317 



(b) Hic:her approximations . — The nth apjiroxLmation is obtained 

 by setting equal to zero the coefficient of oC ~ in Eq. (12), oL*"'''*" in 

 Eq. (13), oC'""' in Eq. (14), and *^^'^ in Eq. (15). Here the strains are, 

 of course, given by the first n terms in Eqs. (9) and (10), 



The reader will observe that in the nth approximation (except for 



.-,,,, , , . ,, _ C1.W-0 Can.) - C*H-x) 



n. a \ ) the dependent variables, z , u. > ^i > ='•"" 



Cj^***"**' , occur linearly^ and that no variables with a higher superscript 

 appear. Consequently, there will be no difficulties associated with nonlin- 

 earity except possibly in the first approximation. The solutions of all 1-1 

 approximations are to be fitted to the appropriate boundary and initio con- 

 ditions before substitution in the nth approximation. 



The foregoing method of successive approximation is applicable 

 only to problems in which the initial conditions are expressible in the 

 manner of Eq. (I6) and the boundary conditions are expansible in oL in 

 such a manner tliat the vanishing of oC corresponds to no motion of the plate. 

 In order to use the initial and boundary conditions in the scheme of suc- 

 cessive approximation, the coefficients of each power of oC are set equal to 

 zero thereby obtaining ctxiditions corresponding to each approximation. 



The bovindary conditions may take varied forms. Boundary condi- 

 tions for the first approximation are considered in more detail in the next 

 section. It is well to consider them separately for each anticipated special 

 application of the general theory presented here. The parameter oc, will 

 usually be assigned a physical meaning (that is, in the case of damage to 

 infinite thin plates by projectiles, aC is the ratio of the initial projec- 

 tile velocity to the velocity of plastic waves), thus establishing a 



