60 
458. Essentially, the method is based on Hamilton's principle, which may be 
regarded as the fundamental basis of classical dynamics, and on an assumption 
of proportional motion for each separate unit of a system, Thus for a plate 
supported on yielding beams, the shape of the deflecting beams would be 
assumed and also the shape of the plate deflection relative to the beams, 
but the overall magnitudes of beam deflection and plate deflection would not 
be restricted and would not neosssarily be in a constant ratio to one 
another. If Hamilton's principle is applied to the motion when so 
restricted, the problem is analogous to one of particle dynamics and the 
approximate solution is given by Lagrangs's equations. A detailed 
description of this method and its application is too lengthy for discussion 
hut the main steps when the mthcod is applied to a simplic elastic problem 
are outlined at Appendix E, 
459. For problems of plastic deformation the application of the method is 
necessarily complicated by consideration of the irreversibility of the 
deformation, but, in general, this is not too difficult, involving minly 
a consideration of the motion in several distinct stages:' The potential 
accuracy of the approximate method has been assessed by applying it to 
elastic problems and comparing the resulting solutions with nominally 
exact solutions. Reasonable agreement was obtained. Further, in one 
plastic problem to which the method has been applied the approximate 
solution was in reasonable agreement with experimental results.** 
460. Summing up, existing theory of the plastic deformation of targets by 
underwater explosions depends to a large extent on approximate plasticity 
assumptions and approximate mthods of analysis. In spite of this, a 
certain masure of agreement between theory and experiment has been obtained 
and, pending more accurate theory, approximate methods may reasonably be 
expected to throw light in particular on the qualitative behaviour of a 
ship's hull structure subjected to underwater explosions. For example, the 
important question of the optimum distribution of weight of steel between 
plating and supporting framework for maximum resistance to impulsive loading 
is essentially a qualitative question in the sense that the answer can 
reasonably be expected not to depend greatly on the precise magnitude of 
the loading. 
Rupture of steel 
461. The onset of rupture in steel undoubtedly depends on many factors suchas 
chemical composition. microstructure, the presence of flaws, size and geometrical 
shape of specimen, rate of loading,and temperature. Much research is at 
present being carried out to investigate the relative importance of these 
factors but at present there is no well-established quantitative theory 
which can be used to prediot rupture. Attention will therefore be 
concentrated on the qualitative points which seem of most importance in 
connection with the problem of underwater explosion damage. 
462. Broadly speaking, two main types of fracture may occur in steel, 
first, ductile fracture in which considerable deformation precedes rupture 
and secondly, brittle fracture in which little deformation occurs. In the 
former type, the energy absorbed in the steel prior to fracture is mich 
greater than in the brittle type of fracture. Since, as previously 
discussed, damage by underwater explosions depends to a large extent on 
the ability of a struoture to absorb energy it is obviously desirable from 
the ship designer's viewpoint that brittle fracture should be avoided, 
The ductile fracture is usually associated with a sliding process in ths 
steel due to shear whereas the brittle fracture is considered to be a 
tearing apart of the mtal due to tension. The type of fracture will 
thus depend not only on the relative inherent strengths of the steel to 
sliding and tearing but also on the type of stress system to which the 
steel is subjected; for example with uniform tension in all directions 
there would be no shear in a uniform material and failure would take 
place by tearing and be of the brittle type. 
