OPTIMESS2009
Buytaert
1
1
2
Full field stress analysis
of window security films
3
by phase

stepping photoelasti
city
4
E. Stoykova
a
,
W. Van Paepegem
b
,
G. Minchev
a
, B. Ivanov
a
,
5
A. J. Degrieck
b
, V
.
Sainov
a
6
7
8
Abstract
9
The paper presents a two

dimensional interferometric photoelastic
phase

10
stepping
measurement in reflection mode with a Mach

Zender interferometer
11
combine
d with a circular polariscope. As such, the paper continues the study
12
on photoelastic analysis of the mechanical behaviour of thermoplastic
13
polyester window security films by including retrieval of isopachics
14
parameter.
A series of six photoelastic fringe
patterns are recorded at
15
different preliminary known orientations of the polarization elements
in the
16
circular polariscope to
build
both isochromatic and isoclinic
phase maps
17
which give
the loci of points with a constant difference of principal stresses
18
an
d constant principal stress direction respectively
.
Holographic recording
19
of four fringe patterns is applied for retrieval of isopachic fringes
which give
20
the sum of principal stresses
.
The coherent length of the used DPSS laser
21
VERDI2 exceeded substantia
lly the optical path difference between the
22
reference and object beam which ensured high contrast of the recorded
23
interference patterns.
Measurement of both isochromatic and isopachic
24
maps permits to separate the stress components all over the sample.
The
25
samples were covered with a photoelastic birefringent PhotoStress coating.
26
T
he change of the photoelastic pattern for film samples with mechanical
27
stress concentrators (holes and
notches
) under tensile load
ing
i
s observed.
28
To study the protective properti
es of the security film fitted to a glass we
29
made a photoelastic
interferometric
measurement of an assembly glass/film
30
with a stress concentrator which had been simulated by sticking two
31
precisely cut glass pieces with parallel edges in contact with each o
ther
32
between two films
. A two

load phase

shifting technique
is
used for
33
unambiguous phase
retrieval of
photoelastic parameters.
34
35
36
Contact information
37
a
elena@optics.bas.bg
38
Central Laboratory of Optical Storage
and Processing of Information (CLOSPI

BAS),
39
Bl. 101, Akad. G. Bonchev St., Sofia 1113, Bulgaria
40
b
Wim.VanPaepegem@UGent.be
41
Ghent University, Dept. of Mechanical Construction and Production,
42
Sint

Pietersnieuwst
raat 41, 9000 Gent, Belgium
43
OPTIMESS2009
Buytaert
2
44
Introduction
45
Photoelasticity is an effective tool for full

field a
nalysis of stress distribution in mechanical
46
parts
with
high
geometrical complexity.
V
arious automated
photoelastic methods in static and
47
dynamic mode of operat
ion
with both monochromatic and white light sources and numerical
48
analysis performed in space or frequency domains
have
been recently proposed
[1

3
]
. A
mong
49
them
the
phase

s
tepping
technique [
4

7
]
has established itself as
a reliable
point

wise
50
approach
whi
ch
manifests good sensitivity and accuracy.
Depending on the optical set

up
51
used, the output of the phase

s
tepping
photoelastic measurement yields the difference or the
52
sum of principal stresses within a sample.
The difference of principal stresses can be
obtained
53
using a circular polariscope (
polarizer
–
quarter wave plate
–
sample
–
quarter wave plate
–
54
analyzer)
.
A series of photoelastic fringe patterns are recorded at different preliminary known
55
orientations of the polarization elements
to retrieve bot
h
isochromatic
and
isoclinic
56
distributions
[8]
which give
the
loci of points with
a
constant difference of principal stresses
57
and constant principal stress direction
respectively
.
The sum of principal stresses is obtained
58
from isopachics by means of holog
raphic recording.
For the purpose, a Mach
–
Zehnder type
59
interferometer can be combined with
the
circular polariscope to enable interference of
60
photoelastic fringes with a reference beam at different phase steps
[
9
]
.
Measurement of both
61
isochromatic and isop
achic maps permits to separate the stress components all over the
62
sample [10].
Nevertheless, retrieval of isopachics is a topic which is
comparatively
rarely
63
addressed.
The photoelastic phase

s
tepping
methods have been applied mostly in transmission
64
type a
rrangement
s
. Only recently, some results of application of this technique in reflection
65
mode have been reported for photoelastic coatings as e.g. adaptation of the two

wavelength
66
polarization

stepping method [11] and usage of a reflective polariscope arran
gement [12].
67
Extraction of all parameters in the
phase

measuring technique is a typical phase retrieval task
68
which involves
inverse trigonometric functions and requires application of unwrapping
69
techniques.
Most of the algorithms
for determination of isoch
romatics
are based on
70
acquisition of six photoelastic fringe patterns. A four

step phase

shifting method
was
71
proposed
at
the assumption
of constant
background and contrast in the intensity equations at a
72
given point for all acquired images
[1
3
]
.
T
he main c
hallenge in the photoelastic fringe
73
analysis is the interaction between the isochromatics and isoclinics/isopachics in the output
74
signal that may lead to ambiguity in evaluation of the measured parameters.
Another obstacle
75
for accurate data processing is c
reated by the isotropic points, in which the difference between
76
principal stresses is zero
[14]
.
In addition, poor signal

to

noise ratios could be observed in
77
some regions.
Different approaches have been developed to ensure unambiguous retrieval of
78
isochro
matics
and
isoclinics
as interpolation for low

modulation zones, multiple wavelength
79
acquisition which relies on the spectral dependence of the isochromatic parameter
[15]
,
80
multiple load stepping
[16,17]
in which, instead of different wavelengths, differen
t loads are
81
used, or
regularized phase

tracking
[18]
.
Using of white light is one of the ways to diminish
82
the interaction between
the
isoclinics and isochromatics
[2]
.
83
This paper presents a two

dimensional interferometric photoelastic measurement of
84
isoch
romatics/isopachics in reflection mode with a Mach

Zender interferometer combined
85
with a circular polariscope. As such, the paper
is a
continuation
of
the
study
started in [
19
]
86
which has been focused on
photoelastic analysis of the mechanical behaviour of
thermoplastic
87
polyester window security films.
In [
19
] t
he
analysis included only retrieval
of
88
isochromatics/isoclinics using
a circular reflection polariscope
at
white light and
89
monochromatic illumination (575 nm).
So, one of the
goal
s
of this report is t
o give
90
experimental verification
of isopachics retrieval.
91
OPTIMESS2009
Buytaert
3
Experimental set

up and light intensity equations
92
93
Experimental set

up
94
The schematic diagram of the
system for
two

dimensional interferometric photoelastic
95
measurement of isochromatics/isopachics
in reflection mode is depicted in Figure 1.
The light
96
source is a DPSS CW generating laser which emits a vertically polarized light at 532 nm.
We
97
use
P
,
Q
, and
A
to denote
a polarizer, a quarter

wave
(retardation)
plate, and an analyzer,
98
respectively; the
subscript indicates the angle formed by the transmission axis of the
99
polarizers or by the fast axes of the retardation plates
with the reference axis X
.
The first
100
quarter

wave plate Q
/4
is positioned at the output of the laser to produce a
circular
ly
101
pola
rized light which is collimated by the lens L
2
after the beam expander (lens L
1
) and the
102
spatial pinhole fi
l
ter (SF).
A
10% beam splitter forms the two arms of a Mach

Zender
103
interferometer with
the
90
percents
of light
used to illuminate the object and
the
remaining
104
10% redirected as a reference beam.
The sample was illuminated
normally to its surface
and
105
observed
using
one and
the same 50% beam splitter
.
The sample was covered with a
106
PhotoStress
birefringent coating.
.
At good adhesion, the displacements
on the surface of the
107
sample are transformed into the coating inducing the birefringence.
When the shutter in the
108
reference beam is closed, the system operates as a circular polariscope for photoelastic
109
measurement of isochromatics and isoclinics. The phas
e steps
were
introduced by rotation of
110
the second quarter

wave plate Q
and the analyzer A
.
For the isopachic measurement the
111
shutter
was
open and the interference patterns of object and reference beams
were
recorded
112
after the second 50% beam splitter, at different phase shifts, produced by
rotation of
Q
and
113
parallel tran
smission axes of
the analyzer
A
and
the
polarizer P
.
The objective L
3
form
ed
114
image plane recording of the fringe patterns.
The whole system
wa
s assembled on a vibro

115
isolated holographic table.
For this reason we designed a
special device
, also positioned
on
116
the holographic table,
to apply increasing loads to the assembly of the sample and the coating.
117
The fringe patterns were recorded with a Baumer Optics CCD camera with 782x582 pixels.
118
The coherent length of the used DPSS laser VERDI2 exceed
ed
substantia
lly the optical path
119
difference between the reference and object beam which ensure
d
high contrast of the recorded
120
interference patterns.
121
122
Figure 1:
Optical arrangement for phase

stepping photoelastic measurement of isochromatics, isoclinics and
123
isopachic
s. L
1
–
beam expander, L
2
–
collimator,
L
3
–
objective, SF
–
spatial filter, BS
–
beam

splitte
r
, M
–
124
mirror, Q
–
quarter

wave plate, P
–
polarizer, A
–
analyzer.
125
126
OPTIMESS2009
Buytaert
4
Light intensity equations
127
To set the necessary phase steps in the two channels of the set

up
in Figure 1, we adopted the
128
approach developed in [
10,20
]. The formulas derived in [
10,20
]
have been
checked by
129
numerical simulation of a full

field stress measurement of a disk under tensile loading.
A
130
birefringent sample
,
R
whose pr
incipal stress (1 direction subtends an isoclinic angle θ with
131
the reference axis X is characterized with the following Jones matrix:
132
2
2
2
2
,
sin
cos
sin
cos
sin
cos
sin
cos
1
2
2
1
2
1
2
1
i
i
i
i
i
i
i
i
e
e
e
e
e
e
e
e
R
(1.1)
133
where
1
and
2
are the phase retardations along the directions of princ
ipal stresses
1
and
134
2
of the specimen, respectively.
Determination of the difference and sum of both phase
135
retardations allows for evaluation of the difference and sum of the principal stresses as
136
follows:
137
2
1
2
1
2
Ct
d
(1.2)
138
139
2
1
2
1
2
Dt
s
(1.3)
140
where
is the laser wavelength,
t
is the thickness of the sample, and
C
and
D
are the optical
141
constants of the sample.
The parallel,
E
p
, and normal,
E
n
, to the analyzer axis components
of
142
the
light vector can be described as
143
t
i
E
P
Q
R
Q
A
E
E
n
p
exp
0
2
/
4
/
,
(
1.4
)
144
where
0
E
is the amplitude of the light vector
,
/
2
. The Jones matrices in (
1.4
) are given
145
by the following expressions:
146
147
cos
sin
sin
cos
A
,
2
cos
1
sin
sin
2
cos
1
2
2
i
i
i
i
Q
,
1
1
2
2
i
i
Q
/4
,
1
0
/2
P
(
1.5
)
148
149
When the interferometric channel is closed,
the intensity registered at a given pixel of the
150
CCD camera is described by the following equation:
151
2
sin
2
cos
sin
2
sin
cos
1
2
1
d
d
I
(
1.6
)
152
where
we have omitted
the backg
round intensity, and
assume that
1
2
0
E
.
Different pha
se
153
steps in the circular polariscope are introduced by rotation of the analyzer and the second
154
quarter

wave plate. The values of
the
angles
and
,
as well as the resulting light inte
nsity
155
equations
,
are given in Table 1.
From these equations one readily obtains the formulas for
156
calculation of
the
wrapped phase maps of
isoclinics and isochromatics
:
157
4
6
5
3
2
1
I
I
I
I
arctg
,
0
sin
d
(1.7)
158
1
2
4
6
5
3
2
cos
2
sin
tan
I
I
I
I
I
I
d
(1.8)
159
The phase maps of the isopachics are calculated
using four interference patterns given by the
160
equations in Table 2. The isopachic parameter is calculated from
161
10
9
8
7
10
9
8
7
arctan
2
1
I
I
I
I
I
I
I
I
s
at
0
2
cos
2
/
cos
d
(1.9)
162
OPTIMESS2009
Buytaert
5
To solve the probl
em with the isoclinic angle ambiguity we applied the two

load method
as is
163
described in [
9,17
]
.
If the phase difference in
retardations
d1
,
d2
obtained at two loads
164
1
2
P
P
is smaller than
, the regions of incorrect determination of is
oclinic and isochromatic
165
parameters are given by
166
1
2
2
1
0
d
d
d
d
(1.10)
167
For illustration Fig
ure
2 shows the
photoelastic and
interference fringes recorded at
the phase
168
steps listed in Table 1 and Table 2
at one of the
tensile
loadings
for a th
in
(0.1 mm)
sheet of a
169
copper alloy with a hole as a stress concentrator.
The copper alloy was chosen as an elastic
170
material for testing the system.
All recorded fringe patterns were filtered using a James

Stein
171
filter [
21
]. The size of the filtering windo
w in the case
of the photoelastic patterns was 5x5
172
pixels. The filter was combined with a homomorphic transformation for processing of the
173
holographic patterns.
Both quality

guided and Goldstein algorithms
[
22,23
]
were used for
174
unwrapping of the isochroma
tic phase maps. In the case of isopachics we applied the
175
unwrapping procedure proposed in [
9
] which permits to eliminate
the reference phase shift.
176
The wrapped and unwrapped phase maps of isochromatics, isoclinics and isopachics for the
177
copper sheet with a
hole are shown in Figure 3 and Figure 4 respectively.
The applied two
178
loads for calculation of the maps were 75N and 120 N.
Although the applied loading is not
179
enough to obtain photoelastic patterns with a regular structure, the photoelastic effect is
180
cle
arly seen.
181
182
Table 1
Light intensity equations for determination of isochromatics and isoclinics
183
184
I
i
0
/4
2
/
cos
1
1
d
I
0
3
/4
2
/
cos
1
2
d
I
0
0
2
/
sin
2
sin
1
3
d
I
/4
/4
2
/
sin
2
cos
1
4
d
I
/2
/2
2
/
sin
2
sin
1
5
d
I
3
/4
3
/4
2
/
sin
2
cos
1
4
d
I
185
186
Table 2
Light intensity equations for determination of isopachics
187
188
189
190
I
i
0
0
2
/
4
cos
2
/
4
sin
7
s
s
s
s
c
c
I
0
⼲
2
/
4
cos
2
/
4
sin
8
s
s
s
s
c
c
I
⼲
0
2
/
4
cos
2
/
4
sin
9
s
s
s
s
c
c
I
⼲
⼲
2
/
4
cos
2
/
4
sin
10
s
s
s
s
c
c
I
2
sin
sin
3
d
Ⱐ
2
/
cos
d
c
Ⱐ
2
/
sin
d
s
Ⱐ
2
/
cos
s
c
Ⱐ
2
/
sin
s
s
OPTIMESS2009
Buytaert
6
191
Figure
2
:
A
thin sheet of copper alloy with a hole under tensile load
ing (P = 75 N)
: photoelastic fringe patterns
192
(a,b,c,d,e,f) correspo
nd to intensities I
1

I
6
in Table1, interferometric fringes (g,h,i,j) corresponds to intensities I
7

193
I
10
in Table2
(
the
width and
the
thickness of the sheet
are 20 mm and 0.1
mm
respectively
)
.
194
195
196
197
198
199
200
201
202
203
Figure
3
:
Wrapped phased maps of isochromatics (left
), isoclinics (middle) and isopachics (right
) for
a copper
204
sheet with a hole under tensile load
ing
(P
1
=
75
N
,
P
2
=
120
N
)
.
205
206
207
208
209
210
211
212
213
214
Figure
4
:
Unwrapped phased maps of isochromatics (left), isoclinics (middle) and isopachics (right
) for
a
215
copper sheet with a
hole under tensile load
ing
(
P
1
=75 N, P
2
=120 N;
t
he scale
s are
in units of
).
216
.
217
218
Experimental
results
219
The
performed photoelastic
experiment included measurement of isochromatics/isopachics of
220
a security film with different stress concentrators at pure tens
ile loading. The thickness of the
221
tested film was 0.1 mm.
We used a thick PhotoStress
coating with the following parameters:
222
thickness
t
= 3 mm, fringe value of the coating
f
= 54700, and strain optical coefficient K =
223
0.009
at
=
575 nm.
Installation of
the coating on the sample was made using adhesives of
224
the producer. Th
e studied samples were
with a length
2
35
mm and width
20
mm.
Having in
225
mind the non

linear mechanical response of the tested polyester materials, a constant strain
226
was maintained during
acquisition of the fringe patterns.
227
228
a
b
a
c
d
e
f
a
g
h
i
j



OPTIMESS2009
Buytaert
7
229
230
231
232
233
234
235
236
237
238
239
240
241
242
(a)
(b)
(c)
243
244
Figure
5
:
Unwrapped phase maps of isochromatics (top) and isopachics (bottom)
for
two
plastic
security films
245
with a hole under
a
tensile load
ing
: (a)
–
sample 1,
P=
45
N
, (b)
–
sample 2
,
P=
45
N, (c)
–
sample 2,
P =
75
N
246
(the scale
s are
in units of
)
.
247
248
Figure
6
:
Unwrapped phase maps of isochromatics (left), isoclinics (middle) and isopachics (
right
) for
a plastic
249
security film
with a hole and two notches under tensile
load
ing
(the scale
s are
in units of
).
250
(a)
(b)
(c)
251
.
252
Figure
7
:
Unwrapped phase maps of isochromatics (top) and isopachics (bottom)
for
an assembly glass/film:
a

253
45N, b
–
60
N, c

75 N
(the scales are in units of
).
The stress concentrat
or is a crack simulated by sticking two
254
precisely cut glass pieces between two plastic security films.
255
256
OPTIMESS2009
Buytaert
8
257
Figure 8:
Unwrapped
phase maps of isochromatics (left
)
, isoclinics (middle)
and isopachics (
right)
for
an
258
assembly glass/film at tensile loading of
60 N
(the scales are in units of
).
The stress concentrator is a crack
259
simulated by sticking two precisely cut glass pieces
between two plastic security
film
s
.
260
261
262
First
,
we measured the patterns for a film with a hole
with a
diameter
of
6 mm in the middle
263
as a stress concentrator. The unwrapped isochromatic and isopachic maps
for two
sample
264
films
are depicted in Figure 5.
The expected symmetric distribution of the principal stresses at
265
pure tensile load can be clearly seen. Even more, despite of the time

d
epending non

linear
266
mechanical response of the tested materials, the results from the holographic recording at
267
static loading are completely satisfactory.
Two notches were added at the opposite edges of
268
the film with a hole normally to the applied loading
for the next measurement. The
269
unwrapped phase maps of isochromatics, isoclinics and isopachics
obtained at 120 N and 140
270
N
are shown in Figure 6.
The obtained isopachic phase map indicates that more sophisticated
271
denoising of the holographic patterns is r
equired.
272
273
To study the protective properties of the security film fitted to a glass we made a photoelastic
274
measurement of an assembly glass/film with a stress concentrator which had been
simulated
275
by two precisely cut glass pieces with parallel edges in c
ontact with each other
. The glass
276
pieces
were st
uck
between two films
. Thus a “crack” was formed perpendicular to the axis of
277
the tensile load. The obtained results
for two such samples
are shown in Fig
ure 7 and Figure
278
8.
As it can be seen, the two samples
demonstrate completely different behavior.
The
279
difference
in
the
obtained
distribution
s
in both cases could be
due to
the non

uniform sticking
280
of the films onto the glasses
and influence of the boundary effects. The results on Figure 8
281
manifest practical
ly uniform stress distribution.
282
283
In summary, we demonstrated
reliable retrieval of isochromatics, isoclinics and isopachics for
284
window security films with different concentrators under tensile loading.
Since the specifics
285
of the studied thermoplastic ma
terials requires simultaneous recording of the phase

shifted
286
patterns in the non

linear part of their loading curve,
a future development
envisages
287
realization of an
optical arrangement for real

time recording as
a
solution
for
non

linear
288
dynamic tasks.
Th
e challenge is to record all the required phase

shifted images
289
simultaneously which is much more complicated task in the case of photoelasticity than in the
290
conventional phase

shifting profilometry due to the necessity to ensure different orientations
291
of t
he optical elements for introducing the phase steps.
292
293
294
Acknowledgment
295
296
The work was made in the frame of bilateral scientific collaboration of CLOSPI

BAS
,
297
Bulgaria,
and Technical Univ
e
rsity of Ghent
, Belgium
.
298
OPTIMESS2009
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9
References
299
300
[1]
RAMESH
,
K.,
Digital photoelas
ticity,
Berlin:
Springer,
2000.
301
[2]
AJOVALASIT A., PETRU
CCI
,
G., SCAFIDI
,
M.,
Phase shifting photoelasticity in white light”,
302
Opt. Las. Eng. 45
:
596
–
611
,
2007.
303
[3]
BEREZHNA
,
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