Constitutive models and finite elements for plasticity in generalised continuum theories

Abstract

The mechanical behaviour of geomaterials (e.g. soils, rocks and concrete) under plastic deformation is highly complex due to that fact that they are granular materials consisting of discrete non-uniform particles. Failure of geomaterials is often related to localisation of deformation (strain-localisation) with excessive shearing inside the localised zones. The microstructure of the material then dominates the material behaviour in the localised zones. The formation of the localised zone (shear band) during plastic deformation decreases the material strength (softening) significantly and initiates the failure of the material. There are two main approaches to the numerical modelling of localisation of deformation in geomaterials; discrete and continuum. The discrete approach can provide a more realistic material description. However, in the discrete approach, the modelling of all particles is complicated and computationally very expensive for a large number of particles. On the other hand, the continuum approach is more flexible, avoids modelling the interaction of individual particles and is computationally much cheaper. However, classical continuum plasticity models fail to predict the localisation of deformation accurately due to loss of ellipticity of the governing equations, and spurious mesh-dependent results are obtained in the plastic regime. Generalised plasticity models are proposed to overcome the difficulties encountered by classical plasticity models, by relaxing the local assumptions and taking into account the microstructure-related length scale into the models. Among generalised plasticity models, Cosserat (micropolar) and stain-gradient models have shown significant usefulness in modelling localisation of deformation in granular materials in the last few decades. Currently, several elastoplastic models are proposed based on Cosserat and strain-gradient theories in the literature. The individual formulation of the models has been examined almost always in isolation and are paired with specific materials in a mostly arbitrary fashion. Therefore, there is a lack of comparative studies between these models both at the theory level and in their numerical behaviour, which hinders the use of these models in practical applications. This research aims to enable broader adoption of generalised plasticity models in practical applications by providing both the necessary theoretical basis and appropriate numerical tools. A detailed comparison of some Cosserat and strain-gradient plasticity models is provided by highlighting their similarities and differences at the theory level. Two new Cosserat elastoplastic models are proposed based on von Mises and Drucker- Prager type yield function. The finite element formulations of Cosserat and strain-gradient models are presented and compared to better understand their advantages and disadvantages regarding numerical implementation and computational cost. The finite elements and material models are implemented into the finite element program ABAQUS using the user element subroutine (UEL) and an embedded user material subroutine (UMAT) respectively. Cosserat finite elements are implemented with different Cosserat elastoplastic models. The numerical results show how the Cosserat elements behaviour in the plastic regime depends on the models, interpolation of displacement and rotation and the integration scheme. The effect of Cosserat parameters and specific formulations on the numerical results based on the biaxial test is discussed. Two new mixed-type finite elements as well as existing ones (C1, mixed-type and penalty formulation), are implemented with different strain-gradient plasticity models to determine the numerical behaviour of the elements in the plastic regime. A detailed comparison of the numerical results of Cosserat and strain-gradient elastoplastic models is provided considering specific strain-localisation problems. Finally, some example problems are simulated with both the Cosserat and strain-gradient models to identify their applicability.