Page 217 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 209

Scharpery (1984) showed that through introducing pseudo-elastic strain, the
hereditary integrals can be converted into elastic stress-strain relations following
Hooke’s Law:
ε t () = σ t () (6-219)
e
E
R
e
e (t)and E R are the pseudo-elastic strain and reference modulus, respectively.
t d ετ()
e
=
ε() E R∫ ( C t − τ) τ d
t
0 τ d (6-220)
t
ε t () = E R ∫ E t ( − τ) d ετ () τ d
−1
e
0 τ d (6-221)
t d ετ()
e
ε t() = E R∫ C ( − τ) kl τ d
t
ij ijkl τ d
0 (6-222)
t d ετ()
ε t() = E R ∫ E ( − τ) kl τ d
−1
e
t
ijkl
ij
0 τ d (6-223)
ε t() = E [( + ν σ − νσ ]
e
−1
)
1
ij R ij kk (6-224)
By defining the following two operators, the above relationships can be concisely
presented as:
t f ∂
{Ddf = E R∫ ( C t − τ ) dτ dτ
}
0 (6-225)
t f ∂
{Edf = E R ∫ ( E t − τ ) dτ dτ
−1
}
0 (6-226)
The correspondence principle by Schapery (1984) states that the elastic solution and
the viscoelastic solution are associated by the following relationships:
σ = σ e ε = { Dd ε } u = { Ddu } T = σ n = σ n (6-227)
e
e
e
n
ij ij ij ij i i i ij i ij j
Therefore, if one has the elastic solution for a boundary value problem, the above
correspondence principle will allow one to obtain the corresponding viscoelastic solu-
tion.
Through those operations, the generalized J integrals are:
⎛ u ∂ e ⎞
J = ⎜ ∫ w dy T i x ∂ i ds ⎟
−
e
Γ ⎝ ⎠ (6-228)
where w = ε kl σε e
e
d
∫ 0 ij ij