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of the subject, consider the forces called into play through the interior of a solid when brought into a condition of strain. We adopt, from Rankine', the term stress to designate such forces, as distingushed from strain defined (§ 135). to express the merely geometrical idea of a change of volume or figure.
630. When through any space in a body under the action of force, the mutual force between the portions of matter on the two sides of any plane area is equal and parallel to the mutual force across any equal, similar, and parallel plane area, the stress is said to be homogeneous through that space. In other words, the stress experienced by the matter is homogeneous through any space if all equal, similar, and similarly turned portions of matter within this space are similarly and equally influenced by force.
631. To be able to find the distribution of force over the surface of any portion of matter homogeneously stressed, we must know the direction, and the amount per unit area, of the force across a plane area cutting through it in any direction. Now if we know this for any three planes, in three different directions, we can find it for a plane in any direction as we see in a moment by considering what is necessary for the equilibrium of a tetrahedron of the substance. The resultant force on one of its sides must be equal and opposite to the resultant of the forces on the three others, which is known if these sides are parallel to the three planes for each of which the force is given.
632. Hence the stress, in a body homogeneously stressed, is completely specified when the direction, and the amount per un area, of the force on each of three distinct planes is given. It is, in the analytical treatment of the subject, generally convenient to take these planes of reference at right angles to one another. But we should immediately fall into error did we not remark that the specification here indicated consists not of nine but in reality only of six, independent elements. For if the equilibrating forces on the six faces of à cube be each resolved into three components parallel to its three edges, ox, OY, OZ, we have in all 18 forces; of which each pair
Z acting perpendicularly on a pair of opposite faces, being equal and directly opposed, balance one another. The twelve tangential components that remain constitute three pairs of couples having their axes in the direction of the three edges,
Y each of which must separately be in
I equilibrium. The diagram shows
X the pair of equilibrating couples having OY for axis; from the consideration of which we infer that the
1 Cambridge and Dublin Mathematical Journal, 1850.
forces on the faces (wy), parallel to OZ, are equal to the forces on the faces (yx), parallel to OX. Similarly, we see that the forces on the faces (yx), parallel to OY, are equal to those on the faces (x2), parallel to OZ; and that the forces on (x2), parallel to Ox, are equal to those on (zy), parallel to OY.
633. Thus, any three rectangular planes of reference being chosen, we may take six elements thus, to specify a stress: P, Q, R_the normal components of the forces on these planes; and S, T, U the tangential components, respectively perpendicular to OX, of the forces on the two planes meeting inox, perpendicular to OY, of the forces on the planes meeting in OY, and perpendicular to OZ, of the forces on the planes meeting in OZ; each of the six forces being reckoned, per unit of area. A normal component will be reckoned as positive when it is a traction tending to separate the portions of matter on the two sides of its plane. P, Q, R are sometimes called simple longitudinal stresses, and S, T, U simple shearing stresses.
From these data, to find in the manner explained in $631, the force on any plane, specified by l, m, n, the direction-cosines of its normal; let such a plane cut OX, OY, OZ in the three points X, Y, Z. Then, if the area XYZ be denoted for a moment by A, the areas YOZ, ZOX, XOY, being its projections on the three rectangular planes, will be respectively equal to Al, Am, An. Hence, for the equilibrium of the tetrahedron of matter bounded by those four triangles, we have, if F, G, H denote the components of the force experienced by the first of them, XYZ, per unit of its area,
F.A=PIA+U.mA + TinA, and the two symmetrical equations for the components parallel to OY and OZ. Hence, dividing by A, we conclude
F=PL+Um + In
(1) H=T1+ Sm+Rn) These expressions stand in the well-known relation to the ellipsoid
Px2 + y +R22 + 2(Syz + Tzx + Uxy)=1, (2) according to which, if we take
x=lr, y=mr, x=nr, and if , M, v denote the direction-cosines and p the length of the perpendicular from the centre to the tangent plane at (x, y, z) of the ellipsoidal surface, we have
pr We conclude that
634. For any fully specified state of stress in a solid, a quadratic surface may always be determined, which shall represent the stress graphically in the following manner :
To find the direction and the amount per unit area, of the force
acting across any plane in the solid, draw a line perpendicular to this plane from the centre of the quadratic to its surface. The required force will be equal to the reciprocal of the product of the length of this line into the perpendicular from the centre to the tangent plane at the point of intersection, and will be perpendicular to the latter plane.
635. From this it follows that for any stress whatever there are three determinate planes at right angles to one another such that the force acting in the solid across each of them is precisely perpendicular to it. These planes are called the principal or normal planes of the stress; the forces upon them, per unit area,-its principal or normal tractions; and the lines perpendicular to them,—its principal or normal axes, or simply its axes. The three principal semi-diameters of the quadratic surface are equal to the reciprocals of the square roots of the normal tractions. If, however, in any case each of the three normal tractions is negative, it will be convenient to reckon them rather as positive pressures; the reciprocals of the square roots of which will be the semi-axes of a real stress-ellipsoid representing the distribution of force in the manner explained above, with pressure substituted throughout for traction.
636. When the three normal tractions are all of one sign, the stress-quadratic is an ellipsoid ; the cases of an ellipsoid of revolution and a sphere being included, as those in which two, or all three, are equal. When one of the three is negative and the two others positive, the surface is a hyperboloid of one sheet. When one of the normal tractions is positive and the two others negative, the surface is a hyperboloid of two sheets.
637. When one of the three principal tractions vanishes, while the other two are finite, the stress-quadratic becomes a cylinder, circular, elliptic, or hyperbolic, according as the other two are equal, unequal of one sign, or of contrary signs. When two of the three vanish,' the quadratic becomes two planes; and the stress in this case is ($ 633) called a simple longitudinal stress. The theory of principal planes, and normal tractions just stated ($ 635), is then equivalent to saying that any stress whatever may be regarded as made up of three simple longitudinal stresses in three rectangular directions. The geometrical interpretations are obvious in all these cases.
638. The composition of stresses is of course to be effected by adding the component tractions thus :- If (P, Q, R, S, T, U,), (P, C, R, S, T, U,), etc., denote, according to $ 633, any given set of stresses acting simultaneously in a substance, their joint effect is the same as that of a simple resultant stress of which the specification in corresponding terms is (EP, EQ, ER, ES, ET, EU).
639. Each of the statements that have now been made (§§ 630, 638) regarding stresses, is applicable to infinitely small strains, if for traction perpendicular to any plane, reckoned per unit of its area, we substitute elongation, in the lines of the traction, reckoned per unit of length; and for half the tangential traction parallel to any
direction, shear in the same direction, reckoned in the manner explained in § 154. The student will find it a useful exercise to study in detail this transference of each one of those statements, and to justify it by modifying in the proper manner the results of $ 150, 151, 152, 153, 154, 161, to adapt them to infinitely small strains. It must be remarked that the strain-quadratic thus formed according to the rule of $ 634, which may have any of the varieties of character mentioned in $$ 636, 637, is not the same as the strain-ellipsoid of $ 141, which is always essentially an ellipsoid, and which, for an infinitely small strain, differs infinitely little from a sphere.
The comparison of $ 151, with the result of $ 632 regarding tangential tractions is particularly interesting and important.
640. The following tabular synopsis of the meaning of the elements constituting the corresponding rectangular specifications of a strain and stress explained in preceding sections, will be found convenient:
Planes; of which Direction
relative motion, or of relative
is reckoned. of force. Р
у 641. If a unit cube of matter under any stress (P, Q, R, S, T, U) experience the infinitely small simple longitudinal strain e alone, the work done on it will be Pe; since, of the component forces, P, UT parallel to OX, U and T do no work in virtue of this strain. Similarly, Of, Rg are the works done if, the same stress acting, the simple longitudinal strains for g are experienced, either alone. Again, if the cube experiences a simple shear, a, whether we regard it ($ 151) as a differential sliding of the planes yx, parallel to y, or of the planes 2x, parallel to %, we see that the work done is Sa: and similarly,
T6 if the strain is simply a shear b, parallel to OZ, of planes zy, or parallel to OX, of planes xy: and Uc if the strain is a shear c, parallel to OX, of planes xz, or parallel to OY, of planes yz. Hence the whole work done by the stress (P, Q, R, S, T, U) on a unit cube taking the strain (e, f, g, a, b, c), is Pe+ Of + Rg + Sa+ Tb + Uc.
(3) It is to be remarked that, inasmuch as the action called a stress is a system of forces which balance one another if the portion of matter experiencing it is rigid, it cannot do any work when the
matter moves in any way without change of shape: and therefore no amount of translation or rotation of the cube taking place along with the strain can render the amount of work done different from that just found.
If the side of the cube be of any length p, instead of unity, each force will be på times, and each relative displacement p times, and, therefore, the work done på times the respective amounts reckoned above. Hence a body of any shape, and of cubic content C, subjected throughout to a uniform stress (P, Q, R, S, T, U) while taking uniformly throughout a strain (e, f, g, a, b, c), experiences an amount of work equal to (Pet Of + Rg + Sa+Tb + Uc)C.
(4) It is to be remarked that this is necessarily equal to the work done on the bounding surface of the body by forces applied to it from without. For the work done on any portion of matter within the body is simply that done on its surface by the matter touching it all round, as no force acts at a distance from without on the interior substance. Hence if we imagine the whole body divided into any number of parts, each of any shape, the sum of the work done on all these parts is, by the disappearance of equal positive and negative terms expressing the portions of the work done on each part by the contiguous parts on all its sides, and spent by these other parts in this action, reduced to the integral amount of work done by force from without applied all round the outer surface.
642. If, now, we suppose the body to yield to a stress (P, Q, R, S, T, U), and to oppose this stress only with its innate resistance to change of shape, the differential equation of work done will [by (4). with de, df, etc., substituted for e, f, etc.] be dw=Pde + Qdf + Rdg + Sda + Tdb + Udc.
(5) If w denote the whole amount of work done per unit of volume in any part of the body while the substance in this part experiences a strain (e, f, g, a, b, c) from some initial state regarded as a state of no strain. This equation, as we shall see later, under Properties of Matter, expresses the work done in a natural fluid, by distorting stress (or difference of pressure in different directions) working against its innate viscosity; and w is then, according to Joule's discovery, the dynamic value of the heat generated in the process. The equation may also be applied to express the work done in straining an imperfectly elastic solid, or an elastic solid of which the temperature varies during the process. In all such applications the stress will depend partly on the speed of the straining motion, or on the varying temperature, and not at all, or not solely, on the state of strain at any moment, and the system will not be dynamically conservative.
643. Definition.-A perfectly elastic body is a body which, when brought to any one state of strain, requires at all times the same stress to hold it in this state; however long it be kept strained, or however rapidly its state be altered from any other strain, or from no strain, to the strain in question. Here, according to our plan