A general framework for softening regularisation based on gradient elasticity

A general non-local approach to regularise strain-softening continua is presented. The key idea is to introduce the gradient-type enrichment at the level of displacements (rather than some internal variable), so the model is formulated with two distinct displacement fields. In fact, gradient models based on two displacement fields are usual in non-local elasticity, where the goal is to avoid the shortcomings of classical (local) elasticity (i.e. strain singularities in statics, non-dispersive behaviour in dynamics). We show that such a gradient elasticity backbone model can be combined with any standard nonlinear constitutive driver to render a regularised model for softening inelasticity. To illustrate the generality of the approach, two prototype models (isotropic damage and von Mises plasticity) are discussed. The numerical examples show that the regularised models exhibit all of the desired features: mesh insensitivity, imperfection size insensitivity and description of size effects.