SUMMARY OF THE GENERAL THEORY OF ELASTICITY.

The elastic properties of an isotropic homogeneous elastic body depend on two qualities of the body-viz. its compressibility and its rigidity. The compressibility determines the alteration in volume due to the action of external forces, the rigidity the alteration in form.

Compressibility and Elasticity of Volume.

Suppose we have a body whose volume is v, and that it is under a hydrostatic pressure P; let the pressure be changed to P+, and the volume in consequence to v-v. Then v/v is the change in unit volume due to the increment of the pressure, and v/(vp) is the change per unit volume due to unit increment of pressure.

This is called the compressibility of the body, which may be defined as the ratio of the cubical compression per unit volume to the pressure producing it. The reciprocal of the compressibility-viz. the value of vp/v-is the elasticity of volume. We shall denote it by k.

Rigidity.

Any alteration of external form or of volume in a body is accompanied by stresses and strains throughout the body. A stress which produces change of form only, without alteration of volume, is called a shearing stress.

Imagine one plane in the body to be kept fixed while all parallel planes are moved in the same direction parallel to themselves through spaces which are proportional to their distances from the fixed plane; the body is said to undergo a simple shear.

Suppose further that this simple shear is produced by the action of a force on a plane parallel to the fixed plane, and uniformly distributed over it; then the ratio of the force per unit of area to the shear produced is defined to be the rigidity of the body.

I

Let T be the measure of the force acting on each unit of area of the plane, and suppose a plane at a distance a from the fixed plane is moved through a distance c; then cla is defined as the measure of the shear, and the rigidity of the body is Ta/c.

Let us call this n. It may be shewn mathematically that, if a circular cylinder of radius and length / be held with one end fixed, the couple required to turn the other πρ4

end through an angle is n

0.

Modulus of Torsion.

The couple required to twist one end of unit length of a wire through unit angle, the other end of the wire being kept fixed, is called the modulus of torsion of the wire.

T

Hence if be the modulus of torsion, the couple required to twist one end of a length / through an angle 0, the other end being kept fixed, is 70/1.

Relation between Modulus of Torsion and Rigidity.

We have given above two expressions for the couple required to twist one end of a length of a wire of circular section through an angle 0, the other end being kept. fixed; equating these two expressions we get for a wire of radius r,

n =

2T

Young's Modulus.

If an elastic string or wire of length / be stretched by a

weight w until its length is l', it is found that

is constant

for that wire, provided that the wire is not strained beyond the limits of perfect elasticity; that is, the weight w must be such that, when it is removed, the wire will recover its original length.

If the cross section of the wire be of unit area, the ratio

of the stretching force to the extension per unit length is called Young's Modulus, for the material of which the wire. is composed, so that if the cross section of the wire be w sq. cm. and we denote Young's Modulus by E, we have

Relation between Young's Modulus and the Coefficients of Rigidity and Volume Elasticity.

We can shew from the theory of elasticity (see Thomson, Ency. Brit. Art. 'Elasticity'), that if E be Young's Modulus,

To determine Young's Modulus for copper, two pieces of copper wire seven or eight metres in length are hung from the same support. One wire carries a scale of millimetres fixed to it so that the length of the scale is parallel to the wire. A vernier is fixed to the other wire,' by means of which the scale can be read to tenths of a millimetre. The wire is prolonged below the vernier, and a scale pan attached to it; in this weights can be placed. The wire to which the millimetre scale is attached should also carry a weight to keep it straight. Let us suppose that there is a weight of one kilogramme hanging from each wire.

Measure by means of a measuring tape or a piece of string the distance between the points of suspension of the

We believe that we are indebted indirectly to the Laboratory of King's College, London, for this elegant method of reading the extension of a wire.

wires and the zero of the scale. Let this be 716.2 centi

metres.

Now put into the pan a weight of 4 kilogrammes, and read the vernier. Let the reading be 2:56 centimetres.

The length of the wire down to the zero of the vernier is therefore 718.76 centimetres.

Now remove the 4 kilogramme weight from the pan. The vernier will rise relatively to the scale, and we shall obtain another reading of the length of the wire down to the zero of the vernier. Let us suppose that the reading is 023 centimetre. The length of the wire to which the millimetre scale is attached is unaltered, so that the new length of the wire from which the 4 kilogramme weight has been removed is 718.53 centimetres.

Thus, 4 kilogrammes stretches the wire from 718.53 centimetres to 718.76 centimetres. The elongation, therefore, is o'23 centimetre, and the ratio of the stretching force to the extension per unit length is

4×71853
23

or 12500 kilogrammes approximately.

We require the value of Young's Modulus for the material of which the wire is composed. To find this we must divide the last result by the sectional area of the wire.

If, as is usual, we take one centimetre as the unit of length, the area must be expressed in square centimetres.

Thus, if the sectional area of the wire experimented on above be found to be oor square centimetre (see § 3), the value of the modulus for copper is

12500

ΟΙ

or 1250000 kilogrammes per square centimetre.

The modulus is clearly the weight which would double the length of a wire of unit area of section, could that be done without breaking it..

Thus, it would require a weight of 1,250,000 kilo.

It is assumed that the zero of the vernier corresponds with its point of attachment to the wire,

grammes to double the length of a copper wire of one square centimetre section.

The two wires in the experiment are suspended from the same support. Thus, any yielding in the support produced by putting on weights below or any change of temperature affects both wires equally.

It is best to take the observations in the order given above, first with the additional weight on, then without it, for by that means we get rid of the effect of any permanent stretching produced by the weight.

The wire should not be loaded with more than half the weight required to break it. A copper wire of oor sq. cm. section will break with a load of 60 kgs. Thus, a wire of o'or sq. cm. section may be loaded up to 30 kgs. The load required to break the wire varies directly as the crosssection.

To make a series of determinations, we should load the wire with less than half its breaking strain, and observe the length; then take some weights off-say 4 or 5 kgs. if the wire be of about o'o1 sq. cm. section, and observe again; then take off 4 or 5 kgs. more, and observe the length; and so on, till all the weights are removed.

The distance between the point of support and the zero of the millimetre scale, of course, remains the same throughout the experiment. The differences between the readings of the vernier give the elongations produced by the corre sponding weights.

The cross-section of the wire may be determined by weighing a measured length, if we know, or can easily find, the specific gravity of the material of which the wire is made. For, if we divide the weight in grammes by the specific gravity, we get the volume in cubic centimetres, and dividing this by the length in centimetres, we have the area in square centimetres.

It may more readily be found by the use of Elliott's wire-gauge (see § 3).