360
6 Visco-Plasticity
• For ˙
vp > 0 (with σ vp + σ hk > +σ y ) the overstress in the viscous damper results
in σ v = [σ vp + σ hk ] − σ y ≡ +|σ vp + σ hk | − σ y and thus the stress σ p + σ hk in the
frictional slider coincides identically with the (positive) yield stress σ p + σ hk ≡
+σ y .
• For ˙
vp < 0 (with σ vp + σ hk < −σ y ) the overstress in the viscous damper results
in σ v = [σ vp + σ hk ] + σ y ≡ −|σ vp + σ hk | + σ y and thus the stress σ p + σ hk in the
frictional slider coincides identically with the (negative) yield stress σ p + σ hk ≡
−σ y .
Finally the above relations may conveniently be summarized as
σ v =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y
for
|σ vp + σ hk | − σ y ]
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.183)
Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.183, the two variants of the associated evolution law for the
visco-plastic strain in Eqs. 6.181a, 6.181b are alternatively expressed in terms of the
visco-plastic stress
˙
vp (σ vp , σ hk ) =
(6.184)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y
for
|σ vp + σ hk | − σ y
η
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
Consequently, based on Eq. 6.181c rendering ˙
hk = ˙
vp , the associated evolution
law for the kinematic-hardening strain follows as
˙
hk (σ vp , σ hk ) =
(6.185)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y
for
|σ vp + σ hk | − σ y
η
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
Obviously, the expressions in Eqs. 6.170, 6.171 and 6.184, 6.185 are inverse relations. With the representation for the evolution of the visco-plastic and kinematichardening strains in Eqs. 6.184 and 6.185, the corresponding (total) dual dissipation
potential π
∗ , as determined from the Legendre transformation