step at right angles to this is i sin 0-j cos 0. One of these receives the elongation e and the other the elongation f, each in its own direction; therefore

(i cos 0+j sin 0) = e(i cos 0 +j sin 0),

(i sin -j cos 0) = ƒ (i sin 0 —j cos 0).

Multiply the first equation by cos 0, the second by sin 0, and add; thus we get

2

pi=(e cos2 0 +ƒsin' 0) i+ (e-f) sin cos 0. j= ai +hj. Similarly, by multiplying the first equation by sin 0, the second by cos 0, and subtracting, we get

bj = (e-ƒ) sin cos 0 . i + (e sin3 0 +ƒ cos3 0). j = hi+bj. It is now necessary to find quantities, e, f, 0, which satisfy the equations

e cos* 0+ƒ sin10 = a,

e sin2 0 +ƒ cos 0=b,

(e-f) sin 0 cos 0,= 1 (e−ƒ) sin 20 = h.

Adding the first two, we have

Compare with this the solution of an analogous problem on p. 131, making in that, 0=π, and k,=k,

When the plane is as much lengthened along one principal axis of the strain as it is shortened along the other, so that (1 + e)(1 +ƒ) = 1, or e+f+ef= 0, the strain is called a shear. In this case it is clear that the area of every figure in the plane remains unaltered.

Let oa be changed into oA, and take ob=0A; then ob will be changed into oB, which is equal to oɑ. Hence the rhomb aba'b' will become the rhomb ABA'B', and ab, which becomes AB, will be unaltered in length. If we combine this pure strain with a rotation, so as to bring ab

to coincide with AB, then a'b' may be brought to A'B' by a sliding motion along its line. Thus all lines parallel to ab will be slid along themselves through lengths proportional to their distances from ab. The amount of sliding per unit distance is called the amount of the shear.

=

Since we have also ab A'B, the shear might also be produced by the sliding of lines parallel to a'b; but then

it would be combined with a different rotation. Thus there are two sets of parallel lines which are unaltered in length, and whose relative motion is a sliding along themselves.

The ratio oA : oa is called the ratio of the shear. If ob=a.oa, the sliding of a'b relative to b'a is 2ab. COS aba' and the distance between a'b and b'a is ab sin aba'. Hence the amount of the shear is 2 cot aba' = 2 cot 20, if 0 = abo, so that cot 0 = a. Now

Thus, if a be the ratio of a shear, its amount s is given by s=a-a2

We have seen that e and ƒ satisfy the equation

e+f+ ef = 0,

in the case of a shear. When e and f are very small fractions, ef is small compared with either of them, and we have approximately e+f=0. The ratio eƒ differs from unity, in fact, by the small fraction e. Thus the displacement-conic is approximately a rectangular hyper

bola.

Now the ratio of the shear is 1+ e, and

Hence the amount is

(1 + e) (1 +ƒ ) = 1.

1+e- (1+f) = e−ƒ;

e:

this is accurate, whether the shear be large or small. But if the shear is very small, ƒ is approximately equal to – e, and thus the amount is approximately = 2e.

COMPOSITION OF STRAINS.

When the displacement of every point, due to a certain strain, is the resultant of its displacements due to two or more other strains, the first strain is said to be the resultant of these latter, which are called its components. If the displacement of the end of p in two strains respec

PRODUCT AND RESULTANT.

169

tively be op and p, the displacement in their resultant is (+)p.

This must be carefully distinguished from the result of making a body undergo the two strains successively. Thus if be changed into p by the first strain, and P intop by the second, the effect of applying the second strain after the first will be to change p into {,(p)} or ψιφιρ. p. To compare this with the preceding expression for the resultant, we must observe that 1=1++ and 1+; so that whereas in the one case the displacement is (+)p, in the other it is (++). In one case only the addition, in the other the multiplication of functions is involved. For this reason we shall speak of the strain, whose effect is the same as that of two other strains successively applied, as the product of the two strains.

=

A strain in which ab=0, and h-h', is called a skew strain, and the displacement-function & a skew function. It is the product of a rotation about the origin and a uniform dilatation; for the displacement of every point p is perpendicular to op and proportional to it.

Every plane strain is the resultant of a pure and a skew strain. For let a, h, h', b have the same meaning as before; these numbers are the sums of

a, 1 (h+h'), 1 (h+h'), b, and 0, †1 (h — h'), 1 (h' — h), 0, of which the former belong to a pure, and the latter to a skew strain.

But every plane strain is the product of a rotation, a uniform dilatation, and a shear. First rotate the plane till the principal axes of the strain are brought into position; then give it uniform dilatation (or compression) till the area of any portion is equal to the strained area; the remaining change can be produced by a pure shear.

When two strains are both very small, their product and resultant are approximately the same strain.

REPRESENTATION OF STRAINS BY VECTORS.

We have seen that if e, f be the principal elongations of a pure strain (a, h, h, b), then e+f=a+b. Hence if a+b=o, we must have e+ƒ=0. Hence the strain is

made by an elongation in one direction, combined with an equal compression in the perpendicular direction. Such a strain is approximately a shear when it is very small; we shall therefore call it a wry shear. Its characteristic is that its displacement-conic is a rectangular hyperbola. A wry shear accompanied by rotation shall be called a wry strain; that is (a, h, h', b) is a wry strain if a+b=0.

Every strain is the resultant of a uniform dilatation and a wry strain. For

(a, h, h', b) = † (a + b, 0, 0, a + b) + 1 (a − b, 2h, 2h', b — a). Every wry strain is the resultant of a skew strain and a wry shear. For

1 (a — b, 2h, 2h', b − a) = 1 (0, h — h', h' — h, 0)

The magnitude of a skew strain (0,h,—h, 0) is h. Being the product of a rotation by a uniform dilatation, it is not specially related to any direction in the plane, and may therefore be represented by a vector of length h perpendicular to the plane.

The wry shear (a, h, h, - a) has for its displacementconic a rectangular hyperbola whose transverse axis makes with oX an angle & such that tan 20=h: a (since in this case a-b2a; the general value being tan 0=2h: a—b). Moreover if e,e are its principal elongations, we have in general (e-ƒ)2 = (a − b)2 + 4h, and therefore in this case e2=a2+h. Hence if a wry shear be represented by a vector in its plane, of length equal to its positive principal elongation, making with oX an angle (20) equal to twice the angle (0) which that elongation makes with it; the components (a, h) of this vector along oX and oY will represent in the same way the wry shears (a, 0, 0, − a) and