Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/848

824 ELASTICITY other strain of equal amount ; and the principal type to which the last corresponds is that of a strain which is maintained by a less stress than any other strain of equal amount in the same body. The stresses corresponding to the fofir other principal strain-types have each the maximum-minimum property in a determinate way. Prop. If a body be strained in the direction of which the concur- . rences with the principal strain-types are I, in, n, , fj., v, and to an amount equal to r, the stress required to maintain it in this state will be equal to dr, where Q = (A*/* + B^m* + Cn s + FX*+G V + II V) , and will be of a type of which the concurrences with the principal types are respectively A.I Bm C FA. Gfji Hi/ Q 12" IF a" IT IT Prop. A homogeneous elastic solid, crystalline or non-crystal line, subject to magnetic force or free from magnetic force, has neither any right-handed or left-handed, nor any dipolar, properties dependent on elastic forces simply proportional to strains. Cor. The elastic forces concerned in the luminiferous vibrations of a solid or fluid medium possessing the right- or left-handed property, whether axial or rotator) , such as quartz crystal, or tartaric acid, or solution of sugar, either depend on the heteroge- neousness or on the magnitude, of the strains experienced. Hence as they do not depend on the magnitude of the strain, they do depend on Us hctcrogencousness through the portion of a medium containing a wave. Cor. There cannot possibly be any characteristic of elastic forces simply proportional to the strains in a homogeneous body, corre sponding to certain peculiarities of crystalline form which have been observed, for instance corresponding to the plagiedral faces dis covered by Sir John Herschel to indicate the optical character, whether right-handed or left-lianded, in different specimens of quartz crystal, or corresponding to the distinguishing characteristics of the crystals of the right-handed and left-handed tartaric acids obtained by M. Pasteur from racemic acid, or corresponding to the dipolar characteristics of form said to have been discovered in electric crystals. CHAPTER XVI. Application of Conclusions to Natural Crystals. It is easy to demonstrate that a body, homogeneous when regarded on a large scale, may be constructed to have twenty-one arbitrarily prescribed values for the coefficients in the expression for its potential energy in terms of any prescribed system of strain co ordinates. This proposition was first enunciated in the paper on the Thermo-elastic Properties of Solids, published April 1855, in the Quarterly Mathematical Journal alluded to above. We may infer the following. Prop. A solid may be constructed to have arbitrarily prescribed values for its six Principal Elasticities and an arbitrary orthogonal system of six strain-types, specified by fifteen independent ele ments, for its principal strains : for instance, five arbitrarily chosen systems of three rectangular axes, for the normal axes of five of the Principal Types ; those of the sixth consequently in general distinct from all the others, and determinate; and the six times two ratios between the three stresses or strnins of each type, also determinate. The fifteen equations expressing (Chap. VI.) the mutual orthogonality of the six types determine the twelve ratios for the six types, and the three quantities specifying the axes of the sixth type in the particular case here suggested : or generally the fifteen equations determine fifteen out of the thirty quantities (viz. twelve ratios and eighteen angular coordinates) specifying six Principal Types. Cor. There is no reason for believing that natural crystals do not exist for which there arc six unequal Principal Elasticities, and six distinct strain-types for which the three normal axes constitute six distinct sets of three principal rectangular axes of elasticity. It is easy to give arbitrary illustrative examples regarding Prin cipal Elasticities : also, to investigate the principal strain-types and the equations of elastic force referred to them or to other iiatural types, for a body possessing the kind of symmetry as to elastic forces that is possessed by a crystal of Iceland spar, or by a crystal of the "tesseral class," or of the included "cubical class." Such illustrations and developments, though proper for a students text book of the subject, are unnecessary here. For applications of the Mathematical Theory of Elasticity to the question of the earth s rigidity and elasticity as a whole, and to the equilibrium of elastic solids in general, which are beyond the scope of the present article, the reader is referred to Thomson and Tait s Natural Philosophy, 588, 740, 832, 849, and Ap pendix C. CHAPTEU XVII. Plane Waves in a Homogeneous ^Eolotropic Solid. A plane wave in a homogeneous elastic solid is a motion in which every line of particles in a plane parallel to one fixed plane ex periences simply a motion of translation but a motion differing from the motions of particles in planes parallel to the same. Let OX, OY, OZ be three fixed rectangular axes ; OX perpendicular to the wave front (as any of the parallel planes of moving particles referred to in the definition is called), and OY, OZ in the wave front. Let x + u, y + v, z + w be the coordinates at time I of a par ticle which, if the solid were free from strain, would be at (a;, y, :K The definition of wave motion amounts simply to this, that u, v, w are functions of x and t. The strain of the solid (Chap. VII. above) is the resultant of a simple longitudinal strain in the direction OX, equal to ~, and two differential slips parallel to OY and OZ, constituting simple distortions of which the numerical magnitudes (Chap. X.) are Put then ---V2, and V - 2. dx Jx du >. dv and let W denote the work per unit of bulk required to produce the strain represented by this notation. We have (Chap. XV. ) where A, B, C, D, E, F denote moduluses of elasticity of the solid. Let p, q, r denote the three components of the traction per unit area of the wave front. We have (Chap. XV.) + En > ...... (3), j Now let |, -i], be taken such that (4) the determinants! cubic gives three real positive values for M, and with M equal to any one of these values, (4) determine the ratios | : 77 : Hence when the solid is strained in any one of the three ways thus determined we have The three components of the whole force due to the tractions on the sides of an infinitely small parallelepiped Sx, oij, S~ of the solid ore clearly dp ^ ^ dq . , dr , ., , ox.oyoz , -- CJC.GHOZ . find o^.cyo^ dx dx dx and therefore, if p be its density, and consequently p$x 5y 8,; its mean, the equations of its motion are rf 2 i dp d?v dq d"ic dr These, putting for;?, q, r their values by (5), become And by (4) and (1) we have (8). (9). E +(00+0)2 = 3110^2 Let MD M 2 , M 3 be the three roots of the dctenmnantal cubic, and b, c l ; b, c. 2 ; 6 3 , c 3 , the corresponding values of the ratios , determined by (9). The complete solution of (8), subject to (Vis U = !< v =i to = c (10), fi, Fj, / F 2 , / 3 , F a denoting arbitary functions. Hence we con clude that there are three different wave-velocities, P P P and three different modes of waves, determined by equations (9). Waves in an Isotropic Solid. If the solid be isotropic, we have B = c )

D = E = F = ... . <i^