Simple shear. Axes of a shear. N K has (1) the property that one set of parallel planes remain each unaltered in itself; (2) that another set of parallel planes remain each unaltered in itself. This other set is found when the first set and the degree or amount of shear are given, thus-Let CC, be the motion of one point of one plane, relative to a plane KL held fixed-the diagram being in a plane of the shear. Bisect CC, in N. Draw NA perpendicular to it. A plane perpendicular to the plane of the diagram, initially through AC, and finally through AC, remains unaltered in its dimensions. A 172. One set of parallel undistorted planes, and the amount of their relative parallel shifting having been given, we have just seen how to find the other set. The shear may be otherwise viewed, and considered as a shifting of this second set of parallel planes, relative to any one of them. The amount of this relative shifting is of course equal to that of the first set, relatively to one of them. 173. The principal axes of a shear are the lines of maximum elongation and of maximum contraction respectively. They may be found from the preceding construction (§ 171), thus-In the plane of the shear bisect the obtuse and acute angles between the planes destined not to become deformed. The former bisecting line is the principal axis of elongation, and the latter is the principal axis of contraction, in their initial positions. The former angle (obtuse) becomes equal to the latter, its supplement (acute), in the altered condition of the body, and the lines bisecting the altered angles are the principal axes of the strain in the altered body. Otherwise, taking a plane of shear for the plane of the diagram, let AB be a line in which it is cut by one of either set of parallel planes of no distortion On any portion AB of this as diameter, describe a semicircle. Through C, its middle point, draw, by the preceding B construction, CD the initial, and CE A E shear. the final, position of an unstretched line. Join DA, DB, EA, Axes of a EB. DA, DB are the initial, and EA, EB the final, positions of the principal axes. a shear. 174. The ratio of a shear is the ratio of elongation or con- Measure of traction of its principal axes. Thus if one principal axis is elongated in the ratio 1:a, and the other therefore (§ 169) contracted in the ratio a : 1, a is called the ratio of the shear. It will be convenient generally to reckon this as the ratio of elongation; that is to say, to make its numerical measure greater than unity. In the diagram of § 173, the ratio of DB to EB, or of EA to DA, is the ratio of the shear. 175. The amount of a shear is the amount of relative motion per unit distance between planes of no distortion. It is easily proved that this is equal to the excess of the ratio of the shear above its reciprocal. specifica shear. 176. The planes of no distortion in a simple shear are Ellipsoidal clearly the circular sections of the strain ellipsoid. In the tion of a ellipsoid of this case, be it remembered, the mean axis remains unaltered, and is a mean proportional between the greatest and the least axis. ple elonga expansion 177. If we now suppose all lines perpendicular to the plane Shear, simof the shear to be elongated or contracted in any proportion, tion, and without altering lengths or angles in the plane of the shear, combined. and if, lastly, we suppose every line in the body to be elongated or contracted in some other fixed ratio, we have clearly (§ 161) the most general possible kind of strain. Thus if s be the ratio of the simple shear, for which case s, 1, 1 are the three principal ratios, and if we elongate lines perpendicular to its plane in the Shear, sim ratio 1 m, without any other change, we have a strain of which the principal ratios are ple elongation, and expansion, combined. If, lastly, we elongate all lines in the ratio 1 n, we have a strain in which the principal ratios are n where it is clear that ns, nm, and may have any values S whatever. It is of course not necessary that nm be the mean principal ratio. Whatever they are, if we call them a, ß, y respectively, we have Analysis of a strain. Displacement of a or not, one point of 178. Hence any strain (2, B, y) whatever may be viewed as compounded of a uniform dilatation in all directions, of linear B Vay ratio Vay, superimposed on a simple elongation in the direction of the principal axis to which ẞ refers, superimposed on a simple shear, of ratio or of amount √(or in the plane of the two other principal axes. 179. It is clear that these three elementary component strains may be applied in any other order as well as that stated. Thus, if the simple elongation is made first, the body thus altered must get just the same shear in planes perpendicular to the line of elongation, as the originally unaltered body gets when the order first stated is followed. Or the dilatation may be first, then the elongation, and finally the shear, and so on. 180. In the preceding sections on strains, we have conbody, rigid sidered the alterations of lengths of lines of the body, and of angles between lines and planes of it; and we have, in partiheld fixed. cular cases, founded on particular suppositions (the principal axes of the strain remaining fixed in direction, § 169, or one which is ment of a or not, one is of either set of undistorted planes in a simple shear remain- Displaceing fixed, § 170), considered the actual displacements of parts body, rigid of the body from their original positions. But to complete point of the kinematics of a non-rigid solid, it is necessary to take a held fixed. more general view of the relation between displacements and strains. It will be sufficient for us to suppose one point of the body to remain fixed, as it is easy to see the effect of superimposing upon any motion with one point fixed, a motion of translation without strain or rotation. 181. Let us therefore suppose one point of a body to be held fixed, and any displacement whatever given to any point or points of it, subject to the condition that the whole substance if strained at all is homogeneously strained. Let OX, OY, OZ be any three rectangular axes, fixed with reference to the initial position and condition of the body. Let x, y, z be the initial co-ordinates of any point of the body, and ,,,,, be the co-ordinates of the same point of the altered body, with reference to those axes unchanged. The condition that the strain is homogeneous throughout is expressed by the following equations:-- where [Xx], [Xy], etc., are nine quantities, of absolutely arbitrary values, the same for all values of x, y, z. They are, of [Xx], [Yx], [Zx] denote the three final co-ordinates of a point originally at unit distance along OX, from 0. course, proportional to the direction-cosines of the tion of the line primitively coinciding with OX. [Xy], [Yy], [Zy], etc. altered posi- Let it be required to find, if possible, a line of the body which remains unaltered in direction, during the change specified by [X], etc. Let x, y, z, and x, y, z,, be the co-ordinates of the primitive and altered position of a point in such a line. We must have y = = 1+, where is the elongation of the From these equations, by eliminating the ratios xyz according to the well-known algebraic process, we find − [Y2][Zy]([Xx] − n) − [Zx][Xz] ([Yy]−n)− [Xy][Yx]([Zz] − n) +[X2] [Yx][Zy] + [Xy] [Yz] [Zx] = 0. This cubic equation is necessarily satisfied by at least one real value of 7, and the two others are either both real or both imaginary. Each real value of 7 gives a real solution of the problem, since any two of the preceding three equations with it, in place of 7, determine real values of the ratios xyz. If the body is rigid (i.e., if the displacements are subject to the condition of producing no strain), we know (ante, § 95) that there is just one line common to the body in its two positions, the axis round which it must turn to pass from one to the other, except in the peculiar cases of no rotation, and of rotation through two right angles, which are treated below. Hence, in this case, the cubic equation has only one real root, and therefore it has two imaginary roots. The equations just formed solve the problem of finding the axis of rotation when the data are the actual displacements of the points primitively lying in three given fixed axes of reference, OX, OY, OZ; and it is worthy of remark, that the practical solution of this problem is founded on the one real root of a cubic which has two imaginary roots. Again, on the other hand, let the given displacements be made so as to produce a strain of the body with no angular displacement of the principal axes of the strain. Thus three lines of the body remain unchanged. Hence there must be three real roots of the equation in 7, one for each such axis; and the three lines determined by them are necessarily at right angles to one another. But if neither of these conditions holds, we may have three real solutions and three oblique lines of directional identity; or we may have only one real root and only one line of directional identity. |