2  xk
xh
 x1


(3.3)
1
x  uh0 ( x) uk0 ( x) 
 ij ( x)  aijhk  x   hk  x   aijhk ( ) 


2
  xk
xh 

x
x 

N1hm ( ) N1km ( )  0

1
x
 
 um ( x)
 aijhk ( ) 

2
   k
 h  x1


(3.4)
Substituting (3.1) into (3.3) and (3.4) one obtains the expression on each component of the strain
tensor inside every cell of the column respecting the symmetry of N1 ( ) and N1 2 ( )
11  
12 
 13
E11
p p



  1
N313  13 N111  23 N 212 

1  E33
E11
E22




p   1
N 323  13 N121  23 N 222 

2 1  E33
E11
E22




p   1

N 313  13 N111  23 N 212 

2  2  E33
E11
E22

13 



p   1
N 333  13 N131  23 N 232 

2 1  E33
E11
E22




p   1

N 313  13 N111  23 N 212 

2 3  E33
E11
E22

 22  
 23 
 23
E22
p p



  1
N323  13 N121  23 N 222 

2  E33
E11
E22




p   1
N 333  13 N131  23 N 232 

2  2  E33
E11
E22




p   1

N 323  13 N121  23 N 222 

2 3  E33
E11
E22

(3.5.1)
(3.5.2)
(3.5.3)
(3.5.4)
(3.5.5)
 33 
where  
x




p
  1
p
N333  13 N131  23 N 232 

E33
3  E33
E11
E22

(3.5.6)
.
Furthermore, from above strains the stresses are evaluated anywhere inside every cell
belonging to the column. Then based on the yield criterion of basic materials, such as matrix,
reinforced fibers and interfaces, the critical point of tensile column of composite materials can be
evaluated.
3.2
Beam bending
1
3
x1
x3
2
M
h
(a)
M
o
x2
L
b
Figure 3. The bending of cantilever made from
composite materials
The bending of the cantilever with rectangular cross section, which is made from composite
materials with periodic configuration, is investigated shown in Figure 3. Let x3  0 denotes
fixed end and at x3  L the bend moment round x2 axis is imposed. From solid mechanics the
bending problem of the cantilever with orthogonal-anisotropic material coefficients has following
solution
 0
M  1 2  13 2  23 2 
x3 
x1 
x2 
u1  

2
I
E
E
E
x2  33
11
22



 23 M
0
x1 x2
u2  
E22 I x2


M
u30 
x1 x3
E33 I x2

(3.6)
where I x2 
bh3
is the moment of inertia round x2 .
12
It is easy to see that the displacements are two-order polynomial. So
 l um0  x 
D u  x  
 0, for l  3 .
x1 x 2 xl
l
0
m
Thus the displacement vector of the bend problem of the cantilever made from composite
materials with periodic configuration can be expressed as
0
2 0
x u  x  0
x  u  x
2
u( x)  u ( x)   N1 ( )
u ( x)   N1 2 ( )
 x1 1
 x1 x2
0
(3.7)
Thus the strains anywhere inside above cantilever can be approximately calculated in following
formulas
1  uh0 ( x) uk0 ( x) 
 hk ( x)  


2  xk
xh 

 2um0 ( x)
 2um0 ( x) 
1
 N1km ( )
 N1hm ( )

2 
x1 xk
x1 xh 
1  N1hm ( ) N1km ( )  um0 ( x)
 


2   k
 h  x1

(3.8)
1  N1 2 hm ( ) N1 2 km ( )   2um0 ( x)



2
 k
 h
 x1 x 2

 ij ( x)  aijhk
 x   hk  x 

 u 0 ( x) uk0 ( x) 
1
aijhk ( )  h


2
xh 
 xk


 2um0 ( x)
 2um0 ( x) 
1
aijhk ( )  N1hm ( )
 N1km ( )

2
x1 xk
x1 xh 

(3.9)
 N1hm ( ) N1km ( )  um0 ( x)
1
 aijhk ( ) 


2
 h  x1
  k
 N1 2 hm ( ) N1 2 km ( )   2um0 ( x)
1
 aijhk ( ) 


2
 k
 h

 x1 x 2
Respecting the symmetry of N1 ( ) and N1 2 ( ) , the components of strain tensor inside
every cell belonging to the cantilever are evaluated in following formulas:
11  

M
I x2
 1



N 313 ( )  13 N111 ( )  23 N 212 ( ) 

E11
E22
 E33




Mx1   1
N 313 ( )  23 N 212 ( )  13 N111 ( ) 

I x2 1  E33
E22
E11




M   1

N 3113 ( )  23 N 2112 ( )  13 N1111 ( ) 

I x 1  E33
E22
E11

(3.10.1)
2

 13 M
x1
E11 I x2
12  
M
2 I x2
 1



N 323 ( )  23 N 222 ( )  13 N121 ( ) 

E22
E11
 E33





Mx1   1
N 323 ( )  23 N 222 ( )  13 N121 ( ) 

2 I x2 1  E33
E22
E11





M   1
N 3123 ( )  23 N 2122 ( )  13 N1121 ( ) 

2 I x 1  E33
E22
E11

(3.10.2)
2




Mx1   1
N 313 ( )  23 N 212 ( )  13 N111 ( ) 

2 I x2  2  E33
E22
E11





M   1
N 3113 ( )  23 N 2112 ( )  13 N1111 ( ) 

2 I x  2  E33
E22
E11

2
13  



M  1
N 333 ( )  23 N 232 ( )  13 N131 ( ) 

2 I x2  E33
E22
E11





Mx1   1
N 333 ( )  23 N 232 ( )  13 N131 ( ) 

2 I x2 1  E33
E22
E11





M   1
N 3133 ( )  23 N 2132 ( )  13 N1131 ( ) 

2 I x 1  E33
E22
E11

(3.10.3)
2




Mx1   1
N 313 ( )  23 N 212 ( )  13 N111 ( ) 

2 I x2 3  E33
E22
E11





M   1
N 3113 ( )  23 N 2112 ( )  13 N1111 ( ) 

2 I x 3  E33
E22
E11

2
 22 




Mx1   1
N323 ( )  23 N 222 ( )  13 N121 ( ) 

I x2  2  E33
E22
E11




M   1
N3123 ( )  23 N 2122 ( )  13 N1121 ( ) 

I x  2  E33
E22
E11

2

 23 Mx1
E22 I x2
(3.10.4)
 23 




Mx1   1
N 333 ( )  23 N 232 ( )  13 N131 ( ) 

2 I x2  2  E33
E22
E11




M   1
N 3133 ( )  23 N 2132 ( )  13 N1131 ( ) 

2 I x  2  E33
E22
E11

2



Mx   1
 1
N 323 ( )  23 N 222 ( )  13 N121 ( ) 

2 I x2 3  E33
E22
E11


(3.10.5)



M   1
N 3123 ( )  23 N 2122 ( )  13 N1121 ( ) 

2 I x 3  E33
E22
E11

2
 33 




Mx1   1
N333 ( )  23 N 232 ( )  23 N131 ( ) 

I x2 3  E33
E22
E22




M   1
N3133 ( )  23 N 2132 ( )  23 N1131 ( ) 

I x 3  E33
E22
E22

(3.10.6)
2
Mx1

E33 I x2
where  
x

.
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 x1 component of macroscopic coordinate
appear in the strain expressions. It means that the strains do not depend on macroscopic coordinate
x2 and x3 . And then the maximum strain occur in the cells located on the above or below
surface of the cantilever, but it is uncertain that the maximum strain occur on the above or below
surface x1   h / 2 , since the strains and stresses change very sharply inside each cell. According
to maximum principal stress and principal strain one can evaluate the elasticity strength limit of
the beam bending of composite materials with any periodic configuration.
It is worthy of note, the basic configuration of the cells of composite materials must lead to
macroscopically orthogonal-anisotropic material coefficients. If not, (3.6) does not hold, herewith,
each formula of (3.10) is wrong.
3.3
Twist of column
The twist of the column with circle cross section, which is made from composite materials
1
3
x3
2
T
x1
(a)
r
T o
x2
L
Figure 4. Twist of the column with circle section
with same periodic configuration, is investigated, shown in Figure 4. Let r denotes the radius of
cross section, L the length of the column, x3  0 fixed end, and on x3  L the twist moment
is imposed. If the column can be considered as that made from orthogonal-anisotropic materials,
from elasticity mechanics the homogenization displacement solution can be expressed as
 0
Tx2 x3  1
1 

u1  


 r 4  G13 G23 


1 
 0 Tx1 x3  1

u2 

4 
 r  G13 G23 


u30   Tx1 x2  1  1 

 r 4  G13 G23 
(3. 11)
where G13 , G23 denote the shear modulus in x1-x3 plain and x2-x3 plain.. It is easy to see that the
displacements are the 2-order polynomial under above supposition. And respecting the symmetry
of N1 ( ) and N1 2 ( ) the components of strain tensor inside the column are expressed as
follows
11 

x
2 T
2T   x1
N 213  4
N 213 ( )  2 N113 ( ) 

4
 r G23
 r 1  G23
G13

(3.12.1)
T
12 
 r 4G23
 N 223 ( )  N113 ( ) 

x
T   x1
N 223 ( )  2 N123 ( ) 

4
 r 1  G23
G13


x
T   x1
 4
N 213 ( )  2 N113 ( ) 

 r  2  G23
G13


13 
T
 r G23
4
N 233 ( ) 
(3.12.2)
Tx2
 r 4G13

x2
T   x1
N
(

)

N
(

)


233
133
 r 4 1  G23
G13


x
T   x1
 4
N 213 ( )  2 N113 ( ) 

 r 3  G23
G13


 22  
 23 
(3.12.3)

x2
2 T
2T   x1
N

N
(

)

N
(

)


123
223
123
 r 4G13
 r 4 2  G23
G23

Tx1
T
 4
N ( )
4
 r G23  r G13 133

x
T   x1
N 233 ( )  2 N133 ( ) 

4
 r  2  G23
G13


x
T   x1
 4
N 223 ( )  2 N123 ( ) 

 r 3  G23
G13


 33 
where  
x

(3.12.4)
(3.12.5)

x
2T   x1
N 233 ( )  2 N133 ( ) 

4
 r 3  G23
G13

(3.12.6)
.
Using the stress-strain relation one can evaluate the stresses inside each cell belonging to the
column. And then according to maximum principal stress and principal strain one can determine
the elasticity critical point of the twist column of composite materials with same basic
configuration.
Remark:
1.
It should be noticed that since the column with circle cross section includes
incomplete cells around its surface, it means that  
c
i
( c i is entire cell)
iT
does not hold, so the expressions (3.12) of the strain components inside the column

are approximate. But if
L
 1 , the strain values evaluated by (3.12) are close to
true values everywhere inside column except the incomplete cells around its surface.
2.
The basic configuration of cells must satisfy such symmetry that it leads to
orthogonal-anisotropic isotropic coefficients. If not, (3.11) does not hold, herewith,
(3.12) is wrong.
Similarly, one can obtain the expressions of strains and stresses for other component of
composite materials with periodic configuration.
4.
Two-Scale Algorithm for Mechanics Parameter Computation
4.1
Approximate formulas for stiffness coefficients
1. FE solutions for N  1m   and the approximate stiffness coefficients
the approximate elasticity stiffness coefficients
solutions N1m  
1, m  1,
h
aˆ 
h
ijhk
aˆ  . From（2.5）
h
ijhk
can be calculated if the approximate
, n  of problem (2.4) are obtained.
As you have known, from PDE theory it follows that problem (2.4) is equivalent to following
virtual work equation

Q
 ij  v  aijhk    hk  N m d     ij  v  aij m   d , v  H 01  Q 
1
1
Q
(4.1)
where
1  vi   v j   


2   j
i 
 ij  v   

Thus approximate N1m  
h
 1 , m=1, 2,3 

can be determined by solving following FE
virtual work equation on FE space S0  Q   v  S
h
(4.2)
h
 Q  / v  Q   0
    v  a     N    d =     v  a    d
ij
eS
h
ijhk
h
hk1
e
Actually N1m  
h
 1 , m=1, 2,3 
approximate stiffness coefficients
aˆ 
h
ijhk
m
ij
eS e
h
ij 1m
v  S0h  Q  (4.3)
are obtained by using general FE software. Then the
can be evaluated in below formulas
h
aˆijhk
  aijhk    aijlm    lm  Nhhk  d .
Q
2. FE computation of
N 1 2m  
(4.4)
N 1 l m   . The FE solutions
and
Nh12m   and Nh1l m   for 3  l  M are obtained by solving following FE virtual
h
work equations on 1-square Q corresponding to FE partition S , respectively

Q
 ij  v  aijhk    hk  Nh  m d     ij  v  a jh   Nh hm d
1
2
2
Q

1
(4.5)

   ai 2 m    ai 2hk    hk Nh m1  aˆih m2  1vi d , v  S0h  Q 
Q

Q
 ij  v  aijhk    hk  Nh
1
   ail hl 1   Nh1
Q
 l 2 hm
d
d     v  a   N
 a     N
 v d , v  S Q 
l m
Q
i l hk
h
ijh l
ij
1  l 1hm
h
1  l 1m
hk
i
(4.6)
h
0
3. Homogenization Solution. From elasticity mechanics the homogenization solution u  x  for
0
typical structure / component can be exactly obtained, and for general structure by solving the FE
virtual work equations corresponding to (2.7) on global  .
4. Approximate displacement, strains and stresses. Two-scale approximate solution anywhere
on structure  can be evaluated in following formulas
l 0
 x   um
x   u x     N1lim  
, x 
   x1 xl
l 1
1 l 1n
M
u
(M )h
i
0
i
l
h
l 0
M
M

 h
1
 x    um
0
l 1


 hk u x     hk  N1l m   
  l

    x1 xl l 1 2
l 1


 hk( M ) h x   


 l 1u m0
 l 1u m0
 x
 x
h
 N h

N




, x  
1 l km
 1l hm    x x xk


x


x

x




h
1
l
1
l






x hk( M )h x ,
 ij( M ) x  aijhk
4.2
(4.7)
x 
(4.8)
(4.9)
Formulas for strength parameters
0
Once the homogenization solution u ( x ) for the component made from composite
materials with small periodicity is obtained the strains and stresses everywhere inside it can be
evaluated in the formulas given in previous section. Thus it is easy that through analyzing the
strain and stress distribution inside each cell belonging to the component the dangerous point of
the strains / or stresses can be determined according with the strength criterion of basic materials.
Since there is always a great difference between the mechanics properties of reinforcement
and matrix of composite materials, so it is necessary to take different strength criterions for
reinforcement and matrix. The facts, which lead to the fracture of the fibers and the yield of the
matrix macroscopically, are that most of fibers used today are brittle, and matrix always has the
characteristic of high ductility. The strength criterions taken in this paper are as follow:
1)
Reinforcement (fiber). Considering the brittle characteristic of the reinforcement (fiber), the
maximum normal stress criterion is adopted. The criterion for homogenous materials says that the
material collapse while one of the principal stresses exceeds the strength of the material. With
respect to the fibers they are always subject to the longitudinal load and fracture along the
transverse section. Thus we take the longitudinal normal stress and the two principal stresses in
the transverse plane instead of the three principal stresses in conventionality. Then the criterion
can be expressed as:
 L  SL ，  T  ST
(4.10)
where  L is the longitudinal normal stress,  T denotes the maximum normal stress in the
transverse plane of the fibers. S L 、 ST denote the longitudinal and transverse limit stresses,
 L can be evaluated through the transformation of
respectively. The longitudinal normal stress
coordinates. As shown in Figure 5, the relation between the stresses vector of the local coordinates
(x'1, x'2, x'3)
 
and
 
global coordinates (x1, x2, x3) is the following:
   T  
where
T
    11  22  33  12  23  13
(4.11)
, the transformation matrix T can be
expressed as follows:
 l12
 2
 l2
 l2
T   3
 l1l2
l2l3

 l3l1
m12
n12
2l1m1
2m1n1
2
2
2
3
2
2
2
3
m
n
2l2 m2
2m2 n2
m
m1m2
n
n1n2
2l3m3
2m3n3
l1m2  l2 m1
m1n2  m2 n1
m2 m3
n2 n3 l2 m3  l3m2
m2 n3  m3n2
m3m1
n3n1
l1m3  l3m1
m1n3  m3n1

2n2l2 
2n3l3 

n1l2  n2l1 
n2l3  n3l2 

n1l3  n3l1 
2n1l1
where li , mi , ni (i  1, 2,3) are the direction cosine between global coordinates and local
coordinates. The third component of
 
is just the longitudinal normal stress  L . The
x'3
x'2
o'
x'1
Fiber
x1
x3
o
x2
Figure 5. The global and local coordinates of
a cell with one fiber
maximum normal stress in the transverse plane of the fibers  T is just the first principal stress in
the transverse plane, which can be evaluated in following formula
T 
 11   22
2
    22 
  11
   12
2


2
(4.12)
If and only if formula (4.10) does not hold, the material begins to stat in nonlinear state.
2)
Matrix. Since the matrix of composite materials is ductile, the Von Mises effective stress
yield criterion is employed. The criterion is following
e 
1
( 1   2 )2  ( 2   3 ) 2  ( 3   1 ) 2  S
2
(4.13)
or
e 
1
2
( 11   22 )2  ( 22   33 )2  ( 33   11 )2  6( 122   23
  312 )  S
2
(4.14)
where  1 、  2 、  3 denote the three principal stresses. S is the tensile (compressive) limit
stress.  e is Von Mises effective stress.  ij (i, j  1, 2,3) are the components of the stress
tensor, which can be obtained from (2.12). If and only if above equation does not hold, the
material begins to yield.
Once the basic configuration of composite materials and the load type of a component / or
structure are given, the critical load of the component / or structure can be determined through the
0
criterions above (4.10) and (4.14). Then the homogenization displacement u ( x ) under the
critical load can be obtained. It’s easy to get the strain and stress state under the critical load
through geometrical and physical equations.
4.3
Algorithm procedure
The algorithm of TSM is following:
1. Determine the composite materials, i.e. the basic configuration of cells, such as the basic
composition, matrix, reinforced fiber and interface materials, and their properties, and the
located position of them in cell. And then verify the distribution functions
a  x  of the

ijhk
material properties of basic configuration.
h
2. Solve FE virtual work equation (4.2) on 1-square Q to obtain N1m   , and then evaluate
the homogenized constitutive coefficients
aˆ 
h
ijhk
in formula (4.4).
3. Design the topology of structure/component  using verified composite materials.
4. Evaluate the FE solutions N1 2m   and N1l m   3  l  M  by solving FE virtual
h
h
work equation (4.5) and (4.6) on 1-square Q , using same FE meshes as in step (2) as well as
the stiffness matrix and its decomposition form.
5. Obtain the homogenization displacement u  x  for typical structure / component, Or FE
0
displacements u
0h
x 
for general structure by using FE software on whole structure  ,
 l u0hm  x 
.
 x  x
0
and then evaluate the high order partial derivatives
1
7. Compute the displacement u
and stresses  hk
(M )h
x 
(M )h
x 
l
(M )h
at arbitrary point of structure  , strains  hk  x 
in (4.7), (4.8) and (4.9).
8. Evaluate constitutive parameters of the composite materials and strength parameters of the
structure / component according to the formulas of previous subsection (4.2).
Remark: if the configuration of composite materials has the characteristic of multi-level and
multi-scale the multi-scale computation is needed, and then above computation should be
repeated.
5.
Numerical Experiments
In this paper the stiffness parameters and strength parameters of unidirectional fiber-reinforced
and 3D orthogonal braided composite materials are predicted through two-scale method. The
results are compared with some theoretical models and experimental data [15-20].
5.1 The mechanics parameter computations of composite materials with unidirectional
fiber-reinforcement
1. Computational model. There are many ways
to choose the basic configuration of cell
compounded into unidirectional fiber reinforced
composite materials, shown in Figure 6. Three
kinds of basic configuration, named C1, C2, C3
shown in Figure 7, respectively, can be chosen in
computation. The computational results obtained
from three kinds of basic configuration, C1, C2
Figure 6. Unidirectional fiber-reinforced
composite structure
and C3, respectively, are very close for the stiffness parameters and the strength parameters in the
tension and bending case of column made from unidirectional fiber reinforced composite materials.
Therefore only the results for the basic configuration C1 are shown below.
C1
C2
C3
Figure 7. Three kinds of basic configurations for
composite structure
In order to compare with the experimental results of [18], two sets of composite materials with
different fraction of fiber volume, respectively, 27% and 23% are investigated. The basic
configuration is denoted by U11 corresponding to 27%, and U21 corresponding to 23%,
respectively. The FE models of basic configuration are shown in Figure 8. The parameters of FE
partition are shown in Table 1, in that tetrahedron element is denoted by “TET”. The elastic
properties of the matrix and the fibers are shown in Table 2.
x2
x1
U11
U12
x3
U13
Figure 8 The FE models of the cells
Table 1 FE models of basic cell（mm）
U11
U21
Element Type
TET
TET
Account of elements
6400
6081
Account of nodes
1415
1355
Table 2 The elastic properties of the matrix and fibers for
unidirectional fiber-reinforced composite material
Matrix (Shell 862)
Fibers (AS-4)
Em
(Gpa)
Gm
(GPa)
Ef1
(GPa)
Ef2
(GPa)
Gf12
(GPa)
Gf23
(GPa)
 12
2.94
1.07
234.6
13.8
13.8
5.5
0.2
2. The computation for stiffness parameters. Once N1m  
h
 1 , m=1, 2,3 
are
obtained by solving FE equation (4.3). Then the approximate elasticity constitutive coefficients
aˆ 
h
ijhk
can
be
evaluated
in
below
formula
(4.4).
Since
the
four-order
tensor
aijhk (i, j , h, k  1, 2,3) and the elastic matrix D have the relationship (5.1). It is easy to calculate
the elastic matrix D from
aˆ  .
h
ijhk

 a1111

a
 2211

 a3311

D
 a2311


 a3111


 a1211
a1122
a1133
a2222
a2233
a3322
a3333
a2322
a2333
a3122
a3133
a1222
a1233
a1123  a1132
2
a2223  a2232
2
a3323  a3332
2
a2323  a2332
2
a3123  a3132
2
a1223  a1232
2
a1131  a1113
2
a2231  a2213
2
a3331  a3313
2
a2331  a2313
2
a3131  a3113
2
a1231  a1213
2
a1112  a1121 

2

a2212  a2221 

2

a3312  a3321 

2
a2312  a2321 

2

a3112  a3121 

2

a1212  a1221 

2
(5.1)
As you have known that the rule of mixture give a good prediction on longitudinal elastic
modulus of unidirectional composites in general case. The formula is as follow:
E11  EmVm  E f 1V f
(5.2)
where E11 denotes the macroscopic longitudinal elastic modulus of the unidirectional composite,
Vm 、V f the volume fraction of matrix and fiber, Em 、E f 1 the elastic modulus of matrix and the
longitudinal elastic modulus of fiber, respectively. To transverse elastic modulus there is not any
effective method for unidirectional composite at all. Furthermore, the results are compared with
experimental results in [18].
Table 3 shows the comparison of the longitudinal modulus calculated by using two-scale
method with experimental results [18] and the results evaluated in formula (5.2). Table 4 shows
the comparison of the transverse modulus calculated by using two-scale method with experimental
results [18]. ELTSM and ETTSM are the results computed by two-scale method, ELMXR the
results calculated from formula (5.2), and ELEXP , ETEXP are experimental results [18]. Table 3
indicates that the error in the prediction from TSM is very small, and smaller than that from the
rule of mixture. The error of 9.620% indicates that TSM can be applied to engineering. All of
these validate TSM proposed in this paper.
Table 3 Comparison of experimental and predicted longitudinal
modulus for the unidirectional composite
Model
Two-Scale Method
Mixture Rule
Exp.
Name
( ELTSM )
( ELMXR )
( ELEXP )[18]
(GPa)
(GPa)
(Gpa)
ELTSM  ELEXP
ELEXP
(%)
U11
64.376
65.5
62.6
2.837
Table 4 Comparison of experimental and predicted transverse
modulus for the unidirectional composite
Model
Name
TSM Results
Exp.[18]
( ETTSM )
( ETEXP )
ETTSM  ETEXP
ETEXP
(GPa)
(Gpa)
(%)
4.989
5.52
9.620
U21
3. The computation of strength parameters. Based on the formulas for the strains and stresses
given in section 3, the tensile (compressive) and bending strength of the composite column with
rectangular cross section, together with the twist strength of the composite column with circle
cross section, are evaluated..
The three model U11 of the unidirectional composite with fiber volume fraction 27% are
employed for the prediction of the longitudinal strength parameters, and the model U21 with fiber
volume fraction 23% for the prediction of the transverse strength parameters. For the longitudinal
strength parameters, three types of loads are imposed on the structure. They are tensile
(compressive) load along x3 axis, bending load around x1 axis and twist load around x3 axis, while
evaluating the transverse strength parameters, the loads are tensile (compressive) load along x1
axis, bending load around x3 axis and twist load around x1 axis, instead. The results are shown in
Table 5 and Table 6. The compressive strength parameters are compared with the theoretical
model and experimental data from [18]. The comparison with experimental results shows that the
longitudinal strength of unidirectional composite predicted by TSM is very accurate.
Table 5 Comparison of predicted, experimental and theoretical
longitudinal strength of unidirectional composite
Load Type
Compressive
Model
U11
TSM
Exp.[18]
( S LTSM )
( S LEXP )
(GPa)
(GPa)
236.2
235.0
Theoretical
Model in [18]
( S LTHY )(GPa)
234.9
S LTSM  S LEXP
S LEXP
(%)
0.664
Table 6 Comparison of predicted by TSM and experimental
transverse strength of unidirectional composite
Load Type
Tension
TSM
Exp.[18]
( STTSM )
( STEXP )
STTSM  STEXP
STEXP
(GPa)
(GPa)
(%)
50.96
76.0
39.0
Model
U21
Table 7 Prediction for the bending and twisting
strength of unidirectional composite
Load Type
Bending
Twisting
Model
U11
U11
Strength by TSM
236.2
40.98
Table 6 shows that there is a big difference between TSM and experimental results for the
transverse strength of unidirectional composite. It means that the transverse strength of
unidirectional composite mainly depends on the strength of matrix. We consider that the
transverse strength predicted by TSM is reasonable since TSM is perfect in theory. In fact, the
experimental results are decentralized for the transverse strength of unidirectional composite.
The prediction for the bending and twisting strength of unidirectional composite is shown in
Table 7. The bending strength indicates the maximum normal stress under critical bending
moment, and the twisting strength the maximum shear stress under limit torque. The result shows
that the bending strength is very close to compressive (tensile) strength. The reason is that the state of
stress of the dangerous point of the structure under bending moment is similar to that under
compressive (tensile) load.
5.2 The mechanics parameter computations of 3-D orthogonal braided composite
1. Computational model. The basic cell of 3-D
orthogonal braided composite is chosen similar
x2
x1
to that in [15] to validate the TSM method
previously. The configuration of the structure in
[15] is shown in Figure 9. A simplified model,
where the volume fractions for x1-fiber, x2-fiber
and x3-fiber are the same as that in [15], is taken
Figure 9. A schematic of the top-view for
3D orthogonal woven CFRP composite
in this paper. The geometric parameter of basic cell is 5mm×6mm×3mm. The overall fiber
volume fraction is 43%, and the nominal proportions of the x1 fiber, x2 fiber and x3 fiber are
1:1.2:0.2. The transverse section shape of x3 fiber is regarded as circle with radius 0.585mm, and
that of x1 fiber and x2 fiber is ellipse with long axial 1.6883mm and short axial 0.6078mm. There
are 27298 tetrahedrons and 5490 nodes in the FE model of basic cell, see Figure 10. The elastic
properties of the basic materials used are shown in Table 8.
Table 8 The elastic properties of basic materials of the 3-D
orthogonal braided composite cell
Matrix (Epicote 828)
Fibers (T-300)
Em
(GPa)
m
Ef1
(GPa)
Ef2
(GPa)
Gf12
(GPa)
Gf23
(GPa)
 f 12
2.2
0.35
220
13.8
11.35
5.5
0.2
x3
x2
Matrix
Fibers
x1
Figure 10 The FE model of 3-D orthogonal braided composite
2. The computation for stiffness parameters. Besides (5.1), the elastic matrix D for
orthogonal-anisotropic materials can be expressed with elastic constants as:
(   232 ) 2

( 12   23 13 )
(   )
D   12 23 13
0
0

0
( 12   23 13 )
( 12 23  13 )
0
0
(  )
( 23  12 13 )
0
0
( 23  12 13  )
(   )
0
0
0
0
G12 0
0
0
0
G23
0
0
0
0
2
13
2
2
12
0 

0 
0 
 (5.3)
0 
0 

G13 
where   E11 / E33 ，   E22 / E33 ，   E33 /(   23    12  2 12 23 13   13 ) ，
2
2
2
2
E11 、E22 、E33 are the longitudinal modulus of three axes, respectively, and  12 、 23 、 13 are
the poison-ratios, and G12 、 G23 、 G13 the shear modulus in the three coordinate planes,
respectively. All the elastic constants can be obtained by comparing the formulas (5.1) and (5.3).
The elastic constants of orthogonal-anisotropic composite are calculated by using two-scale
method, and compared with experimental results from reference [15].
Table 9 shows the prediction results for the elastic modulus of 3D orthogonal braided
composite. The comparison with experimental results indicates that the two-scale method is valid.
It should be noticed that the error of physics experiment for the mechanics parameters of
composite materials are unavoidable.
Table 9 Comparison of experimental and TSM results for elastic
coefficients of 3-D orthogonal braided composite
E11 （Gpa）
E22 （GPa）
 12
Experimental [20]
40.97
47.3
0.0346
TSM Results
43.406
50.570
0.0390
5.95
6.91
12.7
Error (%)
3. The computation of strength parameters. The data in Table 3 show the comparison of the
tensile strength parameters predicted by TSM, experimental results [20] and the data of theoretical
model [15]. It is noted that there is a noticeable difference between the predicted by TSM and
experimental limit stresses. The reasons are following: Firstly, the geometrical simplification of
experimental model may cause the deviation of the results. Secondly, the ignoring waviness of
fibers on the surface of the structure influences the critical stresses in theory. Otherwise, the limit
stresses predicted by TSM are close to the maximum of the experimental results. It seems that the
two-scale method gives reasonable upper bound since the experimental results are decentralized.
Table 3 Comparison of predicted by TSM, experimental and theoretical model in [15] tensile
strength parameters for 3D orthogonal braided composite
Exp.[20]
Failure strength (MPa)
TSM
Theoretical
Model in [15]
avg.
min.
max
 11t
582.84
538.12
483.72
466.68
515.67
11t
0.013427
/
0.012281
0.011173
0.013009
 22t
659.19
711.04
486.24
445.07
509.17
 22t
0.013035
/
0.010249
0.007158
0.012175
The predicted bending and twisting strength of the 3D orthogonal braided composite are listed
in Table 4.  11t ,  22t and  33t are the maximum normal stresses under critical bending
moment around x1, x2 and x3 axes, respectively.  12t ,  23t and  31t are the maximum shear
stresses under limit torque around x2, x3 and x1 axes, respectively. There is no experimental result,
which can be compared with.
Table 4 Prediction for the bending and twisting
strength of 3D orthogonal braided composite
Bending
Load Type
Predicted strength
(MPa)
6.
Twisting
 11t
 22t
 33t
 12t
 23t
 31t
583.22
659.37
179.69
129.88
129.61
69.77
Conclusions
In order to develop new composite materials the mechanics parameter prediction for
composite materials is necessary. In this paper, a two-scale computational method is proposed
based on two-scale asymptotic expression for the displacement solution of the structure made
from composite materials, and the related formulas are given. It can be used to predict the stiffness
parameters and strength parameters of composite materials with small periodic configuration. The
predicting results for the mechanics parameters of unidirectional and 3D orthogonal braided
composites are given. The following conclusions are worked out:
1)
The strength parameters of composite materials depend on the microscopic
configuration of composite materials, macro geometry of the structure (test component) and the
state of loadings.
2)
The two-scale method proposed in this paper is valid by comparing TSM predicting
results with the experimental and some theoretical model. There is a reasonable agreement in the
stiffness and strength parameters of unidirectional composite between TSM and experimental
results. However, in the prediction for the mechanics parameters of 3D orthogonal braided
composite there are some difference from the experimental results, but it still can provide a guide
for design of composites since TSM is a general and perfect method in theory.
References
[1]
T. Iwakuma, S. Koyama, “An estimate of average elastic moduli of composites and
polycrystals”. Mechanics of Materials, 37(2005), 459-472.
[2]
Huiyu Sun, Shenglin Di, Nong Zhang, Ning Pan, Changchun Wu, “Micromechanics of
braided composites via multivariable FEM”. Computers and Structures, 81(2003) 2021-2027.
[3]
Z. J. Wu, D. Brown and J. M. Davies, “An analytical modeling technique for predicting the
stiffness of 3-D orthotropic laminated fabric composites”, Composite Structures, Volume 56, Issue
4, June 2002, Pages 407-412.
[4]
Ju. J.W., Chen. T. M., “Effective elastic moduli of two-phase composites containing
randomly dispersed spherical inhomogeneities”, Acta Mech., 103, 1994, 123-144.
[5]
Pastore, C. M. and Gowayed, Y. A., “A Self-Consistent Fabric Geometry Model:
Modification and Application of a Fabric Geometry Model to Predict the Elastic Properties of
Textile composites”, Journal of Composites Technology & Research, JCTRER, Vol. 16, January
1994, pp. 32-26.
[6]
Chang-Long
Ma,
Jenn-Ming
Yang
and
Tsu-Wei
Chou,
“Elastic
Stiffness
of
Three-Dimensional Braided Textile Structural Composites”, Composite Materials, Testing and
Design (Seventh Conference), ASTM STP 893, J.M. Whiney, Ed., American Society for Testing
and Materials, Philadelphia, 1986, pp. 404-421.
[7]
V. V. Jikov, S. M. Kozlov, O. A. Oleinik, “Homogenization of Differential Operators and
Integral Functionals”, Springer-Verlag, Berlin, 1994.
[8]
J. Z. Cui, Li-qun Cao, “The two-scale asymptotic analysis methods for a class of elliptic
boundary value problems with small periodic coefficients”, Mathematic Numerical Sinica, Vol.
21(1999), No. 1, 29-28.
[9]
J.Z.Cui, “Finite Element Algorithms Based on Two-Scale Analyses Method”, Comp. Mech.
in Struc. Engg., Recent Developments, Edited by Franklin Y. Cheng, Elsevier, 1999, Amsterdam .
New York, Singapore, Tokyo, 31-41.
[10] J. Z. Cui, T. M. Shin and Y. L. Wang, “Two-scale analysis method for bodies with small
period configuration”, Invited Paper in CASCM-97, Feb. 11-14, 1997, Structural Engineering and
Mechanics, Vol.7, No.6, 1999, 601-614.
[11] Y.Y. Li and J.Z. Cui, “Two-scale analysis method for predicting heat transfer performance
of composite materials with random grain distribution”, Science in China Ser. A Mathematics,
2004, Vol.47 Supp. 101-110.
[12] Y.Y. Li and J.Z. Cui, “The multi-scale computational method for mechanics parameters of
composite materials with random grain distribution”, to have been accepted by Journal of
Composites Science & Technology.
[13] Daniel IM, Ishai O., “Engineering mechanics of composite materials”, Oxford: Oxford
University Press, 1994.
[14] Wen-Shyong Kuo, Tse-Hao Ko, “Compressive damage in 3-axis orthogonal fabric
composites. Composites”, Part A: applied science and manufacturing, 31(2000), 1091-1105.
[15] Ping Tan, Liyong Tong, G. P. Steven, “Behavior of 3D orthogonal woven CFRP composites”,
Part II. FEA and analytical modeling approaches. Composites Part A: applied science and
manufacturing. 31(2000), 273-281.
[16] Zh. M. Wang, “Mechanics of composites and structural mechanics of composites”,
Mechanical Industry Press, 1991, p67~100, (in Chinese).
[17] B. Q. Wang. Y. P. Yang, “Experimental study of elastic constants of unidirectional fiber
reinforced composite materials”, Acta Materiae Compositae Sinica. Vol. 3, No. 2, 1986.6. (in
Chinese).
[18] Surya R. Kalidindi and Abdel Abusafieh, “Longitudinal and transverse moduli and strengths
of low angle 3-D braided composites”, J. Composite Mater. 1996. 30(8):885-905
[19] Hirokawa T, Yasuda J., Wasaki Y., “The Characteristics of 3-D Orthogonal Woven Fabric
Reinforced Composites”, In: Stinson J ed. 36th International SAMPE symposium, California, April
15-18, 1991, California, Society for the Advancement of Material and Process Engineering, 1991,
151-159.
[20] Ping Tan, Liyong Tong, G. P. Steven, Takashi Ishikawa, “Behavior of 3D orthogonal woven
CFRP composites”, Part I, Experimental investigation, Composites Part A, applied science and
manufacturing, 31(2000), 259-271.