107 



THE DISTORTION UNDER PRESSURE OF A DIAPHRAGM 

 WHICH IS CLAMPED ALONG ITS EDGE AND STRESSED 

 BEYOND THE ELASTIC LIMIT 



G. I. Taylor 



September 1942 



PART 1. Theoretical considerations. 



It win be assumea that the plastic material has the ideal property that it aeforms 0/ a 

 negligible aimunt until the yield point is reached and that it subsequently flows plastically 

 without further increase in stress. In other words, it will be assumed that no strain hardening 

 occurs. 



In general the criterion for plastic flow is that sone function of the stress components 

 shall exceed a certain limit. Two such criteria have been extensively used, namely (i) that of 

 Mohr. according to which the material flows when the maximum stress difference exceeds a certain 

 limit, (il) that of von xises, according to which flow occurs when the sum of the squaras of the 

 principal stress differences exceeds a certain limit. " "'i aid a, are the principal stress 

 components parallel to the surface of tne diiphr^gm and both are positive (i.e. tensions), and 

 if both are large compared with the stress perpendicular to the diaphragm, then the two criteria 

 for plastic flow are:- 



a^ - f when a > a, [ 

 Mohr. (l) 



0-2='' when a2^ cr. J 



von Mises. '-'^i^ * "'2 " "^l '''2 " ''^ '2) 



where P Is the tensile streigth as measured in a testing machine. 



Circular dia-phragm . 



If a^ is the radial stress and cr, the tangential stress, the equation of equillorium of 

 an element In the radial direction is 



where r is the radial co-ordinate. In the centre of the diaphragm symmetry alone requires that 

 CTj = o-j = P. If Mohr's criterion is accepted three alternatives are possible, either 



(i) a^ = f,o-^< P. This must be rejected because it is inconsistent with (3). 



(ii) o-j < P, °'i-''- In this case D) gives a, = a, 



(iii) o-j = o-j = P. 



Thus only the thi rd- alternative is possible. Similar reasoning gives the same result 

 if von Mises' criterion is accepted so that in either case ct^ = a, and tne stress distribution 

 is like that of a soap film or stretched renCran^". The sheet therefore assumes a spherical 

 form under the action of a uniform pressure p. If h is the displacement of the centre the 

 displacement perpendicular to the plane of the edge at radius r is 



, = ,^ (1 _ r2/r^2) („ 



