Page 22 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 15

The Eulerian Finite Strain Tensor is:
2e = δ − X X = δ − δ − u )( δ − u )
(
ij ij A i , A j , ij Ai A i , Aj A j ,
2e = u + u − u u A j , (1-62)
j i
,
i j
,
A i
,
ij
By neglecting the high order terms, one has:
2E = u + u , 2e = u + u (1-63a, 1-63b)
,
,
AB A B B A ij i j , j i ,
Both reduce to the small strain representation.
1.6.2.3 Rate of Deformation
The previous sections have focused on the deformation gradients and strains that are
based on displacements or locations. They are spatial gradients. This section will dis-
cuss time-dependent characteristics of deformation. Instead of using the displacement,
the velocity v i will be the field variables for the study.
The spatial velocity gradient is defined as:
v ∂
L = i (1-64)
ij x ∂
j
This tensor can be decomposed into a symmetric part and a skew-symmetric part:

L = D + W ij (1-65)
ij
ij
⎛ v ∂ v ∂ ⎞ ⎛ v ∂ v ∂ ⎞
D = 1 ⎜ i + j ⎟ W = 1 ⎜ i − j ⎟ (1-66a, 1-66b)
ij x ∂ x ∂ ij x ∂ x ∂
2 ⎝ i ⎠ 2 ⎝ i i ⎠
j j
The symmetric part D ij is the rate of deformation tensor and the skew-symmetric
part W ij is the spin tensor representing rigid rotation.
The spatial velocity change can be represented as:
v ∂
dv = i dx or dv =• (1-67)
L dx
i x ∂ j
j
Since
∂v ∂v ∂X d ⎛ ∂x ⎞ ∂X
i = j A = ⎜ i ⎟ A (1-68)
∂x ∂X ∂x dt ⎝ ∂X ⎠ ∂x
j A j A j
One has:
F F
L = • −1 (1-69)
or
(1-70)
This relationship is often used for deriving other relationships in the following sec-
tions.