Chapter III: Part 3

Let [rho] denote the density of the body at any point, X, Y, Z, the
components parallel to the axes of x, y, z of the body forces,
estimated as so much force per unit of mass; further let f_x, f_y, f_z
denote the components, parallel to the same axes, of the acceleration
of the particle which is momentarily at the point (x, y, z). The
equations of motion express the result that the rates of change of the
momentum, and of the moment of momentum, of any portion of the body
are those due to the action of all the forces exerted upon the portion
by other bodies, or by other portions of the same body. For the
changes of momentum, we have three equations of the type
_ _ _ _ _ _ _ _
/ / / / / / / /
| | |[rho]Xdx dy dz + | |X_[nu] dS = | | |[rho]f_x dx dy dz, (1)
_/_/_/ _/_/ _/_/_/

in which the volume integrations are taken through the volume of the
portion of the body, the surface integration is taken over its
surface, and the notation X_[nu] is that of S 4, the direction of [nu]
being that of the normal to this surface drawn outwards. For the
changes of moment of momentum, we have three equations of the type
_ _ _ _ _
/ / / / /
| | |[rho](yZ - zY)dx dy dz + | |(yZ_[nu] - zY_[nu])dS =
_/_/_/ _/_/
_ _ _
/ / /
| | |[rho](yf_z - zf_y)dx dy dz. (2)
_/_/_/

The equations (1) and (2) are the equations of motion of any kind of
body. The equations of equilibrium are obtained by replacing the
right-hand members of these equations by zero.

6. These equations can be used to obtain relations between the values
of X_[nu], Y_[nu], ... for different directions [nu]. When the
equations are applied to a very small volume, it appears that the
terms expressed by surface integrals would, unless they tend to zero
limits in a higher order than the areas of the surfaces, be very great
compared with the terms expressed by volume integrals. We conclude
that the surface tractions on the portion of the body which is bounded
by any very small closed surface, are ultimately in equilibrium. When
this result is interpreted for a small portion in the shape of a
tetrahedron, having three of its faces at right angles to the
co-ordinate axes, it leads to three equations of the type

X_[nu] = X_x cos(x, [nu]) + X_y cos(y, [nu]) + X_z cos(z, [nu]), (1)

where [nu] is the direction of the normal (drawn outwards) to the
remaining face of the tetrahedron, and (x, [nu]) ... denote the angles
which this normal makes with the axes. Hence X_[nu], ... for any
direction [nu] are expressed in terms of X_x,.... When the above
result is interpreted for a very small portion in the shape of a cube,
having its edges parallel to the co-ordinate axes, it leads to the
equations

Y_z = Z_y, Z_x = X_z, X_y = Y_x. (2)

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.

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.

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

When this quantity is negative there is "compression."

14. It is important to distinguish between two types of strain: the "rotational" type and the "irrotational" type. The distinction is illustrated in fig. 3, where the figure A"B"C"D" is obtained from the figure ABCD by contraction parallel to AC and extension parallel to BD, and the figure A'B'C'D' can be obtained from ABCD by the same contraction and extension followed by a rotation through the angle A"OA'. In strains of the irrotational type there are at any point three filaments at right angles to each other, which are such that the particles which lie in them before strain continue to lie in them after strain. A small spherical element of the body with its centre at the point becomes a small ellipsoid with its axes in the directions of these three filaments. In the case illustrated in the figure, the lines of the filaments in question, when the figure ABCD is strained into the figure A"B"C"D", are OA, OB and a line through O at right angles to their plane. In strains of the rotational type, on the other hand, the single existing set of three filaments (issuing from a point) which cut each other at right angles both before and after strain do not retain their directions after strain, though one of them may do so in certain cases. In the figure, the lines of the filaments in question, when the figure ABCD is strained into A'B'C'D', are OA, OB and a line at right angles to their plane before strain, and after strain they are OA', OB', and the same third line. A rotational strain can always be analysed into an irrotational strain (or "pure" strain) followed by a rotation.

Analytically, a strain is irrotational if the three quantities

dPw dPv dPu dPw dPv dPu
--- - ---, --- - ---, --- - ---.
dPy dPz dPz dPx dPx dPy

vanish, rotational if any one of them is different from zero. The
halves of these three quantities are the components of a vector
quantity called the "rotation."

15. Whether the strain is rotational or not, there is always one set
of three linear elements issuing from any point which cut each other
at right angles both before and after strain. If these directions are
chosen as axes of x, y, z, the shearing strains e_(yz), e_(zx), e_(xy)
vanish at this point. These directions are called the "principal axes
of strain," and the extensions in the directions of these axes the
"principal extensions."

16. It is very important to observe that the relations between components of strain and components of displacement imply relations between the components of strain themselves. If by any process of reasoning we arrive at the conclusion that the state of strain in a body is such and such a state, we have a test of the possibility or impossibility of our conclusion. The test is that, if the state of strain is a possible one, then there must be a displacement which can be associated with it in accordance with the equations (1) of S 11.

We may eliminate u, v, w from these equations. When this is done we
find that the quantities e_(xx), ... e_(yz) are connected by the two
sets of equations

dP^2e_(yy) dP^2e_(zz) dP^2e_(yz) \
---------- + ---------- = ---------- |
dPz^2 dPy^2 dPydPz |
|
dP^2e_(zz) dP^2e_(xx) dP^2e_(zx) |
---------- + ---------- = ---------- > (1)
dPx^2 dPz^2 dPzdPx |
|
dP^2e_(xx) dP^2e_(yy) dP^2e_(xy) |
---------- + ---------- = ---------- |
dPy^2 dPx^2 dPxdPy /

and

dP^2e_(xx) dP / dPe_(yz) dPe_(zx) dPe_(xy)\ \
2 ---------- = --- ( - -------- + -------- + -------- ) |
dPydPz dPx \ dPx dPy dPz / |
|
dP^2e_(yy) dP / dPe_(yz) dPe_(zx) dPe_(xy)\ |
2 ---------- = --- ( -------- - -------- + -------- ) > (2)
dPzdPx dPy \ dPx dPy dPz / |
|
dP^2e_(zz) dP / dPe_(yz) dPe_(zx) dPe_(xy)\ |
2 ---------- = --- ( -------- + -------- - -------- ) |
dPxdPy dPz \ dPx dPy dPz / /

These equations are known as the _conditions of compatibility of strain-components_. The components of strain which specify any possible strain satisfy them. Quantities arrived at in any way, and intended to be components of strain, if they fail to satisfy these equations, are not the components of any possible strain; and the theory or speculation by which they are reached must be modified or abandoned.

When the components of strain have been found in accordance with these
and other necessary equations, the displacement is to be found by
solving the equations (1) of S 11, considered as differential
equations to determine u, v, w. The most general possible solution
will differ from any other solution by terms which contain arbitrary
constants, and these terms represent a possible displacement. This
"complementary displacement" involves no strain, and would be a
possible displacement of an ideal perfectly rigid body.

17. The relations which connect the strains with each other and with the displacement are geometrical relations resulting from the definitions of the quantities and not requiring any experimental verification. They do not admit of such verification, because the strain within a body cannot be measured. The quantities (belonging to the same category) which can be measured are displacements of points on the surface of a body. For example, on the surface of a bar subjected to tension we may make two fine transverse scratches, and measure the distance between them before and after the bar is stretched. For such measurements very refined instruments are required. Instruments for this purpose are called barbarously "extensometers," and many different kinds have been devised. From measurements of displacement by an extensometer we may deduce the average extension of a filament of the bar terminated by the two scratches. In general, when we attempt to measure a strain, we really measure some displacements, and deduce the values, not of the strain at a point, but of the average extensions of some particular linear filaments of a body containing the point; and these filaments are, from the nature of the case, nearly always superficial filaments.

18. In the case of transparent materials such as glass there is available a method of studying experimentally the state of strain within a body. This method is founded upon the result that a piece of glass when strained becomes doubly refracting, with its optical principal axes at any point in the directions of the principal axes of strain (S 15) at the point. When the piece has two parallel plane faces, and two of the principal axes of strain at any point are parallel to these faces, polarized light transmitted through the piece in a direction normal to the faces can be used to determine the directions of the principal axes of the strain at any point. If the directions of these axes are known theoretically the comparison of the experimental and theoretical results yields a test of the theory.

19. _Relations between Stresses and Strains._--The problem of the extension of a bar subjected to tension is the one which has been most studied experimentally, and as a result of this study it is found that for most materials, including all metals except cast metals, the measurable extension is proportional to the applied tension, provided that this tension is not too great. In interpreting this result it is assumed that the tension is uniform over the cross-section of the bar, and that the extension of longitudinal filaments is uniform throughout the bar; and then the result takes the form of a law of proportionality connecting stress and strain: The tension is proportional to the extension. Similar results are found for the same materials when other methods of experimenting are adopted, for example, when a bar is supported at the ends and bent by an attached load and the deflexion is measured, or when a bar is twisted by an axial couple and the relative angular displacement of two sections is measured. We have thus very numerous experimental verifications of the famous law first enunciated by Robert Hooke in 1678 in the words "_Ut Tensio sic vis_"; that is, "the Power of any spring is in the same proportion as the Tension (--stretching) thereof." The most general statement of Hooke's Law in modern language would be:--_Each of the six components of stress at any point of a body is a linear function of the six components of strain at the point._ It is evident from what has been said above as to the nature of the measurement of stresses and strains that this law in all its generality does not admit of complete experimental verification, and that the evidence for it consists largely in the agreement of the results which are deduced from it in a theoretical fashion with the results of experiments. Of such results one of a general character may be noted here. If the law is assumed to be true, and the equations of motion of the body (S 5) are transformed by means of it into differential equations for determining the components of displacement, these differential equations admit of solutions which represent periodic vibratory displacements (see S 85 below). The fact that solid bodies can be thrown into states of isochronous vibration has been emphasized by G.G. Stokes as a peremptory proof of the truth of Hooke's Law.

20. According to the statement of the generalized Hooke's Law the stress-components vanish when the strain-components vanish. The strain-components contemplated in experiments upon which the law is founded are measured from a zero of reckoning which corresponds to the state of the body subjected to experiment before the experiment is made, and the stress-components referred to in the statement of the law are those which are called into action by the forces applied to the body in the course of the experiment. No account is taken of the stress which must already exist in the body owing to the force of gravity and the forces by which the body is supported. When it is desired to take account of this stress it is usual to suppose that the strains which would be produced in the body if it could be freed from the action of gravity and from the pressures of supports are so small that the strains produced by the forces which are applied in the course of the experiment can be compounded with them by simple superposition. This supposition comes to the same thing as measuring the strain in the body, not from the state in which it was before the experiment, but from an ideal state (the "unstressed" state) in which it would be entirely free from internal stress, and allowing for the strain which would be produced by gravity and the supporting forces if these forces were applied to the body when free from stress. In most practical cases the initial strain to be allowed for is unimportant (see SS 91-93 below).

21. Hooke's law of proportionality of stress and strain leads to the introduction of important physical constants: the _moduluses of elasticity_ of a body. Let a bar of uniform section (of area [omega]) be stretched with tension T, which is distributed uniformly over the section, so that the stretching force is Tw[omega], and let the bar be unsupported at the sides. The bar will undergo a longitudinal extension of magnitude T/E, where E is a constant quantity depending upon the material. This constant is called _Young's modulus_ after Thomas Young, who introduced it into the science in 1807. The quantity E is of the same nature as a traction, that is to say, it is measured as a force estimated per unit of area. For steel it is about 2.04 X 10^12 dynes per square centimetre, or about 13,000 tons per sq. in.

22. The longitudinal extension of the bar under tension is not the only strain in the bar. It is accompanied by a lateral contraction by which all the transverse filaments of the bar are shortened. The amount of this contraction is [sigma]T/E, where [sigma] is a certain number called _Poisson's ratio_, because its importance was at first noted by S.D. Poisson in 1828. Poisson arrived at the existence of this contraction, and the corresponding number [sigma], from theoretical considerations, and his theory led him to assign to [sigma] the value 1/4. Many experiments have been made with the view of determining [sigma], with the result that it has been found to be different for different materials, although for very many it does not differ much from 1/4. For steel the best value (Amagat's) is 0.268. Poisson's theory admits of being modified so as to agree with the results of experiment.

23. The behaviour of an elastic solid body, strained within the limits of its elasticity, is entirely determined by the constants E and [sigma] if the body is _isotropic_, that is to say, if it has the same quality in all directions around any point. Nevertheless it is convenient to introduce other constants which are related to the action of particular sorts of forces. The most important of these are the "modulus of compression" (or "bulk modulus") and the "rigidity" (or "modulus of shear"). To define the _modulus of compression_, we suppose that a solid body of any form is subjected to uniform hydrostatic pressure of amount p. The state of stress within it will be one of uniform pressure, the same at all points, and the same in all directions round any point. There will be compression, the same at all points, and proportional to the pressure; and the amount of the compression can be expressed as p/k. The quantity k is the modulus of compression. In this case the linear contraction in any direction is p/3k; but in general the linear extension (or contraction) is not one-third of the cubical dilatation (or compression).

24. To define the _rigidity_, we suppose that a solid body is subjected to forces in such a way that there is shearing stress within it. For example, a cubical block may be subjected to opposing tractions on opposite faces acting in directions which are parallel to an edge of the cube and to both the faces. Let S be the amount of the traction, and let it be uniformly distributed over the faces. As we have seen (S 7), equal tractions must act upon two other faces in suitable directions in order to maintain equilibrium (see fig. 2 of S 7). The two directions involved may be chosen as axes of x, y as in that figure. Then the state of stress will be one in which the stress-component denoted by X_y is equal to S, and the remaining stress-components vanish; and the strain produced in the body is shearing strain of the type denoted by e _(xy). The amount of the shearing strain is S/[mu], and the quantity [mu] is the "rigidity."

25. The modulus of compression and the rigidity are quantities of the same kind as Young's modulus. The modulus of compression of steel is about 1.43 X 10^12 dynes per square centimetre, the rigidity is about 8.19 X 10^11 dynes per square centimetre. It must be understood that the values for different specimens of nominally the same material may differ considerably.

The modulus of compression k and the rigidity [mu] of an isotropic
material are connected with the Young's modulus E and Poisson's ratio
[sigma] of the material by the equations

k = E/3(1 - 2[sigma]), [mu] = E/2(1 + [sigma]).

26. Whatever the forces acting upon an isotropic solid body may be,
provided that the body is strained within its limits of elasticity,
the strain-components are expressed in terms of the stress-components
by the equations

e_(xx) = (X_x - [sigma]Y_y - [sigma]Z_z)/E, e_(yz) = Y_z/[mu], \
e_(yy) = (Y_y - [sigma]Z_z - [sigma]X_x)/E, e_(zx) = Z_x/[mu], > (1)
e_(zz) = (Z_z - [sigma]X_x - [sigma]Y_y)/E, e_(xy) = X_y/[mu]. /

If we introduce a quantity [lambda], of the same nature as E or [mu], by
the equation

[lambda] = E[sigma]/(1 + [sigma])(1 - 2[sigma]), (2)

we may express the stress-components in terms of the strain-components
by the equations

X_x = [lambda][e_(xx) + e_(yy) + e_(zz)] + 2[mu]e_(xx), Y_z = [mu]e_(yz), \
Y_y = [lambda][e_(xx) + e_(yy) + e_(zz)] + 2[mu]e_(yy), Z_x = [mu]e_(zx), > (3)
Z_z = [lambda][e_(xx) + e_(yy) + e_(zz)] + 2[mu]e_(zz), X_y = [mu]e_(xy); /

and then the behaviour of the body under the action of any forces
depends upon the two constants [lambda] and [mu]. These two constants
were introduced by G. Lame in his treatise of 1852. The importance of
the quantity [mu] had been previously emphasized by L.J. Vicat and G.G.
Stokes.

27. The potential energy per unit of volume (often called the
"resilience") stored up in the body by the strain is equal to

1/2([lambda] + 2[mu])(e_(xx) + e_(yy) + e_(zz))^2 + 1/2[mu][e^2_(yz) + e^2_(zx) +
e^2_(xy) - 4e_(yy)e_(zz) - 4e_(zz)e_(xx) - 4e_(xx)e_(yy)],

or the equivalent expression

1/2[(X^2_x + Y^2_y + Z^2_z) - 2[sigma](Y_yZ_z + Z_zX_x + X_xY_y) +
2(1 + [sigma])(Y^2_z + Z^2_x + X^2_y)]/E.

The former of these expressions is called the
"strain-energy-function."

28. The Young's modulus E of a material is often determined experimentally by the direct method of the extensometer (S 17), but more frequently it is determined indirectly by means of a result obtained in the theory of the flexure of a bar (see SS 47, 53 below). The rigidity [mu] is usually determined indirectly by means of results obtained in the theory of the torsion of a bar (see SS 41, 42 below). The modulus of compression k may be determined directly by means of the piezometer, as was done by E.H. Amagat, or it may be determined indirectly by means of a result obtained in the theory of a tube under pressure, as was done by A. Mallock (see S 78 below). The value of Poisson's ratio [sigma] is generally inferred from the relation connecting it with E and [mu] or with E and k, but it may also be determined indirectly by means of a result obtained in the theory of the flexure of a bar (S 47 below), as was done by M.A. Cornu and A. Mallock, or directly by a modification of the extensometer method, as has been done recently by J. Morrow.

29. The _elasticity of a fluid_ is always expressed by means of a single quantity of the same kind as the _modulus of compression_ of a solid body. To any increment of pressure, which is not too great, there corresponds a proportional cubical compression, and the amount of this compression for an increment [delta]p of pressure can be expressed as [delta]p/k. The quantity that is usually tabulated is the reciprocal of k, and it is called the _coefficient of compressibility_. It is the amount of compression per unit increase of pressure. As a physical quantity it is of the same dimensions as the reciprocal of a pressure (or of a force per unit of area). The pressures concerned are usually measured in atmospheres (1 atmosphere = 1.014 X 10^6 dynes per sq. cm.). For water the coefficient of compressibility, or the compression per atmosphere, is about 4.5 X 10^-5. This gives for k the value 2.22 X 10^10 dynes per sq. cm. The Young's modulus and the rigidity of a fluid are always zero.

30. The relations between stress and strain in a material which is not isotropic are much more complicated. In such a material the Young's modulus depends upon the direction of the tension, and its variations about a point are expressed by means of a surface of the fourth degree. The Poisson's ratio depends upon the direction of the contracted lateral filaments as well as upon that of the longitudinal extended ones. The rigidity depends upon both the directions involved in the specification of the shearing stress. In general there is no simple relation between the Young's moduluses and Poisson's ratios and rigidities for assigned directions and the modulus of compression. Many materials in common use, all fibrous woods for example, are actually _aeolotropic_ (that is to say, are not isotropic), but the materials which are aeolotropic in the most regular fashion are natural crystals. The elastic behaviour of crystals has been studied exhaustively by many physicists, and in particular by W. Voigt. The strain-energy-function is a homogeneous quadratic function of the six strain-components, and this function may have as many as 21 independent coefficients, taking the place in the general case of the 2 coefficients [lambda], [mu] which occur when the material is isotropic--a result first obtained by George Green in 1837. The best experimental determinations of the coefficients have been made indirectly by Voigt by means of results obtained in the theories of the torsion and flexure of aeolotropic bars.

31. _Limits of Elasticity._--A solid body which has been strained by considerable forces does not in general recover its original size and shape completely after the forces cease to act. The strain that is left is called _set_. If set occurs the elasticity is said to be "imperfect," and the greatest strain (or the greatest load) of any specified type, for which no set occurs, defines the "limit of perfect elasticity" corresponding to the specified type of strain, or of stress. All fluids and many solid bodies, such as glasses and crystals, as well as some metals (copper, lead, silver) appear to be perfectly elastic as regards change of volume within wide limits; but malleable metals and alloys can have their densities permanently increased by considerable pressures. The limits of perfect elasticity as regards change of shape, on the other hand, are very low, if they exist at all, for glasses and other hard, brittle solids; but a class of metals including copper, brass, steel, and platinum are very perfectly elastic as regards distortion, provided that the distortion is not too great. The question can be tested by observation of the torsional elasticity of thin fibres or wires. The limits of perfect elasticity are somewhat ill-defined, because an experiment cannot warrant us in asserting that there is no set, but only that, if there is any set, it is too small to be observed.

32. A different meaning may be, and often is, attached to the phrase "limits of elasticity" in consequence of the following experimental result:--Let a bar be held stretched under a moderate tension, and let the extension be measured; let the tension be slightly increased and the extension again measured; let this process be continued, the tension being increased by equal increments. It is found that when the tension is not too great the extension increases by equal increments (as nearly as experiment can decide), but that, as the tension increases, a stage is reached in which the extension increases faster than it would do if it continued to be proportional to the tension. The beginning of this stage is tolerably well marked. Some time before this stage is reached the limit of perfect elasticity is passed; that is to say, if the load is removed it is found that there is some permanent set. The limiting tension beyond which the above law of proportionality fails is often called the "limit of _linear_ elasticity." It is higher than the limit of perfect elasticity. For steel bars of various qualities J. Bauschinger found for this limit values varying from 10 to 17 tons per square inch. The result indicates that, when forces which produce any kind of strain are applied to a solid body and are gradually increased, the strain at any instant increases proportionally to the forces up to a stage beyond that at which, if the forces were removed, the body would completely recover its original size and shape, but that the increase of strain ceases to be proportional to the increase of load when the load surpasses a certain limit. There would thus be, for any type of strain, a _limit of linear elasticity_, which exceeds the limit of perfect elasticity.

33. A body which has been strained beyond the limit of linear elasticity is often said to have suffered an "over-strain." When the load is removed, the _set_ which can be observed is not entirely permanent; but it gradually diminishes with lapse of time. This phenomenon is named "elastic after-working." If, on the other hand, the load is maintained constant, the strain is gradually increased. This effect indicates a gradual flowing of solid bodies under great stress; and a similar effect was observed in the experiments of H. Tresca on the punching and crushing of metals. It appears that all solid bodies under sufficiently great loads become "plastic," that is to say, they take a set which gradually increases with the lapse of time. No plasticity is observed when the limit of linear elasticity is not exceeded.

34. The values of the elastic limits are affected by overstrain. If the load is maintained for some time, and then removed, the limit of linear elasticity is found to be higher than before. If the load is not maintained, but is removed and then reapplied, the limit is found to be lower than before. During a period of rest a test piece recovers its elasticity after overstrain.

35. The effects of repeated loading have been studied by A. Wohler, J. Bauschinger, O. Reynolds and others. It has been found that, after many repetitions of rather rapidly alternating stress, pieces are fractured by loads which they have many times withstood. It is not certain whether the fracture is in every case caused by the gradual growth of minute flaws from the beginning of the series of tests, or whether the elastic quality of the material suffers deterioration apart from such flaws. It appears, however, to be an ascertained result that, so long as the limit of linear elasticity is not exceeded, repeated loads and rapidly alternating loads do not produce failure of the material.

36. The question of the conditions of safety, or of the conditions in which rupture is produced, is one upon which there has been much speculation, but no completely satisfactory result has been obtained. It has been variously held that rupture occurs when the numerically greatest principal stress exceeds a certain limit, or when this stress is tension and exceeds a certain limit, or when the greatest difference of two principal stresses (called the "stress-difference") exceeds a certain limit, or when the greatest extension or the greatest shearing strain or the greatest strain of any type exceeds a certain limit. Some of these hypotheses appear to have been disproved. It was held by G.F. Fitzgerald (_Nature_, Nov. 5, 1896) that rupture is not produced by pressure symmetrically applied all round a body, and this opinion has been confirmed by the recent experiments of A. Foppl. This result disposes of the greatest stress hypothesis and also of the greatest strain hypothesis. The fact that short pillars can be crushed by longitudinal pressure disposes of the greatest tension hypothesis, for there is no tension in the pillar. The greatest extension hypothesis failed to satisfy some tests imposed by H. Wehage, who experimented with blocks of wrought iron subjected to equal pressures in two directions at right angles to each other. The greatest stress-difference hypothesis and the greatest shearing strain hypothesis would lead to practically identical results, and these results have been held by J.J. Guest to accord well with his experiments on metal tubes subjected to various systems of combined stress; but these experiments and Guest's conclusion have been criticized adversely by O. Mohr, and the question cannot be regarded as settled. The fact seems to be that the conditions of rupture depend largely upon the nature of the test (tensional, torsional, flexural, or whatever it may be) that is applied to a specimen, and that no general formula holds for all kinds of tests. The best modern technical writings emphasize the importance of the limits of linear elasticity and of tests of dynamical resistance (S 87 below) as well as of statical resistance.

37. The question of the conditions of rupture belongs rather to the science of the strength of materials than to the science of elasticity (S 1); but it has been necessary to refer to it briefly here, because there is no method except the methods of the theory of elasticity for determining the state of stress or strain in a body subjected to forces. Whatever view may ultimately be adopted as to the relation between the conditions of safety of a structure and the state of stress or strain in it, the calculation of this state by means of the theory or by experimental means (as in S 18) cannot be dispensed with.

38. _Methods of determining the Stress in a Body subjected to given
Forces._--To determine the state of stress, or the state of strain, in
an isotropic solid body strained within its limits of elasticity by
given forces, we have to use (i.) the equations of equilibrium, (ii.)
the conditions which hold at the bounding surface, (iii.) the
relations between stress-components and strain-components, (iv.) the
relations between strain-components and displacement. The equations of
equilibrium are (with notation already used) three partial
differential equations of the type

dPX_x dPX_y dPZ_z
----- + ----- + ----- + [rho]X = 0. (1)
dPx dPy dPz

The conditions which hold at the bounding surface are three equations
of the type

X_x cos(x, [nu]) + X_y cos(y, [nu]) + Z_x cos(z, [nu]) = X`_[nu], (2)

where [nu] denotes the direction of the outward-drawn normal to the
bounding surface, and X`_[nu] denotes the x-component of the applied
surface traction. The relations between stress-components and
strain-components are expressed by either of the sets of equations (1)
or (3) of S 26. The relations between strain-components and
displacement are the equations (1) of S 11, or the equivalent
conditions of compatibility expressed in equations (1) and (2) of S
16.

39. We may proceed by either of two methods. In one method we
eliminate the stress-components and the strain-components and retain
only the components of displacement. This method leads (with notation
already used) to three partial differential equations of the type

dP /dPu dPv dPw\ /dP^2u dP^2u dP^2u\
([lambda] + [mu]) --- ( --- + --- + --- ) + [mu]( ----- + ----- + ----- ) + [rho]X = 0, (3)
dPx \dPx dPy dPz/ \dPx^2 dPy^2 dPz^2/

and three boundary conditions of the type
_
/dPu dPv dPw\ | dPu
[lambda] cos(x, [nu])( --- + --- + --- ) + [mu] | 2 cos(x, [nu])---
\dPx dPy dPz/ |_ dPx
_
/dPv dPu\ /dPu dPw\ |
+ cos(y, [nu])( -- + -- ) + cos(z, [nu])( -- + -- ) | = X`_[nu], (4)
\dPx dPy/ \dPz dPx/ _|

In the alternative method we eliminate the strain-components and the
displacements. This method leads to a system of partial differential
equations to be satisfied by the stress-components. In this system
there are three equations of the type

dPX_x dPX_y dPX_z
----- + ----- + ----- + [rho]X = 0, (1 _bis_)
dPx dPy dPz

three of the type

dP^2X_x dP^2X_x dP^2X_x 1 dP^2
------- + ------- + ------- + ----------- ----- (X_x + Y_y + Z_z) =
dPx^2 dPy^2 dPz^2 1 + [sigma] dPx^2

[sigma] /dPX dPY dPZ\ dPX
- ---------[rho]( --- + --- + --- ) - 2[rho] ---, (5)
1-[sigma] \dPx dPy dPz/ dPx

and three of the type

dP^2Y_z dP^2Y_z dP^2Y_z 1 dP^2
------- + ------- + ------- + ----------- ------ (X_x + Y_y + Z_z) =
dPx^2 dPy^2 dPz^2 1 + [sigma] dPydPz

/dPZ dPY\
- [rho]( --- + --- ), (6)
\dPy dPz/

the equations of the two latter types being necessitated by the
conditions of compatibility of strain-components. The solutions of
these equations have to be adjusted so that the boundary conditions of
the type (2) may be satisfied.

40. It is evident that whichever method is adopted the mathematical
problem is in general very complicated. It is also evident that, if we
attempt to proceed by help of some intuition as to the nature of the
stress or strain, our intuition ought to satisfy the tests provided by
the above systems of equations. Neglect of this precaution has led to
many errors. Another source of frequent error lies in the neglect of
the conditions in which the above systems of equations are correct.
They are obtained by help of the supposition that the relative
displacements of the parts of the strained body are small. The
solutions of them must therefore satisfy the test of smallness of the
relative displacements.

41. Torsion.--As a first example of the application of the theory we take the problem of the torsion of prisms. This problem, considered first by C.A. Coulomb in 1784, was finally solved by B. de Saint-Venant in 1855. The problem is this:--A cylindrical or prismatic bar is held twisted by terminal couples; it is required to determine the state of stress and strain in the interior. When the bar is a circular cylinder the problem is easy. Any section is displaced by rotation about the central-line through a small angle, which is proportional to the distance z of the section from a fixed plane at right angles to this line. This plane is a terminal section if one of the two terminal sections is not displaced. The angle through which the section z rotates is [tau]z, where [tau] is a constant, called the amount of the twist; and this constant [tau] is equal to G/[mu]I, where G is the twisting couple, and I is the moment of inertia of the cross-section about the central-line. This result is often called "Coulomb's law." The stress within the bar is shearing stress, consisting, as it must, of two sets of equal tangential tractions on two sets of planes which are at right angles to each other. These planes are the cross-sections and the axial planes of the bar. The tangential traction at any point of the cross-section is directed at right angles to the axial plane through the point, and the tangential traction on the axial plane is directed parallel to the length of the bar. The amount of either at a distance r from the axis is [mu][tau]r or Gr/I. The result that G = [mu][tau]I can be used to determine [mu] experimentally, for [tau] may be measured and G and I are known.

42. When the cross-section of the bar is not circular it is clear that this solution fails; for the existence of tangential traction, near the prismatic bounding surface, on any plane which does not cut this surface at right angles, implies the existence of traction applied to this surface. We may attempt to modify the theory by retaining the supposition that the stress consists of shearing stress, involving tangential traction distributed in some way over the cross-sections. Such traction is obviously a necessary constituent of any stress-system which could be produced by terminal couples around the axis. We should then know that there must be equal tangential traction directed along the length of the bar, and exerted across some planes or other which are parallel to this direction. We should also know that, at the bounding surface, these planes must cut this surface at right angles. The corresponding strain would be shearing strain which could involve (i.) a sliding of elements of one cross-section relative to another, (ii.) a relative sliding of elements of the above mentioned planes in the direction of the length of the bar. We could conclude that there may be a longitudinal displacement of the elements of the cross-sections. We should then attempt to satisfy the conditions of the problem by supposing that this is the character of the strain, and that the corresponding displacement consists of (i.) a rotation of the cross-sections in their planes such as we found in the case of the circle, (ii.) a distortion of the cross-sections into curved surfaces by a displacement (w) which is directed normally to their planes and varies in some manner from point to point of these planes. We could show that all the conditions of the problem are satisfied by this assumption, provided that the longitudinal displacement (w), considered as a function of the position of a point (x, y) in the cross-section, satisfies the equation

dP^2w dP^2w
----- + ----- = 0, (1)
dPx^2 dPy^2

and the boundary condition

/ dPw \ / dPw \
( --- - [tau]y ) cos(x, [nu]) + ( --- + [tau]x ) cos(y, [nu]) = 0, (2)
\ dPx / \ dPy /

where [tau] denotes the amount of the twist, and [nu] the direction of the normal to the boundary. The solution is known for a great many forms of section. (In the particular case of a circular section w vanishes.) The tangential traction at any point of the cross-section is directed along the tangent to that curve of the family [psi] = const. which passes through the point, [psi] being the function determined by the equations

dPw /dP[psi] \ dPw /dP[psi] \
--- = [tau]( ------- + y ), --- = - [tau]( ------- + x ).
dPx \ dPy / dPy \ dPx /

The amount of the twist [tau] produced by terminal couples of magnitude
G is G/C, where C is a constant, called the "torsional rigidity" of the
prism, and expressed by the formula
_ _ _ _
/ / | /dP[psi]\^2 /dP[psi]\^2 |
C = [mu] | | | ( ------- ) + ( ------- ) | dxdy,
_/ _/ |_ \ dPx / \ dPy / _|

the integration being taken over the cross-section. When the coefficient of [mu] in the expression for C is known for any section, [mu] can be determined by experiment with a bar of that form of section.

43. The distortion of the cross-sections into curved surfaces is shown graphically by drawing the contour lines (w = const.). In general the section is divided into a number of compartments, and the portions that lie within two adjacent compartments are respectively concave and convex. This result is illustrated in the accompanying figures (fig. 4 for the ellipse, given by x^2/b^2 + y^2/c^2 = 1; fig. 5 for the equilateral triangle, given by (x + (1/3)a) [x^2 - 3y^2 - (4/3)ax + (4/9)a^2] = 0; fig. 6 for the square).

44. The distribution of the shearing stress over the cross-section is determined by the function [psi], already introduced. If we draw the curves [psi] = const., corresponding to any form of section, for equidifferent values of the constant, the tangential traction at any point on the cross-section is directed along the tangent to that curve of the family which passes through the point, and the magnitude of it is inversely proportional to the distance between consecutive curves of the family. Fig. 7 illustrates the result in the case of the _equilateral_ triangle. The boundary is, of course, one of the lines. The "lines of shearing stress" which can thus be drawn are in every case identical with the lines of flow of frictionless liquid filling a cylindrical vessel of the same cross-section as the bar, when the liquid circulates in the plane of the section with uniform spin. They are also the same as the contour lines of a flexible and slightly extensible membrane, of which the edge has the same form as the bounding curve of the cross-section of the bar, when the membrane is fixed at the edge and slightly deformed by uniform pressure.