Universität Karlsruhe (TH) - am Institut für Baustatik

E
J 2 -plasticity model for small strains
The applied plasticity model is restricted to small strains, where the additive decomposition
of the Green–Lagrangean strains in elastic an plastic parts is assumed. The strain energy is a
quadratic function of the elastic strains. We use an associative flow rule and a J 2 yield criterion
with linear isotropic hardening. The constitutive equations are summarized as follows:
kinematic assumption E = E e + E p
elastic part of the free energy ψ(E e ) = λ (trE 2 e) 2 + G tr(E 2 e)
Second Piola–Kirchhoff stresses S = ∂ Ee ψ
linear isotropic hardening y(e p ) = y 0 + Ke p
yield criterion Φ = √ 3
2 p)
associative flow rule Ė p = γ∂ S Φ
evolution of e p ė p = √ 2
||Ėp|| 3
loading unloading conditions
γ ≥ 0, Φ ≤ 0, γΦ=0
(70)
Eν
Here, the Lamé parameter λ is related to the elasticity constants by λ = , G denotes
(1+ν)(1−2ν)
the shear modulus, y 0 the initial yield stress and K the plastic tangent modulus. The rate
equations are integrated with a backward Euler algorithm. The stress tensor is linearized to
obtain the consistent tangent matrix ¯C introduced in (60).
Remark:
The definition of the deviatoric part of the Second Piola–Kirchhoff stress tensor for finite
strain kinematics is
Dev S = S − 1 3 [S : Ĉ]Ĉ−1 . (71)
Since Ĉ =(1 +2E), then for small Green–Lagrangean strains (i.e. when ‖E2 ‖≪‖E‖ ≪1)
it follows that
Dev S = S − 1 [S : 1]+O(‖E‖) =devS + O(‖E‖) . (72)
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Here O(‖E‖) denotes the terms which tend to zero as ‖E‖ approaches zero. Thus, when
strains are restricted to be small, an approximate definition of the deviatoric stress tensor
may be employed. Therefore in problems where finite rotations may be present but strains
are restricted to be small, the above constitutive Eqs. (70) describe the small strain von Mises
elasto–plasticity model. In this context we refer to [42].
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