Page 198 - Mechanics of Asphalt Microstructure and Micromechanics

190 Ch a p t e r S i x

If the strain is in pace with stress, the total strain energy stored (the entire loop will
involve unloading) and the total energy would be equal to zero:

/
T 4 dε T 4
/
W = ∫ σ dt = ∫ σε sin ω tcos ω tdt = σε / 2 (6-122)
00
00
0 dt 0
Therefore, the energy dissipation rate is:
ΔW
= 2π sin δ (6-123)
W

6.3.9 Temperature Effect and Time-Temperature Superposition Principle
For thermo-elastic materials, the elastic modulus is a function of temperature and time
E = E(T,t). The relaxation modulus described previously shows a function of time at
constant temperature. The modulus-temperature relationship deserves study. Based on
experimental studies, the Time-Temperature Superposition principle was proposed.
The principle actually states that the relaxation modulus E at temperature T is equal to
the modulus at the reference temperature T 0 at a scaled time by a factor of a T (T) and this
factor is a function of the temperature difference only. Mathematically, this relationship
can be expressed as:
ET t) = ET ( , )ζ
(,
0 (6-124)
ζ = ta T ( )
/
T
Where t is actual time, T is the temperature, and z is the scaled time. A well-known
relationship is the Williams, Landel, and Ferry (Williams, et al., 1955) relationship:
( −
1
( ) log
log aT ≡ t = − kT T 0 ) (6-125)
−
10 T ζ k + (T T )
2 0
Where k 1 and k 2 are material constants. Experimental results indicate that k 1 and k 2
are usually not “true” constants. They also vary with temperature and time, showing
the empirical nature of the above relationship. For transient temperature conditions,
Morland and Lee (1960), proposed the following relationship:
t dt′
ζ() = ∫ aT t′ (6-126)
t
0 T [()]

6.3.10 The Correspondence Principle
For stress analysis of quasi-static viscoelastic problems, the Correspondence Principle
will allow one to obtain viscoelatic solution if the corresponding elastic solution is
known, or directly solving a problem in the transformed space with transformed elas-
ticity components.
For a boundary value problem in viscoelasticity, mathematically it is equivalent to
solving the following set of equations.