A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.