Experimental Stress Analysis Using Photoelasticity Techniques

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GHOUSIA COLLEGE OF ENGINEERING
RAMANAGARAM-562159
EXPERIMENTAL STRESS ANALYSIS
[15ME832]
Dr. MOHAMMED IMRAN
ASST PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING

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Module-3-Part-A
PHOTOELASTICITY
1. NATURE OF LIGHT
It has been observed that the disturbance that is being propagated can be either
perpendicular to the direction of propagation or in the direction of propagation. If the
disturbance is normal to the propagation direction it is called a transverse wave and
when it is in the direction of propagation it is called a longitudinal wave.
Light-waves belong to the class of transverse waves and the disturbance can be
represented by means of a vector called the light-vector. This light vector is
perpendicular to the direction of propagation. Light is known to be an electromagnetic
disturbance propagated through space and two vectors, namely the electric force
vector E and the magnetic force vector H.
H
E
Z
Fig (3) Electric and magnetic vectors
are associated with it. These two vectors are mutually perpendicular as shown in Fig
(3) and either of these can be taken as the fundamental light-vector.
2. PROPERTIES OF LIGHT
The colour of the visible light is determined by the frequency of the components of
the light vector. The colours in the visible spectrum range from deep red to deep
violet with frequencies of 390 × 1012
Hz to 770 × 1012
Hz, respectively. Most
photoelastic studies are made by using light in the visible range.
When the light vector is composed of vibrations, all of them having the same
frequency, it is called monochromatic light, i.e. light of single colour. When the
components of the light vector are of different frequencies, the colours of all the
components are mixed and eye records this mixture as white light.

Ordinary light consists of electromagnetic waves vibrating in directions perpendicular
to the direction of propagation. When the vibration pattern of these waves exhibits a
preference as to the transver
polarized. Two types of light, i.e. (
are used in photoelasticity.
Plane polarized light is obtained by restricting the light vector to vibrate in a single
plane known as the plane of polarization
vector sweeps out a sine curve as it propagates. The light is vibrating in the plane of
polarization.
Plane polarizers are optical elements which absorb the components of the light vector
not vibrating in the direction of the axis of the polarizer. When a light vector passes
through a plane polarizer, this optical element absorbs that component of the light
vector which is perpendicular to the axis of polarization and transmits the component
parallel to the axis of polarization as shown in
vector A = a sin ωt where
angle which the light vector
A0 = Absorbed component=
At = Transmitted component=
In a plane or linear polarizer,
polyvinyl alcohol heated, stretched, and bonded to a supporting sheet of cellulose
acetate butyrate. The polyvinyl face of the assembly is then stained with a liquid rich
in iodine. The amount of iodine diffused into the sheet determines its quality which is
judged by its transmission ratio.
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preference as to the transverse direction of vibration, then the light is said to be
polarized. Two types of light, i.e. (i) plane polarized and (ii) circularly polarized light,
are used in photoelasticity.
plane of polarization. Figure (a) shows that the tip of the light
brating in the direction of the axis of the polarizer. When a light vector passes
llel to the axis of polarization as shown in Figure (b). Say the light
a = amplitude and ω = frequency of light wave, and
angle which the light vector A makes with the axis of polarization. Then
Figure (a) Plane of polarization
= Absorbed component= a sin ωt sinα
= Transmitted component= a sin ωt sin α cosα (1).
plane or linear polarizer, H type polaroid film is used which is a thin sheet of
odine. The amount of iodine diffused into the sheet determines its quality which is
judged by its transmission ratio.
se direction of vibration, then the light is said to be
) circularly polarized light,
shows that the tip of the light
brating in the direction of the axis of the polarizer. When a light vector passes
. Say the light
= frequency of light wave, and α =
makes with the axis of polarization. Then
cosα (1).
type polaroid film is used which is a thin sheet of
odine. The amount of iodine diffused into the sheet determines its quality which is

Circularly polarized light is obtained when the tip of the light vector describes a
circular helix as the lig
Circularly polarized light is obtained with the help of a
made of a double refracting material. It resolves the light vector into two orthogonal
components and transmits each of them at different velocities. The phase difference
between these two components is π/2, i.e. quarter of a cyc
The light vector component transmitted by plane polarizer is
At = a sin ωt sin α cosα
There are two axes 1 and 2 of the QWP shown in
angle β with the axis 1 of the QWP.
axes 1 and 2, i.e. fast and slow axes of the QWP. Component
velocity V1 which is more than the velocity
Figure (c)
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Figure (b) Plane polarizer
circular helix as the light propagates along the z-axis as shown in
Circularly polarized light is obtained with the help of a quarter wave plate (QWP),
between these two components is π/2, i.e. quarter of a cycle.
The light vector component transmitted by plane polarizer is
There are two axes 1 and 2 of the QWP shown in Figure (D)
with the axis 1 of the QWP. At is resolved into two components along two
d 2, i.e. fast and slow axes of the QWP. Component
which is more than the velocity V2 with which the component
Figure (c) Circularly polarized light
axis as shown in Figure (C).
quarter wave plate (QWP),
(D). At makes an
is resolved into two components along two
At travels at a
with which the component At2travels.

Now
Since V1 > V2, the two components emerge from the plate with a phase difference.
Let λ = wave length of light.
Change in refractive index in direction (1)
= n1 – n0
= n2 – n0
Then,
Wave plates employed in a photoelastic study
or calcite cut parallel to the optic axis; a single plate of mica, a sheet of oriented
cellophone, or a sheet of oriented polyvinyl alcohol. QWPs are designed for a
monochromatic light.
When angle β = 45° and δ =
5
Figure (d) Quarter wave plate
components emerge from the plate with a phase difference.
= wave length of light.
Wave plates employed in a photoelastic study may consist of a single plate of quartz
= 45° and δ = , a circularly polarized light is obtained.
components emerge from the plate with a phase difference.
may consist of a single plate of quartz

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The Wave Equation
Since the disturbance producing light can be represented by a transverse wave motion,
it is possible to express the magnitude of the light (electric) vector in terms of the
solution of the one-dimensional wave equation:
E =f(z -ct) + g(z + ct) (1)
E = magnitude of light vector
z = position along axis of propagation
t = time
f(z - ct)1 = wave motion in positive z direction
g(z + ct) = wave motion in negative z direction
Most optical effects of interest in experimental stress analysis can be described with a
simple sinusoidal or harmonic waveform. Thus, light propagating in the positive z
direction away from the source can be represented by Eq. (1) as
E =f(z - ct) =(K / z) z cos/(z - ct) (2)
where K is related to the strength of the source and K/z is an attenuation coefficient
associated with the expanding spherical wave front. At distances far from the source,
the attenuation is small over short observation distances, and therefore it is frequently
neglected. For plane waves, the attenuation does not occur since the beam of light
maintains a constant cross section. Equation (2) can then be written as
E = a cos(2 / )(z - ct) (3)
Where a is a constant known as the amplitude of the wave. A graphical representation
of the magnitude of the light vector as a function of position along the positive z axis,
at two different times, is shown for a plane light wave in Fig. a. The length from peak
to peak on the magnitude curve for the light vector is defined as the wavelength A.
The time required for passage of two successive peaks at some fixed value of z is
defined as the period T of the wave and is given by
T = / c (4)

Fig: a Magnitude or the light vector as a
or position along the axis or propagation at two
different times.
The frequency of the light vector is defined as the number
Thus, the frequency is the reciprocal of the period, or
The terms angular frequency and wave number are frequently used to simplify the
argument in a sinusoidal representation of a light wave. The
the wave number are given by
=
Substituting Eqs. (5) and (6) into Eq. (3) yields
Two waves having the same wavelength and ampl
shown in Fig b. The two waves can be
E1= a cos (z + 1 - ct) 2n E
Where 1= initial phase of wave
2= initial phase of wave E
= 2- 1= the linear phase difference
The linear phase difference
wave 1.
The magnitude of the light vector can also be plotted as a function of time at a
position along the beam. This representation is useful for many applications since the
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Magnitude or the light vector as a function
along the axis or propagation at two
Fig: b Magnitude of the light vector as a function
of position along the axis of propagation for two
waves with different initial phases.
of the light vector is defined as the number of oscillations per second.
Thus, the frequency is the reciprocal of the period, or
argument in a sinusoidal representation of a light wave. The angular frequency
are given by
= = 2 (5)
= (6)
Substituting Eqs. (5) and (6) into Eq. (3) yields
E = a cos( z - t) (7)
Two waves having the same wavelength and amplitude but a different phase are
shown in Fig b. The two waves can be expressed by
ct) 2n E2= a cos (z + 2- ct)
= initial phase of wave E1
E2
linear phase difference between waves
The linear phase difference 15is often referred to as retardation since wave 2 trails
The magnitude of the light vector can also be plotted as a function of time at a
Magnitude of the light vector as a function
of position along the axis of propagation for two
waves with different initial phases.
of oscillations per second.
angular frequency cu and
itude but a different phase are
since wave 2 trails
The magnitude of the light vector can also be plotted as a function of time at a fixed

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eye, photographic films, and other light-detecting devices are normally located at
fixed positions for observations.
2. STRESS-OPTIC LAW-TWO DIMENSIONAL CASE:
Consider a model of uniform thickness made of glass, epoxy or some
transparent high polymer material. Let the model be loaded such that it is in a plane
state of stress. Then the state of stress at any point can be characterized by the three
rectangular stress components σx, σxandτxy or by the principal stresses σ1, σ2and their
orientations with reference to a set of axes. The situation is as shown in figure (1).
Let n0 be the refractive index of the material when it is in free (i.e. unstressed)
state. When the model is put in a state of stress, experiments show that:
Fig (1) Plane stress state and principal stresses.
(i) The model becomes doubly refractive;
(ii) the directions of the polarizing axes in the plane of the model at any
point P coincide with the directions of the principal stress axes at that
point; and
(iii) if n1 and n2 are the refractive indices for vibrations corresponding to
these two directions, then
– = –
– = – (1)
c1is called the direct stress –optic coefficient and c2 the transverse stress-optic
coefficient. Since the stress vary uniformly, i.eσ1, σ2and θ are continuously distributed
functions over the model in the xy-plane, the directions of the polarizing axes as well
as the values of n1 and n2 vary uniformly over the xy-phase of the model.

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If a plane polarized light is incident normally at any point P of the model, then the
incident light vector gets resolved along the directions of σ1, σ2 and these two
vibrating components travel through the thickness of the model with different
velocities. The velocities of propagation of these two components are governed by
Equation (1). When they emerge, there will be a certain amount of relative phase
difference between these two components. The relative phase difference ε is given by
= ( – ) (2)
Using Equation (1).
=
2
( – − – ) =
2
[ ( – ) + ( – )]
. . =
2
( + ) ( – )
If c1 + c2 is set equal to c, the stress-optic coefficient, the relative retardation ε is then
given by
= ( – ) (3)
The number of wavelengths of relative path difference is given by
=
=
( – ) (4)
Equations (3) & (4) are known as stress-optic relations. They relate the stresses to the
optical behavior of the model. According to these equations, the relative phase
difference is directly proportional to (σ1-σ2) and model thickness d,and inversely
proportional to the wavelength of light used. In photoelastic analysis we try to
evaluate the value of (σ1-σ2) at a point from the measured value of ε or N.
from Eq. (4), therefore,
( – ) = (5)
Denoting λ/cd by f, the principal difference is given by
( – ) = (6)
‘f ’is called the model fringe constant. Putting N =1, we can see that f expresses the
value of (σ1-σ2) necessary to cause a relative path difference of one λ in a model of
given thickness d. This is also equal to the value (σ1-σ2) necessary to cause a relative
phase difference of 2π radians in given model. Equation (5) can also be written in the
form
( – ) = (7)

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F represents the material fringe constant. If d = 1, F becomes equal to f. Hence, F
represents the model fringe constant per unit thickness. From Equations (6) and (7)
= =
3. POLARIZATION:
Light-waves belong to the class of transverse waves and the disturbance can
be represented by means of a vector called the light vector. In ordinary light, the tip of
the light-vector describes a random vibratory motion in a plane transverse to the
direction of propagation as shown in figure(4). If the tip of the light-vector is forced to
follow a definite law (or a pattern), the light is said to be polarized.
Light vector
(a)
(b)
Fig (4) Ordinary light - random vibratory motion
Types of polarization
i)Plane polarization or linear polarization
If thelight-vector is parallel to a given direction in the wave-front, it is said tobe
linearly or plane polarized as shown in Fig (5 c).
(a) (b) (c)
Fig (5) Circular, elliptical and linear polarizations

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ii) Circular polarization
If the tip of the light is constrained to lie on circumference of a circle, it is said to be
circularly polarized. If the tip of the Light-vector in Fig. 5(a) describes the circle in a
counter-clockwise directionas shown, then it is said to be right-handedly circularly
polarized. Onthe other hand, if the path is traversed in a clockwise direction, then it
isleft-handedly circularly polarized. This notation is adopted so as to beconsistent
with the right-handed coordinate system. The positive z-axis is away from the source
and the vibrations are in planes parallel to the xyplane.
iii) Elliptical polarization
If the light beam is constrained to movein a pattern such that the tip of the
light vectordescribes an ellipse then the light is said to be elliptically polarized.
One can speak of right-handedly or left-handedly ellipticallypolarized light. It is
easily observed that an elliptically polarized lightis the most general form of polarized
light since a circle can be consideredas an ellipse with the major and minor axes being
equal. Similarly, astraight line is a degenerated form of an ellipse with the minor axis
being equal to zero.
4. PLANE POLARISCOPE, ISOCHROMATICS AND ISOCLINICS
Fig (2) Plane polariscope, Isochromatics and Isoclinics
Consider the arrangement shown in figure (2). S is a source of monochromatic
light, P is a polarizer, M is the model under a plane state of stress, A, called the

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analyzer, is a second polarizing element kept at 900
to the polarizer and B is the
screen. We shall assume that through a suitable optical arrangement, the image of the
model is projected on the screen. We shall trace the passage of a typical ray of light
through the various optical elements in the assembly.
The arrangement shown in figure (2) is known as a plane polariscope. The
polarizer and the analyzer are always kept crossed, but their combined orientation can
be arbitrary. Now we can make a few important observations as follows.
When the model is stressed, it behaves as a crystal and at the point where the
ray passes, the polarizing axes coincide with the principal stress axes σ1,σ2 at that
point. In general, the polarizer (i.e the axis of the polarizing element) makes an angle
Φ with theσ1axis. If Φhappens to be zero (or π/2), i.e. if the polarizer coincides with
either σ1 (or σ2), then a plane polarized light incident on the model at that point will
emerge as a plane polarized light. Since the analyzer is kept crossed with respect to
the polarizer, the light coming out of the analyzer is zero. Consequently, at all those
points of the model, where the directions of the principal stresses happen to coincide
with the particular orientation of the polarizer-analyzer combination, the light coming
out of the analyzer will be zero. If the polarizer-analyzer combination happens to
coincide with the directions of σ1,σ2stresses at one point of the model, then in general,
there will be a locus of points in the model along which this condition is satisfied.
This is so because, in general, the stresses are disturbed in a continuous manner in the
model.
The locus of points where the directions of the principal stresses coincide with
a particular orientation of the polarizer-analyzer combination is known as an isoclinic
(meaning same inclination). For example, if the polarizing element is kept vertical and
the analyzer is kept horizontal, then on the screen, a dark band will be seen which is
the locus of the points where the σ1,σ2directions happen to be vertical and horizontal.
If one measures angles from the vertical reference axis, this isoclinic will be called the
00
-isoclinic. If now, the polarizer is turned through say, 300
and the analyzer is also
rotated through an equal amount (so that the analyzer is always kept crossed with
respect to the polarizer) then the previously observed 00
-isoclinic vanishes and a new
dark band is observed on the screen. This is the 300
-isoclinic and it represents the
locus of points in the model where the principal stress axes are oriented at 300
and
300
+ (π/2) with respect to the vertical. In figure (2) one such isoclinic is marked on
the image.

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Let us now consider another situation. Suppose at a particular point of the
model, the values of σ1and σ2 are such as to cause a relative phase difference of 2πm
where m is integer. The relative phase difference is related to σ1 - σ2. When the
relative phase difference is 2mπ, the model behaves as a full-wave plate at that
particular point. An incident linearly polarized light on a full-wave plate emerges as a
linearly polarized light and is cutoff by the analyzer, because of its crossed position.
Therefore, at all those points of the model where the values of σ1- σ2 are such to cause
a relative phase difference of 2mπ (m = 0, 1, 2…), the intensity of light on the screen
will be zero. On the screen, a series of dark bands corresponding to the loci of these
points are observed. These dark bands or fringes are known as isochromatics. An
isochromatic is a locus of points where the values of σ1- σ2 are such to cause a relative
phase difference of 2mπ (m = 0, 1, 2…), when the background is dark. The locus of
points where the values of σ1- σ2 are such to cause zero radians of phase difference (or
equivalently, zero number of wavelengths of relative path difference) is called the
zero-order fringe. The locus of points where the values of σ1- σ2 are such as to cause
2π radians phase difference (equivalently to a relative path difference of λ) is known
as the first-order fringe. Similarly, on the screen one can observe the second-order
fringe, third-order fringe, and so on. These are shown qualitatively in fig (2) on the
screen.
It should be observed that the background on the screen (i.e. the region outside
the image of the model) is dark, since the light coming out of the polarizing element is
cut off by the analyzer. Hence, the dark background corresponds to the zero-order
fringe.
ANALYSIS THROUGH TRIGONOMETRIC RESOLUTIONS
(Effects of a stressed model in a plane polariscope)
The concept of Isoclinic’s and Isochromatics can also be explained through
trigonometric resolution process as discussed below.
Consider fig (3a), which is equivalent to fig (2). Let the linearly polarized light
coming out of the polarizer will be
A1 = a cosωt

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Fig (3) Passage of polarized light through stresses model
Upon entering the model, the light-vector gets resolved along the principal stress axes.
Thus,
A2 = a cosΦcosωt
A3 = a sin Φcosωt
Upon leaving the model, the two vibrating components acquire a relative phase
difference δ (which depends upon the value of σ1- σ2 at that point, thickness d of the
model, wavelength λ of the light used and the model material. We shall assume that
A2 leadsA3. Hence, upon leaving the model, the vibrating components are
A4 = a cosΦcos (ωt + δ)
A5 = a sin Φcosωt
We should observe that both A4 and A5 will have certain absolute phase differences.
However, in photoelasticity, we confine ourselves essentially to relative phase
differences. Hence, only A4 is given the additional phase δ. One can, of course, add
additional equal phase values say ε, to the components A4 and A5. The final result,
however depends only on the relative phase difference and not on the absolute phase
values.
On entering the analyzer, only the components along A6 are allowed to emerge.
Hence, from Fig. (3b),
A6 =A4 sin Φ – A5
= a cosΦsin Φcos (ωt + δ) - a cosΦsin Φcosωt
= a/2 sin 2Φ [cos (ωt + δ) - cosωt]
=a/2 sin 2Φ [cosωt(cos δ – 1) – sin ωtsin δ]
= a/2 sin 2Φ [- 2 cosωt sin2
δ /2 – 2 sin ωt sin δ /2 cos δ /2]

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= - a sin 2Φsinδ /2 [cosωtsinδ /2 - sin ωtcos δ /2]
= - a sin 2Φsinδ /2 sin (ωt + δ /2)
= -b sin (ωt + δ /2)
Where b = a sin 2Φsinδ /2 is the amplitude of the emerging light vector. A measure of
the intensity of light is given by the square of the amplitude. In our case, the intensity
is, therefore,
I = a2
sin2
2Φsin2
δ /2
The intensity of light coming out of the analyzer is zero under two conditions.
(i) When Φ= 0 or π/2; or/and
(ii) When δ = 2mπ(m = 0, 1, 2…)
Condition (i) tells that light extinction occurs at a point when the direction of the
principal stresses coincide with the directions of the polarizer and the analyzer. The
locus of points where this happens is called the isoclinic.
Condition (ii) tells that light extinction occurs at a point when the relative phase
difference is equal to 2mπ. The locus of points where this occurs is called the
isochromatic.
5. CIRCULAR POLARISCOPE
The optical arrangement is shown in figure (4). A quarter-wave plate is placed
after the polarizing element with its axes at 450
to the polarizing axis. The model is
placed after the first-quarter-wave-plate. The fast and slow axes of the λ/4-plate are
marked respectively as F and S. A second quarter-wave plate is introduced after the
model and is kept crossed with respect to the first-quarter-wave-plate. i.e. the fast axis
of the first-quarter-wave-plate and the slow axis of the second-quarter-wave-plate are
kept parallel. The last optical element in the set-up is the analyzer kept crossed with
respect to the polarizer. This arrangement is called the circular-polariscope with dark
field. The reason for calling this arrangement as a circular polariscope is based on the
fact that the light incident on the model in this set-up is circularly-polarized, whereas,
in the plane Polariscope arrangement, the light incident on the model is linearly
polarized. We know that a linearly polarized light incident on a λ/4-plate at 450
to the
axes of the plate emerges as a circularly polarized light. Hence, the light incident on
the model is circularly polarized. Further, in the absence of the model (or

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equivalently, the region outside the model on the screen), the effect of the first λ/4-
plate is cancelled by the λ/4-plate since their axes are kept crossed; and the intensity
of light reaching the screen is zero. Hence, the background is dark and the set-up is
called a dark-field set-up. The fringe pattern observed on the screen will consist of
only iscohromatics and no isoclinic's. Since the source is monochromatic, all the
isochromatic fringes appear as dark fringes. Figure shows such an isochromatic
fringe pattern.
Fig. (4) Arrangements for a circular polariscope.
a. Circular Polariscope – Dark-Field Arrangements
There are two possible arrangements for a circular polariscope set-up to give a
dark background (i.e. dark field set-up). These are shown in fig (5) and (6).
Fig (5, 6) Circular polariscope setups for dark field.

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The first set-up is identical to figure (4). In this the two quarter-wave plate are
kept crossed, i.e. the fast axis of the first quarter-wave plate is kept parallel to the
slow axis of the second quarter-wave plate. The polarizer and analyzer are kept
crossed. In the second arrangement, the elements are kept parallel. That the
background in both cases are dark can easily be understood. In the first arrangement,
the incident polarized light becomes circularly polarized after passing through the λ/4-
plate. The second λ/4-plate being crossed, the circularly polarized light becomes
linearly polarized (i.e. the two λ/4-plates cancel each other) and is cut off by the
analyzer. In the second arrangement, the two λ/4-plates being parallel, act as a half-
wave (λ/2) plate, and the incident linearly polarized light gets rotated by 900
, and gets
cut off by the analyzer, which is kept parallel to the polarizer.
b. Circular polariscope – light field arrangements
Figure (7) and (8) shows the Optical arrangement for a circular polariscope with a
light field i.e. bright background.
Figure (7, 8) Circular polariscope setups for light field.
That the backgrounds in the two arrangement are bright can easily be seen by
removing the model and analyzing the behavior of a ray of light. In the absence of the
model, the two λ/4-plates in the arrangement of fig (7) act as a half-wave plate, and
the incident linearly polarized light gets rotated by 900
and is allowed through the

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analyzer. In case fig (8) the two λ/4-plates cancel each other as the polarizer is parallel
to the analyzer, the light is admitted through.
6. ISOCLINIC AND ISOCHROMATIC FRINGE ORDER AT A POINT
We have observed that using a plane polariscope and white light source, the
isoclinic can be easily distinguished. In order to determine the direction of the
principal stresses at a desired point of stressed model, the following steps are
followed.
(i) The model is kept between the crossed polarizer and analyzer of a
plane polariscope.
(ii) The polarizer and analyzer are rotated in unison until the dark band
representing the isoclinic passes through the point of interest.
(iii) The orientations of the polarizer and analyzer coincide with the
principal stress directions at the point.
Regarding the fringe order at the point (i.e. the relative phase difference or relative
path difference), an isochromatic may not exactly pass through the point. Using a
dark-field set-up and a bright-field set-up, two sets of isochromatic fringe patterns
over the field of view can be recorded. The dark-field set-up gives fringes
representing an integral number of wave-lengths of retardations. The bright-field set-
up gives fringes representing odd multiples of half-wavelengths of retardations. With
these, one can determine the retardation existing at the point either by interpolation or
extrapolation. However, situations often exist where the pattern (either in the dark-
field or bright-field set-up) will not reveal a sufficient number of fringes to enable one
to adopt either the interpolation or extrapolation technique to evaluate accurately the
relative retardation at the desired point. In such situations, one adopts what is
generally known as the compensation technique to determine the fractional fringe
order existing at a point.
a. Compensation Technique
Suppose at a point of interest in the model, the relative retardation is between
3λ and 3.5λ as observed by the bright – and dark-field set-ups. Let us assume that the
value is 3.36λ. This is equal to 3.36 fringe order. The decimal part i.e. 0.36 of this
value, is called the fractional fringe order at the point under consideration. The

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method of compensation consists in bringing the existing fringe order to an integral
value in the case of a dark-field set-up or to an odd number of half-wave lengths in
the case of a bright-field set-up. Since a dark-field set-up is commonly used in
practice, we shall confine our decision to this particular arrangement. However, the
analysis is equally applicable to a bright-field set-up. In our present case,
compensation means raising the existing value of 3.36λ to 4.0λ or reducing the value
to 3.0λ. The retardation that is necessary to be added (=0.64λ) or subtracted (=0.36λ)
determines the existing fractional fringe order.
Methods of Compensation
We shall discuss three methods that are commonly employed in practice. In
the first method a known or measurable amount of retardation is either added or
subtracted to make the final retardation value an integral value. This is done by
putting a crystal combination (called the Babinet-Soleil compensator) in front of the
model and suitably adjusting the value of retardation given by the crystal
combination. In the second method, a quarter-wave plate is used to reduce the ellipse
of light coming out of the model into a linearly polarized light and determining the
orientation of this by means of the analyzer. This is known as Tardy’s method of
compensation. A variation of this, known as Friedel’s method of compensation is
also discussed.
(i) Babinet-Soleil Compensator
The Babinet-Soleil compensator consists of two quartz wedges cut similarly
with respect to their optical axes, as shown in figure (9). A and B are the two wedges
with their fast axes similarly oriented, so that the two wedges together form one
rectangular piece of uniform thickness over a limited portion. By moving one wedge
with respect to the other, the thickness of the combination over this portion can be
varied. Hence, the two wedge combination forms a crystal plate whose thickness can
be varied. Next to the wedge-combination is a quartz plate C of uniform thickness.
The fast axis of this plate is at right angles to the first axis of the wedge-combination.
These are shown in figure 9 (a) and 9 (b).

Fig (9)
20
Fig (9) Elements of Babinet – Soleil compensatorSoleil compensator

21
The retardation given by the plate Ccan be cancelled partially or fully by
varying the thickness of the wedge combination. Hence, by adjusting the overall
thickness of the wedge-combination of the compensator can add or subtract relative
retardations with a given range. The micrometer screw is calibrated in number of
wavelengths of retardation added or subtracted along a marked axis of the
compensator. In practice, the compensator is kept before or after the model and is
oriented along the principal stress axis at the point of interest in the model. From the
zero position, the micrometer head is turned either one way or the other until a dark
fringe passes through the point of interest. Observation will generally indicate
whether the higher-order fringe or lower-order fringe has moved to the point of
interest, thus indicating whether the integral value has been obtained (i.e.
compensation has been achieved) by addition or subtraction of retardation given by
the compensator.
(ii) Tardy’s Method of Compensation
The arrangement of the optical elements involved in Tardy’s method is shown in
figure (10) which are similar to a circular polariscope setup.
Fig (10) Arrangement for Tardy’s method of compensation
The passage of a ray of light can easily be traced through the optical elements
involved.
(i) The light coming out of the polarizer is incident at 450
to the axes of the
first quarter-wave plate. The light emerging from the first quarter-wave

22
plate is circularly polarized and can be represented by Acosωtalong the S-
axis and Acos (ωt + π/2) along the F-axis.
(ii) Since a circularly polarized light is incident on the model, the amplitudes
of vibrations along σ1, σ2 axes of the model are equal and the vibrations can
be represented by
Acos (ωt + π/2) along σ1 - axis
and
Acosωtalong σ2 - axis
(iii) Assuming that the σ1 – axis is the fast axis and that the relative retardation
added by the model is (2mπ + δ), the emerging light vectors are
Acos (ωt + π/2 +2mπ + δ) = Acos (ωt + π/2+ δ) along σ1 - axis
and
Acosωt along σ2 - axis
This in general represents an elliptically polarized light with axes at β and β +
π/2to the σ1 – axis such that
tan 2β = 2A1A2cos ε / (A2
1 - A2
2)
= [2A2
/ (A2
- A2
)] cos (π/2+ δ)
Therefore β = 450
and 450
+ π/2.
The axes of the ellipse will be at 450
to the principal stress axes. Further, the semi –
axes of the ellipse are given as
a2
= A2
1cos2
β + A2
2 sin2
β + 2A1A2cosβ sin β cos ε
b2
= A2
1cos2
β + A2
2 sin2
β + 2A1A2cosβ sin β cos ε
With β = 450
and A1= A2= A, ε = (π/2+ δ).
a2
= A2
+A2
cos(π/2+ δ) = A2
(1 + sin δ) ------ (1)
Similarly,
b2
= A2
-A2
cos(π/2+ δ) = A2
(1 - sin δ) ------ (2)
The identification of the axes as major or minor depends on the value of δ.
(iv) The incident light ellipse on the second quarter wave-plate will have its axes
parallel to the axes of the quarter wave – plate. Hence, the vibratory components
along these axes will have amplitude a andb. whether a is along the S – axis or F –
axis depends on the magnitude of δ. The relative phase difference will be π/2. We

23
know that a light ellipse described with respect to its own axes will have a relative
phase difference equal to π/2. Hence, the vibratory components are:
acos (ωt + π/2) or b cos (ωt + π/2) along S– axis
and
bcosωt or a cosωt along F – axis
(v) The quarter-wave plates plate adds π/2retardation along the F – axis, which is
equivalent to removing π/2along the S – axis. Hence, the vibrating components
coming out the second quarter-wave plate are:
acosωt or b cosωt along S – axis
bcosωt or a cosωt along F – axis
These two vibrations are in phase and are therefore the components of a linearly
polarized light which is inclined at an angle Фto the S – axis, such that
tanФ= a/b or b/a
From Esq. (1) and (2)
tan2
Ф= a2
/b2
= (1 + sin δ) / (1 - sin δ)
= (1 + 2 sin δ /2 cos δ) /2 / (1 - 2 sin δ /2 cos δ /2)
= (sec2
δ /2 + 2 tan δ /2) / (sec2
δ /2 - 2 tan δ /2)
= (1+ tan δ /2)2
/ (1- tan δ /2)2
Or
tanФ= + (1+ tan δ /2) / (1- tan δ /2) = + tan (450
+ δ /2)
Similarly using tan Ф= b/a, and substituting for a and b, one gets the alternative value
tanФ= + tan (450
- δ /2)
(vi) The light coming out of the second quarter-wave plate is therefore linearly
polarized and is at an angle of + tan (450
+ δ /2) to the S – axis, or equivalently at
+(π/2 +δ /2) to the horizontal axis. Thus, by turning the analyzer through an angle θ =
+ δ /2 to the horizontal, the light can be cut off. Hence the relative retardation given
by the model is equal to δ = + 2 θ, neglecting the integral fringe value of 2mπ. The
ambiguous sign (+) just indicates that compensation can be obtained either by
increasing the existing fringe order to the next higher integral value or decreasing it to
the next lower integral value.
The light ellipses coming out of each optical element in the set-up for Tardy’s method
of compensation are shown in figure (11).

24

F S
/4 - plate
aorb
bora

Analyzer
2
1
Model /4 - plate Polarizer
S F
Fig (11) Light ellipses emerging from optical elements
(iii) Friedel’s Method of Compensation
This is similar to Tardy’s method without the first quarter wave-plate. The
object of using the first quarter wave-plate is to obtain a circularly polarized light
which when enters the stressed model will give equal amplitudes along σ1, σ2 axes of
the model. When these two amplitudes are equal, then irrespective of the retardation
given by the model, the axes of the light ellipse coming out of the model will always
be at 450
to the σ1, σ2 – axes. Since the axes of the second quarter wave-
plate are at 450
to the σ1, σ2 – axes (i.e. along the axes of the light
ellipse), addition or subtraction of π/2 retardation will always produce a linearly
polarized light which can be cut off by the analyzer.
The same result can be obtained by orienting the polarizer at 450
to the
σ1, σ2 – axes of the model, without the first quarter wave-plate. Since the polarizer is
equally inclined to the σ1, σ2 – axes of the model, the amplitudes along these axes are
equal and the emerging vibrations from the model will be
A'cos (ωt + δ) along σ1 - axis
A'cosωt along σ2 – axis
Where A' = √2Ais the amplitude of the incident linearly polarized light. The light
coming out of the model is elliptically polarized with its axes at 450
to the σ1, σ2 –
axes. The semi-axis a and b of this ellipse are such that
a2
= A2
(1 + cos δ) and b2
= A2
(1 - cos δ)
The vibrating components incident on the second quarter wave-plate are:
acos (ωt + π/2) along S– axis

25
bcosωt along F – axis
The emerging components are therefore
acos (ωt + π/2) and b cos (ωt + π/2)
These two, being in phase, yield a linearly polarized light at an angle Фto the S– axis
of the λ/4- plate fig (14) such that
tan2
Ф= b2
/a2
= (1 – cos δ) / (1 + cos δ)
= tan2
δ / 2
Or
Ф= δ / 2
This linearly polarized light can now be cut off by rotating the analyzer through an
angle θ = (450
– Ф) clockwise from the horizontal. The relative retardation given by
the model is now equal to (900
– 2θ), where θis measured clockwise from the
horizontal.
Fig (12)Arrangement for Friedel’s method of compensation
Figure (12) shows the set-up for Friedel’s method of compensation and figure (13)
shows the light ellipses coming out of each optical element of the set-up.

26
Fig (13) Light ellipses emerging from optical elements
7. FRINGE MULTIPLICATION BY PHOTOGRAPHIC METHODS
By employing normal photo-elastic procedures, two photographs (one light-field, the
other dark-field) are obtained which permit the determination of the order of the
fringes in the following sequence: N = 0, 1/2, 1, 1(1/2), 2, 2(1/2), .... In certain photo-
elastic applications it is desirable to improve the accuracy of the determination of
fractional fringe orders which are between those previously listed. This objective can
be accomplished in a number of ways. In this section a photographic technique is
described which gives, through superposition of ordinary light- and dark-field
isochromatic fringe patterns, a new fringe pattern (mixed-field). This mixed-field
pattern has fringes at the N /4 and 3N /4 positions. Use of the mixed-field fringe
pattern coupled with ordinary light- and dark-field fringe patterns permits the orders
of the fringes to be determined in the 0, 1/4,1/2,3/4, 1, 5/4,...sequence, and thus
represents a factor of 2 increase in the number of countable fringes. The proof of this
photographic method can easily be established by drawing From Eqs. (1), which
describe the brightness of a negative obtained with a dark-field polariscope,
⎩
⎪
⎨
⎪
⎧=
1
⁄ (∆ 2⁄ )
≥ (∆ 2⁄ ) ≥ ( )
= 1 (∆ 2⁄ ) ≤ ( )
→ 0 (∆ 2⁄ ) ≥ ( ) (13)

=
∆
2
(14)
Dark field negative

27
= =
∆
2
=
∆
2
(15)
Similarly by combining Eqs. (14) and (15) with Eq. (12), the brightness ratio for a
negative in a light-field polariscope is
⎩
⎪
⎨
⎪
⎧=
1
⁄ (∆ 2⁄ )
≥ (∆ 2⁄ ) ≥ ( )
= 1 (∆ 2⁄ ) ≤ ( )
→ 0 (∆ 2⁄ ) ≥ ( ) (16)

The effect of superimposing light and dark field negatives is obtained by multiplying
the expressions for the brightness ratios. Thus, the brightness ratio mfor the mixed
field associated with superimposed light- and dark-field negatives is
= (18)
Equation (g) leads to four nonzero expressions for Pm
⎩
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎧
= 1 . ( ) ( )
=
1
⁄ (∆ 2⁄ )
. ( ) ( )
=
1
⁄ (∆ 2⁄ )
. ( ) ( )
=
1
⁄ ⁄ (∆ 2⁄ ) (∆ 2⁄ )
. ( ) ( ) (16)
Of these four solutions for ,the solution = 1 locates the fringe position on the
superimposed negative. Note that = 1 only in regions where both
∆
2
≤
∆
2
≤ (20)
With equal light- and dark-field exposures
= = (21)
whereMis the exposure multiple. Combining Eqs.(20) and (21) yields the condition M
~ 2 to obtain any region on the superimposed negative where Pm = I. With M= 2, Eqs.
(20) become
∆
2
≤
1
2

∆
2
≤
1
2

It is evident then that
∆
2
=
(2 + 1)
4
ℎ = 0,1,2,3, … ….

=
∆
2
=
With Epd= Epland M = 2, the superimposed negatives yield a fringe pattern where the
1/4-order fringes are displayed with
brightness ratio mas a function of retardation
PHOTO ELASTIC MODEL MATERIAL
By employing the plane polariscope and white light source, the directions of the
principal stresses at a desired point in a model under a plane state of stress can be
determined. The use of circular polariscope (with monochromatic light source) and
one of the compensation methods will enable one to determine accurately the relative
retardation at the point of interest. The relative retardation can be expressed either in
the form of fringe order (integral and fractional) or in terms of wavelengths. If
the number of wavelengths of relative retardation, then as per stress optic law
Where ƒ is the model fringe constant and
model fringe constant is defined as the value of (σ
retardation of 1λ in the model of given thickness
defined as the value of (σ1
thickness. The method to determine either ƒ or
called the calibration method. There are three methods of calibration which are
commonly used.
28
=
(2 + 1)
4
(22)
and M = 2, the superimposed negatives yield a fringe pattern where the
order fringes are displayed with N = ¼, ¾ , 5/4, .... A schematic illustration of the
as a function of retardation /2 is shown in Fig. (e).
8. CALIBRATION METHODS OF
PHOTO ELASTIC MODEL MATERIAL
at a desired point in a model under a plane state of stress can be
nt of interest. The relative retardation can be expressed either in
the form of fringe order (integral and fractional) or in terms of wavelengths. If
σ1- σ2 = ƒNor σ1- σ2 = F/dN
Where ƒ is the model fringe constant and F the material fringe constant. The
model fringe constant is defined as the value of (σ1- σ2) necessary to cause a relative
retardation of 1λ in the model of given thickness d. The material fringe constant
1- σ2) to cause a relative retardation of 1λ in a model of unit
thickness. The method to determine either ƒ or F for the given model material is
Figure (e) Mixed-field brightness ratio
as a function of retardation.
)
and M = 2, the superimposed negatives yield a fringe pattern where the
, .... A schematic illustration of the
.
CALIBRATION METHODS OF
at a desired point in a model under a plane state of stress can be
nt of interest. The relative retardation can be expressed either in
the form of fringe order (integral and fractional) or in terms of wavelengths. If N is
the material fringe constant. The
) necessary to cause a relative
. The material fringe constant F is
) to cause a relative retardation of 1λ in a model of unit
for the given model material is
field brightness ratio
as a function of retardation.

29
(i) Use of a Tension Specimen
Consider a uniform tension specimen of width w and thickness h subjected to
a load P. The stress in the central zone, far from the loading points is given by σ = P/
(wh). The member is in a plane state of stress with
=
( ℎ)
; = 0
Hence,
σ1- σ2 = σ1 = P/ (wh)
The tension specimen can be viewed in a circular polariscope and the formation of
successive integral number of fringe orders as the load P is continuously varied can be
observed. A curve of the load P is plotted as a function of N for five or six fringe
orders as shown in fig (1). The slope of the straight line gives the average value of
load P* per fringe.
Fig. (1) Calibration curve
Hence, the value of (σ1- σ2) necessary to give one fringe order in a calibration model
of thickness h is
∗
. We have F= ƒ h. Hence the material fringe value F is equal to
=
∗

If d is the thickness of the model under investigation, then the model fringe value is
=
∗

30
(b) Use of a Circular Disc
A circular disc under diametral compression as shown in fig (2) is frequently
used as a calibration specimen. A circular disc can be easily machined and the loading
is also simpler. If D is the diameter of the disc and h its thickness, the stress
distribution along the horizontal diameter is given by
= =
(2 )
( ℎ )
– 4
( + 4 )
= = −
(2 )
( ℎ )
4
( + 4 )
− 1
Where P is the diametral load applied along the y-axis and x is measured along the
diameter from the centre of the disc. At the centre of the disc,
– =
8
ℎ

The formation of fringes at the centre can easily be observed and a curve as shown in
fig (2) can be plotted and the average load P* to obtain one fringe at the centre
determined. Hence the value of (σ1 – σ2) necessary to give one fringe in a calibration
model of thickness h is 8P* / (πDh). The material fringe value F is then given by
=
8 ∗

Fig (2) Calibration using beam under pure bending

31
(c) Use of Rectangular Beam under Pure Bending
A beam with rectangular cross-section subjected to four-point loading, i.e.
pure bending, is used for calibration purposes. Figure (3) shows a beam under four-
point loading. The middle section of the beam is subjected to a pure bending moment
M = Pl, where l is the distance shown in the figure. Assuming that only σx is acting,
one has from mechanics of solids, σ1 – σ2 = σx = My/ I, where I is the area moment of
inertia for the beam section and y is the distance from the neutral axis. If the stressed
beam is observed in a circular polariscope, a series of horizontal fringes are seen at
the mid section as shown in fig (3a). The trace of the neutral surface is represented by
the zero-order fringe. The other fringes are equally spaced above and below the zero-
order fringe. If y1,y2 … are the distances of the fringes of first, second ,…. order
from the zero-order fringe, one can plot a curve of fringe order N versus σx = σ1 – σ2
as shown in figure (3).
Fig (3)
Theoretically, this should be a straight line. From this, the average value of σ1 –
σ2 = σx necessary to produce a retardation of one fringe order in a calibration model of
thickness h can be obtained. The advantage of this method is that a calibration plot as
shown in figure (3b) can be obtained with one loading system unlike in other methods
cases, where three or four different loads are required to obtain a calibration plot.

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