240 Mr. C. Chree on some Applications of 



One of the theories referred to above is that when the 

 algebraic difference between the greatest and least of the 

 principal stresses at any point — a pressure being reckoned 

 negative — attains a certain value, rupture will ensue at this 

 point. Thus, if in descending order of magnitude the prin- 

 cipal stresses at a point be T 1? T 2 , T 3 , then T L — T 3 is the stress- 

 drffierence* &t this point, and the theory asserts that rupture 

 will ultimately ensue if the stress- difference anywhere equals 

 L 3 , the load for ultimate rupture of a bar of the material by 

 longitudinal traction ; while if the stress-difference anywhere 

 equals L 4 , the load for immediate rupture by longitudinal 

 traction, then immediate rupture will ensue. 



The second theory, which is supported by the great 

 authority of de Saint- Venant f, replaces the stress-difference 

 of the first theory by the greatest strain. It thus asserts 

 that the condition for rupture is found by equating the 

 largest value found anywhere for the greatest strain to the 

 longitudinal strain answering to longitudinal traction L 3 , or 

 to that answering to the traction L 4 , according as the rupture 

 is ultimate or immediate. This theory maintains that extension 

 in some direction is necessary for rupture. 



The tw T o theories may, as in the case of pure longitudinal 

 traction, lead to the same result ; but in general they do not, 

 so one at least of them must be wrong. When we examine 

 the theories, still supposing the mathematical and physical 

 limits of perfect elasticity the same, a very obvious difficulty \ 

 presents itself. It is assumed that the stress-difference and 

 greatest strain are derived by the mathematical theory ; but 

 that theory applies only so long as the material is everywhere 

 perfectly elastic, whereas rupture, at least when immediate, 

 presents itself after the elastic limit has been passed. Thus 

 if the application of the mathematical theory lead to values 

 for the maximum stress-difference and greatest strain equal to 

 the values of these quantities answering to rupture, at all 

 events when immediate, the true conclusion would seem to 

 be that the fundamental hypothesis on which the treatment 

 proceeds, viz. that the material follows the laws assumed by 

 the mathematical theory, has been shown to be incorrect. 

 Nothing has been proved except that the elastic limit must 

 be passed and that the mathematical theory does not apply. 



The only logical way of interpreting the theories is to 



* See Professor Darwin, Phil. Trans. 1882, pp. 220-1, &c. ; also 

 Thomson and Tait's Nat. Phil. vol. i. Part ii. p. 423. 



t See Pearson's ' The Elastical Researches of Bane de Saint- Venant,' 

 Arts. 5 (c), &c. 



% Ibid. Arts. 4 (y), 5 {a), &c. 



