168 a, 169] HETEROGENEOUS STRAIN. 445 



Thus the relative displacements are given as linear functions of the 

 relative coordinates f, g, li whose coefficients are the values of the 

 nine first derivatives at the point P, that is to say, constants for all 

 points Q in the neighborhood of P, consequently the relative strain 

 of the portion of the body in the neighborhood of P is homogeneous. 

 Thus we say that any continuous heterogeneous strain is homogeneous 

 in its smallest parts. 



Comparing with equations 49) and 51) we find the stretches, 

 dilatation, slides and rotations at any point to be respectively 



A \ du dv dw 



74) s x = -, Sy = -, S , = -, 



75) 

 ' 



ox 

 1 dw . dv\ 1 du dw\ l dv . du 



1 (w . v\ 1 (u , w\ l (dv . du 



?.-W + J^ ^ = \^ + W 9* = *(dx+Jy 



1 /dw dv\ 1 (du dw\ l (dv du 



X == ~ \-~ -- -x ? 



2 V^i/ W 



= ^ -- -o > G) * == ^ -- -K 

 2 \^^ &ar/ 2 



Thus the volume dilatation is equal to the divergence of the dis- 

 placement, while the rotation is equal to one half its curl. 



We might have obtained the value of by the divergence 

 theorem. Consider any closed surface S fixed in space so that por- 

 tions of the deformable body flow through it daring the strain, and 

 let us find the volume of the matter which passes outward through S. 

 Through an element dS at which the displacement is q there passes 

 out a quantity filling a prism of slant- height q and base dS whose 

 volume is therefore qcos(nq)dS, where n is the outward normal 

 to S. Through the whole surface there accordingly issues an amount 

 whose volume is 



78) Q = I I qcos(nq)dS 



= I I [u cos (nx) + v cos (ny) -f w cos (n&)} dS 



by the divergence theorem. This is accordingly the increase in volume 

 of the portion of substance originally included by the surface S. 

 The ratio of this to the original volume is accordingly the mean 

 value of the divergence in the volume in question, and making the 

 volume infinitesimal, this becomes the dilatation 6. 



In order that a strain shall be everywhere irrotational we must 

 have the curl components of the displacement vanish everywhere. 



