G. F. Becker — Finite Elastic Stress-Strain Function. 341 



In many cases it is convenient to abbreviate the strain pro- 

 ducts. Thus if one writes hjiji % — h^ qr/p 1 = a and pq/r 9 = fi, 

 the product are h/a, haft and A//3. 



Inferences from the table. — It is at once evident that the 

 load sums correspond to the products of the strain ratios, and 

 that zero force answers to unit strain ratios. There are also 

 several reciprocal relations which are not unworthy of atten- 

 tion. If P=0 and Q=—P, the strain reduces to a pure shear. 

 But the positive force, say Q, would by itself produce a dila- 

 tion A 2 , while the negative force, minus P, would produce 

 cubical compression of ratio h 1 <1. Now a shear is by defini- 

 tion undilatational and therefore, in this case, h/i^—1. Hence 

 equal initial stresses of opposite signs produce dilatations of 

 reciprocal ratios. The same two forces acting singly would 

 each produce two shears while their combination produces but 

 one. Q would contract lines parallel to oz in the ratio 1/q 

 while minus P would elongate the same lines in the ratio p/1. 

 Since the combination leaves these lines unaltered, p/q=l. 

 Hence equal loads of opposite signs produce shears of recipro- 

 cal ratios. It is easy to show by similar reasoning that equal 

 loads of opposite signs must produce pure distortions and ex- 

 tensions of reciprocal ratios. 



Strain as a function of load. — One may at will regard 

 strains as functions of load or of final stress ; but there seem 

 to be sufficient reasons for selecting load rather than final stress 

 as the variable. To obtain equations giving results applicable 

 to different substances, the equations must contain constants 

 characteristic of the material as well as forces measured in an 

 arbitrary unit. In other words the forces must be measured 

 in terms of the resistance which any particular substance pre- 

 sents. Now these resistances should be determined for some 

 strain common to all substances for forces of a given intensity. 

 The only such strain is zero strain corresponding to zero force. 

 Hence initial stresses or loads are more conveniently taken as 

 independent variables.* 



Argument based on small strains. 



Physical hypothesis. — In the foregoing no relation has been 

 assumed connecting stress and strain. The stresses and strains 

 corresponding to one another have been enumerated, but the 

 manner of correspondence has not been touched upon. One 

 may now at least imagine a homogeneous elastic substance of 

 such a character as to offer equal resistance to distortion in 



* When the strains are infinitesimal, it is easy to see that load and final stress 

 differ from one another by an infinitesimal fraction of either. 



