Plane strain problems in second-order elasticity theory

The equations of second-order elasticity are developed in polar coordinates R, θ for plane strain deformations of incompressible isotropic elastic materials. By considering a ‘displacement function’ the second-order problem is reduced to the solution of an equation of the form ⁴ψ = g(R, Θ) where ² is Laplace's differential operator and g(R, Θ) depends only on the first-order solution. The displacement function technique is then applied to obtain a second-order solution to the problem of an elastic body contained between two concentric rigid circular boundaries, when the outer boundary is held fixed and the inner is subjected to a rigid body translation.     

The axisymmetric contact problem of the theory of elasticity for a half-space in the presence of body forces

We consider the axisymmetric problem of determination of the stress-strain state in an elastic half-space in the case of a circular line of separation of the boundary conditions on the boundary plane z = 0. We assume that on the entire boundary z = 0 the tangential stress _rz = 0, while inside the circle r a (z = 0) the normal displacement u_z is known and in its exterior the normal stress _z is given. In addition, we assume that body forces are acting in the half-space. The investigation of problems of similar kind presents interest in connection with the application of A. A. Il'yushin's method of elastic solutions to the problem of the indentation of punches into a nonlinear-elastic, in particular, into an elastoplastic half-space.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 110–115, July–August, 1971.The author wishes to thank M. Ya. Leonov for his valuable suggestions during the preparation of this paper.     

Nonlinear laws of dry friction in contact problems of linear theory of elasticity

A problem of elastic body deformation with conditions of dry friction imposed at the boundary is considered. Various friction laws are studied, including linear oneparameter and nonlinear twoparameter laws. A general view of a nonlinear function with two constants is suggested, which determines the friction force as a function of normal pressure. The problem of elastic plate compression by rough infinite plates for a variety of friction conditions on the contact surfaces is solved. The plate equations are employed, which make it possible to specify arbitrary conditions on the front faces without reducing the order of differential equations. The unknown boundary of ideal contact zones and sliding zones is determined. Solutions obtained by using various friction conditions on the contact surfaces are compared.     

Solution of contact problems of elasticity theory for an anisotropic body by the method of similarity

Moscow Institute of Radio Engineering, Electronics, and Automation. Translated from Prikladnaya Mekhanika, Vol. 26, No. 7, pp. 17–23, July, 1990.     

A method of solving plane problems of the moment theory of elasticity for multiply connected regions

Summary The proposed method can be used to investigate problems for plates weakened by a finite number of arbitrarily located curvilinear openings at given stresses or displacements, and also in the presence of reinforcing rings or elastic cores.Without serious modification the method can also be used by analogy with [4] and [5], for the solution of periodic, doubly periodic and cyclosymmetric problems.Prikladnaya Mekhanika, Vol. 2, No. 1, pp. 3–19, 1966