1881.] 



The Action of Cutting Tools. 



133 



thickness of the parallelopiped ; hence when BD is nothing the work 

 is also nothing. The rate of distortion, i.e., the distortion produced 

 by a definite motion of AD, is inversely as BD, and is therefore in- 

 finite when BD=0. 



Thus, when the rate of distortion is infinite and the limit of distor- 

 tion is finite, no work is required to effect the rupture. 



In the case of the tool it is, of course, only actually at the edge that 

 the rate of distortion is infinite ; but if it were possible to apply suit- 

 able forces simultaneously at every point of a solid through which a 

 surface of separation was desired to pass, that separation could be 

 effected with no expenditure of work whatever. 



The resistance due to elastic and permanent distortion is ^ , 9 



1 cos 6 



being the angle DEB (fig. 11), Q the integral of all the reaction due 



to distortion along the line EC at the moment sliding begins. 



The work done in internal friction is EC X distance through which 



the lamina slides per unit travel of tool X by the pressure under 



which the sliding takes place x by the coefficient of friction of the 



material on itself. 



Fig. 11. 



Let £=BC, the thickness of the cut, 

 t' — thickness of shaving, 



fx— coefficient of friction of the material on itself, 

 ^^coefficient of friction of material on the tool, 

 6>=DEB, 



0=FEA, viz., the angle which the face of the tool makes with 

 AB, 



then , = f«n(0 + 0) # . 



sin 



ED=^L- 



sin 9 



(2). 



