330
6 Visco-Plasticity
Figure 6.17b showcases the resulting strain history (t) that displays an increasing signal with (t) → 469. ¯
3. The holding phase with ˙
σ(t) = 0 is characterised by
extensive creep with ˙
(t) = [5 − 1]/0.075 = 53. ¯
3. (Here the elastic contribution to
the strain with at most σ a /E = 5 is negligible in the eye-ball-norm.)
The resulting σ = σ() diagram is highlighted in Fig. 6.17c. The creep towards
≈ 450 (from visual inspection) during the holding phase is clearly visible at σ = 5.
Figure 6.17d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 469. ¯
3. (In the eye-ball-norm the difference to the strain signal is
negligible due to the smallness of the elastic strain e ≤ 5.)
Finally, the strain arc-length κ(t) in Fig. 6.17e follows from integrating ˙
κ(t) =
|˙ (t)| over two and a half periods and approaches κ max = 469. ¯
3.
Figure 6.17d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.9d and e since they are both exposed to the
identical stress history.
6.2.4 Generic Perzyna Model: Formulation
A generic formulation of the Perzyna model can be obtained from generalizing the
specific Perzyna model in Fig. 6.10 by assuming the elastic spring or/and the viscous
dashpot or/and the frictional slider as nonlinear.
For the generic Perzyna model the free energy density ψ is expressed as a nonquadratic but convex function of − vp (the elastic strain e )
ψ(, vp = ψ( − vp ).
(6.108)
Note that ψ(, vp ) and ψ( − vp ) are different functions that return, however, the
same function value for the same values of and vp . Then the energetic stress σ
and the energetic visco-plastic stress σ
vp follow as
σ
(, vp ) = ∂ ψ(, vp ) = ∂ ψ( − vp ),
(6.109a)
σ
vp (, vp ) = ∂ vp ψ(, vp ) = ∂ vp ψ( − vp ).
(6.109b)
Recall that the total stress σ (that enters the equilibrium condition) coincides
identically with the energetic stress σ
≡ σ and the negative of the energetic viscoplastic stress −σ
vp ≡ σ. Moreover the energetic and the dissipative visco-plastic
stresses are constitutively related by σ
vp + σ
vp = 0, thus the notion of visco-plastic
stress defined as σ vp := σ
vp = −σ
vp will exclusively be used in the sequel. Note
moreover that the visco-plastic stress σ vp in the viscous frictional slider decomposes
additively into the viscous stress σ v in the viscous dashpot and the plastic stress σ p
in the frictional slider.
Furthermore, for the generic Perzyna model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙ vp ) and π
∗
= π
∗
(σ vp ),
respectively, are related via corresponding Legendre transformations
6 Visco-Plasticity
Figure 6.17b showcases the resulting strain history (t) that displays an increasing signal with (t) → 469. ¯
3. The holding phase with ˙
σ(t) = 0 is characterised by
extensive creep with ˙
(t) = [5 − 1]/0.075 = 53. ¯
3. (Here the elastic contribution to
the strain with at most σ a /E = 5 is negligible in the eye-ball-norm.)
The resulting σ = σ() diagram is highlighted in Fig. 6.17c. The creep towards
≈ 450 (from visual inspection) during the holding phase is clearly visible at σ = 5.
Figure 6.17d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 469. ¯
3. (In the eye-ball-norm the difference to the strain signal is
negligible due to the smallness of the elastic strain e ≤ 5.)
Finally, the strain arc-length κ(t) in Fig. 6.17e follows from integrating ˙
κ(t) =
|˙ (t)| over two and a half periods and approaches κ max = 469. ¯
3.
Figure 6.17d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.9d and e since they are both exposed to the
identical stress history.
6.2.4 Generic Perzyna Model: Formulation
A generic formulation of the Perzyna model can be obtained from generalizing the
specific Perzyna model in Fig. 6.10 by assuming the elastic spring or/and the viscous
dashpot or/and the frictional slider as nonlinear.
For the generic Perzyna model the free energy density ψ is expressed as a nonquadratic but convex function of − vp (the elastic strain e )
ψ(, vp = ψ( − vp ).
(6.108)
Note that ψ(, vp ) and ψ( − vp ) are different functions that return, however, the
same function value for the same values of and vp . Then the energetic stress σ
and the energetic visco-plastic stress σ
vp follow as
σ
(, vp ) = ∂ ψ(, vp ) = ∂ ψ( − vp ),
(6.109a)
σ
vp (, vp ) = ∂ vp ψ(, vp ) = ∂ vp ψ( − vp ).
(6.109b)
Recall that the total stress σ (that enters the equilibrium condition) coincides
identically with the energetic stress σ
≡ σ and the negative of the energetic viscoplastic stress −σ
vp ≡ σ. Moreover the energetic and the dissipative visco-plastic
stresses are constitutively related by σ
vp + σ
vp = 0, thus the notion of visco-plastic
stress defined as σ vp := σ
vp = −σ
vp will exclusively be used in the sequel. Note
moreover that the visco-plastic stress σ vp in the viscous frictional slider decomposes
additively into the viscous stress σ v in the viscous dashpot and the plastic stress σ p
in the frictional slider.
Furthermore, for the generic Perzyna model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙ vp ) and π
∗
= π
∗
(σ vp ),
respectively, are related via corresponding Legendre transformations