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to the plane of the diagram, cuts the ellipsoid in an ellipse of which the two principal axes are equal, that is to say, in a circle. Hence the elongations along all lines in either of these planes are equal to the elongation along the mean principal axis of the strain ellipsoid.
148. The consideration of the circular sections of the strain ellip soid is highly instructive, and leads to important views with reference to the analysis of the most general character of a strain. First let us suppose there to be no alteration of volume on the whole, and neither elongation nor contraction along the mean principal axis.
Let OX and OZ be the directions of maximum elongation and maximum contraction respectively. Let A be any point of the body
z in its primitive condition, and A, the same point of the altered body, so that 0A, = a.OA.
Now, if we take OC=0A,, and if C, be the position of that point 8
-X of the body which was in the position C initially, we shall have OC, = OC, and therefore 00,04. Hence the two triangles COA and COA, are equal and similar.
Hence CA experiences no alteration of length, but takes the altered position CA, in the altered position of the body. Similarly, if we measure on Xo produced, OA and OA, equal respectively to OA and 04,, we find that the line CA' experiences no alteration in length, but takes the altered position C, A.
Consider now a plane of the body initially through CA perpendicular to the plane of the diagram, which will be altered into a plane through C, A, also perpendicular to the plane of the diagram. All lines initially perpendicular to the plane of the diagram remain so, and remain unaltered in length. AC has just been proved to remain unaltered in length. Hence ($ 139) all lines in the plane we have just drawn remain unaltered in length and in mutual inclination. Similarly we see that all lines in a plane through CA, perpendicular to the plane of the diagram, altering to a plane through CA'y per. pendicular to the plane of the diagram, remain unaltered in length and in mutual inclination.
149. The precise character of the strain we have now under cop sideration will be elucidated by the following:
Produce CO, and take OC and OC', respectively equal to OC and OC Join C'A, C'A', C',A,, and C, A 1, by plain and dotted lines as in the diagram. Then we see that the rhombus CAC'A' (plain lines) of the body in its initial state becomes the rhombus C A, C, A (dotted) in the altered condition. Now imagine the body thus strained to be moved as a rigid body (i e. with its state of strain kept unchanged) till A, coincides with A, and C', with C, keeping all the lines of
the diagram still in the same plane. A,C, will take a position in CA' produced, as shown in the new diagram, and the original and the altered
parallelogram will be on the same base A C', and A'
A between the same parallels AC and CA's, and
their other sides will be equally inclined on the two sides of a perpendicular to them. Hence, irrespectively of any rotation, or other absolute motion of the body not involving change of form.
or dimensions, the strain under consideration may be produced by holding fast and unaltered the plane of the body through AC', perpendicular to the plane of the diagram, and making every plane parallel to it slide, keeping the same distance, through a space proportional to this distance (i.e. different planes parallel to the fixed one slide through spaces proportional to their distances).
150. This kind of strain is called a simple shear. The plane of a shear is a plane perpendicular to the undistorted planes, and parallel to the lines of the relative motion. It 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
got when the first set and the degree or K
amount of shear are given, thus:--Let CC, be the motion of one point of one plane, relative to a plane KL held fixedthe 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 L
finally through AC, remains unaltered in
its dimensions. 151. 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.
152. 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 ($ 150), 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 D body.
Otherwise, taking a plane of shear for the plane of the diagram, let AB be a
С line in which it is cut by one of either
B 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 construction, CD the initial
, and CE the final, position of an unstretched line. Join DA, DB, EA, EB. DA, DB are the initial, and EA, EB the final, positions of the principal axes.
153. The ratio of a shear is the ratio of elongation and contraction of its principal axes. Thus if one principal axis is elongated in the ratio 1: a, and the other therefore ($ 148) 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 $ 152, the ratio of DB to ÉB, or of EA to DA, is the ratio of the shear.
164. 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.
155. The planes of no distortion in a simple shear are clearly the circular sections of the strain ellipsoid. In the ellipsoid of this case, be it remembered, the mean axis remains unaltered, and is a mean proportional between the greatest and the least axis.
156. If we now suppose all lines perpendicular to the plane of the shear to be elongated or contracted in any proportion, without altering lengths or angles in the plane of the shear, and if, lastly, we suppose every line in the body to be elongated or contracted in some other fixed ratio, we have clearly (8 142) the most general possible kind of strain.
157. Hence any strain whatever may be viewed as compounded of a uniform dilatation in all directions, superimposed on a simple elongation in the direction of one principal axis superimposed on a simple shear in the plane of the two other principal axes.
158. 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.
159. When the axes of the ellipsoid are lines of the body whose direction does not change, the strain is said to be pure, or unaccom. panied by rotation. The strains we have already considered were pure strains accompanied by rotations.
160. If a body experience a succession of strains, each unaccompanied by rotation, its resulting condition will generally be producible by a strain and a rotation. From this follows the remarkable corol. lary that three pure strains produced one after another, in any piece of matter, each without rotation, may be so adjusted as to leave the body unstrained, but rotated through some angle about some axis. We shall have, later, most important and interesting applications to fluid motion, which will be proved to be instantaneously, or differ: entially, 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 ($ 150), 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 CD; and in other planes parallel to it let there be motions proportional to their distances from 0. Consider first a shear from P to Py; then from Pi on to Ps; and let O be taken in a line through Pa, perpendicular to CD.
During the shear from P to P, -D a pointe moves of
course to l through a
distance 9 = PP. E Choose midway be
tween P and Pu, so that PQ- QP= PP. Now, as we have seen above ($ 152), the line of the body, which is the principal axis of contraction in the shear from ļ to lı, is OA, bisecting the angle QoE at the beginning, and 04, bisecting GoE at the end, of the whole motion considered. The angle between these two lines is half the angle lol, that is to say, is equal to P,OQ. Hence, if the plane cd is rotated through an angle equal to POQ, in the plane of the diagram, in the same way as the hands of a watch, during the shear from Q to li, 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 nonrotational shear.)
Now, let the second step, P, to Pq, be made so as to complete the whole shear, P to Ps, 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 preceding parenthesis) of CD, and to make it also non-rotational, as its predecessor has been made, we, must turn further round, 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 loQ. 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 QOQ by the excess of POQ above OOP.. Hence the resultant of the two shears, PP, P,Pe, 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.
161. 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.-PO, in the same direction as the hands of a watch.]
162. As there can be neither annihilation nor generation of matter in any natural motion or action, the whole quantity of a fluid within any space at any time must be equal to the quantity originally in that space, increased by the whole quantity that has entered it, and diminished by the whole quantity that has left it. This idea, when expressed in a perfectly comprehensive manner for every portion of a Huid in motion, constitutes what is commonly called the equation of continuity.'.
163. Two ways of proceeding to express this idea present themselves, each affording instructive views regarding the properties of fluids. In one we consider a definite portion of the fluid ; follow it in its motions; and declare that the average density of the substance varies inversely as its volume. We thus obtain the equation of continuity in an integral form.
The form under which the equation of continuity is most commonly given, or the differential equation of continuity, as we may call it, expresses that the rate of diminution of the density bears to the density. at any instant, the same ratio as the rate of increase of the volume of an infinitely small portion bears to the volume of this portion at the same instant.
164. To find the differential equation of continuity, imagine a space fixed in the interior of a fuid, and consider the fluid which flows into this space, and the fluid which flows out of it, across different parts of its bounding surface, in any time. If the fluid is of the same density and incompressible, the whole quantity of watter in the space in question must remain constant at all times, and therefore the quantity flowing in must be equal to the quantity flowing out in any time. If, on the contrary, during any period of motion, more fluid enters than leaves the fixed space, there will be condensa. tion of matter in that space; or if more Auid leaves than enters, there will be dilatation. The rate of augmentation of the average density