185 



The Exact Solution of the Ek^uations of Motion for a Material 

 vfith no Work-41ardeninR. 



Although the differential equations (B) are non-linear, still 



they may be solved explicitly in a fLnite form involving a single quadrature, 



at least in the case for which there is no work-hardening. Let us introduce 



the non-dimensional parameters 



p=R ^=vt=Z 



a a a 



V Pv2 



n-H K =-lkcr cr(i-v) 



^ h \jpy2^ E 



where, for t-e values suggested in a previous section d= 320, and 

 K = or .675. 



After a certain amount of algebra, equations (3), reduce to 



2 (30) 



2^?' " y- - 1 (b) 



(2 - ;^ ) P' * p;^/*-'= (3 -S)/x (c) 



where a prime means differentiation with respect to $ « These equations are 

 to be solved subject to the initial conditions (C) which may be rewritten: 



p=1|=l, ;a=X, when 5 = 0. (*) (30) 



By dividing the sides of (30) (c) by the corresponding sides of (30) (b), the 

 variable ^ is eliminated, and we are left with an integrable equation whose 

 variables are separated, relating P, and M-. Similarly by multiplying the 

 right side of (3O) (a) by the left side of (3O) (b) and conversely, and 

 employing the previous integrable expression to eliminate P, we find an 

 integrable relation involving Y) and %*-, Finally C is foimd in quadrature 

 form in terms of i^, and P(M-) from (30)(b). This explicit solution may be 

 exhibited as 



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