Page:Elementary Text-book of Physics (Anthony, 1897).djvu/124

110 of which the radius is $$r$$, becomes by the strain the ellipsoid, of which the axes are $$r (1 + e_{1}), r (1 + e_{2}), r (1 + e_{3})$$; the increase in volume of the sphere by the strain is therefore The quantity $$e_{1} + e_{2} + e_{3}$$ is called the coefficient of expansion of the body.

Two cases of strain need to be specially examined—the pure expansion or dilatation, and the shear or shearing strain. A dilatation occurs if the three coefficients of elongation are equal; in this case the strained cube remains a cube, the strained sphere remains a sphere, and the change of volume in each case is $$3e$$ times the original volume. A shear occurs when one of the coefficients, say $$e_{3}$$, equals zero, and when $$e_{1}$$ equals $$-e_{2}$$; in this case the expansion is zero.

The shear may be defined from another point of view. For, consider a body subjected to a shear and suppose a section made in it by the plane containing the elongations $$e$$ and $$-e$$: it is clear that the shear will be completely described if we describe the deformation of a figure in this plane. We select for this purpose a rhombus, $$ABCD$$, of which the diagonals $$AB$$ and $$BC$$ are so related that after the shear we have $$AD (1 + e) = BC$$ and $$BC (1 - e) = AD$$. If the rhombus produced by the shear be turned until one of its sides coincides with $$AB$$, we shall have the original rhombus and the one produced by shear in the relation shown in Fig. 34. The new rhombus $$AC'D'B$$ may manifestly be produced from the original rhombus by the displacement of all its lines parallel to the fixed base $$AB$$, each line being displaced by an amount proportional to its distance from the line $$AB$$. The ratio of this displacement to the distance of the displaced line from the base $$AB$$ is called the amount of the shear; that is, $$\frac{DD'}{EB}$$ is the amount of the shear.