ELASTICITY 821 at right angles to one another are equivalent to a single dilatation or condensa tion tqual in all directions. The single stress equivalent to three equal tensions or pressures in directions at right angles to one another is a negative or positive pressure equal in all directions. (5) If a certain stress or infinitely small strain be defined (Chapter III. Cor. 3, or Chapter IV.) lay the ellipsoid -r(l + B)Y2+(H-C)Z 2 +DYZ+EZX+FXY=l, and another stress or infinitely small strain by the ellipsoid; where A, B, C, D, E, F, &c., are all infinitely small, their resultant stress or strain is that represented by the ellipsoid CHAPTER Till. Specification of Strains and Stresses by their Components according to chosen Types. Prop. Six stresses or six strains of six distinct arbitrarily chosen types may be determined to fulfil the condition of having a given stress or a given strain for their resultant, provided those six types are so chosen that a strain belonging to any one of them cannot be the resultant of any strains whatever belonging to the others. For, just six independent parameters being required to express any stress or strain whatever, the resultant of any set of stresses or strains may be made identical with a given stress or strain by ful filling six equations among the parameters which they involve ; and therefore the magnitudes of six stresses or strains belonging to the six arbitarily chosen types may be determined, if their resultant be assumed to be identical with the given stress or strain. Cor. Any stress or strain may be numerically specified in terms of numbers expressing the amounts of six stresses or strains of six arbitrarily chosen types which have it for their resultant. Types arbitrarily chosen for this purpose will be called types ot reference. The specifying elements of a stress or strain will be called its components according to types of reference. The specifying elements of a strain may also be called its coordinates, with reference to the chosen types. Examples. (1) Six strains in each of which one of the six edges of a tetra hedron of the solid is elongated while the others remain unchanged, may be used as types of reference for the specification of any kind of strain or stress. The ellipsoid representing any one of those six types will have its two circular sections parallel to the faces of the tetrahedron which do not contain ;he stretched side. (2) Six strains consisting, any one of them, of an infinitely small alteration either of one of the three edges, or of one of the three angles between the faces, of a parallelepiped of the solid, while the other five angles and edges remain un changed, may be taken as types of reference, for the specification of either stresses or strains. In some cases, as for instance in expressing the probable elastic properties of a crystal of Iceland spar, it might possibly be convenient to use an oblique parallelepiped for such a system of types of reference; but more frequently it will be convenient to adopt a system of types related to the defor mations of a cube of the solid. CHAPTER IX. Orthogonal Types oj Reference. Def. A normal system of types of reference is one in which the strains or stresses of the different types are all six mutually ortho gonal (fifteen conditions). A normal system of types of reference may also be called an orthogonal system. The elements specify ing, with reference to sxich a system, any stress or strain, will be called orthogonal components or orthogonal coordinates. Examples. (1) The six types described in Example (2) of Chapter VIII. are clearly orthogonal, if the parallelepiped referred to is rectangular. Three of these are simple longitudinal extensions, parallel to the three sets of rectangular edges of the parallelepiped. The remaining three are plane distortions parallel to the faces, their axes bisecting the angles between the edges. They constitute the system of types of reference uniformly used hitherto by writers on the theory of elasticity. (2) The six strains in which a spherical portion of the solid is changed into ellipsoids having the following equations X+T"+(l-t-C)Z*=l X 2 +Y 2 +Z 2 +DYZ = 1 X 2 +Y 2 +Z 2 +FXY = 1, are of the same kind as those considered in the preceding example, and there fore constitute a normal system of types of reference. The resultant of the strains specified, according to those equations, by the elements A, B, C, D, E, F, is a strain in which the sphere becomes an ellipsoid whose equation see above, Chapter VII. Ex. (5) is (l + A)X2+(l-l-B)Y-l-(l+C)Z2+DYZ-fEZX- r .FXY = l. (3) l A compression equal in all directions (I.), three simple distortions hav ing their planes at right angles to one another and their axes 2 bisecting the angles between the lines of intersection of these planes (II.) (III.) (IV.), any simple or compound distortion consisting of a combination of longitudinal strains parallel to those lines of intersections (V.), and the distortion (VI.), con stituted from the same elements which is orthogonal to the last, afford a system of six mutually orthogonal types which will be used as types of reference below in expressing the elasticity of cubiciilly isotropic solids. (Compare Chapter X. Example 7 below.) , 1 This example, as well as (7) of Chapter X.. (5)"oi XL, and the example of Chapter XII, are intended to prepare for the application of the theory of .Principal Elasticities to cubically and spherically isotropic bodies, in Part II. Chapter XV. 2 The "axes of a simple distortion" are the lines of its two component longi tudinal strains. CHAPTER X. On the Measurement of Strains and Stresses Def. Strains of any types are said to be to one another in the same ratios as stresses of the same types respectively, when any particular plane of the solid acquires, relatively to another plane parallel to it, motions in virtue of those strains which are to one another in the same ratios as the normal components of the forces between the parts of the solid on the two sides of either plane due to the respective stresses. Def. The magnitude of a stress and of a strain of the same type are quantities which, multiplied one by the other, give the work done on unity of volume of a body acted on by the stress Avhile acquiring the strain. Cor. 1. If a;, y, z, |, ri, denote orthogonal components of a certain strain, and if P, Q, R, S, T, U denote components, of the same type respectively, of a stress applied to a body while acquiring that strain, the work done upon it per unit of its volume will be Cor. 2. The condition that two strains or stresses specified by (x, y, z, , n, ) and (xf, y , z , , 17 , ), in terms of a normal system of types of reference, may be orthogonal to one another is Cor. 3. The magnitude of the resultant of two, three, four, five, or six mutually orthogonal strains or stresses is equal to the square root of the sum of their squares. For if P, Q, &c., denote several orthogonal stresses, and F the magnitude of their resultant ; and x, y, &c., a set of proportional strains of the same types respectively, and r the magnitude of the single equivalent strain, the resultant stress and strain will be of one type, and therefore the work done by the resultant stress will be Fr. But the amounts done by the several components will be PJC, Qy, &c., and therefore Now we have, to express the proportionality of the stresses and strains, Each member must be equal to P 2 + Qg + .fee. . PJT + Qy + &c. and also equal to + Qy + *c. Hence an<i p2 _1_ Q2 -L ( tc , = jfc -- . which gives F 2 - P 2 + Q 2 + &c. - r = x -i + y i + &( ;. which gives r 2 = x 1 + y" + &c. Cor. 4. A definite stress of some particular type chosen arbi trarily may be called unity ; and then the numerical reckoning of all strains and stresses becomes perfectly definite. Def. A uniform pressure or tension in parallel lines, amounting in intensity to the unit of force per unit of area normal to it, will be called a stress of unit magnitude, and will be reckoned as positive when it is tension, and negative when pressure. Examples. (1) Hence the magnitude of a simple longitudinal strain, in which lines of the body parallel to a certain direction experience elongation to an ex tent bearing the ratio K to their^original dimensions, must be called K. (2) The magnitude of the single stress equivalent to three simple pressures in directions at right angles to one another each unity is ^/S; a uniform com pression in all directions of unity per unit of surface is a negative stress equal to v 3 in absolute value. (3) A uniform dilatation in all directions, in which lineal dimensions are aug mented in the ratio 1 : 1 + x, is a strain equal in magnitude to x/3 ; or a uni form " cubic expansion" E is a strain equal to -75 . (4) A stress compounded of unit pressure in one direction and an equal tension in a direction at right angles to it, or which is the same thing, a stress com pounded of two balancing couples of unit tangential tensions in planes at angles of 45 to the direction of those forces, and at right angles to one another amounts in magnitude to /2. (5) A strain compounded of a simple longitudinal extension r, and a simple longitudinal condensation of equal absolute value, in a direction perpendicular to it, is a strain of magnitude x -v/2; or, which is the same thing (if <r = 2.r), a simple distortion such that the relative motion of two planes at unit distances parallel to either of the planes bisecting the angles between the two planes mentioned above is a motion or parallel to themselves, is a strain amounting in magnitude to ^ .
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(6) If a strain be such that a sphere of unit radius in the body becomes an ellipsoid whose equation is (1 - A)X2 + (1 - B)Y 2 + (1 - C)Z2 - DYZ - EZX - FXY = 1 , the values of the component strains corresponding, as explained in Example (2 of Chap. IX., to the different coefficients respectively,>re JA,JB,iC,j--,-, 2-^-, 2 . For the components corresponding to A, B, C are simple longitudinal strains, in which diameters of the sphere along the axes of coordinates become elongated from 2 to 2 + A, 2 + B, 2 + C respectively: D is a distortion in which diameters in the plane YOZ, bisecting the angles YOZ and Y OZ, become respectively elongated and contracted from 2 to 2 + $D, and from 2 to 2 - JD; and so for
the others. Hence, if we take x, >, t, , tj, to denote the magnitudes of six