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