as in case of a stretched wire, it is said to be homogeneous. In such a case the strain ellipsoids are all alike and similarly situated, as shown in figure 130.

When a fluid is compressed the strain is homogeneous and the ellipsoids are spheres slightly smaller than in the unstrained state.

The distribution of strain in a bent beam is shown by the ellipsoids in figure 131. The strain in this case is not homogeneous and there is a surface of no strain indicated by the dotted line.

237. Resistance to Strain.-A body is said to be elastic if after having been strained it springs back to its original form when the stress is removed. If the stress is the same for a given amount of strain whether the strain is increasing or diminishing, the body is said to be perfectly elastic.

When strained beyond a certain point called the limit of elasticity, substances yield permanently and do not return to the original state when the straining forces are removed. In this case there may be a great internal stress while the body is being strained, but on a very slight diminution of strain the stress entirely disappears. Putty, wet clay, and lead all exhibit this permanent distortion under comparatively small forces and even when the strain is small; while india-rubber is remarkable for the great strain which it can experience without passing its elastic limit. It is said to have a wide limit of elasticity.

Even within the limits of elasticity most substances show a time lag in returning to their original state after having been strained. Thus when a steel wire is firmly clamped at its upper end, if the lower end is twisted through an arc well within its limit of elasticity, the wire when set free returns at once nearly to its original position, but creeps very slowly back through the

remaining distance. This lag is found in metals and in glass, but quartz fibers are remarkably free from it.

238. Hooke's Law. In small strains of elastic bodies the stress is proportional to the strain. This is known as Hooke's law, having been enunciated by him in 1676. According to this law, a long spring when stretched two centimeters will exert twice the force that it would if stretched one centimeter, and the tension required to stretch a spring a small distance is equal to the pressure when the spring is compressed an equal amount.

Careful experiment, however, shows that the law is not exactly true. Most substances offer slightly more resistance to a given small compression than to an equal extension.

An illustration of this law is afforded by the ordinary spring balance in which equal divisions of the scale correspond to equal increments of weight. In this case the elongation or compression of the helical spring may be relatively very great, yet because of its shape the distortion or strain of any little portion is extremely minute and Hooke's law holds very nearly true.

239. Elasticity. In elastic bodies the elasticity is measured by the ratio of the stress to the corresponding strain.

In bodies which are homogeneous and isotropic there are two principal kinds of elasticity, that in virtue of which the body resists change of volume and that resisting change of shape.

The first is called volume elasticity and the second rigidity. Volume elasticity is possessed by all bodies, fluids as well as solids, but rigidity is a characteristic of solids.

In some strains both of these elasticities are involved; for instance, when a wire is stretched there is a sidewise contraction as well as an elongation, so that the resistance to stretching depends on both the rigidity and volume elasticity of the substance. The elasticity of stretching or compression is so important in engineering that it has received a special name. and is known as Young's modulus.

240. Volume Elasticity.-When a body is so strained that every little cubical portion is compressed into a smaller cube

the corresponding stress must be a pressure equal in all directions, provided the substance is isotropic or equally compressible in every direction.

This kind of stress is called hydrostatic pressure because it is the only kind of stress that can exist in fluids at rest.

The volume elasticity or bulk modulus of a substance is the ratio of the increase in pressure to the corresponding compression per unit volume.

Thus this elasticity will be represented by

E=Volume Elasticity=

pressure increase change in unit volume

Р

υ

V

=

where p is the increase of pressure causing a contraction v in a total volume V.

The elasticity of solids may be found by subjecting a long

FIG. 132.-Oersted's piezometer.

bar of the substance to hydrostatic pressure in a strong tube having thick glass windows through which its changes in length may be observed by fixed microscopes.

241. Compressibility of Liquids.— Liquids are, as a rule, somewhat more compressible than solids, but on the other hand so great is their resistance to compression that for most practical purposes they may be treated as if incompressible.

The compressibility of a liquid may be measured by the apparatus shown in figure 132, known as Oersted's piezometer. In this instrument the liquid to be tested is contained in a bulb of glass terminating in a long narrow tube of uniform diameter, open at the end and carefully graduated. This bulb A is surrounded by water in a stout cylindrical vessel of glass and subjected to pressure by means of a piston forced in by a screw. A globule of mercury in the narrow tube

separates the liquid in the bulb from the surrounding water. From the number of scale divisions through which the mercury moves down toward the bulb as pressure is applied, the apparent compressibility of the contained liquid is determined, the relation between the volume of the bulb and the volume contained in one division of the capillary tube having been previously ascertained. Although the pressure is the same on the outside of the bulb as on the inside, its volume diminishes in consequence of the compression of the glass of which it is made, so that the experiment gives the difference between the compressibility of the liquid and that of the glass bulb.

A thermometer gives the temperature of the liquid examined, and the pressure may be determined from the amount of compression observed in a tube M containing air and placed open end downward in the cylinder.

242. Elasticity of Gases.-In case of a gas it is necessary to distinguish between its elasticity when the temperature is kept constant during the compression, and its elasticity when compressed so suddenly that there is no time for the flow of heat to take place. The first is called isothermal elasticity and the second adiabatic elasticity; the latter is always greater, being, in case of air, oxygen, hydrogen, and nitrogen, about 1.40 times as great as the isothermal elasticity.

The isothermal elasticity of a gas may be calculated from Boyle's law. Suppose the pressure is increased from p to p', the decrease in volume will be v-v' and we have

But the difference between p and p' is supposed extremely small, so that for gases kept at constant temperature the volume elasticity is equal to the pressure.

243. Rigidity. If a cylindrical rod or wire is twisted about its axis without change of length it may be imagined divided into sections of equal thickness, in each of which there has been no change in volume but simply a distortion of the little elements of which it may be conceived as made up.

Α Α'

F

B BI

Take such a little block as that represented in figure 133. If the base CD is firmly fixed, a force F applied to the upper surface will strain it into the position A'B', just as a thick book lying on a table may be pushed out of shape by force applied to the upper cover. The strain in this case is a pure distortion without any change in volume and is called a shear, and the forces bringing it about constitute a shearing The strain is measured by the ratio of the displacement AA' or x to the height h, while the stress is the force applied per unit area; or if S is the area of the upper

FIG. 133.

stress.

F

surface of the block the stress is

The rigidity n, or elastic

S

resistance to distortion, may therefore be expressed thus: