Shear stability and strain, strain-rate and temperature-dependent “cold” work

We present the stability and instability conditions of a thermoviscoplastic material with the assumption that the fraction ζ of plastic work stored as energy of crystal defects (“cold” work) is a function of strain, strain-rate and temperature. The results comply with the stability and instability criteria of a material with power constitutive law and constant ζ, provided that the function ζ satisfies an additional condition, valid for many used materials. However, the maximum and minimum values of the function ζ may, respectively, affect the critical time at which shear banding is manifested and the range of bounds of the shear stress. Moreover, we present the role of the function ζ in the stability of the uniform shearing of a mild steel characterized by a Costin et al. flow stress [Inst. Phys. Conf. Ser. No. 47, Adam Hilger, Bristol, 1979, Chapter 1, p. 90]. We show that, if ζ is sufficiently increasing with respect either to strain or to temperature and, at the same time, its slope of decrease with respect to strain-rate is moderate, then the uniform shearing is stable.