MAE 244 LAB 4
Optical Methods PHOTOELASTICITY
Basic Optics
For a light beam, when it travels in material:
E k sin(t 1 )
when it travels to different material, it will subject to refraction:
refrraction from material 1 to material 2 :
sin n2
n21 , where :
sin
n1
Where n1 and n2 are the refraction indexes of material 1 and material 2, respectively
n1
v1
n2
,
2
Where is the velocity of light in the vacuum, 1 is the velocity of light in
material 1, and 2 is the velocity of light in material 2.
Re lative angular phase shift caused by birefringence in a WAVE PLATE of thickness " h":
2
2 h
(n2 n1 ), where wavelength,
2
We can define the stress-optical coefficient C, such that
n2 n1 c( 1 2 )
, wave number
Nf
2 hc
( 1 2 ), 1 2
h
Basic Optics
For a light beam, when it travels in material:
E k sin(t 1 )
E k sin(t 2 )
2 1
Phase shift caused by birefringence in wave
plate of thickness, h
2
2 h
(n2 n1 ),
Where, is wavelength
The refraction index is dependent on the stress status
n2 n1 c(1 2 ) Stress Optic Law (Maxwell, 1853)
c is define the stress-optical coefficient,
2 hc
( 1 2 )
Dr. Charles Vest (1941-2013), MAE Distinguished Alumni (BSME,
1963), Ph.D. (1967, Univ. of Michigan), President, MIT (2004-2007),
President, National Academy of Engineering (2007-2013)
Stress-optic law
2 hc
f
N
c
2
( 1 2 ), 1 2
Nf
h
f is material fringe constant
is the fringe order;
The stress status is correlated with the optical response (phase
shift) via this equation-stress-optic law
Plane polariscope
I K sin 2 2 sin 2 ( / 2)
Where is the principle stress direction of 1. is the phase difference
2h
(n2 n1 )
2 hc
( 1 2 )
The intensity will become zero (I = 0, black fringes occur) in the following two
cases:
sin 2 (2 ) 0
(which is related to the principle stress direction), Isoclinic fringe
patterns will occur.
2 =n, n=0, 1, 2,…, a whole family of isoclinics may be obtained.
Plane polariscope
I K sin 2 2 sin 2 ( / 2)
sin 2 ( / 2) 0 (which is related to the principle stress difference), Isochromatic
fringe patterns will occur.
/2=n, n=0,1,2,…
such isochromatic fringe is related to the stress level at any particular
point along that fringe by the stress-optic law
Circular polariscope
S
F
P
A
W
M
In circular polariscope, the isoclinics are
removed, leaving only the isochromatics
for the analysis of stress levels
/4 L
L /4
Dark field: If the polarizer is perpendicular
to the analyzer (noted as “PA”),
I K sin 2 ( )
2
n ,
2
for
n 0,1,2,3,.......
or
N
0,
2
1,
2, n
Circular polariscope
S
F
P
A
W
M
L /4
/4 L
Bright filed: If the polarizer is parallel to the analyzer (noted as “P//A”)
I K cos2 ( / 2), which shows that extinctionoccurs when
1 2n
,
2
2
for
n 0,1,2,3,.......
or
1
N
n
2 2
N=0.5, 1.5, 2.5, ...
Four-point beam bending- to determine stress by
experimental approach
-1.5
- 0.5
b
max
f N
b
Bright filed: If the polarizer is parallel to the analyzer (noted as “P//A”)
Four-point beam bending- to determine stress by
theoretical approach
M
P
P
(x d ) x 0
2
2
Hence, the moment acting is
M
Pd
2
, at distance y from the neutral axis is
(x) = My / I
(y) = 0
(xy) = 0
Where I is the beam moment of inertia.
I
1 3
bh
12
P
To determine stress concentration by experiment
Stres s Concentration Factor=
Maximum Stres s
Nominal Stress
max
nom
max
K tg
,
norm
Kt
gross
nom
gross
max
P
hw
is the gross remote stress
N
f N
h
3
2
1
Distance from center
K tn
max
,
norm
net
where nomnet
P
is net remote stress
h( w 2r )
P
To determine the net stress concentration by
calculation
K tn 3.065 3.472(
2r
2r
2r
) 1.009( ) 2 0.405( ) 3
w
w
w
MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
Theoretical Background
Two-Dimensional Photoelasticity
Photoelasticity is a nondestructive, optical technique for experimental stress analysis that is
particularly useful for structural components with complex geometric configurations, or
subjected to complex loading conditions. Analytical methods of stress analysis are very
cumbersome, and often unavailable for such cases, thus amplifying the importance and the need
for a suitable experimental approach. Photoelasticity has been used widely, over an extended
period of time, for problems in which stress distributions have to be investigated over large
sections, or regions, of the structure. It provides quantitative information on highly stressed
areas and the associated peak stresses. Equally important is the capability offered by
photoelasticity to discern areas of low stress levels, where structural materials are utilized
inefficiently.
The method of photoelasticity can be applied in various forms to a wide variety of problems
ranging from stress wave propagation to fracture mechanics, to three-dimensional studies. The
applications illustrated and practiced in the following experiments are restricted, however, to
two-dimensional static problems.
Plane Polariscope
The photoelastic effect is based on the temporary double refraction or temporary
birefringence phenomenon, which is exhibited by many transparent polymeric materials when
their deformed configurations under stress are observed in polarized light. The birefringence
phenomenon is due to the fact that a stressed model of such a material allows two light waves
to propagate through it, in other words two different waves are refracted from one incident
wave. The directions of the refracted waves are oriented along the principal stress directions,
based on the equivalence between the STRESS ELLIPSOID for principal stresses, and the
INDEX ELLIPSOID for principal indexes of refraction. The two refracted waves travel at
different speeds, one in the FAST direction, and another in the SLOW direction, which creates a
RELATIVE RETARDATION (phase difference) that is proportional to the difference
between principal stresses, in accordance with the stress-optic law. An instrument called a
polariscope can be used to polarize the light in a desired direction, and detect the phase
difference introduced by the model between the two refracted waves of polarized light.
For a light beam, when it travels in material:
E k sin(t 1 )
When it travels to different material, it will subject to refraction:
refrraction from material 1 to material 2 :
sin n2
n21 , where :
sin
n1
1
2
Where n1 and n2 are the refraction indexes of material 1 and material 2, respectively.
1
MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
v1
2
Where is the velocity of light in the vacuum, 1 is the velocity of light in material 1, and 2 is
the velocity of light in material 2.
n1
n2
,
Re lative angular phase shift caused by birefringence in a WAVE PLATE of thickness " h":
2 h
2
2
(n2 n1 ), where wavelength,
, wave number
An applied stress can result in a change in the refractive index, n, of a transparent substance. If a
general system of stresses is applied in a plane, the optical birefringence, n2-n1 will be
proportional to the difference, 1-2 between the two principal stresses in the plane. We can
define the stress-optical coefficient C, such that
n2 n1 c( 1 2 )
For a sample of uniform thickness, regions in which 1-2 is constant show the same interference
color when viewed between crossed polars.
2 hc
( 1 2 ), 1 2
Nf
h
(1)
where:
N
is the fringe order;
2
c is the relative stress-optic coefficient, expressed in units of "length2/force", or "brewsters";
, which is expressed in units of "force/length".
f is material fringe constant,
f
c
The basic components of a plane polariscope are a light source and two flat plates that form the
Linear (Plane) Polarizer/Analyzer combination, consisting of the POLARIZER, placed in front
(before) of the model, and the ANALYZER, placed behind (after) the loaded structural model.
Each linear polarizer/analyzer unit is made, usually, of a thin Polaroid H film of polyvinyl
alcohol, which is heated, stretched, and immediately bonded on a supporting sheet of cellulose
acetate butyrate.
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MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
Since the intensity of light is proportional to the square of the wave amplitude, the light that
emerges from the analyzer of a plane polariscope can be described by the equation:
I K sin 2 (2 ) sin 2 ( )
2
(2)
Where is the principle stress direction of 1. is the phase difference and expressed by,
2h
(n2 n1 )
2 hc
( 1 2 )
Equation (2) indicates that the intensity will become zero (I = 0, black fringes occur) in the
following two cases:
a) When sin 2 (2 ) 0 (which is related to the principle stress direction), Isoclinic fringe
patterns will occur.
b) When sin 2 ( / 2) 0 (which is related to the principle stress difference), Isochromatic
fringe patterns will occur.
Two families of optical fringes are observed through the plane polariscope as a result of
the birefringence phenomenon: Isoclinics and Isochromatics.
Black fringes on the ruler [1]
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MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
Isoclinics
In order to determine the directions of the principal stress it is necessary to use isoclinic lines
as these dark fringes occur whenever the direction of either principal stress aligns parallel to the
analyzer or polariser direction. The "isoclinics" are black fringes that describe the loci of
constant principal directions, i.e. the lines joining all the points in the model where the
orientation of principal stresses is the same. The specific orientation of principal directions
corresponding to a certain isoclinic is determined by the specific orientation of the
polarizer/analyzer combination, since "" is the angle between the axis of the polarizer and the
principal 1 direction. Thus by rotating, in increments, the polarizer/analyzer pair of the
polariscope in order to reach the conditions of 2 =n, n=0, 1, 2,…, a whole family of
isoclinics may be obtained.
The principal direction corresponding to a certain isoclinic angle is related by Eq.(1) below to
the stress components that define the state of stress at any point along that isoclinic:
Tan2
2 x y
x x y y
(3)
Isochromatics
The "isochromatics" are lines of constant color which are obtained when a source of white
light is used in the polariscope, and they are related to the level of loading. When a
monochromatic source is used in the setup, only black fringes are observed, which are labeled by
the fringe order, N=0,1,2,...., and they are caused by the extinction of the light emerging from
the analyzer, as a result of a RELATIVE RETARDATION that meets the condition: /2=n,
n=0,1,2,…The fringe order, "N", of any such isochromatic fringe is related to the stress level at
any particular point along that fringe by the stress-optic law:
1 2 f
N
h
(4)
where
and 2 are the principal stresses at that point, f is the material fringe constant, N is
the fringe order and h is the thickness of the plastic model . Contours of constant principal stress
difference are therefore observed as 1-2 isochromatic lines. It is obvious from Eq.(4) that a
large number of fringes (large fringe orders) indicate regions of high stress in the model. In
general, the principal-stress difference and the principal-stress directions vary from point to point
in a photoelastic model. As a result the isoclinic fringe pattern and the isochromatic fringe
pattern are SUPERIMPOSED when the model is viewed through a PLANE polariscope.
Circular Polariscope
In many cases, simultaneous occurrence of both isoclinics and isochromatics sets of fringes can
obscure the interpretation of the fringe pattern. A refinement of the plane polariscope is obtained
by inserting quarter wave (/4) plates before and after the model, oriented at an inclination of
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MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
45˚ with respect to the axis of the polarizer. This arrangement, which is illustrated in Fig.1
below, is called a circular polariscope, where the isoclinics are removed, leaving only the
isochromatics for the analysis of stress levels.
S
P
F
A
W
M
L
/4 L
/4
Fig.1 – Common Schematic of Circular Polariscope
S - Light Source
F- Filter
L - Field Lens
M - Test Specimen
P - Polarizer
4 - Quarter-Wave Plate
A - Analyzer
W - Wall or camera
There are two arrangements for the orientation between the polarizer and the analyzer in order to
generate two modes of birefringence: “dark field” and “bright field”.
Dark field: If the polarizer is perpendicular to the analyzer (noted as “PA”), the "QuarterWave Plate", which is a particular type of "Wave Plate", is designed such that the birefringence
effect generates (in the ABSENCE of any stress) an angular retardation of =/2 (Wave Plates
that produce angular retardations of =, or 2, are known as "Half, and Full-Wave" Plates,
respectively).
polarizer
analyzer
The light emerging from the analyzer of a Circular Polariscope in a Dark Field arrangement, is
given by the following equation:
I K sin 2 ( )
2
5
MAE244
n ,
2
OPTICAL Methods of Stress Analysis - Photoelasticity
for
n 0,1,2,3,....... or
N
0,
2
1,
Lab-4A
2, n
(5)
which is clearly a function ONLY of the principal-stress DIFFERENCE, since the angle ""
does not appear in the amplitude of the wave. The last equation indicates that the extinction that
creates the dark fringes occurs, as in the case of a plane polariscope, when /2=n, n=0,1,2,….
Bright filed: If the polarizer is parallel to the analyzer (noted as “P//A”) in a CIRCULAR
Polariscope, similar to that depicted in Fig.1, A bright filed will be generated. And then the black
fringes observed in the camera "W" correspond to different fringe orders, N=0.5, 1.5, 2.5, ...
since the "fringe order" N, and the integer "n" no longer coincide. Indeed, the light intensity is
proportional to the square of the amplitude of the light-wave, the light emerging from the
analyzer in this type of arrangement is given by:
I K cos 2 ( / 2), which shows that extinction occurs when
1 2n
1
, for n 0,1,2,3,....... or N
n
2
2
2 2
(6)
polarizer
analyzer
Intermediate data can thus be obtained, for smaller increments of stress levels. When BOTH the
light- and the dark field arrangements are used in a Circular Polariscope, it is possible to obtain
TWO photographs of the resulting fringe patterns, that provide a WHOLE-FIELD
representation of the fringe orders, to the nearest ½ order.
Calibration
Any photoelastic material must be calibrated before an experiment, in order to obtain the
material fringe constant, f., the simplest approach to the calibration of a new model is the
uniaxial tension test of a corresponding specimen, and the following analysis of its results:
P
, 2 0
wh
giving
P = f wN
1
w = width
(7)
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MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
Therefore, "f " can be obtained from the slope of the graph of applied load versus fringe order.
Photoelastic Analysis
A photoelastic model that is geometrically similar to the actual structural component is made
of a proper transparent polymer, and then observed in the polariscope, under the proper loading
conditions. The fringes of integer order are sketched or photographed, and labeled. If the
accuracy of the measured stress levels is important, then the half order fringe data are also
collected. Based on the stress-optic relationship, the difference between the principal stresses
(or the maximum shear stress) can be determined at any point in the body.
Additional information is required if the principal stress components have to be determined
separately (rather than merely the difference between them). At free boundaries the separated
stresses are obtained immediately since one of the principal stresses is zero. A beam in pure
bending can be assumed to experience only uniaxial stress, since the bending moment is uniform.
Stress Concentrations
Since the method of photoelasticity is not restricted to simple geometric configurations, nor
to regular boundary shapes, it is applied widely and effectively to analyzing stress
concentrations, and calculating stress concentration factors around holes and other geometric
discontinuities. Furthermore, the point of maximum stress does not have to be known a priori,
since photoelasticity provides stress distributions over a finite area. The separate components of
principal and maximum shear stresses can be obtained directly from the fringe data, without the
need for tedious strain/stress transformations.
A geometric discontinuity (hole or notch, for example, Fig.2) in a body causes a localized
increase in stress, even though the externally applied loads remain unchanged. The stress
concentration factor is defined as the ratio between the maximum local stress around the region
of geometric discontinuity, and to the nominal, reference or undisturbed stress in the structure, in
regions that are sufficiently remote from the discontinuity. It is calculated through the following
equations:
Stress Concentration Factor =
Kt
Maximum Stress
Nominal Stres s
ma x
nom
(8)
Specially,
K tg
max
norm
where
,
(9)
gross
f N
h
P
nom gross
hw
max
is the gross remote stress
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MAE244
K tn
OPTICAL Methods of Stress Analysis - Photoelasticity
max
,
norm
where normnet
net
P
is the net remote stress
h( w 2r )
Lab-4A
(10)
max is determined from fringe patterns. Hence, Ktn can be determined by the photoelastic
experiment
From equation (9) and (10),
K tg K tn
w
w 2r
(11)
In the above equations, P is the applied load, h is the specimen thickness, w is the specimen
width and r is the radius of the notch. The choice of reference stress is not always obvious or
unique, so when a stress concentration factor is stated, it is also necessary to define the state of
reference stress.
For a strip with double notched in tension (Fig.2), the theoretical stress concentration Ktn is
expressed by:
K tn 3.065 3.472(
2r
2r
2r
) 1.009( ) 2 0.405( )3
w
w
w
0.465"
d
h = 0.125"
Tensile Specimen
w=2.000”
h=0.120”
d=0.885”
r=(w-d)/2=0.5575’’
w
r
thickness, h
Grooved Specimen
Fig.2 - Photoelastic Test Specimens Made of Polycarbonate
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MAE244
OPTICAL Methods of Stress Analysis - Photoelasticity
Lab-4A
Figure 3 Four-Point Bending Beam Specimen
Another useful test configuration is Four-Point Bending Beam system (Fig.3), since the bending
moment is constant between the innermost loads. To show this we consider moment balance in
the region between the innermost loads:
M
P
P
(x d ) x 0
2
2
Hence, the moment acting is M
Pd
2
The beam experiences a pure bending moment. The magnitude of the bending stress, , at
distance y from the neutral axis is
x = - My / I
y = 0
τxy = 0
Where I is the beam moment of inertia.
1 3
bh
12
In the beam, the longitudinal stress (x) is a principal stress, there are no vertical or horizontal
shear stresses and no transverse normal stresses. Therefore, if at a point P a distance y from the
neutral axis there exists a stress which would cause a phase difference of one wave length, then
all points on the horizontal line through P parallel to the longitudinal axis of the beam would
cause the same phase difference.
I
[1] http://www.doitpoms.ac.uk/tlplib/photoelasticity/index.php
9
Introduction
Photoelasticity describes the changes in
the optical properties of a translucent material
under deformation. It is often used to
determine the stress distribution in a material.
Photoelastic tests are also a strong and useful
tool in determining and evaluating critical
stress points of a material. These critical
points can be formed by any unique property
on the body, such as a hole, crack, filet, etc.
When observing the cross section of a
specimen under certain stresses, light is
passed through the body and filtered through
lenses to block out certain wavelengths of
light to be able to see clearly, the fringes that
light to be able to see clearly, the fringes that
form due to this stress. In this lab we will be
attempting to replicate this method and
accurately calculate the stress elements within
a specimen.

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