MECHANICS Pure .strain. Sacces- sive applica tion of conju gate strains. Recipro cal strain. When two only of the changes of length are equal, the ellipsoid becomes one of rotation, oblate or prolate as the case may be; and if the radius of the sphere be inter mediate in value to the axes of this rotation-ellipsoid, we have a right cone of rays unaltered in length. When all three changes of length are equal we have the simplest possible case, which has been already treated. 88. The essential element in these particular cases is that three lines at right angles to one another are unaltered in direction by the strain. Here there is a mere change of form, and the strain is said to be " pure," or " free from rotation." Such a strain, in its most general form, is fully characterised by six independent numbers. For a system of three mutually perpendicular lines is fully given in direction by three numbers, and three more are required for the changes of length which they severally undergo. But, in general, a strain is not pure. It will be shown, however, below that the principal axes of the ellipsoid into which a sphere is changed by any strain, and which is called the " strain ellipsoid," were originally radii of the sphere at right angles to one another. Hence the strain may be looked upon as made up of two operations, viz., a pure strain, and a rotation through a definite angle about an axis in a definite position in space. The axes of the pure strain are lines fixed in the figure. 89. It is useful, in farther considering the subject, to introduce along with the original strain (thus analysed), another which is called its "conjugate." This is defined as composed of an equal pure strain with the first with an equal but opposite rotation. And the separate component operations must be taken in the opposite order in the strain and in its conjugate. The successive application of the strain and its conjugate thus necessarily leads to the reduplication (or squaring) of the pure part of the strain, and to the annihilation of the rotation. For, call the parts, as operators, P and R. The strain and its conjugate, referred to axes fixed in space, may be either EP and PR- 1 or PiRandR- 1 ?!, according as the pure strain or the rotation is first applied. The operations in each group are written, from right to left, in the order in which they are performed. Thus RP means the pure strain P, followed by the rotation R. 90. The final results are P 2 and P*. In the first case we have the pure strain, followed by the rotation ; then (by the conjugate) the rotation is undone, and the pure strain reapplied. In the second we rotate first, then apply the pure strain twice, and finally undo the rotation. Thus the student must be cautioned against the error of sup posing that the results of applying PR and RP separately are generally the same. Perhaps it will be easier for the reader to consider the "reciprocal," instead of the con jugate, of a strain. For if the strain be RP, the reciprocal is obviously P^R" 1 ; if it be PR, the reciprocal is R" 1 ?" 1 . Either pair of these, taken in either order, restores the figure to its primitive form. The one point to be noticed is that, in whatever order the direct com ponent operations are supposed to occur in the strain, their reciprocals must bo taken in the opposite order in the reciprocal strain. The reciprocal strain simply undoes the strain, and therefore differs from the conjugate by a factor, the square of the pure part of the strain. From this we have at once, as will be seen later, the means of decomposing a given strain into its pure and its rotational factors. This is effected as soon as we can form the expression for the conjugate strain in terms of that for the strain itself.
- 91. As, in general, any strain converts a spherical
portion of a figure into an ellipsoid, and as an ellipsoid has Shear, two series of parallel circular sections, it appears that in every strain there are two series of planes, of no distortion. 1 The consideration of these planes leads us to a second and very different mode of analysing a strain into simpler components. Perhaps the most elementary mode of con sidering this subject is by thinking of the motion of water flowing slowly down a uniform channel. We know that water, at ordinary pressures, is practically incompressible ; also that the upper layers of the water in a canal flow faster than those below them. Hence the definition of a " simple shear." Let one plane of a figure be fixed, and let the various planes parallel to it slide over it and over one another, all in the same direction, and with velocities pro portional to their distances from the fixed plane. It is clear that this shear produces homogeneous strain in the figure, but it is mere change of form without change of volume. The fixed plane and all those parallel to it, are planes of no distortion. But we have seen that there must be two sets of such planes. To find the second set, let us suppose the plane of fig. 30 to be parallel to the M P . 0- common direction of sliding, and perpendicular to the fixed plane. This plane, so defined, / / is the plane of the shear. Let AB be the trace on it of the fixed plane, PQ that of one of the sliding planes, PP the ^ g amount of its sliding. Bisect PP in M by a perpendicular, meeting AB in A. Join AP, take AB = AP, and draw BQ parallel to AP. Consider the strain of the rhomboidal portion APQB of the figure. P moves to P , and Q to Q , where QQ = PP . Hence the rhombus remains a rhombus, for AP = AP = AB. But the lengths of its diagonals have been interchanged. It has been subjected to an elongation of AQ, and a contraction of BP, each in the same ratio (so that their product, i.e., double the area of the rhombus, remains unaltered), while all lines perpendicular to the plane of the figure remain unaltered. From the symmetry of the rhombus it is obvious that AQ and BP are the greatest and least axes of the strain ellipsoid, while AB and AP are parallel to its circular sections. Planes originally parallel to AP and perpendicular to the paper are therefore the second set of planes of no distortion. The rotational part of the shear may be measured in terms of the angle PAM, but is given directly by PBP , and its axis is perpendicular to the plane of the figure. The most convenient measure of the shear is the ratio of PP to AM, or, what involves the same, the angle PAP . Another mode of measuring it is by means of the ratio BP/AQ, = 1 + e, suppose. If e be a small quantity, as is usually the case with solids, we may write 1 e for the measure of the shear. Here e indicates the extension per unit of length along one diagonal of the rhombus, and the contraction per unit of length along the other. 92. It is quite clear from what has been said that we can analyse a strain by the help of simple shears com pounded with different forms of pure strain. For the shears may be taken in an infinite number of ways so as to produce the rotational part of the given strain, while also producing deformation without change of volume. The final adjustment is to be made by a pure strain, whose axes are those of the strain ellipsoid clue to the shears. As a shear depends on four quantities only, two shears and a dilatation furnish the nine constants required for a homo geneous strain. 1 We here exclude spheres and ellipsoids of rotation. The latter have only one series of circular sections, the former an infinite number. Decom- positioi of a
strain.