Strain gradient plasticity based on saturating variables

Abstract

Standard scalar-based strain gradient plasticity models generally incorporate the gradient of the cumulative plastic strain into the constitutive modeling. The following limitations of this approach have been pointed out in the literature: Unlimited enhanced hardening at large strains, unlimited cyclic hardening in bending, strain localization band broadening at large applied strains for softening materials with saturation, and possible vanishing of the yield stress for too high values of the Laplace term in the enhanced hardening function. The present work proposes an alternative scalar-based micromorphic approach that solves most of the listed difficulties. The theory incorporates the effect of the gradient of saturating variables instead of the gradient of ever-increasing cumulative plastic strain. A phenomenological theory is presented that includes an exponentially saturating internal variable and its gradient. The approach is also applied to dislocation density based plasticity where the dislocation density is known to saturate at large strains. The benefits of the new theory are demonstrated by finite element simulations of various boundary value problems: Monotonic and cyclic tension and simple glide of an infinite strip, strain localization in a plate, bending and torsion of bars. Both hardening and softening plasticity laws are addressed illustrating the size-dependent hardening in the former case and the evolution of the finite width of localization bands in the latter. Weak points of the approach, namely the possibility of negative dissipation and of negative yield stress, are also identified.