Extended orthogonality relationships for some problems of elasticity theory: PMM vol. 34, n≗5, 1970, pp. 945–951

Abstract

In connection with the solution of boundary-value problems of the theory of elasticity for an orthotropic strip the problem of expanding two different limiting functions in series in terms of the characteristic elements of the generalized eigenvalue problem is considered. A system of functions biorthogonal to the system of characteristic elements is constructed. The double completeness of characteristic elements is proved. It is shown that the biorthogonality condition is equivalent to a generalized orthogonality relation of Papkovich type. The form of systems of biorthogonal functions is established. For expansions of a special form the biorthogonal systems are identical with the systems of characteristic elements. Biorthogonal systems of functions are constructed corresponding to expansions of a general form. Using the biorthogonal systems obtained, explicit expressions for the expansion coefficients are found. An example demonstrating the existence of a non-trivial double null expansion is given.