Page:Notes on HA Bethes Theory of Armor Penetration 1. Static Penetration.pdf/4

In the range $r_{1}>r>r_{2}$, when the maximum stress difference is that between the two principal stresses in the plane of the sheet, the equation of equilibrium is sufficient, with Mohr's strength condition prescribing a constant difference between them, to determine the stress. Inside the radius $r_{2}$, i.e. when $r_{2}>r>b$ where $b$ is the radius of the hole, the tangential stress $\sigma _{\theta }$ cannot remain positive (tensile). Two alternatives remain -

(a) $\sigma _{\theta }$ becomes negative (i.e. there can be a compressive tangential stress) or

(b) $\sigma _{\theta }=0.$

Bethe rejects alternative (a) because in that case $\sigma _{\theta }$ would be the intermediate principal stress and by his strain assumption it would be necessary that no strain could take place in the tangential direction. This would preclude any radial displacement. He is left with (b) as the only possible alternative consistent with his strain assumption, namely $\sigma _{\theta }=0.$ This alternative, however, suffers from very severe disadvantages. The stress at every point is one which is symmetrical about the radial direction, i.e. the stress ellipsoid at any point is a spheroid whose axis of symmetry is along a radius. On the other hand the plastic strain which according to Bethe's calculation results from this symmetrical or uni-directional stress is very far from symmetrical and is variable along the radius. Expressed in terns of Lode's variables the stress in the range $r_{2}>r>b$ is represented by $\mu =1$ while the strain is indeterminate and covers a range of the line $\mu =1$ in Fig.1.

Since the alternative (a) that $\sigma _{\theta }$ becanes a ccmpressive stress when $r<r_{2}$ is perfectly possible if other stress-strain assumptions are used, it will be seen that the sole reason for Bethe's conclusion that $\sigma _{\theta }=0$ is that he assumes that when a stress is applied in one direction (e.g. a pure pressure or tension unaceanpanied by transverse stresses) the strain is Completely indetermninate. A round. bar, for instance, when stretched in an ordinary testing ma.chine,, would,, it it obeyed Bethe's stress-strain law, in general acquire an elliptical section and it is this assumed asymmetrical property of plastic material which alone is res~ible tor Bethe's conclusion that $\sigma _{\theta }=0$.

It would seem better to abandon the attempt to give a reasoned justification of the assumption that $\sigma _{\theta }=0$ when $r_{2}>r>b$ and to fall back on the fact that this assumption enables a stress distribution to be determined without reference to the strain. The equilibrium equation then suffices to determine the thickness of the plate. Comparison between the results obtained by assuming that $\sigma _{\theta }=0$ and those observed experimentally might then afford a justification for this assumption as being adequate tor demonstrating the features of the mechanics of the problem which do not depend on the relationship between plastic stress and strain.

Though Bethe manages, by endowing his plastic material with the ability to suffer unsymnetric strains when subjected to a symnetrical stress, to avoid all consi'deration of successive steps by whioh any given configuration of finite strain is attained, this simplification cannot in general be made. In fact, so far as I am aware, no problem of plastic flow which involves finite displacements has ever been obtained ucept in cases such as the expansion of an infinite cylindrical tube by intern.al pressure, where symnetry alone enables the strain to be determined. For this reason it seems desirable to to:nnulate the equations for plastio radial flow round a hole in a sheet in a form which can be applied to any desired law ot strength such as Mohr's or v Mises' or any desired relationship between Lode's variables $\mu$ and $\nu$.

Analysis of strain round an expanding radial hole in a sheet.

When a hole is enlarged the finite strain at any stage ia made up of infinitesimal elements of strain which vary as the enlargement proceeds. Thus when a small pin hole in a plate is enlarged we must study the small strain produced in an element of the sheet which was originally at radius $s$ from the pinhole, when the hole enlarges fran radius $b$ to radius $b+\delta b$.