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in mutual inclination. geodetic lines.

Hence, also, geodetic lines must remain

135. We have now to consider the very important kinematical conditions presented by the changes of volume or figure experienced by a solid or liquid mass, or by a group of points whose positions with regard to each other are subject to known conditions.

Any such definite alteration of form or dimensions is called a Strain.

Thus a rod which becomes longer or shorter is strained. Water, when compressed, is strained. A stone, beam, or mass of metal, in a building or in a piece of framework, if condensed or dilated in any direction, or bent, twisted, or distorted in any way, is said to experience a strain. A ship is said to 'strain' if, in launching, or when working in a heavy sea, the different parts of it experience relative motions.

136. If, when the matter occupying any space is strained in any way, all pairs of points of its substance which are initially at equal distances from one another in parallel lines remain equidistant, it may be at an altered distance; and in parallel lines, altered, it may be, from their initial direction; the strain is said to be homogeneous.

137. Hence if any straight line be drawn through the body in its initial state, the portion of the body cut by it will continue to be a straight line when the body is homogeneously strained. For, if ABC be any such line, AB and BC, being parallel to one line in the initial, remain parallel to one line in the altered, state; and therefore remain in the same straight line with one another. Thus it follows that a plane remains a plane, a parallelogram a parallelogram, and a parallelepiped a parallelepiped.

138. Hence, also, similar figures, whether constituted by actual portions of the substance, or mere geometrical surfaces, or straight or curved lines passing through or joining certain portions or points of the substance, similarly situated (i.e. having corresponding parameters parallel) when altered according to the altered condition of the body, remain similar and similarly situated among one another.

139. The lengths of parallel lines of the body remain in the same proportion to one another, and hence all are altered in the same proportion. Hence, and from § 137, we infer that any plane figure becomes altered to another plane figure which is a diminished or magnified orthographic projection of the first on some plane.

The elongation of the body along any line is the proportion which the addition to the distance between any two points in that line bears to their primitive distance.

140. Every orthogonal projection of an ellipse is an ellipse (the case of a circle being included). Hence, and from § 139, we see that an ellipse remains an ellipse; and an ellipsoid remains a surface of which every plane section is an ellipse; that is, remains an ellipsoid.

141. The ellipsoid which any surface of the body initially spherical

becomes in the altered condition, may, to avoid circumlocutions, be called the Strain Ellipsoid.

142. In any absolutely unrestricted homogeneous strain there are three directions (the three principal axes of the strain ellipsoid), at right angles to one another, which remain at right angles to one another in the altered condition of the body. Along one of these the elongation is greater, and along another less, than along any other direction in the body. Along the remaining one the elongation is less than in any other line in the plane of itself and the first mentioned, and greater than along any other line in the plane of itself and the second.

Note.-Contraction is to be reckoned as a negative elongation: the maximum elongation of the preceding enunciation may be a minimum contraction: the minimum elongation may be a maximum contraction. 143. The ellipsoid into which a sphere becomes altered may be an ellipsoid of revolution, or, as it is called, a spheroid, prolate, or oblate. There is thus a maximum or minimum elongation along the axis, and equal minimum or maximum elongation along all lines perpendicular to the axis.

Or it may be a sphere; in which case the elongations are equal in all directions. The effect is, in this case, merely an alteration of dimensions without change of figure of any part.

144. The principal axes of a strain are the principal axes of the ellipsoid into which it converts a sphere. The principal elongations of a strain are the elongations in the direction of its principal axes.

145. When the positions of the principal axes, and the magnitudes of the principal elongations of a strain are given, the elongation of any line of the body, and the alteration of angle between any two lines, may be obviously determined by a simple geometrical construction.

146. With the same data the alteration of angle between any two planes of the body may also be easily determined, geometrically. 147. Let the ellipse of the annexed diagram represent the section of the strain ellipsoid through the greatest and least principal axes.

X

Z

Let S'OS, T'OT be the two diameters of this ellipse, which are equal to the mean principal axis of the ellipsoid. Every Xplane through O, perpendicular to the plane of the diagram, cuts the ellipsoid in an ellipse of which one principal axis is the diameter in which it cuts the ellipse of the diagram, and the other, the mean principal diameter of the ellipsoid. Hence a plane through either SS or TT", perpendicular 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 ellipsoid 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.

Z

Let OX and OZ be the directions of maximum elongation and maximum contraction respectively. Let A be any point of the body in its primitive condition, and A, the same point of the altered body, so that OA, = a.OA.

Now, if we take OC = OA,, and if C, be the position of that point of the body which was in the position C initially, we shall have

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OA. Hence the two triangles COA and COA, are equal and similar.

X

C'

Z

A

-X

Hence CA experiences no alteration of length, but takes the altered position C,A, in the altered position of the body. Similarly, if we measure on XO produced, OA' and OA, equal respectively to OA and OA,, 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 CA, 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 C,A',, perpendicular 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 consideration will be elucidated by the following:-Produce CO, and take OC and OC, respectively equal to OC and OC1. Join C′A, C'A', C'1 A, and C, 4',, by plain and dotted lines as

1

1

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 A1
coincides with A, and C with C', keeping all A
the lines of the diagram still in the same plane.
A', C, will take a position in CA' produced,

1

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as shown in the new diagram, and the original and the altered

parallelogram will be on the same base AC', and between the same parallels AC and CA, 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)

L

A

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 CC1 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 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 body.

N

E

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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 EB, or of EA to DA, is the ratio of the shear.

154. 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 (§ 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 unaccompanied 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 corollary that three pure strains produced one after another, in any piece of matter, each without rotation, may be so adjusted as to leave the

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