Page 222 - Mechanics of Asphalt Microstructure and Micromechanics

214 Ch a p t e r Sev e n

where s ij , e kl = stress and strain tensors
E ijkl (t) = the relaxation modulus tensor
t = physical time
x = t/a T = reduced time
a T = the time-temperature shift factor
t = the integration variable
R
Through introducing the reference modulus E R and pseudostrain e kl (Schapery,
1984):
−
ε = 1 ∫ ξ E ( ξ τ) ε ∂ kl τ d (7-3)
R
kl E 0 ijkl τ ∂
R
The stress and pseudostrain relationship can be written in the same format as in
elasticity:
σ
σ = E ε R or ε = ij (7-4)
R
ij R kl kl E
R
Clearly, if the relaxation modulus is constant E 0 and E R = E 0 , the above relationship
reduces to those of linear elasticity.
In the above formulations, the reduced time is calculated as follows for transient
and steady phenomena respectively:

ξ = t dt ξ = t (7-5a, b)
∫ 0 a a
T T
The pseudostrain can be obtained through a piecewise integration:
1 ⎡ t 1 ε d t 2 ε d t n dε ⎤
ε = ⎢∫ Et − τ) 1 τ d + t ∫ E t − τ) 2 τ d + ... + ∫ E Et − τ ) n dτ ⎥ (7-6)
R
(
(
(
E R ⎣ 0 τ d 1 τ d t n−1 dτ ⎦

Through the Prony representation of the relaxation modulus:
∞ ∑
Et () = E + m E e − t/ρ i (7-7)
i
i=1
The pseudostrain can be obtained as:
1 ⎡ m ⎤
ε Rn+1 ) = ⎢ η n+1 + ∑ η n+1 ⎥ (7-8)
(
E R ⎣ 0 i ⎦
i=1
Where h 0 and h i are the internal state variables for the elastic responses and for the
specific Maxwell element I at time step n + 1.
For the general case, that is viscoelastic body with damage, the stress-pseudostrain
relationship can be written as:

(
σ = CS E ε R (7-9)
)
m R
Where C (S m ) is a function of damage parameters S m .