Theory of Elasticity. 273 



results first in three differential (interior) and three ordinary 

 (surface) equations between the changes in strain and the 

 bodily forces and surface tractions, and thence in three dif- 

 ferential (interior) and three ordinary (surface) equations 

 between the displacements and the bodily forces and surface 

 tractions, which equations then permit of definite solution. 

 These equations for isotropic bodies subject to " flooke's law " 

 are particularly well known.* They are the subject of all the 



* Here the strains are expressed in terms of the stresses through the relations 



1 



p 



e =E 



Q + R 



* E 



S 

 a = — 



J E 



R + P 

 ° E 



f 1 



R 

 g =E 



a P+Q 

 E 



u 



\ (30 

 I 



J 



where efg are the changes in elongation per unit in the directions x, y and z ; 

 abc are the changes in the shears in the planes of x, y and z; E is "Young's 

 modulus;" a is the ratio of lateral contraction to longitudinal extension or " Pois- 

 son's ratio " and fi is the shearing modulus or modulus of rigidity. Between 

 these quantities and two others, — k the modulus of volumetric change or bulk 

 modulus and the coefficient A = /c— ■§-// we have the relations 



„ ii(Bl + 2ii) 9utc , x 1 

 E=-.'-\ CL= Ul ( 4 ) and a = — (5 



2, + fi 3/c + // v ' 2(l + fi) 



whence denotiDg the sum e +/+ g by A (the change in the dilatation), we have 

 the converse solutions of (3') giving stresses in terms of strains 



P = 1A + 2 fie S = fia) 



Q = 2,b + 2fif T = fib[ (60 



R = AA + 2fig U= fie ) 



Introducing these converse solutions of the properties (30 into the equations 

 (10 and (20 we deduce what are known as the equations of equilibrium of an 

 isotropic elastic solid in terms of the strains (actually the changes in strain). 

 These are 



For interior points. For surface points. 



d& / de dc db\ 

 dx \ dx dy dz ) 

 ,M ( dc 6f 6a\ 

 dy \ dx dy dz J 

 d A / db da dg \_ 

 dz V 6x 6y 6z J 



-pX 

 - P T 

 - P Z 



A/A + ii(2le + mc + nb) = F 



<■ (7') AmA+ fi(lc +2mf+na)= G 



ln& + fi{lb + ma + 2ng)= H 



The changes in strain are related to the displacements by the equations 



du 



' — 6x 



dw dv 



a = t- + ~r~ 



dy 6 z 





6v 

 f ~dy~ 



6u 6w 



b = —. — 1 — — - 

 6z dx 



> (90 



dw 

 g =dz~ 



c = 



6v 6xi 

 dx dy 





(80 



