MECHANICS 693 Sym- 82. The process above is essentially unsymmetrical. metrical The first, suggestion of a symmetrical system is due to process. j u i erj anc i depends upon the general proposition of 75. What we must seek is the single axis, and the angle of rotation about it, which (by one operation) will bring the system or figure from its initial state determined by X, Y, Z to its state at time t, determined by A, B, C. Let /, m, n be the direction cosines of this axis, w the angle of rotation about it. Then by the elementary theorems of spherical trigonometry we find cosXA= Z 2 +(1 - P) cos-sr, cosYB = ra 2 -t- (1 m-)cosT*r, cos ZC = 2 +(1 -n") cosw. Thus, as we have an independent relation among Z 2 , m 2 , n 2 , these quantities, as well as w, can all be expressed in terms of the cosines of the three angles between the original and final directions of the three axes severally. We have other six equations, of which only one need be written, viz. : Rod- t S3. If we put rigues s t0 = cos4w, x=lsin^w, y = msin^zr, z=nsin%-ar, uates. which involve the equation of condition Iff 2 + 3? + ?/ 2 + 2 2 = 1 , the nine direction cosines of the new positions OA, OB, OC, referred to the fixed lines OX, OY, OZ, become w- + x 2 -y 2 -z 2 , 2(wz + xy) , 2(xz-wy), 2(yx-wz), w 2 -x 2 + 7/ 2 -s 2 , 2(yz + wx), 2(xz + wy) , 2(yz - wx] , v? -x*- y 2 + z 2 . These expressions, rational in terms of the four quanti ties w, x, y, z, are due to Rodrigues, who, however, gave them in a slightly different form. If coj, &> 2 , 3 be the angular velocities about OA, OB, OC, respectively, we have 2y = 2o> 1 + Wca 2 - xca 3 , 2z = -l/coj + Xu 2 +Wa> 3 . If ca x , wy, ta, be the angular velocities about OX, OY, OZ, respectively, we have 2w = -xa> x -yuy-zwf, 2Jc = W(i> x + Z(>> wz Each of these sets is equivalent to three independent equations only, on account of the relation Kinematics of a Deformable Figure. Strain. Strain. 84. So far we have considered change of position of a figure of invariable form. We must now consider changes of form and volume in the figure itself. This is required for application to physical problems, such as compression of a liquid or gas, the distortion of a piece of india-rubber. &c. Any such change of volume or form is called a "strain." The treatment of strains is entirely a kinematical question, until we come to regard them as produced in physical bodies, and consider their cause. Stress. The system of forces which is said to produce a strain is called a " stress." But, just as we study velocity as a preparation for the discussion of the effect of force on a free body, so we study strains as a preparation for the discussion of the effects of stress. Hetero- 85. In order to fix the ideas, it is convenient to suppose 5ra US P 1 ? *J8 ure wnic k is to undergo strain to be cut up into an infinite number of similar, equal, and similarly situated parellelepipeds. This is effected at once by supposing it to be cut by three series of planes, those of each series being parallel to one another, and equidistant. No two of these three series must be parallel, but the distance from plane to plane need not be the same in any two of the series. If the strain be continuous there will be no finite difference of effect upon any two neighbouring parallelepipeds ; but in general their edges, which originally formed three series of parallel straight lines, will become series of curves. No two parallelepipeds of the system will in general be altered in precisely the same manner. This is called "heterogeneous strain." j 86. We found it convenient to study uniform speed Homo- before proceeding to consider variable speed, and so we geneous. find it convenient to take up first what is called strain, Homogeneous Strain. A figure is said to be homo geneously strained when all parts of it originally equal, similar, and similarly situated remain equal, similar, and similarly situated, however much they may individually have been altered in form, volume, and position. Now recur to our set of parallelepipeds. After a homogeneous strain these remain equal, similar, and similarly situated. Hence they must remain parallelepipeds, for they must together still continuously make up the volume of the altered figure. Thus planes remain planes, and straight lines remain straight lines. Equal parallel straight lines remain equal and parallel. Parallel planes remain parallel, ellipses remain ellipses (as is obvious from their properties relative to conjugate diameters), ellipsoids remain ellipsoids, <fec. But we can now easily see how many conditions fully determine a homogeneous strain. For if we know how each of three conterminous edges of any one of the original parallelepipeds is altered in length and direction, we can build up the whole altered system. Hence, to fully requires describe a homogeneous strain, we require merely to know n " ie cou what changes take place in the lengths and directions of stants> three unit lines not in one plane. Three numbers are required for the altered lengths, and two (analogous, say, to altitude and azimuth, or latitude and longitude, or R.A. and N.P.D.) for each of the altered directions. Hence, in general, a homogeneous strain depends upon, and is fully characterized by, nine independent numbers. 87. The simplest form of strain is that which is due Uniform to uniform hydrostatic stress acting on a homogeneous dilata- isotropic body, Here directions remain unaltered, and tlon> the lengths of all lines are altered in the same ratio. Every portion of the original figure remains similar to itself, its linear, superficial, and volume dimensions being altered as the first, second, and third powers of that ratio. Next in order of simplicity is the case in which there Pure are three directions, at right angles to one another, which strain, suffer no change except as regards length. This state of things would be produced in a homogeneous isotropic body by three longitudinal extensions or compressions in lines at right angles to one another, or by hydrostatic pressure in a homogeneous non-isotropic solid. In this case, if the changes of length above spoken of are all different, an originally spherical figure becomes an ellipsoid, with three unequal axes parallel respectively to the lines whose directions remain unaltered. Every line in the body not originally parallel to one of these is altered in direc tion. If one of the principal changes of length be an extension, and another a shortening, there will be a cone formed of lines which are not altered in length. This is seen at once by describing from the centre of the ellipsoid a sphere equal to the original sphere. One axis of the ellipsoid being greater than the radius of the sphere, and another less, the ellipsoid and sphere must intersect ; and all lines drawn from the common centre to the curve of intersection are unaltered in length (though all altered, as
before remarked, in direction).