Page:Notes on HA Bethes Theory of Armor Penetration 1. Static Penetration.pdf/2



The first part of this paper describes the static stresses in a long cylindrical hollow cylinder and in a flat sheet when a concentric hole is opened out by radial pressure applied over its surface. Within a certain radius the material is assumed to be overstrained and to flow radially. Outside this radius the conditions are elastic. For the thick cylinder, where it is assumed that there is no extension parallel to the axis of symmetry, the problem and its solution are identical with those given in text books of gunnery in connection with the autofrettage of guns and with those which have been used in designing cylinders for high pressure work. In this case the type of the strain can be related immediately to a single variable, namely the radial displacement which is a function of one independent variable the radius, and one parameter, the radial displacement of the inner surface. This consideration remains true when, as in the case considered by Dr. Bethe, the strains in the inner plastic region are not small.

The hole in a thin plate is more interesting and more difficult to analyse because it is no longer possible to treat the strain as two dimensional, so that the relationship between plastic strain and stress must be considered. It is usually assumed that hydrostatic pressure merely compresses a plastic material without altering its strength to resist shear stresses. For this reason it is sometimes convenient in comparing various, theories of plasticity to use reduced principal stresses $$\sigma_1'=\sigma_1-p$$, $$\sigma_2'=\sigma_2-p$$, $$\sigma_3'=\sigma_3-p$$, where $$3p=\sigma_1+\sigma_2+\sigma_3$$, so that $$\sigma_1'+\sigma_2'+\sigma_3'=0$$. Similarly reduced principal strains $$e_1'=e_1-e$$, $$e_2'=e_2-e$$, $$e_3'=e_3-e$$ where $$e_1+e_2+e_3=3e$$and $$e$$ represents the volumetric strain. The plasticity relations are concerned firstly with the maximum values which the stresses $$\sigma_1'$$, $$\sigma_2'$$, $$\sigma_3'$$ can attain before plastic flow occurs and, secondly, with the dependence of $$e_1'$$, $$e_2'$$, $$e_3'$$ on $$\sigma_1'$$, $$\sigma_2'$$, $$\sigma_3'$$. These two kinds of plasticity condition are quite unrelated to one another. Of the first type two alternative hypotheses are mentioned by Dr. Bethe, namely those of Mohr and v. Mises, and he points out that there is but little difference between them.

For two dimensional problems, where if the compressibility be neglected we may take $$e_3=0$$, the second type of plasticity condition does not affect the distribution of stress in the plane to which the displacements are confined. This is because when $$e_3'=0$$, $$e_1'=-e_2'$$, so that only one kind of strain is possible when the directions of the principal strains are assumed to coincide with those of principal stresses.

The case is very different when the strain is not two dimensional. Here it is necessary to choose sane arbitrary law or to use experimental data. The problem can be visualised. by thinking of the relationship between the stress ellipsoid and the strain ellipsoid.

The following points may be noticed:-

(1) The absolute magnitude of the stress ellipsoid is related to the strength criterion, e.g. the Mohr or v. Mises criteria.