Papkovich-Fadle Eigenfunctions Research Articles

EXACT SOLUTIONS TO INHOMOGENEOUS BOUNDARY VALUE PROBLEMS OF THE THEORY OF ELASTICITY IN A RECTANGLE

A method is proposed for constructing exact solutions to boundary value problems of the theory of elasticity in a rectangle with stiffeners located inside the region (an inhomogeneous problem). The solutions are represented by series in Papkovich–Fadle eigenfunctions, the coefficients of which are determined explicitly. The method is based on the Papkovich orthogonality relation and the theory of expansions in Papkovich–Fadle eigenfunctions developed by the authors in homogeneous boundary value problems of the theory of elasticity in a rectangle (the biharmonic problem). The solution sequence is demonstrated with the example of an even-symmetric problem for a rectangle the sides of which are free, and an external load acts along the stiffener located on the axis of symmetry of the rectangle.

A strip with constant stresses on the cut: Exact solutions

AbstractWe construct exact solutions of three boundary value problems in the theory of elasticity for an infinite strip with a central transverse cut on which constant normal stresses are specified (even‐symmetric deformation). We consider three variants of homogeneous boundary conditions on the strip sides: (1) free sides, (2) firmly clamped sides, and (3) there are identical stiffeners on the strip sides. The solutions of all problems are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple closed formulas.

Examples of exact solutions to boundary value problems of the theory of elasticity for a truncated wedge

The paper presents examples of exact solutions to boundary value problems of the theory of elasticity for a truncated wedge. The straight sides of the wedge are free, and normal stresses are given on its circular “end” (the tangential stresses are considered equal to zero). The solutions are sought in the form of expansions in Papkovich–Fadle eigenfunctions. The coefficients of the expansions are determined in a simple explicit form.

An exact solution for a partially clamped rectangle with a crack

The article deals with a boundary value problem for a rectangle whose horizontal sides are rigidly clamped, and the ends are free. In the centre of the rectangle, a vertical cut is made on which a discontinuity of the longitudinal displacements is given. An exact solution to the problem is constructed in the form of series in Papkovich–Fadle eigenfunctions. First, the corresponding boundary value problem for an infinite clamped strip is solved, then the solution for a rectangle is superimposed on this solution, with the help of which the boundary conditions at its ends are satisfied. Examples are given in which discontinuities of three types are considered which differ in the smoothness of the discontinuity contour near its ends.

A temperature problem for a square: An exact solution

We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this solution. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas.

Exact Solution for a Rectangle with Rigidly Clamped Horizontal Sides

In this paper, in the form of expansions in Papkovich–Fadle eigenfunctions, an exact solution to the boundary value problem of the theory of elasticity is constructed for a rectangle whose horizontal sides are rigidly clamped. The expansion coefficients are determined with the help of biorthogonal functions by simple formulas.

Thermoelastic Problem for a Free Square Plate: Exact Solution

The paper presents a method for determining thermal stresses in an elastic free square plate (the plane problem). First, we construct the solution to the nonhomogeneous temperature problem for an infinite plane. Then, we add to this solution the solution for a square, with the help of which the required boundary conditions on its sides are satisfied. The temperature factors are obtained in the form of explicit series in Papkovich–Fadle eigenfunctions.

Thermal Stresses in an Elastic Clamped Square: Exact Solution

This paper presents a method for determining thermal stresses in an elastic clamped square with a given temperature distribution (the plane problem). First, the solution to the temperature problem for an infinite plane is constructed. Then, the solution for a square is added to this solution, with the help of which the boundary conditions on its sides are satisfied. The thermal stresses have been obtained in the form of series in Papkovich–Fadle eigenfunctions, the coefficients of which are determined explicitly. The final formulas are simple and can easily be used in engineering.

Exact Solution for an Elastic Square Plate Loaded with Tangential Stresses

This paper provides an exact solution to the boundary value problem of elasticity theory for a square loaded with tangential stresses along all its sides (even-symmetric deformation with respect to the central axes). The solution is represented as series in Papkovich–Fadle eigenfunctions. The coefficients of the series are determined by simple closed formulas.

Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half‐strip with stiffeners

AbstractWe have constructed an exact solution to the boundary value problem in the theory of elasticity for a half‐strip with identical stiffeners along its long sides and a load acting at its end (even‐symmetric deformation). The solution is represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from closed formulas. The solution includes two easily variable parameters that characterize the relative tensile‐compressive and flexural rigidities of the stiffeners. The final formulas for the stresses and displacements in the plate as well as for the longitudinal and transverse forces and bending moment in the stiffeners are simple and can be easily used in engineering.

An inhomogeneous problem for an elastic half-strip: An exact solution

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

A boundary value problem in the theory of elasticity for a rectangle: exact solutions

We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas. The obtained formulas remain the same for other types of homogeneous boundary conditions, for example, when the horizontal sides of the rectangle are firmly clamped, have stiffening ribs that work in tension–compression and/or bending, etc. Obviously, in this case, the Papkovich–Fadle eigenfunctions will change, as well as the corresponding biorthogonal functions and normalizing factors. To solve a specific boundary value problem, it is enough to find the Lagrange coefficients, which are determined from simple formulas, as Fourier integrals of boundary functions specified at the ends of the rectangle, and then substitute them into the necessary formulas. Examples of solving two problems (even-symmetric deformation relative to the central coordinate axes) are given: (a) The normal stresses are known at the rectangle ends, and the tangential ones are zero; and (b) the longitudinal displacements conditioned by the action of some normal stresses are known at the rectangle ends (the tangential stresses are zero). These solutions are compared with the known solutions in trigonometric Fourier series. The basis of the exact solutions obtained is the theory of expansions in Papkovich–Fadle eigenfunctions based on the Borel transform in the class of quasi-entire functions of exponential type (developed by the authors in their previous studies).

Inhomogeneous Problem of the Theory of Elasticity in a Half-Strip. Exact Solution

In the paper, in the form of series in the Papkovich–Fadle eigenfunctions, exact solutions of two inhomogeneous boundary value problems of elasticity theory for a half-strip with free long sides are constructed: (1) the half-strip end is free, (2) the half-strip end is rigidly clamped. To solve the inhomogeneous problem in a free strip, the Papkovich orthogonality relation is used. The exact solution for the half-strip is then added to it.

Formation of Discontinuities in Rectangular Plates as a Result of Residual Stress Relief

The paper deals with the problem of relieving residual stresses in an elastic domain of rectangular shape with free sides as a result of the formation of a discontinuity of particular shape. First, we construct the solution to the problem of residual stresses in an infinite strip with free sides and with a central transverse cut on which a discontinuity of displacements is known. Then, the solution for a rectangle is added to this solution, with the help of which the boundary conditions at the ends are satisfied. The formulas for the residual stresses and for the corresponding displacements are represented in the form of series in Papkovich-Fadle eigenfunctions. The expansion coefficients (Lagrange coefficients) have the form of simple Fourier integrals.

Eigenfunction Expansion for the Elastic Rectangle

In the paper, we construct an exact solution to a boundary value problem of the theory of elasticity for a rectangle in which the longitudinal sides are free, while normal and tangential stresses are given at the ends (even-symmetric deformation with respect to the central axes). The solution is represented in the form of series in Papkovich–Fadle eigenfunctions. The coefficients are determined explicitly by using functions biorthogonal to the Papkovich–Fadle eigenfunctions. We give the final formulas which have a simple appearance and can easily be used in engineering practice. The obtained solution is compared with the solution to the corresponding boundary value problem for a half-strip.

Two Nonhomogeneous Boundary Value Problems for a Rectangle: Exact Solutions

In the paper, for the first time we give exact solutions to two nonhomogeneous boundary value problems of the theory of elasticity for a rectangle with free long sides. Inside the rectangle there are applied two equal concentrated forces directed oppositely along the horizontal axis (even-symmetric deformation). The method of solution is based on the use of the solution to the biharmonic problem for a smooth semi-strip and the method of the integral Fourier transform. In the first problem, the short sides of the rectangle are free; in the second, they are rigidly clamped. The solutions to both problems are constructed on the superposition principle in the form of the sum of integrals and series in trigonometric functions and Papkovich–Fadle eigenfunctions. The coefficients of these expansions are determined by simple formulas as the Fourier integrals of given boundary functions.

Method of initial functions and integral Fourier transform in some problems of the theory of elasticity

In the 1950s, the method of initial functions (MIF) was developed in the Soviet Union. It rapidly became popular among research scientists, civil engineers, and, later, among strength engineers engaged in the aerospace industry. Since the MIF is known comparatively little to the Western reader, a brief overview of the publications devoted to the formation and development of the MIF and its application to the solution of various engineering problems is given in the Introduction. In this paper, the MIF is considered in the space of Fourier transforms. This allows its use to be facilitated still further by reducing the solution of boundary value problems to simple algebraic transformations. The final solutions are represented either as improper Fourier integrals or as expansions into series in eigenfunctions of the boundary value problem, Papkovich–Fadle eigenfunctions. Using various examples, we show the basic techniques of working with the MIF in the space of Fourier transforms. In the final section, the method is applied to the solution of problems for a plane with displacement discontinuities. The solutions are obtained rapidly, easily, and, in contrast to the classical solution, without using the theory of functions of a complex variable.

Exact solutions of boundary value problems in the theory of plate bending in a half-strip: basics of the theory

Using the boundary value problem on the bending of a thin elastic semi-infinite plate in which the long sides are free, while a self-balanced bending moment and a generalized shearing force are specified at its end as an example, we consider the main steps in constructing exact solutions to the boundary value problems of bending of thin elastic rectangular plates. The solutions are constructed in the form of series in Papkovich–Fadle eigenfunctions. The unknown expansion coefficients are determined in the same way as in classical periodic solutions in trigonometric series and have the same structure, i.e., are expressed via the Fourier integrals of the boundary functions specified at the half-strip end. The systems of functions biorthogonal to the Papkovich–Fadle eigenfunctions constructed here are used in this case. The exact solutions possess properties that are not inherent in any of the known solutions in the theory of plate bending. Some of them are discussed in the paper. The final formulas describing the exact solution of the boundary value problem are simple and can be easily used in engineering practice. The results that we have obtained previously when solving the boundary values problems of the plane theory of elasticity in a rectangular region form the basis for our work.

Two Problems for a Strip with a Transverse Crack: Exact Solutions

In this paper for the first time we have constructed the exact solutions of two boundary value problems of the theory of elasticity for an infinite strip with a central transverse crack on which a constant normal stress is given (even-symmetric deformation). In the first problem the sides of the strip are free, while in the second they are rigidly clamped. The solution is represented in the form of series in Papkovich–Fadle eigenfunctions. The expansion coefficients (Lagrange coefficients) have the form of simple Fourier integrals. The final formulas are simple and can easily be used in engineering.

On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip

We have constructed the solution of the first basic odd-symmetric boundary value problem in the theory of elasticity in a half-strip with free longitudinal sides. The solution is represented as series in Papkovich–Fadle eigenfunctions whose coefficients are found in an explicit form by using functions biorthogonal to the Papkovich–Fadle eigenfunctions. It is shown that the obtained solution describes residual stresses in an infinite strip with zero boundary conditions on its sides and the displacements that arise when the residual stresses are released as a consequence of the formation of a discontinuity. The same formulas give the exact solution of a boundary value problem for the half-strip with stresses specified at its end in the traditional statement, and only the displacements should now be taken with the opposite sign. In the constructed solutions, the angular points have the properties of infinitesimal elements, where, for the uniqueness of the solution, the boundary functions must be specified together with all their derivatives. In this, they are different from an angular point in an infinite wedge. The final formulas are simple and can easily be used in engineering.