Interaction between different internal length scales for strain localisation analysis of single phase materials

Abstract

It is well known that both rate dependent and gradient-dependent constitutive models introduce internal length scales in dynamic initial value problems. As a result, numerical solutions of such initial value problems involving strain-softening no longer exhibit excessive mesh dependence. In this paper, the length scales included in a solid model which exhibits both above mentioned constitutive behaviours are discussed. The internal length scales derived from damping effects, which are typical for the viscoplastic models, and the wave length, obtained from the critical wave number for which the wave speed is not imaginary, are used together to give a prediction of the internal length scale of the combined model. The approach proposed here for prediction of the internal length scale is more general than commonly used procedures and permits to explain phenomena observed in viscoplastic and gradient dependent models. A one dimensional example is given to illustrate the theoretical findings

A multiscale overlapped coupling formulation for large-deformation strain localization

“…To circumvent the loss of material stability, a localization limiter must be introduced. This localization limiter can be (i) a nonlocal or gradient model in which the constitutive response is governed by the gradient or the integral of at least one local internal variable [8], (ii) a rate dependent constitutive law [10], (iii) a multiphysics model that provides regularization through coupling with heat or pore-fluid diffusion [1,27,33,34,39] or (iv) a formulation that allows displacement discontinuities [9,15,16,28,38]. These methods may help to maintain the uniqueness of the governing equations and eliminate the pathological dependence of the numerical solutions on the mesh size.…”

Interaction between different internal length scales in strain localization analysis of fully and partially saturated porous media—the 1‐D case

“…This way of determining the upper and lower bound values can also be used for the internal length prediction of single-phase problem with both the visco-plastic model and gradientdependent model, as in Reference [25]. (32), i.e.…”

“…To predict the internal length scale in the gradient-dependent multiphase model it is necessary to consider both the damping effects by the negative part of the complex roots and the wavelength corresponding to the critical wave number from which the wave speed becomes real (this method is used in Reference [10] for a single-phase viscoplastic material and in Reference [25] for viscoplastic gradient-dependent single-phase material).…”