On the stressed state of the joint upon bending beam with an elastic coating

Joint deformation of two elastic layers is considered, one of them being a coating applied to the surface of another thicker layer. It is assumed that slippage is absent in the plane of layer contact. External load is normal to the coating surface and unchanging in width. In the absence of lateral loading, the junction can be considered a two-layer beam and described by the equations of the plane elasticity using the asymptotic method. The required functions, normal and tangential stresses, and also the components of the displacement of an arbitrary point of the beam and coating are expanded in power series of the small parameter (accepted as a half thickness of the corresponding layer). In contrast to the known asymptotic expansions, an alternative asymptotic method is proposed in which all the required functions are asymptotically equivalent, i.e., expanded in asymptotic series of the same structure. When all positive powers of the small parameter are present in the series, the asymptotic algorithm leads to the appearance of two independent recurrent systems of linear equations, which greatly simplifies their solution. In each approach for one elastic layer the algorithm generates five indefinite functions of the longitudinal coordinate. Due to a rapid convergence of the asymptotic series, the first asymptotic approximation is used to construct the mathematical model of the deformation of a beam-to-coating coupling. Ten indefinite functions of the coordinate x provide fulfilling of eight conditions of continuity of the stress-strain state in the linkage depth, two functions remaining indefinite. Equilibrium equations (a system of two linear differential equations of the sixth order) are derived for them proceeding from the principle of minimum potential energy of deformation and corresponding boundary conditions (that can be realized both in static and kinematic variants). Numerical results are obtained in static approximation for two types of surface loading — steady and sinusoidally varying — along the length of the coating. An emphasis is made on the necessity of allowing for tangential stresses when studying the stress state of the coating as they can be rather significant upon bending.