Elastic limit

“ It is not considered good practice to stress a material, especially a brittle material, repeatedly to a point half-way to its elastic limit. (The elastic limit is that stress that may be given to a body without producing a permanent set; below which it will return to its original shape upon the removal of the stress.) ”

“ On again applying the stress there is further rapid lengthening up to twenty-five per cent, this yielding causing a decrease of stress till the rod breaks at a limit below fifty-eight thousand pounds. The elastic limit and the breaking limit are thus widely different. In the case of steel the elastic limit is not reached until the stress becomes fifty thousand pounds. Its elastic limit is thus double that of wrought iron. Further increase of stress now causes the steel to increase its rate of stretching, and permanent strain results. ”

“ The yield-point may be raised, as, for instance, in wire which has been hardened by stretching, but the elasticity is much impaired, and it is only within very narrow limits, if at all, that proportionality between stress and strain is found. Subsequent prolonged rest gradually restores the elasticity, and after a sufficient number of weeks or months the metal is found to be elastic up to a point which may be much higher than the original elastic limit. ”

“ It is in fact the elastic limit rather than the ultimate strength of the material on which the question mainly depends of how high the working stress may safely be allowed to rise in any particular conditions as to mode of loading, and accordingly it becomes a matter of much practical importance to determine by tests the amount of stress which can be borne without permanent strain. ”

“ If the engineers of those stupendous structures had had at their disposal a metal of forty tons strength and twenty-eight tons elastic limit, instead of thirty tons strength and seventeen tons elastic limit in the one case and, say, twenty-two tons strength and fourteen to sixteen tons elastic limit in the other, how many difficulties would have been reduced in magnitude as the weight of materials was reduced! ”

“ If the load be increased to a value exceeding the elastic limit, and then kept constant, the metal will be seen to draw out (if the stress be one of pull) , at first rapidly and then more slowly. When the applied load is considerably less than the ultimate strength of the piece (as tested in the ordinary way by steady increment of load) it appears that this process of slow extension comes at last to an end. On the other hand, when the applied load is nearly equal to the ultimate strength, the flow of the metal continues until rupture occurs. ”

“ The true margin for safety is not the difference between the working-strain and the breaking-strain, but between the working-strain and that strain which will in any way unfit the material for use. Now, [Pg 34] any material is unfitted for use when it is so far distorted by overstraining that it cannot recover, or, technically speaking, when its elastic limit has been exceeded. The elastic limit of the best grades of iron is just about half the breaking-weight, or from 25,000 to 30,000 pounds per inch. ”

“ The maximum resistance of a tube or hollow cylinder to external stresses will be attained when all the layers are expanded simultaneously to the elastic limit of the material employed. In that case, observing the same notation as that already adopted, we have—
But since the initial internal stresses before firing, that is previous to the action of the pressure inside the bore, should not exceed the elastic limit,2 the value of R will depend upon this condition. ”

“ The most important point to consider when calculating shrinkage fits is the stress in the hub at the bore, which depends chiefly upon the shrinkage allowance. If the allowance is excessive, the elastic limit of the material will be exceeded and permanent set will occur, or, in extreme cases, the ultimate strength of the metal will be exceeded and the hub will burst. ”

“ In some substances the limit is reached with slight distortion, while others are perfectly elastic even when greatly stretched. India rubber is an example of a body having perfect elasticity through wide limits. Glass has great elastic force but its limit of elasticity is soon reached. Substances like India rubber may be said to have great "stretchability," but little elastic force. In physics, elasticity refers to the elastic force rather than to ability to endure stretching. ”

“ If the applied stress is less than a certain limit, the strain is in general small in amount, and disappears wholly, or almost wholly when the stress is removed. If the applied stress exceeds this limit, the strain is, in general, much greater than before, and most of it is found, when the stress is removed, to consist of permanent set. The limits of stress within which strain is wholly or almost wholly elastic are called limits of elasticity. ”

“ The total work done upon the specimen is equal to the area of the stress-strain diagram plus the effect of local inertia of the molecules at point of contact.
The stresses involved in impact are complicated by the fact that there are various ways in which the energy of the striking body may be spent:
(a) It produces a local deformation of both bodies at the surface of contact, within or beyond the elastic limit. In testing wood the compression of the substance of the steel striking-weight may be neglected, since the steel is very hard in comparison with the wood. ”

“ A body which has been strained beyond the limit of linear elasticity is often said to have suffered an “over-strain.” When the load is removed, the set which can be observed is not entirely permanent; but it gradually diminishes with lapse of time. This phenomenon is named “elastic after-working.” If, on the other hand, the load is maintained constant, the strain is gradually increased. This effect indicates a gradual flowing of solid bodies under great stress ”

“ Here, then, out of matter possessing rigidity, but absolutely devoid of elasticity, we have made a perfect model of a spring in the form of a spring balance. Connect millions of millions of particles by pairs of rods such as these of this spring balance, and we have a group of particles constituting an elastic solid; exactly fulfilling the mathematical ideal worked out by Navier, Poisson, and Cauchy, and many other mathematicians, who, following their example, have endeavored to found a theory of the elasticity of solids on mutual attraction and repulsion between a group of material particles. ”

“ It has been the common practice to give the poles sufficient strength to withstand this pull without failing. The maximum amount of the pull may be safely taken at the sum of the elastic limits of the wires, since it is unsafe so to design the spans as to be subject to larger stresses. ”

“ It possesses so-called elasticity of bulk. In the case of water having a free surface there is, as we have seen, a resistance to any change of level in the surface. This may be called an elasticity of surface form. Whenever an elastic material is strained or deformed, energy has to be expended on it to create the deformation. Thus to wind up a watch-spring, stretch a piece of indiarubber, compress some air, or bend a bow, requires an energy expenditure. ”

“ The elasticity of all the diameters of any plane perpendicular to the axis of a prism of rock crystal, may be considered as being sensibly the same.
2nd. All the planes parallel to the axis are far from possessing the same elastic state; but if any three of these planes be taken, restricting ourselves only to this condition, that the angles which they form with each other are equal, then their elastic state is the same. ”

“ In the case of a uniformly distributed stress, ab is a plane surface parallel to AB, and c is the centre of gravity of the surface AB. When a bar is subjected to simple pull applied axially—that is to say, so that the resultant stress passes through the centre of gravity of every cross-section —the stress may be taken as (sensibly) uniformly distributed over any section not near a place where the form of the cross-section changes, provided the bar is homogeneous in respect of elastic quality and is initially in a state of ease and the stress is within the limits of elasticity. ”

“ At the moment when the weight falls and the cord is stretched, the balance inclines, stretching the elastic spring, but the mass of one hundred grammes does not yet move; it is only when this spring is stretched that the mass, obedient to the prolonged action of this elastic spring, begins to move and rises, representing a certain amount of work accomplished.
Thus the suppression of shock in traction economizes a certain part of the moving labor; it is then advantageous to give to the traces of a carriage a certain elasticity. ”

“ A medium, in order to be available at all, must transmit actions across it in time ; therefore there must be postulated for it the property of inertia, — of the same kind as ordinary matter possesses, for there can hardly be a more general kind, — and also the property of elasticity or statical resistance to change either of position or of form. In ordinary matter the elasticity has reference solely to deformation; while in the constitution here assumed for the æther there is perfect fluidity as regards form, but elastic resistance to rotational displacement. ”

“ It therefore requires energy to bend a bow, just as truly and visibly as it does to raise a weight above the earth, and elasticity is, therefore, as truly a species of force as gravity is. We shall not here attempt to discuss the various ways in which this force may act, or in which a soUd elastic substance will resist all attempts to deform it; but in all cases it is clearly manifest that work must be spent upon the body, and the force of elasticity must be encountered and overcome throughout a certain space before any sensible deformation can take place. ”

“ Space is necessary simply to allow this tension to expend itself in work. It is this tension, not space, which reappears as work or kinetic energy." [8] In the case of a sling with a perfectly' elastic cord whose elongation is proportional to the strain, the tension or potential element does not " decrease as work is performed ; " on the contrary, it Increases directly with the work ; and whatever the centrifugal velocity of the ball, the period of revolution remains constant. [9] The elasticity does not " expend itself in work." ”

“ This power, whereby bodies are inabled to regain their first figure, is usually called their elasticity, and when it acts, it forces the bodies from each other, and causes them to separate. Now the effect of this elasticity in the present case is such, that if the bodies are perfectly elastic, so as to recoil with as great a force as they are bent with, that they recover their figure in the same space of time, as has been taken up in the alteration made in it by their compression together ”

“ What is the elastic curve?—the curve into which an elastic line forms itself under the pressure of a weight—is a problem which had been proposed by Galileo, and was fully solved, as a mathematical problem, by Euler and others.
But beams in practice are not mere lines: they are solids. And their resistance to flexure, and the amount of it, depends upon the resistance of their internal parts to extension and compression, and is different for different substances. ”

“ When a man stands at the center of a plank supported at its ends, the action of the man's weight bends the plank until the elastic force developed [Pg 94] in the plank equals the weight applied. Further, when a train or a wagon is on a bridge the bridge yields until it has developed an elastic reaction equal to the weight applied. If a person stands in the center of a room, the floor beams yield until the third law is satisfied. In fact, whenever a force acts, a contrary equal force always acts. ”

“ The elastic state of these plates cannot present such remarkable differences as those we have observed in the preceding series; for, being all cut round the axis of greatest elasticity, they can only contain in their plane that of least or that of mean elasticity, or lastly, those intermediate between these limits, which do not vary greatly from each other. ”

“ All the consequences of the kinetic theory of gases are already built on the assumption of elasticity: the application of this principle to the smaller particles is therefore to be viewed as a perfectly natural and logical consequence. In fact the gravitation pressure by the encounters of elastic [158] particles is explained quite as completely as the air-pressure is explained by the encounters of such particles (molecules) . ”

“ And therefore in mediums of equal density and elastic force, all the pulses are equally swift. Now if the density or the elastic force of the medium were augmented, then, because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of the density, the time which is necessary for producing the same motion as before will be increased in the subduplicate ratio of the density, and will be diminished in the subduplicate ratio of the elastic force. ”

“ To derive the play of internal force, this energy function must be varied with respect to all these independent quantities; the result is elastic tractional stress in the medium across every ideal interface, together with forcive tending to displace each strain-centre, which we can consider either as resisted by extraneous constraint preventing displacement of the strain-centre, or as compensated by the reaction of the inertia of the strain-form against acceleration. ”