σ 11 ¼ μ 0 λ
2
À
1
λ
ð67Þ
In the same manner by means of Eq. (56) the unidimensional stress-strain
relation can be obtained for Mooney-Rivlin and Yeoh model. These results are
shown in Table 4.
Compressible Materials
In the compressible case the relation (61) is not anymore valid, thus the orthogonal
stretch λ 2 must be derived from the implicit relation:
σ 22 λ 1 ; λ 2
ð
Þ¼0
ð68Þ
or equivalently σ 33 (λ 1 , λ 2 ) ¼ 0, which follows from the condition that the components of the stress σ 22 and σ 33 vanish. Eq. (68) has to be solved numerically for
each λ 1 .
3.6.2 Simple Shear
Another example of a homogeneous deformation state is a simple shear defined by
x 1 ¼ X 1 þ γX 2 , x 2 ¼ X 2 , x 3 ¼ X 3
ð69Þ
where γ is the amount of shear strain. Thus,
F ¼
1 γ 0
0 1 0
0 0 1
2
4
3
5 , B ¼
1 þ γ
2
γ 0
γ
1 0
0
0 1
2
4
3
5 , C ¼
1
γ
0
γ 1 þ γ
2 0
0
0
1
2
4
3
5
ð70Þ
And
I 1 C
ð Þ ¼ 3 þ γ
2 , I 2 C
ð Þ ¼ 3 þ γ
2 , I 3 C
ð Þ ¼ 1
ð71Þ
that is the strain invariants are even function of the shear strain.
Table 4 Simple shear test
results for some of the most
used material models
Model
σ 12
Neo-Hooke
σ 12 ¼ 2c 10 γ
Mooney-Rivlin
σ 12 ¼ 2(c 10 À c 01 )γ
Yeoh (c 30 ¼ 0)
σ 12 ¼ 2[c 10 + 2c 20 (I 1 À 3)]γ
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
237
2
À
1
λ
ð67Þ
In the same manner by means of Eq. (56) the unidimensional stress-strain
relation can be obtained for Mooney-Rivlin and Yeoh model. These results are
shown in Table 4.
Compressible Materials
In the compressible case the relation (61) is not anymore valid, thus the orthogonal
stretch λ 2 must be derived from the implicit relation:
σ 22 λ 1 ; λ 2
ð
Þ¼0
ð68Þ
or equivalently σ 33 (λ 1 , λ 2 ) ¼ 0, which follows from the condition that the components of the stress σ 22 and σ 33 vanish. Eq. (68) has to be solved numerically for
each λ 1 .
3.6.2 Simple Shear
Another example of a homogeneous deformation state is a simple shear defined by
x 1 ¼ X 1 þ γX 2 , x 2 ¼ X 2 , x 3 ¼ X 3
ð69Þ
where γ is the amount of shear strain. Thus,
F ¼
1 γ 0
0 1 0
0 0 1
2
4
3
5 , B ¼
1 þ γ
2
γ 0
γ
1 0
0
0 1
2
4
3
5 , C ¼
1
γ
0
γ 1 þ γ
2 0
0
0
1
2
4
3
5
ð70Þ
And
I 1 C
ð Þ ¼ 3 þ γ
2 , I 2 C
ð Þ ¼ 3 þ γ
2 , I 3 C
ð Þ ¼ 1
ð71Þ
that is the strain invariants are even function of the shear strain.
Table 4 Simple shear test
results for some of the most
used material models
Model
σ 12
Neo-Hooke
σ 12 ¼ 2c 10 γ
Mooney-Rivlin
σ 12 ¼ 2(c 10 À c 01 )γ
Yeoh (c 30 ¼ 0)
σ 12 ¼ 2[c 10 + 2c 20 (I 1 À 3)]γ
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
237