Page 181 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 173

The engineering stress is:
1 ∂ W
(
T = σ / λ = 2 λ − )
11 11 λ 3 I ∂
1
Neo-Hookean Solid Model
The model of neo-Hookean solid assumes that the extra stresses due to deformation are
proportional to the Left Cauchy-Green deformation tensor:
T =− p + GB (6-45)
I

Where T = stress tensor, p = pressure, I = the unity tensor, G = a constant equal to
shear modulus, B = the Left Cauchy-Green deformation tensor (sometimes referred as
Finger tensor).
The strain energy for this model is:
1
W = GI (6-46)
2 B
Where W is potential energy and I B = tr(B) is the trace (or first invariant) of the Left
Cauchy-Green deformation tensor B.
Usually the model is used for incompressible media.

Uniaxial Tension
Under uniaxial extension from the definition of the Left Cauchy-Green deformation
tensor:
+
T =− p Gα , T = T =− + G (6-47)
2
p
11 1 22 33 α
1
Where a 1 is the elongation in the stretch ratio in the 1 direction.
Assuming no traction on the sides, T22 = T33 = 0, so:
ε
3 + 3ε 2 + ε 3
T = G(α 2 −α 1 − ) = G
11 1 1 1+ ε
Where e = a 1 is the strain.
The equation above is for the true stress (ratio of the elongation force to deformed
cross-section), for engineering stress the equation is:

T = G(α −α − 2 )
11 eng 1 1
For small deformations e < < 1 we will have:
T = 3 Gε
11
Thus, the equivalent Young’s modulus of a neo-Hookean solid in uniaxial extension
is 3G (shear modulus). For simple shear:
T = Gγ
11
T − T = Gγ 2 (6-48)
11 22
T − T = 0
22 33