Periodic System of Closely Spaced Holes in Elastic Plane

Abstract

The sixth chapter deals with the periodic elasticity theory problem for a plane weakened with an infinite row of closely spaced identical curvilinear holes. Stress concentration factors in the tips of bilateral parabolic or rounded V-shaped notches were found for a limiting case of infinitesimal hole spacing. These results are compared with known expressions for hyperbolic notches. Using the limit transition to zero tip-rounding radius, a solution is derived for bilateral sharp V-shaped notches. Periodic problems of elasticity theory for a plane weakened with an infinite row of closely spaced identical curvilinear holes had been studied by many researchers [8, 9, 14, 15, 16, 17, 18, 19]. However, numerical results concerning stress concentration factors were obtained mainly for far enough spaced circular or elliptical holes. This situation was caused generally by the strong stress concentration at contours of closely spaced holes. It is known that the stress concentration creates great difficulties of computational nature in studying the stress distributions. Nevertheless, up-to-date computing techniques and computer hardware allow for numerically determining both the order of maximal stress singularity and the factor at the singularity for closely spaced holes with different geometry. Understanding the stress singularity is of great importance in deriving the direct numerical methods for solution of such problems. This knowledge is useful also in constructing numerical solutions of many other problems based on limit transitions. This chapter outlines the method of singular integral equations in application to solution of the periodic problem of elasticity theory for a plane containing an infinite row of closely spaced curvilinear holes [6, 25, 26]. We have derived stress intensity factors in sharp hole tips using the unified approach to stress concentration in sharp or rounded hole tips [22, 24] and starting from solution to the problem for a smooth boundary contour. Again using the limit transition, we have found stress concentration and stress intensity factors in rounded and sharp tips of bilateral curvilinear notches in elastic plane as well.