Mathematical insights for rate-dependent slip model in crystal plasticity

It is usual to employ the power type relationship between the shear rate and shear stress on a slip system for the calculation of deformation texture evolution with the rate-dependent crystal plasticity model. In this paper, we discuss the rational mechanical meanings for formulations of the rate-dependent crystal plasticity theory with particular emphasis on mathematical principle, in order to elucidate a theoretical foundation for the rate-dependent slip model. The mathematical and rational mechanical arguments have been done based on "the linear vector space and functional analysis", in which plasticity theories can be considered as a mathematical problem in "the finite dimensional Banach space" consisting of the sets whose members are shear rates and shear stresses on crystallographic slip systems. As the result of the analysis, it has been theoretically shown that the power type constitutive law for rate -dependent materials can be derived as a kind of topological property which characterizes a metric relationship between the linear vector space of shear rate and its dual space (the shear stress space). Furthermore, the limiting case of the rate sensitivity index approaching zero which corresponds to the rate-independent theory is also discussed in this study.