General spatial static contact problem for a prestressed elastic half-space

Abstract

The general spatial static contact problem for an elastic half-space with initial stresses is considered. Exact solutions are constructed for an arbitrary structure of the elastic potential, which are twice continuously differentiate functions of components of Green's strain tensor. The investigation is carried out in general form for compressible and incompressible bodies. Questions associated with contact problems for bodies with initial stresses were examined in /1–4/ for particular forms of the elastic potential. Problems on the vibration of a rigid stamp on the surface of an initially stressed half-space and cylinder were examined from the aspect of the linearized theory of elastic wave propagation /7/ in /5, 6/. Contact problems for bodies with initial stresses were investigated within the framework of the linearized theory /7/ in /8, 9/ for an arbitrary structure of the elastic potential in general form for the theory of large (finite) initial strains and different modifications of the theory of small initial deformations. A formulation is given in /8, 9/ for contact problems for elastic bodies with initial stresses and torsion contact problems are examined. A number of contact problems for an elastic half-plane with initial stresses is examined in /10–12/ by dusing complex potentials of plane static linearized problems /13, 14/. Investigations are performed in general form for compressible and incompressible bodies. Complex potentials are introduced /15–18/ for plane dynamic problems and the plane dynamic contact problem for a prestressed half-plane is solved on their basis /19, 20/, when the initial problem allows transformation to a stationary problem in a moving coordinate system. The complex potentials introduced in the absence of initial stresses reduce to the complex S.G. Lekhnitskii potentials /21/ for an orthotropic linear elastic body in the case of unequal roots of the governing equation, and into the complex Kolosov-Muskhelishvili potentials for an isotropic linear elastic body in the case of equal roots.