Full field stress analysis of window security films by phase-stepping photoelasticity

tobascothwackUrban and Civil

Nov 15, 2013 (3 years and 11 months ago)

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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

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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

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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

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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

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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
















2
/
4
cos
2
/
4
sin
8
















s
s
s
s
c
c
I




0













2
/
4
cos
2
/
4
sin
9
















s
s
s
s
c
c
I



















2
/
4
cos
2
/
4
sin
10
















s
s
s
s
c
c
I






2
sin
sin
3
d






2
/
cos
d
c







2
/
sin
d
s







2
/
cos
s
c






2
/
sin
s
s




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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




-





-


-




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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

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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
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