G. F. Becker — Finite Elastic Stress-Strain Function. 343 



initial stress, the corresponding extension will vary with 2Q/3c. 



In simple extension all faces of the unit cube remain parallel 



to their original positions, and the principle of superposition is 



applicable throughout the strain. Hence the total variable 



1 1 2 \ 

 may be written Q[-^- + x- I. The intensity of Q will not 



\oCl oC I 



affect the values of the constants a and c which indeed should 

 be determined for vanishing strain as has been pointed out. 



The quantities a and c have been intentionally denoted by 

 unusual letters. In English treatises it is usual to indicate the 

 modulus of cubical dilation by k and the modulus of distor- 

 tion by n. With this nomenclature a—Zk and c=2n. Using 

 the abbreviation M for Young's modulus the variable then 

 becomes 



«(a + L) = «* 



Since this is the form of the variable whether Q is finite or 

 infinitesimal, the length of the strained cube according to the 

 postulate of continuity must be developable in terms of Q/M 

 and cannot consist, for example, solely of a series of terms in 

 powers of Q/9k plus a series of powers of Q/B?i ; in other 

 words the general term of the development must be of the 

 form A m {Q/M) m and not A m (Q/9k) m + B m (Q/Sn) m . 



Form of the functions. — If a is the ratio of shear produced 

 by the traction Q in the ideal isotropic solid under discussion, 

 a must be some continuous function of Q/Sn. So too if h is 

 the ratio of linear dilation, h is some continuous function of 

 Q/9k. The length of the strained mass is a 2 /), and this must 

 be a continuous function of Q/M. If then f <p and (p are 

 three unknown continuous functions, one may certainly write 



«•=/(?> Mi> — Ki> « 



It also follows from the definitions of a and h that 



l=/(0); 1 = ^(0); 1=^(0). (2) 



For the sake of brevity let Q/3n=v and Q/9k=x. Then v 

 and x may be considered algebraically as independent of one 

 another even if an invariable relation existed between n and 

 k; for since in simple traction, the faces of the isotropic cube 

 maintain their initial direction, the principle of superposition 

 is applicable ; and to put n=oo or k — co is merely equivalent 

 to considering only that part of a strain due respectively to 



Am. Jocr. Sci.— Third Series, Vol. XLVI, No. 275. — Nov., 1893. 

 24 



