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Journal of Mechanical Science and Technology 25 (10) (2011) 2535~2543
www.springerlink.com/content/1738494x
DOI 10.1007/s1220601107401
Hybrid determination of mixedmode stress intensity factors on discontinuous
finitewidth plate by finite element and photoelasticity†
Tae Hyun Baek1,*, Lei Chen2 and Dong Pyo Hong3
1School of Mechanical and Automotive Engineering, Graduate School, Kunsan National Univeristy, Gunsan, 573701, Korea
2Mechanical Engineering Department, Graduate School, Kunsan National Univeristy, Gunsan, 573701, Korea
3School of Mechanical Engineering System, Chonbuk National Univeristy, Jeonju, 561756, Korea
(Manuscript Received February 10, 2011; Revised June 1, 2011; Accepted June 18, 2011)

Abstract
For isotropic material structure, the stress in the vicinity of crack tip is generally much higher than the stress far away from it. This
phenomenon usually leads to stress concentration and fracture of structure. Previous researches and studies show that the stress intensity
factor is one of most important parameter for crack growth and propagation. This paper provides a convenient numerical method, which
is called hybrid photoelasticity method, to accurately determine the stress field distribution in the vicinity of crack tip and mixedmode
stress intensity factors. The model was simulated by finite element method and isochromatic data along straight lines far away from the
crack tip were calculated. By using the isochromatic data obtained from finite element method and a conformal mapping procedure,
stress components and photoelastic fringes in the hybrid region were calculated. To easily compare calculated photoelastic fringes with
experiment results, the fringe patterns were reconstructed, doubled and sharpened. Good agreement shows that the method presented in
this paper is reliable and convenient. This method can then directly be applied to obtain mixed mode stress intensity factors from the
experimentally measured isochromatic data along the straight lines.
Keywords: Photoelasticity; Polariscope; Stress intensity factor; Isochromatics; Isoclinics; Inclined crack; Mixedmode stress intensity factor; Photoelastic
fringe doubling; Fringe sharpening

1. Introduction
Due to irregular geometries and complicated work condi
tion, structure problems can not be easily solved by numerical
method. It is necessary to investigate the stress distribution in
a machine element or a structural part by experiment when it
is under various loads and boundary conditions.
Photoelasticity is one kind of experimental methods which
can be used to obtain isochromatics and isoclinics which ap
pear through the specimen setup in a polariscope [13].
Isoclinics are the locus of the points in the specimen along
which the principal stresses are in the same direction.
Isochromatics are the locus of the points along which the
difference in the first and second principal stress,
remains the same. Thus, they are the lines which join the
points with equal maximum shear stress magnitude. For these
facts, photoelasticity is used to determine stress distribution in
a material. The method is mostly used in cases where
12
σσ
−
,
mathematical techniques become quite cumbersome. Unlike
the analytical methods for stress determination, photoelasticity
gives a fairly and visually accurate picture of stress
distribution even around abrupt discontinuities in a material [1,
2]. Although photoelasticity serves as an important tool for
determining the critical stress points in a material and is often
used for determining stress concentration factors in irregular
geometries, the photoelastic fringe patterns around high stress
concentrated region become blur and ambiguous due to
optical caustics. As such, it may not provide an accurate data
and stress distribution for any location of highly stress concen
trated region.
Hence, the hybrid method [413], which combined the advan
tages of mathematical analysis and experimental measurements,
was developed. In this paper, the hybrid photoelasticity method
[6, 9, 10] is employed. At first, the isochromatic data of given
points are calculated by finite element method and are used as
input data of complex variable formulations. Then the numeri
cal model of specimen is transformed from the physical plane to
the complex plane by conformal mappings. The stress field is
analyzed and mixedmode stress intensity factors are calculated
on this complex plane. The results are also calculated by finite
† This paper was recommended for publication in revised form by Associate Editor
Vikas Tomar
*Corresponding author. Tel.: +82 63 469 4714, Fax.: +82 63 469 4727
Email address: thbaek@kunsan.ac.kr
© KSME & Springer 2011
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T. H. Baek et al. / Journal of Mechanical Science and Technology 25 (10) (2011) 2535~2543
element method and empirical equation and compared with
each other.
2. Theoretical formulation
2.1 Equations of hybrid photoelasticity method
The present technique employs general expressions for the
stress functions with tractionfree conditions which are satis
fied at the geometric discontinuity using conformal mapping
and analytical continuation [14].
As shown in Fig. 1, the inverse of the mapping function ω ,
namely
ω−, maps the geometry of interest from the physical
 z plane into the

ζ plane (
j
ζ
terials, the conformal transformations between unit circle in
the

ζ plane and the inclined crack at an angle α to the
 x axis and total length 2a (a = a half crack length in the
 z plane in Fig. 1) are given by Eqs. (1) and (2) [4, 14].
a
e
ωαµα
=+
1
j
ξ µ η
=+
). For isotropic ma
1
(cos sin )()
2
j
ii
jjj
e
αα
ζζ
−−
+
(1)
()
()
2
2
j
2cos sin
cos
sin
i
j
j
j
j
ea
a
αω
ωαµα
ζ
αµα
±−+
=
+
(2)
where
chosen so that 
Then, general stress functions can be expressed in the

ζ plane. In the absence of body forces and rigid body motion,
the stresses under isotropy plane can be written as [4, 14]
()
2Re
()
ω ζω ζ
′′
φ ζ ψ ζ
σ
ω ζω ζ
′′
φ ζψ ζ
τµµ
ω ζ ω ζ
′
where
11
()/,dd
φ ζφζ
′
=
ψ ζ
/,dd
ωζ
222
()/
dd
ωζωζ
=
and ‘Re’ stands for real part of
the function. Complex material parameters
the roots of the characteristic Eq. (6) for an isotropic material
1
i =− . The branches of the square root of Eq. (2) are
 1
j
ζ
≥ (j =1, 2).
2
1
2
2
12
112
2
(
(
)
)
x
φ ζ
′
ψ ζ
′
σµ
µ
=+
(3)
12
1122
(
(
)
)
(
(
)
)
2Re
y
′′
=+
(4)
12
12
112
′
2
(
(
)
)
(
(
)
)
2Re
xy
′′
= −+
(5)
22
()/,
dd
φζ
′
=
11
()
ω ζ
′
=
1
′
(1,2)
jj
µ=
are
under plane stress.
4
11
S
µ
+
where
i j
S
The two complex stress functions
related to each other by the conformal mapping and analytic
continuation. For a tractionfree physical boundary, the two
functions within subregion Ω of Fig. 1 can be written as
Laurent expansions, respectively [4, 14]
m
k
k
km
=−
m
k
k
k
km
=−
Complex quantities B and C depend on material properties
and are defined as
µµµµ
µµµµ
−−
The coefficients of Eqs. (7) and (8) are
where bk and ck are real numbers. In addition to satisfying the
tractionfree conditions on the crack boundary Γ, the stresses
of Eqs. (3)(5) associated with these stress functions
and
2
()
ψ ζ
satisfy equilibrium and compatibility.
Combining Eqs. (1)(9) gives the stress components through
regions Ω of Fig. 1 in matrix form [4] as
{ } [ ]{ }
V
σβ
=
where { } {,,}
xy xy
σ σ σ τ
=
, { } {
β
[V] is a rectangular coefficient matrix whose size depends on
material properties, positions and the number of terms m of the
power series expansions of Eqs. (7) and (8) as below [4]:
ζ
µ
ω ζ
+= −−
2.2 Nonlinear leastsquares method
2
126622
(2)0
SSS
µ
++= (6)
( ,
i j =
1,2,6)
are the elastic compliances.
1
()
φ ζ
and
2
()
ψ ζ
are
11
()( 0)
k
φ ζ β ζ
=≠
∑
(7)
222
()() (0).
k
BCk
ψ ζβζβ
+
ζ
−
=≠
∑
(8)
1
212
22
22
,.
BC
−−
==
(9)
kkk
bic
β =+
,
1
()
φ ζ
(10)
,,,,}
mmmm
bcbc
−−
=
⋯
, and
111
111
1
'
1
22
12
'
212
()
( , )V i j( 1)
= −
(2 )Rek
()()
kkk
iii
BC
ζζ
µ
ω ζ
−− −−
−−−
−+
+
(11)
1
)
(12)
11
111
1
'
1
22
12
'
212
(
( , 1)( 1)(2 )Im
k
.
()()
kkk
iii
BC
V i j
ζ
ω ζ
ζζ
µµ
ω ζ
− − −−
−−−
−−
+
By substituting the stress components {
(3)(5) into Eq. (13), one obtains the basic relationship be
tween isochromatic fringe order N and the inplane stress
components,
x
σ ,
y
σ and
xy
τ
,,}
xyxy
σ σ τ
of Eqs.
, as below [1].
2
2
2
{}
22
xy
xy
Nf
t
σ
σ
σ
τ
−
+=
(13)
Fig. 1. Conformal mapping of an inclined crack.
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T. H. Baek et al. / Journal of Mechanical Science and Technology 25 (10) (2011) 2535~2543
2537
where fσ is a material fringe constant and t is the thick
ness of the specimen.
Arranging the above expression, an arbitrary function G as
in Eq. (14), whose value should be zero, ideally, is obtained as
follows [15]:
2
2
{ }{}
2
n
A truncated Taylor series expansion of the unknown pa
rameters can linearize Eq. (14) with respect to unknown con
stants { }
β and an iterative procedure is developed with
m
n
n in i
km
=−
Knowing { }
σ at various “n” locations enables one to solve
for the best values of the unknown coefficients {β} in the
nonlinear leastsquares sense from Eq. (15). The subscript i
indicates the number of iteration step. For measured fringe
orders and a predetermined value of m of Eqs. (7) and (8), the
coefficients {β} in Eq. (10) are obtained by nonlinear least
square method [15].
2.3 Doubling and sharpening techniques for isochromatic
images
2
0.
2
xy
n xy n
τ
n
Nf
G
t
σ
σ
σ
β
−
=+−=
(14)
1
()().
n
i
G
∂
GG
c
β
+
∂
≅+∆
∑
(15)
The techniques of fringe doubling and sharpening were em
ployed in order to obtain accurate isochromatic fringe patterns
[10, 1618]. For fringe doubling technique [16], two images
are used as below:
cos(2)
RLD
IIIAN
π
=−=
where
L I and
D
I are the light intensities of the light field
and darkfield isochromatic fringe patterns, respectively. In
order for
R
I of Eq. (16) to be zero, cos(2πN) should be zero.
In a circular polariscope arrangement, dark and light fringes
appear as a halforder interval alternately (N = 0, 1/2, 1, 3/2, 2,
…). However, after fringe multiplication, dark and light
fringes, whose fringe orders are N = 0, 1/4, 2/4, 3/4, 1, 5/4, etc.,
appear as a quarterorder interval alternately. As a result,
fringe patterns processed by Eq. (16) are twicemultiplied
images.
The sharpening technique [10, 16, 18] described here comes
from the proportions of the gradient vector. To sharpen photo
elastic fringes, measured changes in the gradient direction
throughout an area are used. The operator T, which is used for
sharpening fringes, is given by Eq. (17)
∇∑+∇∑
yx
(16)
∇∑+∇∑
−=
yx
AT1
(17)
where A is a proportionality constant,
x
∇ and
y
∇ are x
and y directional components of the photoelastic fringe gra
dient vector, respectively.
2.4 Stress intensity factor
The coordinate system of an inclined crack whose length is
2a in the plate is shown in Fig. 2. The inclination of the
crack is positioned as angle α with respect to x
dinates.
As shown in Fig. 2, the crack lies along the x’axis in the
physical zplane and a point (r, θ) are the local polar coordi
nates measured from the crack tip.
When θ=0 and r<<a, where a is the half of crack length,
the stress intensity factor of mode I and Mode II is determined
as follows:
y
−
coor
'2
Iy
Kr
σπ
=
(18a)
' '2
x y
τ
II
Kr
π
=
(18b)
where
(5) and coordinate transformation.
'
x
σ
'
y
σ
and
' '
x y
τ
are obtained from Eqs. (3) through
3. Experiment and analysis
3.1 Model geometry and specimen
In this photoelasticity experiment, to obtain the reference
fringes which are compared with those of finite element
analyses, a PSM1* plate shown in Fig. 3 was subjected to the
uniaxial tensile load. This photoelastic material used in our
Fig. 2. Coordinate system of the inclined crack.
Fig. 3. Uniaxially loaded finitewidth tensile plate containing an in
clined crack.
* Photoelastic Division, Measurement Group, Inc., Raleigh, NC 27611, USA.
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T. H. Baek et al. / Journal of Mechanical Science and Technology 25 (10) (2011) 2535~2543
experiment is characterized by excellent transparency, easy
machinability and highstress optic constant. This material is
nonbrittle and shows free from timeedge effects.
The inclination angle of a crack is ranged from 0° to 60° by
15 degrees interval and the width of crack is 0.5 mm. The
crack tip was machined to Vshape so that it simulated a natu
ral crack tip. The material properties and dimensions of
specimen are given by Table 1.
3.2 Photoelastic fringes obtained by experiment
By changing optical arrangement of circular polariscope,
dark and lightfield images were captured by CCD camera.
Figs. 4(a)(e) show the dark and lightfield fringe patterns of
the loaded tensile plate containing an inclined crack.
Experiment fringe patterns were digitized as 640*480 pixel
bmp files and grey level ranged from 0 to 255. For the com
parison of reconstructed fringes obtained by hybrid FEM
method with experimentally measured fringes, these images
were then processed to have doubled and sharpened fringes by
the inhouse developed image processing programs [16, 17].
3.3 FEM analysis
In order to calculate isochromatic fringe orders of given
points around the crack tip in uniaxially loaded finite width
tensile plate by finite element method, a commercial software
was used. ABAQUS [19] is a kind of widely used FEM soft
ware and its analysis results are known to be reliable. As
shown in Fig. 5, the tensile loaded finitewidth plate was
simulated by ABAQUS.
The specimen was discretized into two kinds of elements,
CPS3 (3node linear plane stress triangle element) and CPS4R
(4node bilinear plane stress quadrilateral element).
Both the isochromatic data along the lines around the crack
tip as shown in Fig. 3 and the mixed mode stress intensity
factors of each specimen were calculated by ABAQUS.
The vonMises stress distribution of ABAQUS model for
the crack inclination angle of 45 degrees is shown as Fig. 6.
3.4 Hybrid photoelasticity analysis
In order to obtain the input data of hybrid method, the
isochromatic fringe orders of given points along the lines of A
B, BC and CD as shown in Fig. 3 are required. According to
the stressoptic law of Eq. (13), the isochromatic fringe orders
(
inp
N
) at those points can be expressed by using the stress
Table 1. Material properties of PSM1* nd geometries of the specimen.
Description Symbol Value
Elastic modulus E 2482 MPa
Poisson's ratio ν 0.38
Photoelasticity constant
fσ
σ
7005 N/m
Tensile stress 3.05 MPa
Initial crack length 2a 12.7 mm
Width of specimen W 38.1 mm
Thickness of specimen t 3.175 mm
(i) Darkfield (ii) Lightfield
Fig. 4(a). Dark and light field fringe patterns of inclined crack (
o
0
α =
).
(i) Darkfield (ii) Lightfield
Fig. 4(b). Dark and light field fringe patterns of inclined crack (
o
15
α =
).
(i) Darkfield (ii) Lightfield
Fig. 4(c). Dark and light field fringe patterns of inclined crack (
o
30
α =
).
(i) Darkfield (ii) Lightfield
Fig. 4(d). Dark and light field fringe patterns of inclined crack (
o
45
α =
).
(i) Darkfield (ii) Lightfield
Fig. 4(e). Dark and light field fringe patterns of inclined crack (
o
60
α =
).
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T. H. Baek et al. / Journal of Mechanical Science and Technology 25 (10) (2011) 2535~2543
2539
components of
For given isochromatic fringe orders calculated by FEM
software (ABQAQUS) and a predetermined value of “m” in
Eqs. (7) and (8), coefficients { }
by leastsquares method [9, 10] as below:
(
Thus, a stress component at any point in the hybrid region
can be calculated by using Eq. (10). Also, the isochromatic
fringe orders (
cal
N
) at the same given points along the lines
of AB, BC and CD can then be computed [20].
Then, the percentage error ( E ) between the input fringes
(
inp
N
) and the calculated fringes (
()
100 %.
inp
N
Table 2 shows the comparison of input and calculated
fringe orders along the lines of AB, BC and CD around the
crack tip as shown in Fig. 3. As shown in Table 2, the maxi
mum percentage error ( E ) between the input fringes (
and the calculated fringes (
cal
N
To show the physical effect, full fringes were reconstructed
using the stress components (
shown in Figs. 7(a)(e). In order to conveniently compare
calculated results with actual fringes obtained from photoelas
tic experiment, both of darkfield fringes and lightfield
fringes are presented. Also, doubled and sharpened fringes by
x
σ ,
y
σ and
xy
τ
.
β of Eq. (10) were obtained
)
1
{ }
β
[ ] [ ]
V
[ ] { }.
V
TT
V
σ
−
=
(19)
cal
N
) at any point is
calinp
NN
E
−
=×
(20)
inp
N
)
) at any point is 4.54%.
x
σ ,
xy
σ
,
xy
τ
) and were
digital image processing [16, 17] which uses Eqs. (16) and
(17) are plotted. Hybrid method with m=1 in Eqs. (7) and (8)
was used in all the reconstructed fringes as shown in Figs.
7(a)(e).
In order to compare the reconstructed fringes analyzed by
hybrid FEM with actual fringes obtained from photoelastic
experiment, darkfield fringes of hybrid FEM (i) and experi
ment (ii) as shown in Figs. 8(a)(e) are presented. Also, sharp
ened fringes from hybrid FEM (iii) and sharpened fringes
from experiment (iv) are compared.
The fringes of hybrid FEM (i) of Figs. 8(a)(e) are the same
fringes (i) of Figs. 7(a)(e). The sharpened fringes from
Fig. 5. FEM model of specimen in ABAQUS (crack inclination angle
o
45
α =
).
Fig. 6. VonMises stress distribution of the loaded tensile plate ob
tained by ABAQUS discretization and analysis (crack inclination angle
o
45
α =
).
Table 2. Comparison of input and calculated fringe orders along the
lines of AB, BC , CD and DA around the crack tip.
No
x (mm) y (mm)
inp
N
cal
N
E (%)
1
7.000 7.000 1.381 1.425 3.199
2
7.000 5.250 0.589 0.598 1.474
3
7.000 3.500 0.203 0.206 1.190
4
7.000 1.750 0.996 0.982 1.402
5
7.000 0.000 1.789 1.730 3.253
6
7.000 1.750 2.582 2.484 3.788
7
7.000 3.500 3.376 3.251 3.696
8
7.000 5.250 4.170 4.051 2.844
9
7.000 7.000 4.964 4.898 1.332
10
5.444 7.000 4.259 4.334 1.766
11
3.889 7.000 3.554 3.626 2.030
12
2.333 7.000 2.849 2.952 3.642
13
0.778 7.000 2.144 2.113 1.407
14
0.778 7.000 1.439 1.413 1.781
15
2.333 7.000 0.734 0.724 1.325
16
3.889 7.000 0.029 0.028 0.912
17
5.444 7.000 0.676 0.666 1.520
18
7.000 7.000 1.381 1.398 1.222
19
7.000 5.250 0.589 0.603 2.310
20
7.000 3.500 0.203 0.207 1.747
21
7.000 1.750 0.996 1.016 2.023
22
7.000 0.000 1.789 1.818 1.660
23
7.000 1.750 2.582 2.598 0.632
24
7.000 3.500 3.376 3.420 1.310
25
7.000 5.250 4.170 4.235 1.565
26
7.000 7.000 4.964 5.049 1.714
27
5.444 7.000 4.259 4.341 1.931
28
3.889 7.000 3.554 3.631 2.184
29
2.333 7.000 2.849 2.978 4.544
30
0.778 7.000 2.144 2.233 4.196
31
0.778 7.000 1.439 1.468 2.076
32
2.333 7.000 0.734 0.738 0.565
33
3.889 7.000 0.029 0.029 0.866
34
5.444 7.000 0.676 0.685 1.217