Integral equations of contact problems of elasticity theory for stratified media

Abstract The formulation of an integral equation for the contact problem for a stratified multi-layered medium is considered using the example of a pure shear of a multi-layered (stratified) coating, bonded to a rigid base, subjected to the action of a rigid strip indenter. The thickness and shear modulus of each coating layer are known in advance. The solution to the contact problem is constructed using the Fourier integral transform, applied to the equilibrium differential equations for each layer and to the interface boundary conditions for layer couplings. Interface boundary conditions are written in a matrix form as recurrent relations. Taking into account the coating support conditions and mixed contact boundary conditions, the solution to the contact problem (contact stresses) is reduced to a Fredholm integral equation of the first kind of the Fourier convolution type. The transform of the integral equation kernel is expressed in a matrix form and contains all the mechanical and geometric parameters of the multilayered coating. An analytical solution to the integral equation is constructed in the form of a functional series in a small geometric parameter. Formulas to determine the stress-strain state in an arbitrary layer (in any of the layers) of a stratified medium are obtained for such cases as layered rock, underground structures, mines, etc., as well as in microelectronics, where the number of stratified material layers may exceed one hundred, and in biomechanics, where the layering of biomaterials ranges from ten or more layers of elements within the human or animal body.