Page 178 - Mechanics of Asphalt Microstructure and Micromechanics

170 Ch a p t e r S i x

Where N, m P , and a P are material constants. Under the assumption of incompress-
ibility one can rewrite as:
2 ∑
W(,λλ = N μ P (λ α P + λ α P + λ − α P λ − α P − ) 3 (6-22)
)
1 p α 1 2 1 2
=
1 P
In general, the shear modulus results from:
N
2μ = ∑ μ α p (6-23)
p
p = 1
With N = 3 and by fitting the material parameters, the material behavior of rubbers
can be described very accurately. For particular values of material constants, the Ogden
model will reduce to either the Neo-Hookean solid (N = 1, a = 2) or the Mooney-Rivlin
material (N = 2, a 1 = 2, a 2 = −2).
Using the Ogden material model, the three principal values of the Cauchy stresses
can now be computed as:
σ =+ λ α ∂W (6-24)
p
α
λ ∂
α
Where use is made of s a = l a P a .
Uniaxial Tension
An incompressible material under uniaxial tension is now considered, with the stretch
l
ratio given as λ = . The principal stresses are given by:
l
0 N
σ =+ ∑ μ λ α p (6-25)
p
α p p
p =1
The pressure p is determined from incompressibility and boundary condition
s 2 = s 3 = 0, yielding:
α ∑
σ = N ( μ λ α p − μ λ p − 2 1 α p ) (6-26)
p
p
p
p =1
Yeoh Model
The original model proposed by Yeoh had a cubic form with only I 1 dependence and is
applicable to purely incompressible materials. The strain energy density for this model
is written as:
3
W = ∑ C I −( 1 ) 3 i (6-27)
i
i=1
Where C i are material constants. The quantity 2C 1 can be interpreted as the initial
shear modulus.
Today, a slightly more generalized version of the Yeoh model is used. This model
includes n terms and is written as:
n
W = ∑ C I −( 1 ) 3 i (6-28)
i
i=1