454 IX. DYNAMICS OF DEFORMABLE BODIES. 



172. Work of Stress in producing 1 Strain. If every point 

 in a body move a distance dq, whose components are 8ti, dv, dw, 

 and if there act upon every unit of mass of the body the external 

 forces X, Y, Z, and upon each unit of surface the forces X n , Y n , Z n , 

 the work done by all the forces in the displacement is 



118) dW= I UXndu + Y n dv + Z n dw}d8 



+ / / I Q[Xdu + Ydv + Zdw}dr, 

 which becomes by equations 86), 



119) d W = I j {[_X X cos (nx) + X y cos (ny) + X z cos (ngj] du 



+ [Y x cos (nx) + Y y cos (ny) + Y z cos (nz)\ dv 



+ [Z x cos (wfl?) 4- Z y cos (wy) + ^ cos (mi)] dw} dS 



and transforming surface integrals into volume integrals by differentia- 

 tion in the manner of the divergence theorem and making use of 

 equations 94), 



YydV 



w + F^i; + Zdw) 



" 



az 0Z f , dZ 

 ' ' + ^ 



+ X^+Y 1 r^ + Z, 



TT /^^'^ _j_ ^V\ , ^ (W<* . 



~\ J-z \ "iaTT ' Q * / ~t~ ^a; I o 



