P. W. Bridgman — Crystalline Cylinders. 26!) 



Art. XVIII. — Stress-Strain Relations in Crystalline Cylinders,' 

 by P. W. Bridgman. 



In a preceding paper I have made experimental examination 

 of the manner of failure under high pressure of cavities in var- 

 ious materials of geological interest. In order to simplify the 

 question as much as possible a number of the specimens were 

 cut from single crystals ; these were made in the form of circular 

 cylinders, pierced axially with circular holes concentric with the 

 outer surface. It appeared, however, on assembling the results, 

 that the mathematical problem of the elastic behavior of crystal- 

 line material, even under such simple conditions as above, had 

 never been discussed. I have been able to obtain an approxi- 

 mate solution, and the results for several systems of crystals are 

 recorded here. 



In the following mathematical analysis there are two main 

 guiding ideas. It appeared very soon that an exact solution was 

 out of the question ; the form of the solution could be deter- 

 mined, but in order to compute the numerical coefficients it was 

 necessary to solve a system of infinitely many linear algebraic 

 equations. However, an approximate solution, which would be 

 expected to give the most important terms, was obtained by the 

 device of varying the elastic constants, allowing them to approach 

 infinitely close to those of an isotropic body, and finding the in- 

 finitesimal terms which must in consequence be added to the 

 well known solution for an isotropic body. The second guiding 

 idea was to discover the nature of the terms present by consider- 

 ing the symmetry relations of the crystals. This saved an enor- 

 mous amount of time, for if an ordinary series were assumed for 

 the solution most of the coefficients would be found to be zero. 



In the following the solution is obtained to first order terms 

 for cubic and tetragonal crystals, and to second order terms for 

 trigonal crystals. For the latter the solution has been carried 

 through numerically in the case of quartz, which was one of the 

 materials experimented upon, and the closeness of the approxi- 

 mation is discussed. 



The mathematical problem consists in finding such a set of 

 displacements that the set of strains determined by them will in 

 turn determine such a set of stresses that the equations of equili- 

 brium and the boundary conditions (which are conditions on the 

 stresses) shall be satisfied. The boundary conditions are that all 

 components of stress on the inner curved surface of the cylinder 

 vanish, and on the exteraalsurface the stresses reduce to a uni- 

 form normal pressure. With regard to the stresses on the in- 

 finitely distant ends the following method of procedure, which 

 is usual for isotropic solids, is applicable. Since the cylinder is 

 infinitely long, the stress and strain must be independent of z. 



