Laws of Elasticity

LAWS OF ELASTICITY Boyle's Law.—The first law of elasticity was published in 1662 by Boyle, and in 1676 by Edme Mariotte. It is now called the law of the isothermal expansion or contraction of a "perfect" gas, and is generally written in the form pv =f (e) , where v is the volume of unit mass of the gas and p the pressure intensity (see HYDROMECHANICS). The quantity on the right hand side depends only on the temperature 0, which was kept constant in Boyle's experiments. When 0 is the absolute tem perature, as measured by an air-thermometer, f (0) becomes simply RO, where R is a constant depending on the nature of the gas. The quantity v is the reciprocal of the density (p), and is frequently called the specific volume, while p is called simply the pressure. The law indicates that, under isothermal conditions, the pressure is proportional to the density. For a real gas this law has only a limited validity, it fails at high pressures and low temperatures, i.e., under conditions close to those at which liquefaction occurs.

If dp and dv denote small changes in p and v at constant tern perature, Boyle's law gives the equation p= —v dp The quantity on the right is called the volume-elasticity ; under isothermal conditions it is equal to the pressure, but under adia batic conditions, i.e., when there is neither gain nor loss of heat it is equal to -y p where y , the ratio of the specific heats at con stant pressure and constant volume, is always greater than r and is about r -4 for air.

In Boyle's law the pressure is supposed to be constant through out the unit mass of gas, but, since the choice of this unit is arbitrary, the law may be applied to a small portion of the gas, whose size may be diminished indefinitely. The limiting value of the ratio dv/v is called the dilatation (v), and—A is called the compression. The volume-elasticity k is thus the ratio of the in crease of pressure to the resulting compression. With a suitable definition of pressure this law is also applicable to solids, and to prevent misunderstanding the pressure so defined is often called hydrostatic pressure.

Hooke's Law.

The first law of elasticity for solid bodies was discovered by Robert Hooke in r66o, and published in 1676 in the form of an anagram meant to represent the words Ut tensio sic vis. Tensio is understood to mean what is now called extension. This, in the case of a wire stretched to a new length L, is the ratio of the gain in length, L—1, to the original length 1. Interpreting vis as the tension in the wire, the law states that the tension is pro portional to the extension. If the tension is regarded as a force, the factor of proportionality is found, not to be a physical con stant for the substance of which the wire is made, but to depend upon the area of the cross-section of the wire as well as upon the properties of the substance. For this reason it is more convenient to regard tension as a stress, i.e., the ratio of a force to the area across which it is transmitted. A more general and precise defini tion of stress, due to Augustin Louis Cauchy, will be given later, but the present one will suffice to indicate the physical dimensions of a stress and the most convenient units of stress. Extension will be regarded as a particular type of strain, a physical quantity which is always the ratio of two lengths and therefore a mere number, so no unit of strain is needed.

Hooke's law may be expressed in the form of an equation S=Es, where S is the stress, s the strain and E a modulus of elasticity, now called Young's modulus, in honour of Thomas Young, who gave it a physical meaning. This law does not hold for very great values of S, and even with small values of S it does not give a complete description of the elastic behaviour of the substance, as there is a lateral contraction when a rod or wire is stretched. For a long time, however, problems in the theory of elasticity were treated with the aid of this law alone, the first use of the law being made by Mariotte, who discovered it independently. He pointed out that, when a beam is bent, some of the longitudinal fibres are stretched and others contracted.

Bending of Rods.

This idea was developed by Jacques (Jakob, James) Bernoulli and Leonhard Euler into a theory of thin elastic rods, which involved the idea of a couple whose moment resists the bending moment and is proportional at each place to the curvature of the rod when bent. If the deflection y at a distance x from one end of the rod is small, the equation for the bending moment M is approximately where El is a constant called the flexural rigidity of the beam, E is Young's modulus for the material and I is a constant depend ing on the shape of the cross section of the rod. In the case of a rod, of length 1, free to turn about its ends and bent by forces acting along the line joining these ends, the bending moment (fig. r) is —yP, and this is balanced by the resisting moment if Since y is zero when x = o and when x= r, an appropriate solution is y = A sin ax, where = P and A sin al = o.

The last equation gives either A = o or al = n'ir, where n is an integer. A more complete theory, in which an exact expression for the curvature is used, indicates that the straight form, given by A = o, is stable if < and unstable if In the latter case a bent form, such as that shown in fig. r is stable. This result can be extended, with suitable modifications, to columns and frameworks supported and loaded in various ways. The fact that the criterion for the failure of a rod, through in stability of the straight form, depends upon the modulus E adds to the importance of this modulus as a physical quantity. The theory has been confirmed by the very careful experiments of Andrew Robertson. In practice a rod generally buckles under a load less than Euler's critical load, because the load is badly centred or because the rod has a slight initial curvature.

In the theory of the flexure of a thick beam, given by C. A. Coulomb, a line through the centroid of the cross-section, called the neutral line, is found to possess the property that the extension of a longitudinal fibre through a point Q of the section is repre sented by Cy, where y is the distance of Q from the neutral line and C is the curvature of the line of centroids at the position of this section. This relation is derived on the assumption that the cross-sections remain plane after bending.

The couple required to maintain flexure is then parallel to the plane of bending if f f xydxdy = o, where x is measured parallel to the neutral line. The couple is then given by the formula M = ff (ECy) ydxdy = ECI, where I is the moment of inertia of the cross-section of the beam about the neutral axis. The stress in the fibre under consideration is a tension or compression, according as y is positive or negative; it is given by the quantity ECy which may be expressed in the more convenient form My/I. There may be an additional tudinal stress in the fibre due to longitudinal end loads. If the total stress in a fibre is greater than the material can sustain there will be rupture. When the problem of flexure is treated by more exact methods, it is found that the bending moment is not always proportional to the curvature of the line of centroids. This was first noted by Karl Pearson, for a special distribution of lateral loads. In the case treated by Barre de Saint Venant, in which the proportionality was confirmed, the bending was supposed to be produced by forces applied to the terminal sections. The propor tionality fails even in this case if the beam is a very thin tube and is consequently capable of large displacements. A theory devel oped for these conditions by L. G. Brazier gives a relation between bending moment and curvature of the type shown in fig. 2. If the bending moment is increased above the maximum value, corre sponding to the point A, the beam must collapse. This is a new form of instability, and is an indication of the new results which may be expected from the general theory of elastic stability which has been developed by R. V.

Southwell and applied to rods, tubes and plates.

Torsion and Shearing Stress.

—The advantage of using the idea of stress instead of that of force is seen again from a consid eration of the torsion of a shaft of length 1. If 0 is the angular displacement of one end relative to the other, and T the total twisting couple transmitted along the shaft, there is, for small values, of 0, a relation of type where K is a constant which may be called the torsional rigidity of the shaft. This constant K is not, however, a physical constant for the material ; it depends in a rather complicated way on the shape and size of the cross-section. Cauchy's theory showed that, in the case of a cylindrical shaft of uniform cross-section, K could be written in the form µJ, where J is the moment of inertia of the cross-sectional area about the axis of the cylinder and µ is a physical constant called the rigidity of the material. Saint Venant later gave a general theory of torsion, in which K has the same form as before except that J is now a geometrical constant, the determination of which for a given section depends on the solu tion of a problem of potential theory. The stress which, in this theory, is supposed to be transmitted across any element of area of a cross-section, is a shearing stress, i.e., a stress derived from a force tangential to the area. This stress is expressed in terms of geometrical quantities by means of an extension of Hooke's Law appropriate for shearing stresses. This and other extensions of Hooke's Law will be given later when the ideas of stress and strain are made more definite. A mechanical method of determining K by means of a soap film stretched across a hole whose shape is the same as that of the cross-section of the shaft has been developed by A. A. Griffith and Geoffrey I. Taylor from a suggestion made by Ludwig Prandtl.

The Spiral Spring.—In 1848 James Thomson pointed out that the action of an ordinary spiral spring of circular cross-section depends mainly on torsion. The spring consists of a thin wire coiled into a helix of small pitch and radius e. When one end is fixed at a point 0 and the other attached to a weight W, it may be assumed that the axis of the coil is vertical and along the line OW. The twisting moment is then We, and the twist for a length 1 of coil is given approximately by the equation lWVc=KO.

The vertical displacement due to a twist OSs/l in an element of length Ss is cOSs/l. The total vertical displacement is therefore c0=/WO/K.

There is no appreciable horizontal displacement of the weight, because the horizontal components due to the different elements will on the whole neutralize one another. Unless the pitch is small it is necessary in a more exact theory to take into consideration the bending moment and the flexural rigidity. The spring then tends to uncoil when stretched.

Units of Stress.

The c.g.s. unit of one dyne per square cen timetre is rather small, and so a unit of one kilogram per square centimetre is used by many scientists. This unit will be denoted here by the symbol K. Other convenient symbols which will also be used are T for a stress of one ton per square inch, 7r for a stress of one pound per square inch and a for a unit pressure of one atmosphere. The transformation from one unit to another may be made with the aid of the relations I000a = 6.56r IT = 147007= Io33K = I '0I3 X C.G.S. units.

The unit used in the International Critical Tables is one kilo megabyre, a megabyre being equal to 14.57r. Young's modulus for steel is a little greater than 13,200T, consequently a tensional stress of 13.2T will produce an extension of only about o•ooi.