Composi tion of pure strains. which (Chap. II.) will be proved to be instantaneously, or differentially, irrotational; but which may result in leaving a whole fluid mass merely turned round from its primitive position, as if it had been a rigid body. The following elementary geometrical investigation, though not bringing out a thoroughly comprehensive view of the subject, affords a rigorous demonstration of the proposition, by proving it for a particular case. Let us consider, as above (§ 171), a simple shearing motion. A point O being held fixed, suppose the matter of the body in a plane, cutting that of the diagram perpendicularly in CD, to move in this plane from right to left parallel to DC; and in other planes parallel to it let there be motions proportional to their distances from 0. Consider first a shear from P to P,; then from P, on to P,; and let O be taken in a line through P1, perpendicular to C_P2 PA CD. During the shear from P to P, a point Q1 2 P -D 1 1 PP,. Choose Q midway between P and P1, so that PQ = QPP,P. Now, as we have seen above (§ 152), the line of the body, which is the principal axis of contraction in the shear from Q to Q1, is OA, bisecting the angle QOE at the beginning, and OA,, bisecting Q,OE at the end, of the whole motion considered. The angle between these two lines is half the angle Q1OQ, that is to say, is equal to P,OQ. Hence, if the plane CD is rotated through an angle equal to P,OQ, in the plane of the diagram, in the same way as the hands of a watch, during the shear from Q to Q,, or, which is the same thing, the shear from P to P,, this shear will be effected without final rotation of its principal axes. (Imagine the diagram turned round till OA, lies along OA. The actual and the newly imagined position of CD will show how this plane of the body has moved during such non-rotational shear.) Now, let the second step, P, to P,, be made so as to complete the whole shear, P to P2, which we have proposed to consider. Such second partial shear may be made by the common shearing process parallel to the new position (imagined in the preced 1 tion of pure ing parenthesis) of CD, and to make itself also non-rotational, Composi as its predecessor has been made, we must turn further round, strains. in the same direction, through an angle equal to Q,OP,. Thus in these two steps, each made non-rotational, we have turned the plane CD round through an angle equal to Q2OQ. But now, we have a whole shear PP,; and to make this as one non-rotational shear, we must turn CD through an angle P ̧OP only, which is less than Q,OQ by the excess of P,OQ above QOP. Hence the resultant of the two shears, PP,, P1P, each separately deprived of rotation, is a single shear PP,, and a rotation of its principal axes, in the direction of the hands of a watch, through an angle equal to QOP ̧-POQ. 185. Make the two partial shears each non-rotationally. Return from their resultant in a single non-rotational shear: we conclude with the body unstrained, but turned through the angle QOP-POQ, in the same direction as the hands of a watch. x1 = Ax + cy + bz y1 = cx + By + az ≈1 = bx+ay + Cz is (§ 183) the most general possible expression for the displace- Similarly, if the body thus strained be again non-rotationally Substituting in these, for x,, y,, %,, their preceding expressions, x2 = (4 ̧A + c ̧c + b,b) x + (4 ̧c + c ̧ B + b,a) y + (A ̧b + c ̧a + b,C) z Composition of pure strains. Displacement of a curve. Tangential displacement. The resultant displacement thus represented is not generally of the non-rotational character, the conditions (18) of § 183 not being fulfilled, as we see immediately. Thus, for instance, we see that the coefficient of y in the expression for x is not necessarily equal to the coefficient of x in the expression for y, Cor. If both strains are infinitely small, the resultant displacement is a pure strain without rotation. For A, B, C, A ̧, B ̧, C, are each infinitely nearly unity, and a, b, etc., each infinitely small. Hence, neglecting the products of these infinitely small quantities among one another, and of any of them with the differences between the former and unity, we have a resultant displacement x= A ̧Ax + (c + c,) y + (b + b,)z Y1 = (c ̧ + c) x + B ̧By +(a+a,)≈ ≈2 = (b ̧ + b) x + (a, + a) y + C' ̧Cz, which represents a pure strain unaccompanied by rotation. 186. The measurement of rotation in a strained elastic solid, or in a moving fluid, is much facilitated by considering separately the displacement of any line of the substance. We are therefore led now to a short digression on the displacement of a curve, which may either belong to a continuous solid or fluid mass, or may be an elastic cord, given in any position. The propositions at which we shall arrive are, of course, applicable to a flexible but inextensible cord (§ 14, above) as a particular case. It must be remarked, that the displacements to be considered do not depend merely on the curves occupied by the given line in its successive positions, but on the corresponding points of these curves. What we shall call tangential displacement is to be thus reckoned:-Divide the undisplaced curve into an infinite number of infinitely small equal parts. The sum of the tangential components of the displacements from all the points of division, multiplied by the length of each of the infinitely small parts, is the entire tangential displacement of the curve reckoned along the undisplaced curve. The same reckoning carried out in the displaced curve is the entire tangential displacement reckoned on the displaced curve. ings of tan placement 187. The whole tangential displacement of a curve reckoned Two reckonalong the displaced curve, exceeds the whole tangential dis- gential displacement reckoned along the undisplaced curve by half the compared. rectangle under the sum and difference of the absolute terminal displacements, taken as positive when the displacement of the end towards which the tangential components are if positive exceeds that at the other. This theorem may be proved by a geometrical demonstration which the reader may easily supply. Analytically thus :-Let x, y, z be the co-ordinates of any point, P, in the undisplaced curve; x, y, z,, those of P, the point to which the same point of the curve is displaced. Let dx, dy, dz be the increments of the three co-ordinates corresponding to any infinitely small arc, ds, of the first; so that ds = (dx2 + dy2+ dz2)3, and let corresponding notation apply to the corresponding D= {(x,-x)2+(y,− y)2 + (≈, — 2)2}}, being the absolute space of displacement. Hence Similarly we have D cos 0,ds, = (x, x) dx ̧ + (y,− y) dy, + (z ̧ − z) dz ̧, and therefore D cos 0 ds-D cos Ods = (x ̧ − x)d(x ̧ − x) + (y, − y) d (y, − y) or D cos 01ds, - D cos @ds = 1⁄2 d (D3). To find the difference of the tangential displacements reckoned JD cos 0,ds, - SD cos @ds = }(D"2 – D′) = }(D′′ + D'′) (D′′ – D′), where D' and D' denote the displacements of the two ends. Tangential displace 188. The entire tangential displacement of a closed curve ment of a is the same whether reckoned along the undisplaced or the displaced curve. closed curve. Rotation of a rigid 189. The entire tangential displacement from one to another of two conterminous arcs, is the same reckoned along either as along the other. 190. The entire tangential displacement of a rigid closed closed curve. curve when rotated through any angle about any axis, is equal to twice the area of its projection on a plane perpendicular to the axis, multiplied by the sine of the angle. Tangential displacement in a solid, in terms of components of strain. (a) Prop.-The entire tangential displacement round a closed curve of a homogeneously strained solid, is equal to 2(Pw + Qp + Ro), where P, Q, R denote, for its initial position, the areas of its projections on the planes YOZ, ZOX, XOY respectively, and , p, σ are as follows: Hence, according to the previously investigated expression, we have, for the tangential displacement, reckoned along the undisplaced curve, {(x,-x)dx + (y,-y)dy + (z, −z) dz} = ƒ[}d{(4 − 1) x2 + (B − 1) y2 + (C − 1) z2 + 2 (ayz + bzx + cxy)} +w (ydz − zdy) + p (zdx − xdz) + σ (xdy — ydx)]. The first part, d{ }, vanishes for a closed curve. |