Journal of Mechanics, Vol. 32, No. 1, February 2016 1 ON THE BOUNDARY VALUE KIRSCH’S PROBLEM D. Rezini  A. Khaldi Y. Rahmani Université des Sciences et de la Technologie Mohamed Boudiaf Oran, Algérie ABSTRACT Analytical closed-form solution to the stress distribution associated with a hole in finite plates subject- ed to tension has not been obtained yet. Wherefore, a method developed in this paper is based on a Bel- trami-Michell methodology analyzing the Kirsch’s problem under finite dimensions conditions of both plane stress and plane strain. This aimed ability is achieved by combining the Beltrami-Michell plane equations, isochromatic information on the boundaries only; and the finite difference method into an ef- fectual hybrid method for analyzing rectangular plates of finite width with circular holes. Furthermore, the Beltrami-Michell methodology suggested may be applied on other plate and cut-out forms. Keywords: Kirsch’s problem, Dirichlet, Photoelasticity, Hybrid method. 1. INTRODUCTION Analytical and experimental analysis showed that near openings in a loaded flat member a peak stress occurs, which reaches much larger magnitudes than does the average stress over the section. Therefore, a stress field disturbance occurs round about. First of all, Kirsch [1] published a pioneering work on the stress state of a uniaxially tensioned elastic plane which is weakened by a circular hole. Acknowledged as Kirsch’s problem, this work was the beginning of a new direction in the plane theory of elasticity. His ideas were further developed by the Russian mathematician Kolosov [2] who initiated the stress concentration study around arbitrary shaped holes and size in elastic medi- um, which is subjected to diverse loading conditions. Hereafter, Inglis [3] accomplished the work of Kolosov and has obtained an analytical solution associated with an elliptical hole. Finally, two further important problems are treated, which are fundamental ideas of fracture mechanics. Earlier on, Howland [4] analyzed the stresses in the neighbourhood of a circular hole, and he suggested a formulation for calculation of the stress concentration factors of an isotropic strip with circular holes. Over the years different techniques have been developed for the study of such weakened plates. Thereafter, the stress distribution was studied around both circular (Durelli et al. [5]) and elliptical disconti- nuity (Durelli et al. [6]) using, among others, photoe- lasticity. Additionally, much experimental work re- lated to this topic has contributed to validate the ana- lytical solutions [7]. Most recently, for other shapes of holes Kawadkar [8], for example, proposes an evalua- tion of stress concentration in plate with cut-out and its experimental verification. Furthermore, a study on the effect of stress concentration on cut-out orientation of plates with various cut-outs and bluntness was con- ducted by Mohan [9]. Numerous of papers can by cited concerning solutions for stress concentration around holes [10,11]. Studies of elastic stress in infi- nite plate weakened by a single circular hole have been published by Singh [12], in comparison with those weakened by multiple circular holes [13]. Further- more, finite element analyses are performed to investi- gate elastic stresses around holes in plate subjected to uniform tensile loading, as published by Shastry et al. [14]. Semi-analytical and experimental work also exists for finite width plates [15], and for finite thick- ness plates. Indeed, on the basis of the plane strain theory, Kotousov [16] presented analytical solutions for a wide range of thickness to radius ratios around typical stress concentrators in an isotropic plate with arbitrary thickness. However, Yang et al. [17] showed that the location of stress raiser occurs on the middle surface only in thin plates and it moves away from the mid plane of plate by increasing the plate thickness. To abbreviate this bibliographical review, an exhaustive review on stress analysis of an infinite plate with cut-outs is edited by Dharmin et al. [18], where contri- butions of work from several researchers have been recapitulated. Besides, an extensive collection of stress concentration factors, in table-forms ready for use, is provided in textbooks, as compiled by Pilkey [19]. There are various theories and techniques commonly used for this purpose, some of which have been listed above. Some problems have been solved using series method [20], as introduced early by Love [21]. Abdou and Monaquel have used an integral method to deter- mine the stress components of stretched infinite plate with a curvilinear hole [22]. Various numerical pro- * Corresponding author (redjellou@yahoo.fr) DOI : 10.1017/jmech.2015.93 Copyright © 2015 The Society of Theoretical and Applied Mechanics, R.O.C.