About Kolovos’s formulas in in plane problem of the theory of elasticity in the presence of periodical cuts

Abstract

The solutions of the plane theory of elasticity in terms of the complex functions Φ(z) and Ψ(z) (two Kolosov’s formulas) are analyzed. The right side of the first of these formulas is the integral of the equation of indissolubility, and the right side of the second formula is the integral of two equations of equilibrium. Besides the classical form, two different versions of Kolosov’s relations are found. It is also shown that Westergaard’s formulas are the special case of Kolosov’s relations. The formulas for displacements are derived. The exact solution of the periodic problem for a plane with infinite number of rectilinear cracks on the horizontal line OX is obtained. It is assumed, that stresses vanish at infinity and the opposite crack edges are loaded with normal concentrated forces. The analytical solution of crack theory exists, if the following conditions are fulfilled: 1) the region with a straight crack is the boundless plane; 2) the main vector of external stresses applied to all cracks equals zero; 3) the stress and strain conditions are symmetrical regarding the axis OX. It should be noted, that solutions in functions Φ(z) and Ψ(z) (these functions are simple in the multiply-connected domain) are more preferable in comparison with solutions in terms of the multivalued function ϕ(z) and ψ(z).

Keywords:

theory of elasticity, complex variable, Kolosov’s formulas, mathematical theory of cracks