5.3 Prandtl Hardening Model
227
Then the energetic stress σ
conjugated to the total strain and the energetic plastic
stress σ
p conjugated to the plastic strain p together with the isotropic-hardening
stress σ
hi conjugated to the isotropic-hardening strain
hi follow as
σ
(, p
) = ∂ ψ(, p , hi ) = E [ − p ],
(5.105a)
σ
p (, p
) = ∂ p ψ(, p , hi ) = −E [ − p ],
(5.105b)
σ
hi (
hi ) = ∂ hi ψ(, p , hi ) = H hi
.
(5.105c)
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 isotropic-hardening frictional slider,
also with the negative of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl isotropic hardening model the convex but
non-smooth dissipation potential π is chosen as
π(˙ p , ˙
hi ) = [σ y + H hi ] |˙ p | − H hi ˙
hi .
(5.106)
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 , ˙
hi } and is obviously non-smooth at the origin {˙ p , ˙
hi } = {0, 0}. Consequently,
the dissipative plastic stress σ
p and the dissipative isotropic-hardening stress σ
hi
compute as some sub-derivatives of the dissipation potential with respect to their
conjugated variables
σ
p (˙ p , ˙
hi ) ∈ d ˙
p π(˙ p , ˙
hi ),
σ
hi (˙ p , ˙
hi ) ∈ d ˙
hi π(˙ p , ˙
hi ),
(5.107)
with
d ˙
p π(˙ p , ˙
hi ) =
⎧
⎨
⎩
+[σ y + H hi ]
˙
p > 0
−[σ y + H hi ], +[σ y + H hi ]
for ˙
p = 0
−[σ y + H hi ]
˙
p < 0
⎫
⎬
⎭
,
d ˙
hi π(˙ p , ˙
hi ) =
− H hi ,
(5.108)
whereby d ˙
p π and d ˙
hi π denote the sets of sub-derivatives, i.e. the sub-differentials
of π with respect to ˙
p and ˙
hi , respectively.
Recall that the energetic and the dissipative plastic as well as isotropic-hardening
stresses are constitutively related by σ
p + σ
p = 0 and σ
hi + σ
hi = 0, respectively,
thus the notions of plastic stress and isotropic-hardening stress defined as the values
σ p := σ
p = −σ
p ,
(5.109a)
σ hi := σ
hi = −σ
hi ,
(5.109b)
227
Then the energetic stress σ
conjugated to the total strain and the energetic plastic
stress σ
p conjugated to the plastic strain p together with the isotropic-hardening
stress σ
hi conjugated to the isotropic-hardening strain
hi follow as
σ
(, p
) = ∂ ψ(, p , hi ) = E [ − p ],
(5.105a)
σ
p (, p
) = ∂ p ψ(, p , hi ) = −E [ − p ],
(5.105b)
σ
hi (
hi ) = ∂ hi ψ(, p , hi ) = H hi
.
(5.105c)
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 isotropic-hardening frictional slider,
also with the negative of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl isotropic hardening model the convex but
non-smooth dissipation potential π is chosen as
π(˙ p , ˙
hi ) = [σ y + H hi ] |˙ p | − H hi ˙
hi .
(5.106)
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 , ˙
hi } and is obviously non-smooth at the origin {˙ p , ˙
hi } = {0, 0}. Consequently,
the dissipative plastic stress σ
p and the dissipative isotropic-hardening stress σ
hi
compute as some sub-derivatives of the dissipation potential with respect to their
conjugated variables
σ
p (˙ p , ˙
hi ) ∈ d ˙
p π(˙ p , ˙
hi ),
σ
hi (˙ p , ˙
hi ) ∈ d ˙
hi π(˙ p , ˙
hi ),
(5.107)
with
d ˙
p π(˙ p , ˙
hi ) =
⎧
⎨
⎩
+[σ y + H hi ]
˙
p > 0
−[σ y + H hi ], +[σ y + H hi ]
for ˙
p = 0
−[σ y + H hi ]
˙
p < 0
⎫
⎬
⎭
,
d ˙
hi π(˙ p , ˙
hi ) =
− H hi ,
(5.108)
whereby d ˙
p π and d ˙
hi π denote the sets of sub-derivatives, i.e. the sub-differentials
of π with respect to ˙
p and ˙
hi , respectively.
Recall that the energetic and the dissipative plastic as well as isotropic-hardening
stresses are constitutively related by σ
p + σ
p = 0 and σ
hi + σ
hi = 0, respectively,
thus the notions of plastic stress and isotropic-hardening stress defined as the values
σ p := σ
p = −σ
p ,
(5.109a)
σ hi := σ
hi = −σ
hi ,
(5.109b)