A strong discontinuity finite element model

In strong discontinuity models the presence of a discontinuity such as a crack or a slip surface within a finite element (embedded discontinuities) is modelled as a jump in the displacement field. Additional displacement values are thus introduced, which constitute degrees of freedom that can be incorporated into the model using various schemes, several examples of which exist in the literature. In these, either or both kinematic and static conditions are adjusted so as to correctly link the reciprocal influence of displacement and traction vectors between the bulk material and the part directly influenced by the discontinuity, thus deriving basic equations which combine into a stiffness matrix that is generally non-symmetric. In order to complete these basic equations, some kind of traction-separation law for the discontinuity surface must be assumed. In the formulation here presented, an original material law is used to model the discontinuity and its influence on the bulk material. This law, derived from a model existing in the literature for describing the stress-displacement behaviour in a medium due to an embedded crack, is based on the theory of dislocations, and describes the discontinuity as a continuous distribution of point dislocations making use of appropriate influence functions. Material behaviour related to the presence of a discontinuity is hence analytically expressed, and it is possible to associate a specific stress-displacement field in the medium to the displacement discontinuity values. In this way a strong discontinuity formulation is developed in which the displacement discontinuity field is directly related to the stress and displacement fields in the medium. As a consequence, a relatively simple constitutive law can be used for the rest of the material. This mathematical approach is closely linked and wholly justified by actual behaviour observed in certain classes of geomaterials and real-life configurations: in fact, it is often the case in soil mechanics that the whole stress-deformation behaviour of a large soil mass is determined by the presence of discontinuities in its interior, rather than by some complex constitutive law pertaining to the entire soil volume. Progressive failure of slopes due to shear band propagation is a significant example of this phenomenon. In this work, the details of the material law linking the displacement jump to the traction on the discontinuity surface are illustrated, and examples of validation are proposed. Some aspects regarding the way in which this material law can be incorporated into a finite element formulation are also introduced and commented.