

458 Mr. A. Mallock. Instability of India-rubber [Api 



II. " Note 011 the Instability of India-rubber Tubes and Balloons 

 when distended by Fluid Pressure." By A. MALLC 

 Communicated by LORD RAYLEIGH, Sec. R.S. Receive 

 March 16, 1891. 



When an india-rubber tube is expanded by internal fluid present 

 it preserves its cylindrical fom> until the increase in its diamett 

 bears a certain proportion to its diameter when unstrained ; but, whe 

 more fluid is introduced, the condition of the tube becomes unstabl 

 and the internal fluid pressure diminishes. 



When more fluid, therefore, is introduced into a length of tnl 

 than will suffice to expand it to its stable limit, it no longer rei 

 cylindrical throughout its length, but assumes the form of a cylindt 

 with one or more bulbous expansions ; and the diameter of the 

 which remains cylindrical, though greater, of course, than the ui 

 strained diameter, is less than that attained at the stable limit ii 

 fig. 1. 



FIG. 1. 



In the case of an elastic hollow sphere, although the spheric 

 form is retained, whatever be the amount of fluid introduced, there ifi 

 a similar limit to the pressure which the elastic reaction of its 

 can cause within it. 



If the thickness of the walls of the tube or sphere is small ooi 

 pared with the radius, and if, further, the material of which they 

 composed be considered as incompressible, while the other elastic 

 constants are invariable for such extensions as are involved (assnmj: 

 tions which are approximately true for india-rubber), the value of the 

 radius when instability begins, may readily be found. Taking 

 &c, cy, z as the sides of any small cube of the material of the walls, 

 (x and ?y being parallel to the tangent plane of the surface and 2f 

 normal to it, let a stretching force act in the direction of x causing & 

 to become pSx. 



Since the material is incompressible, y and :, under the 

 influence of this force, will become respectively (l/v'jOfy and 



