This thesis is devoted to the mathematical study of two kinds of junction problems. The first model is obtained as the variational limit of a junction problem in the scope of elasticity where the adhesive occupying the junction possesses a stiffness of the order of the thickness. One adds a smooth or non smooth convex surface energy density h to the elastic density W of the adhesive. This additional density conveys the fact that the joint and the adherents are not perfectly stuck together. We show that the asymptotic model consists in replacing the joint by a constraint which now is the inf-convolution of h and the limit density of W. In the scalar case we analyse the gradient concentration phenomenon at the interface by means of recent tools stemming from measure theory. In a second model, the stiffness of the adhesive is of the order of the inverse of the thickness of the junction. The bulk energy density of the adherents grows superlinearly while that of the adhesive grows linearly. Following the strategy used in the first problem, we propose a simplified but accurate model by studying the asymptotic behavior when the thickness goes to zero through a variational convergence method. At the limit the intermediate layer is replaced by a pseudo-plastic interface which allows cracks to appear.