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doubly refracting power communicated to glass and other elastic solids by com-
pression, and the pressure which produces it^^" ; but the phenomena of bent glass
seem to prove, that, in homogeneous singly-refracting substances exposed to
pressures, the principal axes of pressure coincide with the principal axes of
double refraction ; and that the diflference of pressures in any two axes is
proportional to the difference of the velocities of the oppositely polarised rays
whose directions are parallel to the third axis. On this principle I have
calculated the phenomena seen by polarised light in the cases where the solid
is bounded by parallel planes.

In the following pages I have endeavoured to apply a theory identical
with that of Stokes to the solution of problems which have been selected on
account of the possibility of fulfilling the conditions. I have not attempted to
extend the theory to the case of imperfectly elastic bodies, or to the laws of
permanent bending and breaking. The solids here considered are supposed not
to be compressed beyond the limits of perfect elasticity.

The equations employed in the transformation of co-ordinates may be found
in Gregory's Solid Geometry.

I have denoted the displacements by Zx, By, Bz. They are generally denoted
by a, /8, y ; but as I had employed these letters to denote the principal axes
at any point, and as this had been done throughout the paper, I did not alter
a notation which to me appears natural and intelligible.

The laws of elasticity express the relation between the changes of the
dimensions of a body and the forces which produce them.

These forces are called Pressures, and their effects Compressions. Pressures
are estimated in pounds on the square inch, and compressions in fractions of the
dimensions compressed.

Let the position of material points in space be expressed by their co-ordinates
X, y, and z, then any change in a system of such points is expressed by giving
to these co-ordinates the variations Bx, By, Bz, these variations being functions of

X, y, 2.

* See note C.

THE EQUILIBRIUM OF ELASTIC SOLIDS. 35

The quantities Sx, Sy, 8z, represent the absolute motion of each point in
the directions of the three co-ordinates ; but as compression depends not on
absolute, but on relative displacement, we have to consider only the nine
quantities —

dSx

dSx

dhx

dx '

dy'

dz'

dSy
dx '

dhy
dy'

dSij

dz '

dSz
dx'

dhz

dy-

dBz

dz '

Since the number of these quantities is nine, if nine other independent
quantities of the same kind can be found, the one set may be found in terms
of the other. The quantities which we shall assume for this purpose are—

1. Three compressions, — , —■ , — , in the directions of three principal

a Id y

axes a, yS, y.

2. The nine direction-codnes of these axes, with the six connecting equa-
tions, leaving three independent quantities. (See Gregory's Solid Geometry.)

3. The small angles of rotation of this system of axes about the axes of
x, y, z.

The cosines of the angles which the axes of x, y, z make with those of
a, ^, y are

cos(aOa-)=aj, cos {^Ox) = \, co%(yQ)x) = c,,
cos (aOy) = tto, _cos {fiOy) = h„, cos (yO^/) = c.,
cos (aOz) =a3, cos (/SOz) =63, cos {yOz) = c,.

These direction-cosines are connected by the six equations,
a^ + h{ + Ci' = 1 , «i«s + ^h + CjC, = 0,

a./ -I- h^ + c,' = 1 , a^a^ + h.h^ + cx^ = 0,

a; + 63' + Gj' = 1 , a/t, + bj), + c^c, = 0.

The rotation of the system of axes a, 13, y, round the axis of
x, from y to z, =B0^,
y, from z to x, =S^j,
z, from x to y, =^0/,

36

THE EQUILIBRIUM OF ELASTIC SOLIDS.

By resolving the displacements 8a, h/S, By, B6„ B9.„ Z6„ in the directions
of the axes x, y, z, the displacements in these axes are found to be
hx = a,8a + h,Bp + c3y -Be^ + Bd,y,
By = aM + h,Bl3 -f c,By - Bd,x + Bd.z,
Bz = a,Ba + hM + CsBy - BO^ + Bd,x.
Sa .^ ^Si8

But

B^^rf, and 8y = y^,

and Q. = a^x + a^ + a.^, /3 = b,x + h^ + h.^, and y = c,x + c,y -h c^z.

Substituting these values of Sa, Sy8, and By in the expressions for Bx, By,
Bz, and differentiating with respect to x, y, and z, in each equation, we obtain
the equations

dBx Ba, ,. 8/8,2 , ^y

dy a ^ y

dBz _ Ba
dz

a p y

(1)-

dBx Ba B^ T J By ,5s/,

dy a ' ^ y

a

dBx Ba
dz a

Ba

BI3

J'

8^

dz a p y

dBy Ba BB T ^ By

dx a p y

Be,

c.f^ + Bdi

Be,

-J— = — ctjCti + -^ 6361 + -^ C3C1 + 8^2

dZz
dx

dBz

8^
a

Sa

8^
S/8

r

Be,

Equations of
compression.

{2).

Equations of the equilibnum of an element of the solid.
The forces which may act on a particle of the solid are : —

1. Three attractions in the direction of the axes, represented by X, Y, Z.

2. Six pressures on the six faces.

THE EQUILIBRIUM OF ELASTIC SOLIDS.

37

3. Two tangential actions on each face.

Let the six faces of the small parallelopiped be denoted by x^, 3/,, z„ x^ y„
and z,, then the forces acting on x^ are : —

1. A normal pressure jp, acting in the direction of x on the area dydz,

2. A tangential force g, acting in the direction of y on the same area.

3. A tangential force q^ acting in the direction of z on the same area,
and so on for the other five faces, thus : —

Forces which act in the direction of the axes of

a; 2/ z

On the face a:,

— 'p^dydz

- q^dydz

-q.'dydz

^.

{P^'r J^dx)dydz

(^3 + 7^ ^^) c?yc?x

(q.'+-^^dx)dydz

2/1

— q^dzdx

—p^dzdx

— q.dzdx

y-x

{q\ + ^dy)dzdx

{p.+^dy)dzdx

(q, + ^dy)dzdx

Zi

— q^dxdy

— q^dxdy

—p^dxdy

^2

fe+ -4^dz)dxdy

(q^+^dz)dxdy

(p. + ^dz)dxdy

Attractions,

pXdxdydz

p Ydxdydz

pZdxdydz

Taking the moments of these forces round the axes of the particle, we find

?i' = ?i, q^=q.^ qz=qz',

and then equating the forces in the directions of the three axes, and dividing
by dx, dy, dz, we find the equations of pressures,

dy dz dx ^
dz dx dy '^

Equations of Pressures.

(3).

38

THE EQUILIBRIUM OF ELASTIC SOLIDS.

The resistance which the sohd opposes to these pressures is called Elasticity,
and is of two kinds, for it opposes either change of volume or change of Jigure.
These two kinds of elasticity have no necessary connection, for they are possessed
in very different ratios by different substances. Thus jelly has a cubical elas-
ticity little different from that of water, and a linear elasticity as small as we
please ; while cork, whose cubical elasticity is very small, has a much greater
Imear elasticity than jelly.

Hooke discovered that the elastic forces are proportional to the changes
that excite them, or as he expressed it, " Ut tensio sic vLs."

To fix our ideas, let us suppose the compressed body to be a parallelepiped,
and let pressures Pi, Pj, P3 act on its faces in the direction of the axes
a> A y, which will become the principal axes of compression, and the com-

pressions will be

So. 8^ Sy
a' ^' y

The fundamental assumption from which the following equations are deduced
is an extension of Hooke's law, and consists of two parts.

I. The sum of the compressions is proportional to the sum of the pressures.

II. The difference of the compressions is proportional to the difference of
the pressures.

These laws are expressed by the following equations

I. (P. + P, + P.) = 3,(^ + f + ^

(4).

II.

(P,-P,) = m

(P._p.) = „,g_^

(P.-P,) = m

rv ^rts T Equations of Elasticity.

h
7

By Ba

(5).

The quantity fj. is the coefiicient of cubical elasticity, and m that of linear
elasticity.

THE EQUILrBRIUM OF ELASTIC SOLmS.

39

By solving these equations, the values of the pressures P„ P,, P„ and the

8a 8^ Sy , r J

compressions — ' ~S ' ^^7 ^^ found.

a \9/x 3m/ ^ ^ m

! = (!_ M(p. + P, + p.) + lp,

j3 \9/x 3 m/ ^ * ^ ?7i '

?r = (_L_ i\(P_+P_+P_) + ip_

y \9/z 3m/ ^ ^ m

(6).

(7).

From these values of the pressures in the axes a, )8, y, may be obtained..
the equations for the axes x, y, z, by resolutions of pressures and compressions*.

For

and

q = aaP^ + hhP, + ccP, ;
, . . IdZx , d%y , d8z\ . d8x'

, . V IdZx . d8y , d8z\ dBy

, , , fdSx , d8y , rfSj\ , dSz

m /c?Sz c?Sx

(8)-

2 Vo?a; c?2

.(9).

See the Memoir of Lame and Clapeyron, and note A.

40

THE EQUIUBRIUM OP ELASTIC SOLIDS.

d\$X /I 1 \ , , , N , 1

(10).

dy * ax ' m^
dz dy m ^

d^
dx

dz m^

(11).

By substituting in Equations (3) the values of the forces given in Equa-
tions (8) and (9), they become

(12).

These are the general equations of elasticity, and are identical with those
of M. Cauchy, in his Exercices d' Analyse, Vol. ni., p. 180, published in 1828,

where h stands for m, and K for ft - o" > and those of Mr Stokes, given in the

Cambridge Philosophical Transactions, Vol. viii., part 3, and numbered (30);

in his equations ^ = 3/x, B = — .

If the temperature is variable from one part to another of the elastic

soHd, the compressions -y- , -r^, -J^ , at any point will be diminished by a

quantity proportional to the temperature at that point. This prmciple is applied
in Cases X. and XI. Equations (10) then become

THE EQUILIBRIUM OF ELASTIC SOLIDS.

41

dy

^ = fe - 3mj (P^-^P^+P^) + '^^^^P^

(13).

CfV being the linear expansion for the temperature v.

Having found the general equations of the equilibrium of elastic solids, I
proceed to work some examples of their application, which afford the means of
determining the coefficients /t, m, and o), and of calculating the stiffness of
solid figures. I begin with those cases in which the elastic soHd is a hollow
cylinder exposed to given forces on the two concentric cylindric surfaces, and
the two parallel terminating planes.

In these cases the co-ordinates x, y, z are replaced by the co-ordinates
x = x, measured along the axis of the cylinder.
2/ = r, the radius of any point, or the distance from the axis.
z — rd, the arc of a circle measured from a fixed plane passing
through the axis.

Px = o, are the compression and pressure in the direction of the
axis at any point.

-^ = -J— , Pi =p, are the compression and pressure in the direction of the

dBz dhrd Br . . _ . , ,. . - 1

~dz~'db¥~l^' JP8 = ?, are the compression and pressure m the direction of the

tangent.

Equations (9) become, when expressed in terms of these co-ordinates —

m doO

dZx
dx

dSx
dx

m dB0

m dSx
dr

*=2

.(14).

The length of the cylinder is h, and the two radii a, and a, in every
VOL. I. G

42 THE EQUIUBRnJM OF ELASTIC SOLIDS.

Case I.

The first equation is applicable to the case of a hollow cylinder, of which
the outer surface is fixed, while the inner surface is made to turn through
a small angle Bd, by a couple whose moment is M.

The twisting force M is resisted only by the elasticity of the solid, and
therefore the whole resistance, in every concentric cylindric surface, must be equal
to M.

The resistance at any point, multiplied into the radius at which it acts, is
expressed by

m „ dhd

Therefore for the whole cylindric surface

ar
Whence 8,=_^^ (1,_1.) ,

^^ "' = 2^&-i) ('«>■

The optical effect of the pressure of any point is expressed by

I=<oq,b = <o.^^ (15).

Therefore, if the solid be viewed by polarized light (transmitted parallel to
the axis), the difference of retardation of the oppositely polarized rays at any
point in the solid will be inversely proportional to the square of the distance fi-om
the axis of the cylinder, and the planes of polarization of these lays will be
inclined 45" to the radius at that point.

The general appearance is therefore a system of coloured rings arranged
oppositely to the rings in uniaxal crystals, the tints ascending in the scale as
they approach the centre, and the distance between the rings decreasing towards
the centre. The whole system is crossed by two dark bands inclined 45* to the
plane of primitive polarization, when the plane of the analysing plate is perpen-
dicular to that of the first polarizing plate.

THE EQUILIBRIUM OF ELASTIC SOLIDS. 43

A jelly of isinglass poured when hot between two concentric cylinders forms,
when cold, a convenient solid for this experiment ; and the diameters of the rings
may be varied at pleasure by changing the force of torsion appUed to the interior
cylinder.

By continuing the force of torsion while the jeUy is allowed to dry, a hard
plate of isinglass is obtained, which still acts in the same way on polarized light,
even when the force of torsion is removed.

It seems that this action cannot be accounted for by supposing the interior
parts kept in a state of constraint by the exterior parts, as in, unannealed and
heated gla^s ; for the optical properties of the plate of isinglass are such as
would indicate a strain preserving in every part of the plate the direction of
the original strain, so that the strain on one part of the plate cannot be main-
tained by an opposite strain on another part.

Two other uncrystallised substances have the power of retaining the polariz-
ing structure developed by compression. The first is a mixture of wax and resin
pressed into a thin plate between two plates of glass, as described by Sir David
Brewster, in the Philosophical TransoLctions for 1815 and 1830.

When a compressed plate of this substance is examined with polarized light,
it is observed to have no action on light at a perpendicular incidence ; but when
inclined, it shews the segments of coloured rings. This property does not belong
to the plate as a whole, but is possessed by every part of it. It is therefore
similar to a plate cut from a uniaxal crystal perpendicular to the axis.

I find that its action on light is like that of a jpositive crystal, while that
of a plate of isinglass similarly treated would be negative.

The other substance which possesses similar properties is gutta percha. This
substance in its ordinary state, when cold, is not transparent even in thin films;
but if a thin film be drawn out gradually, it may be extended to more than
double its length. It then possesses a powerful double refraction, which it
retains so strongly that it has been used for polarizing light""'. As one of its
refractive indices is nearly the same as that of Canada balsam, while the other
is very different, the common surface of the gutta percha and Canada balsam
will transmit one set of rays much more readdy than the other, so that a film
of extended gutta percha placed between two layers of Canada balsam acts like

* By Dr Wright, I believe.

44 THE EQUILIBRIUM OF ELASTIC SOLIDS.

a plate of nitre treated in the same way. That these films are in a state of
constraint may be proved by heating them slightly, when they recover their
original dimensions.

As all these permanently compressed substances have passed their limit of
perfect elasticity, they do not belong to the class of elastic solids treated of in
this paper ; and as I cannot explain the method by which an imcrystallised body
maintains itself in a state of constraint, I go on to the next case of twisting,
which has more practical importance than any other. This is the case of a
cylinder fixed at one end, and twisted at the other by a couple whose moment
is M.

Case II.

In this case let hB be the angle of torsion at any point, then the resistance
to torsion in any circular section of the cylinder is equal to the twisting force M,

The resistance at any point in the circular section is given by the second
Equation of (14).

?2 = 1^^

dx '

This force acts at the distance r from the axis ; therefore its resistance to torsion
will be q.r, and the resistance in a circular annulus will be

q^r^Ttrdr = mirr' -r- dr

and the whole resistance for the hollow cylinder will be expressed by

„, mn dS6 , ^ ,. /,^v

720 M

^(-1-] (17).

In this equation, m is the coefl&cient of linear elasticity; a^ and a^ are the
radii of the exterior and interior surfaces of the hollow cyUnder in inches ; M is
the moment of torsion produced by a weight acting on a lever, and is expressed

THE EQUILIBRIUM OF ELASTIC SOLIDS. 45

bj the product of the number of pounds in the weight into the number of inches
in the lever; b is the distance of two points on the cylinder whose angular
motion is measured by means of indices, or more accurately by small mirrors
attached to the cylinder ; n is the difference of the angle of rotation of the two
indices in degrees.

This is the most accurate method for the determination of m independently
of /x, and it seems to answer best with thick cylinders which cannot be used
with the balance of torsion, as the oscillations are too short, and produce a
vibration of the whole apparatus.

Case III.

A hollow cylinder exposed to normal pressures only. When the pressures
parallel to the axis, radius, and tangent are substituted for p^, p^, and pt,
Equations (10) become

S = (i-34)(^+^-^^) + ^ (^«)-

^^t^(±-±]io+p + q) + :^q (20).

By multiplying Equation (20) by r, differentiating with respect to r, and

comparing this value of —j— with that of Equation (19),

p-q _(J__ _1\ /^ . ^ . ^\ _ i ^
rm " \9/x 3m/ \dr dr drj m dr '

The equation of the equilibrium of an element of the solid is obtained by
considering the forces which act on it in the direction of the radius. By
equating the forces which press it outwards with those pressing it rnwarde, we
find the equation of the equiHbrium of the element,

ir£ = 4 (21).

r dr

46 THE EQUILIBRIUM OF ELASTIC SOLIDS.

By comparing this equation witli the last, we find

\9fi Zmj dr \9/i ^ 3m/ \dr ^ drj
Integrating,

Since o, the longitudinal pressure, is supposed constant, we may assume

c -(^-^]o
' \9u, 3m/ . , .

c. = 12 =(^ + g)-

9/x, 3 m
Therefore q—p = c^ — 2p, therefore by (21),

a linear equation, which gives

1 ^c,
^ = ^3^ + 2-

The coefficients Cj and Cj must be found from the conditions of the surface
of the soHd. If the pressure on the exterior cylindric surface whose radius is a,
be denoted by A,, and that on the interior surface whose radius is a^ by A,,

then p = h^ when r = ai
and p = h.j when r = a^
and the general value of p is

_a^h^ — a^\ a^a^ h^ — h^ /22\

^" a,' -a,' ^ oT^^ ^ ^'

2-i'=2i^ ^73^- ''y (21).

*= «.'-«.' +^^57::^' (^^^■

/=5<.(^-2)=-26<.^"A^. (24).

This last equation gives the optical eflfect of the pressure at any point. The
law of the magnitude of this quantity is the inverse square of the radius, as in

THE EQUILIBRIUM OF ELASTIC SOLIDS. 47

Case I. ; but the direction of the principal axes ia different, as in this case they
are parallel and perpendicular to the radius. The dark bands seen by polarized
Ught wiU therefore be parallel and perpendicular to the plane of polarisation, in-
stead of being inclined at an angle of 45", as in Case I.

By substituting in Equations (18) and (20), the values of p and q given in
(22) and (23), we find that when r = a,.

hx (l\( ^aX-ct'h-X . 2 / a,%-a,%\ ]

X \9/x

= o(^ + ~] + 2{Ka,^-Ka,^)

1/1 1

.(25).

,9/x 3m/ ' ^ ' ' ' 'Ui,'-a,'\9fj, 3mJ
r 9/x \ a/ — a/ / 3?

When r = a., - ^ ^ fo4-2 ^4-^) + ^^^ ( - ^._^. ' ' -o

(26).

~ VSft 3my "^ ' a; - a,' \ 9/x ^ 3m / ^ cv - a,' 1,9/x "^ 3m/ J

From these equations it appears that the longitudinal compression of cylin-
dric tubes is proportional to the longitudinal pressure referred to unit of surface
when the lateral pressures are constant, so that for a given pressure the com-
pression is inversely as the sectional area of the tube.

These equations may be simplified in the following cases : —

1. When the external and internal pressures are equal, or h^ = h^.

2. When the external pressure is to the internal pressure as the square of
tlie interior diameter is to that of the exterior diameter, or when a^-h^ = a^-h^.

3. When the cylinder is soHd, or when a. = 0.

4. When the solid becomes an indefinitely extended plate with a cylindric
hole in it, or when a^ becomes infinite.

5. When pressure is applied only at the plane surfaces of the solid cylinder,
and the cylindric surface is prevented from expanding by being inclosed in a

strong case, or when — = 0.

6. When pressure is applied to the cylindric surface, and the ends are
retained at an invariable distance, or when — = 0.

X

48

THE EQUILIBRIUM OF ELASTIC SOLIDS.

1. When ^ji = A„ the equations of compression become

\9fi'*"3mj"'"^ '\9ij. 3m

(27).

7 = i('>+2^) + 3i(^-<')

When hi = hi = o, then

Zx _hr _ \
X ~ r " Sfi'

The compression of a cylindrical vessel exposed on all sides to the same
hydrostatic pressure is therefore independent of m, and it may be shewn that
the same is true for a vessel of any shape.

2. When a,% = a^%

^ \9yx "^ 3m/

Bx

X

7 = |w + 3l(3^ - »)^

(28).

In this case, when o = 0, the compressions are independent of /x.
3. In a solid cylinder, aj = 0,

The expressions for — and — are the same as those in the first case, when
h^ — hf

When the lon^tudinal pressure o vanishes,

Bx

X

r ' \9/x 3m/ '

THE EQUILIBRIUM OF ELASTIC SOLIDS.

49

When the cylinder ia pressed on the plane sides only,

8x

r \9fi dmj

4. When the solid is infinite, or when a, is infinite,

p = K - ._a-(\-K)

I=<o{p-q)=-^a.;{h,-h,)
r 9/x ^ ' 3m ^ '

(29).

5. When 8r = in a solid cylinder,

Zx Zo

6. When

X 2m + 3/A

So; _ hr _ 2>h
x~ * r ~ m + 6iM

.(30).

Since the expression for the efiect of a longitudinal strain is
Bx

if we make

VOL. I.

-=o(— + —)
X \9/i, 3m/ '

r, 9mu, ^, 8x 1

E = ^ , then — = o ^^

m + 6/x cc E

(31).

50 THE EQUILIBRIUM OF ELASTIC SOLIDS.

The quantity E may be deduced from experiment on the extension of wires
or rods of the substance, and /x is given in terms of m and E by the equation,

„ = _^!!L_ (32),

^^^ ^ = S (^^)'

P being the extending force, h the length of the rod, s the sectional area,
and Bx the elongation, which may be determined by the deflection of a wire,
as in the apparatus of S' Gravesande, or by direct measurement.

Case IV.

The only known direct method of finding the compressibihty of liquids is
that employed by Canton, (Ersted, Perkins, Aime, &c.

The liquid is confined in a vessel with a narrow neck, then pressure is
applied, and the descent of the liquid in the tube is observed, so that the
difference between the change of volume of liquid and the change of internal
capacity of the vessel may be determined.

Now, since the substance of which the vessel is formed is compressible, a
change of the internal capacity is possible. If the pressure be applied only to
the contained liquid, it is evident that the vessel will be distended, and the
compressibihty of the liquid will appear too great. The pressure, therefore, is
commonly applied externally and internally at the same time, by means of a
hydrostatic pressure produced by water compressed either in a strong vessel or
in the depths of the sea.

As it does not necessarily follow, from the equality of the external and
internal pressures, that the capacity does not change, the equilibrium of the
vessel must be determined theoretically. (Ersted, therefore, obtained from Poisson
his solution of the problem, and applied it to the case of a vessel of lead.
To find the cubical elasticity of lead, he appUed the theory of Poisson to the
numerical results of Tredgold. As the compressibility of lead thus found was
greater than that of water, (Ersted expected that the apparent compressibility
of water in a lead vessel would be negative. On making the experiment the
apparent compressibihty was greater in lead than in glass. The quantity found

THE EQUILIBRrcrM OF ELASTIC SOLIDS. 51

by Tredgold from the extension of rods was that denoted by E, and the value
of ft deduced from E alone by the formulae of Poisson cannot be true, unless

— = |-; and as — for lead is probably more than 3, the calculated compressi-
bility is much too great.

A similar experiment was made by Professor Forbes, who used a vessel of
caoutchouc. As in this case the apparent compressibility vanishes, it appears
that the cubical compressibihty of caoutchouc is equal to that of water.

Some who reject the mathematical theories as unsatisfactory, have conjec-
tured that if the sides of the vessel be sufficiently thin, the pressure on both
sides being equal, the compressibility of the vessel will not affect the result.

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