APPLICATION OF SINGULAR INTEGRAL EQUATIONS IN SOLVING SOME CONTACTPROBLEMS IN THEORY OF ELASTICITY

Abstract

In this paper the modification can be used to find an expression of the unbounded contact stress in the theory of elasticity. The vertical vibrations of a punch lying on an elastic isotropic rectangle under harmonic force are considered. This mixed boundary value problem is reduced to a discrete Riemann problem connecting the Fourier components of the required extension of the unknown functions. Separation of the singular part of the discrete problem leads to the Hilbert type singular integral equation, rather than of the Cuchy’s kernel one.The unknown function in this integral equation is normal contactstress. Thus, it is possible to search for the physically interesting unbounded solution. The inversion of the integral equation provided an expression of that solution in terms of its Fourier components. To complete the definition of this expression, that is, to determine these Fourier components presented therein, it is further reduced to an infinite system of algebraic equations through application of the Fourier transforms. The solution of this system completes the definition with the aid of the physical conditions. The solution of this system can be obtained approximately by means of the truncation. The truncation of the algebraic system of equations is justified by using the corresponding theorem. It is supposed that the appropriate homogeneous systemhas only trivial solution in some space. Then the infinite system has a unique solution. The truncated system has a unique solution and the error is estimated. We assume that the frequency differs from those values for which the homogeneous system corresponding to the given system of equations has nontrivial solutions. On choosing the number of equation and using the formula for determination of the contact stress we can get an approximate solution of the contact problem for rectangle to find a contact stress up to any prescribe accuracy. The resonance frequencies are the real roots of some resonance equations. In the dimensionless frequency, time and coordinate the table of the values of the contact stress corresponding to different values of coordinate is given. The values of the contact stress increase unboundedly at the vicinities of the end points of the contact interval.