6.3 Perzyna Hardening Model
355
here with the kinematic-hardening modulus K and the kinematic-hardening strain
hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear kinematic-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 kinematic-hardening spring with
stiffness K (the kinematic-hardening modulus).
For the specific Perzyna kinematic hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − vp (the elastic strain e )
and hk (the kinematic-hardening strain)
ψ(, vp , hk ) =
1
2
E [ − vp ]
2
+
1
2
K
2
hk .
(6.165)
Then the energetic stress σ
conjugated to the total strain and the energetic viscoplastic stress σ
vp conjugated to the visco-plastic strain vp together with the kinematichardening stress σ
hk conjugated to the kinematic-hardening strain
hk follow as
σ
(, vp
) = ∂ ψ(, vp , hk ) = E [ − vp ],
(6.166a)
σ
vp (, vp
) = ∂ vp ψ(, vp , hk ) = −E [ − vp ],
(6.166b)
σ
hk (
hk ) = ∂ hk ψ(, vp , hk ) = K hk
.
(6.166c)
Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ
≡ σ, and, due
to the serial arrangement of the elastic spring and the kinematic-hardening viscous
frictional slider, also with the negative of the energetic visco-plastic stress, −σ
vp ≡ σ.
Furthermore, for the specific Perzyna kinematic hardening model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous
dissipation potential π v , together with a convex and non-smooth plastic contribution,
the plastic dissipation potential π p , i.e.
π(˙ vp , ˙
hk ) = π v (˙ vp ) + π p (˙ vp , ˙
hk ).
(6.167)
Thereby the viscous and plastic contributions π v and π p to the (total) dissipation
potential π are chosen as
π v (˙ vp ) =
1
2
η |˙ vp |
2 and π p (˙ vp , ˙
hk ) = σ y |˙ vp | + K hk [˙ vp − ˙
hk ]. (6.168)
Observe that (i) π = π v + π p does not depend on ˙
, thus the dissipative stress
σ
= σ − σ
≡ 0 vanishes identically, and that (ii) π v is positively homogenous of
degree two in ˙
vp and obviously smooth at the origin ˙
vp = 0, and that (iii) π p is
positively homogenous of degree one in {˙ p , ˙
hk } and obviously non-smooth at the
origin {˙ p , ˙
hk } = {0, 0}. As a consequence of the additive structure of the (total)
dissipation potential the dissipative visco-plastic stress σ
vp consists of a viscous
355
here with the kinematic-hardening modulus K and the kinematic-hardening strain
hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with
stiffness E and (2) a linear kinematic-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 kinematic-hardening spring with
stiffness K (the kinematic-hardening modulus).
For the specific Perzyna kinematic hardening model the free energy density ψ is
expressed as a quadratic (and thus convex) function of − vp (the elastic strain e )
and hk (the kinematic-hardening strain)
ψ(, vp , hk ) =
1
2
E [ − vp ]
2
+
1
2
K
2
hk .
(6.165)
Then the energetic stress σ
conjugated to the total strain and the energetic viscoplastic stress σ
vp conjugated to the visco-plastic strain vp together with the kinematichardening stress σ
hk conjugated to the kinematic-hardening strain
hk follow as
σ
(, vp
) = ∂ ψ(, vp , hk ) = E [ − vp ],
(6.166a)
σ
vp (, vp
) = ∂ vp ψ(, vp , hk ) = −E [ − vp ],
(6.166b)
σ
hk (
hk ) = ∂ hk ψ(, vp , hk ) = K hk
.
(6.166c)
Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ
≡ σ, and, due
to the serial arrangement of the elastic spring and the kinematic-hardening viscous
frictional slider, also with the negative of the energetic visco-plastic stress, −σ
vp ≡ σ.
Furthermore, for the specific Perzyna kinematic hardening model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous
dissipation potential π v , together with a convex and non-smooth plastic contribution,
the plastic dissipation potential π p , i.e.
π(˙ vp , ˙
hk ) = π v (˙ vp ) + π p (˙ vp , ˙
hk ).
(6.167)
Thereby the viscous and plastic contributions π v and π p to the (total) dissipation
potential π are chosen as
π v (˙ vp ) =
1
2
η |˙ vp |
2 and π p (˙ vp , ˙
hk ) = σ y |˙ vp | + K hk [˙ vp − ˙
hk ]. (6.168)
Observe that (i) π = π v + π p does not depend on ˙
, thus the dissipative stress
σ
= σ − σ
≡ 0 vanishes identically, and that (ii) π v is positively homogenous of
degree two in ˙
vp and obviously smooth at the origin ˙
vp = 0, and that (iii) π p is
positively homogenous of degree one in {˙ p , ˙
hk } and obviously non-smooth at the
origin {˙ p , ˙
hk } = {0, 0}. As a consequence of the additive structure of the (total)
dissipation potential the dissipative visco-plastic stress σ
vp consists of a viscous