Page 35 - Mechanics of Asphalt Microstructure and Micromechanics

28 Ch a p t e r O n e

1.6.4.7 Objectivity
The objectivity of constitutive modeling includes the Principle of Equipresence, which
states that an independent variable assumed to be present in one constitutive equation of
a material should be assumed to be present in all constitutive equations of the same mate-
rial, unless its presence contradicts an assumed symmetry of the material, or contradicts
the principle of material frame-indifference or some other fundamental principle.
There are three other fundamental postulates including the Principle of Determin-
ism for stress, the Principle of Local Action, and the Principle of Material Frame-Indif-
ference (Malvern, 1969). The Principle of Material Frame-Indifference is described as
follows.
The material frame-indifference principle actually states that an event {x,t} (location
and time) in frame x should be observed the same by the observers in another frame
x*as (vector transformation law):
x = c t () + Q t ()• ⎫ ⎪ ⎬
*
x
t =− ⎭ ⎪ (1-137)
*
t a

Vectors v
v = Q t ()• v (1-138)
*

Second-order tensors T or S regarded as linear vector transformations:
=
•
•
uT v or u = v S
T = Q t ()• T Q t ()• T ⎫ ⎪
*
⎬ (1-139)
S = Q t ()• S Q t ()• T ⎭ ⎪
*

Deformation gradient F(X,t)
F = Q t ()• F (1-140)
*

(This two-point tensor transforms like a vector under change of frame at time t)
Motion of a medium:
x = χ(,
X t)

•

x =+ • (1-141)
*
c Q x

Velocity:
v = dx * = + • + dx (1-142)
cQ x Q •
*
dt dt