572 hersey: stiffness of elastic systems 



duced in each case the same relative variation of elastic con- 

 stants over the cross section. When any elastic constant of a 

 series of bodies is represented by a single symbol, it is to be 

 understood that this refers to its mean value, or to its value 

 at a particular point and in a particular direction, and that all 

 bodies in the series are of the same generalized shape. Fin- 

 ally, if the bodies in such a series, although not violating Hooke's 

 law at any point within the material, are so greatly deformed 

 that there is no longer a direct proportionality between load and 

 deflection, it is to be understood that the bodies remain geo- 

 metrically similar to each other while being deformed. 



With this understanding, it may be shown by dimensional 

 reasoning that the stiffness of any of a series of perfectly elastic 

 bodies of different sizes or materials but of the same generalized 

 shape is given by 



S = LE-4>{a) = L M ^(a) = L-f(E^) (3) 



in which L denotes any chosen linear dimension, E Young's 

 modulus, ij, the rigidity (i.e., shear modulus) and o- Poisson's 

 ratio. The functions 4>, \p, and / are to be found, if they need 

 to be known at all, by detailed calculations employing the 

 conventional theory of elasticity, or by model experiments. 



Thus to find <p (<r) we need only plot observed values of — — 



LE 



against known values of a for a series of models of the same gen- 

 eralized shape, but having any convenient values of L, E, and a 

 covering the desired range along the a scale. In practice, the 

 method of model experiments will be limited by the difficulty 

 of procuring materials, which, if intended to be isotropic, are 

 sufficiently so; or which, if intended to be anisotropic, are suffi- 

 ciently similar in internal structure sensibly to satisfy the re- 

 quirement of having the same generalized shape. 



The change in stiffness with temperature, and criteria for com- 

 pensation. Differentiating (3) gives, for the fractional change 

 in stiffness with temperature, 6, 



I^=Aa + £/3 + 7 (4) 



S d9 



in which 



