Page:Elementary Text-book of Physics (Anthony, 1897).djvu/123

§ 98] for every external stress in which its molecules are in equilibrium. Any change of the stress in the body is accompanied by a readjustment of the molecules, which is continued until equilibrium is again established.

98. Strains.—The complete geometrical representation of the changes of form which occur when a body is strained is in general impossible, or at least exceedingly complicated. In the theory of elasticity it is generally possible to avail ourselves of a simplification in the character of the strain, which facilitates its geometrical representation, by assuming that the strain is such that a line in the body which was straight in its unstrained position remains straight after the strain: such a strain is called a homogeneous strain. It may be shown, by an argument too extended for presentation here, that in any case of homogeneous strain there are always three directions in the strained body, at right angles to one another, in which the only change produced by the strain is a change in length and not a change in relative direction. Thus, if the strained body be originally a cube, with its sides parallel to these three directions, the cube will strain into a rectangular parallelepiped. If the strained body be originally a sphere, it will strain into an ellipsoid, the three axes of the ellipsoid being the three directions already mentioned. These three directions are called the principal axes of strain.

The increase in length of a line of unit length by strain is called its elongation. Evidently, from the description of the relations of a homogeneous strain to the principal axes, the whole strain will be described if the elongations along the principal axes be given. Let us denote by $$e_{1}, e_{2}, e_{3}$$ the elongations, which may be either positive or negative, along the three principal axes. These elongations are assumed to be so small in comparison with the unit line that their squares or products may be negkcted. Then, in the examples just given, if $$a$$ represent a side of the cube before strain and $$a^3$$ its volume, the increase in volume of the cube by the strain is given by $$a^3(1 + e_{1}) (1 + e_{2}) (1 + e_{3}) - a^3 = a^3 (e_{1} + e_{2} + e_{3}),$$ since the products of the $$e$$'s may be neglected. Similarly, the sphere,