G. F. Becker — Finite Elastic Stress-Strain Function. 347 



from being fulfilled by all substances of sensibly symmetrical 

 properties. This is substantially a demonstration that the 

 molecular constitution of matter is very complex,* but pro- 

 vided that the mass considered is very large relatively to the 

 distances between molecules this complexity does not interfere 

 with the hypothesis that pure shear and simple dilation can 

 each be characterized by one constant only. 



The continuity of the load-strain function both for loads of 

 the same sign and from positive to negative loads is regarded 

 as established by experiment for many substances ; and equally 

 well established is the conclusion that for small loads, load and 

 strain are of the same order. f In other words Hooke's law is 

 applicable to minute strains. Perfect elastic recovery is prob- 

 ably never realized, but it is generally granted that some sub- 

 stances approach this ideal under certain conditions so closely 

 as to warrant speculation on the subject. 



These results appear to justify the assumptions made in the 

 paragraph headed "physical hypothesis" as representing the 

 most important features of numerous real substances. On the 

 other hand viscosity, plasticity and ductility have been entirely 

 ignored ; so that the results are applicable only to a part of 

 the phenomena of real matter. 



Stress-strain function. — It is perfectly easy to pass from 

 the load-strain function to the stress-strain function for the 

 ideal solid under discussion. The area of the extended cube is 

 its volume divided by its length or h 3 /d 2 k. Hence if Q' is the 

 stress, or force per unit area, Q'h'/a' — Q. Therefore the 

 stress-strain function is 



ia% f/v = s Q'/M 



an equation which though explicit in respect to stress and very 

 compact is not very manageable. If one writes a 2 h = y and 



h/a = x, the first member of this equation becomes y l ' x . Here 

 x and y are the coordinates of the corner of the strained cube. 

 Verbal statement of law. — If one writes d 2 h—l=f the last 

 of equations (5) gives 



df=(l+f)dQ/M 



or the increment of strain is proportional to the increment of 

 load and to the length of the strained mass. This is of course 

 the " compound interest law" while Hooke's law answers to 

 simple interest. 



* Compare Lord Kelvin's construction of the system of eight molecules in a 

 substance not fulfilling Poisson's hypothesis in his Lectures on Molecular 

 Dynamics. 



f Compare B. de Saint-Venant in his edition of Navier's Lecons, 1864, p. 14, 

 and Lord Kelvin, Encyc. Brit., 9th ed.. Art. Elasticity, Section 37. 



