44 CHAPTER V. STRESS, STRAIN, AND RESILIENCE. 47. IN the nomenclature introduced by Rankine and adopted by Thomson and Tait, any change in the shape or size of a body is called a strain, and an action of force tending to produce a strain is called a stress. We shall always suppose strains to be small; that is, we shall suppose the ratio of the initial to the final length of every line in the strained body to be nearly a ratio of equality. 48. A strain changes every small spherical portion of the body into an ellipsoid; and the strain is said to be homogeneous when equal spherical portions in all parts of the body are changed into equal ellipsoids with their corresponding axes parallel. When the strain consists in change of volume, unaccompanied by change of shape, the ellipsoids are spheres. When strain is not homogeneous, but varies continuously from point to point, the strain at any point is defined by attending to the change which takes place in a very small sphere or cube having the point at its centre, so small that the strain throughout it may be regarded as homogeneous. In what follows we shall suppose strain to be homogeneous, unless the contrary is expressed. 49. The axes of a strain are the three directions at right angles to each other, which coincide with the directions of the axes of the ellipsoids. Lines drawn in the body in these three directions will remain at right angles to each other when the body is restored to its unstrained condition. A cube with its edges parallel to the axes will be altered by the strain into a rectangular parallelepiped. Any other cube will be changed into a parallelepiped not in general rectangular. When the axes have the same directions in space after as before the strain, the strain is said to be unaccompanied by rotation. When such parallelism does not exist, the strain is accompanied by rotation, namely, by the rotation which is necessary for bringing the axes from their initial to their final position. The numbers which specify a strain are mere ratios, and are therefore independent of units. 50. When a body is under the action of forces which strain it, or tend to strain it; if we consider any plane section of the body, the portions of the body which it separates are pushing each other, pulling each other, or exerting some kind of force upon each other, across the section, and the mutual forces so exerted are equal and opposite. The specification of a stress must include a specification of these forces for all sections, and a body is said to be homogeneously stressed when these forces are the same in direction and intensity, for all parallel sections. We shall suppose stress to be homogeneous, in what follows, unless the contrary is expressed. 51. When the force-action across a section consists of a simple pull or push normal to the section, the direction of this simple pull or push (in other words, the normal to the section) is called an axis of the stress. A stress (like a strain) has always three axes, which are at right angles to one another. The mutual forces across a section not perpendicular to one of the three axes are in general partly normal and partly tangential—one side of the section is tending to slide past the other. The force per unit area which acts across any section is called the intensity of the stress on this section, or simply the stress on this section. The dimensions of therefore call the dimensions of stress. 52. The relation between the stress acting upon a body and the strain produced depends upon the resilience of the body, which requires in general 21 numbers for its complete specification. When the body has exactly the same properties in all directions, 2 numbers are sufficient. These specifying numbers are usually called coefficients of elasticity; but the word elasticity is used in so many senses that we prefer to call them coefficients of resilience. A coefficient of resilience expresses the quotient of a stress (of a given kind) by the strain (of a given kind) which it produces. A highly resilient body is a body which has large coefficients of resilience. Steel is an example of a body with large, and cork of a body with small, coefficients of resilience. In all cases (for solid bodies) equal and opposite strains (supposed small) require for their production equal and opposite stresses. 53. The coefficients of resilience most frequently referred to are the three following: (1) Resilience of volume, or resistance to hydrostatic compression. If V be the original and V - the strained. volume, is called the compression, and when the body V is subjected to uniform normal pressure Р per unit area over its whole surface, the quotient of P by the compression is the resilience of volume. This is the only kind of resilience possessed by liquids and gases. (2) Young's modulus, or the longitudinal resilience of a body which is perfectly free to expand or contract laterally. In general, longitudinal extension produces lateral contraction, and longitudinal compression produces lateral extension. Let the unstrained length be L and the strained length L ± 7, then is taken as the measure L of the longitudinal extension or compression. The stress on a cross section (that is, on a section to which the stress is normal) is called the longitudinal stress, and Young's modulus is the quotient of the longitudinal stress by the longitudinal extension or compression. If a wire of cross section A sq. cm. is stretched with a force of F dynes, and its length is thus altered from L to L7, the value F L of Young's modulus for the wire is A' 7' (3) "Simple rigidity" or resistance to shearing. This requires a more detailed explanation. 54. A shear may be defined as a strain by which a sphere of radius unity is converted into an ellipsoid of semiaxes Ï, 1+e, 1−e; in other words, it consists of an extension in one direction combined with an equal compression in a perpendicular direction. 55. A unit square (Fig. 1) whose diagonals coincide |