6.3 Perzyna Hardening Model
333
Note that the visco-plastic strain vp together with the hardening strain ε h measuring the elongation of the hardening spring denote the only elements contained in
the set of internal variables α = { vp , ε h } for the Perzyna hardening model.
6.3.1 Specific Perzyna Isotropic Hardening Model:
Formulation
The specific Perzyna isotropic hardening model, similar to that displayed in Fig. 6.18
(however with the hardening modulus H and the hardening strain ε h coinciding
here with the isotropic-hardening modulus H and the isotropic-hardening strain hi ,
respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear isotropic-hardening viscous frictional slider consisting of
a parallel arrangement of (i) a linear frictional slider with threshold σ y , (ii) a linear
viscous dashpot with viscosity η, and (iii) a linear isotropic-hardening spring with
stiffness H (the isotropic-hardening modulus).
For the specific Perzyna isotropic hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − vp (the elastic strain e )
and hi (the isotropic-hardening strain)
ψ( vp , hi ) =
1
2
E [ − vp ]
2
+
1
2
H
2
hi .
(6.119)
σ
σ
e
vp
ε h
E
H
η
σ y
Fig. 6.18 Specific Perzyna hardening model
333
Note that the visco-plastic strain vp together with the hardening strain ε h measuring the elongation of the hardening spring denote the only elements contained in
the set of internal variables α = { vp , ε h } for the Perzyna hardening model.
6.3.1 Specific Perzyna Isotropic Hardening Model:
Formulation
The specific Perzyna isotropic hardening model, similar to that displayed in Fig. 6.18
(however with the hardening modulus H and the hardening strain ε h coinciding
here with the isotropic-hardening modulus H and the isotropic-hardening strain hi ,
respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear isotropic-hardening viscous frictional slider consisting of
a parallel arrangement of (i) a linear frictional slider with threshold σ y , (ii) a linear
viscous dashpot with viscosity η, and (iii) a linear isotropic-hardening spring with
stiffness H (the isotropic-hardening modulus).
For the specific Perzyna isotropic hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − vp (the elastic strain e )
and hi (the isotropic-hardening strain)
ψ( vp , hi ) =
1
2
E [ − vp ]
2
+
1
2
H
2
hi .
(6.119)
σ
σ
e
vp
ε h
E
H
η
σ y
Fig. 6.18 Specific Perzyna hardening model