Page 176 - Mechanics of Asphalt Microstructure and Micromechanics

168 Ch a p t e r S i x

σ = λε δ + 2 με (6-4)
ij kk ij ij
ε = 1 [( 1 + ν σ −) νσ δ ] (6-5)
ij E ij kk ij

σ = 3 K ε , s = 2 μe (6-6)
kk kk ij ij
6.1.2 Hyperelasticity (Bower, 2010)
Hyperelasticity is used to describe the constitutive relationship of elastic materials at
large strains. It is often non-linear. The non-linearity may result in directly the large-
strain definition and strain energy dependency on strains of higher order terms.
If W(F) is the specific strain-energy, the first Piola-Kirchoff (PK) stress tensor is con-
jugated to the deformation gradient F and therefore:
∂ W ∂ W
P = or P = (6-7)
F ∂ iK F ∂
iK
If the strain energy is represented as a function of the Lagrangian Green strain E:
∂ W ∂ W
P = F ⋅ or P = F (6-8)
∂ E iK iL ∂ E
LK
In the case that the strain energy is a function of the right Cauchy-Green deforma-
tion tensor (C), then:
∂ W ∂ W
P = 2 F ⋅ ∂ or P = 2 F iL ∂ (6-9)
iK
C C LK
Considering the definition of the second PK tensor S and its relationship with the
first PK tensor, the following relationships will be valid:
∂ W ∂ W
S = F ⋅ or S = F −1 (6-10)
−1
F ∂ IJ IK F ∂
kJ
∂ W ∂ W
S = or S = (6-11)
∂ E IJ ∂ E
IJ
∂ W ∂ W
S = 2 or S = 2 (6-12)
∂ C IJ ∂ C
IJ
It can be proved (see also Chapter 1), for small strains, one can use Cauchy stress.

6.1.2.1 Hyperelastic Models
Saint Venant-Kirchhoff Model
The Saint Venant-Kirchhoff model is the simplest hyperelastic material model. It is an
extension of the linear elastic material model to the nonlinear regime. This model has
the form:
E + μI 2
S = λtr() E (6-13)