5.3 Prandtl Hardening Model
263
σ
(, p
) = ∂ ψ(, p , hi , hk ) = E [ − p ],
(5.176a)
σ
p (, p
) = ∂ p ψ(, p , hi , hk ) = −E [ − p ],
(5.176b)
σ
hi (
hi
) = ∂ hi ψ(, p , hi , hk ) = H hi
,
(5.176c)
σ
hk (
hk ) = ∂ hk ψ(, p , hi , hk ) = K hk .
(5.176d)
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 mixed-hardening frictional slider,
also with the negative of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl mixed hardening model the convex but nonsmooth dissipation potential π is chosen as
π(˙ p , ˙
hi , ˙
hk ) = [σ y + H hi ] |˙ p | − H hi ˙
hi + K hk [˙ p − ˙
hk ].
(5.177)
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 , ˙
hk } and is obviously non-smooth at the origin {˙ p , ˙
hi , ˙
hk } = {0, 0, 0}. Consequently, the dissipative plastic stress σ
p and the dissipative isotropic- and kinematichardening stresses σ
hi and σ
hk compute as some sub-derivatives of the dissipation
potential with respect to their conjugated variables
σ
p (˙ p , ˙
hi , ˙
hk ) ∈ d ˙
p π(˙ p , ˙
hi , ˙
hk ),
σ
hi (˙ p , ˙
hi , ˙
hk ) ∈ d ˙
hi π(˙ p , ˙
hi , ˙
hk ),
σ
hk (˙ p , ˙
hi , ˙
hk ) ∈ d ˙
hk π(˙ p , ˙
hi , ˙
hk ),
(5.178)
with
d ˙
p π(˙ p , ˙
hi , ˙
hk ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+
[σ y + H hi ] + K hk
˙
p > 0
−
[σ y + H hi ] − K hk
,
for ˙
p = 0
+
[σ y + H hi ] + K hk
−
[σ y + H hi ] − K hk
˙
p < 0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
d ˙
hi π(˙ p , ˙
hi , ˙
hk ) =
− H hi ,
d ˙
hk π(˙ p , ˙
hi , ˙
hk ) =
− K hk .
(5.179)
whereby d ˙
p π, d ˙
hi π and d ˙
hk π denote the sets of sub-derivatives, i.e. the subdifferentials of π with respect to ˙
p , ˙
hi and ˙
hk , respectively.
Recall that the energetic and the dissipative plastic as well as isotropic- and
kinematic-hardening stresses are constitutively related by σ
p + σ
p = 0, σ
hi + σ
hi =
0 and σ
hk + σ
hk = 0, respectively, thus the notions of plastic stress and isotropicand kinematic-hardening stresses defined as the values
263
σ
(, p
) = ∂ ψ(, p , hi , hk ) = E [ − p ],
(5.176a)
σ
p (, p
) = ∂ p ψ(, p , hi , hk ) = −E [ − p ],
(5.176b)
σ
hi (
hi
) = ∂ hi ψ(, p , hi , hk ) = H hi
,
(5.176c)
σ
hk (
hk ) = ∂ hk ψ(, p , hi , hk ) = K hk .
(5.176d)
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 mixed-hardening frictional slider,
also with the negative of the energetic plastic stress, −σ
p ≡ σ.
Furthermore, for the specific Prandtl mixed hardening model the convex but nonsmooth dissipation potential π is chosen as
π(˙ p , ˙
hi , ˙
hk ) = [σ y + H hi ] |˙ p | − H hi ˙
hi + K hk [˙ p − ˙
hk ].
(5.177)
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 , ˙
hk } and is obviously non-smooth at the origin {˙ p , ˙
hi , ˙
hk } = {0, 0, 0}. Consequently, the dissipative plastic stress σ
p and the dissipative isotropic- and kinematichardening stresses σ
hi and σ
hk compute as some sub-derivatives of the dissipation
potential with respect to their conjugated variables
σ
p (˙ p , ˙
hi , ˙
hk ) ∈ d ˙
p π(˙ p , ˙
hi , ˙
hk ),
σ
hi (˙ p , ˙
hi , ˙
hk ) ∈ d ˙
hi π(˙ p , ˙
hi , ˙
hk ),
σ
hk (˙ p , ˙
hi , ˙
hk ) ∈ d ˙
hk π(˙ p , ˙
hi , ˙
hk ),
(5.178)
with
d ˙
p π(˙ p , ˙
hi , ˙
hk ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+
[σ y + H hi ] + K hk
˙
p > 0
−
[σ y + H hi ] − K hk
,
for ˙
p = 0
+
[σ y + H hi ] + K hk
−
[σ y + H hi ] − K hk
˙
p < 0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
d ˙
hi π(˙ p , ˙
hi , ˙
hk ) =
− H hi ,
d ˙
hk π(˙ p , ˙
hi , ˙
hk ) =
− K hk .
(5.179)
whereby d ˙
p π, d ˙
hi π and d ˙
hk π denote the sets of sub-derivatives, i.e. the subdifferentials of π with respect to ˙
p , ˙
hi and ˙
hk , respectively.
Recall that the energetic and the dissipative plastic as well as isotropic- and
kinematic-hardening stresses are constitutively related by σ
p + σ
p = 0, σ
hi + σ
hi =
0 and σ
hk + σ
hk = 0, respectively, thus the notions of plastic stress and isotropicand kinematic-hardening stresses defined as the values