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The method of initial functions and fourier integral transform in the boundary value problem of the elasticity theory for an infinite strip

Kerzhaev A.P., Kovalenko M.D., Menshova I.V.

#### Abstract:

In the 50th years of the last century by two outstanding scientists-mechanics Vasily Zakharovich Vlasov and Anatoly Isaakovich Lurie independently proposed a method of reducing the solutions of differential equations to ordinary differential equations of infinite order, later called the method of initial functions. In its final form the method of initial functions is described in the book of V.V.Vlasov [1]. Method of initial functions is very convenient for solution various boundary value problems, since most of the intermediate computations is already done and included in the statements of the method. Method of initial functions are widely used and applied in engineering calculations (see e.g. reference [1]). Development and generalization of the method of initial functions devoted several original works, in particular, the unique work Agarev V.A. [2]. In this article, the method of initial functions recorded in the space of Fourier transforms applies to the solution of the boundary value problem for infinite strip. Final formulas for stresses and displacements can be represented as in the form of improper integrals – inverse Fourier transforms and by the residue theorem in the form of a series of functions of Fadle-Papkovich. The last representation is then convenient to use in solution boundary value problems in semi-strip or rectangle, overlaid on the solution in an infinite strip corresponding solution for the semi-strip (rectangle) with homogeneous boundary conditions on the sides [3,4]. The method for solving boundary value problems of elasticity theory for an infinite strip with the help of the Fourier transforms is well known [5]. Using the method of initial functions in Fourier transforms space allows, in contrast to [5], to make the scheme boundary problem solution is formally independent of the type of boundary conditions on the longitudinal sides of the strip (stress, displacement, or the boundary conditions of mixed type, but without points of change of boundary conditions type). Method of initial functions operators in the space of Fourier transforms are algebraic expressions that are easy to use symbolic transformations MathCad. Therefore, the boundary problem of multilayered plates with different physical and mechanical properties of plates, with large sets of longitudinal stiffeners, etc. the proposed unit, based on symbolic MathCad mathematics, extremely easy to use, and sometimes simply irreplaceable.

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