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Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification. (English) Zbl 1481.74084

Summary: In this work the strain localization analysis of Hill’s orthotropic plasticity is addressed. In particular, the localization condition derived from the boundedness of stress rates together with Maxwell’s kinematics is employed. Similarly to isotropic plasticity considered in our previous work, the plastic flow components on the discontinuity surface vanish upon strain localization. The resulting localization angles in orthotropic plastic materials are independent from the elastic constants, but rather, they depend on the material parameters involved in the plastic flow in the material axes. Application of the above localization condition to Hill’s orthotropic plasticity in 2-D plane stress and plane strain conditions yields closed-form solutions of the localization angles. It is found that the two discontinuity lines in plane strain conditions are always perpendicular to each other, and for the states of no shear stresses, the localization angle depends only on the tilt angle of the material axes with respect to the global ones. The analytical results are then validated by independent numerical simulations. The \(B\)-bar finite element is employed to deal with the incompressibility due to the purely isochoric plastic flow. For a strip under vertical stretching in plane stress and plane strain as well as Prandtl’s problem of indentation by a flat rigid die in plane strain, numerical results are presented for both isotropic and orthotropic plasticity models with or without tilt angle. The influence of various parameters is studied. In all cases, the critical angles predicted from the localization condition coincide with the numerical results, giving compelling supports to the analytical prognoses.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74E10 Anisotropy in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
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