6.2 Perzyna Model
315
Remarkably, since at yield the plastic stress satisfies |σ p | = σ y , the viscous overstress σ v = σ vp − σ p allows representation in terms of the yield condition that is,
however, evaluated in terms of the visco-plastic stress σ vp
σ v =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
+|σ vp | − σ y
0
−|σ vp | + σ y
if
σ vp > +σ y
else
σ vp < −σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.81)
The reasoning for the representation in Eq. 6.81 is highlighted in Fig. 6.11 and
follows as:
• For ˙
vp = 0 the overstress in the viscous damper is identically zero, i.e. σ v ≡ 0
and thus the stress σ p in the frictional slider coincides identically with the viscoplastic stress σ p ≡ σ vp . Consequently, and again since ˙
vp = 0, the visco-plastic
stress satisfies |σ vp | ≤ σ y .
• For ˙
vp > 0 (with σ vp > +σ y ) the overstress in the viscous damper results in σ v =
σ vp − σ y ≡ | + σ vp | − σ y and thus the stress σ p in the frictional slider coincides
identically with the (positive) yield stress σ p ≡ +σ y .
• For ˙
vp < 0 (with σ vp < −σ y ) the overstress in the viscous damper results in σ v =
σ vp + σ y ≡ −|σ vp | + σ y and thus the stress σ p in the frictional slider coincides
identically with the (negative) yield stress σ p ≡ −σ y .
Finally the above relations may conveniently be summarized as
σ v =
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp | < σ y
for
|σ vp | − σ y
σ vp
|σ vp |
|σ vp | ≥ σ y
⎫
⎪ ⎬
⎪ ⎭
.
(6.82)
Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.82, the two variants of the associated evolution law for the
visco-plastic strain in Eqs. 6.80a, 6.80b are alternatively expressed in terms of the
visco-plastic stress
˙
vp (σ vp ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp | < σ y
for
|σ vp | − σ y
η
σ vp
|σ vp |
|σ vp | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.83)
Obviously, the expressions in Eqs. 6.71, 6.72 and 6.83 are inverse relations. With
the representation for the evolution of the visco-plastic strain in Eq. 6.83, the corresponding (total) dual dissipation potential π
∗ , as determined from the Legendre
transformation
π
∗
(σ vp ) = max
˙
vp
σ vp ˙
vp −
1
2
η |˙ vp |
2
− σ y |˙ vp |
(6.84)
315
Remarkably, since at yield the plastic stress satisfies |σ p | = σ y , the viscous overstress σ v = σ vp − σ p allows representation in terms of the yield condition that is,
however, evaluated in terms of the visco-plastic stress σ vp
σ v =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
+|σ vp | − σ y
0
−|σ vp | + σ y
if
σ vp > +σ y
else
σ vp < −σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.81)
The reasoning for the representation in Eq. 6.81 is highlighted in Fig. 6.11 and
follows as:
• For ˙
vp = 0 the overstress in the viscous damper is identically zero, i.e. σ v ≡ 0
and thus the stress σ p in the frictional slider coincides identically with the viscoplastic stress σ p ≡ σ vp . Consequently, and again since ˙
vp = 0, the visco-plastic
stress satisfies |σ vp | ≤ σ y .
• For ˙
vp > 0 (with σ vp > +σ y ) the overstress in the viscous damper results in σ v =
σ vp − σ y ≡ | + σ vp | − σ y and thus the stress σ p in the frictional slider coincides
identically with the (positive) yield stress σ p ≡ +σ y .
• For ˙
vp < 0 (with σ vp < −σ y ) the overstress in the viscous damper results in σ v =
σ vp + σ y ≡ −|σ vp | + σ y and thus the stress σ p in the frictional slider coincides
identically with the (negative) yield stress σ p ≡ −σ y .
Finally the above relations may conveniently be summarized as
σ v =
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp | < σ y
for
|σ vp | − σ y
σ vp
|σ vp |
|σ vp | ≥ σ y
⎫
⎪ ⎬
⎪ ⎭
.
(6.82)
Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.82, the two variants of the associated evolution law for the
visco-plastic strain in Eqs. 6.80a, 6.80b are alternatively expressed in terms of the
visco-plastic stress
˙
vp (σ vp ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp | < σ y
for
|σ vp | − σ y
η
σ vp
|σ vp |
|σ vp | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.83)
Obviously, the expressions in Eqs. 6.71, 6.72 and 6.83 are inverse relations. With
the representation for the evolution of the visco-plastic strain in Eq. 6.83, the corresponding (total) dual dissipation potential π
∗ , as determined from the Legendre
transformation
π
∗
(σ vp ) = max
˙
vp
σ vp ˙
vp −
1
2
η |˙ vp |
2
− σ y |˙ vp |
(6.84)