When we substitute in the general equations the particular results
which are thus obtained, we find that the equations of motion take
such forms as
dPX_x dPX_y dPZ_x
[rho]X + ----- + ----- + ----- = [rho] f_x, (3)
dPx dPy dPz
and the equations of moments are satisfied identically. The equations
of equilibrium are obtained by replacing the right-hand members by
zero.
7. A state of stress in which the traction across any plane of a set of
parallel planes is normal to the plane, and that across any
perpendicular plane vanishes, is described as a state of "simple
tension" ("simple pressure" if the traction is negative). A state of
stress in which the traction across any plane is normal to the plane,
and the traction is the same for all planes passing through any point,
is described as a state of "uniform tension" ("uniform pressure" if the
traction is negative). Sometimes the phrases "isotropic tension" and
"hydrostatic pressure" are used instead of "uniform" tension or
pressure. The distinction between the two states, simple tension and
uniform tension, is illustrated in fig. 1.
[Illustration: FIG. 1.]
A state of stress in which there is purely tangential traction on a
plane, and no normal traction on any perpendicular plane, is described
as a state of "shearing stress." The result (2) of S 6 shows that
tangential tractions occur in pairs. If, at any point, there is
tangential traction, in any direction, on a plane parallel to this
direction, and if we draw through the point a plane at right angles to
the direction of this traction, and therefore containing the normal to
the first plane, then there is equal tangential traction on this second
plane in the direction of the normal to the first plane. The result is
illustrated in fig. 2, where a rectangular block is subjected on two
opposite faces to opposing tangential tractions, and is held in
equilibrium by equal tangential tractions applied to two other faces.
[Illustration: FIG. 2.]
Through any point there always pass three planes, at right angles to
each other, across which there is no tangential traction. These planes
are called the "principal planes of stress," and the (normal) tractions
across them the "principal stresses." Lines, usually curved, which have
at every point the direction of a principal stress at the point, are
called "lines of stress."
8. It appears that the stress at any point of a body is completely
specified by six quantities, which can be taken to be the X_x, Y_y, Z_z
and Y_z, Z_x, X_y of S 6. The first three are tensions (pressures if
they are negative) across three planes parallel to fixed rectangular
directions, and the remaining three are tangential tractions across the
same three planes. These six quantities are called the "components of
stress." It appears also that the components of stress are connected
with each other, and with the body forces and accelerations, by the
three partial differential equations of the type (3) of S 6. These
equations are available for the purpose of determining the state of
stress which exists in a body of definite form subjected to definite
forces, but they are not sufficient for the purpose (see S 38 below). In
order to effect the determination it is necessary to have information
concerning the constitution of the body, and to introduce subsidiary
relations founded upon this information.
9. The definite mathematical relations which have been found to connect
the components of stress with each other, and with other quantities,
result necessarily from the formation of a clear conception of the
nature of stress. They do not admit of experimental verification,
because the stress within a body does not admit of direct measurement.
Results which are deduced by the aid of these relations can be compared
with experimental results. If any discrepancy were observed it would not
be interpreted as requiring a modification of the concept of stress, but
as affecting some one or other of the subsidiary relations which must
be introduced for the purpose of obtaining the theoretical result.
10. _Strain._--For the specification of the changes of size and shape
which are produced in a body by any forces, we begin by defining the
"average extension" of any linear element or "filament" of the body. Let
l0 be the length of the filament before the forces are applied, l its
length when the body is subjected to the forces. The average extension
of the filament is measured by the fraction (l - l0)/l0. If this
fraction is negative there is "contraction." The "extension at a point"
of a body in any assigned direction is the mathematical limit of this
fraction when one end of the filament is at the point, the filament has
the assigned direction, and its length is diminished indefinitely. It is
clear that all the changes of size and shape of the body are known when
the extension at every point in every direction is known.
The relations between the extensions in different directions around
the same point are most simply expressed by introducing the extensions
in the directions of the co-ordinate axes and the angles between
filaments of the body which are initially parallel to these axes. Let
e_(xx), e_(yy), e_(zz) denote the extensions parallel to the axes of
x, y, z, and let e_(yz), e_(zx), e_(xy) denote the cosines of the
angles between the pairs of filaments which are initially parallel to
the axes of y and z, z and x, x and y. Also let e denote the extension
in the direction of a line the direction cosines of which are l, m, n.
Then, if the changes of size and shape are slight, we have the
relation
e = e_(xx)l^2 + e_(yy)m^2 + e_(zz)n^2 + e_(yz)mn + e_(zx)nl + e_(xy)lm.
The body which undergoes the change of size or shape is said to be
"strained," and the "strain" is determined when the quantities e_(xx),
e_(yy), e_(zz) and e_(yz), e_(zx), e_(xy) defined above are known at
every point of it. These quantities are called "components of strain."
The three of the type e_(xx) are extensions, and the three of the type
e_(yz) are called "shearing strains" (see S 12 below).
11. All the changes of relative position of particles of the body are
known when the strain is known, and conversely the strain can be
determined when the changes of relative position are given. These
changes can be expressed most simply by the introduction of a vector
quantity to represent the displacement of any particle.
When the body is deformed by the action of any forces its particles
pass from the positions which they occupied before the action of the
forces into new positions. If x, y, z are the co-ordinates of the
position of a particle in the first state, its co-ordinates in the
second state may be denoted by x + u, y + v, z + w. The quantities, u,
v, w are the "components of displacement." When these quantities are
small, the strain is connected with them by the equations
e_(xx) = dPu/dPx, e_(yy) = dPv/dPy, e_(zz) = dPw/dPz, \
|
dPw dPv dPu dPw dPv dPu >(1)
e_(yz) = --- + ---, e_(zx) = --- + ---, e_(xy) = --- + --- . |
dPy dPz dPz dPx dPx dPy /
12. These equations enable us to determine more exactly the nature of
the "shearing strains" such as e_(xy). Let u, for example, be of the
form sy, where s is constant, and let v and w vanish. Then e_(xy) = s,
and the remaining components of strain vanish. The nature of the strain
(called "simple shear") is simply appreciated by imagining the body to
consist of a series of thin sheets, like the leaves of a book, which lie
one over another and are all parallel to a plane (that of x, z); and the
displacement is seen to consist in the shifting of each sheet relative
to the sheet below in a direction (that of x) which is the same for all
the sheets. The displacement of any sheet is proportional to its
distance y from a particular sheet, which remains undisplaced. The
shearing strain has the effect of distorting the shape of any portion of
the body without altering its volume. This is shown in fig. 3, where a
square ABCD is distorted by simple shear (each point moving parallel to
the line marked xx) into a rhombus A'B'C'D', as if by an extension of
the diagonal BD and a contraction of the diagonal AC, which extension
and contraction are adjusted so as to leave the area unaltered. In the
general case, where u is not of the form sy and v and w do not vanish,
the shearing strains such as e_(xy) result from the composition of pairs
of simple shears of the type which has just been explained.
13. Besides enabling us to express the extension in any direction and
the changes of relative direction of any filaments of the body, the
components of strain also express the changes of size of volumes and
areas. In particular, the "cubical dilatation," that is to say, the
increase of volume per unit of volume, is expressed by the quantity
dPu dPv dPw
e_(xx) + e_(yy) + e_(zz) or --- + --- + ---.
dPx dPy dPz