5.3 Prandtl Hardening Model
245
stress σ
hk conjugated to the kinematic-hardening strain
hk follow as
σ
(, p
) = ∂ ψ(, p , hk ) = E [ − p ],
(5.141a)
σ
p (, p
) = ∂ p ψ(, p , hk ) = −E [ − p ],
(5.141b)
σ
hk (
hk ) = ∂ hk ψ(, p , hk ) = K hk .
(5.141c)
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 frictional slider,
also with the negative of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl kinematic hardening model the convex but
non-smooth dissipation potential π is chosen as
π(˙ p , ˙
hk ) = σ y |˙ p | + K hk [˙ p − ˙
hk ].
(5.142)
Observe that (i) π does not depend on ˙
, thus the dissipative stress σ
= σ − σ
≡
0 vanishes identically, and that (ii) π is positively homogenous of degree one in
{˙ p , ˙
hk } and is obviously non-smooth at the origin {˙ p , ˙
hk } = {0, 0}. Consequently,
the dissipative plastic stress σ
p and the dissipative kinematic-hardening stress σ
hk
compute as some sub-derivatives of the dissipation potential with respect to their
conjugated variables
σ
p (˙ p , ˙
hk ) ∈ d ˙
p π(˙ p , ˙
hk ),
σ
hk (˙ p , ˙
hk ) ∈ d ˙
hk π(˙ p , ˙
hk ),
(5.143)
with
d ˙
p π(˙ p , ˙
hk ) =
⎧
⎨
⎩
+[σ y + K hk ]
˙
p > 0
−[σ y − K hk ], +[σ y + K hk ]
for ˙
p = 0
−[σ y − K hk ]
˙
p < 0
⎫
⎬
⎭
,
d ˙
hk π(˙ p , ˙
hk ) =
− K hk ,
(5.144)
whereby d ˙
p π and d ˙
hk π denote the sets of sub-derivatives, i.e. the sub-differentials
of π with respect to ˙
p and ˙
hk , respectively.
Recall that the energetic and the dissipative plastic as well as kinematic-hardening
stresses are constitutively related by σ
p + σ
p = 0 and σ
hk + σ
hk = 0, respectively,
thus the notions of plastic stress and isotropic-hardening stress defined as the values
σ p := σ
p = −σ
p ,
(5.145a)
σ hk := σ
hk = −σ
hk ,
(5.145b)
will exclusively be used in the sequel for convenience of exposition.