# A strip with constant stresses on the cut: exact solutions

The Wiley-VCH journal “Zeitschrift für Angewandte Mathematik und Mechanik” (Web of Science Q3, Scopus Q2, Impact Factor JCR: 1.103) published an article “A strip with constant stresses on the cut: exact solutions”. Among the authors are employees of the IEPT RAS: senior researcher, Ph.D. A.P. Kerzhaev and senior researcher, Ph.D. I.V. Menshov.

We 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.

Boundary value problems in the theory of elasticity for an infinite strip with various boundary conditions on its sides and with a central transverse cut on which constant normal stresses are specified have been the subject of numerous studies. However, as far as we know, no exact solutions have been constructed. Almost all of the solutions were initially reduced to singular integral equations and subsequently to an approximate solution of the corresponding infinite system of algebraic equations.

In this paper, following the methodology discussed in previous publications by the authors, 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. All of the solutions are represented by series in Papkovich–Fadle eigenfunctions that exactly satisfy the specified homogeneous boundary conditions on the strip sides. 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). The minimal system of functions is separated out from the non-minimal 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.

It can be noticed that the curves corresponding to the solution for a strip with stiffeners are intermediate between the solutions for free and firmly clamped strips.

As the crack length (i.e., the parameter a) increases, the convergence of the series for the stresses deteriorates, which is particularly clearly seen at small |x|. Furthermore, at x very close or equal to zero and significant a (beginning approximately from a > 0) a high accuracy of determining the eigenvalues λk is important. At a ≤ 0.5 the differences between p and sx(0.01) are very small.

We considered a simplified model of a stiffened strip with a transverse crack that has tensile-compressive stiffeners. The flexural rigidity of the stiffener can also be taken into account.

The research was carried out with the support of the Russian Foundation for Basic Research in cooperation with the National Natural Science Foundation of China (project No. 20-51-53021).

Source: Kovalenko, M.D., Menshova, I.V., Kerzhaev, A.P., Yu, G.: A strip with constant stresses on the cut: exact solutions. Z. Angew. Math. Mech. e202100431 (2022).

DOI: 10.1002/zamm.202100431.