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for it will be immediately seen, that this latter fissure together with AB, would destroy all tension at Q, and would of course prevent the possibility of the formation of any other fissure through that point".
30. Hence it follows, that in any system of parallel fissures which are not remote from each other, the fissures could not be formed in succession. It will be easy however to understand how, in the case above assumed, of two systems of tension perpendicular to each other, any number of parallel fissures may be formed simultaneously. Let AB, A'B' be two such fissures, and let GH be parallel to and equidistant from them.
Now if the two fissures begin simultaneously at A and A', (the line AA' being perpendicular to the direction of propagation,) and be propagated with equal velocity, it is obvious that no point in the physical line GH will have any motion communicated to it by the relaxation of the portion of the lamina between the fissures. Hence, if the line GH were to become absolutely fixed, the formation of the fissures would not be affected; but in this case the portions of the lamina on opposite sides of GH might be regarded as two absolutely distinct laminae, having that line for a common fixed boundary. Consequently it is as easy to understand the simultaneous formation of any number of parallel fissures, under the circumstances supposed, as that of a single fissure.
31. Let us assume the two systems of tension not to be perpendicular to each other, and suppose AB, A'B', two parallel fissures of which the directions are perpendicular to the maximum resultant tension. These fissures would not necessarily be continued parallel to each other. For let YY be parallel to the direction of one system, XX" (meeting the fissure A'B') to that of the other. The opening of A'B' will relax
* It must be recollected that the impossibility here spoken of assumes the tensions not to be produced by impulsive forces acting on the mass, the intensity of these tensions being always supposed to increase continuously, till sufficient to produce the fissure, and not to acquire that requisite intensity instantaneously, as previously stated in the Introduction, p. 11.
the tension along XX", while that along YY will not be affected. Consequently the ratio of the tensions at Q will not be the same as originally, when AB, A'B' began to be formed. The direction of propagation of the former will evidently deviate towards perpendicularity with YY', and that of the latter in the same manner more nearly to perpendicularity with XX'. They will not therefore in such case preserve their parallelism.
A finite time, however, will be necessary to produce the relaxation at Q, after the opening of A'B', and therefore if the distance between the fissures be not too small, and the velocity of propagation very great, as we have shewn it may be (Art. 13) AB may be propagated through Q before the relaxation is produced there, and the fissures might under such circumstances preserve, at least approximately, their parallelism.
32. It is evident, however, that in whatever manner a system of parallel fissures may be produced, that, after their formation, the only tension of the mass between them must be in a direction parallel to them. Consequently, should any other system be subsequently formed, it must necessarily be in a direction perpendicular to that of the first system. No two systems of parallel fissures, not perpendicular to each other, could be formed by causes similar to those of which we have been investigating the effects. It will appear also, as in Art. 30, that this second system must be of simultaneous origin.
33. From our assumptions respecting the variable cohesive power of the mass, it is manifest that different fissures might commence simultaneously at different points, and be propagated in opposite directions.
Thus, suppose the fissure CD to commence at D, when AB and EF commence at A and E respectively. When the first of these arrives at C, as the two others arrive respectively at B and F, the further propagation of each of them may be prevented by the relaxation of the mass. Consequently a system of fissures might thus be formed similar to that
represented in the above figure.
§. Application of the previous Propositions to a Mass of three dimensions.
34. These investigations have been applied immediately to the case of a thin lamina, to avoid the complexity which would necessarily have been introduced in their immediate application to a mass of three dimensions. The extension of the preceding propositions, however, to this latter case is sufficiently obvious to require little more than an enunciation of the results, which may also serve as a summary of the most important of those at which we have arrived in this section.
A slight inspection of what has been advanced in Art. 15, will shew that the existence of a line of less resistance in a thin lamina, will have no effect on the propagation of a fissure in a direction perpendicular to it; and similarly, if we suppose any mass acted on by horizontal tensions, it is manifest that a horizontal plane of less resistance will have no effect on the verticality or horizontal direction of the vertical fissures resulting from such tensions. Consequently, the tensions being horizontal, the cohesive power of the mass may be sup
posed to vary continuously or discontinuously along any vertical line, and, as explained in Art. 8, it may vary according to any continuous law in any horizontal lamina of the mass. The same assumptions are made respecting the continuous but rapid increase of the tensions, as in Art. 12.
I. If this mass be acted on by a single system of horizontal parallel tensions, a fissure beginning at any point will be propagated in a vertical plane perpendicular to the direction of the system. (Art. 2).
II. If the mass be subjected to any number of systems of parallel tensions, the fissure will be propagated through any point in a direction perpendicular to the maximum resultant tension at that point, at the instant the fissure reaches it, (Art. 12.) the horizontal direction being determined by equation (2), (Art. 7). If the ratios of the tensions at each point at the instant of propagation through it be the same, the fissure will, in general, be formed in one vertical plane. (Art. 14.)
III. If there be only two systems of horizontal tensions, and these be perpendicular to each other, the fissure will lie in one vertical plane perpendicular to the direction of the system of the greatest intensity, whatever be the ratio of the tensions at each point in the two systems, provided the tension at each point always remain the greatest in the same system. (Arts. 6, 14.)
IV. Each fissure under the conditions assumed, will be propagated with extreme velocity. (Art. 13.)
V. The tendency of the tensions to propagate the fissure in one particular direction rather than in any other, or the permanence of the permanent direction of cleavage, depends on the rapidity with which the magnitude of the resultant tension, estimated in a particular direction, decreases as that direction deviates from that of the maximum resultant tension; or generally, on the ratio which the maximum bears to the minimum resultant tension, which is perpendicular to it. (Art. 18.) VI. If in addition to a system of horizontal tensions, there be also a force acting on the opposite sides of the fissure, perpendicularly to its direction, and tending to increase its width”, the permanence of direction in the progressive formation of the fissure will be diminished, but the permanent direction will remain the same as if there were no other force than the system of horizontal tensions, i. e. if the direction in which the propagation of the fissure is taking place be disturbed by any partial cause, it will still constantly tend again to perpendicularity with the directions of the system of tensions; but this tendency will be less than if the force always acting perpendicularly to the fissure did not exist. (Art. 20.) Consequently, deviations from the permanent direction of cleavage will, in the case we have supposed, be greater than if the sides of the fissure were not subjected to the action of this last-mentioned force.
VII. If there be no tension acting on the mass, and a fissure be formed solely by this force, acting perpendicularly to its sides, the fissure will be propagated in the plane in which it begins to be formed, if the cohesive power of the mass vary according to any continuous law. There will be, however, but little permanence in its direction, so that if it be turned from its original direction by planes of less resistance, there will be little tendency to resume that direction, and the fissure may thus assume any form of irregular curvature. (Art. 20.)
VIII. If a fissure commence at, or in the course of its progressive formation meet, a partial plane of less resistance at an acute angle, it will, under certain conditions, be propagated along it; but when from any cause this ceases to be the case, the fissure will almost immediately resume a direction parallel to its original one, supposing it produced by tensions, which, independently of the existence of planes of less resistance, would produce rectilinear fissures. (Arts. 17, 18.)
IX. If the mass be subjected to two systems of parallel tensions, of which the directions are perpendicular to each other, two systems of
* This will be the case if the fissure be filled with any kind of fluid subjected to a great pressure from some external cause.