6.5.2 Ensemble-Average Stress Norm in the Matrix
According to the Eshelby theory, the perturbed stress σ
0 at any point in the matrix
due to the presence of the particles and voids can be written as
σ
0 x
ð Þ ¼ C 0 :
Z
V
G x 2 x
0
ð
Þ: ε
Ã
1 þ ε
T
1
À
Á
dx
0
þ C 0 :
Z
V
G x 2 x
0
ð
Þ
: ε
Ã
2 þ ε
T
2
À
Á
dx
0
ð6:183Þ
The eigenstrain in a single ellipsoidal inclusion is uniform for the interior points
of an isolated (no interacting) inclusion. Therefore, the perturbed stress for any
matrix point x due to a typical isolated q-phase inhomogeneity centered at x q takes
the form
σ
0 xjx q
À
Á ¼ C 0 Á G x 2 x q
À
Á
Â
à : ε
Ã0
q þ ε
T
q
ð6:184Þ
where G x À x q
À
Á
is given by Eq. (6.39), and
ε
Ã0
q ¼ À A q þ S
À
Á À1 : ε
0
þ A q þ S
À
Á À1 Á I 2 S
ð
Þ: ε
T
q
ð6:185Þ
We denote by H(x| ℘) the square of the current stress norm at the local point x,
which determines the plastic strain for a given phase configuration ℘. Since there is
no plastic strain in the elastic particles or voids, H(x| ℘) can be written as
H xj℘
ð Þ ¼
σ xj℘
ð Þ : I d : σ xj℘
ð Þ x in the matrix
0
otherwise
(
ð6:186Þ
In addition, hHi m (x) is defined as the ensemble average of H(x| ℘) over all
possible points where x is in the matrix phase. Matrix point receives perturbations
from particles and voids. Therefore, the ensemble-average stress norm for any matrix
point x can be evaluated by collecting and summing up all the current stress norm
perturbations produced by any typical particle centered at x 1 in the particle domain
and any typical void centered at x 2 in the void domain, and averaging over all
possible locations of x 1 and x 2
302
6 Unified Micromechanics of Particulate Composites