5.2 Prandtl Model
223
The resulting σ = σ() diagram is highlighted in Fig. 5.11c. The expected
parallelogram-type format of the σ = σ() diagram is captured exactly, whereby
the slopes at = 0 and = 5 obviously coincide with the elastic modulus E = 1.
Figure 5.11d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| = 5 (or ˙
p (t) = 0 in
the holding phase), whereas p (t) stays constant with p (t) = 4 (or as initial value
p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.11e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ (t)| = {0, 5, 0, 5} over the time interval t ∈
[0, t max = 10], thus κ max = 4 + 3 = 7.
5.2.4 Generic Prandtl Model: Formulation
A generic formulation of the Prandtl model can be obtained from generalizing the
specific Prandtl model in Fig. 5.7 by assuming the elastic spring or/and the frictional
slider as nonlinear.
For the generic Prandtl model the free energy density ψ is expressed as a nonquadratic but convex function of − p (the elastic strain e )
ψ(, p ) = ψ( − p ).
(5.91)
Note that ψ(, p ) and ψ( − p ) are different functions that return, however, the same
function value for the same values of and p . Then the energetic stress σ
and the
energetic plastic stress σ
p follow as
σ
(, p ) = ∂ ψ(, p ) = ∂ ψ( − p ),
(5.92a)
σ
p (, p ) = ∂ p ψ(, p ) = ∂ p ψ( − p ).
(5.92b)
Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ
≡ σ and the negative of the energetic plastic
stress −σ
p ≡ σ. Moreover the energetic and the dissipative plastic stresses are
constitutively related by σ
p + σ
p = 0, thus the notion of plastic stress defined as
σ p := σ
p = −σ
p will exclusively be used in the sequel.
Furthermore, for the generic Prandtl model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙ p ) and π
∗
= π
∗
(σ p ),
respectively, are related via corresponding Legendre transformations
π ( ˙
p ) = max
σ p
{σ p ˙
p − π
∗
(σ p )},
(5.93a)
π
∗
(σ p ) = max
˙
p
{σ p ˙
p − π ( ˙
p )}.
(5.93b)
223
The resulting σ = σ() diagram is highlighted in Fig. 5.11c. The expected
parallelogram-type format of the σ = σ() diagram is captured exactly, whereby
the slopes at = 0 and = 5 obviously coincide with the elastic modulus E = 1.
Figure 5.11d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| = 5 (or ˙
p (t) = 0 in
the holding phase), whereas p (t) stays constant with p (t) = 4 (or as initial value
p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.11e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ (t)| = {0, 5, 0, 5} over the time interval t ∈
[0, t max = 10], thus κ max = 4 + 3 = 7.
5.2.4 Generic Prandtl Model: Formulation
A generic formulation of the Prandtl model can be obtained from generalizing the
specific Prandtl model in Fig. 5.7 by assuming the elastic spring or/and the frictional
slider as nonlinear.
For the generic Prandtl model the free energy density ψ is expressed as a nonquadratic but convex function of − p (the elastic strain e )
ψ(, p ) = ψ( − p ).
(5.91)
Note that ψ(, p ) and ψ( − p ) are different functions that return, however, the same
function value for the same values of and p . Then the energetic stress σ
and the
energetic plastic stress σ
p follow as
σ
(, p ) = ∂ ψ(, p ) = ∂ ψ( − p ),
(5.92a)
σ
p (, p ) = ∂ p ψ(, p ) = ∂ p ψ( − p ).
(5.92b)
Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ
≡ σ and the negative of the energetic plastic
stress −σ
p ≡ σ. Moreover the energetic and the dissipative plastic stresses are
constitutively related by σ
p + σ
p = 0, thus the notion of plastic stress defined as
σ p := σ
p = −σ
p will exclusively be used in the sequel.
Furthermore, for the generic Prandtl model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙ p ) and π
∗
= π
∗
(σ p ),
respectively, are related via corresponding Legendre transformations
π ( ˙
p ) = max
σ p
{σ p ˙
p − π
∗
(σ p )},
(5.93a)
π
∗
(σ p ) = max
˙
p
{σ p ˙
p − π ( ˙
p )}.
(5.93b)