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النشر الإلكتروني

SECTION I.

1. THE simplest form of the mass in which we have to consider the formation of fissures, is obviously that of a thin lamina. The investigations therefore of this section will be applied directly to this case, from which the results applicable to a mass of three dimensions are immediately deducible. It will appear that its cohesive power may vary according to any continuous law.

§. Lamina subjected to one System of Tensions.

2. Suppose the lamina acted on by external forces, which shall place it in a state of tension, such that the direction of the tension at every point shall be parallel to a given line CD*. Let AB be any

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D

proposed line in the lamina; P any point in this line. Also assume F to be the tension at P, estimated by the force which the tension at that point would produce, if it acted uniformly on a line of which the length should be unity, and which should be perpendicular to CD, the common direction of the forces of tension. Then if we take Pp a small and given element of the line AB, and draw PQ parallel to CD, and pm perpendicular to PQ, the force of tension on Pp in the direction PQ will be measured by F.pm, or F. Pp. sin ↓ (BPQ = √); or the

* The reference will always be made to the figure in the same page, unless stated to the contrary.

tendency of the forces of tension to separate the particles which are contiguous, but on opposite sides of the geometrical line Pp, by causing them to move parallel to CD, will be measured by

pm. F sin,

dx. Fsin (if AP = x).

The tendency to separate the particles at P, by causing them to move in a direction making an angle ✪ with CD or PQ, will be estimated by

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Sx. cos. sin. F.

π

This is greatest when = 0, and ↓ =, as of course it ought to be,

AB being then perpendicular to CD.

2

6. Lamina acted on by two or more Systems of Tensions-Direction in which their tendency to produce a fissure is greatest.

3. Let us next suppose a second system of parallel tensions superimposed on the former, their common direction making an angle ß with that of the first system. Let PQ, PQ, in the following figure, be the directions of the tensions acting on the element da, of the line AB at

B

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P, and therefore QPQ = B; and let the intensities of these tensions (estimated as in Art. 2.) be represented by F and f. Then if QPB =y, , and therefore QPB = -ß, we shall have the forces 8x. F. sin, and da.f. sin (B) acting on the element da; and to find

the tendency of these forces to separate the contiguous particles on opposite sides of the elementary portion da, of the geometrical line AB, estimated by their tendency to give an opposite motion to these particles along any assigned line r PR, we must resolve the forces in the direction of that line. Let RPQ=0; then will the sum of the resolved parts of our forces in the directions PR and Pr be

Sx. F sin cos 0+8x.f sin (4-8) cos (3-0) ............(A).

If the value of this expression, considered as a function of the independent variables and be made a maximum, we shall manifestly obtain from the corresponding value of that angular direction of the line AB along which the two sets of tensions we are considering have the greatest tendency to form a fissure.

Differentiating the expression with regard to 0, we have

Sx. Fsin sine-dx.fsin (4-6) sin (3-0)=0.

The left-hand side of this equation is the expression for the sum of the resolved parts of the forces da. F sin and Safsin (↓ — ß), perpendicular to the line PR. Consequently the equation expresses the condition that PR must coincide with the direction of the resultant of the above forces.

F

Again, differentiating with respect to y, we obtain

Fcos. cose+fcos (4-B) cos (3-0)=0.

From the above equations we must determine and e. If we put =μ, we obtain from thence

1+μ (cosẞ-sinß cot) (cosß- sin ẞ. cot 0) = 0,

1 +μ (cosß + sinß. tan↓) (cosß + sinẞ tan 0) = 0,

or, putting cos ẞ=c, sin ß=s, cotex, cot↓ = 2,

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From the inspection of these equations it is manifest that x=

putting this value for %, and therefore also

1

1; for

for x, the two equa

tions will only be converted into each other. Substituting in the first equation, we have

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4. Let 1, be the two values of given by this equation. Then, since the last term is -1,

coty, cot = -1;

which shews that the difference between, and 1⁄2 is, or

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and if 0, 0, be the values of e corresponding respectively to , and Y. we have

2

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The angle BPR is consequently a right angle.

5. Since PR coincides, as shewn above, with the direction of the forces of tension acting on the element dx of the line AB, the expression (A) is the value of that resultant. Consequently, and 2, which correspond to the maximum and minimum values of the quantity (A), determine the position of the line AB in which the resultant of the above-mentioned forces of tension is a maximum or minimum.

6. If the two systems of tensions be equal and perpendicular to each other, equation (1) becomes

sin.cot-2 (1 + cos π) cot-sinπ = 0,

In this case,

and is satisfied independently of particular values of y. therefore, there is no greater tendency to form a fissure in one direction than another. If F be greater than f, the equation becomes

sin. coty-2-1) cot – sin π = 0,

of which the two roots are 0 and ∞, which shews that the greatest tendency is to form a fissure in a direction perpendicular to that of F.

7. The above investigation easily admits of generalization for any number of systems of parallel tensions superimposed upon each other. Let F denote, as before, the intensity of the tension in the direction from which and are measured; fi, f, &c. the tensions in directions making respectively angles B1, B., &c. with the direction of the tension F. Then shall we have

dx. {Fsin ycose+fsin (y-ẞ1) cos (0–ẞ1) +ƒ1⁄2sin (y-ẞ2) cos (0— B2) + &c. } = max.; and proceeding exactly in the same manner as in the previous investigation, and adopting an analogous notation, we shall manifestly obtain the following equations:

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and putting, for the same reason as before, x=—

we obtain

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+ {1 + μ‚ (C,2 − 8,2) + μ2 (C22 − 822) + &c. } ≈ + μ1 C1 S1 + μ22 C2 S2 + &C. = 0;

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