G. F. Becker — Finite Elastic Stress-Strain Function. 345 



Here h/M is the tangent of the load-strain curve for vanish- 

 ing strain, and this by definition is 1/M, so that 5 = 1. 



It appears then that the equations sought for the load- 

 strain functions are 



«• = ««/*»; ft=eW*; a>h= &* ■ ( 5 ) 



a result which can also be reached from (4) without the aid of 

 Taylor's theorem. 



Tests of the equations. — These equations seem to satisfy all 

 the kinematical conditions deduced on preceding pages. It is 

 evident that opposite loads of equal intensity give shears, dila- 

 tions and extensions of reciprocal ratios and that the products 

 of the strain ratios vary with the sums of the loads. It is also 

 evident that infinite forces and such only will give infinite 

 strains. A very important point is that these equations repre- 

 sent a shear as held in equilibrium by the same force system 

 whether this elementary strain is due to posit ve or negative 

 forces. If any other quantity (not a mere power of Q or the 

 sum of such powers), such as the final stress were substituted 

 for the load Q, a pure shear would be represented as due to 

 different force systems in positive and negative strains which 

 would be a violation of the conditions of isotropy.* One 

 might suppose more than two independent moduluses to enter 

 into the denominator of the exponent ; but this again would 

 violate the condition of isotropy by implying different resist- 

 ances in different directions. Any change in the numerical 

 coefficients of the moduluses would imply a different partition 

 of the load between dilation and distortion, which is inadmis- 

 sible. It would be consistent with isotropy to suppose the 

 exponent of the form {Q/Mf +2c ; but then, if c exceeds zero, 

 the development of the function would contain no term in the 

 first power of the variable and the postulate that strains and 

 loads are to be of the same order would not be fulfilled. The 

 reciprocal relations of load and strain would be satisfied and 

 the loads would be of the same order as the strains, if one were 

 to substitute a series of uneven powers of the variables for v 

 and x Such series are for example the developments of tan v 



* Let a shearing strain be held in equilibrium by two loads, Q/3 and minus 

 Q/B. If a second equal shear at right angles to the first is so combined with it 

 that the tensile axes coincide, the entire tensile load is 2 Q/'i. If on the other 

 baud the two shears are combined by their contractile axes, the total pressure is 

 2 Q/?>. In the first case the area of the deformed cube measured perpendicularly 

 to the direction of the tension is 1/a' 2 , and if Q' is the final stress, Q' '/a? = 20/3 

 or Q' = 2Qa 2 /3. In the second case the area on which the pressure acts is a 2 

 and if the stress is Q\ Q" = — 2Q/la' 2 . Thus Q' = — Q"a A . Hence equal final 

 stresses of opposite signs cannot produce shears of reciprocal ratios in an iso- 

 tropic solid. The same conclusion is manifestly true of any quantity excepting Q 

 or an uneven power of Q or the sum of such uneven powers. 



