Cell-substrate interaction is examined in a two-dimensional mechanics model. The cell and substrate are treated as a shell and an elastic solid, respectively. Their interaction through adhesion is treated using nonlinear springs. Compared to previous cell mechanics models, the present model introduces a cohesive force law that is dependent not only on cell-substrate distance but also on internal cell-membrane stress. It is postulated that a living cell would establish focal adhesion sites with density dependent on the cell-membrane stress. The formulated mechanics problem is numerically solved using coupled finite elements and boundary elements for the cell and the substrate, respectively. The nodes in the adhesion zone from either side are linked by the cohesive springs. The specific cases of a cell adhering to a homogeneous substrate and a heterogeneous bimaterial substrate are examined. The analyses show that the substrate stiffness affects the adhesion behavior significantly and regulates the direction of cell adhesion, in good agreement with the experimental results in the literature. By introducing a reactive parameter (i.e., cell-membrane stress) linking biological responses of a living cell to a mechanical environment, the present model offers a unified mechanistic vehicle for characterization and prediction of living cell responses to various kinds of mechanical stimuli including local extracellular matrix and neighboring cells.