LIMITS OF ELASTICITY Behaviour of a Substance Under Great Stress.—When a rod or wire has been stretched by a load, it recovers its original length and elasticity after the removal of the load, provided the stress produced by the load does not exceed a certain threshold value. This value is called the elastic limit; it generally gives very closely the limit of the range for which the strain is proportional to the stress, but may be slightly less than the proportional limit. With other kinds of stress the reverse may be the case. In Albert A. Michelson's experiments with twisted prisms the elastic strain produced by a stress p was found to be given by an expression of type Cpe' P, where C and h are constants, while in P. W. Bridg man's experiments with substances subjected to hydrostatic pres sures up to 12,000K no elastic limit was detected, though there was always a limit of proportionality. As the pressure increases the compressibility decreases, the decrease for solids being gen erally less than for liquids.
With stresses beyond the elastic limit the behaviour is differ ent for different substances, but generally the material becomes plastic when the load is applied, and the magnitude of the exten sion corresponding to a given load is indefinite and varies with the time. With a sufficiently large load the extension may increase with time in such a manner as to lead to rupture; with a smaller load the rate of increase, after quickly reaching a maximum value, may gradually decrease to zero, just as if the motion were resisted by a kind of viscous drag. In this case a state of plastic equilib rium is approached. On the removal of the load the rod may not return to its original length, but may assume a slightly greater length. Sometimes the old or a new length is approached grad ually, as if a viscous drag were again operative. This gradual creeping to a different length is called elastic a f terworking; it was discovered by Wilhelm Weber in If, after receiving a permanent set or deformation, a rod is reloaded, the strain may not be exactly proportional to the stress, but in some cases it is, and the proportional limit may be either greater or less than before. Sometimes the proportional limit is surprisingly low, but if the rod be allowed to rest for some days after the overstrain it recovers the property of linear elasticity. The stress at which the strain begins to increase very rapidly as the load is increased is called the yield point. For mild steel it may be about 18T. The cross-section of the bar now diminishes in area, and just before rupture a narrow neck forms. The exten sion may then be about 30.6% and the reduction in area 67.4%.
In order to avoid calculation of the stress at each stage, engi neers generally use a load-extension diagram in preference to a stress-strain diagram, and the stress at rupture, or ultimate strength, is a nominal stress calculated on the basis of the original cross-section. The actual stress is much greater than this; and it increases right up to the moment of rupture, while the load may actually decrease after the yield point has been passed. The ulti mate strength of ordinary mild steel is about 30T, that of piano wire about 120T, while the ultimate strength of wood is generally less than 8 T.
The stresses thus found do not necessarily form a physically possible system, for it may happen that the calculated stress in one or more of the rods exceeds the proportional limit. One method of treating the problem in such a case is to make the simplifying assumption that, when the stress in a rod reaches a certain yield point, it cannot exceed this value. Assuming that certain rods behave as plastic and the rest as elastic members, the stresses in the plastic rods are given by the known yield points, while the stresses in the elastic rods may be determined by the methods of statics or by an application of the principle of least work.
The stresses determined by the ordinary methods of statics or by the principle of least work may fail to represent a physical possible system for another reason. If the stress in a member is compressive and is greater than a certain critical value, the mem ber will buckle. This critical value depends on the modulus E, and may be less than the yield point of the material. For some purposes it is useful to compare an elastic solid with a structure of elastic rods, and one of the important problems of the theory of elasticity is the determination of the critical loads for struts, structures and elastic plates of various shapes. These loads may be expected in general to depend, not only upon E, but also upon the other elastic constants just as in the case of a structure they depend upon the geometrical form. In some cases the problem can be treated by the elementary methods, in which the displace ments of particles are treated as small, but for a full elucidation of some points a more exact theory is needed.
Following up Boyle's idea of particles of air endowed with the properties of a spring, Sir Isaac Newton endeavoured to explain the simple proportionality between air pressure and density by introducing the idea of a repulsive force between two molecules depending only on their mutual distance. Newton actually used an inverse distance law, but was careful to state that the force must be regarded as operative only when the distance does not exceed a certain length. An atom or molecule was thus regarded as having a more or less definite sphere of action, and attractive forces of various intensities were regarded as the cause of chemical action. The idea of molecular attractive forces with a limited range of action was used also by P. S. Laplace in his theory of surface tension.
The use in physics of forces depending only on the distance and acting only over a definite range had thus already received some sanction when C. L. M. H. Navier used the idea in 1821 to obtain a set of general equations for the equilibrium and vibration of an elastic solid. Navier's molecules, like those of R. G. Boscovich, were practically centres of force, and his equations for an iso tropic substance involved only a single elastic constant analogous to Young's modulus. The theory was carried a step further by Cauchy, who imagined the molecules to form a homogeneous assemblage or lattice. He thus obtained a generalization of Hooke's law involving 15 elastic constants, and regarded it as applicable to any crystalline or aeolotropic substance.
The need for a greater number of elastic constants for both crystalline and isotropic bodies was realized by S. D. Poisson who, prior to Navier's work, had used the Newtonian ideas in some work on the equilibrium of an elastic surface. Poisson proposed to go back to the Newtonian conception of a molecule with a definite shape and size, and to consider the effect of changes in the orienta tion as well as the position of the molecules of an elastic body. The calculations were completed by Woldemar Voigt in 1887. Shortly afterwards Lord Kelvin (William Thomson), in an attempt to raise the number of elastic constants for a crystalline substance to 21, represented a molecule by a pair of centres of force. This work has been extended by Max Born and his co-workers. By using a number of centres of force as the structural element, it is possible to calculate the elastic constants with fair accuracy and to estimate also the thermal characteristics of a crystalline sub stance. The hypotheses upon which the structure theory is based are only rough approximations to the truth, and so the general equations of elasticity have also been derived from general mechanical and thermodynamical principles, a solid body being treated as continuous.