A Two-order and Two-scale Computation Model for
the Damage of Composite Materials
Fang Su ， J.Z.Cui
(Academy of Mathematics and System Science, Chinese
Academy of Sciences, Beijing, 100080 China)
Abstract: In this paper, we present a newly two-order and two-scale method to account for damage
effects in heterogeneous media. In our damage model, we employ incremental theory avoiding nonlinear
character of the original problem. The damage parameter is regarded as a nonlocal function, and related to
the strain energy storage of last increment step. The solution using this method is equivalent to the true
solution of original problem, not only in the homogenization sense, but also in the nearly pointwise sense
with

−approximation. We also obtained the convergence of this method, and the mathematical
convergence proofs are given at last. Several numerical results are found to be in good agreement with our
conclusions.
Keywords: Homogenization; Quasi-periodic elastic structure; Composite material
1
Introduction
The damage of composite materials occurs through different mechanisms usually
involve the interaction between micro-constituents. During the past two decades, a number
of models have been developed to simulate the damage and failure process of composite
materials, among which the damage mechanics approach is particularly attractive in the
sense, which provides a feasible framework for the description of distributed damage
including material stiffness degradation, initiation, growth and coalescence of micro-cracks
and voids. The damage effects in heterogeneous media has been studied by
homogenization method or one-order multiscale approximation, but the works are few.
Homogenization method for the periodic structure of composite materials has been
discussed in [1,2,3,4]. In damage mechanics of composite materials elasticity is no longer
whole-periodic, but local-periodic. In some sence, It is quasi-periodic
structures.Homogenization theory for the quasi-periodic structures of composite materials
has been presented in [3]. Cao has given the first-order Approximation and several basic
stimations of this problem in [5]. But it suggested that the original problem and the
obtained solution were equivalent only in integral sence.
______________________
Email: [email protected]
1
In this paper, we present a newly two-order and two-scale method to account for
damage effects in heterogeneous media. In our damage model, we employ incremental
theory avoiding nonlinear character of the original problem. The damage parameter is
regarded as a nonlocal function, and related to the strain energy storage of last increment
step. The solution using this method is equivalent to the true solution of original problem,
not only in the homogenization sense, but also in the nearly pointwise sense with
 −approximation. We also obtained the convergence of this method, and the mathematical
convergence proofs are given at last. This method is suitable for quasi-periodic microstructures, such as accumulation of damage for multi-phase materials.
2
Bsaic Idea of Correct Two-order and Two-scale Computation
Method
Consider the boundary value problem of second-order elliptic equation with quasi-
periodic coefficients as follows:
x
where  is a bounded domain with Lipschitz continuous boundary. aij ( x, ) =

aij ( x,  ) ,    1 x (i,j =1,…,n) are the n  n elements of matrix A and satisfy the
following conditions:
(A1) aij ( x,  ) be 1-periodic functions in 
(A2) aij ( x,  ) = a ji ( x,  )
(A3) 1   aij ( x,  )i j  2 
2
2
1 , 2  0
(1 ,
,n )  R n
To this problem, we will seek an approximation solution of of the form:
Inserting the approximation solution (2.2) into equilibrium equation (2.1), and taking into
account




1 
xi xi  i
. Equating the coefficients of the same powers  , we have
2
where
and
The homogenized equations associated with the Eqs. (2.1) can be represented as follows:
To obtained, we require correct term auxiliary function must satisfy
equation as follows:
In this situation, we will obtained from the following equations:
To demonstrate our approximation solution approaching effectively to the solution of
(2.1),we must have convergence results.
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Error Estimation
We continue to use the sign in Section 1
Lemma 3.1 Let  be a bounded domain with a smooth boundary and B ={x 
3
 ,  ( x, )   },   0 . Then there exists  0 >0 such that for every   (0,  0 ) and every
v  H 1 (  ) we have
where C is a constant independent of  and v.
Theorem 3.1 Let u  (x) is the weak solution of the (2.1), u 0 (x)  H 4 (  ), then following
estimation holds:
where C > 0 is nothing with  , u  , u 0 .
4 Numerical results
Consider the Dirichlet boundary value problem of second order elliptic type equation
with highly oscillatory coefficients as follows:
where  as shown in Fig.1a. The periodicity cell Q as shown in Fig.1b,  =1/8.
a
b
Fig.3.(a) Domain  and (b) unit cell Q = [0; 1]2.
Since it is difficult to find the analytic solution of (4.1), we have to replace u  (x) with
its FE solution in a very refined mesh. Now we implement the quadrangle partition for  ,
which is such that the discontinuities of the coefficients aij coincide with sides of the
quadrangle. The number of quadrangle is 2304.
where  ij =1, if i=j;  ij =0, if i  j. Where e0 = u  - u 0 , e1 = u  - U 1 , e2 = u  - U 2 , u 0 (x) is the
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FE solution of the homogenized equation, and U 1 , U 2 are the first-order and the secondorder multiscale FE solutions respectively. new U 2 is the correct second-order multiscale
FE solution.
a
b
c
d
Fig.2 Two order error
e2 = u  -new U 2 (a) Case1
(b) Case2
(c)Case3
(d)Case4
Appendix: Computation Method for damaged composites
Using this method, we consider the corresponding boundary value problem governed
by the following set of equations:
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where   is a scalar damage parameter;  ij and  ij are components of stress and strain
tensors; Lijkl represents components of elastic stiffness; bi is a body force; ui denotes the
components of the displacement vector.
In this section, we describe computational aspects of the nonlocal piecewise constant
damage model for two-phase materials. The stress update (integration) problem can be
stated as follows:
Given: displacement vector t um ; overall strain t  mn ; strain history parameter t  ( m ) ;
damage parameter t  ( m ) ; and displacement increment um calculated from the finite
element of the macro problem.
(m)
Find: displacement vector um = t t um ; overall strain  mn ; nonlocal phase strains  mn
and
(f)
 mn
; nonlocal strain history parameter  ( m) ; nonlocal phase damage parameter  ( m ) ;
(m)
(f)
overall stress  mn and nonlocal phase stresses  mn
and  mn
.
The stress update procedure consists of the following steps:
i.) Calculate macroscopic strain increment,  mn = u( m, xn ) , and then update macroscopic
strains through  mn = t  mn +  mn .
ii.) Compute the damage equivalent strain 
(m)
in terms of t  ( m ) and  mn .
iii.) Check the damage evolution conditions. Note that  ( m) =  ( m) - t  ( m )
If damage process, i.e. 
(m)
> t  ( m ) , then  ( m) = 
(m)
and update for  ( m ) .
Otherwise for elastic process:  ( m ) = t  ( m ) .
(m)
(f)
vi.) Update the nonlocal strains  mn
,  mn
and update the nonlocal strain history
parameter  ( m) .
v.) Update macroscopic stresses  ij and calculate nonlocal phase stresses  ij( m ) and  ij( f ) .
References
[1] J.Z.Cui and Li-qun Cao, Two-scale asymptotic analysis methods for a class of elliptic boundary value
problems with small periodic coefficients, Math. Numer. Sinica, Vol.21:1 (1999), 19-28.
[2] Doma Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford, 1999.
[3] Bensoussan,A., Lions,J.L., and Papanicolaou,G., Asymptotic Analysis for Periodic Structures, NortthHolland Amsterdam, 1978.
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[4] O.A.Oleinik, A.S.Shamaev, Mathematical problems in elasticity and homogenization, 1992.
[6] Liqun Cao and Junzhi Cui Homogenization method for the quasi-periodic structures of composite
materials, Math.Num.Sin, Vol.21 (1999), No.3, 331-344.
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