Int. J. Mech. $ci. Vol. 32, No. 7, pp. 551-563, 1990 0020-7403/90 $3.00+ .00 Printed in Great Britain. © 1990 Pergamon Press pie ELASTIC-PLASTIC BEHAVIOUR OF A THICK-WALLED TUBE WITH GENERAL NONLINEAR HARDENING PROPERTIES M. M. MEGAHED Department of Mechanical Design and Production, Cairo University, Giza, Egypt (Received 20 June 1989; and in revised form 30 December 1989) Abstract The paper presents analytical solutions for the elastic-plastic behaviour of thick-walled tubes under internal pressure. The hardening law employed in the investigation is: a = Y + A. ~ in unia×ial form. Closed-form analytical solutions are developed for four particular cases of the hardening exponent n; n = i, i/2, I/3 and I/4. The solutions enable characterization of the tube behaviour in terms of a hardening function which incorporates both elastic and plastic constants of the material. Furthermore, an approximate solution which can be used for any value of the strain hardening exponent n is developed and it is shown how its validity can be verified once the constants describing material hardening are specified. a,b A C,D H I, I o P r X Y 8p r, O, Z NOTATION inner and outer radii of the tube material constant, equation (10) integration constants, equation (5a) hardening function defined by equation (1 lb) integrals defined by equations (5b) and (6d) respectively internal pressure generic radius = a/r an initial yield stress hardening functions defined by equations (13b), (14b), (15b) and (16b) respectively total and plastic strain components respectively effective plastic strain stress component and effective stress respectively subscripts denoting radial, hoop and axial directions respectively. INTRODUCTION Interest in the elastic-plastic behaviour of thick spherical and cylindrical shells under internal and external pressures, radial thermal gradients and body forces has never ceased [1-5]. Plasticity problems with spherical or cylindrical symmetry can usually be treated analytically particularly when simplifying assumptions are made regarding material hardening and the reader is referred to [1] for various interesting assumptions in this regard. Bland [2] developed analytical solutions for thick-walled tubes made from mater- ials with linear hardening properties and subject to internal and external pressures and to temperature gradients. Carroll [3] studied the problem of large radial expansions of spherical shells assuming incompressibility in both elastic and plastic domains and utilizing various forms of material hardening. Gamer [4] considered a hardening law similar to the one employed here and developed analytical solutions for the elastoplastic behaviour of a thick spherical shell under internal pressure using two values for the exponent n, viz; n = 1/2 and n = 2. It is rather strange to take the exponent n = 2 since it is known that n is usually less than unity for engineering metals. In the present work, the elastic-plastic behaviour of a thick cylindrical shell under internal pressure is considered assuming that the shell material hardens according to a = Y + A. e~, with n < 1. The problem is formulated in a manner which isolates the effect of plastic strains and hence analytical treatment of higher order of nonlinearity is made possible. Also, compressibility is retained in the elastic range. Firstly, analytical solutions are developed for the radial distributions of plastic hoop strain, radial and tangential 551