315 



3. Expansion in Terms of an Auxiliary Parameter and a Method of Suc - 

 cessive Approximation 



Since there is little hope of solving exactly the equations pre- 

 sented in assumptions (l) to (U), the desirability of a method of successive 

 approximation is apparent. Follovdng the introduction of a parameter «?t into 

 the initial conditions and external forces in such a way that the dependent 



, variables are analytic in ot , the dependent variables in each equation are 

 expanded in terms of o(. , and coefficients of the powers of o< are collected, 

 A method of successive approximation is immediately obtained by setting 

 equal to zero the coefficient of each power of oC . 



'I The parameter aC is introduced as follows. 



■x.ir,o) = O, 



Now the independent variables *• , u., 5, , and S^are functions of oC and 

 furthermore they are analytic in «t . Consequently, they can b e expanded in 

 power series in ct with a radius of convergence hoped to be sufficiently large. 

 It is apparent that changing the sign of oC is equivalent to a reflection 

 through the r, <^ -plane, and that consequently z must be odd in oC and 

 **• , 5| , and S must be even in oL • Both r and u. vanish with cc , 

 whereas 5, and S.^ approach a finite limit according to the idealized stress- 

 strain r elationship assumed in this treatment. In view of these considera- 

 tions, the expansions take the form. 



