Mixed boundary value problems in the theory of elasticity in an infinite strip

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

“…In the exact solution constructed in this paper the type of boundary conditions changes at the point of intersection of two coordinate straight lines. In this case, as was shown in Kovalenko et al [18], a singularity at the corner points of the half-strip can appear only if the type of boundary conditions changes at them when continuing the solution from the right half-strip to the left one with different homogeneous boundary conditions on its long sides. The existence of a singularity will inevitably manifest itself in the fact that the functions biorthogonal to the Papkovich–Fadle eigenfunctions s x ( λ k , y ) , t xy ( λ k , y ) will have the corresponding singularities at y = ± 1 , which, in fact, as follows from Kovalenko et al [12], are absent.…”

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

“…If it changed, for example, if the solution from the right halfstrip were continued to the left half-strip with free long sides, then the characteristic function ( ) in Equations ( 8) would have as its zeros both zeros of the function (2) with Re < 0 and zeros of the entire function + sin cos (corresponding to the even-symmetric boundary value problem for a half-strip with free long sides) with Re > 0. In this case, the finite parts of the biorthogonal functions and, hence, the solution itself would have the same singularities at the ends of the segment | | ≤ 1 as does the solution for an infinite wedge (for more details, see [17]). 4.…”

“…Multiplying the first equation by ( ) and the second equation by ( ) and integrating along the contour 1 on the right and along an infinite straight line on the left, we will find the Lagrange coefficients and based on the biorthogonality relations (13) and (14). Substituting them into (17) gives the Lagrange expansions of the generating functions (in order to save space, below we will use the designation 2Re for the sum of two complex-conjugate expressions):…”

A strip with constant stresses on the cut: Exact solutions

“…In this case, two complete minimal systems of basis functions, whose union is not minimal and, consequently, has no biorthogonal system of functions, will be involved in the boundary conditions at the joint between the half-strips (on the right and the left). Following Kovalenko et al [12], the minimal system of functions is separated out from the nonminimal one by introducing two analytic functions. A biorthogonal system is constructed to it with which the unknown coefficients of the expansions into series in Papkovich-Fadle eigenfunctions can be easily found.…”

“…In this paper, following the methodology [12], the problem is considered as a problem of contact between two half‐strips, when a discontinuity of the longitudinal displacements is specified at their joint. The long sides of the strip are (a) free, (b) firmly clamped, and (c) have tensile–compressive stiffeners.…”