Page 187 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 179

6.2.4.3 Principle of Maximum Plastic Resistance
For the actual stress (s ij ) on the yield surface and any other stress (s* ij ) within the yield
surface (the stress state outside the yield surface is impossible), the Principle of Maxi-
mum Plastic Resistance states:
ij (
ij)
σ − σ d ε ≥ 0or σ ε ≥ σ ε p (6-80)
p
*
*
p
d
d
ij ij ij ij ij
That is the actual stress state that will offer the maximum plastic resistance.
6.2.4.4 Drucker’s Postulate T i
Druker proposed the following conditions for:
⎧ ≥ 0, hardening material
p ⎪
σε ⎨ 0, perfectly plastic material (6-81)
p
⎪ ≤
⎩ ⎩ 0, softening material
For a stable plastic material, if the surface traction has an increase of ΔT i and its cor-
responding displacement increase is Δu i , the following inequality will hold.
⎧ Δu ⎫ ⎪
ΔW = ⎨∫∫ ΔT i i ⎬ dt ≥ 0 (6-82)
A ⎩ ⎪ dt ⎭
The implications of the inequality are:

• The yield surface f(s ij ) must be convex;
f ∂
• The plastic strain rate must be normal to the yield surface dε = dε p ∂ σ ;
p
ij
dY ij
• The rate of strain hardening must be positive or zero ≥ 0 .
dε p

6.3 Viscoelasticity (Findley et al., 1989)
6.3.1 Creep, Recovery, and Relaxation
For a material of linear, isotropic elasticity, Hook’s law applies. The 1-D stress-strain
relation follows the linear relation presented in Equation 6-83.
σ
ε = (6-83)
E

Where e and s are correspondingly the uniaxial strain and stress, E is the Young’s
modulus. Obviously, when the stress is a constant, the strain is also a constant and vice
versa. Where the stress is removed, the strain will return to zero immediately. Figure 6.2
illustrates this relationship.
However, for a material of viscoelasticity, the straining process is quite different
under the same stressing process. First, the strain is not a constant under the same con-
stant stress s 0 ; it increases with time. Second, when the stress is removed, the strain
does not return to zero immediately; instead, it gradually returns to zero (Figure 6.3).