34:0 G. F. Becker — Finite Elastic Stress-Strain Function. 



Application to system of forces. — Without any knowledge 

 of the relations between stress and strain, the foregoing an- 

 alysis can be applied to developing corresponding systems of 

 stress and strain. Let a unit cube of an elastic substance pre- 

 senting equal resistance in all directions be subjected to axial 

 loads P, Q, B. Suppose these forces to produce respectively 

 dilations of ratios A a , A 3 , h 3 and shears of ratios j?, q, r. Then 

 the following table shows the effects of each axial force on 

 each axial dimension of the cube in any pure strain. 



Active force 



P 



Q 



R 



Axis of strain 



x y z 



x y z 



x y z 



Dilation 



Shear 



Shear 



h 1 h 1 h 1 



p 1/p 1 

 p 1 1 lp 



K K K 



\jq q 1 

 1 q 1/q 



K K K 



l/r 1 r 



1 l/r r 



Grouping the forces and the strains by axes, it is easy to see 

 that the components may be arranged as in the following table, 

 which exhibits the compound strains in comparison with the 

 compound loads which cause them, though without in any way 

 indicating the functional relation between any force and the 

 corresponding strain. 



Pure Strains. 



Axes 



X 



y 



z 



Dilation 

 Shear 



Shear 



h x hji z 



qr 



1 



hJi % H % 



qr 



P 



pq 



KKK 



i 



r 2 

 pq 



Products 



q r 



hjijijf 

 pr 



pq 





Loads or Initi 



al Stresses. 





Axes 



X 



y 



z 



Dilation 



Shear 



Shear 



P+Q + R 



3 

 Q + R-2P 



3 







P+Q + R 



3 

 Q + R-2P 



3 

 P+ Q-2R 



3 



P+Q + R 



3 







P+ Q-2R 



3 



Sums 



p 



Q 



R 



