Deformation of a plane modelled by John’s material with a rigid elliptical inclusion loaded by force and moment

Abstract

An exact analytical solution is obtained for a non-linear problem of elasticity theory for a plane with a rigid elliptical inclusion. A concentrated force and a moment are applied at the center of inclusion. The elastic properties of the plane are modeled by John’s harmonic material. For this material methods of the theory of functions of a complex variable are using for solving nonlinear plane problems. Nominal stresses and displacements are expressed in terms of two analytical functions of a complex variable, which are determined from the boundary conditions on the contour of inclusion. The problems of the action of a concentrated force and moment on an elliptical core in a plane are considered separately. A comparison with a similar linear problem is made. The influence of the applied force and moment on the magnitude of stresses is studied depending on various parameters of the problem. Calculations of nominal stresses on the contour joining the plane with inclusion are performed.