34:8 G. F. Becker — Finite Elastic Stress-Strain Function. 



Curves of absolute movement. — Let o be Poisson's ratio 



3lc—2n 



Or 6——r -i — ■ r. 



2(3& + 2ra) 



Let x y be the original positions of a particle in an unstrained 

 bar, and let xy be their positions after the bar has been ex- 

 tended by a load Q. Then x=x h/a and y—y Q a^h. It also 

 follows from (5) that a en =h 9k , whence it may easily be shown 

 that the path of the particle is represented by the extraordin- 

 arily simple equation* 



x y G = x yo . ( 6 ) 



If one defines Poisson's ratio as the ratio of lateral contrac- 

 tion to axial elongation, its expression is by definition 



_ dx ,dy ydx 



x y xdy' 



and this, when integrated on the hypothesis that a is a con- 

 stant, gives (6). Thus for this ideal solid, the ratio of lateral 

 contraction to linear elongation is independent of the previous 

 strain. 



The equation (6) gives results which are undeniably correct 

 in three special cases. For an incompressible solid a= 1/2, and 

 (6) becomes x 2 y= constant, or the volume remains unchanged. 

 For a compressible solid of infinite rigidity a— — 1 and (6) 

 becomes a?/y= constant so that only radial motion is possible. 

 For linear elongation unaccompanied by lateral extension <r=0, 

 and (6) gives a?=constant.f 



* On Cauchy's hypothesis (7=1/4, which, introduced into this equation, implies 

 that the volume of the strained cube is the square root of its length. 



f It seems possible to arrive at the conclusion that a is constant by discussion of 

 these three cases Let e and — / be small axial increments of strain due to a small 

 increment of traction applied to a mass already strained to any extent. Let it 

 also be supposed that the moduluses are in general functions of the coordinates, 

 so that n and k are only limiting values for no strain. Then, by the ordinary 

 analysis of a small strain (Thomson and Tait, section 682), one may at least 

 write for an isotropic solid 



e=p( I + 1_ 



\3n[l +/!(*)] 9k[l+f 2 (x)] 



-f = p( l - L v 



where /(a?) is supposed to disappear with the strain. These values represent each 

 element of the axial extension and each element of the lateral contraction as 

 wholly independent. The value of a is — f/e. Now for an incompressible sub- 

 stance, as mentioned in the text, (7=1/2 and the formula gives 



(7 = 1. i±^) , so that/ 1 (^)=/ 3 (a-). 

 2 1 +f 3 (x) 



Again for n = oo only dilation is possible, or o = — 1, while the formula gives 



Si)' 



