BOOK III. STRAINS.

CHAPTER I. STRAIN-STEPS.

STRAIN IN STRAIGHT LINE.

WE have hitherto studied the motion of rigid bodies, which do not change in size or shape. We have now to take account of those strains, or changes in size and shape, which we have hitherto neglected.

The simplest kind of strain is the change of length of an elastic string when it is stretched or allowed to contract. When every portion of the string has its length altered in the same ratio, the strain is called uniform or homogeneous. Thus if apb is changed into a'p'b' by a uniform strain, ap: a'p' = ab a'b'. The ratio

ar

α'ι

b

p

a'p' ap, or the quantity by which the original length must be multiplied to get the new length, is called the ratio of the strain. The ratio of the change of length to the original length, or a'p' - ap ap, is called the elongation; it is reckoned negative when the length is diminished. A negative elongation is also called a compression.

Let e be the elongation, s the unstretched length ap, ☛ the stretched length a'p', then σ − s = es, or σ = 8 (1+e). Thus 1e is the ratio of the strain.

In general, a solid body undergoes a strain of simple elongation e, when all lines parallel to a certain direction. are altered in the same ratio 1 : 1+e, and no lines perpendicular to them are altered in magnitude or direction.

STRAIN OF PLANE FIGURE.

159

The strain is then entirely described if we describe the strain of one of the parallel lines.

HOMOGENEOUS STRAIN IN PLANE.

The kind of strain next in simplicity is that of a flat membrane or sheet. Suppose this to be in the shape of a square; we may give it a uniform elongation e parallel to one side, and then another uniform elongation ƒ parallel to the other side. It is now converted into a rectangle, whose sides are proportional to 1+e, 1+f. By each of these operations two equal and parallel lines, drawn on the membrane, will be left equal and parallel; though, if not parallel to a side of the square, they will be altered in direction.

We may prove, conversely, that every strain which leaves straight lines straight, and parallel lines parallel, is a strain of this kind combined with a change of position of the membrane in its plane. Such a strain is called uniform or homogeneous.

Since a parallelogram remains a parallelogram, equal parallel lines remain equal. Then it is easy to shew, by the method of equi-multiples, that the ratio of any two parallel lines is unal

tered by the strain. Next, if we draw a circle on the unstrained membrane, this circle will be altered by the strain into an ellipse. For in the unstrained figure

L

B

B

A'M. MA: CA2=MP2 : CB2,

b

a

C

and since these ratios of parallel lines are unaltered, it follows that in the strained figure also

a'm.ma: ca2=mp2 : cb2.

Hence the strained figure is an ellipse, whose conjugate diameters are the strained positions of perpendicular diameters of the circle.

It follows that there are two directions at right angles to one another, which remain perpendicular after the

strain; namely those which become the axes of the ellipse into which a circle is converted. If these lines remain parallel to their original directions, the strain is produced by two simple elongations along them respectively; in that case it is called a pure strain. If they are not parallel to their original directions, the strain is compounded of a pure strain and a rotation.

Two lines drawn anywhere in the strained membrane parallel to the axes of the ellipse into which a circle is converted, or in the unstrained membrane parallel to the unstrained position of those axes, are called principal axes of the strain. The elongations along them are called principal elongations; the ratios in which they are altered are called principal ratios.

REPRESENTATION OF PURE STRAIN BY ELLIPSE.

When the strain is pure, the new position of any step may be conveniently represented by means of a certain ellipse. Let the principal ratios be p, q, so that every line parallel to oX is altered in the ratio 1: p, and every line parallel to oY in the ratio 1 : q. Take two lengths oa, ob, along oX, oY respectively, such that oa: obq p, and let m be the positive geometric mean of p, q, so that m2: = =pq. Then we shall have, so far as length is concerned, p. oa = m. ob, and q. ob = m. oa. Hence, taking account of direction, oa becomes im. ob', and ob becomes im, oa, in consequence of the strain,

:

Now construct an ellipse having oa, ob for semi-axes; then if p be any point on it and qq' the diameter conjugate

DISPLACEMENT-CONIC.

161

to op, the strain will turn op into im. og. For since it turns oa into im. ob', it will turn on into im. rq, because on: oarq': ob' (p. 129). And since it turns ob into im. oa, it will turn np into im. or, because np: obor : oa. Therefore it will turn op, which is on +np, into im (rq′ + or), that is, into im. oq'.

Hence we see that the strained position of any vector is perpendicular to the conjugate diameter of a certain ellipse, having that vector as diameter, and is proportional to the conjugate diameter in length. For the ellipse used in this representation may be of any size, since all that is necessary for it is that its axes should be parallel to the principal axes of the strain, and inversely proportional to the square roots of the principal ratios.

REPRESENTATION OF THE DISPLACEMENT.

The displacement of any point is the step from its old position to its new one. Thus if a vector op is turned by the strain into op', the displacement of p is pp'.

When the two principal elongations e, f are of the same sign, the displacement may be represented by an ellipse, in the same way as we have represented the new position of any vector. The only difference is that we are now to draw an ellipse whose axes are inversely proportional to the square roots of the elongations, so that oa2 : ob2 = ƒ : e, and to make m2 = ef, giving to m the same sign as e or f. Then the displacement of a will be im . ob', and the displacement of b will be im. oa. Hence it follows (as before) that the displacement of p will be im. og'. In this case therefore the displacement of every point on the ellipse is perpendicular and proportional to that diameter which is parallel to the tangent at the point.

But when e and ƒ are of different signs, it is necessary to use a hyperbola to represent the displacement. Let m2 = ef, and oa2: ob2 = -ƒ: e; and let m be taken of the same sign as f. Then the displacement of a will be im. ob, and the displacement of b will be im. oa. If then a hyperbola be described with oa and ob as axes, and

op, og be a pair of conjugate semi-diameters, the displacement of p will be im. oq and that of q will be im.op. The

proof is the same as for the ellipse, depending on the property that пр : ob = or oa, and rq: ob= on: oa. The ellipse or hyperbola which is thus used to represent the displacement is called the displacement-conic of the strain.

LINEAR FUNCTION OF A VECTOR.

One vector is said to be a function of another, when its components are functions of the components of the other; so that, for every value (including magnitude and direction) of one of them, there is a value or values of the other. Thus pi+qj+rk is a function of xi+yj+zk if P, q, r are functions of x, y, z. We may express this relation between them thus: pi+qj +rk = $ (xi +yj+zk).

A function of a vector is said to be linear when that function of the sum of two vectors is the sum of the functions of the vectors. Thus the function is linear when $(a + B) = pa +$8.

At present we shall consider only linear functions of vectors which are all in one plane. It is clear that when a plane figure receives a homogeneous strain, the strained position of any vector is a linear function of the vector. For the triangle made of two vectors a, ẞ and their sum a+B becomes after the strain a triangle made of the $x, $ß, & (a + B) ;

vectors

and consequently

φ (α + β) = φα + φβ.