The present contribution focuses on the mechanobiological aspects related to cell-substrate adhesion and their involvement in positive durotaxis as well as in the recently discovered negative durotaxis [6]. A key component of this kind of migration is the adhesion between the cell and the extracellular matrix performed through mechanosensitive cell structures called focal adhesion complexes. These structures grow and disrupt during their life cycle undergoing a chemo-physical degradation process which is here modeled by means of an elastic-damaging cohesive law. The resulting traction-sliding law was first applied to a simplified two-element tensegrity model [1, 2, 8] and then exploited in a fully three-dimensional continuum model of a single cell [3]. The application of a contractile pre-stretch to the cell shows the inference of the elastic-damaging law. The reversal of positive into negative durotaxis is modelled as a consequence of a force-driven degradation process of the adhesion structures. An optimal stiffness range of the substrate at which the cell deploys its maximum traction force can also be detected.