An analysis of nonuniform and localized deformation in ductile single crystals

Abstract

The nonuniform and localized deformations of ductile single crystals subject to tensile loading are analyzed numerically. The crystal is modelled by a rate independent, elastic-plastic relation based on Schmid's law which precisely accounts for lattice rotations. Both self hardening and latent hardening of the slip systems are included in the model. The crystal geometry is idealized in terms of a planar double slip model. Initial imperfections are specified in the form of slight thickness inhomogeneities and the calculations follow the crystal deformation through diffuse necking and the formation of shear bands. The pattern of shear bands depends on the initial imperfection, but, independent of the particular small imperfection, the material planes of the bands are inclined at a characteristic angle to the slip planes. Also, the lattice misorientation across the shear band, which is such as to cause geometrical softening of the bands, is not sensitive to the imperfection form. For high strength, low hardening crystals a comparison with existing experimental data shows remarkably good qualitative and quantitative agreement between the calculations and observations. We also model a relatively soft high hardening crystal which undergoes more diffuse necking than the strong high hardening crystal. Diffuse necking leads to lattice rotations which produce geometrical softening and hence promote shear band formation. Furthermore, we carry out a calculation for a high strength low hardening crystal with the latent hardening rate prescribed somewhat larger than for isotropic hardening. In this case a ‘patchy’ pattern of slip emerges. However, the course of shear band development is unaffected.