Page 179 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 171

When n = 1 the Yeoh model reduces to the neo-Hookean model for incompressible
materials.
The Cauchy stress for the incompressible Yeoh model is given by:
n
+
σ =−p1 2 ∂W B; ∂W = ∑ iC I ( − 3) − i 1 (6-29)
∂I ∂I i 1
1 1 = i 1
Uniaxial Extension
For uniaxial extension in the n 1 direction, the principal stretches are l 1 = l,l 2 = l 3 . From
incompressibility l 1 = l 2 = l 3 = 1. Hence, l 2 = l 3 = 1/l. Therefore,
I = λ 2 + λ 2 + λ 2 = λ 2 + 2 (6-30)
1 1 2 3 λ
The left Cauchy-Green deformation tensor can then be expressed as:
B = λ 2 n n + 1 n ( n + n n ) (6-31)
1 1 λ 2 2 3 3
If the directions of the principal stretches are oriented with the coordinate basis vec-
tors, it follows:
σ =− + 2 λ 2 ∂W ; σ =− + 2 ∂W = σ (6-32)
p
p
11 ∂I 22 λ ∂I 33
1 1
Since s 22 = s 33 = 0, the resulting situation is:
p = 2 ∂ W (6-33)
λ I ∂
1
Therefore,
1 ∂W
σ = 2( λ − ) (6-34)
2
11 λ ∂I
1
The engineering strain is l−1. The engineering stress is:
1 ∂ W
T = σ / λ = 2 λ − ) (6-35)
(
11 11 λ 2 I ∂
1
Equibiaxial Extension
For equibiaxial extension in the n 1 and n 2 directions, the principal stretches are l 1 = l 2 = l.
2
From incompressibility l 1 = l 2 = l 3 = 1. Hence, l 3 = 1l . Therefore,
2
I = λ 2 + λ 2 + λ 2 = 2λ 2 + (6-36)
1 1 2 3 λ 4
The left Cauchy-Green deformation tensor can then be expressed as:
1
B = λ 2 n n + λ 2 n n + n n (6-37)
1 1 2 2 λ 4 3 3