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

Abstract

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.