Page 23 - Mechanics of Asphalt Microstructure and Micromechanics

16 Ch a p t e r O n e

1.6.2.4 Stretch Ratio
The stretch ratio a is defined as the ratio between the length dx and that of dX measured
in the reference direction N. It is a scalar.
dx
α = (1-71)
dX
One may use a definition that can be more conveniently implemented.
dx 2
2
α = ( )
dX

(1-72)

One can define the stretch ratio in the n direction.
1 = dX (1-73)
β dx

1 dX ) dx • • dx
2
nc
β 2 = ( dx dX c dX = • • n (1-74)

Due to the deformation, n may not be a unit vector and therefore typically α ≠ β .
1.6.2.5 Material-Time Derivative
For any quantity (may be a scalar, vector, or tenor),

X t) or
x t)
B ij... = B (, B ij... = B ( , (1-75a, 1-75b)
ij...
ij...
Its time variation rate can be represented either as:
d ⎡ B (, ⎤ = ∂ ⎡ B (, ⎤
dt ⎣ ij... X t) ⎦ t ∂ ⎣ ij... X t) ⎦ (1-76)
or in the spatial reference as:
d ⎡ B (, ⎤ = ∂ ⎡ B (, ⎤ + ∂ ⎡ B (, ⎤ dx k (1-77)
xt)
x t)
x t)
dt ⎣ ij... ⎦ t ∂ ⎣ ij... ⎦ x ∂ ⎣ i ij... ⎦ dt
k
d ⎡ B (, ⎤ = ∂ ⎡ B (, ⎤ + ∂ ⎡ B (, ⎤
)
⎦
dt ⎣ ij... x t) ⎦ t ∂ ⎣ ij... x t) ⎦ x ∂ ⎣ i ij... xt v k (1-78)
k
Therefore, one has the following operator:
d = ∂ + v ∂ (1-79)
dt t ∂ k x ∂
k
or
d = ∂ +•
v ∇
dt t ∂ x (1-80)