On a contact problem of stringer–wedge interaction in antiplane deformation mode

Under antiplane deformation, the problem of the contact interaction of a thin infinite stringer of finite width with an elastic 3D wedge-shaped body of an arbitrary opening angle is considered. It is assumed that one face of the wedge is rigidly clamped while a stringer is attached to the other face, which does not extend to the top of the wedge. The deformation of a stringer is described by the well-known Melan model for antiplane deformation, and the stress–strain state of an elastic wedge is described by the equations of elasticity theory. Two formulations of the contact problem are discussed according to the two models of stringer deformation. In the first formulation, as usual, external tangential loads and concentrated forces acting on the stringer are specified, and it is required to determine the tangential contact stresses arising under the stringer and axial stresses in it. In the second formulation of the problem, the elastic displacements of the stringer points are predetermined; it is required to determine the force factors acting on the stringer that provide the specified mode of elastic displacements. Using the Mellin integral transform, solving the contact problem under consideration is reduced to solving a singular integral equation (SIE) with a kernel in the form of a sum of the Cauchy kernel and a regular kernel in the first formulation and to solving a Fredholm integral equation (IE) of the first kind with a symmetric logarithmic kernel in the second formulation. The solution of both integral equations is constructed by the method of mechanical quadratures, based on the use of well-known Gauss-type quadrature formulas for calculating singular integrals at Chebyshev nodes, as well as Gauss-type quadrature formulas for calculating integrals containing logarithmic and signum regular kernels. Ultimately, solving the equation is reduced to solving a finite system of linear algebraic equations (SLAE). A numerical analysis of the problem was carried out, and regularities in the change in the main mechanical quantities of the problem were revealed in a quite wide range of changes in the characteristic physical and geometric parameters.