Page:Notes on HA Bethes Theory of Armor Penetration 1. Static Penetration.pdf/3

(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^[1], and for copper, iron and nickel by Lode^[2]. 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.

1. Phil. Trans. Roy. Soc., 230, 1931.
2. "Versuche uber den Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle, Eisen, Kupfer und Nickel", Z. Physik, vol.36 (1926).