4UG 29 l924
Thesis presented to the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
In partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
by
WILLIAM PHELPS ALLIS
Prepared under the direction of
PROFESSOfl PAUL HEYXjAXS
Signature redacted
,Signature redacted
THE PHOTOELASTIC PROPERTIES
OF
CELLULOID, GLASS, AND FUSED QUARTZ
1 40273
ri
PREFACE
-
In order to become a candidate for the degree
of Master of Science it is necessary to have the degree
of Bachelor of Science.
As I had not yet presented a
thesis in completion of the requirements for the latter
degree I presented that part of this work relating to
the production of uniform tension in glass (pp. 5 to 27).
It is again included here for the sake of completeness,
but should not be considered as counting towards the
Master's degree.
I wish to express here my thanks to the
Massachusetts Institute of Technology for its liberality
in supplying all the material that I needed for this
work.
I also wish to thank Professor Heymans and
Messfrs T.Frost and W.Dietz for their many usefull suggestions and kind help.
Signature redacted.
Nancy, France
December 5, 1923.
A
INTRODUCTION
When an isotropic transparent body is stressed
unequally in different directions it becomes doubly
refracting.
Let p, q, and r be the principal stresses,
and let us consider two rays of light travelling along
r, and polarized, respectively, in the planes r,p and
r,q.
These rays will travell at different speeds, the
differencebeing proportional to p - q.
After going
through a given thickness of material one of these rays
will have fallen behind the other, and the relative
retardation will be proportional to p - q.
Conversely
we can obtain the difference p - q of the principal
stresses in the material from a measurement of the relative retardation of two rays such as those described
above.
I will not go into the details of photoelasti-
cimetry, it will suffice to say that models of individual pieces of ma.chineryor of complete structures, are
made out of transparent materials, and then loaded as
2
the originals would be loaded in practice.
Under suit-
able conditions the stresses in the models are proportional to those in the original.
By studying the dou-
ble refraction produced in these models valuable infor-
t
mation is obtained as to the distribution of stresses
in the originals.
The success of the method depends to
a great extent upon the choice of the proper material
out of which to make the models.
The investigations
presented in this thesis were originally undertaken, at
the suggestion of Professor Paul Heymans, to measure
those properties of fused quartz which relate to its
use in photoelasticime ry.
Fused quartz is now being
produced sufficiently homogenious and free from internal
strains by the General Electric Co.at Lynn, Mass.
As
it is at present, owing to the small demand, very
expensive, all the tests were first made on plate glass,
which I obtained from the Standard Plate Glass Co.
In
order to avoid unnecessary breaking of glass, test
pieces were cut out of camphor celluloid, produced by
the Dupont Co., and used in setting up the apparatus
I
and in making the first trial tests.
While the work was in progress a new kind of
celluloid, deposited from a solution in naphtha, was
obtained from Paris, and it seemed desirable to test it
also.
The tests were accordingly made on all four sub-
stances, and this thesis is now presented as a comparison of their respective adaptability to photoelasticimetry.
The stresses in the models can only be proportional to those in the originals if the deformations
are similar.
This requires that the materials obey
Hooke's law and that they have the same value for
Poisson's ratio.
The latterfortunately, does not vary
greatly from one material to anotherso that the error
introduced by this factor is usually less that the
accuracy of the.method.
Its value lies in the determi-
nation of p + q by lateral extensometer measurements.
I therefore measured Poisson's ratio for the materials
studied.
I also took the stress-strain curves in order
to determine Young's modulus and the limit of Hooke's
4
law.
The stress-optical coefficient is of course a
primary consideration, and the transparency is also
important.
The following pages contain a description
of the measurement of these properties.
5
TESTS UNDER TENSION
ATTEPT TO OBTAIN STRESS-STRAIN CURVE
The maik difficulty in obtaining the stressstrain curve of glass, or any other brittle material,
under tension, is in obtaining pure tension in a test
piece up to the breaking point.
Brittleness is due to
a very small strain at rupture;
only half a thousandth
in glass.
The yielding of materials when stressed
tends to render the distribution of the stresses more
uniform.
Thus, with a test piece in a tension machine,
the jaws of the machine at first make contact at only
a few points.
The material at these points yields, so
that the areas of contact widen, and the pressure
becomes more uniformly distributed.
Also, if the test
piece is not well centered it will bend,-and this bending will tend to render the distribution of the stress
in the shaft more uniform.
If the material of the test
6
piece is brittle two difficulties will be encountered:
First, the test piece will tend to break where the jaws
come in contact with the glass.
Second, it will be
found difficult to obtain uniform stress in the shaft.
Method of holding test piece
The ordinary types of holders are inapplicable to glass because they would crush it or because
they would require it to be threaded.
Attempts have
been made to hold glass test pieces by boring holes in
the ends and passing pins through these holes and
pulling on these pins.
would break at the hole.
In every case the test piece
The surface of fracture would
frequently have several times the area of the minimum
cross section of the shaft.
This is an indication that
glass does not deviate much from Hooke's law even up to
the breaking point.
of the holes.
High stresses arise at the surfaces
If there were any deviation from Hook6's
law the over-stressed material would yield, and in doing
so would pass some of the stress on to the material
7
behind it.
If there were enough material behind it the
stress would be withstood at this point and rupture
would occur in the shaft.
If there is no deviation
from Hooke's law the surface material will continue to
bear the greater part of the load, until it breaks down.
At the bottom of the crack thus formed there is at once
a very high stress, and the material there fails.
This
repeats itself, the crack cutting its wiay through whatever thickness of material there may happen to be.
The shape of test piece finally adopted is
shown in figures 1 and 2.
Figure 3 is a photograph of
a test piece in the holders, ready to be put in the
testing machine.
Pins pass through the holes in the
two holders and it is through these two pins that the
tension is applied.
The test pieces were ground by the
Pynkham and Smith Co. of Boston.
out of half inch steel plate.
The holders were cut
1/4 inch pressure tubing
slit down the side and slipped over the sides of the
test piece prevented direct contact between glass and
steel.
I found it necessary to have the test piece and
9
83Piece ini Holde
1/
m4
11
holders fit pretty closely.
If the angle of the sides
of the head of the test piece was but slightly less than
the corresponding angle in the holders, so that contact
came near the lower end of the head, the fracture would
be produced at the point of contact, as several of the
test pieces photographed in figure 12 show.
The angle
of the sides of the head might be slightly greater than
the corresponding angle of the holders without harm.
Use of double refraction in centering
The fracture of glass which has failed under
uniform tension has the appearance of finely ground
glass.
If the stress is not uniform the glass will
the stZesJ
fail first there where AM is greatest.
Usually the
stress distribution at this point will be close enough
to uniformity to give to this portion of the fracture
the frosted appearance.
But as soon as some of the
material has failed, and a crack has been formed, there
will be a concentration of the stress at the bottom of
the crack.
The high stress at the surface of the crack
I
12
falling off very rapidly towards the interior of the
material.
The glass at the bottom of the crack will
fail next, forming a deeper crack with high stress at
the bottom of it.
The glass here will fail, and so
forth, the crack cutting its way through the material
with a region of high stress always in front of it.
The glass is then failing in a region in which the
gradient of the stress is high;
and it will produce
the familiar "glassy" fracture.
The true tensile
strength of the glass is somewhere between the tension
divided by the cross section, and the tension divided
by that portion of the cross section which showgs the
frosted appearance.
To get the true tensile strength
the glass must be broken under pure tension and the
frosted appearance obtained over the whole fracture.
In-order to verify, before fracture, whether
the stress in the loest ri!eo wl
niirformI made use of
the property of glass to become double refracting,
stressed.
when
Figure 7 shows the apparatus ueed.
The light from an arc (a) was focused on a
13
nicol prism (d), which polarized it in a plane inclined
at 45Q to the horizontal.
it was then rendered parallel
and directed on the test piece (g).
(f) will be explaned later.
The mirror system
One of the colors of the
beam incident on the test piece can be represented by
the vector I, figure 4, vibrating in a plane perpendicular to the plane of polarization.
This vector can be
decomposed into its horizontal and vertical components
I
and lb.
When the test piece is under tension, the
direction of the stress being vertical, Ib will travel
slower through the glass than I. 'If we consider the
two rays corresponding to these two vectors at the same
will
instant, Iq will have gone farther than Ib, or lb have
suffered a relative retardation with respect to Ia..
If we consider conditions at the same point in the glass,
Ia and Ib will be out of phase.
In either way of
looking at it the phenomenon is proportional to .the
stress and to the distance travelled through the glass.
As Ia and Ib are out of phase at the farther surface of
the glass, their resultant is not, in general, a vector
14
15
vibrating in a plane, but one whose extremity describes
an elliptic helix (II,Fig. 4).
The beam of light issu-
ing from the test piece is elliptically polarized.
I
now focus it on a second nicol (i, Fig. 7) crossed with
the first.
through;
Only vibrations perpendicular to I can get
in this case represented by vector III.
The
length of this vector, that is the intensity of the
light that gets through, depends upon the eccentricity
of the ellipse II.
The ray incident on the test piece is in
rheality composed of a great number of vectors such as
I, each corresponding to a different color and therefore vibrating with a different frequency.
Upon pass-
ing through the glass their components such as Io and Ib
will have the same relative retardation, very nearly,
and therefore quite different phase differences.
The
phase difference qis related to the relative retarda__
lion p by the formula
4
rr
as can be readily verified.
X
The vectors representing
the different wave lengths will 6ppcribe ellipses such
16
as II, but each with a different eccentricity.
If the
light incident on the test piece is white, that issuing
from the analyzer-will appear colored, for some of the
wave lengths will have been reduced in intensity more
than others.
The color depends on the value of
p,
which in turn depends upon the stresseand on the thickness oi glass through which the beam has passed.
If a
screen is placed in the focal plane of the image of the
test piece, the image received on it will be colored.
If the stress is uniform the color will be uniform.
If
the stress is not uniform the color at each point of
the image will depend on the stresses at the points in
the test piece through which the ray reaching that
point of the image has passed.
The colors of the image
represent the stresses in the test piece integrated in
the direction of the beam of light.
An idea of the
distribution of stresses in the test piece can thus be
had from the colors of the image on the screen, (j), fig 7.
17
Procedure
The first three tests were made in an Amsler,
5000 pound, tension and compression machine.
It was
found that the distribution of the stress in the test
piece could be changed, while the load was on, by
moving a nut, similar in function to (e) fig.10, by
hitting it with a mallet.
As soon as the tension was
the nut was moved in such a direction as to make the
-
sufficient to produce colors in the image on the screen
color more uniform.
The first test piece used was not quite symmetrical (1, fig. 12).
tension with it.
I was unable to obtain uniform
The fracture had a small frosted area
barely visible to the unaided eye.
The next test piece was more nearly symmetrical, and I obtained uniform stress up to 500 pounds, or
3600 lbs./ina.
I was measuring the index of double
refraction with this load on when the test piece broke,
without warning.
The effect of time on the condition
under
of glass
this tension is, therefore, quite marked.
I--
4>
t
'a
b
19
The glass had been under tension fifteen to twenty
minutes when it
broke.
The fracture was plane, perpendicular to the
axis of the test piece, and with only some small chips
along the edges missing.
the frosted appearance.
Almost half the fracture had
This frosted area was symmet-
rical about the short axis of the cross section, the
direction of the beam of polarized light, but not about
the long axis.
This showed that the stress had been
symmetrical about the shirt axis, as had been observed
in the image on the screen.
It also showed that the
stress had not been symmetrical about the long axis.
This could not have been observed in the image for, as
was shown above, the colors depend on the integrated
stresses along the short axis.
In order to observe
bending about the long axis it would be necessary to
pass a beam of polarized light through the test piece
in the direction of the long axis of the cross section.
The test piece must be observed in two directions at
right angles simultaneously.
(%4
F
A
21..
SI
j
1 ~
1 ~ ~1
4
''t
Eq
|I
/
41
I,
Ir
9
21
I then designed several mirror systems which
would divide the beam of light issuihg from the polarif
zer into two, pass these through the test piece in two
directions at right angles, and then combine them again
to pass them together through the analyzer.
My first attempt is shown in figure 5.
It
was never used, as I thought of a better design before
finishing this one.
The main objection to it was that
the two images formed could not be focused on the same
screen.
The second design is shown in figure 6.
Mir*
rors (a) and (f) reflect half and transmit half the
light incident on them.
Mirrors (b), (c), (d), and (e)
are surface silvered, that is they reflect on the side
which is silvered.
This device was used, but was not
very satisfactory because there were too many mirrors
to get out of adjustment.
In the last design, which is shown in figures
7 and 8 together with the complete optical system used,
mirrors (b) and (d), and (c) and (e) of the previous
modelwre, respectively, combined into single mirrors.
*
Lo
li
.45
4:4
rrtTLs S
w
a
In)
n 77sr
i
44
LJ~adiP
24
Also, (a) and (f)were fully silvered over half their
surface.
This throw,
two semi-circular spots of light
ed
on ths screen which joixj along a common diameter,
instead of two circular spots of half the brightness, as
the previous model did.
Besides the increased bright-
ness, this produced a field more nearly the same shape
as the image, and enabled the images to be brought
closer together.
This made it easier to pass both
beams through the same analyzer.
This mirror system is
also shown in figure 9 with a test piece in place.
The first glass tested while being observed
with the aid of one of these mirror systems is shown in
(3), figure 12.
The nut (e)(Fig. 10) is free to move
in all directions in a horizontal plane within the
limits imposed upon rod (d) by the hole in plate (g).
A similar condition of affairs existed in the Amsler
machine, in which test piece (3)
was tested.
The
limits of the hole were reached before uniformity of
stress was attained.
Although the stress was visibly
not uniform I increased the tension until the test
N
A.
Tenr.ion
Mahine.
'S
NNW
to- Igjure
26
piece broke, at a tension of 910 pounds.
This would
have corresonded to a stress of 6300 lbs./inz had the
stress been uniform.
The test piece broke in several
pieces, one of the fractures having a considerable area
showing the frosted appearance.
This fracture had a
radial structure indicating the probable existance of
some twisting.
This form of stress cannot be observed
optically with the system used.
TQ 4etect it it would
be necessary to pass a beam of polarized light lengthwise through the test piece.
6300 lbs./in. is doubtless
far below the true tensile 6trength of the glass.
After this test I changed to the tension
machine shown in figures 8, 9, and 10;
which was
designed by Mr. Kimball of the General Llectric Co. to.
produce strictly uniforan tension.
Other reasons for
the change were that a device for moving nut (e) and a
new mirror system could be made iore eaaily for this
machine than for the Amaler.
The machine needed, be ore it
could be used,
the two chucks (c),figure 10, so 1 made these.
also
made the device shown in figures 10 and 11 for adjusting
Xx
Ic
0
b
c
Te-t Piece
14olier s,. edjst
C- AA Orass
Nt
Uprn
sc/-Cw
L ower
14e
t~ft,
r
-d
-5
W ir7"
-7
krN
1w?
4
~r
4-
r '
- me-
=
29
the tension.. By turning the screws kf), figure 10, nut
ie) in moved in whatever direction is desired.
This
displaces rod (d) roughly parallel to itself, carrying
with it chuck (c).
Holder kb/ turns slightly on pin
ki), changing the points of contact between ho.der ano.
pin,and pin and chuck.
There can be no couple in the
plane uf the holder acting on the test piece, for this
would produce rotation of the holder around the pin ki)
Nor can there be a couple acting on the test piece in
the vertical plane perpendicular to the holders, for
the test piece is not held firmly in the holders and is
free to turn in this plane.
There may be a couple in
the horizontal plane producing twisting, but if we neglect this, the forces acting on one end of the test
piece can be reduced to a simple force.
it will suf-
fice that this resultant force pass through the center
of the test piece in order that there be uniform stress
over the middle cross section.
By turning screws (f)
it was shown that the points of contact between chuck,
I
I
pin, and holder are changed.
This changes the point of
application of the resultant force, and it
can be
31
made to epproach the center of the test piece.
A ball
bearing between rod id) and chuck (c) would eliminatd
the possibility of twisting.
Until this time I had not been troubled by
failures occuring in the head, and had not realized the
importance of the shape of this part of the test piece.
The next four test pieces were defective in this respect,
the sides of their heads making angles smaller than
the corresponding angle in the holders.
7)p figure 12, show the result.
(4), (5), and
I tried holding test
piece (Y) by' casting fusible alloy around the heads.
This does Lot allow the test piece to move in the holders, and therefore bending in a plane perpendicular to
the holders is not eliminated.
The test piece broke as
shown. the fracture having no frosted area. at all.
Care was exercised in grinding the next test
pieces to give the sides of the heads the proper
slopes.
I could not obtain uniform tension up to the
breaking point, however, because, as in the Amsler
machine, the hole in plate (g) was not big enough.
.32
Time did not permit me to continue the tests,
but I believe that, by the method used, uniform tension
can be obtained up to the breaking point in a machine
providing for sufficient adjustment.
33
TESTS UNDER TENSION
TKE STRESS-OPTICAL COEFFICIENT
The prkperty upon which the photoelastic
method of stress analysis depends is the double refraction incident to stress.
The amount of this double
refraction is proportional to the stress, and for the
same atress differs in different substances.
The coef-
ficient of proportionality between the double refraction and the stress is known as the stress-optical
coefficient.
It has the dimensions of the reciprocal
of a stress, and is measured in square centimeters per
gram weight.
It is considered positive if, in a body
under tension, the ray polarised in the plane containing the direction of the stress is retarded relatively
to a ray polarised in the plane perpendicular to the
stress.
The stress-optical coefficient is usually
negative,
The stress-optical coefficient was measured
34
12
4
0
Babinet compensator
Figure 13
-
-
35
in test pieces that were used in the tension tests.
Referring back to figure 4, 1 represents a ray polarised
in a plane inclined at 45*.
It can be considered as
composed of two components I4 and 16 polarised, respecIn the
tively, in the vertical and horizontal planes.
test piece 14 travels faster than 16.
Therefore in the
ray issuibg from the test piece Ib is behind I..
The
dimensions of the test piece being known, it remains to
measure this relative retardation.
Babinet compensator.
This is done with a
This instrument, figures 13 and
14, consists of a plate P and a wedge W cut out of
crystals of quartz, a nicol prism N, and a lens L.
The
wedge can slide parallely to itself over the plate, its
motion being controlLed by a graduated screw.
The
wedge and plate are placed with their crystal axes at
right angles so that the double refraction of one is
partially compensated for in the other.
The wedge has
the same thickness as the plate at the middle, so here
the double refraction is completely compensated.
The
total relative retardation produced by klate and wedge
varies uniformly with distance along the face of the
I
wedge, passing through zero at the middle.
The are
light (a, fig.7) is replaced by a sodium flame and the
compensator is placed in the beam issuing from the test
piece.
Then, if the test piece is unstressed and the
nicol N is crossed with the polarizer, the surface of
the wedge as seen through the lens L will be crossed by
dark bands.
The central one corresponds to the middle
of the wedge where there is no total relative retardation, and the others to parts of the wedge where the
37
total relative retardation is a multiple of the wave
length.
In these places the vectors I
and I
are in
phase when they issue from the wedge and have no component III.
When the test piece is stressed these bands
move, parallelly to themselves, to parts of the wedge
where the relative retardation produced in the compensator is equal in magnitude but of opposite sign to that
produced in the test piece.
The bands can be brought
back to their original positions by moving the wedge by
means of the screw.
The relative retardation in the
test piece is then proportional to the turns of the
screw.
It remains only to determine the factor of pro-
portionality between the graduations on the screw and
the relative retardation in centimeters.- This is done
by turning the screw until the bands have moved a distance equal to the separation between two successive
bands.
The number of graduations passed on the screw
corresponds to a relative retardation of one wave
length of the D-line of sodium.
The relative retarda-
tion measured with the compensator divided by the stress
38
andIthe thikness of the test piece is the stressoptical coefficient.
I measured the stress-optical coefficient of
the celluloids, of glass, and of fused quarts in this
way.
For tensions of about 200 pounds, at which these
measurements were made, I was able to obtain, as nearly
as could be determined, uniform tension.
are given in the table on page 67.
The results
39
TESTS UNDER BE-DING
POISSON'S RATIO
Let us consider a rectangular bar under
uniform bending;
that is, subjected to forces whose
bending moment is uniform over the length considered.
Such a system of forces is shown in figure 15.
The
bending moment being uniform, so will be the curvature,
and the bar will be bent in an arc of a circle.
The
40
upper half of the bar will be under tension, and the
lower half under compression.
In accordance with the
property of matter of contracting laterally when it is
under tension and of expanding laterally when it is
under compression, and of which Poisson's ratio is a
measure, the upper half of the bar will contract
laterally and the lower half will expand laterally.
These lateral deformations can be derived by multiplying the longitudinal deformations by Poisson's ratio.
They will therefore be proportional to distance from
the neutral layer, and will give to the bar a lateral
curvature of opposite direction to the longitudinal
curvature. (Fig. 16)
anticlastic.
The surfaces of the bar will be
This double curvature can be clearly
observed on a glass bar.
The upper surface is ground
"plano" and an optical flat is placed upon it and
illuminateo. normally with monochromatic light.
Where
"Plano" is a term used by opticians to designate
a surface which has been ground mechanically as flat as
possible, but without the repeated hand grinding and
testing which is necessary in producing a "flat" surface.
I
I
41
the distance between the lower surface of the flat and
the upper surface of the bar can be expressed
by (+
n
n being an integer and X the wave length of the light
used, the light reflected from the lower surface of the
flat and that reflected from the upper surface of the'
bar will interfere.
Where the distance is n& they
will reinforce each other.
Interference bands will thus
be formed and will follow the curves obtained by cutting
42
the upper surface of the bar by equidistant planes
parallel to the flat (Fig. 17).
These bands will
resemble a family of hyperbolae. (Fig. 18)
Let us make the assumptions usually made in
the theory of beams that the bar 'is long enough so that
the effects of the compressions produced at the points
of application of the forces can be neglected, and let
us consider a longitudinal section of the bar (Fig. 19).
43
Let d be the thickness of the bar, 1 the length considered, r the radius of the neutral axis.
The portion of
the upper surface which fqrmerly had the length 1 now
1 + el
-. I
whence the elongation
r
+
has the length 1 + el, and we have, from the figure:
e = L
rI
2r
44
M
45
I will chose as coordinate axes the tangents
in the longitudinal and transverse planes and the
normal to the upper surface of the bar.
When the bar
is bent, the equation of the intersection of the upper
surface and the xz-plane is:
X2 + (r +
whence
and
A22a
z)2
(r + d)2
x 2 + Z2 =(2r + d)z
r = X 2 +z 2
2z
_
d
2
The measurement is taken in an interference
band so z can be obtained by counting the interference
bands.
x can be measured directly.
In the measurements
made .x was about half an inch and z less than 15 wave
lengths of the sodium D-line.
z2 is therefore entirely
negligible in comparison with x 2 and
A,
about 1/8 inch,
2
is also negligible in comparison with A . The expression for r then simplifies to
x2
r 7=zg-
46
e = dxz
x
whence
Similarly the lateral contraction is given by:
c = d/2r'
where r' is the radius of curvature in a transverse
r'Y=11
2z'
section.
c =
whence
Poisson's ratio is
1= E = z'x 2
in which x, z, y', and z' can be measured.
With the measurement of Poisson's ratio in
view, simultaneously with the work on pure tension, I
designed an apparatus for producing uniform bendin,.
The bars used were 1/4 by 1 1/4 inches in cross section
by 4 inches lorA.
The objection may be raised that the
width of the bar is not small in comparison with the
length and, therefore, the deviations from the theory
given above arising where the knif6 edges come in
contact with the glass, and annul the lateral curvature,
47
are not negligible even in the middle section where the
measurements were made.
I at first used a narrower bar.
This gave me only one interference band in which I
could measure y', and that, being near the center where
the slope of the surface is small, was too broad, so
that the error in estimating the middle was quite
large in comparison with the distance from the center of
the bar.
The error in C was even larger because this
distance figures in the square.
wider bar it was tOO I#&e U
tus also.
When I changed to a
change the bending appara-
To justify the results I took measurements
near the center as well as near the edges and,also,for
different values of the load.
The values for Poisson's
ratio computed from these different measurements were
all in good agreement.
The bar was supported on two knife edges,and
two other knife edges were made to bear down on the
extremities of the bar.
Several devices were designed
and constructed to hold the knives, without being used,
because it was found difficult to hold the knives
properly parallel.
The last device constructed, and
the one actually used, is shown disassembled in figure
o7F
~
-
K
L
~
L~4~
49
20.
The same with some minor improvements is outlined
in figure 21.
A wire passes around rod (c) on which
hangs a weight W.
Notch (b), in which the wire passes,
is in the middle of the bar, so that the forces transmitted'to each of the nuts (d) is the same.
These forces
are transmitted to the upper plate (h) by the rods (e).
The upper knives (j), attached to the upper plate by
the cleats (i), are equidistant from the rods (e), and
hence each must exert a force W/2 plus half the weight
of the system (c d g h i j m) on the bar tested (k).
Bar (k) rests on the two lower knife edges.
These are
equidistant from the holes in the lower plate (a)
through which the rods (e) pass, and are hence equidistant from the upper knife edges.
They therefore each
exert an upward force on the bar equal to the downward
forces exerted by the upper knife edges.
The weights
of the bar (k) and the flat (1) are negligible.
Buffers (f) receive the upper plate when the bar (k)
breaks.
A glass plate km) inclined at 45* reflects the
light from a sodium flame down normally to the bar.
The
reflected light passes up through kim) again and into a
411
TIu
T C7
51
microscope.
This microscope,is shown in figure 21,
is mounted on a bench and is moved by means of a screw
graduated in thousandths of a millimeter.
measured with it
directly.
x and y are
The distance actually
measured is from the nth band on one side of the center
of the interference pattern to the nth band on the other,
that is 2x and 2y.
The flat does not touch the center of the bar,
but rests on its edges, on account of the lateral
curvature.
The distance of the flat from the center of
the bar increases with the lateral curvature as W is
increased.
W can be adjusted so that the center of the
interference pattern appears dark.
It is then known
that the distance of the flat from the bar at the
center is a whole number of half wave lengths plus X/4.
The nth band from the center, not including the one
passing through the center, is n half wave lengths
above or below this level.
Substituting this
Therefore
z
n>/2
e
d xn
a
Fi
~I1
i
I
I
-4
['
iLOfr
22
N
53
c
C
And Poisson's ratio
e
e
n
n
(2y )2
2x
x
The values obtained by this method are given
in the table on page 67.
In order to have comparable values, the same
method was used in measuring Poisson's ratio for the
celluloids.
Here interference plenomena cannot be
produced, but the curvature is sufficient to be measured
directly with a Geneta Lens Measurer.
This instrwuent
is shaped like a watch, ,and has three steel pointe
projecting in the same plane.
The outer ones are fixed.
The middle one is connected, th1rough a system of levers,
to a pointer moving on a dial.
When the outer points
rest on a surface the middle one is forced in an amount
The pointer
varying with the curvature of the surface.
then indicates the curvature in diopters.
One diopter
is a curvature whose radius is half a meter.
If0r is
the curvature in diopters the radius of curvature is 1/2 a-
54
Substituting in the formulae given above
e
d cr
c = dcr'
andt
55
TESTS UNDER BENDING
STRESS-STRAIN CURVE
I have given a formula for the elongation of
The stress at the
the upper surface of a bent bar.
upper surface, assuming that the moduli of elasticity
in tension and in compression are the same, and that
Hooke's law holds, is given by
f = M/S
where M is the bending moment and S the section modulus.
For a rectangular cross section
S =
If g,
d
6
is the distance between the upper knife
edges and g, that between the lower knife edges, and if
the rods (e), fig. 19, pass accurately through the
centers of the holes in (a), then
2
2
where V is the weight of the system (c d g h i j i).
If the rods (e) are not quite centered in the holes in
56
(a) M remains practically unchanged, for let them be
Let Ua and U& be
off-center by an amount e6g, fig 23.
the weights supported by the lower knife edges.
U6 + Ua = W + V
Taking moments about the center of the bar, half way
between the lower knife edges
4
-
(
g)
-
=
-
-U4=
U
+ ag) - Ubi
(f
(W + V, 4
Ub = W +L
2
(1 + 2)
92
At the middle of the rod
IL
W + V(
2
2
+ g)
-
U6 a =
2
IL
2
W+ V
2
(
2
+4g)-
(
2 2
+4g)
-L.
g,
2
M, instead of being constant over the rod, varies uniformly, passing by the same value at the middle of the
bar that it would have had were the upper plate properly
centered.
As readings were taken only near therniddle,
the small variation allowed by the holes in (a) could
not affect the results.
We can therefore write:
57
6
(
bd
+V)
2
(g,
2
2
,
bd
(w+v)
which gives the stress in terms of quantities which can
be measured.
In measuring the factors in the formula for e
4
new difficulty enters.
Readings to be used in calcu-
lating Poisson's ratio were taken when the center of
the interference pattern was dark, and the position of
the center of the bar was known, plus or minus a half
a wave length.
In taking a stress.strain curve,read-
ings were also taken when the center was dark, but this
58
did not give a sufficient number of points.
For inter-
mediate points it was assumed that, between successive
readings taken when the center was dark, the flat was
raised proportionally to the increments in weight.
z is thus a multiple of a half wave length minus a
z = (n where W, and W.
frZ)
-
fraction of a half wave length.
are the preceeding and succeeding
weights which produce a dark center, and n is the
number of bands counted from the center.
The formula used in calculating the stress is
correct only if the substance obeys Hooke's law.
The
formula for the strain on the other hand holds, for a
sufficiently long bar, even aboe the elastic limit,
for the only assumption made in establishing it is that
a cross section through the bar remains plane after
the bar is bent.
This is true, for there is no more
reason for the cross section to bend one way than the
other as the bending moment is the same on both sides
of it.
Let us consider the strain e as the independent
59
variable.
Below the limit of Hooke's law W varies
linearly with e.
When the limit is reached for any
part of the bar, tkis part will not, for a given strain
e, support its proportional share of the stress.
W
will therefore fall below its linear value in terms of
e.
f, as calculated by the formula given above, varies
linearly with W.
Therefore,after the limit of Hooke's
law is reached,the stress-strain curve calculated by
this formula will bend down.
The true stress-strain
curve will bend down even more, but the important
point is that the limit of Hooke's law will be indicated on the curve calculated by the above formulae.
As
the curves for glass and for fused quartz, figure 25,
calculated by these formulae did not deviate from a
straight line, these substances obey Hooke's law up to
rupture;
at least if they are not left loaded too long.
A time effect was quite noticeable at the
higher loads, and present, though less pronounced, at
quite small loads.
The interference bands would creep
across the field of the microscope, very slowly, but
fast enough so that successive readings on the same
YT4
4
62
band had a tendency to vary in the same direction.
The actual change in the value of e during a measurement
was too small to show on the curves, and it is probable
that two curves in which the loads were left on different lengths of time would differ more markedly in the
position of the breaking point than in the slope.
The
measurements for each of the curves given took about
three hours to perform.
Curves for celluloid, figure 24, were also
taken by this method, using a Geneve Lens 2easurem.
Be
Because of the greater curvatures, and therefore greater
departures from ideal conditions assumed in the theory,
these curves are much less reliable than those for
glass and for fused quarts.
in neither case were the
curves continued till rupture.
ihey were discontinued
when the curvature had reached tne maximum allowed by
the apparatus.
PHOTOMETRIC
MEASUPMeJNTS
COEFFICIENT OF ABSORPTION
When light travels through a material medium
it
is progressively decreased in intensity by absorp-
tion.
The amount absorbeddiin a small length, dt,
is a constant fraction of the total intensity i.
As
di is negative we must Write
di =
-
kidt
The factor k is called the coefficient of absorption.
It has the dimensions of an inverse length, and is
measured in centimeters
1
.
The above equation inte-
i = i ekt
grates into
When light passes from one medium to another
there is a loss by reflection.
The fraction lost at
the front and back faces of a slab of material is given
by Fresnel's formula (n
n + 11)2, where n is the index of
refraction.
The light which gets through a slab of
material is then
i = cci
=
i.ekt (1
.(n - 1) 2
n + 1
1
2 ekt
4n
(P n + 17
64
The instrument used, a Lummer-Brodhun photometer,
of
figure 26, consists of a graduated rod, at each end
which is an electric light,
Moveable on the rod is a
box in the middle of which is a screen, and in the
opposite sides of which are holes of equal rizo.
Both
sides of the screen can be observed simultaneously
through a telescope.
A slab of the material whose
holes,
absorption is sought is placed over one of the
until both
and the box moved along the graduated bod
Let x, be
sides of the screen appear equally bright.
placed
the distance read on the rod. The slab is then
65
over the other hole and a balance again obtained.
x. be this reading.
Let
If S and S' are the strengths of
the lights and 1 their distance, then, if both holes
are free, the intensities of illumination on the two
sides of the screen are respectively
-
x
and
St
(l - x)z
When a slab is placed over one of the holes, the illumination of this side of the screen is reduced in the
ratio cx.
This is compensated for by moving the box
nearer one of the lights.
Writing that the illumination
of both sides of the screen was the same when the readings were taken
Cw
=
(1-X,
x
S
x
Dividing
A
S'
-
C=2
x,
whence
OL=
But
Cn
)2
(1-
1
X)2
-
n +
x
ek
e
66
e
=23026
log(l
-
x,
)
-
+
xZ
X,
Xf'
4n
4n
2.
- xz n
+
kt = log (1
k
- xx n
ekt = 1 x1
- X,
hence
log x, + log x7 - log(l - xZ)
+ log 4n - 2 log(n + 1)
In this formula everything was measured except
n.
For this I assumed the following values:
Fused quartz
n = 1.47
Glass
n = 1.51
For the celluloids the terms in n were omitted because
n was not known.
For camphor celluloid these terms are
certainly negligible besides the high absorption.
For
Camphor celluloid they would only produce a small difference.
.
The results are given in the table on page67
L
TABLE OF RESULTS
m0
//
.C+
10
0p0
WS6.
72
0
t
m
noe
A7e
Naphtha
Kg/cm2
Fused
Quartz
C+
0
C
.214
H.3
lbs/in lbs/ fin 2
m"C+H2
.1 8703-
m
1
x
300k
1.87
n2
.195
0C0
4
-r&r0
428
2.505
-
--
428
10 000 000
6
090 A
6 tress-stra pls strai ht at 1 ast up
185.
-
320AD
3020
0(0
2.8
522
68.86
7800
cm
1-
1x1
429
.lu i312.17
274
1
.
---W
25j 00 30 P-
25 000
f; 0fl
_
x
9 I- 5.
000
Z6
Glass
_
32P
.0
Y-/7-P2
728 000
_
t-4-2-H 4 C 0
(I 1iI
25 000
Celluloid
_
0
totoCD
none
lbs/in 2zahh
Units
0
0
R
.108
-0_
_
3 3 71
_O__
_
this
1
p-t.
7.4
_
7
1
-
0t
68
CONCLUSIONS
In measuring a stress by the photoelastic
method, the relative retardation of the two rays is
They are made to pass through
not actually measured.
a comparison member which is stressed until its double
refraction exactly compensates that of the member
studied.
Then, if the thicknesses are the same, the
stress in the model and in the comparison member are
the same.
The comparison member is usually subjected
to tension, which is measured directly on a spring
balance.
Where there is compensation the image of the
member appears dark.
The precision of a measurement
depends on how small a variation in stress will produce
a noticeable departure from darkness.
Let p be the relative retardation in the
beam issuing from the model, and let ap be the smallest
variation in retardation which occasions a noticeable
departure from darkness.
The relative error ist6p/p.
bp is independent of p for p is compensated and therefore
69
4? is always from a total relative retardation equal
to zero. 4 P being independent of p the only way to
reduce the relative error is to increase p.
But
P = tp - q) Ct
in which p and q are the principal stresses, C the
stress-optical coefficient, and t the thickness of the
member examined.
possible.
These quantities must be as great as
p and q are limited by the limit of Hooke's
law, for above this limit the proportionality between
the stresses in tne model and in the original no longer
holds.
The greater the thickness t used the less
light gets through.
Two materials should be compared
in thicknesses which let through the same fraction of
the light incident on them.
proportional to l/k.
and we can write
f
Such thicknesses are
is therefore limited by LC/k,
p
k/LC.
LC/k can be taken as a measure of the precision which
can be attained when using models made of a substance
having these constants as properties.
Its value for
the four substances considered is tabulated in the last
70
columin of the table on page 67.
Measured by this
factor naphtha-celluloid is far superior to any of the
other three.
Other considerations ented, however, such as
the internal strains present in the material.
Non-
crystalline substances have to be used, and these are
either super-cooled liquids or colloids deposited from
solutions.
in the first case internal strains are
produced in cooling.
They are functions of the coeffi-
cient of thermal expansion, the teaparature of fusion
or of softening, the thermal conductivity,and the specific heat.
The approximate values of these functions
for glass and for fused quartz are given in the table
below
Property
Coef. of exp. X
Fusion point F
Thermal cond.
Specific heat
4cf
Fused quartz
9 x 10-8
1658
3.3 x 10.15
1.49 x 10-
Glass
7 x 10-6
500
2.9 x 10-3
.20
35. x 104
t71
The product xT is also given because, other thipgs
being equal, the strains would be proportional to this
product.
This product is favorable to quartz as is
also the higher conductivity and the lower specfic
heat.
Similarly, uneven drying produces internal
stresses in the celluloids.
They are functions of the
contraction of the celluloid on drying and of the rate
of diffusion of the liquid through the celluloid.
The internal stresses in glass and in fused
quartz can be largely eliminated by prolonged annealing.
To eliminate the internal stresses in the celluloids by
slow
cring would require much too long.
For thicknes-
ses above an inch the time of drying would extend into
years.
Even the lesser thicknesses do not attain A
permanent state but are continually "ageing",, and this
has to be taken account of.
The internal stresses in
naphtha-celluloid appear to be particularly high,
producing first order blue in a thickness of half an
inch.
Another consideration is the ease of construction of the model.
Celluloid has the great advantage
72
of being easy to machine.
the shape desired.
Glass has to be ground to
Fused quartz may, because of its
better thermal properties, be cast to approximately
the shape desires and then the surface ground off.
From the above it appears that fused quartz
has every advantage over the glass tested.
Naphtha-
celluloid is very much better than camphor-celluloid,
except for the very high internal stresses.
The rela-
tive merits of naphtha-celluloid and fused quartz are
more difficult to decidc.
They depend mainly on
whether the high internal stresses ii the celluloid
are sufficient to aniiul the higher accuracy that could
otherwise be attained with it.
I consider naphtha-
celluloid the best on account of the exceptionally high
value of the function LC/k and because of the greater
ease in making the models out of this substance.