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 oba. 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: = α. Now Thus, if a be the ratio of a shear, its amount s is given by s=a-a ̄2. We have seen that e and ƒ satisfy the equation e+f+ ef = 0, in the case of a shear. When e and ƒ are very small fractions, ef is small compared with either of them, and we have approximately e+f=0. The ratio e: ƒ 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 +ƒ) = e−ƒ ; 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 pp 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 p be changed into p by the first strain, and intop by the second, the effect of applying the second strain after the first will be to change p into {6,(p)} or ψιφερ. e. To compare this with the preceding expression for the resultant, we must observe that p=1+ and =1+; so that whereas in the one case the displacement is (+)p, in the other it is (++) p. 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, (h+h'), (h+h'), b, and 0, (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=0, we must have e+f=o. 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) = 1 (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) + 1 (a − b, h + h', h + h', b − a). 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-b=2a; 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+h2. 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 SYMBOL OF A PLANE STRAIN. 171 (0, h, h, 0), having oX and of respectively for axes and asymptotes, of which the given wry shear is the resultant. Let such a vector be called the base of the wry shear; then our proposition is that the base of the resultant of two wry shears is the resultant of their bases. This is obvious, because the base of (a, h, h, — a) is ai+hj. This mode of representation is to a certain extent arbitrary, because it depends upon the position of oX. It will, however, be found useful in many ways. Combining this with our previous representation of a skew strain, we see that a wry strain in general may be represented by a vector not necessarily in its plane, the normal component of which represents the skew part of the strain, while the component in the plane represents the wry shear. When a figure receives a uniform dilatation, without. rotation, we may regard it as merely multiplied by a numerical ratio or scalar quantity. Thus the whole operation of any plane strain may be regarded as the sum of a scalar and a vector part. If we write, for example, 1=(1,0, 0, 1) ... (leaves the figure unaltered) I = (1, 0, 0, -1)... (turns it over about oY) 0)... (interchanges oX and oY) 0)... (turns counter clock-wise through a right angle) (a, h, h', b)=(a+b) + (a−b) 1 + 1 (h+h') J + † (h — h') K, and it will be easy, by combining these operations, to verify that I2 = 1, J2 = 1, K2 = − 1, JK = I = − KJ, KI=J=−IK, IJ=K=-JI. GENERAL STRAIN OF SOLID. PROPERTIES OF THE When a solid is so strained that the lengths of all parallel lines in it are altered in the same ratio, it is said to undergo uniform or homogeneous strain. It follows easily, as before, that all parallel planes remain parallel planes, and undergo the same homogeneous strain, besides being altered in their aspect. A sphere is changed into a surface which is called an ellipsoid, having the property that every plane section of it is an ellipse. We may easily obtain its principal properties from those of the sphere, if we remember only that the ratios of parallel lines are unaltered by the strain. Thus we know that if a plane be drawn through the centre of a sphere, the tangent planes at all points where it cuts the sphere are perpendicular to it, and therefore parallel to the normal to it through the centre; this normal meets the sphere in two points where the tangent planes are parallel to the first plane. A plane A drawn through the centre of the ellipsoid (a point such that all chords through it are bisected at it) is called a diametral plane. The tangent planes at all points where it cuts the surface are parallel to a certain line through the centre, called the diameter conjugate to the given plane; this line cuts the surface in two points where the tangent planes are parallel to the given plane A. Any two conjugate diameters of the ellipse in which the ellipsoid is cut by the plane A, together with the diameter conjugate to that plane, form a system of three conjugate diameters; each of them is conjugate to the plane containing the other two. They correspond to three diameters of a sphere at right angles to one another. The |