Page:Elementary Text-book of Physics (Anthony, 1897).djvu/125

§ 100] '''99. The Superposition of Strains.'''— We will now show that two elongations, applied successively or simultaneously in the same direction, are equivalent to a single elongation equal to their sum. This follows from the assumption already made, that the elongations are so small that their squares or products may be neglected. For, suppose a line of unit length to receive the elongation $$e_{1}$$; its length becomes $$1 + e_{1}$$. If it then receive the elongation $$e_{2}$$, its length becomes $$1 + e_{1} + e_{2} (1 + e_{1}) = 1 + e_{1} + e_{2}$$, because the product $$e_{1}e_{2}$$ may be neglected. This principle is called the principle of the superposition of strains.

By its help we may show that a simple elongation may be produced by the combination of a dilatation and two equal shears in planes at right angles to each other. In the case of a simple elongation, the elongations along the principal axes are $$e, 0, 0$$. Let this suppose a dilatation of which the elongations are $$\frac{e}{3}, \frac{e}{3}, \frac{e}{3},$$; a shear of which the elongations are $$\frac{e}{3}, - \frac{e}{3}, 0,$$; and a shear of which the elongations are $$\frac{e}{3}, 0, - \frac{e}{3},$$. By the principle of the superposition of strains we find the elongations produced if these three strains be superposed by adding the three elongations along the three axes. Carrying out this operation we obtain $$e, 0, 0$$ as the elongations produced by the superposition, that is, the superposition of these three strains is equivalent to a simple elongation. Since all homogeneous strains may be produced by three simple elongations at right angles to each other, any homogeneous strain may be produced by a combination of dilatations and shears.

100. Stresses.—If a body be maintained in equilibrium by forces applied to points on its surface, and if we conceive it divided into two parts, $$A$$ and $$B$$, by an imaginary surface drawn through it, and if we assume, for the present, the molecular structure of matter, it is clear that the forces applied to the portion $$A$$ of the body are in equilibrium with the forces which act between the molecules of $$A$$ lying near the surface which divides it from $$B$$, and the