to the glass transition temperature T g of the PMMA. These stresses can be quantified
by calculation or by direct measurement and are of the order between 15% and 75%
of the tensile strength of the composite. Therefore, the thermal stress associated with
the CTE mismatch between the matrix and particles is an important factor for the
failure of particulate composites subjected to thermo-mechanical loads.
6.2 Ensemble-Volume Averaged Micromechanical Field
Equations
In order to obtain effective constitutive equations and properties of random heterogeneous composites, one typically performs the ensemble-volume averaging process
within a mesoscopic representative volume element (RVE). The volume-averaged
stress tensor is defined as
σ ¼
1
V
Z
V
σ x
ð Þdx ¼
1
V
Z
V 0
σ x
ð Þdx þ
X n
q¼1
Z
V q
σ x
ð Þdx
2
6
4
3
7
5
ð6:11Þ
where V is the volume of a RVE, V 0 is the volume of the matrix, V q is the volume of
the qth phase particles, and n denotes the number of particulate phases of different
material properties (excluding the matrix).
Similarly, the volume-averaged strain tensor is defined as
ε ¼
1
V
Z
V
ε x
ð Þdx ¼
1
V
Z
V 0
ε x
ð Þdx þ
X n
q¼1
Z
V q
ε x
ð Þdx
2
6
4
3
7
5
1
V
V 0 ε 0 þ
X n
q¼1
V q ε q
"
#
ð6:12Þ
According to Eshelby’s equivalence principle, the perturbed strain field ε
0 (x)
induced by inhomogeneity can be related to specified eigenstrain ε
à (x) by replacing
the inhomogeneities with the matrix material. Sometimes the inhomogeneities may
also involve its own eigenstrain caused by, for example, phase transformation,
precipitation, plastic deformation, or CTE mismatch between different constituents
of the composites. However, it is not necessary to attribute the eigenstrain to any
specific source. Then the total perturbed stress is the sum of the two parts, one caused
by the inhomogeneity and the other one is the eigenstress caused by eigenstrain. For
the domain of the qth-phase particles with elastic stiffness tensor C q , Mura (1987)
provided the following relation:
280
6 Unified Micromechanics of Particulate Composites
by calculation or by direct measurement and are of the order between 15% and 75%
of the tensile strength of the composite. Therefore, the thermal stress associated with
the CTE mismatch between the matrix and particles is an important factor for the
failure of particulate composites subjected to thermo-mechanical loads.
6.2 Ensemble-Volume Averaged Micromechanical Field
Equations
In order to obtain effective constitutive equations and properties of random heterogeneous composites, one typically performs the ensemble-volume averaging process
within a mesoscopic representative volume element (RVE). The volume-averaged
stress tensor is defined as
σ ¼
1
V
Z
V
σ x
ð Þdx ¼
1
V
Z
V 0
σ x
ð Þdx þ
X n
q¼1
Z
V q
σ x
ð Þdx
2
6
4
3
7
5
ð6:11Þ
where V is the volume of a RVE, V 0 is the volume of the matrix, V q is the volume of
the qth phase particles, and n denotes the number of particulate phases of different
material properties (excluding the matrix).
Similarly, the volume-averaged strain tensor is defined as
ε ¼
1
V
Z
V
ε x
ð Þdx ¼
1
V
Z
V 0
ε x
ð Þdx þ
X n
q¼1
Z
V q
ε x
ð Þdx
2
6
4
3
7
5
1
V
V 0 ε 0 þ
X n
q¼1
V q ε q
"
#
ð6:12Þ
According to Eshelby’s equivalence principle, the perturbed strain field ε
0 (x)
induced by inhomogeneity can be related to specified eigenstrain ε
à (x) by replacing
the inhomogeneities with the matrix material. Sometimes the inhomogeneities may
also involve its own eigenstrain caused by, for example, phase transformation,
precipitation, plastic deformation, or CTE mismatch between different constituents
of the composites. However, it is not necessary to attribute the eigenstrain to any
specific source. Then the total perturbed stress is the sum of the two parts, one caused
by the inhomogeneity and the other one is the eigenstress caused by eigenstrain. For
the domain of the qth-phase particles with elastic stiffness tensor C q , Mura (1987)
provided the following relation:
280
6 Unified Micromechanics of Particulate Composites