On the Strength of Materials, 71 



cavities, which may be of use as a general indication of the 

 order of magnitude involved in other similar cases also. Even 

 for the actual cases worked out the result is not, however, to 

 be interpreted exactly. For, in the first place, to make cal- 

 culation possible the proportionality of stress to strain (Hooke's 

 law) is assumed, and this ceases to hold, the material some- 

 times even begins to flow, before the critical condition is 

 attained ; and, secondly, the conditions that produce a break- 

 down of the material are but vaguely understood. 



A spherical portion of the mass becomes, when strained, an 

 ellipsoid of which the principal axes determine the three prin- 

 cipal elongations which constitute the strain. Now it is 

 sometimes assumed that a simple change of volume by 

 compression or expansion cannot produce or affect rupture, 

 and therefore this ellipsoid need only be compared with the 

 sphere of equal volume from which it is derived by three 

 simple shears in mutually rectangular planes. The value of 

 the greatest of these shears may then perhaps be taken to be 

 the circumstance determining the limiting strength of the 

 material. It may, however, be remarked that, as the forces 

 of cohesion between the elements of the material are not 

 infinite, it must be possible to break it down or pull it asunder 

 by a tension uniform in all directions (say a negative hydro- 

 static pressure) ; and it is quite conceivable that a pressure 

 equal in all directions may by the opposite displacement loosen 

 the bonds of cohesion and so produce a plastic condition which 

 will give other forces play to act. The experiments of 

 W. Spring, in which an intimate mixture of two solid sub- 

 stances which do not combine chemically under ordinary 

 circumstances is caused to combine by the application of 

 great pressure, may have a bearing on this question. The 

 fact that cast iron supports compression much better than 

 tension is also in point. If it is, however, the case with any 

 material that the range of tension uniform in all directions 

 which it can stand is very much greater than the range 

 of stresses involving shear, the rupture would depend for 

 that material on the shears only, and the greatest of them 

 might be taken to be its determining cause. Thus rupture 

 would be determined by the difference between the greatest 

 and least axes of the ellipsoid into which a sphere of unit 

 radius is strained. When this supposition is not valid the 

 greatest elongation would be a more likely criterion ; bur. in 

 any case the assumed law will be a sufficient indication for 

 our purpose, because any more precise specification would be 

 vitiated in its application by the causes above mentioned. 

 which render elastic calculations illustrative rather than exact 



