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since the resultant tension is a maximum in the direction PR, and a minimum in that perpendicular to PR. (Art. 5). Consequently, the greater the ratio which the former of these resultants bears to the latter, the more rapidly will R.' decrease while RPR' increases, and the smaller will be the angle EPR, within which the above condition will be satisfied, and the narrower therefore will the angular limits, within which a line of less resistance must be situated, in order that it may cause a fissure proceeding in any assigned direction to deviate from its course. A line through P perpendicular to PR, may be
termed a permanent line of cleavage. If the ratios % % &c. be the
same at every point of the lamina, all such lines will be straight lines (Art. 14) and parallel to each other. A fissure will always have a tendency to resume this direction, when made by any partial cause to deviate from it, and will resume it taking our assumptions respecting the impressed tensions, (Art. 12); almost immediately after the cessation of such cause. It will be well to examine this tendency in a few particular cases. It may be considered as measuring what may be termed the permanence of the fissure's general direction.
19. Let there be two systems of tensions, the directions of which are perpendicular to each other, and of which the intensities are F and f respectively, at any proposed point, when they become sufficient to form the fissure there. The greatest of these (F) will be the maximum, and f the minimum resultant tension, (Art. 6), and therefore the less f is, the greater will be the permanence of the permanent direction, perpendicular to that of F. If f = F, there will be no permanence in any particular direction. We have already seen (Art. 6), that there is, in fact, no greater tendency in this case to form a fissure in one direction than another.
20. Again, let us suppose in addition to the systems of tensions, of which the intensities are f, f, &c., and which have determinate directions, a force acting within the fissure perpendicularly to its direction, and with equal intensity on its opposite sides, exactly as a fluid would act when forcibly injected into a fissure formed in a solid mass. Vol. VI. PART I. D
Let PP be the fissure. It is manifest that this force (p) will produce a tension on the mass contiguous to the extremity of the fissure, in a direction Po perpendicular to PP, and must therefore tend to propagate the fissure along PP produced. Hence it will follow that such
a force cannot affect the permanent direction of cleavage as determined by the tensions f, f, &c. alone. For, suppose PR the direction of the maximum resultant (R) of these tensions, it is manifest that the whole resultant tension (including that produced by p) immediately beyond the extremity P of the fissure, must be in a direction PR, between Po and PR; consequently, the direction of propagation from P will deviate from PPN, and approximate more nearly to perpendicularity with PR, and therefore also with PR. For the same reason, the direction of its further propagation will approximate still more nearly to a line perpendicular to PR, till it coincide with it. The permanent direction will therefore be the same as if the force p did not exist.
If however p be large compared with R, it is manifest that the angle p PR will be very small, and that the tendency to resume the permanent direction, when the fissure has been obliged by any partial cause to deviate from it, will be much less than if p were relatively smaller.
21. If the lamina be subjected to no tension, and the fissure be produced entirely by p, the tendency will be to propagate the fissure in the direction in which it may originally be formed. Suppose AP, to be its original direction, but that from P. it follows a line P. P.
of less resistance; then if ‘we suppose the force p not to act effectively in propagating the fissure, except near its extremity”, its action will not extend beyond the portion P. P., of the fissure, and consequently
its tendency will be to propagate it in the direction of PP, produced, after it has reached the termination of the line of less resistance. There will be no tendency, as in the former cases, to resume any particular direction.
§. Modification of the Tensions in the vicinity of a Fissure.
22. Let us now suppose a fissure to have been formed in the manner above described, and extending between two points in the lamina, where we may conceive its propagation to have been arrested either by an increased cohesive power, or by a diminution of intensity in the tensions. It is manifest that the state of tension in the vicinity of this fissure, will become entirely different from that which existed previously to its formation; and that the subsequent formation of any other fissure not very remote from the first, must therefore be influenced by the modification of the original tensions thus produced. It will now therefore be our object to examine this consequence of the existence of a fissure. For the greater simplicity, we may suppose it to be rectilinear. It will also suffice for our immediate purpose, to suppose the lamina subjected to two sets of tensions acting perpendicularly to each other, the direction of the fissure being perpendicular to that of the system of the greater intensity.
* This will be true in the actual case to which it is intended to apply this part
of the investigation.
23. Let AB represent the fissure, AU; BP, the physical lines perpendicular to it, and passing close to its extremities, and for the greater distinctness, let us suppose the boundary of the lamina along UP to be parallel to the fissure. Let EF be a physical line originally parallel to the straight line AB. After the formation of the fissure,
it will evidently assume a curved form resembling that of APB ; but its curvature will be less than that of the latter line, since the curvature of all such lines must obviously be smaller the nearer they are situated to the fixed straight line UP, along which it becomes evanescent. If, however, the length of the fissure be considerable, the curvature of APB will be very small, and therefore the variation of curvature in successive physical lines such as EF, will be extremely slow, AU being very large *.
Also, let PQS be a physical line, parallel before the opening of the fissure, to AU. If the form of every such line as EQF were exactly the same, this line would still be accurately straight and parallel to AU, and consequently in the case we are supposing it will be approximately so. The tension of all such lines will evidently be much affected by the opening of the fissure. Since there is no force acting at P, the tension of SP in the direction of its length, will, at that extremity, become evanescent; but since the line is extended, though not by a force at its extremity P, it must at every other point be subjected to a certain tension, and our object will be to compare this tension at any point Q with that acting in the direction EQF at the same point, with the view of ascertaining within what limits another fissure might be formed subsequently to the formation of AB, and parallel to it between the lines AU and BP. Such a fissure could not be formed through Q, by the tensions to which we are supposing the lamina subjected, if the tension in the direction EQF at that point should be greater than that in the direction PS, since the fissure must necessarily be formed perpendicular to the greater of these tensions (Art. 6).
* If the boundary of the lamina be not parallel to the fissure, UV may be conceived to be a physical line in the lamina, very distant from and parallel to the line AB, previously to the formation of the fissure, since the position or rectilinearity of such a line will not be sensibly affected by the opening of the fissure, as appears from the text.
24. In the first place, let us suppose a physical line of indefinitely small width to be attached at its extremities to the fixed points A, B, and then conceive parallel forces to act on each element of this line,
with the same or different intensities at different points, and in directions perpendicular to AB. The line will thus be made to assume a curvilinear form, and if the extensibility be small, as we shall suppose it to be, the curvature will be small, so that if AQ=s, and a be the original length of AQ, a and s may be considered as very approximately equal. Let T denote the tension at Q, p the radius of curvature, and p the intensity of the force at that point, p being any function of a. Then the force on the element 3s, will be p. 8a, and the nor
mal force produced by the tension r, will = 1 estimated by the effect - p it would produce, if it acted uniformly on a unit of the line, so that
the normal force acting on the element 3s, will = +. 3s, or ; 3a very