boundary value problem, strip, half-strip, displacement discontinuity, Fourier series.

The article is devoted to solutions in trigonometric series of four boundary value problems of the theory of elasticity for an infinite horizontal strip with a vertical section. On a section can be set discontinuities longitudinal or transverse displacements (symmetric and antisymmetric deformations). On the sides of а strip the periodicity conditions are performed: shear stresses and transverse (perpendicular to the axis of a strip) displacements are equal zero. Unknown expansions coefficients are found from the conditions of a joint on the section of two analytical functions to the right of the section (in the right half-strip) and the corresponding two analytic functions on the left of the section (in the left half-strip). These functions were first introduced in work [1] and then were used, in particular, at the solution of boundary value problems for a rectangle with free sides and discontinuities longitudinal and transverse displacements [2]. It is shown that the solution for the strip with discontinuity longitudinal (along the axis of the strip) displacements is equivalent to solution for the half-strip, which are given at the end of the longitudinal displacements and zero shear stresses. And the solution for the strip with a transverse discontinuity equivalent to solution for the half-strip with specified at its end transverse displacement and zero normal stress. The examples illustrating behavior of stresses depending on smoothness of the curve near the tip of the discontinuity are reviewed. Solutions is simple and does not require knowledge of special branches of mathematics - just be aware of the Fourier series. The analysis of solutions shows that in sufficiently wide area, adjacent to the axis of the strip, and for not too long section received solutions can be used for preliminary estimates of stress state even in those cases, when on the longitudinal sides of the strip other boundary conditions are given, for example, when sides of the strip are free. 

1. Kovalenko, Mikhail Denisovich Dr. Sci. Phys. & Math., Professor, Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow

2. Menshova, Irina Vladimirovna PhD, Senior Researcher at the Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow

3. Kerzhaev, Alexandr Petrovich PhD, Senior Researcher, Laboratory of Geodynamics, Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow