A finite element displacement formulation for gradient elastoplasticity A. Zervos National Technical University of Athens, Greece Schlumberger Cambridge Research, U.K. P. Papanastasiou Schlumberger Cambridge Research, U.K. I. Vardoulakis National Technical University of Athens, Greece ABSTRACT: We present a gradient elastoplasticity model extending the idea of gradient plasticity, by assuming that the underlying elasticity is of the gradient type as well. The numerical implementation is based entirely on a finite element displacement formulation. The model is developed to regularise the ill-posedness caused by strain-softening material behaviour. The displacement field is the only field needed to be discretised using a continuity element. Mesh insensitivity is demonstrated by modelling localisation of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear-band zone are insensitive to the mesh directionality and refinement. The inclusion of gradient terms allows robust modelling of the post-peak material behaviour leading to localisation of deformation. 1 INTRODUCTION Failure of geomaterials is commonly associated with localisation of deformation in zones of intense shear- ing such as shear bands or shear interface layers. Near a failure state the material inside the localisation zones undergoes significant deformation and degra- dation while the rest of the material remains rather inert. Experiments show that the regions of locali- sation have a characteristic finite dimension (thick- ness) (M¨ uhlhaus & Vardoulakis 1987). Modelling of progressive localisation requires the incorporation of material strength loss with straining (softening). Mathematically, the introduction of soft- ening leads to loss of ellipticity of the governing equa- tions (Vardoulakis & Sulem 1995). Consequently, nu- merical computations based on classical theories give mesh dependent results. The governing equations can be regularised to re- main elliptic by resorting to higher order continuum theories. These theories take into account the micro- structure of the deforming solid introducing an extra parameter of material length. This parameter is typ- ically related either to the mean grain diameter or to some other larger characteristic scale, e.g. the length of microcracks. The shear band thickness scales with this material length and is no longer indeterminate. In addition, the introduction of the material length per- mits modelling of the scale effect which is often ob- served in many geomechanics problems and cannot be modelled by the classical theories. Higher order theories presented in the literature are the Cosserat continuum theory, the Mindlin theory, the non-local continuum and gradient plasticity. In the Cosserat approach, micro-rotation is intro- duced as an additional degree of freedom at each material point, leading to a non-symmetric stress tensor (M¨ uhlhaus & Vardoulakis 1987) (Cosserat & Cosserat 1909) (Papanastasiou & Vardoulakis 1992) (Tejchman & Wu 1993). A higher order couple stress tensor arises as an extra state variable. The Cosserat theory can be seen as a special case of the general the- ory for microstructure developed by Mindlin (Mindlin 1964), where two inter-linked levels of deformation are assumed: micro-deformation at the particle level and macro-deformation at the structural level. In non-local continua, the response of the mate- rial at a point is determined not only by the state at that point but also from the deformation of its ‘neigh- bourhood’. The neighbourhood’s contribution is ex- pressed through an integral equation (Pijaudier-Cabot 1