Theoretical considerations upon the MK model for limit strains prediction: The plane strain case, strain-rate effects, yield surface influence, and material heterogeneity

The paper presents a study of the Marciniak and Kuczynski (MK for short) model for the prediction of limit strains of orthotropic sheet metal under in-plane proportional biaxial stretching. In two particular cases analytical results can be obtained if the groove of the MK model is oriented along one of the in-plane symmetry axes. The first case is the plane strain loading mode. Necessary and sufficient conditions are derived for the MK-predicted plane strain limit strain to match exactly the experimentally measured limit strain. An example of material, the AA5182-O aluminum alloy, that does not satisfy these conditions is discussed. It is shown then that if a power-law strain rate sensitivity is included in the hardening law then the MK-model can match exactly any target plane strain limit strain. The second case is the non-hardening case for positive strain ratios. This case allows for an insight into the way the MK-predicted limit strains depend upon the yield function. Based on the theory developed for the plane strain case, material heterogeneity as a possible cause for unstable plastic flow is further discussed. It is shown that such heterogeneities can be modeled by perturbing the rate of deformation with an eigenstrain. This allows for an extension of the MK-model to sheets of uniform thickness.