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I. GENERAL THEORY OF Ti€ AXIALI.Y SYNJ^TtaC DEFORMATIOK' OF THIN PLATES 



1. Introduction 



The purpose of this part is the general formulation of the theory 

 of the a>d.ally symmetric deformation of plates for the dyna.Tiic as well as 

 static case. The treatment is limited to ductile materials whose stress- 

 strain relations may be adequately described by incorapressibility, a some- 

 what modified von Mises plasticity condition, and the principle of the 

 similarity of Mohr circles for any nonvanishing strain. This description 

 neglects the elastic domain of strain and hence corresponds to infinite 

 elastic constants with a finite yield stress. Since terms proportional 

 to powers of the thickness higher than th^j first are neglected, the theory 

 is strictly applicable only to parts of the plate where the radii of ciu*va- 

 ture are much larger than the thickness. In the specific cases to be con- 

 sidered later the regions of inapplicability will be of negligible extent. 

 By the use of a parameter development of the nonlinear equations 

 resulting from the above assumptions, relatively tractable equations are 

 obtained as a first approximation. These equations are quite adequate for 

 cases in which the angles between elements of plate in the initial and 

 deformed states are not too large, 



2. Model and Basic Assumptions 



For the description of the deformation process cylindrical coordi- 

 nates { r ^ -z, ) are used with an orientation such that the middle surface 

 of the plate is initially coincident with r, ^ -plane (see figure 1). Sin'-.e 

 aixial symmetry is preserved, the coordinate ^ is redundant. Overly specific 

 statements about the boundary conditions are avoided as much as possible in 



