MECHANICS 695 om- The successive application of two pure strains does not, osition except in special cases, give rise to a pure strain. This fpure j gj physical a most important proposition. Thus, for instance, the instantaneous strains of each element of a perfect fluid in which there is no vortex motion are pure ; and yet, if the element be followed in its motion, it will be found in general to rotate. Its motion is said to be "differentially irrotational." To prove this proposition by the help of a particular case is simple enough. Take, for instance, a compression in one direction, followed by an equal extension in a different direction. Only when these directions are at right angles to one another is the resultant strain pure. .nalysis. 93. The analytical theory of strains is, at least in its elements, an immediate application of the properties of determinants, usually of the third order. We subjoin a slight sketch of it. "We have seen that it is only necessary, for the full characterizing of a strain, that we should know what becomes of three unit lines not originally coplanar. Take these parallel to the axes (generally oblique) of x, y, z. Then if the x unit becomes a line which is the diagonal of a parallelepiped with sides a, d, g parallel to the axes, y similarly that of b, c, h, and z of c, f, i, we see at once that the coordinates of the point originally at x, y, z become x = ax + by + cz (A). Here it is obvious, from the premises, that the nine quantities a, b, c; d, c, f; g, h, i are all real, and altogether independent, at least so far as kinematics is concerned. 1 To obtain an idea of their nature from another point of view, let us suppose the axes to be rectangular. Let unit parallel to x become e lt in the direction given by the cosines l l} m^, n^ Similarly, let e 2 , 1%, m. 2 , n 2 belong to a unit originally parallel to y, and %, ? 3 , m 3 , n 3 to a unit parallel Then the broken line x, y, z becomes x , y , z 1 , where x = c i l^x + e. 2 l z y + e 3 l 3 z y ^e^n-^x + e^n^y + e z m s z . . . (A ). to z. Though we have introduced three numbers e along with nine direction cosines, no greater generality is secured, for there are three necessary relations, one among each set of cosines. 3harac- To find the characteristic property of a pure strain, let us take
- eristic it in its most general form. Thus let l lt ??i 1 , % now denote a line
>f pure which, without change of direction, has its length altered by the train. strain in the ratio c^: 1. Let Z 2 , m 2 , n 2 , c 2 and Z 3 , 1 3 , n 3 , e 3 be similar data for the other two of the system of rectangular axes of the pure strain. Then to these axes the coordinates of x, y, z are = lyK + m 3 y + n 3 z . The strain converts into | = Cj|, r) into ij = c 2 i7, and into C =e 3C- Hence the final coordinates (to the original axes) of the point originally at x, y, z arc or, in terms of x, y, z, If we compare this with the general expression above given for a strain, we see that the coefficient of y in the value of a; is equal to that of x in the value of y . Similarly that of z in x is equal to that of x in z 1 ; and that of z in y is equal to that of y in z ; or finally b = d, c = g, /=/*. 1 But when strain is produced in a piece of matter, a limitation comes in. For, to take the simplest case, the strain x = x, y = - y, Z 1 z implies that the figure to which it is applied has been "perverted," i.e., changed into its image as seen in a plane mirror. Conversely, when those three conditions are satisfied, and not Pure otherwise, the strain is pure. It is to be observed that f, 77, form strain a rectangular system, and thus the nine direction cosines (usually depends involving six arbitrary numbers) depend here on three numbers on six alone. Thus there are six independent numbers, corresponding to condi- a, e, i, b, c, f, in the general expression for the. strain. tions. It is clear, from the elements of coordinate geometry, that the Change of determinant volume a b c by strain d e f g h i represents the ratio in which the volume is increased by the strain. 2 Let us now introduce, in succession to the strain Conju- a b c 8 ate strains. the connected strain h i b e d g h (1), c f i which obviously produces an equal change of volume with the former. Applying these strains in succession, ve have as the final result x" = ax + dy + gz , v y" = bx + ey + hz , z" = ex +fy + iz , or, substituting for x , y , z 1 their values in terms of x, y, z, x" = (a? + d 2 + g^)x +(ab + de + gh)y + (ac + df+ gi)z , y" = (ba + ed + hg)x + (b z + c 2 + h*)y + (be + cf + M)z , z" = (ca +fd + ig)x + (cb +fe + ih)y + (c 2 +/ 2 + fi}z . Thus the resultant strain is a 2 + d 2 + g" ab + de + gh ac+ df +gi ba + cd+hg 6 2 + e 2 + A 2 bc+ cf + hi ca +fd + iy cb +fe + ih c 2 + / 2 + i? which, for simplicity, we will write as a 5 y S e (2). It will be observed that this group of nine numbers, if treated as a determinant, constitutes the product of the determinants formed of the two systems above. This satisfies the criterion of a "pure strain," as given above ; and we thus see that in the successive application of the strains a b c a d g d e f and b e h g h i c f i the rotation produced by the first is annihilated by the second. , Let A, B, C, &c. , be the minors of a b c d e f g h i corresponding to a, b, c, &c. Then by our original equations we have Ax^Ax +Dy + Gz , Thus the reciprocal of the strain a b c A/A D/A G/A d c f is B/A E/A H/A g h i C/A F/A I/A. This is evident from the formula} just written. For they express that the new strain converts x , y , z into x, y, z. Apply the resultant strain in succession to this reciprocal. The result is easily foreseen from separate terms like the following :- A( rt 2 + d i + gt) + B(fflZ> + de + gh) + C(ac + df+ gi) = a(Aa + B& + Cc) + d(Ad + Be + C/) + g(Ag + Kh + Ci) = ffA;&c. ^ 2 When this strain is produced in a piece of matter, the numerical value of the determinant obviously cannot be negative. Recipro cal of
strain.