β 1 ¼ 2 5v m À 1
ð
Þþ10 1 À v m
ð
Þ
k m
k
Ã
À k m
À
μ m
μ Ã À μ m
ð6:262Þ
β 2 ¼ 2 4 À 5v m
ð
Þþ15 1 À v m
ð
Þ
μ m
μ Ã À μ m
ð6:263Þ
and
γ 1 ¼
5φ f
8β 2
2
13 À 14v m
ð
Þ v m À
8β 1
3β 1 þ 2β 2
1 À 2v m
ð
Þ1 þ v m
ð
Þ
&
'
ð6:264Þ
γ 2 ¼
1
2
þ
5φ f
16β 2
2
 25 À 34v m þ 22v
2
m
À
Á À
6β 1
3β 1 þ 2β 2
1 À 2v m
ð
Þ1 þ v m
ð
Þ
&
'
ð6:265Þ
where k
à and μ
à are the effective bulk modulus and shear modulus of the composite
spherical filler assemblage, respectively, and φ f is the particle volume fraction.
Example
Nie (2005) studied lightly cross-linked poly-methyl methacrylate (PMMA) filled
with alumina trihydrate (ATH). The following phase properties were used for the
numerical analysis:
Equivalent
Continuum
Medium
Fig. 6.12 Modeling procedures—step 4: micromechanics (Nie 2005)
6.7 Micromechanical Constitutive Model of the Particulate Composite
319
ð
Þþ10 1 À v m
ð
Þ
k m
k
Ã
À k m
À
μ m
μ Ã À μ m
ð6:262Þ
β 2 ¼ 2 4 À 5v m
ð
Þþ15 1 À v m
ð
Þ
μ m
μ Ã À μ m
ð6:263Þ
and
γ 1 ¼
5φ f
8β 2
2
13 À 14v m
ð
Þ v m À
8β 1
3β 1 þ 2β 2
1 À 2v m
ð
Þ1 þ v m
ð
Þ
&
'
ð6:264Þ
γ 2 ¼
1
2
þ
5φ f
16β 2
2
 25 À 34v m þ 22v
2
m
À
Á À
6β 1
3β 1 þ 2β 2
1 À 2v m
ð
Þ1 þ v m
ð
Þ
&
'
ð6:265Þ
where k
à and μ
à are the effective bulk modulus and shear modulus of the composite
spherical filler assemblage, respectively, and φ f is the particle volume fraction.
Example
Nie (2005) studied lightly cross-linked poly-methyl methacrylate (PMMA) filled
with alumina trihydrate (ATH). The following phase properties were used for the
numerical analysis:
Equivalent
Continuum
Medium
Fig. 6.12 Modeling procedures—step 4: micromechanics (Nie 2005)
6.7 Micromechanical Constitutive Model of the Particulate Composite
319