ON STRESS DISTRIBUTION IN ENGINEERING MATERIALS. 475 



closely. A point of inflexion is very readily determined by the optical 

 method, the appearances in such a case being as shown in fig. 10. A 

 wrong calibration factor for the test piece used in the optical measurements 

 might account for the discrepancies of figs. 6 to 8, but this, again, seems 

 unlikely, as it would involve a progressive variation of this calibration, 

 which is hardly reconcilable with a numerical oversight. 



Passing now to the discrepancies with the theory. Major Low has 

 given (fig. 11) a diagram showing the observed deflections of the nodes : 

 this makes it clear that the fundamental assumption of the theory, that 

 the nodes remain collinear, is not tenable, and this is probably quite suffi- 

 cient to account for the observed divergencies, about which — especially 

 as regards the shift of the point of inflexion — there seems to be no doubt. 



Note on the Principle of Dynamical Similarity applied to Deformable 



Elastic Str lectures. 



By Professor L. N. G. Filon, F.R.S. 



1. General case of structure of finite thickness. 



Consider any deformable structure subjected to strain. Let a mechani- 

 cal model of this structure be made accurately to scale, but of different 

 material. "We will investigate the relations which have to be satisfied 

 by the elastic constants and the stresses applied if the deformations of the 

 original system and of the model are to be geometrically similar. Let k 

 be the ratio of similarity of the model and the original, so that if I be any 

 length in the original, kt is the corresponding length in the model. 



If X, y, z are the co-ordinates of any point of the original {kx, ky, kz) 

 are the co-ordinates of the corresponding point of the model. 



If (u, V, w) are the displacements at {x, y, z),(ku,kv,kw) are the displace- 

 ments at (kx, ky, kz). 



The strains s __du^ s do s div 



X 9.r y dif' z dz' 



a- dv dw a- dw da <r du dv 



yz dz''dy ' zx dx dz ' xy dy dx 



are identical in the model and original. 



The stresses are given by 



XX 



» /du , dv dw\ da ■ ... 



= \ a~ '" a — '" ?P / I f^ 7i~ ' ^^d. two similar equations 



yz =z fj.l^--\- r-— I ; and two similar equations, X and /*, being the 



elastic constants of Lame. 



Thus, in general, for dynamical similarity, all the elastic constants have 

 to be altered in the same ratio q, and the applied stresses must be altered 

 also in the ratio q. 



