We consider some applications of domain decomposition methods with nonmatching meshes. Of special interest are coupling techniques for overlapping and nonoverlapping decompositions based on the weak mortar surface coupling. We focus on the coupling of mechanical structures, and we consider different material laws (linear and nonlinear) and material parameters. We address numerical stability problems in case of curvilinear interfaces, nearly incompressible materials or contact problems and propose suitable local modifications in terms of quadrature formulas. All theoretical results are illustrated by numerical simulations in two and three dimensions including multigrid convergences rates, Newton iterations and a priori convergence rates.