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The concept of bi-potential, introduced by de Saxcé, extends the notion of pseudo-potential and enables to write the evolution laws as implicit normality rules.
The plastic flow of matrix and the void growth are dealt with separately after we show that two separated loading sub-surfaces and corresponding normality rules for matrix and damage exist.
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The corresponding plastic strain vector respects the normality rule and can vary substantially with stress state.
It was also shown that the direction of the vector of generalized strain rates obeys the normality rule.
The good properties of GSM result from the generalized normality rule and the convexity of the associated pseudo-potential.
The constitutive law is based on applying the von Mises yield criterion in association with the normality rule.
In accordance with the normality rule, evolution laws for the damage variables are obtained to complete the proposed plastic-damage model.
Some existing formulations of anisotropic plasticity are compared, discussing in particular the formulations of the normality rule and the assumptions regarding the elastic behaviour.
Under a standard derivation, the stiffness modulus for the non-associated flow rule is asymmetric since its plastic potential (for the plastic strain increment under the normality rule) differs from the plastic yield stress function (to define the elastic range).
The evolution equations governing the overall dissipative phenomena are deduced from the generalized normality rule applied to the plastic potential, while the consistency condition is still deduced from the yield function.
For micro-porous viscoplastic material described via micro-mechanics based elliptic potentials (and obeying the normality rule) this 'back mean stress' concept allows reproducing the inelastic dilatancy due to void growth, in addition to the isochoric plastic deformation due to dislocation glide, at zero and low negative stress triaxiality.
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