382
6 Visco-Plasticity
Remarkably, since at yield the plastic stress satisfies |σ p + σ hk | = σ y − σ hi , 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 hk | − σ y + σ hi
0
−|σ vp + σ hk | + σ y − σ hi
if
σ vp + σ hk > +σ y − σ hi
else
σ vp + σ hk < −σ y + σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.227)
The reasoning for the representation in Eq. 6.227 is highlighted in Fig. 6.33 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 kinematic-hardening frictional slider coincides identically
with the visco-plastic stress σ p ≡ σ vp . Consequently, and again since ˙
vp = 0, the
visco-plastic stress satisfies |σ vp + σ hk | ≤ σ y − σ hi .
• For ˙
vp > 0 (with σ vp + σ hk > +[σ y − σ hi ]) the overstress in the viscous damper
results in σ v = [σ vp + σ hk ] − [σ y − σ hi ] ≡ +|σ vp + σ hk | − σ y + σ hi and thus the
stress σ p + σ hk in the isotropic-hardening frictional slider coincides identically
with the (positive) current yield stress σ p + σ hk ≡ +[σ y − σ hi ].
• For ˙
vp < 0 (with σ vp + σ hk < −[σ y − σ hi ]) the overstress in the viscous damper
results in σ v = [σ vp + σ hk ] + [σ y − σ hi ] ≡ −|σ vp + σ hk | + σ y − σ hi and thus the
stress σ p + σ hk in the isotropic-hardening frictional slider coincides identically
with the (negative) current yield stress σ p + σ hk ≡ −[σ y − σ hi ].
Finally the above relations may conveniently be summarized as
σ v =
(6.228)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
|σ vp + σ hk | − [σ y − σ hi ]
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.228, the two variants of the associated evolution law for the
visco-plastic strain in Eqs. 6.226a, 6.226b are alternatively expressed in terms of the
visco-plastic stress
˙
vp (σ vp , σ hi , σ hk ) =
(6.229)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
|σ vp + σ hk | − [σ y − σ hi ]
η
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
6 Visco-Plasticity
Remarkably, since at yield the plastic stress satisfies |σ p + σ hk | = σ y − σ hi , 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 hk | − σ y + σ hi
0
−|σ vp + σ hk | + σ y − σ hi
if
σ vp + σ hk > +σ y − σ hi
else
σ vp + σ hk < −σ y + σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.227)
The reasoning for the representation in Eq. 6.227 is highlighted in Fig. 6.33 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 kinematic-hardening frictional slider coincides identically
with the visco-plastic stress σ p ≡ σ vp . Consequently, and again since ˙
vp = 0, the
visco-plastic stress satisfies |σ vp + σ hk | ≤ σ y − σ hi .
• For ˙
vp > 0 (with σ vp + σ hk > +[σ y − σ hi ]) the overstress in the viscous damper
results in σ v = [σ vp + σ hk ] − [σ y − σ hi ] ≡ +|σ vp + σ hk | − σ y + σ hi and thus the
stress σ p + σ hk in the isotropic-hardening frictional slider coincides identically
with the (positive) current yield stress σ p + σ hk ≡ +[σ y − σ hi ].
• For ˙
vp < 0 (with σ vp + σ hk < −[σ y − σ hi ]) the overstress in the viscous damper
results in σ v = [σ vp + σ hk ] + [σ y − σ hi ] ≡ −|σ vp + σ hk | + σ y − σ hi and thus the
stress σ p + σ hk in the isotropic-hardening frictional slider coincides identically
with the (negative) current yield stress σ p + σ hk ≡ −[σ y − σ hi ].
Finally the above relations may conveniently be summarized as
σ v =
(6.228)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
|σ vp + σ hk | − [σ y − σ hi ]
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.228, the two variants of the associated evolution law for the
visco-plastic strain in Eqs. 6.226a, 6.226b are alternatively expressed in terms of the
visco-plastic stress
˙
vp (σ vp , σ hi , σ hk ) =
(6.229)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
|σ vp + σ hk | − [σ y − σ hi ]
η
σ vp + σ hk
|σ vp + σ hk |
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.