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Simple shear.

Axes of a shear.



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.




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



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.

Since DCA2DBA, and tan DBA





we have tan DCA

DE=2CN tan DCN = 2CN cot DCA.
DE a2 - 1 1
CN 2a

= α



176. The planes of no distortion in a simple shear clearly the circular sections of the strain ellipsoid. In ellipsoid of this case, be it remembered, the mean axis remains unaltered, and is a mean proportional between the greatest and the least axis.

are Ellipsoidal


the tion of a


ple elonga


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

ple elongation, and expansion, combined.

which the principal ratios are

Analysis of a strain.

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If, lastly, we elongate all lines in the ratio 1n, we have a strain in which the principal ratios are


s, m,



where it is clear that ns, nm, and may have any values

whatever. It is of course not necessary that nm be the mean principal ratio. Whatever they are, if we call them a, B, y respectively, we have

n ns, nm, 9 S



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178. Hence any strain (2, B, y) whatever may be viewed as compounded of a uniform dilatation in all directions, of linear B ratio Vay, superimposed on a simple elongation in the Nay direction of the principal axis to which ẞ refers, superimposed


on a simple shear, of ratio

√ (or

√ - √2)


in the plane of the two other principal axes.

or of amount

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.

Displacement of a

180. In the preceding sections on strains, we have con

body, rigid sidered the alterations of lengths of lines of the body, and of

or not, one point of which is

angles between lines and planes of it; and we have, in parti

held fixed. cular cases, founded on particular suppositions (the principal

axes of the strain remaining fixed in direction, § 169, or one

ment of a

or one

which 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 x1, y, z, 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:

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where [Xx], [Xy], etc., are nine quantities, of absolutely arbitrary values, the same for all values of x, y, z.

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[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.


Let it be required to find, if possible, a line of the body which remains unaltered in direction, during the change specified by [xx], 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. must have Y, x y line in question.


1+, where is the elongation of the

They are, of
altered posi-
Similarly for

Displacement of a body, rigid or not, one point of which is held fixed.

Thus we have x, = (1 + €)x, etc., and therefore if n = 1 + €

{[Xx] −n}x

+ [Xy]y

+ [X2] = 0, [Yx]x + {[Yy] - ny + [Y2]z = 0, [Zx]x + [Zy]y + {[Z≈] − n}z = 0.



From these equations, by eliminating the ratios xy: z according
to the well-known algebraic process, we find

([Xx] − n) ([Yy] − n) ([Z≈] − n)

- [Y2][Zy]([Xx] – n) − [Zx][Xz]([Yy] −n) − [Xy][Yx]([Zz] − n)
+[X≈] [Yx][Zy] + [Xy][Y≈] [Zx] = 0.

This cubic equation is necessarily satisfied by at least one real
value of
η, and the two others are either both real or both ima-
ginary. 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 imagi-
nary 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.


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