Adiabatic Shear Banding for Metals at High velocity Impact

Abstract

The localization of deformation into narrow bands of intense straining is a characteristic feature of inelastic deformation. The high concentration of strains in small areas of engineering structures and laboratory specimens is a physical phenomenon observed, in particular, for materials with a softening portion due to elevated temperatures, high strain rates, etc. Even in cases when the increase of loads is slow, the increase of strains in the areas of strain concentration is fast and associated with higher than average dissipation of energy. Therefore, plastic strain localization may be considered as a symptom of the initiation of structural failure. In fact, the problem of strain localization still remains a challenging problem for future research. It has been proved that using rate independent models in the analysis of structures made of materials that exhibit softening leads to ill posed problems. Theoretically, for rate independent solids, localization is associated with a change in the character of the governing equations. That is, the type of the partial differential equations changes from elliptic into hyperbolic in statics and conversely in dynamics. Consequently, numerical solutions to localization problems for rateindependent solids exhibit inherent mesh dependence. The minimum width of the band of localized deformation is set by the mesh spacing. Furthermore, global quantities such as the overall stiffness characteristics of the body depend on the mesh size used to resolve the band of localized deformations. Therefore, rate independent theories without any type of regularization cannot be used in the analysis where strain localization appears. A possible way to overcome this problem is to introduce rate dependent models into the formulation in which the type of governing equations does not change during the expected strain localization which leads to well posed problem. This may be viewed in terms of material rate dependence implicitly introducing a length scale into the boundary value problem formulation. Abed proposed a physically based multiscale-viscoplastic model for capturing dynamic localizations of different types of metals (bcc, fcc, and hcp) at low and high strain rates and temperatures. He utilized new definitions for the static (athermal) and dynamic (thermal) yield functions which are derived based on thermal activation analysis as well as dislocation interactions mechanisms as elaborated by Voyiadjis and Abed. In this work we present some localization results (shear bands and necking) using the aforementioned model for Vanadium, Tantalum, Niobium and OFHC Copper subjected to different displacement velocities. The effect of initial temperature on the initiation and development of shear bands for a simple tensile plane strain problem are also introduced. The constitutive equations are integrated in order to obtain a framework suitable for the application of a displacement finite element method at finite strains. A corotational formulation procedure in which all the fields of interest are transformed into the