Hybrid determination of mixed-mode stre ss intensity ... · However, after fringe multiplication, dark and light fringes, whose fringe orders are N = 0, 1/4, 2/4, 3/4, 1, 5/4, etc.,

Journal of Mechanical Science and Technology 25 (10) (2011) 2535~2543

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0740-1

Hybrid determination of mixed-mode stress intensity factors on discontinuous

finite-width 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, 573-701, Korea 2Mechanical Engineering Department, Graduate School, Kunsan National Univeristy, Gunsan, 573-701, Korea

3School of Mechanical Engineering System, Chonbuk National Univeristy, Jeonju, 561-756, 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 mixed-mode 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; Mixed-mode 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 [1-3]. 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, 1 2σ σ− , 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

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 [4-13], 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 mixed-mode 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 E-mail address: [email protected]

© KSME & Springer 2011

2536 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 traction-free 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 1ω− , maps the geometry of interest from the physical

-z plane into the -ζ plane ( j jζ ξ μ η= + ). For isotropic ma-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].

1(cos sin )( )2 j

i ij j j

a e eα αω α μ α ζ ζ− −= + + (1)

( )( )

22 2 cos sin

cos sin

ij j j

jj

e a

a

α ω ω α μ αζ

α μ α

⎧ ⎫± − +⎨ ⎬

⎩ ⎭=+

(2)

where 1i = − . The branches of the square root of Eq. (2) are chosen so that | | 1jζ ≥ (j =1, 2).

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]

2 21 21 2

1 1 2 2

( ) ( )2Re( ) ( )x

φ ζ ψ ζσ μ μω ζ ω ζ

⎡ ⎤′ ′= +⎢ ⎥′ ′⎣ ⎦

(3)

1 2

1 1 2 2

( ) ( )2Re( ) ( )y

φ ζ ψ ζσω ζ ω ζ⎡ ⎤′ ′

= +⎢ ⎥′ ′⎣ ⎦ (4)

1 21 2

1 1 2 2

( ) ( )2Re( ) ( )xy

φ ζ ψ ζτ μ μω ζ ω ζ

⎡ ⎤′ ′= − +⎢ ⎥′ ′⎣ ⎦

(5)

where 1 1( ) / ,d dφ ζ φ ζ′ = 2 2( ) / ,d dψ ζ φ ζ′ = 1 1( )ω ζ′ =

1/ ,d dω ζ 2 2 2( ) /d dω ζ ω ζ′ = and ‘Re’ stands for real part of the function. Complex material parameters ( 1,2)j jμ = are the roots of the characteristic Eq. (6) for an isotropic material

under plane stress.

4 211 12 66 22(2 ) 0S S S Sμ μ+ + + = (6)

where ( , 1,2,6)i jS i j = are the elastic compliances.

The two complex stress functions 1( )φ ζ and 2( )ψ ζ are related to each other by the conformal mapping and analytic continuation. For a traction-free physical boundary, the two functions within sub-region Ω of Fig. 1 can be written as Laurent expansions, respectively [4, 14]

1 1( ) ( 0)m

kk

k m

kφ ζ β ζ=−

= ≠∑ (7)

2 2 2( ) ( ) ( 0).m

k kkk

k m

B C kψ ζ β ζ β ζ−

=−

= + ≠∑ (8)

Complex quantities B and C depend on material properties

and are defined as

12 1 2

2 22 2, .B C

μ μ μ μμ μ μ μ

− −= =

− − (9)

The coefficients of Eqs. (7) and (8) are k k kb i cβ = + ,

where bk and ck are real numbers. In addition to satisfying the traction-free conditions on the crack boundary Γ , the stresses of Eqs. (3)-(5) associated with these stress functions 1( )φ ζ 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

{ } [ ]{ }Vσ β= (10)

where { } { , , }x y xyσ σ σ τ= , { } { , , , , }m m m mb c b cβ − −= , and [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]:

1 1 1

1 1 11 2 21 2' '

1 1 2 2

( )( , ) ( 1) (2 )Re( ) ( )

k k ki i i B CV i j k ζ ζ ζμ μ

ω ζ ω ζ

− − − −− − −⎧ ⎫− +⎪ ⎪= − +⎨ ⎬

⎪ ⎪⎩ ⎭

(11) 1 1 1

1 1 11 2 21 2' '

1 1 2 2

( )( , 1) ( 1) (2 ) Im .( ) ( )

k k ki i i B CV i j k ζ ζ ζμ μ

ω ζ ω ζ

− − − −− − −⎧ ⎫− −⎪ ⎪+ = − − +⎨ ⎬

⎪ ⎪⎩ ⎭

(12)

2.2 Nonlinear least-squares method

By substituting the stress components { , , }x y xyσ σ τ of Eqs. (3)-(5) into Eq. (13), one obtains the basic relationship be-tween isochromatic fringe order N and the in-plane stress components, xσ , yσ and xyτ , as below [1].

2 2

2{ }2 2

x yxy

Nftσσ σ

τ−⎧ ⎫⎪ ⎪ ⎧ ⎫+ =⎨ ⎬ ⎨ ⎬

⎩ ⎭⎪ ⎪⎩ ⎭ (13)

Fig. 1. Conformal mapping of an inclined crack.

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{ } { } 0.2 2

x yn xy n

nn

NfGtσσ σ

β τ−⎧ ⎫⎪ ⎪ ⎧ ⎫= + − =⎨ ⎬ ⎨ ⎬

⎩ ⎭⎪ ⎪⎩ ⎭ (14)

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

1( ) ( ) .m

nn i n i n

k m i

GG Gc

β+=−

∂⎛ ⎞≅ + Δ⎜ ⎟∂⎝ ⎠∑ (15)

Knowing { }σ at various “n” locations enables one to solve

for the best values of the unknown coefficients {β} in the nonlinear least-squares 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

The techniques of fringe doubling and sharpening were em-ployed in order to obtain accurate isochromatic fringe patterns [10, 16-18]. For fringe doubling technique [16], two images are used as below:

cos(2 )R L DI I I A Nπ= − = (16)

where LI and DI are the light intensities of the light field and dark-field isochromatic fringe patterns, respectively. In order for RI of Eq. (16) to be zero, cos(2πN) should be zero. In a circular polariscope arrangement, dark and light fringes appear as a half-order 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 quarter-order interval alternately. As a result, fringe patterns processed by Eq. (16) are twice-multiplied 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

yxAT 1 (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 y− coor-dinates.

As shown in Fig. 2, the crack lies along the x’-axis in the physical z-plane 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:

' 2I yK rσ π= (18a)

' ' 2II x yK rτ π= (18b)

where 'xσ 'yσ and ' 'x yτ are obtained from Eqs. (3) through (5) and coordinate transformation.

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 PSM-1* plate shown in Fig. 3 was subjected to the uni-axial tensile load. This photoelastic material used in our

Fig. 2. Coordinate system of the inclined crack.

Fig. 3. Uni-axially loaded finite-width tensile plate containing an in-clined crack.

* Photoelastic Division, Measurement Group, Inc., Raleigh, NC 27611, USA.


experiment is characterized by excellent transparency, easy machinability and high-stress optic constant. This material is non-brittle and shows free from time-edge 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 V-shape 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 light-field images were captured by CCD camera. Figs. 4(a)-(e) show the dark- and light-field 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 in-house 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 uni-axially 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 finite-width plate was simulated by ABAQUS.

The specimen was discretized into two kinds of elements, CPS3 (3-node linear plane stress triangle element) and CPS4R (4-node 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 von-Mises 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, B-C and C-D as shown in Fig. 3 are required. According to the stress-optic law of Eq. (13), the isochromatic fringe orders ( inpN ) at those points can be expressed by using the stress

Table 1. Material properties of PSM-1* nd geometries of the specimen.

Description Symbol Value

Elastic modulus E 2482 MPa

Poisson's ratio ν 0.38

Photoelasticity constant fσ 7005 N/m

Tensile stress σ 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) Dark-field (ii) Light-field Fig. 4(a). Dark and light field fringe patterns of inclined crack ( o0α = ).

(i) Dark-field (ii) Light-field Fig. 4(b). Dark and light field fringe patterns of inclined crack ( o15α = ).

(i) Dark-field (ii) Light-field Fig. 4(c). Dark and light field fringe patterns of inclined crack ( o30α = ).

(i) Dark-field (ii) Light-field Fig. 4(d). Dark and light field fringe patterns of inclined crack ( o45α = ).

(i) Dark-field (ii) Light-field Fig. 4(e). Dark and light field fringe patterns of inclined crack ( o60α = ).


components of xσ , yσ and xyτ . For given isochromatic fringe orders calculated by FEM

software (ABQAQUS) and a predetermined value of “m” in Eqs. (7) and (8), coefficients { }β of Eq. (10) were obtained by least-squares method [9, 10] as below:

( ) 1{ } [ ] [ ] [ ] { }.T TV V Vβ σ

−= (19)

Thus, a stress component at any point in the hybrid region

can be calculated by using Eq. (10). Also, the isochromatic fringe orders ( calN ) at the same given points along the lines of A-B, B-C and C-D can then be computed [20].

Then, the percentage error ( E ) between the input fringes ( inpN ) and the calculated fringes ( calN ) at any point is

( )

100 %.cal inp

inp

N NE

N−

= × (20)

Table 2 shows the comparison of input and calculated

fringe orders along the lines of A-B, B-C and C-D around the crack tip as shown in Fig. 3. As shown in Table 2, the maxi-mum percentage error ( E ) between the input fringes ( inpN ) and the calculated fringes ( calN ) at any point is 4.54%.

To show the physical effect, full fringes were reconstructed using the stress components ( xσ , xyσ , xyτ ) and were shown in Figs. 7(a)-(e). In order to conveniently compare calculated results with actual fringes obtained from photoelas-tic experiment, both of dark-field fringes and light-field fringes are presented. Also, doubled and sharpened fringes by

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, dark-field 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

o45α = ).

Fig. 6. Von-Mises stress distribution of the loaded tensile plate ob-tained by ABAQUS discretization and analysis (crack inclination angle

o45α = ).

Table 2. Comparison of input and calculated fringe orders along the lines of A-B, B-C , C-D and D-A around the crack tip. No x (mm) y (mm) inpN calN 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


(i) Dark-field fringes (ii) Light-field fringes

(iii) Doubled fringes (iv) Sharpened fringes Fig. 7(a). Reconstructed fringes of inclined crack ( o0α = ).

(i) Dark field fringes (ii) Light field fringes

(iii) Doubled fringes (iv) Sharpened fringes Fig. 7(c). Reconstructed fringes of inclined crack ( o30α = ).


(iii) Doubled fringes (iv) Sharpened fringes Fig. 7(b). Reconstructed fringes of inclined crack ( o15α = ).


(iii) Doubled fringes (iv) Sharpened fringes Fig. 7(d). Reconstructed fringes of inclined crack ( o45α = ).


(iii) Doubled fringes (iv) Sharpened fringes Fig. 7(e). Reconstructed fringes of inclined crack ( o60α = ).


(i) Hybrid FEM (ii) Experiment

(iii) Sharpened from hybrid FEM (iv) Sharpened from experiment Fig. 8(a). Comparisons of (i) reconstructed fringes and (iii) sharpenedfringes by Hybrid FEM with (ii) experimental fringes and (iv) sharp-ened fringes from experiment of inclined crack ( 0 ).α = °


(iii) Sharpened from hybrid FEM (iv) Sharpened from experiment Fig. 8(c). Comparisons of (i) reconstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharp-ened fringes from experiment of inclined crack ( 30 ).α = °


(iii) Sharpened from hybrid FEM (iv) Sharpened from experiment

Fig. 8(e). Comparisons of (i) reconstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharpenedfringes from experiment of inclined crack ( 60 ).α = ° .


(iii) Sharpened from hybrid FEM (iv) Sharpened from experiment Fig. 8(b). Comparisons of (i) reconstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharp-ened fringes from experiment of inclined crack ( 15 ).α = °


(iii) Sharpened from hybrid FEM (iv) Sharpened from experiment Fig. 8(d). Comparisons of (i) reconstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharp-ened fringes from experiment of inclined crack ( 45 ).α = °


hybrid FEM (iii) of Figs. 8(a)-(e) are the same sharpened fringes (iv) of Figs. 7(a)-(e). Also, the fringes of photoelastic experiment (ii) of Figs. 8(a)-(e) are the same dark-field fringes (i) of Figs. 4(a)-(e).

As shown in Figs. 8(a)-(e), all the reconstructed and sharp-ened fringes analyzed by hybrid FEM are quite comparable to actual fringes obtained from the photoelastic experiment.

Table 3 shows the comparison of normalized Mode I stress intensity factor, 0/IK aσ π , obtained from hybrid photoe-lasticity, FEM and empirical approximation for the plate with inclined crack.

Toward a relationship with mixed-mode stress intensity fac-tor, Table 4 shows comparison of the ratio of stress intensity factors ( /I IIK K ) obtained from hybrid photoelasticity, FEM and empirical approximation for the plate with inclined crack.

4. Discussion and conclusions

In this study, we used stress components analyzed by finite element method with their respective coordinates on the straight lines far away from the tip of inclined crack in a finite width plate to calculate isochromatic data. Using these isochromatic data and conformal mapping procedure, a hybrid method was employed to reconstruct photoelastic fringes. These reconstructed fringes were compared to the actual fringes obtained by photoelastic experiment. Also sharpened fringes obtained by hybrid FEM method were compared to those of experimental fringes. All the reconstructed fringes by hybrid FEM method are quite comparable to experimental ones.

From Tables 3 and 4 and Figs. 8(a)-(e) presented above, we can see that the results of mixed mode stress intensity factors calculated by three kinds of different method are well agreed with each. Considering the experimental and calculated errors, the technique presented in this paper is effective and reliable.

Here we utilized FEM to obtain the isochromatic data and coordinate information of given points along the straight lines far away from the crack tip. This method can then be directly applied to obtain mixed mode stress intensity factors from the experimentally measured isochromatic data along the straight lines. It may be convenient to use photoelastic phase-shifting method to obtain isochromatic data along a straight line [7, 22]. The use of hybrid method has a potential future and the results attained in this study can be used for bench mark test in theoretical simulation and experiment.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2010-0021248).

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Table 3. Comparison of normalized Mode I stress intensity factors obtained from hybrid photoelasticity, FEM and empirical approxima-tion for the plate with inclined crack.

Stress intensity

factor Angle Hybrid FEM [19] Equation [21]

0 1.075 1.062 1.065

15 0.982 0.986 0.994

30 0.782 0.804 0.799

45 0.492 0.526 0.532 0

IKaσ π

60 0.248 0.264 0.267

Table 4. Comparison of the ratio of stress intensity factors ( /II IK K ) obtained from hybrid photoelasticity, FEM and empirical approxima-tion for the plate with inclined crack.

Stress intensity

factor Angle Hybrid FEM [19] Equation [21]

0 0 0 0

15 0.274 0.262 0.268

30 0.558 0.574 0.577

45 1.062 1.006 1.000

III

KK

60 1.648 1.746 1.732


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[13] J. S. Hawong, C. H. Lin, S. T. Lin, J. Rhee and R. E. Row-lands, A hybrid method to determine individual stresses in orthotropic composites using only measured isochromatic data, Journal of Composite Material, 29 (18) (1995) 2366-2387.

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[15] R. J. Sanford, Application of the least squares method to the photoelastic analysis, Experimental Mechanics, 20 (6) (1980) 192-197.

[16] T. H. Baek and J. C. Lee, Development of image process-ing technique for photoelastic fringe analysis, Transactions of Korean Society for Mechanical Engineers, 18 (10) (1994) 2577-2584.

[17] T. H. Baek, Digital image processing technique for photoe-lastic isochromatic fringe sharpening, Journal of the Korean Society for Precision Engineering, 10 (3) (1993) 220-230.

[18] T. H. Baek and C. P. Burger, Accuracy improvement tech-nique for measuring stress intensity factor in photoelastic experiment, KSME International Journal, 5 (1) (1991) 22-27.

[19] ABAQUS analysis user’s manual, ABAQUS Inc., Pro-

vidence, RI 02909, USA. [20] L. Chen, S. Jin, B. H. Lee, M. S. Kim and T. H. Baek,

Analysis of photoelastic stress field around inclined crack tip by using hybrid technique, Transactions of Korean Society for Mechanical Engineers A, 34 (9) (2010) 1287-1292.

[21] T. L. Anderson, Fracture mechanics, 3rd Ed., CRC Press Taylor & Francis Group, USA (2005).

[22] T. H. Baek, H. Panganiban, C. T. Lee and T. J. Chung, Hybrid full-field stress analysis around holes in tensile-loaded plates by Phase-shifting photoelasticity, Key Engi-neering Materials, 345-346 (2007) 865-868.

Tae Hyun Baek received a B.S degree in Mechanical Engineering from Han-yang University in 1971. Then he re-ceived his M.S degree in 1984 and Ph. D degree in 1986 from Iowa State Uni-versity in USA, respectively. Dr. Baek is currently a professor of Mechanical and Automotive Engineering at Kunsan

National University, Jeonbuk, Korea. Prof. Baek’s research interests include experimental stress analysis, finite element method and numerical analysis, etc.

Lei Chen received a B.S degree in Me-chanical Engineering from Qingdao Ocean University in 2002. Then he received his M.S degree from Ocean University of China in 2006. Mr. Chen is currently a Ph.D. candidate at the graduate school of Kunsan National University, in Kunsan city, Korea. Mr.

Chen’s research interests include stress analysis, experimental mechanics and topology optimization, etc.

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Hybrid determination of mixed-mode stre ss intensity ... · However, after fringe multiplication, dark and light fringes, whose fringe orders are N = 0, 1/4, 2/4, 3/4, 1, 5/4, etc.,