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



Limitation of %armonic vibrations. — While the theory of 

 harmonic vibrations is applicable to very small vibrations on 

 any theory in which the load strain curve is represented as 

 continuous and as making an angle with the axes whose tan- 

 gent is finite, it appears to be inapplicable in all cases where 

 the excursions are sufficient to display the curvature of the 

 locus. If the attraction toward the position of no strain in 

 the direction of oy is proportional to y— 1, then in an isotropic 

 mass there will also be an attraction in the direction of ox 

 which will be proportional to 1— x. The path of the particle 

 at the corner of a vibrating cube will therefore be the resultant 

 of two harmonic motions whose phases necessarily differ by 

 exactly by one-half of the period of vibration, however great 

 and however different the amplitudes may be. This resultant 

 is well known to be a straight line. Hence the theory pre- 

 cludes all displacements excepting those which are so small 

 that the path of the corner of the cube may properly be 

 regarded as rectilinear. It seems needless to insist that such 

 cannot be the case for finite strains in general. 



There is at least one elastic solid substance, vulcanized india 

 rubber, which can be stretched to several times its normal 

 length without taking a sensible permanent set. Now if the 

 ideal elastic solid stretched to double its original length (or 

 more) were allowed to vibrate, the hypothesis of simple har- 

 monic vibration implies that this length would be reduced to 

 zero (or less) in the opposite phase of the vibration, a manifest 

 absurdity. 



Variation of pitch by Hookers taw. — It remains to be shown 

 that if the commonly accepted law were applicable to finite 

 strain, sonorous vibrations would be accompanied by changes 

 of pitch which could scarcely have escaped detection by musi- 

 cians and physicists. Experiments have shown that the elon- 

 gation of steel piano wire may be pushed to 0'0115 before the 

 limit of elasticity is reached.* Since virtuosos not infre- 

 quently break strings in playing the piano, it is not unreason- 

 able to assume that a one per cent elongation is not seldom 

 attained. In simple longitudinal vibration the frequency of 

 vibration is expressed by 1/4 of v / M/p, and if according to 

 Hooke's law, s = Q/3f, where Q is the load, the number of 

 vibrations, v, may be written 



w 



sp 



If, on the other hand, according to the theory of this paper, 

 ln(l+s) — Q/M the number of vibrations, u, may be written 



* From experiments on Knglish steel piano wire by Mr. D. McFarlane. 



