($$ 396, 401) for Abstract Dynamics, we ignore variation of temperature in the body. If, however, we add a condition of absolutely no variation of temperature, or of recurrence to one specified temperature after changes of strain, we have a definition of that property of perfect elasticity towards which highly elastic bodies in nature approximate; and which is rigorously fulfilled by all fluids, and may be so by some real solids, as homogeneous crystals. But inasmuch as the elastic reaction of every kind of body against strain varies with varying temperature, and (a thermodynamic consequence of this, as we shall see later) any increase or diminution of strain in an elastic body is necessarily accompanied by a change of temperature ; even a perfectly elastic body could not, in passing through different strains, act as a rigorously conservative system, but on the contrary, must give rise to dissipation of energy in consequence of the conduction or radiation of heat induced by these changes of temperature. But by making the changes of strain quickly enough to prevent any sensible equalization of temperature by conduction or radiation (as, for instance, Stokes has shown, is done in sound of musical notes travelling through air); or by making them slowly enough to allow the temperature to be maintained sensibly constant? by proper appliances; any highly elastic, or perfectly elastic body in nature may be got to act very nearly as a conservative system. 644. In nature, therefore, the integral amount, w, of work defined as above, is for a perfectly elastic body, independent (§ 246) of the series of configurations, or states of strain, through which it may have been brought from the first to the second of the specified conditions, provided it has not been allowed to change sensibly in temperature during the process. When the whole amount of strain is infinitely small, and the stresscomponents are therefore all altered in the same ratio as the straincomponents if these are altered all in any one ratio ; w must be a homogeneous quadratic function of the six variables e, f, g, a, b, c, which, if we denote by (e, e), (8,5)... (e, f)... constants depending on the quality of the substance and on the directions chosen for the axes of co-ordinates, we may write as follows:w={{[e,e) e? +($.$$+(8,8)go +(a,a)a2 + (6,6) 62+(C,c)c +2(ef)ef+2 (0,8) cg+2 (e, a) e a +2 (e, b) e 6+2 (e, chec +2(8,8) fg+2 f, a) fa + 2 (8,6) f6+2 (5.c)fc +2 (ga) ga+2(g,b)gb+2 (8,0) gc +2 (a,b) ab +2 (a,c) ac +2b,c) bc} The 21 co-efficients (e, e), (1,1)... (6, c), in this expression constitute the 21 co-efficients of elasticity,' which Green first showed to be proper and essential for a complete theory of the dynamics of an elastic solid subjected to infinitely small strains. The only condition 1. On the Thermoelastic and Thermomagnetic Properties of Matter' (W. Thomson). Quarterly Journal of Mathematics, April 1857. 6 2 Ibid. that can be theoretically imposed upon these co-efficients is that they must not permit w to become negative for any values, positive or negative, of the strain-components e, f, ...l. Under Properties of Matter, we shall see that a false theory (Boscovich's), falsely worked out by mathematicians, has led to relations among the co-efficients of elasticity which experiment has proved to be false. 645. The average stress, due to elasticity of the solid, when strained from its natural condition to that of strain (e, f, g, a, b, c) is (as from the assumed applicability of the principle of superposition we see it must be) just half the stress required to keep it in this state of strain. 646. A body is called homogeneous when any two equal, similar parts of it, with corresponding lines parallel and turned towards the same parts, are undistinguishable from one another by any difference in quality. The perfect fulfilment of this condition without any limit as to the smallness of the parts, though conceivable, is not generally regarded as probable for any of the real solids or fluids known to us, however seemingly homogeneous. It is, we believe, held by all naturalists that there is a molecular structure, according to which, in compound bodies such as water, ice, rockcrystal, etc., the constituent substances lie side by side, or arranged in groups of finite dimensions, and even in bodies called simple (i.e. not known to be chemically resolvable into other substances) there is no ultimate homogeneousness. •In other words, the prevailing belief is that every kind of matter with which we are acquainted has a more or less coarse-grained texture, whether having visible molecules, as great masses of solid stone or brick-building, or natural granite or sandstone rocks; or, molecules too small to be visible or directly measurable by us (but not infinitely small) in seemingly homogeneous metals, or continuous crystals, or liquids, or gases. We must of course return to this subject under Properties of Matter; and in the meantime need only say that the definition of homogeneousness may be applied practically on a very large scale to masses of building or coarse-grained conglomerate rock, or on a more moderate scale to blocks of common sandstone, or on a very small scale to seemingly homogeneous metals 2; or on a scale of extreme, undiscovered fineness, to vitreous bodies, continuous crystals, solidified gums, as India rubber, gum-arabic, etc., and Auids. 647. The substance of a homogeneous solid is called isotropic when a spherical portion of it, tested by any physical agency, exhibits no difference in quality however it is turned. Or, which amounts to the same, a cubical portion cut from any position in an isotropic body exhibits the same qualities relatively to each pair of parallel faces. Or two equal and similar portions cut from any positions Probably not undiscoverably small, although of dimensions not yet known to us. 2 Which, however, we know, as recently proved by Deville and Van Troost, are porous enough at high temperatures to allow very free percolation of gases. in the body, not subject to the condition of parallelism ($ 646), are undistinguishable from one another. A substance which is not isotropic, but exhibits differences of quality in different directions, is called aeolotropic. 648. An individual body, or the substance of a homogeneous solid, may be isotropic in one quality or class of qualities, but aeolotropic in others. Thus in abstract dynamics a rigid body, or a group of bodies rigidly connected, contained within and rigidly attached to a rigid spherical surface, is kinetically symmetrical ($ 239) if its centre of inertia is at the centre of the sphere, and if its moments of inertia are equal round all diameters. It is also isotropic relatively to gravitation if it is centrobaric ($ 542), so that the centre of figure is not merely a centre of inertia, but a true centre of gravity. Or a transparent substance may transmit light at different velocities in different directions through it (that is, be doubly refracting), and yet a cube of it may (and generally does in natural crystals) absorb the same part of a beam of white light transmitted across it perpendicularly to any of its three pairs of faces. Or (as a crystal which exhibits dichroism) it may be aeolotropic relatively to the latter, or to either, optic quality, and yet it may conduct heat equally in all directions. 649. The remarks of § 646 relative to homogeneousness in the aggregate, and the supposed ultimately heterogeneous texture of all substances however seemingly homogeneous, indicate corresponding limitations and non-rigorous practical interpretations of isotropy. 650. To be elastically isotropic, we see first that a spherical or cubical portion of any solid, if subjected to uniform normal pressure (positive or negative) all round, must, in yielding, experience no deformation : and therefore must be equally compressed (or dilated) in all directions. But, further, a cube cut from any position in it, and acted on by tangential or distorting stress (§ 633) in planes parallel to two pairs of its sides, must experience simple deformation, or shear ($ 150), in the same direction, unaccompanied by condensation or dilatation, and the same in amount for all the three ways in which a stress may be thus applied to any one cube, and for different cubes taken from any different positions in the solid. 651. Hence the elastic quality of a perfectly elastic, homogeneous, isotropic solid is fully defined by two elements ;—its resistance to compression, and its resistance to distortion. The amount of uniform pressure in all directions, per unit area of its surface, required to produce a stated very small compression, measures the first of 1 It must be remembered that the changes of figure and volume we are concerned with are so small that the principle of superposition is applicable; so that if any distorting stress produced a condensation, an opposite distorting stress would produce a dilatation, which is a violation of the isotropic condition. But it is possible that a distorting stress may produce, in a truly isotropic solid, condensation or dilatation in proportion to the square of its value: and it is probable that such effects may be sensible in India rubber, or cork, or other bodies susceptible of great deformations or compressions, with persistent elasticity. these, and the amount of the distorting stress required to produce a stated amount of distortion measures the second. The numerical measure of the first is the compressing pressure divided by the diminution of the bulk of a portion of the substance which, when uncompressed, occupies the unit volume. It is sometimes called the elasticity of volume, or the resistance to compression. Its reciprocal, or the amount of compression on unit of volume divided by the compressing pressure, or, as we may conveniently say, the compression per unit of volume, per unit of compressing pressure, is commonly called the compressibility. The second, or resistance to change of shape, is measured by the tangential stress (reckoned as in § 633) divided by the amount of the distortion or shear ($ 154) which it produces, and is called the rigidity of the substance, or its elasticity of figure. 652. From § 148 it follows that a strain compounded of a simple extension in one set of parallels, and a simple contraction of equal amount in any other set perpendicular to those, is the same as a simple shear in either of the two sets of planes cutting the two sets of parallels at 45°. And the numerical measure (§ 154) of this shear, or simple distortion, is equal to double the amount of the elongation or contraction (each measured, of course, per unit of length). Similarly, we see ($ 639) that a longitudinal traction (or negative pressure) parallel to one line, and an equal longitudinal positive pressure parallel to any line at right angles to it, is equivalent to a distorting stress of tangential tractions ( 632) parallel to the planes which cut those lines at 45°. And the numerical measure of this distorting stress, being (§ 633) the amount of the tangential traction in either set of planes, is equal to the amount of the positive or negative normal pressure, not doubled. 653. Since then any stress whatever may be made up of simple longitudinal stresses, it follows that, to find the relation between any stress and the strain produced by ^}P it, we have only to find the strain produced by a single longitudinal 3P stress, which may do at 13P once thus:-A simple longitudinal stress, P, is equivalent to a uniform dilating tension iP in all directions, compounded with two Р Р. JAPP distorting stresses, each equal to } P, and having a common axis in the line of the given longitudinal stress, and their other two axes any two lines at right angles 3P to one another and to it. The 3P diagram, drawn in a plane through one of these latter lines, and the VIP former, sufficiently indicates the synthesis; the only forces not shown being those perpendicular to its plane. we Hence if n denote the rigidity, and k the resistance to dilatation [being the same as the reciprocal of the compressibility ($ 651)], the effect will be an equal dilatation in all directions, amounting, per unit of volume, to АР (1) k compounded with two equal distortions, each amounting to АР (2) n and having ($ 650) their axes in the directions just stated as those of the distorting stresses. 654. The dilatation and two shears thus determined may be conveniently reduced to simple longitudinal strains by still following the indications of $ 652, thus : The two shears together constitute an elongation amounting to IP in the direction of the given force, P, and equal contraction АР amounting to in all directions perpendicular to it. And the n n implies a lineal dilatation, equal in all directions, cubic dilatation k IP amounting to R 9 On the whole, therefore, we have 1 1 stress, and linear elongation=P + ), in the direction of the applied (3n 9k (3) 1 1 linear contraction=P(; „), in all directions perpendicular 6n 9 to the applied stress. 655. Hence when the ends of a column, bar, or wire, of isotropic material, are acted on by equal and opposite forces, it experiences 3k— 2n a lateral lineal contraction, equal to of the longitudinal 2(3k + n) dilatation, each reckoned as usual per unit of lineal measure. One specimen of the fallacious mathematics above referred to (§ 644), is a celebrated conclusion of Navier's and Poisson's that this ratio is , which requires the rigidity to be of the resistance to compression, for all solids: and which was first shown to be false by Stokes from many obvious observations, proving enormous discrepancies from it in many well-known bodies, and rendering it most improbable that there is any approach to a constancy of ratio between 1. On the Friction of Fluids in Motion, and the Equilibrium and Motion of Elastic Solids.' Trans. Camb. Phil. Jour., April 1845. See also Camb. and Dub. Math. Four., March 1848. |