Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/829

ELASTICITY 805 the stress is such that it produces a strain of its own type. (1.) An isotropic solid has two principal moduluses a modulus of compression and a rigidity. (2.) A crystal of the cubic class (fluor-spar, for instance) has three principal moduluses, one modulus of com pression and tivo rigidities. (3.) An seolotropic solid having (what no natural crystal has, but what a draivn ivire has) perfect isotropy of physical qualities relative to all lines perpendicular to a certain axis of its substance has three principal moduluses, two determiuable from its different compressibilities along and perpendicular to the axis, or from one compressibility a,nd the "Young s modulus" (section 42) of an axial bar of the substance, or determinable from two com pressibilities; and one rigidity determinable by measure ment of the torsional rigidity of a round axial bar of the substance. (4.) A crystal of Iceland spar has four principal moduluses, three like those of case (3), aud another rigidity depending on (want of complete circular symmetry, and) possession of triple symmetry of form, involving sextuple elastic symmetry, round the crystalline axis. (5.) A crystal of the rectangular parallelepiped (or "tessaral") class has six distinct principal moduluses which, when the directions of the principal axes are known, are determinable by six single observations, three, of the three (generally unequal) compressibilities along the three axes; and three, of the three rigidities (no doubt generally unequal) relatively to the three simple distortions of the parallelepiped, in any one of which one pair of parallel rectangular faces of the parallelepiped become oblique parallelograms. (6.) An aiolotropic solid generally has six principal moduluses, 1 which, when a piece of the solid is presented without information, and without any sure indication from its appearance of any particular axis or axes of <symmetry of any kind, require just twenty-one independent observations for the determination of the fifteen quantities specifying their types, and the six numerical values of the moduluses themselves. 42. " Young s Modulus," or Modulus of Simple Longi tudinal Stress. Thomas Young called the -modulus of elasticity of an elastic solid the" amount of the end-pull or end-thrust required to produce any infinitesimal elongation or contraction of a wire, or bar, or column of the substance multiplied by the ratio of its length to the elongation or contraction. In this definition the definite article is clearly misapplied. There are, as we have seen, two moduluses of elasticity for an isotropic solid, one measuring elasticity of bulk, the other measuring elasticity of shape. An interesting and instructive illustration of the confusion of ideas so often rising in physical science from faulty logic is to be found in " A.n Account of an Experiment on the Elasticity of Ice : By Benjamin Bevan, Esq.,, in a letter to Dr Thomas Young, Foreign Sec. R. S." and in Young s " Note " upon it, both published in the Transac tions of the Royal Society for 1826. Bevan gives an interesting account of a well-designed and well-executed experiment on the flexure of a bar, 3 97 inches thick, 10 inches broad, and 100 inches long, of ice on a pond near Leighton Buzzard (the bar remaining attached by one end to the rest of the ice, but being cut free by a saw along its sides and across its other end), by which he obtained a fairly accurate determination of " the modulus of ice "; 2 and says that he repeated the experiment in various ways on ice bars of various dimensions, some remaining attached by 1 Mathematical Theory, chap. xvi. 2 The result is given in the Table of Moduluses, sec. 77, below. one end, others completely detached, and found results agreeing with the first as nearly " as the admeasurement of the thickness could be ascertained." He then proceeds to compare " the modulus of ice " which he had thus found with " the modulus of water," which he quotes from Young s Lectures as deduced from Canton s experiments on the compressibility of water. Young in his " Note " does not point out that the two moduluses were essentially different, and that the modulus of his definition, the modulus determinable from the flexure of a bar, is essentially zero for every fluid. We now call " Young s modulus " the particular modulus of elasticity defined as above by Young, and so avoid all confusion. 43. Modulus of Rigidity. The " modulus of rigidity " of an isotropic solid is the amount of tangential stress divided by the deformation it produces, the former being measured in units of force per unit of the area to which it is applied in the manner indicated by the annexed diagram (fig. 3), and the latter by the variation of each of the four right angles reckoned in fraction of the radian. By drawing either diagonal of the square in the diagram we see that the distorting stress repre sented by it gives rise to a normal traction on every sur face of the substance perpen dicular to the square and parallel to one of its diagonals, and an equal normal pi*essure on every surface of the solid perpendicu lar to the square and parallel to T Fig. 3. the other diagonal; and that the amount of each of these normal forces 3 per unit of area is equal to the amount per unit area of the tangential forces which the diagram indicates. The corresponding 4 geometrical proposition, also easily proved, is as follows : A strain compounded of a simple extension in one set of parallels, and a simple con traction 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 measuring of this shear or simple distortion is equal to double the amount of the elongation or contrac tion, each reckoned per unit of .length. Hence we have another definition of " modulus of rigidity " equivalent to the preceding : The modulus of rigidity of an isotropic substance is the amount of normal traction or pressure per unit of area, divided by twice the amount of elongation in the direction of the traction or of contraction in the direction of the pressure, when a piece of the substance is subjected to a stress producing uniform distortion. 44. 5 Conditions fulfilled in Elastic Isotropy. To be elas- tically isotropic, a spherical or cubical portion of any solid, if subjected to uniform normal pressure (positive or nega tive) all round, must, in yielding, experience no deforma tion, 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 in planes parallel to two pairs of its sides, must experience simple deformation, or " shearing" parallel to either pair of these sides, unaccompanied by condensation or dilatation, 6 3 The directions of these forces are called the " axes" of the stress. The corresponding directions in the corresponding strain are called the axes of the strain. 4 Mathematical Theory, chap. vi. 6 This, with several of the following sections, 44-51, is borrowed, with but slight change, from the first edition of Thomson and Tait s Natural Philosophy, by permission of the authors. 6 It must be remembered that the changes of figure and volume we

are concerned with are so small that the principle of superposition is