ADIABATIC SHEAR BAND LOCALIZATION IN ELASTIC PLASTIC SINGLE-CRYSTALS

The main objective of the paper is the investigation of shear band localization criteria for finite elastic-plastic deformations of a single crystal subjected to an adiabatic process. The next objective is to focus attention on the temperature dependent plastic behaviour of the single crystal considered. A constitutive model is developed within the thermodynamic framework of the rate type covariance constitutive structure, i.e. it is invariant with respect to diffeomorphism. To achieve this aim a multiplicative decomposition of the deformation gradient is adopted and the Lie derivative is used to define all objective rates for introduced vectors and tensors. Thermomechanical couplings are investigated and a method is developed which allows us to use the standard bifurcation procedure in the examination of the adiabatic shear band localization. The general evolution equation for the Kirchhoff stress tensor is obtained. The fundamental matrix in this evolution equation describes thermomechanical couplings as well as local lattice deformation and rotation. For the particular elastic properties of the single crystal and for some simplified case of the coupling effects the criteria for adiabatic shear band localization are obtained in their exact analytical form. The influence of two important thermal effects, namely thermal expansion and thermal plastic softening on the criteria of localization is investigated. The similar influence of spatial covariance effects (which arise from the difference between the Lie derivative and the material rate of the Kirchhoff stress tensor) is also examined. It has been shown that by incorporating the thermomechanical effects and the spatial covariance effects into a constitutive law of the elastic-plastic single crystal, the plastic hardening modulus h(crit) at the inception of localization is in fact small but positive. It has also been proved that this thermomechanical theory of single crystals can describe the misalignment of the shear bands from the active slip systems in the crystal's matrix. The computed critical value of the strain-hardening rate h(crit) as well as the difference between the direction of the macroscopic shear band and the primary slip systems of the single crystal appeared to be in accord with recent experimental observations [cf. Chang and Asaro (1981, Acta Metall. 29, 241-257) for Al-Cu single crystals and Spitzig (1981, Acta Metall. 29, 1359-1377) for Fe-Ti-Mn single crystals].