Periodic problem of concentrated forces in an elastic half-plane with holes.

Abstract

The plane problem of the theory of elasticity for the linearly elastic half-plane with an infinite periodical series of free-form holes bounded by identical smooth contours is investigated. The halfplane is considered to be subjected to the tension at infinity, periodic external load at the rectilineal boundary and at the surface of cavities. Periodic concentrated forces or different types of singularities are also supposed to be applied within the solid concerned. The periods of all the systems are taken to be equal. The problem has been formulated using the Kolosov-Muskhelishvili complex stress potential technique. The results have been obtained by the superposition of two auxiliary problems. The first of them is the problem of the intact semiplane (without holes) under given periodic outside load at the straight boundary, tension at infinity and under known periodic concentrated forces. The second one is the problem of the intact semiplane under unknown periodic inside load (applied within the body) to be defined. Applying formulas of summation of series to complex potentials for single and distributed forces at points within a half-plane, the solution for the periodic problem has been written. The solution found thoroughly satisfies boundary conditions at the straight-line border of the semiplane and at infinity. For the surface of the cavities resolving Fredholm integral equations of the first kind in unknown load have been derived. Further, using the concentrated fictitious forces, the problem has been directly reduced to the system of linear algebraic equations. The system of the equations solved, stress-components at any points within the body can be defined by Kolosov's formulas. A worked out example for the semiinfinite plane with elliptic cuts is presented. Calculations have shown that increasing the period of series of holes may lead to both rise and fall of stresses in the half-plane depending on external load.