210
5 Plasticity
Table 5.3 Summary of the generic St. Venant model
(1) Strain
≡ p
(2) Potential π = π(˙
(3) Stress
σ ∈ d ˙
π σ
or
(2) Yield
φ = φ(σ) ≤ 0
(3) Evolution ˙
= λ ∂ σ φ
(4) KKT
λ ≥ 0, φ ≤ 0, λ φ = 0
d = π(˙ ≥ 0.
(5.55)
Finally for an equivalent stress that is homogeneous of degree one in the stress (thus
σ ∂ σ ϕ = ϕ), the dissipation d = σ ˙
is exclusively given in terms of the Lagrange
multiplier λ and the yield limit σ y , since then
d = λ σ ∂ σ ϕ = λ ϕ = λ σ y .
(5.56)
The generic St. Venant model is summarized in Table 5.3.
5.2 Prandtl Model
The Prandtl model of a perfect elasto-plastic solid (in short the Prandtl model) consists
of a serial arrangement of (1) an elastic spring and (2) a frictional slider (see the sketch
of the specific Prandtl model in Fig. 5.7).
σ
σ
e
p
E
σ y
Fig. 5.7 Specific Prandtl model
5 Plasticity
Table 5.3 Summary of the generic St. Venant model
(1) Strain
≡ p
(2) Potential π = π(˙
(3) Stress
σ ∈ d ˙
π σ
or
(2) Yield
φ = φ(σ) ≤ 0
(3) Evolution ˙
= λ ∂ σ φ
(4) KKT
λ ≥ 0, φ ≤ 0, λ φ = 0
d = π(˙ ≥ 0.
(5.55)
Finally for an equivalent stress that is homogeneous of degree one in the stress (thus
σ ∂ σ ϕ = ϕ), the dissipation d = σ ˙
is exclusively given in terms of the Lagrange
multiplier λ and the yield limit σ y , since then
d = λ σ ∂ σ ϕ = λ ϕ = λ σ y .
(5.56)
The generic St. Venant model is summarized in Table 5.3.
5.2 Prandtl Model
The Prandtl model of a perfect elasto-plastic solid (in short the Prandtl model) consists
of a serial arrangement of (1) an elastic spring and (2) a frictional slider (see the sketch
of the specific Prandtl model in Fig. 5.7).
σ
σ
e
p
E
σ y
Fig. 5.7 Specific Prandtl model