J2 plasticity

The J2 plasticity are based in the following considerations:

i-) Additive decomposition of the strain tensor . One asumes that the strain tensor ε is decomposed into an elastic and plastic part, denoted by ε e and ε p respectively, according to the relationship.

ε=ε e + ε p

ii-) Elastic stress response . The stress tensor σ is related to the elastic strain by means of a stored-energy function W according to the relationship σ =∂W/ε e

For linearized elasticity, W is a quadratic form in the elastic strain, i.e. W= 1/2 ε e : C : ε e , where C is the tensor of elastic moduli wich is asumed constant. Then the stress tensor is written σ =C :(ε-ε p ).

iii-) Yield condition. We define a function f(σ ,q ) called yield criterion where q is a vector of interanal variables. The admisible states { σ ,q } are constraint by f(σ ,q )≤0.

A choice of internal variables wich is typically of metal plasticity is q ={ ξ, β }. Here, ξ is the equivalent plastic strain that defines isotropic hardening of the von Mises yield surface, and β defines the center of the von Mises yield surface in stress deviator space. The resulting J2-plasticity model has the following yield condition:

η =dev [σ ]-β , tr[β ]=0

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f(σ ,q )=√η·η - √2/3 K(ξ)

iv-) Flow rule and hardening law. The flow rule and hardening law for a J2-plasticity model are:

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dε p = γ η/ η·η

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dβ =γ 2/3 H(ξ) η/ η·η

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dξ=γ √2/3

Further explanations can be found in Simo and Hughes (1997)