Notes on H.A. Bethe's "Theory of armor penetration". 1. Static penetration

by G. I. Taylor

CONFIDENTIAL. R.C. 279.

MINISTRY OF HOME SECURITY.

CIVIL DEFENCE RESEARCH COMMITTEE.

Notes on H.A. Bethe's "Theory of armor penetration".

I. Static penetration.

By Professor G. I. Taylor, F.R.S.

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.^[1] 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. (2) The absolute magnitude of the strain ellipsoid bears no relationship to the stresses if the plastic body is assumed to possess the property tba t flow will occur when the yield a tress is reached.

(3) It is a necessary condition of isotropy of the plastic material that the directions of the principal axes of the stress and strain ellipsoids shall coincide.

(4) Owing to the fact that $e_{1}'+e_{2}'+e_{3}'=0$ and $\sigma _{1}'+\sigma _{2}'+\sigma _{3}'=0$, it is necessary to know only one relationship in order to determine the ratios $e_{1}':e_{2}':e_{3}'$ when the ratio of any pair of $\sigma _{1}'$, $\sigma _{2}'$, $\sigma _{3}'$ is known. This relationship can conveniently be defined in terms of two non-dimensional variables $\mu$ and </math>\nu</math> (Lode's variables)

$\mu =2{\frac {\sigma _{2}-\sigma _{1}}{\sigma _{1}-\sigma _{3}}}-1,\qquad \nu =2{\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}-1$

where $\sigma _{1}>\sigma _{2}>\sigma _{3}$. These variables are chosen for convenience so that $\mu$ lies between $-1$ and $+1$. It seems that all plastic materials must satisfy the relationship $e_{1}>e_{2}>e_{3}$ when $\sigma _{1}>\sigma _{2}>\sigma _{3}$, so that $\nu$ also lies between $-1$ and $+1$. The observed relationship between $\mu$ and $\nu$ for mild steel, soft iron and copper is given in a paper by Taylor and Quinney^[2], and for copper, iron and nickel by Lode^[3]. For all these metals the relationship is substantially that shown in Fig.1, which also contains Taylor and Quinney's experimental results. This experimental relationship may be compared with that which exists in all Newtonian viscous fluids, namely $\mu =\nu$. It seems unlikely that the divergence between the observed relationship and the assumed $\mu =\nu$, though used by v. Mises, is quite unrelated to v. Mises' criterion of strength. The relationship $\mu =\nu$ could equally well be used with Mohr's strength relationship, namely that flow begins when $\sigma _{1}-\sigma _{3}=$constant$=Y$.

Bethe's stress-strain assumption.

Bethe considers two regions of plastic flow, the outer one extending inwards from the outer limit of plastic flow $r=r_{1}$ to the radius $r=r_{2}$ at which the tangential stress ceases to be a tension. In this region the radial stress must be taken as $\sigma _{2}$, the tangential tension as $\sigma _{1}$, and $\sigma _{2}$, the intermediate stress normal to the sheet, is zero. Between $r=r_{1}$ and $r=r_{2}$, therefore, Lode's variables $\mu$ is positive but $<1$. At $r+r_{2}$, $\mu =1$ since at that point $\sigma _{2}=\sigma _{1}=0$. In this region Bethe's assumption (which he attributes to Mohr) is that the plastic flow is limited to the plane of the sheet, no thickening occurring (see p.9 of Bethe's report). If the strain is limited to the plane of the sheet $e_{2}=0$ and if the effect of compressibility is neglected $e_{3}=-e_{1}$. Thus in the region $r_{1}>r>r_{2}$, $\nu =0$. This is shown in Fig.1 by means of the line AB.

Though Bethe's strain assumption is very far from what is observed in experiments in which plastic strains are measured, yet this does not necessarily detract from the value of his calculation of stress distribution in the region $r_{1}>r>r_{2}$, because with the "ideal" plastic body, which begins to flow as soon as the stress reaches a given value and continues flowing until the stress is reduced, an infinitesimal plastic strain may enable the equilibrium stress distribution to be attained. In other words, if only a negligible small thickening of the sheet does occur it will produce only a negligible effect on the stress distribution. 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 $b+\delta b$. i” i

a

In the more general case when the initial radius of the hole in the unstretched sheet is not zero this is very difficult to analyse, but when the expansion starts from a small pinhole it may be expected that the configur- ation when the hole has radius b. will be similar to that round the hole when its radius is b, except that the radii where any given thickness occurs will be changed in’the ratio b,/b,. Thus if h is the thickness and u the radial displacement, it may be assumed that h/ho and u/b and also the stresses are functions of s/b only: where h, is the initial thickness of the sheet.

To simplify matters I have assumed that the canpressibility is so small that it may be neglected and the material taken as incanpressible. The relationship between the small strain which occurs at any radius during the expansion of the hole through a small increase in radius fram b to b + § b can be understood by referring to Fig.2. Here the ordinates represent u and the abscissae r.

The initial radial distance s of the element which at a subsequent stage in the opening out of the hole is at radius r is related to u by the equation

roe a+. Snes oT)

In Fig.2, therefore, the displacement of a particle fram its initial radius s ig represented by a line drawn at 45° to the axes. In particular the displacement of the particles which were initially at the pinpoint where the hole began is represented by the 45 line ORP,. The curved line P,AQ, represents the relationship between r and u which it is the object of the analysis to calculate. At a subsequent stage of the expansion, when the hole has expanded fran radius b to radius b + 5b, the curve P,BCQ, representing displacement is similar to R AQ, but with its linear dimensions increased in the ratio (b+§b) : b; thus in Fig.2 P.Po . AC . AD - Sb so that OP, AO r b

AD = r5b/b. saree

If Sr is the change in r for a given particle of material when the hole

expands fram b to b + Sb, $r is found by drawing the line AB at 45° to the axes to meet the curve P,BCQ, in B. If $b/b is small enough, the arc CB may be taken as straight so that if 17 -d is the slope of CB to the axis

oa = -tana. conte Cay

If S is the angle AOQ., tan 3 = u/r. Pram the geometry of the figure ABCD (Pig. 2)

Sr = AP = BP = CE tana + DA tan = (DA~ir)tana +DA tang

reeacsths

Hence | ee (tone + tam) on ea rgd § 3 |

+ tand |

and fran (2) : u_

Sr = es r 2 seenes tS) i

or |

|

The radial strain canponent during the expansion of the hole fran b tob+ $b is 2 ) and differentiating (6) with respect to r keeping $b constant, ar (er pouce - 5 -

fr (Sr)

• • • • • (7)

=

Since the strain during expansion of the hole f'rcm b to b + ~ b ia proportional to ~b/b, it is convenient to detine strain canponenta C.,., and Bz. so that straina during the small enlargement Sb are €,. ~b/b, E,, ~ b/b, Cz. ~b/b. With this definition

£9

• • . • • ( 8)

The tangential strain is simply

_ -

[

~r -~] ~r

••••• (9)

1 - ~ 'ar

and the strain perpendicular to the sheet is

....

(10)

The thickness h at any stage can be found simply tran the equation of'

continuity:

it is given by h

• • • • ( 11)

ho = where h 0 is the initial thickness of the sheet.

It is a simple matter to verify that (10) is consistent with (11). These expressions for strain take simple :f Oima when expressed in tenns of a new independent variable '8 = r"' and a new dependent variable '? = a :a. = (r-u) .l.. Making these transfonnati ems and writing p

d'f =~

•••• { 12)

,

(8) and (9) becane • • • • {13) E~

=

1 -

....

i

~p

while {11) reduces to the ·simple form

h/ho

= p.

{14)

.... ( 15)

It is a simple matter to deduce ( 14) directly tran ( 15). The stress equilibrium equation for a thin sheet is

fr {ha;) +

h(o)"; o-;)

•

o

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1. not counting as parameters the compressibility of the material or the yield strength.
2. Phil. Trans. Roy. Soc., 230, 1931.
3. "Versuche uber den Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle, Eisen, Kupfer und Nickel", Z. Physik, vol.36 (1926).