Spline-Interpolation Solution of One Elasticity Theory Problem

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We present the methods of approximate solution of the 3D basic problem of elasticity for the solids of the special types in this work. The classic formulation of the problem is the following: given the boundary displacements it should be possible to find the displacements in the whole elastic body which satisfy the equilibrium equations. This problem is named the second basic problem of elasticity [7]. There exist the well-known exact solutions of this problem in the symmetrical cases (e.g. the solids of revolution with the symmetrical stresses) [6]. The exact solution for the general case has not been found yet, so the engineers apply the approximate methods (Finite Element Method, Boundary Element Method). The application of these methods for the solids with the certain singularities (e.g. cones in the neighborhood of the vertex) or asymmetrical boundary conditions often fail to be correct.

Note that Kolosov-Muskhelishvili method based on the application of the complex variables and analytic functions yields the exact solutions for the wide range of the plane problems [7]. There were numerous attempts of the generalisation of Kolosov-Muskhelishvili method for the 3 -dimensional solids, for example, by A.F. Tsalik, A. Alexandrov and F.A. Bogashov [14, 1, 3]. But their methods imply either solution of very large systems of symbol equations ([14, 3]) or transform the original problem to the different one (which is not equivalent to the given problem in the general case [1]). We might also recall the quaternion matrix representation method developed in [2], which also implies necessity of some complicated non-commutative calculations.....

This chapter is devoted to solution of the system of equilibrium equations. We present the general form of the solution as a polynomial in the variable h.

In this chapter we present the interpolation solution of the 3D second basic elasticity problem for the circular cylinder and explore some of its features. We present the solution of the first type which uses only the data on directrices of the cylindrical surface. Then we present the interpolation solution of the second type which uses the data at the ends of the circular cylinder.

In this chapter we present the interpolation solution of two elasticity problems for the tube: with the given displacements and with the given stresses at the exterior surface. We introduce some additional conditions at the interior surface in order to obtain the solution.

In this chapter we present the interpolation solution of the 3D second basic elasticity problem for non-circular cylinder. We apply the rational conform mapping in order to reduce the boundary value problem in a non-circular domain to the boundary value problem in the unit disk.

In this chapter we construct the spline-interpolation solution for the cylindrical solids.

In this chapter we present the spline-interpolation solution of the 3D second basic elasticity problem for a solid of revolution. The methods of gluing the displacements at the ends are analogous to that for the circular cylinder.

In this chapter we construct the spline interpolation solution of the second basic problem of elasticity theory for asymmetric solids.

The interpolation and spline-interpolation solutions of the 3D basic problem of elasticity are the approximate solutions which satisfy the given boundary displacements only at a subset of the boundary surface of an elastic body. So these solutions are compared with the exact solution in this chapter.

In this chapter we present the most natural generalisation of the interpolation and spline-interpolation solutions of the static problems to dynamic ones. At first we present the interpolation solution of the 2D dynamic problem. And for the 3D case we simply add the time variable to the solutions of static problems and construct the solution in the form of polynomial in two variables, namely h and t in each layer of solid M.

In the first appendix we give the thorough method for solution gluing on the common section of adjacent layers presented in general form in Chapter 2. The second appendix contains general solution of the 3D second basic elasticity theory problem for conoid of revolution with one shifted point at one of the bases. The main idea of this solution is given in Chapter 6. The third appendix gives the solution algorythm.

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