Page 180 - Mechanics of Asphalt Microstructure and Micromechanics

172 Ch a p t e r S i x

If the directions of the principal stretches are oriented with the coordinate basis vec-
tors, then:
σ =− + 2 λ 2 ∂W = σ ; σ = − + 2 ∂W (6-38)
p
p
11 ∂I 22 33 λ ∂I
4
1 1
Since s 33 = 0, the following results:
p = 2 ∂ W (6-39)
λ 4 I ∂
1
Therefore,
σ = 2( λ − 1 ∂W = σ (6-40)
2
)
11 λ 4 ∂I 22
1
The engineering strain is l−1. The engineering stress is:
1 ∂ W
T = σ / λ = 2 λ − ) = T (6-41)
(
11 11 λ 5 I ∂ 22
1
Planar Extension
Planar extension tests are carried out on thin specimens that are constrained from de-
forming in one direction. For planar extension in the n 1 directions with the n 3 direction
constrained, the principal stretches are l 1 = l, l 3 = 1. From incompressibility l 1 = l 2 =
l 3 = 1. Hence, l 2 = 1/l. Therefore,

I = λ 2 + λ 2 + λ 2 = λ 2 + 2 + 1 (6-42)
1 1 2 3 λ 2
The left Cauchy-Green deformation tensor can then be expressed as:
B = λ 2 n n + 1 n n + n n (6-43)
1 1 λ 2 2 2 3 3
If the directions of the principal stretches are oriented with the coordinate basis vec-
tors, it follows:
W
σ =− + 2 λ 2 ∂W , σ =− + 2 ∂W , σ =− + 2 ∂W (6-44)
p
p
p
2
11 ∂I 22 λ ∂I 33 I ∂
1 1 1
Since s 22 = 0, then:
p = 2 ∂ W
λ 2 I ∂
1
Therefore,
σ = 2( λ − 1 ∂W
2
)
11 λ 2 ∂I
1
σ = 0
22
21−
σ = ( 1 ∂W
)
33 λ 2 ∂I
1