« السابقةمتابعة »
contiguous to those points, consisting of that superimposed as above described, as well as of that impressed generally on the lamina; and consequently, if we conceive the latter of these tensions, (and therefore also the former) to increase till the resultant tension is sufficient to overcome the cohesive power at A or B, the fissure will not necessarily be continued in the same direction, as if its continuation were independent of the partial system superimposed about its extremities.
It will be observed, however, that in the case just considered, in which the forces are not producing motion in the mass, the whole force exerted by gC, and k K, and similar lines is effective in producing the superimposed system of tension about the extremity of the fissure. We shall shew however, that such is not generally the case during the propagation of the fissure, if propagated in the manner we shall suppose it to be, and that consequently this force will have no material effect on the direction in which the fissure will be continued, and which will therefore be very approximately determined by equation (2).
11. For this purpose, let us suppose in the first place, any systems of tensions impressed on the lamina, of which the resultant tension (R) shall be less than the cohesive power (II), at any proposed point P; and let us then conceive subsequently superimposed on these another system of which the direction is different to that of R, and of which the intensity ‘p shall increase continuously with the time t, till the resultant of R and p shall be equal to II, so that a fissure shall then begin to be formed at P. Its direction will evidently depend on R, and the value (p), which p shall have acquired at the instant the fissure commences. If R differ but little from II, p, will be generally small*, and cannot (however the forces producing 4 may subsequently act on the lamina), produce any material influence on the direction of the fissure, which will therefore, in such case, nearly coincide with the direction in which the tensions whose resultant is R may have the greatest tendency to form it, i. e. it will be nearly perpendicular to the direction of that resultant.
* If the direction of p coincided with that of R, the fissure would manifestly begin to be formed when R + p, should = II, or b. = II – R, which by hypothesis is small. If the angle between the directions of R and b be not too near a right angle, it is equally manifest that q), must be small. In the actual case considered in the text, this angle obviously cannot be
12. Let us now suppose P, to designate a point in the lamina, at which a fissure shall begin, and P. another point through which it shall be subsequently propagated; and let II, II, denote the cohesive powers of the lamina at those points respectively, II, being the least. It has been already stated, (Introd. p. 11.) that in the case to which these investigations are to be applied, the intensity of the elevatory force, and therefore, of the tensions produced by them, will be assumed to increase continuously from the commencement of the action of this force, to the formation of the fissures; we shall here also make an additional assumption, viz., that this intensity shall increase rapidly, so that a very small time shall elapse between the commencement of the elevatory action, and the instant when the fissures shall begin to be formed*. The tensions therefore to which our lamina is subjected, will be assumed to increase in the same manner. Let R, denote the intensity of their resultant at the time t; then if t, be the time when the fissure begins at P, R, must be equal to the cohesive power at P = III. When the fissure is thus begun to be formed, the partial system of tensions described in Art. 9., will be superimposed about its extremities. Let p, denote its intensity at the time t, and at any proposed point. As the fissure in its progressive formation approaches P, this force will be superimposed on the lamina there, in addition to the force R, previously acting there, so that if t, be the time when the fissure is first formed at P., we must have at P., the resultant of R, and of p, as II. Now, if during the time t, t, R, increases from R, or II, so that R, nearly = II, blo must be small at P., and therefore can have but little influence on the direction of the fissure through that point, whatever be the direction of that tension, or the intensity it might acquire if the cohesive power at P, were sufficient to prevent the propagation of the fissure beyond that point (Art. 11.) In such case therefore the direction of the fissure will be at least very approximately determined by equation (2), p. 18, in which the values of u do not include the tension .p, * This assumption is not absolutely necessary for the truth of the approximation we but only the values F, f, f, &c. of the general tensions, at the instant when the fissure is propagated through the proposed point".
have to establish or for the proof of it. It renders however the approximation more accurate, and the proof much more simple.
13. Under the circumstances here supposed, the fissure will be propagated from P to P., nearly in the time t, —t, during which R, increases from II, to II. Consequently, if the difference between these latter quantities be not great, i. e. if the cohesive power do not vary rapidly; or if R (heretofore assumed to be the same at the same time at different points of the lamina) increase with rapidity, it follows that the velocity of propagation will be extremely great, becoming infinite, when the cohesive power, and the tension R, are accurately uniform throughout the lamina.
If R be not uniform, it is easy to see that reasoning similar to the above will hold equally true, with respect to the progressive formation of any fissure.
14. The fissure will be propagated in a straight line, if the values of u in equation (2) remain the same, i. e. if the ratios of the tensions at different points be the same at the instant the fissure is propagated through them. If these ratios be different for different points, the fissure will generally be curvilinear; there is, however, an important exception to this rule, when there are only two systems of tension, of which the directions are perpendicular to each other; for in this case it appears by Art. 6, that the direction of the fissure will always be perpendicular to that of the greater of these two tensions.
t Effect of Lines of Less Resistance on the Direction of a Fissure. Permanent Direction of Cleavage.
15. In the preceding articles, we have supposed the cohesive power of the lamina to vary according to some continuous law. Let us now consider the effect of the existence of lines of less resistance in the lamina, in which case the continuity above assumed will no longer exist along these lines.
* When the cohesive power of the lamina is not sufficient to prevent the propagation of the fissure, the problem presented to us is no longer a statical one. In the case above considered, a small portion only of the extraneous forces producing the tension p, is effective in causing an additional tension of the lamina before the formation of the fissure. The greater part is effective in communicating motion to those parts of the mass, the receding of which from each other causes the opening of the fissure. On the contrary, when the formation of the fissure is arrested, the whole of these forces is effective in producing this partial system of tensions.
Let DE be a line of this description, along which the cohesive power estimated in a direction perpendicular to it = II", that of the lamina near to DE being = II. Also let R, acting in the direction
PR, be as before, the resultant at the time t and at the point P, of the general systems of tensions impressed upon the lamina; and let R! denote the tension along PR perpendicular to DE at the time t.
TF be > TT it is manifest that the fissure will begin to be formed along the line DE, rather than in a direction perpendicular to R, in which it would
16. Let us now suppose this line to terminate at D and E. When the fissure has been propagated to those points, its progress will be arrested till the tension R, and that superimposed just beyond the extremities of the fissure, and before denoted by p, (Art. 11), produce a resultant tension greater than the cohesive power II. The direction in which the fissure will be then immediately continued, will not be known, P, being unknown; but without staying to enquire what this may be, we may observe, that the fissure must very soon in its pro
* It is assumed in the above condition, that if the fissure be formed along DE, the particles on opposite sides of the fissure in separating would move in lines perpendicular to DE. This would be only approximately true.
gressive formation, arrive at a point at which R, will be very nearly equal to the cohesive power, since that force by hypothesis increases rapidly with t, (Art. 12)}, and where, consequently, the direction of the fissure must necessarily be very approximately that determined by equation (2), as explained in Art. 12. Hence then we may conclude, that under the hypotheses we are taking, whatever may be the direction first given to the fissure by any local cause, its subsequent direction will soon become independent of that cause.
17. If the fissure, instead of beginning at some point in a line of less resistance meet it, in its progressive formation, it will pass along it, or will cross it, according as a condition exactly similar to that given above (Art. 15), be satisfied or not. At the termination of this line, the fissure will soon resume the direction given to it by the general systems of tensions to which the lamina is subjected, as just explained. Such also will be the case at the point at which the line of less resistance, should it be a curved or broken line, may assume a direction in which the condition just referred to is no longer satisfied.
18. The condition given in Art. 15 gives us
R. T. II The first of these ratios will in each particular case be a function
of the angle RPR or EPB, the angle between the line of less resistance and the direction AB, (perpendicular to PR) in which the