Three-Dimensional Contact Problem for an Elastic Layer and a Cylindrical Punch with Prestresses

Abstract

An axisymmetric mixed problem for an elastic cylindrical punch pressed into a layer with initial (residual) stresses is solved using the linearized theory of elasticity. The analysis is performed in general form for the theory of large (finite) deformations and various theories of small initial deformations for an arbitrary elastic potential Keywords: linearized theory of elasticity, initial (residual) stresses, Fredholm equation, method of successive approximations Introduction. Results related to the three-dimensional linearized theory of stability of deformable bodies and the three-dimensional linearized theory of propagation of elastic waves in bodies with initial (residual) stresses are analyzed from a contemporary standpoint in [2,8]. The approaches [2,8] were used to analyze results on a number of problems in linearized solid mechanics: (i) contact interaction of elastic bodies with initial (residual) stresses [2, 6] and (ii) stability of local equilibrium of rock near mine workings [2] (these are problems for inhomogeneous subcritical states). There are also other reviews on linearized solid mechanics [4,5,7,[9][10][11][12][13][14][15]. The publications cited above are fully or partially related to the subject of the present paper: contact interaction of elastic punches and elastic bodies, both with initial (residual) stresses.The effect of a finite cylindrical punch on a half-space with initial (residual) stresses was analyzed in [3]. A general solution to the three-dimensional linearized axisymmetric problem for a rigid circular punch pressed without friction into a prestressed layer is presented in [1].In what follows, we will use the linearized theory of elasticity [2, 6] to solve a mixed axisymmetric problem for an elastic cylindrical punch pressed into a layer with initial (residual) stresses. Two cases will be examined: (i) a layer lies without friction on a rigid foundation and (ii) a layer is bonded to a rigid foundation. The analysis will be performed in general form for the theory of large (finite) initial deformations and various theories of small initial deformation for an arbitrary elastic potential.We will use the coordinates Oy i of the initial deformed configuration, which are related to the Lagrangian coordinates (natural configuration) by y i = l i x i ( , ) i = 1 3 , where l i are the elongations (the displacements of the initial configuration). Let the