NATIONAL TRIBOLOGY CONFERENCE 24-26 September 2003 THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS“ OF GALAŢI FASCICLE VIII, TRIBOLOGY 2003 ISSN 1221-4590 345 INTERACTION BETWEEN TWO IDENTICAL CIRCULAR HOLES IN AN ELASTIC PLAN STRETCHED AT INFINITY Stelian Alaci, Emanuel Diaconescu, Ilie Musc ă University "Ştefan cel Mare" of Suceava, Romania stalaci@yahoo.com ABSTRACT The paper presents a comparison between analytically, experimentally and numerically determined stress states, produced in an elastic plane stretched at infinity, which contains two holes. The holes have equal radii and do not overlap. KEYWORDS: elastic plane, two circular holes, stress concentration factor. 1. THEORETICAL REMARKS In his book [1], Savin states that this problem was solved by Ling and gives the formulae for the stresses. But, these stresses do not satisfy boundary conditions. In this paper, authors solved the problem using bipolar co-ordinate, α and β. The relations for changing co-ordinates are: x a ch y a sh ch = − = − sin cos ; cos β α β α α β (1) where a is a positive constant. To the stress function for the plane without holes, written in bipolar co- ordinate and expanded in Fourier series, U 0 ( , ) α β , an auxiliary potential, χ α β ( , ) was added. The form of auxiliary potential was given by Jeffery, [2]. χαβ α α β α β α β α α β α α β α α α α β α ( ) [ ( ) cos( )] ln[ ( ) cos( )]}[ ( ) cos( )] [ ( ) ( )] cos( ) [ ' ( ) ' ( )] sin( ) [ [( ) ] [( ) [( ) ] [( ) ] cos( ) [ ' [( ) ] ' , = − + + − − + + + + + + + + + − + + + − + + + B ch K ch ch A ch B C sh A ch C sh A ch k B ch k C sh k D sh k k A ch k B ch k k k k k k 0 1 1 1 1 2 2 2 2 1 1 1 1 1 [( ) ' [( ) ] ' [( ) ] sin( ) k C sh k D sh k k k k k − + + + −               = ∞ ∑ 1 1 1 2 α α α β (2) From total potential U U = + 0 χ one can obtain the stresses: a ch sh ch U σ α β ∂ ∂β α ∂ ∂α β ∂ ∂β α α = − − − +         ( cos ) sin 2 , a ch sh U σ α β ∂ ∂α α ∂ ∂α β ∂ ∂β β β = − − − +       ( cos ) sin cos 2 , a ch U σ α β ∂ ∂α∂β α =− − ( cos ) 2 , (3) which must satisfy boundary conditions: σ τ αα α αβ α α = = = = 0 0 0 0 ; ; σ τ αα α αβ α α =− =− = = 0 0 0 0 ; , (4) where α α = ± 0 are equations of hole contours. As shown by Jeffery, boundary conditions may be required directly on the stress function U, because the hole contours are unloaded For photoelasic studies it is needed to calculate the stresses in Cartesian co-ordinate. For this propose the following formulae are to be used: σ α β σ α β σ α β α β τ α β α β β x a sh ch sh ch ch = + − − − −         − ( ) sin( ) [ ( ) cos( ) ] ( ) sin( )[ ( ) cos( ) ] [ ( ) cos( )] 2 2 2 2 1 2 1 , σ α β σ α β σ α β α β τ α β α β β y a ch sh sh ch ch = − + + + −         − [ ( ) cos( ) ] ( ) sin( ) ( ) sin( )[ ( ) cos( ) ] [ ( ) cos( )] 1 2 1 2 2 2 2 , τ α β α β σ σ α β α β τ α β α β αβ xy sh ch ch sh ch = − − − − − −         − ( ) sin( )[ ( ) cos( ) ]( ) {[ ( ) cos( ) ] ( ) sin( ) } [ ( ) cos( )] 1 1 2 2 2 2 . (5) Three situations have been considered: plane stretched along the centre axis, normal to this axis and hydrostatic. For these cases, patterns of principal stress are presented: