Page:American Journal of Mathematics Vol. 2 (1879).pdf/21

If this assumption were a cause of the approximation in the results, those results would not be essentially changed thereby in all ordinary cases of engineering practice as the compression is very slight.

In the case of glass the experiments of the late Mr. Louis Nickerson, C. E., of St. Louis, would seem to show that a high intensity of local pressure at the point where the external force is applied causes the neutral surface to move toward that point through an appreciable distance.

The general equations of equilibrium, however, do not indicate such a result, and there are strong resons for believing that his experiments may have indicated something different.

The first assumption made renders it possible to make uses of Lamé's general equations for homogeneous solids of constant elasticity. These are found on page 65 of his "Leçons sur le théorie mathématique de l'elasticité des corps solides," and are the following. Let $u,$ $v$ and $w$ be the actual displacements of any molecule along any assumed three rectangular axes of $x,$ $y$ and $z;$ then $N_{1},$ $N_{2}$ and $N_{3}$ represent the three normal intensities of stresses along these axes respectively, and $T_{1},$ $T_{2}$ and $T_{3}$ the intensities of tangential stresses producing moments around the same axes, i. e. $T_{1}$ around $x,$ $T_{2}$ around $y,$ and $T_{3}$ around $z.$ Let $\lambda$ and $\mu$ represent empirical constants depending on the nature of the material, and let $\theta ={\tfrac {du}{dx}}+{\tfrac {dv}{dy}}+{\tfrac {dw}{dz}}.$ This quantity $\theta$ will be recognized as the dilitation per unit of volume. Using this notation, the general equation for a homogeneous solid are

$N_{1}=\lambda \theta +2\mu {\tfrac {du}{dx}},$$T_{1}=\mu \left({\tfrac {dv}{dz}}+{\tfrac {dw}{dy}}\right)$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$......(1)
$N_{1}=\lambda \theta +2\mu {\tfrac {dv}{dy}},$$T_{1}=\mu \left({\tfrac {dw}{dx}}+{\tfrac {du}{dz}}\right)$
$N_{1}=\lambda \theta +2\mu {\tfrac {dw}{dz}},$$T_{1}=\mu \left({\tfrac {du}{dy}}+{\tfrac {dv}{dx}}\right)$

No demonstration of these equations is given, for it is difficult to conceive of one more elegant or more general than that given by Lamé.

Neglecting the effect of forces emanating from an exterior centre, the conditions of equilibrium are involved in the following equations, also given by Lamé,

${\tfrac {dN_{1}}{dx}}+{\tfrac {dT_{3}}{dy}}+{\tfrac {dT_{2}}{dz}}=0$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$........(2)
${\tfrac {dT_{3}}{dx}}+{\tfrac {dN_{2}}{dy}}+{\tfrac {dT_{1}}{dz}}=0$
${\tfrac {dT_{2}}{dx}}+{\tfrac {dT_{1}}{dy}}+{\tfrac {dN_{2}}{dz}}=0$