342 G. F Becker — Finite Flastic Stress- Strain Function. 



every direction and equal resistance to dilation in every direc- 

 tion. The two resistances may also be supposed independent 

 of one another — for this is a more general case than that of 

 dependence. The resistance finally may be supposed continuous 

 and everywhere of the same order as the strains. 



In such an ideal isotropic substance it appears that the num- 

 ber of independent moduluses cannot exceed two ; for a pure 

 shear irrespective of its amount is the simplest conceivable dis- 

 tortion and no strain can be simpler than dilation, while to as- 

 sume that either strain involved more than one modulus would 

 be equivalent to supposing still simplier strains, each dependent 

 upon one of the units of resistance. It is undoubtedly true 

 that, unless the load-strain curve is a straight line, finite strains 

 involve constants of which infinitesimal strains are independent ; 

 but these constants are mere coefficients and not moduluses: 

 for the function being continuous must be developable by 

 Taylor's Theorem, and the first term must contain the same 

 variable as the succeeding terms, this variable being the force 

 measured in terms of the moduluses. In this statement it must 

 be understood that the moduluses are to be determined for 

 vanishing strain.* 



One can determine the general form of the variable in terms 

 of the resistances or moduluses for the ideal isotropic solid 

 defined above. The load effecting dilation in simple traction, 

 as was shown above, is exactly one third of the total load, or 

 say Q/S ; and if a is the unit of resistance to linear dilation, 

 Q/Sa is the quantity with which the linear dilation will vary. 

 The components of the shearing stresses in the direction of the 

 traction are each Q/3, and, if c is the unit of resistance to this 



* One sometimes sees the incompleteness of Hooke's law referred to in terms 

 such as " Young's modulus must in reality be variable." This is a perfectly legit- 

 imate statement provided that Young's modulus is defined in accordance with it; 

 but the mode of statement does not seem to me an expedient one to indicate the 

 failure of linearity. Let fi represent Young's modulus regarded as variable and 

 Fa force or a stress measured in arbitrary units. Then if y is the length of a 

 unit cube when extended by a force, the law of extension may be written in the 

 form y— 1 4- F/ju. Now let if be the value of Young's modulus for zero strain, 

 and therefore an absolute constant. Then, assuming the continuity of the func- 

 tions, one may write // in terms of M thus, 



1 _ l ■ ( F \— ] (\ + AF BF ' 2 \ 



fI~M 9 \Mj~M\ If ^"JP^ /* 



But this gives 



F AF' 2 BF* 

 y M M* iF 



so that 1/fi merely stands for a development in terms of F/M. If therefore one 

 defines Young's modulus as the tangent of the curve for vanishing strain, the 

 fact of curvature is expressed by saying that powers of the force (in terms of 

 Young's modulus) higher than the first enter into the complete expression for 

 extension. 



