SURFACE AND OTHER EFFECTS 



409 



if the radius of curvature r is assumed independent of the direction of a 

 tangent Hne to the surface. The normal stress thus increases with the 

 curvature. If a circular diaphragm of radius a is assumed to be de- 

 formed into a spherical profile, as shown in Fig. 10.5b, the radius of 

 curvature is for small central deflection Zc given by a^/22c, and the nor- 

 mal stress becomes 



(10.11) 



4:(roh 



Zc 



(0) STRESSES IN (b) SPHERICAL 



AN ELEMENT DEFORMATION 



Fig. 10.5 Tangential and normal stress for a plate in "plastic tension." 



The plate is therefore in equilibrium with an outward normal pressure 

 equal to an- If this pressure is released the plate will not, however, re- 

 turn to its original undeformed profile, but will acquire a permanent set 

 less than the plastic deformation by the amount of elastic recovery from 

 the stress cr„. The initial displacement of the plate will also of course 

 be in the domain of elastic strain, but since the cases of interest involve 

 large plastic displacements, the elastic effects can be neglected in an 

 approximate treatment. 



The permanent plastic deformation is the result of plastic work Wp 

 done by the applied forces, and if the increase in area as a result of 

 tension doh is AA this work is given by 



Wp = (To/iAA 



