A usual trick in various modelling approaches for computational fracture (i.e. damage models, phase-field approaches) is to include an artificial residual stiffness. This is done to avoid singularity of the tangent stiffness matrix. The goal of this MSc thesis is to assess how this numerically motivated parameter affects the mechanical response. Alternative ways to include the desired residual stiffness without an accompanying, undesired, residual load-carrying capacity will also be explored, and tested by means of one- and two-dimensional numerical examples.