An analysis of strain localization in a shear layer under thermally coupled dynamic conditions. Part 1: Continuum thermoplastic models
ABSTRACT The work presented in this two-part paper investigates the characteristics of strain localization in a one-dimensional shear layer with thermomechanical softening behaviour. We consider in this first part the case of a classical continuum model characterized with a stress–strain relation incorporating the inelastic effects through plastic strains. The thermomechanical coupling effects in the infinitesimal shear problem of interest here include the thermal softening of the material with the increase of the temperature, and the heat source generated by the plastic work in the equation of conservation of energy. We first present a spectral analysis of the linearized problem, characterizing its stability and well-posedness. We consider the general problem including a viscoplastic regularization in this coupled thermomechanical context, without neglecting the elastic effects. This analysis identifies, in particular, the ill-posedness of the local continuum model under certain conditions, most notably in the inviscid problem with strain softening. The coupled thermal effects are shown then not to regularize the coupled problem in these circumstances. The lack of an internal length scale associated with the strain localization is concluded. The analytical closed-form solution of the full non-linear problem is also presented for the identified ill-posed problems, revealing the lack of physical significance of these models in these cases. The implications of this analysis for the finite element simulations in the form of pathologically mesh-size dependent solutions is also concluded and illustrated with a number of representative numerical simulations. Copyright © 2003 John Wiley & Sons, Ltd.