Page 199 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 191

A) The equilibrium equations:
(,
∂σ xt)
ij
(,
+ Fx t) = 0 (6-127)
∂x j
i
B) The time-dependent constitutive equations:
t
st () = 2 ∫ G t ( − ξ d ) ( )ξ dξ
ij ij
0 (6-128)
t
σ t () = 3 ∫ K t ( −ξ )ε ξ ξ d()
ii ii
0
C) The strain-displacement relations (small strain):
ε t() = 1 [ ∂ ut() + ∂ ut() ] (6-129)
j
i
ij 2 x ∂ x ∂
j i
D) The compatibility relations:
2
2
t
∂ ε () ∂ ε () ∂ ε ()
t
2
t
2 xy = xx + yy (6-130)
∂∂xy ∂y 2 ∂x 2

E) Boundary conditions:
Tx t) = σ (,
(,
x t n )
j ij i
Ux t) = u x t) (6-131)
(,
(,
j j
A solution using the elastic-viscoelastic correspondence principle applies the La-
place transform to the above equations and obtains a set of equations in the transformed
space.
σ
∂ ˆ (, )
xs
ˆ
Fx
A) ij + (, ) s = 0 (6-132)
∂x j
i
ˆ ˆ ˆ
s
()
B) Ss = 2 sG () ()
s
d
ij ij
σ sK ˆ ˆ () s (6-133)
ε
ˆ () s = 3
ii ij
ˆ ()
1 ∂ us ∂ us
ˆ ()
C) ε s = [ i + ] (6-134)
j
ˆ ()
ij
2 x ∂ x ∂
j i
ε
ε
ε
2
2
2
∂ ˆ () ∂ ˆ () ∂ ˆ () s
s
s
D) 2 xy = xx + yy (6-135)
∂∂xy ∂y 2 ∂x 2
(, ) = σ
ˆ
E) Tx s ˆ (, )x s n i
j
ij
ˆ
(, ) =
ˆ (, )x s
Ux s u (6-136)
j ij
By comparing this set of equations with the equations in small-strain linear elastic-
ˆ
ˆ
ity, one can notice that if G is replaced with sG(s)K, with 3sK the set of equations is the
same as the set of equations in elasticity. This will allow one to solve the viscoelasticity