An analysis of strain localization in a shear layer under thermally coupled dynamic conditions. Part 1: Continuum thermoplastic models

The work presented in this two-part paper investigates the characteristics of strain localization in a onedimensional shear layer with thermomechanical softening behaviour. We consider in this rst part the case of a classical continuum model characterized with a stress–strain relation incorporating the inelastic e ects through plastic strains. The thermomechanical coupling e ects in the in nitesimal 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 rst 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 e ects. This analysis identi es, 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 e ects 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 identi ed ill-posed problems, revealing the lack of physical signi cance of these models in these cases. The implications of this analysis for the nite element simulations in the form of pathologically meshsize dependent solutions is also concluded and illustrated with a number of representative numerical simulations.