The method of volterra integral equations in contact problems for thin-walled structural elements

Abstract

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) The properties of the statement and solution of contact problems for thin-walled structure elements as functions of the theory that describes their stress-strain state have been noted by many authors. Different theories and methods have been used in the investigation: hypothetical methods, the operator method, the method of expansion in polynomials. In specific problems the moment-free stressed state has been studied, shear strains and reducing have been taken into account, asymptotic methods and the method of piecing solutions together have been used, etc. The absence of a unified approach causes certain difficulties in the analysis and comparison of the results. In the present paper we reduce the solution of contact problems on the interaction of rigid bodies with thin-walled elements to Volterra integral equations.Suppose that a rigid smooth die with a base described by the function f(x) (Fig. 1) strikes with force P a plate whose edges are restrained. We denote by 2a the region of contact, by 2h the thickness of the plate, and by et the displacement of the die. To describe the stress-strain state of the plate we use the model of a plate taking account of shear strain and normal reducing, whose relations are obtained by approximation of the state characteristics by Legendre polynomials [1]. The material is transversally isotropic, and the model of approximation is N = 1.In accordance with the approximation just adopted we have the following expressions for the displacements and stresses, equilibrium equation, and elasticity relations:

Waer of composite thin-walled structural elements with consideration of thermal effects

“…According to [11], its solution has the form (f(x) = lx2/2R) It should be noted that in the first step we have an integral equation similar to Eq. (18), whereas in solution (19) the quantity ;~" must be replaced by ~."…”

Specific Features of the Contact Interaction and Wear of Thin-Walled Structural Elements

“…The same method was also used to obtain the equation of nonstationary heat conduction for thin transversely isotropic plates with mixed boundary conditions on the faces. The indicated equations are basic for the construction of the mathematical models and methods aimed at the solution of the problems of contact interaction with wear of the material [13,27,47,64,69,74,81].…”

A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation

“…Volterra integral equation aries in many physical applications, e.g., heat conduction problem [1], concrete problem of mechanics or physics [2], on the unsteady poiseuille flow in a pipe [3], diffusion problems [4], electroelastic [5], contact problems [6], etc.…”

“…Consider the Volterra integral equation of the second kind(6) . Assume that k(x, t) is continuous on the square [0, 1] 2 , and the solution of the equation belong to (C a T…”