1021] on Elasticity 381 



constant distance from one another. It is evident that the volume 

 of the cube is unchanged by this process, but the lengths of the 

 diagonals of two of the faces are changed, one being shortened and 

 the other lengthened. 



If a square be inscribed on one of the faces before distortion 

 with its edges parallel to the diagonals, the subsequent distortion 

 changes it to a rectangle whose length is 1 + e and breadth ] - e. 

 If F is the force required to produce the alteration, the rigidity is 

 defined by the relation F = ne, or the coefficient of rigidity, n, is 

 equal to F/e, and is the force which if applied as a pull over one 

 pair of opposite faces of a cube and a push over another opposite 

 pair would (if strain and stress remained directly proportioned to 

 one another) double its length. 



In ordinary solids strain and stress are only proportional within 

 generally rather narrow limits, which, however, vary widely in 

 different materials, as, for instance, in glass compared with india- 

 rubber. 



When these limits are exceeded the material is either ruptured or 

 permanently distorted. 



The point to which attention should be directed is that the 

 limits for rigidity are quite different and apparently independent of 

 those for dilatation. 



As has been said before, volume elasticity and rigidity are the 

 fundamental qualities which regulate the elastic behaviour of a solid, 

 but the quality most ordinarily in evidence is the elastic resistance 

 opposed to a direct pull. 



This is known as Young's Modulus, and may be defined as the 

 direct pull which would be required to double the length of a rod of 

 uniform section, assuming strain and stress to be always proportional. 

 If the modulus is denoted by E, and an extension e is caused by a 

 direct pull F, 



E = F/e. 



E differs from the rigidity constant n in that no force is applied at 

 right angles to the pull, and it involves the volume elasticity Jc, as 

 may be shown by the diagram (practically the same as that given in 

 Thomson and Tait's " Natural Philosophy ") in slide (3). Let the 

 direct pull at the two opposite faces of the cube be represented by P, 

 and let P be divided into three equal parts as shown. On the other 

 four faces of the cube let forces each equal to -^- P act in opposite 

 directions. 



The joint action of all these forces is a direct pull equal to P 

 tending to increase the distance between the first-named pair of 

 faces, while no force is exerted on the other two pairs. This is 

 equivalent to two shearing stresses at right angles to one another, 

 each equal to ^ P, combined with dilating stress equal to ^ P in 

 all directions. 



2 d 2 



