Page 177 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 169

Where S is the second PK stress and E is the Lagrangian Green strain, and l and m
are the Lame constants.
The specific strain-energy for the St. Venant-Kirchhoff model is:
λ
WE() = [ tr E()] + μ tr E ) (6-14)
2
2
(
2
And the second PK stress can be derived from the relation:
∂ W
S = (6-15)
∂ E
Mooney-Rivlin Model
In continuum mechanics, a Mooney-Rivlin solid is a generalization of the Neo-Hookean
solid model, where the strain energy W is a linear combination of two invariants of
Finger tensor {B}:
W = C I − 3) + C I − 3) (6-16)
(
(
1 1 2 2
– –
Where C 1 and C 2 are constants; I 1 and I 2 and are the first and second invariant of the
deviatoric component of the Finger tensor:
I = λ 2 + λ 2 + λ 2
1 1 2 3
I = λλ 2 + λλ 2 + λλ 2 (6-17)
2
2
2
2 1 2 2 3 3 1
I = λλλλ 2
2
2
3 1 2 3
Where λ = ω , λ = ω , λ = ω , and w 1 , w 2 and w 3 and the eigenvalues of B.
3
3
1
2
1
2
If C = 1 G (where G is the shear modulus) and C 2 = 0, we obtain a Neo-Hookean
1
2
solid, a special case of a Mooney-Rivlin solid.
The stress tensor T depends upon the left Cauchy-Green deformation tensor and is
related to B by the following equation:
p +
+
T =− I 2 C B 2 C B −1 (6-18)
1 2
Uniaxial Tension
For the case of uniaxial elongation, true stress can be calculated as:
2 C 1 −
T = ( 2 C + 2 )(α 2 −α ) (6-19)
11 1 α 1 1
1
And the engineering stress can be calculated as:
2 C 2 −
T = 2 ( C + 2 )(α −α ) (6-20)
11 eng 1 α 1 1
1
The Mooney-Rivlin solid model usually fits experimental data better than the Neo-
Hookean solid model does, but requires an additional empirical constant.
Ogden Material Model
In the Ogden Material model, the strain energy density is expressed in terms of the
principal stretches l j , j = 1,2,3 as:
3 ∑
W(,λλ λ = N μ P (λ α P + λ α P + λ α P − ) 3 (6-21)
,
)
1 2 p α 1 2 3
= 1 P