1,08 SURFACE AND OTHER EFFECTS 



or plate may be broadly grouped under determination of the elastic or 

 plastic resistance to deformation, and analysis of the disturbances in the 

 pressure field resulting from the finite area of the structure. Each of 

 these affects and is affected by the other and the two must therefore be 

 considered together in obtaining solutions for the motion, but we shall 

 begin by first considering the resistance of the structure. 



A. Plastic deformation. The elastic properties of a material such 

 as steel are characterized by the relation of applied forces to the charges 

 in shape which they produce. If a plate clamped at its edges undergoes 

 an increase in area by load applied on one face, standard methods of the 

 theory of elasticity can be applied to formulate the differential equa- 

 tions of its motion in terms of its elastic moduli and inertia, and the ap- 

 plied load. The cases of most interest, however, are those in which the 

 material is stressed beyond its elastic limit and a permanent defor- 

 mation results, which may become so large that the plate thins to the 

 point of rupturing. The plastic state, considered in its full generality, 

 is complicated and incompletely understood, particularly under the 

 conditions of dynamic loading of interest here. Fortunately, a simple 

 first approximation to the true situation can be made which is illumi- 

 nating and reasonably accurate. In this approximation, it is assumed 

 that a definite elastic limit exists, below which Hooke's law applies and 

 above which the plate acts like a membrane under a constant tension 

 determined by the yield stress under tension at the elastic limit. 



The nature of this plastic region depends on the material, but steel 

 and copper can withstand very large strains before finally rupturing. 

 The assumption that a definite value of the yield stress ao exists and is 

 further independent of the rates of strain is an idealization. It is well 

 established that, at the large strain rates of interest in explosion dam- 

 age, (To is much greater than for statically applied loads, but this value 

 does not increase rapidly for large strain rates. It is reasonable to as- 

 sume a figure for ao which is independent of this rate and somewhat 

 greater than the static value, and such an assumption is common to 

 virtually all the theoretical treatments. 



With the assumption that the plastic deformation can be described 

 by a tangential stress equal to the yield stress, the behavior of a plate 

 becomes the same as that of a film under tension of magnitude aoh for 

 unit length of a cut in the plate, h being its thickness. The deformation 

 of the plate into a curved profile therefore develops a normal stress com- 

 ponent an for unit area of the plate as shown in Fig. 10.5a. The mag- 

 nitude of o-„ can be calculated in the same way as for a fihn with surface 

 tension, with the result 



2aoh 

 r 



