C q : ε
0
þ ε
0 x
ð Þ À ε
T
q x
ð Þ
h
i
¼ C 0 : ε
0
þ ε
0 x
ð Þ À ε
T
q x
ð Þ À ε
Ã
q x
ð Þ
h
i
ð6:13Þ
where C 0 is the stiffness tensor of the matrix, ε
0 is the uniform elastic strain field
induced by far-field loads for a homogeneous matrix material only, ε
T
q is its own
eigenstrain associated with the qth particle, ε
Ã
q is the fictitious equivalent eigenstrain
by replacing the qth particles with the matrix material, ε
0
(x) is the perturbed strain
due to distributed eigenstrain ε
T , and ε
à is associated with all the particles in the
RVE. The stress-strain relation is given in Newtonian mechanics formulation by
σ
0
¼ C 0 : ε
0
ð6:14Þ
The strain at any point within an RVE is decomposed into two parts, the uniform
strain and the perturbed strain, due to the distributed eigenstrain. It is emphasized
that the eigenstrain ε
à and ε
T are nonzero in the particle domain and zero in the
matrix domain, respectively. The perturbed strain field induced by all the distributed
eigenstrains ε
à and ε
T can be expressed by (Mura 1987)
ε
0 x
ð Þ ¼
Z
V
G x À x
0
ð
Þ: ε
T x
0
ð Þ þ ε
à x
0
ð Þ
Â
Ã
dx
0
ð6:15Þ
where x, x
0
2 V, and G are Green’s function in a linear elastic homogeneous matrix.
For a linear elastic isotropic matrix, the fourth-rank tensor Green’s function is
(Ju and Chen 1994a)
G ijkl x 2 x
0
ð
Þ¼
1
8π 1 À v 0
ð
Þr 3 F ijkl À15, 3v 0 , 3, 3 À 6v 0 , À1 þ 2v 0 , 1 À 2v 0
ð
Þ ð 6:16Þ
where r x À x
0
, r j rj and v 0 is Poisson’s ratio of the matrix. The components of
the fourth-rank tensor F are defined by
F ijkl B m
ð Þ ¼B 1 n i n j n k n l þ B 2 δ ik n j n l þ δ il n j n k þ δ jk n i n l þ δ jl n i n k
À
Á þ B 3 δ ij n k n l þ B 4 δ kl n i n j
þ B 5 δ ij δ kl þ B 6 δ ik δ jl þ δ il δ jk
À
Á
ð6:17Þ
with the unit normal vector n r/r and index m ¼ 1 to 6.
From Eqs. (6.13) and (6.15), we arrive at
ÀA q : ε
Ã
q x
ð Þ ¼ ε
0
À ε
T
q x
ð Þ þ
Z
V
G x 2 x
0
ð
Þ: ε
T x
0
ð Þ þ ε
à x
0
ð Þ
Â
Ã
dx
0
x
0
2 V ð6:18Þ
where
6.2 Ensemble-Volume Averaged Micromechanical Field Equations
281
0
þ ε
0 x
ð Þ À ε
T
q x
ð Þ
h
i
¼ C 0 : ε
0
þ ε
0 x
ð Þ À ε
T
q x
ð Þ À ε
Ã
q x
ð Þ
h
i
ð6:13Þ
where C 0 is the stiffness tensor of the matrix, ε
0 is the uniform elastic strain field
induced by far-field loads for a homogeneous matrix material only, ε
T
q is its own
eigenstrain associated with the qth particle, ε
Ã
q is the fictitious equivalent eigenstrain
by replacing the qth particles with the matrix material, ε
0
(x) is the perturbed strain
due to distributed eigenstrain ε
T , and ε
à is associated with all the particles in the
RVE. The stress-strain relation is given in Newtonian mechanics formulation by
σ
0
¼ C 0 : ε
0
ð6:14Þ
The strain at any point within an RVE is decomposed into two parts, the uniform
strain and the perturbed strain, due to the distributed eigenstrain. It is emphasized
that the eigenstrain ε
à and ε
T are nonzero in the particle domain and zero in the
matrix domain, respectively. The perturbed strain field induced by all the distributed
eigenstrains ε
à and ε
T can be expressed by (Mura 1987)
ε
0 x
ð Þ ¼
Z
V
G x À x
0
ð
Þ: ε
T x
0
ð Þ þ ε
à x
0
ð Þ
Â
Ã
dx
0
ð6:15Þ
where x, x
0
2 V, and G are Green’s function in a linear elastic homogeneous matrix.
For a linear elastic isotropic matrix, the fourth-rank tensor Green’s function is
(Ju and Chen 1994a)
G ijkl x 2 x
0
ð
Þ¼
1
8π 1 À v 0
ð
Þr 3 F ijkl À15, 3v 0 , 3, 3 À 6v 0 , À1 þ 2v 0 , 1 À 2v 0
ð
Þ ð 6:16Þ
where r x À x
0
, r j rj and v 0 is Poisson’s ratio of the matrix. The components of
the fourth-rank tensor F are defined by
F ijkl B m
ð Þ ¼B 1 n i n j n k n l þ B 2 δ ik n j n l þ δ il n j n k þ δ jk n i n l þ δ jl n i n k
À
Á þ B 3 δ ij n k n l þ B 4 δ kl n i n j
þ B 5 δ ij δ kl þ B 6 δ ik δ jl þ δ il δ jk
À
Á
ð6:17Þ
with the unit normal vector n r/r and index m ¼ 1 to 6.
From Eqs. (6.13) and (6.15), we arrive at
ÀA q : ε
Ã
q x
ð Þ ¼ ε
0
À ε
T
q x
ð Þ þ
Z
V
G x 2 x
0
ð
Þ: ε
T x
0
ð Þ þ ε
à x
0
ð Þ
Â
Ã
dx
0
x
0
2 V ð6:18Þ
where
6.2 Ensemble-Volume Averaged Micromechanical Field Equations
281