Page 203 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 195

may reach the yield surface at different locations. Naghdi and Murch (1963) proposed
the so-called flow surface concept.
κ
f = (σε p , , )
β
f
,
ij ij (6-155)
In addition to the stress state s ij , plastic strain e ij and the strain hardening parame-
p
v
ter k, a parameter b, a function of the viscoelastic strain b = b(e kl ) was introduced. In-
troduction of b makes the yield surface “flow” with time. An examination of the time
rate of the yield function will allow one to define loading and unloading.
f ∂ f ∂ f ∂ f ∂
f = σ + ε p + κ + β (6-156)
∂σ ij ∂ε p ij ∂κ ∂β
ij kl
· ·
If ƒ < 0 then, ƒ + ƒdt < 0, leading to the viscoelastic state, and therefore the rate of the
·
plastic strain (no incremental plastic strain), or e ij = 0. This will also lead to k = 0. Under
these conditions, one can have:
f ∂ f ∂
f = σ + β = ζ σ ( ) (6-157)
∂σ ij ∂β ij
ij
·
ƒ = 0 z(s ij < 0), unloading
·
ƒ = 0 z(s ij = 0), neutral process
·
ƒ = 0 z(s ij > 0), process yielding plastic strains
Unlike the traditional yielding, where neutral loading will be tangent to the yield
∂f ∂f ∂f
surface, for the viscoplastic situation σ + β = 0 does not require σ = 0 .
∂σ ij ∂β ∂σ ij
ij ij
It should be noted that the above formulations indicate that loading, unloading, and
·
·
neutral loading conditions are related to strain rates of both s ij and b.
Like plasticity theories, the viscoplasticity theory also requires the satisfaction of
the conditions for stable materials, and convexity conditions.
Naghdi and Murch (1963) proposed for the following relationship (please note its
similarity to the incremental plastic strain formulation, but here it is time rate of the
plastic strain):
f ∂
ε = Λ (6-158)
p
ij ∂ σ
ij
·
Through the use of the condition ƒ = 0, one can obtain:
∂f ∂f ∂f ∂f ∂f ∂f
Λ= −( σ + β )[ + κ( ( )] −1 (6-159)
∂σ kl ∂β ∂ε p ∂σ ∂κ ∂σ
kl mn mn pq
f ∂
Where κε( p ) = Λ κ( ) .
kl ∂ σ
pq
By introducing the viscoplastic potential function h, Perzyna arrived at the follow-
ing general formulation for the plastic deformation rate:
h ∂ f ∂ f ∂ h ∂
ε = N < σ + β >= N < ζ(σ ) > (6-160)
p
ij ∂ σ ∂ σ ij β ∂ ∂ σ ij
ij ij ij