(1) radial contraction of amount
_ _
3 + [sigma] | a^2 |
----------- | (1 + [sigma])--- - (1 - [sigma]) | [omega]^2[rho]b^2,
8E |_ r^2 _|
(2) extension along the circular filaments of amount
_ _
3 + [sigma] | a^2 |
----------- |(1 + [sigma])--- + (1 - [sigma]) | [omega]^2[rho]b^2.
8E |_ r^2 _|
(3) contraction of the filaments parallel to the axis of amount
[sigma](3 + [sigma])
-------------------- [omega]^2[rho]b^2.
4E
Again, the greatest extension is the circumferential extension at the
inner surface, and, when the hole is very small, its amount is nearly
double what it would be for a complete disk.
82. In the problem of the rotating shaft we have the following
stress-system:
(1) radial tension of amount
(1/8)[omega]^2[rho](a^2 - r^2)(3 - 2[sigma])/(1-[sigma]),
(2) circumferential tension of amount
(1/8)[omega]^2[rho]{(a^2(3 - 2[sigma])/(1-[sigma])
- r^2(1 + 2[sigma])/(1 - [sigma])},
(3) longitudinal tension of amount
1/4[omega]^2[rho](a^2 - 2r^2)[sigma]/(1 - [sigma]).
The resultant longitudinal tension at any normal section vanishes, and
the radial tension vanishes at the bounding surface; and thus the
expressions here given may be taken to represent the actual condition
at all points which are not very close to the ends of the shaft. The
contraction of the longitudinal filaments is uniform and equal to
1/2[omega]^2[rho]a^2[sigma]/E. The greatest extension in the rotating
shaft is the circumferential extension close to the axis, and its
amount is (1/8)[omega]^2[rho]a^2(3 - 5[sigma])/E(1 - [sigma]).
The value of any theory of the strength of long rotating shafts
founded on these formulae is diminished by the circumstance that at
sufficiently high speeds the shaft may tend to take up a curved form,
the straight form being unstable. The shaft is then said to _whirl_.
This occurs when the period of rotation of the shaft is very nearly
coincident with one of its periods of lateral vibration. The lowest
speed at which whirling can take place in a shaft of length l, freely
supported at its ends, is given by the formula
[omega]^2[rho] = 1/4Ea^2([pi]/l)^4.
As in S 61, this formula should not be applied unless the length of
the shaft is a considerable multiple of its diameter. It implies that
whirling is to be expected whenever [omega] approaches this critical
value.
83. When the forces acting upon a spherical or cylindrical body are not
radial, the problem becomes more complicated. In the case of the sphere
deformed by any forces it has been completely solved, and the solution
has been applied by Lord Kelvin and Sir G.H. Darwin to many interesting
questions of cosmical physics. The nature of the stress produced in the
interior of the earth by the weight of continents and mountains, the
spheroidal figure of a rotating solid planet, the rigidity of the earth,
are among the questions which have in this way been attacked. Darwin
concluded from his investigation that, to support the weight of the
existing continents and mountain ranges, the materials of which the
earth is composed must, at great depths (1600 kilometres), have at least
the strength of granite. Kelvin concluded from his investigation that
the actual heights of the tides in the existing oceans can be accounted
for only on the supposition that the interior of the earth is solid, and
of rigidity nearly as great as, if not greater than, that of steel.
84. Some interesting problems relating to the strains produced in a
cylinder of finite length by forces distributed symmetrically round
the axis have been solved. The most important is that of a cylinder
crushed between parallel planes in contact with its plane ends. The
solution was applied to explain the discrepancies that have been
observed in different tests of crushing strength according as the ends
of the test specimen are or are not prevented from spreading. It was
applied also to explain the fact that in such tests small conical
pieces are sometimes cut out at the ends subjected to pressure.
85. _Vibrations and Waves._--When a solid body is struck, or otherwise
suddenly disturbed, it is thrown into a state of vibration. There always
exist dissipative forces which tend to destroy the vibratory motion, one
cause of the subsidence of the motion being the communication of energy
to surrounding bodies. When these dissipative forces are disregarded, it
is found that an elastic solid body is capable of vibrating in such a
way that the motion of any particle is simple harmonic motion, all the
particles completing their oscillations in the same period and being at
any instant in the same phase, and the displacement of any selected one
in any particular direction bearing a definite ratio to the displacement
of an assigned one in an assigned direction. When a body is moving in
this way it is said to be _vibrating in a normal mode_. For example,
when a tightly stretched string of negligible flexural rigidity, such as
a violin string may be taken to be, is fixed at the ends, and vibrates
transversely in a normal mode, the displacements of all the particles
have the same direction, and their magnitudes are proportional at any
instant to the ordinates of a curve of sines. Every body possesses an
infinite number of normal modes of vibration, and the _frequencies_ (or
numbers of vibrations per second) that belong to the different modes
form a sequence of increasing numbers. For the string, above referred
to, the fundamental tone and the various overtones form an harmonic
scale, that is to say, the frequencies of the normal modes of vibration
are proportional to the integers 1, 2, 3, .... In all these modes except
the first the string vibrates as if it were divided into a number of
equal pieces, each having fixed ends; this number is in each case the
integer defining the frequency. In general the normal modes of vibration
of a body are distinguished one from another by the number and situation
of the surfaces (or other _loci_) at which some characteristic
displacement or traction vanishes. The problem of determining the normal
modes and frequencies of free vibration of a body of definite size,
shape and constitution, is a mathematical problem of a similar character
to the problem of determining the state of stress in the body when
subjected to given forces. The bodies which have been most studied are
strings and thin bars, membranes, thin plates and shells, including
bells, spheres and cylinders. Most of the results are of special
importance in their bearing upon the theory of sound.
86. The most complete success has attended the efforts of
mathematicians to solve the problem of free vibrations for an
isotropic sphere. It appears that the modes of vibration fall into two
classes: one characterized by the absence of a radial component of
displacement, and the other by the absence of a radial component of
rotation (S 14). In each class there is a doubly infinite number of
modes. The displacement in any mode is determined in terms of a single
spherical harmonic function, so that there are modes of each class
corresponding to spherical harmonics of every integral degree; and for
each degree there is an infinite number of modes, differing from one
another in the number and position of the concentric spherical
surfaces at which some characteristic displacement vanishes. The most
interesting modes are those in which the sphere becomes slightly
spheroidal, being alternately prolate and oblate during the course of
a vibration; for these vibrations tend to be set up in a spherical
planet by tide-generating forces. In a sphere of the size of the
earth, supposed to be incompressible and as rigid as steel, the period
of these vibrations is 66 minutes.
87. The theory of free vibrations has an important bearing upon the
question of the strength of structures subjected to sudden blows or
shocks. The stress and strain developed in a body by sudden applications
of force may exceed considerably those which would be produced by a
gradual application of the same forces. Hence there arises the general
question of _dynamical resistance_, or of the resistance of a body to
forces applied so quickly that the inertia of the body comes sensibly
into play. In regard to this question we have two chief theoretical
results. The first is that the strain produced by a force suddenly
applied may be as much as twice the statical strain, that is to say, as
the strain which would be produced by the same force when the body is
held in equilibrium under its action; the second is that the sudden
reversal of the force may produce a strain three times as great as the
statical strain. These results point to the importance of specially
strengthening the parts of any machine (e.g. screw propeller shafts)
which are subject to sudden applications or reversals of load. The
theoretical limits of twice, or three times, the statical strain are not
in general attained. For example, if a thin bar hanging vertically from
its upper end is suddenly loaded at its lower end with a weight equal to
its own weight, the greatest dynamical strain bears to the greatest
statical strain the ratio 1.63 : 1; when the attached weight is four
times the weight of the bar the ratio becomes 1.84 : 1. The method by
which the result just mentioned is reached has recently been applied to
the question of the breaking of winding ropes used in mines. It appeared
that, in order to bring the results into harmony with the observed
facts, the strain in the supports must be taken into account as well as
the strain in the rope (J. Perry, _Phil. Mag._, 1906 (vi.), vol. ii.).
88. The immediate effect of a blow or shock, locally applied to a body,
is the generation of a wave which travels through the body from the
locality first affected. The question of the propagation of waves
through an elastic solid body is historically of very great importance;
for the first really successful efforts to construct a theory of
elasticity (those of S.D. Poisson, A.L. Cauchy and G. Green) were
prompted, at least in part, by Fresnel's theory of the propagation of
light by transverse vibrations. For many years the luminiferous medium
was identified with the isotropic solid of the theory of elasticity.
Poisson showed that a disturbance communicated to the body gives rise to
two waves which are propagated through it with different velocities; and
Sir G.G. Stokes afterwards showed that the quicker wave is a wave of
irrotational dilatation, and the slower wave is a wave of rotational
distortion accompanied by no change of volume. The velocities of the two
waves in a solid of density [rho] are [root]{([lambda] + 2[mu])/[rho]}
and [root]([mu]/[rho]), [lambda] and [mu] being the constants so denoted
in S 26. When the surface of the body is free from traction, the waves
on reaching the surface are reflected; and thus after a little time the
body would, if there were no dissipative forces, be in a very complex
state of motion due to multitudes of waves passing to and fro through
it. This state can be expressed as a state of vibration, in which the
motions belonging to the various normal modes (S 85) are superposed,
each with an appropriate amplitude and phase. The waves of dilatation
and distortion do not, however, give rise to different modes of
vibration, as was at one time supposed, but any mode of vibration in
general involves both dilatation and rotation. There are exceptional
results for solids of revolution; such solids possess normal modes of
vibration which involve no dilatation. The existence of a boundary to
the solid body has another effect, besides reflexion, upon the
propagation of waves. Lord Rayleigh has shown that any disturbance
originating at the surface gives rise to waves which travel away over
the surface as well as to waves which travel through the interior; and
any internal disturbance, on reaching the surface, also gives rise to
such superficial waves. The velocity of the superficial waves is a
little less than that of the waves of distortion: 0.9554
[root]([mu]/[rho]) when the material is incompressible
0.9194[root]([mu]/[rho]) when the Poisson's ratio belonging to the
material is 1/4.
89. These results have an application to the propagation of earthquake
shocks (see also EARTHQUAKE). An internal disturbance should, if the
earth can be regarded as solid, give rise to three wave-motions: two
propagated through the interior of the earth with different velocities,
and a third propagated over the surface. The results of seismographic
observations have independently led to the recognition of three phases
of the recorded vibrations: a set of "preliminary tremors" which are
received at different stations at such times as to show that they are
transmitted directly through the interior of the earth with a velocity
of about 10 km. per second, a second set of preliminary tremors which
are received at different stations at such times as to show that they
are transmitted directly through the earth with a velocity of about 5
km. per second, and a "main shock," or set of large vibrations, which
becomes sensible at different stations at such times as to show that a
wave is transmitted over the surface of the earth with a velocity of
about 3 km. per second. These results can be interpreted if we assume
that the earth is a solid body the greater part of which is practically
homogeneous, with high values for the rigidity and the resistance to
compression, while the superficial portions have lower values for these
quantities. The rigidity of the central portion would be about
(1.4)10^12 dynes per square cm., which is considerably greater than that
of steel, and the resistance to compression would be about (3.8)10^12
dynes per square cm. which is much greater than that of any known
material. The high value of the resistance to compression is not
surprising when account is taken of the great pressures, due to
gravitation, which must exist in the interior of the earth. The high
value of the rigidity can be regarded as a confirmation of Lord Kelvin's
estimate founded on tidal observations (S 83).
90. _Strain produced by Heat._--The mathematical theory of elasticity as
at present developed takes no account of the strain which is produced in
a body by unequal heating. It appears to be impossible in the present
state of knowledge to form as in S 39 a system of differential equations
to determine both the stress and the temperature at any point of a solid
body the temperature of which is liable to variation. In the cases of
isothermal and adiabatic changes, that is to say, when the body is
slowly strained without variation of temperature, and also when the
changes are effected so rapidly that there is no gain or loss of heat by
any element, the internal energy of the body is sufficiently expressed
by the strain-energy-function (SS 27, 30). Thus states of equilibrium
and of rapid vibration can be determined by the theory that has been
explained above. In regard to thermal effects we can obtain some
indications from general thermodynamic theory. The following passages
extracted from the article "Elasticity" contributed to the 9th edition
of the _Encyclopaedia Britannica_ by Sir W. Thomson (Lord Kelvin)
illustrate the nature of these indications:--"From thermodynamic theory
it is concluded that cold is produced whenever a solid is strained by
opposing, and heat when it is strained by yielding to, any elastic force
of its own, the strength of which would diminish if the temperature were
raised; but that, on the contrary, heat is produced when a solid is
strained against, and cold when it is strained by yielding to, any
elastic force of its own, the strength of which would increase if the
temperature were raised. When the strain is a condensation or
dilatation, uniform in all directions, a fluid may be included in the
statement. Hence the following propositions:--
"(1) A cubical compression of any elastic fluid or solid in an ordinary
condition causes an evolution of heat; but, on the contrary, a cubical
compression produces cold in any substance, solid or fluid, in such an
abnormal state that it would contract if heated while kept under
constant pressure. Water below its temperature (3.9 deg. Cent.) of
maximum density is a familiar instance.
"(2) If a wire already twisted be suddenly twisted further, always,
however, within its limits of elasticity, cold will be produced; and if
it be allowed suddenly to untwist, heat will be evolved from itself
(besides heat generated externally by any work allowed to be wasted,
which it does in untwisting). It is assumed that the torsional rigidity
of the wire is diminished by an elevation of temperature, as the writer
of this article had found it to be for copper, iron, platinum and other
metals.
"(3) A spiral spring suddenly drawn out will become lower in
temperature, and will rise in temperature when suddenly allowed to draw
in. [This result has been experimentally verified by Joule
('Thermodynamic Properties of Solids,' _Phil. Trans._, 1858) and the
amount of the effect found to agree with that calculated, according to
the preceding thermodynamic theory, from the amount of the weakening of
the spring which he found by experiment.]
"(4) A bar or rod or wire of any substance with or without a weight hung
on it, or experiencing any degree of end thrust, to begin with, becomes
cooled if suddenly elongated by end pull or by diminution of end thrust,
and warmed if suddenly shortened by end thrust or by diminution of end
pull; except abnormal cases in which with constant end pull or end
thrust elevation of temperature produces shortening; in every such case
pull or diminished thrust produces elevation of temperature, thrust or
diminished pull lowering of temperature.
"(5) An india-rubber band suddenly drawn out (within its limits of
elasticity) becomes warmer; and when allowed to contract, it becomes
colder. Any one may easily verify this curious property by placing an
india-rubber band in slight contact with the edges of the lips, then
suddenly extending it--it becomes very perceptibly warmer: hold it for
some time stretched nearly to breaking, and then suddenly allow it to
shrink--it becomes quite startlingly colder, the cooling effect being
sensible not merely to the lips but to the fingers holding the band. The
first published statement of this curious observation is due to J. Gough
(_Mem. Lit. Phil. Soc. Manchester_, 2nd series, vol. i. p. 288), quoted
by Joule in his paper on 'Thermodynamic Properties of Solids' (cited
above). The thermodynamic conclusion from it is that an india-rubber
band, stretched by a constant weight of sufficient amount hung on it,
must, when heated, pull up the weight, and, when cooled, allow the
weight to descend: this Gough, independently of thermodynamic theory,
had found to be actually the case. The experiment any one can make with
the greatest ease by hanging a few pounds weight on a common
india-rubber band, and taking a red-hot coal in a pair of tongs, or a
red-hot poker, and moving it up and down close to the band. The way in
which the weight rises when the red-hot body is near, and falls when it
is removed, is quite startling. Joule experimented on the amount of
shrinking per degree of elevation of temperature, with different weights
hung on a band of vulcanized india-rubber, and found that they closely
agreed with the amounts calculated by Thomson's theory from the heating
effects of pull, and cooling effects of ceasing to pull, which he had
observed in the same piece of india-rubber."
91. _Initial Stress._--It has been pointed out above (S 20) that the
"unstressed" state, which serves as a zero of reckoning for strains and
stresses is never actually attained, although the strain (measured from
this state), which exists in a body to be subjected to experiment, may
be very slight. This is the case when the "initial stress," or the
stress existing before the experiment, is small in comparison with the
stress developed during the experiment, and the limit of linear
elasticity (S 32) is not exceeded. The existence of initial stress has
been correlated above with the existence of body forces such as the
force of gravity, but it is not necessarily dependent upon such forces.
A sheet of metal rolled into a cylinder, and soldered to maintain the
tubular shape, must be in a state of considerable initial stress quite
apart from the action of gravity. Initial stress is utilized in many
manufacturing processes, as, for example, in the construction of
ordnance, referred to in S 79, in the winding of golf balls by means of
india-rubber in a state of high tension (see the report of the case _The
Haskell Golf Ball Company_ v. _Hutchinson & Main_ in _The Times_ of
March 1, 1906). In the case of a body of ordinary dimensions it is such
internal stress as this which is especially meant by the phrase
"initial stress." Such a body, when in such a state of internal stress,
is sometimes described as "self-strained." It would be better described
as "self-stressed." The somewhat anomalous behaviour of cast iron has
been supposed to be due to the existence within the metal of initial
stress. As the metal cools, the outer layers cool more rapidly than the
inner, and thus the state of initial stress is produced. When cast iron
is tested for tensile strength, it shows at first no sensible range
either of perfect elasticity or of linear elasticity; but after it has
been loaded and unloaded several times its behaviour begins to be more
nearly like that of wrought iron or steel. The first tests probably
diminish the initial stress.
92. From a mathematical point of view the existence of initial stress
in a body which is "self-stressed" arises from the fact that the
equations of equilibrium of a body free from body forces or surface
tractions, viz. the equations of the type
dPX_x dPX_y dPZ_x
----- + ----- + ----- = 0,
dPx dPy dPz
possess solutions which differ from zero. If, in fact, [phi]1, [phi]2,
[phi]3 denote any arbitrary functions of x, y, z, the equations are
satisfied by putting
dP^2[phi]3 dP^2[phi]2 dP^2[phi]1
X_x = ---------- + ----------, ..., Y_z = - ----------, ...;
dPy^2 dPz dPydPz
and it is clear that the functions [phi]1, [phi]2, [phi]3 can be
adjusted in an infinite number of ways so that the bounding surface of
the body may be free from traction.
93. Initial stress due to body forces becomes most important in the case
of a gravitating planet. Within the earth the stress that arises from
the mutual gravitation of the parts is very great. If we assumed the
earth to be an elastic solid body with moduluses of elasticity no
greater than those of steel, the strain (measured from the unstressed
state) which would correspond to the stress would be much too great to
be calculated by the ordinary methods of the theory of elasticity (S
75). We require therefore some other method of taking account of the
initial stress. In many investigations, for example those of Lord Kelvin
and Sir G.H. Darwin referred to in S 83, the difficulty is turned by
assuming that the material may be treated as practically incompressible;
but such investigations are to some extent incomplete, so long as the
corrections due to a finite, even though high, resistance to compression
remain unknown. In other investigations, such as those relating to the
propagation of earthquake shocks and to gravitational instability, the
possibility of compression is an essential element of the problem. By
gravitational instability is meant the tendency of gravitating matter to
condense into nuclei when slightly disturbed from a state of uniform
diffusion; this tendency has been shown by J.H. Jeans (_Phil. Trans_. A.
201, 1903) to have exerted an important influence upon the course of
evolution of the solar system. For the treatment of such questions Lord
Rayleigh (_Proc. R. Soc. London_, A. 77, 1906) has advocated a method
which amounts to assuming that the initial stress is hydrostatic
pressure, and that the actual state of stress is to be obtained by
superposing upon this initial stress a stress related to the state of
strain (measured from the initial state) by the same formulae as hold
for an elastic solid body free from initial stress. The development of
this method is likely to lead to results of great interest.
AUTHORITIES.--In regard to the analysis requisite to prove the results
set forth above, reference may be made to A.E.H. Love, _Treatise on
the Mathematical Theory of Elasticity_ (2nd ed., Cambridge, 1906),
where citations of the original authorities will also be found. The
following treatises may be mentioned: Navier, _Resume des lecons sur
l'application de la mecanique_ (3rd ed., with notes by Saint-Venant,
Paris, 1864); G. Lame, _Lecons sur la theorie mathematique de
l'elasticite des corps solides_ (Paris, 1852); A. Clebsch, _Theorie
der Elasticitat fester Korper_ (Leipzig, 1862; French translation with
notes by Saint-Venant, Paris, 1883); F. Neumann, _Vorlesungen uber die
Theorie der Elasticitat_ (Leipzig, 1885); Thomson and Tait, _Natural
Philosophy_ (Cambridge, 1879, 1883); Todhunter and Pearson, _History
of the Elasticity and Strength of Materials_ (Cambridge, 1886-1893).
The article "Elasticity" by Sir W. Thomson (Lord Kelvin) in 9th ed. of
_Encyc. Brit_. (reprinted in his _Mathematical and Physical Papers_,
iii., Cambridge, 1890) is especially valuable, not only for the
exposition of the theory and its practical applications, but also for
the tables of physical constants which are there given.
(A. E. H. L.)