Hybrid determination of mixed-mode stress intensity factors on discontinuous finite-width plate by finite element and photoelasticity

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1 Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~ DOI /s Hybrid determinatin f mixed-mde stress intensity factrs n discntinuus finite-width plate by finite element and phtelasticity Tae Hyun Baek 1,*, Lei Chen 2 and Dng Py Hng 3 1 Schl f Mechanical and Autmtive Engineering, Graduate Schl, Kunsan Natinal Univeristy, Gunsan, , Krea 2 Mechanical Engineering Department, Graduate Schl, Kunsan Natinal Univeristy, Gunsan, , Krea 3 Schl f Mechanical Engineering System, Chnbuk Natinal Univeristy, Jenju, , Krea (Manuscript Received February 10, 2011; Revised June 1, 2011; Accepted June 18, 2011) Abstract Fr istrpic material structure, the stress in the vicinity f crack tip is generally much higher than the stress far away frm it. This phenmenn usually leads t stress cncentratin and fracture f structure. Previus researches and studies shw that the stress intensity factr is ne f mst imprtant parameter fr crack grwth and prpagatin. This paper prvides a cnvenient numerical methd, which is called hybrid phtelasticity methd, t accurately determine the stress field distributin in the vicinity f crack tip and mixed-mde stress intensity factrs. The mdel was simulated by finite element methd and ischrmatic data alng straight lines far away frm the crack tip were calculated. By using the ischrmatic data btained frm finite element methd and a cnfrmal mapping prcedure, stress cmpnents and phtelastic fringes in the hybrid regin were calculated. T easily cmpare calculated phtelastic fringes with experiment results, the fringe patterns were recnstructed, dubled and sharpened. Gd agreement shws that the methd presented in this paper is reliable and cnvenient. This methd can then directly be applied t btain mixed mde stress intensity factrs frm the experimentally measured ischrmatic data alng the straight lines. Keywrds: Phtelasticity; Plariscpe; Stress intensity factr; Ischrmatics; Isclinics; Inclined crack; Mixed-mde stress intensity factr; Phtelastic fringe dubling; Fringe sharpening Intrductin Due t irregular gemetries and cmplicated wrk cnditin, structure prblems can nt be easily slved by numerical methd. It is necessary t investigate the stress distributin in a machine element r a structural part by experiment when it is under varius lads and bundary cnditins. Phtelasticity is ne kind f experimental methds which can be used t btain ischrmatics and isclinics which appear thrugh the specimen setup in a plariscpe [1-3]. Isclinics are the lcus f the pints in the specimen alng which the principal stresses are in the same directin. Ischrmatics are the lcus f the pints alng which the difference in the first and secnd principal stress, σ 1 σ 2, remains the same. Thus, they are the lines which jin the pints with equal maximum shear stress magnitude. Fr these facts, phtelasticity is used t determine stress distributin in a material. The methd is mstly used in cases where This paper was recmmended fr publicatin in revised frm by Assciate Editr Vikas Tmar * Crrespnding authr. Tel.: , Fax.: address: thbaek@kunsan.ac.kr KSME & Springer 2011 mathematical techniques becme quite cumbersme. Unlike the analytical methds fr stress determinatin, phtelasticity gives a fairly and visually accurate picture f stress distributin even arund abrupt discntinuities in a material [1, 2]. Althugh phtelasticity serves as an imprtant tl fr determining the critical stress pints in a material and is ften used fr determining stress cncentratin factrs in irregular gemetries, the phtelastic fringe patterns arund high stress cncentrated regin becme blur and ambiguus due t ptical caustics. As such, it may nt prvide an accurate data and stress distributin fr any lcatin f highly stress cncentrated regin. Hence, the hybrid methd [4-13], which cmbined the advantages f mathematical analysis and experimental measurements, was develped. In this paper, the hybrid phtelasticity methd [6, 9, 10] is emplyed. At first, the ischrmatic data f given pints are calculated by finite element methd and are used as input data f cmplex variable frmulatins. Then the numerical mdel f specimen is transfrmed frm the physical plane t the cmplex plane by cnfrmal mappings. The stress field is analyzed and mixed-mde stress intensity factrs are calculated n this cmplex plane. The results are als calculated by finite

2 2536 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~2543 under plane stress. 4 2 S11μ + (2 S12 + S66) μ + S22 = 0 (6) Fig. 1. Cnfrmal mapping f an inclined crack. element methd and empirical equatin and cmpared with each ther. 2. Theretical frmulatin 2.1 Equatins f hybrid phtelasticity methd The present technique emplys general expressins fr the stress functins with tractin-free cnditins which are satisfied at the gemetric discntinuity using cnfrmal mapping and analytical cntinuatin [14]. As shwn in Fig. 1, the inverse f the mapping functin ω, 1 namely ω, maps the gemetry f interest frm the physical z- plane int the ζ - plane ( ζ j = ξ + μjη ). Fr istrpic materials, the cnfrmal transfrmatins between unit circle in the ζ - plane and the inclined crack at an angle α t the x- axis and ttal length 2a ( a = a half crack length in the z- plane in Fig. 1) are given by Eqs. (1) and (2) [4, 14]. a iα iα 1 j = (cs + jsin )( e j + e ) (1) j 2 ω α μ α ζ ζ i 2 e α ωj ± ωj α + μ j α ζ j = a 2 a 2 ( cs sin ) ( csα + μ j sinα) where i = 1. The branches f the square rt f Eq. (2) are chsen s that ζ j 1 (j =1, 2). Then, general stress functins can be expressed in the ζ - plane. In the absence f bdy frces and rigid bdy mtin, the stresses under istrpy plane can be written as [4, 14] 2 φ ( ζ1) 2ψ σx = 2Re μ1 + μ2 ω1 ( ζ1) ω2 φ ( ζ1) ψ σ y = 2Re + ω1 ( ζ1) ω2 φ ( ζ1) ψ τxy = 2Re μ1 + μ2 ω1 ( ζ1) ω2 where φ ( ζ1) = dφ/ dζ1, ψ = dφ/ dζ2, ω 1( ζ1) = dω / dζ 1, ω 2 = dω / dζ 2 and Re stands fr real part f the functin. Cmplex material parameters μ j ( j = 1,2) are the rts f the characteristic Eq. (6) fr an istrpic material (2) (3) (4) (5) where Sij(, i j = 1,2,6) are the elastic cmpliances. The tw cmplex stress functins φ( ζ1) and ψ ( ζ 2) are related t each ther by the cnfrmal mapping and analytic cntinuatin. Fr a tractin-free physical bundary, the tw functins within sub-regin Ω f Fig. 1 can be written as Laurent expansins, respectively [4, 14] m k φζ ( 1) = βζ k 1 ( k 0) (7) k= m m k k ψζ ( 2) = ( βk Bζ2 + βk Cζ2) ( k 0). (8) k= m Cmplex quantities B and C depend n material prperties and are defined as μ μ μ μ B =, =. μ μ μ μ C The cefficients f Eqs. (7) and (8) are β k = b k + ic k, where b k and c k are real numbers. In additin t satisfying the tractin-free cnditins n the crack bundary Γ, the stresses f Eqs. (3)-(5) assciated with these stress functins φ( ζ 1) and ψ ( ζ 2) satisfy equilibrium and cmpatibility. Cmbining Eqs. (1)-(9) gives the stress cmpnents thrugh regins Ω f Fig. 1 in matrix frm [4] as { σ} = [ V ]{ β} (10) where { σ} = { σx, σy, τxy}, { β} = { b m, c m,, bm, cm}, and [V] is a rectangular cefficient matrix whse size depends n material prperties, psitins and the number f terms m f the pwer series expansins f Eqs. (7) and (8) as belw [4]: k 1 k 1 k 1 i 1 i 1 ζ1 i 1( Bζ2 + Cζ2 ) Vij (, ) = ( 1) (2 k)re μ1 + μ ' 2 ' ω1( ζ1) ω2 (11) k 1 k 1 k 1 i 1 i 1 ζ1 i 1( Bζ2 Cζ2 ) Vij (, + 1) = ( 1) (2 k)im μ1 + μ ' 2. ' ω1( ζ1) ω2 (12) 2.2 Nnlinear least-squares methd By substituting the stress cmpnents { σ x, σy, τxy} f Eqs. (3)-(5) int Eq. (13), ne btains the basic relatinship between ischrmatic fringe rder N and the in-plane stress cmpnents, σ x, σ y and τ xy, as belw [1] Nf + { τ xy} = σ σx σy 2 2t (9) (13)

3 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~ where f σ is a material fringe cnstant and t is the thickness f the specimen. Arranging the abve expressin, an arbitrary functin G as in Eq. (14), whse value shuld be zer, ideally, is btained as fllws [15]: Nf τ σ xy n t n n σx σy Gn{ β} = + { } = (14) A truncated Taylr series expansin f the unknwn parameters can linearize Eq. (14) with respect t unknwn cnstants { β } and an iterative prcedure is develped with Fig. 2. Crdinate system f the inclined crack. m G ( ) 1 ( ) n Gn i+ Gn i+ Δβn. c k= m i (15) Knwing { σ} at varius n lcatins enables ne t slve fr the best values f the unknwn cefficients {β} in the nnlinear least-squares sense frm Eq. (15). The subscript i indicates the number f iteratin step. Fr measured fringe rders and a predetermined value f m f Eqs. (7) and (8), the cefficients {β} in Eq. (10) are btained by nnlinear leastsquare methd [15]. 2.3 Dubling and sharpening techniques fr ischrmatic images The techniques f fringe dubling and sharpening were emplyed in rder t btain accurate ischrmatic fringe patterns [10, 16-18]. Fr fringe dubling technique [16], tw images are used as belw: I = I I = Acs(2 π N) (16) R L D where I L and I D are the light intensities f the light field and dark-field ischrmatic fringe patterns, respectively. In rder fr I R f Eq. (16) t be zer, cs(2πn) shuld be zer. In a circular plariscpe arrangement, dark and light fringes appear as a half-rder interval alternately (N = 0, 1/2, 1, 3/2, 2, ). Hwever, after fringe multiplicatin, dark and light fringes, whse fringe rders are N = 0, 1/4, 2/4, 3/4, 1, 5/4, etc., appear as a quarter-rder interval alternately. As a result, fringe patterns prcessed by Eq. (16) are twice-multiplied images. The sharpening technique [10, 16, 18] described here cmes frm the prprtins f the gradient vectr. T sharpen phtelastic fringes, measured changes in the gradient directin thrughut an area are used. The peratr T, which is used fr sharpening fringes, is given by Eq. (17) x + y T = A 1 x + y where A is a prprtinality cnstant, x and (17) y are x Fig. 3. Uni-axially laded finite-width tensile plate cntaining an inclined crack. and y directinal cmpnents f the phtelastic fringe gradient vectr, respectively. 2.4 Stress intensity factr The crdinate system f an inclined crack whse length is 2a in the plate is shwn in Fig. 2. The inclinatin f the crack is psitined as angle α with respect t x y crdinates. As shwn in Fig. 2, the crack lies alng the x -axis in the physical z-plane and a pint (r, θ) are the lcal plar crdinates measured frm the crack tip. When θ=0 and r<<a, where a is the half f crack length, the stress intensity factr f mde I and Mde II is determined as fllws: KI σ y' 2π r KII τ x' y ' 2π r = (18a) = (18b) where σ x ' σ y ' and τ x' y ' are btained frm Eqs. (3) thrugh (5) and crdinate transfrmatin. 3. Experiment and analysis 3.1 Mdel gemetry and specimen In this phtelasticity experiment, t btain the reference fringes which are cmpared with thse f finite element analyses, a PSM-1* plate shwn in Fig. 3 was subjected t the uni-axial tensile lad. This phtelastic material used in ur * Phtelastic Divisin, Measurement Grup, Inc., Raleigh, NC 27611, USA.

4 2538 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~2543 Table 1. Material prperties f PSM-1* nd gemetries f the specimen. Descriptin Symbl Value Elastic mdulus E 2482 MPa Pissn's rati ν 0.38 Phtelasticity cnstant fσ 7005 N/m Tensile stress σ 3.05 MPa Initial crack length 2a 12.7 mm Width f specimen W 38.1 mm Thickness f specimen t mm experiment is characterized by excellent transparency, easy machinability and high-stress ptic cnstant. This material is nn-brittle and shws free frm time-edge effects. The inclinatin angle f a crack is ranged frm 0 t 60 by 15 degrees interval and the width f crack is 0.5 mm. The crack tip was machined t V-shape s that it simulated a natural crack tip. The material prperties and dimensins f specimen are given by Table 1. Fig. 4(a). Dark and light field fringe patterns f inclined crack ( α = 0 ). Fig. 4(b). Dark and light field fringe patterns f inclined crack ( α = 15 ). 3.2 Phtelastic fringes btained by experiment By changing ptical arrangement f circular plariscpe, dark- and light-field images were captured by CCD camera. Figs. 4(a)-(e) shw the dark- and light-field fringe patterns f the laded tensile plate cntaining an inclined crack. Experiment fringe patterns were digitized as 640*480 pixel bmp files and grey level ranged frm 0 t 255. Fr the cmparisn f recnstructed fringes btained by hybrid FEM methd with experimentally measured fringes, these images were then prcessed t have dubled and sharpened fringes by the in-huse develped image prcessing prgrams [16, 17]. Fig. 4(c). Dark and light field fringe patterns f inclined crack ( α = 30 ). 3.3 FEM analysis In rder t calculate ischrmatic fringe rders f given pints arund the crack tip in uni-axially laded finite width tensile plate by finite element methd, a cmmercial sftware was used. ABAQUS [19] is a kind f widely used FEM sftware and its analysis results are knwn t be reliable. As shwn in Fig. 5, the tensile laded finite-width plate was simulated by ABAQUS. The specimen was discretized int tw kinds f elements, CPS3 (3-nde linear plane stress triangle element) and CPS4R (4-nde bilinear plane stress quadrilateral element). Bth the ischrmatic data alng the lines arund the crack tip as shwn in Fig. 3 and the mixed mde stress intensity factrs f each specimen were calculated by ABAQUS. The vn-mises stress distributin f ABAQUS mdel fr the crack inclinatin angle f 45 degrees is shwn as Fig Hybrid phtelasticity analysis In rder t btain the input data f hybrid methd, the ischrmatic fringe rders f given pints alng the lines f A- Fig. 4(d). Dark and light field fringe patterns f inclined crack ( α = 45 ). Fig. 4(e). Dark and light field fringe patterns f inclined crack ( α = 60 ). B, B-C and C-D as shwn in Fig. 3 are required. Accrding t the stress-ptic law f Eq. (13), the ischrmatic fringe rders ( Ninp ) at thse pints can be expressed by using the stress

5 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~ Table 2. Cmparisn f input and calculated fringe rders alng the lines f A-B, B-C, C-D and D-A arund the crack tip. Fig. 5. FEM mdel f specimen in ABAQUS (crack inclinatin angle α = 45 ). Fig. 6. Vn-Mises stress distributin f the laded tensile plate btained by ABAQUS discretizatin and analysis (crack inclinatin angle α = 45 ). cmpnents f σ x, σ y and τ xy. Fr given ischrmatic fringe rders calculated by FEM sftware (ABQAQUS) and a predetermined value f m in Eqs. (7) and (8), cefficients { β} f Eq. (10) were btained by least-squares methd [9, 10] as belw: ( T ) 1 T { β} = [ V] [ V] [ V] { σ}. (19) Thus, a stress cmpnent at any pint in the hybrid regin can be calculated by using Eq. (10). Als, the ischrmatic fringe rders ( N cal ) at the same given pints alng the lines f A-B, B-C and C-D can then be cmputed [20]. Then, the percentage errr ( E ) between the input fringes ( N inp ) and the calculated fringes ( N cal ) at any pint is ( Ncal Ninp) E = 100 %. (20) N inp Table 2 shws the cmparisn f input and calculated fringe rders alng the lines f A-B, B-C and C-D arund the crack tip as shwn in Fig. 3. As shwn in Table 2, the maximum percentage errr ( E ) between the input fringes ( N inp ) and the calculated fringes ( N cal ) at any pint is 4.54%. T shw the physical effect, full fringes were recnstructed using the stress cmpnents ( σ x, σ xy, τ xy ) and were shwn in Figs. 7(a)-(e). In rder t cnveniently cmpare calculated results with actual fringes btained frm phtelastic experiment, bth f dark-field fringes and light-field fringes are presented. Als, dubled and sharpened fringes by N x (mm) y (mm) N inp N cal E (%) digital image prcessing [16, 17] which uses Eqs. (16) and (17) are pltted. Hybrid methd with m=1 in Eqs. (7) and (8) was used in all the recnstructed fringes as shwn in Figs. 7(a)-(e). In rder t cmpare the recnstructed fringes analyzed by hybrid FEM with actual fringes btained frm phtelastic experiment, dark-field fringes f hybrid FEM (i) and experiment (ii) as shwn in Figs. 8(a)-(e) are presented. Als, sharpened fringes frm hybrid FEM (iii) and sharpened fringes frm experiment (iv) are cmpared. The fringes f hybrid FEM (i) f Figs. 8(a)-(e) are the same fringes (i) f Figs. 7(a)-(e). The sharpened fringes frm

6 2540 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~2543 fringes fringes (i) Dark field fringes (ii) Light field fringes (iii) Dubled fringes Fig. 7(a). Recnstructed fringes f inclined crack ( α = 0 ). (iii) Dubled fringes Fig. 7(b). Recnstructed fringes f inclined crack ( α = 15 ). (i) Dark field fringes (ii) Light field fringes (i) Dark field fringes (ii) Light field fringes (iii) Dubled fringes Fig. 7(c). Recnstructed fringes f inclined crack ( α = 30 ). (iii) Dubled fringes Fig. 7(d). Recnstructed fringes f inclined crack ( α = 45 ). (i) Dark field fringes (ii) Light field fringes (iii) Dubled fringes Fig. 7(e). Recnstructed fringes f inclined crack ( α = 60 ).

7 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~ (iii) Sharpened frm hybrid FEM (iv) Sharpened frm experiment (iii) Sharpened frm hybrid FEM (iv) Sharpened frm experiment Fig. 8(a). Cmparisns f (i) recnstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharpened fringes frm experiment f inclined crack ( α = 0 ). Fig. 8(b). Cmparisns f (i) recnstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharpened fringes frm experiment f inclined crack ( α = 15 ). (iii) Sharpened frm hybrid FEM (iv) Sharpened frm experiment (iii) Sharpened frm hybrid FEM (iv) Sharpened frm experiment Fig. 8(c). Cmparisns f (i) recnstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharpened fringes frm experiment f inclined crack ( α = 30 ). Fig. 8(d). Cmparisns f (i) recnstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharpened fringes frm experiment f inclined crack ( α = 45 ). (iii) Sharpened frm hybrid FEM (iv) Sharpened frm experiment Fig. 8(e). Cmparisns f (i) recnstructed fringes and (iii) sharpened fringes by Hybrid FEM with (ii) experimental fringes and (iv) sharpened fringes frm experiment f inclined crack ( 60 ). α =.

8 2542 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~2543 Table 3. Cmparisn f nrmalized Mde I stress intensity factrs btained frm hybrid phtelasticity, FEM and empirical apprximatin fr the plate with inclined crack. Stress intensity factr K I σ0 π a Angle Hybrid FEM [19] Equatin [21] Table 4. Cmparisn f the rati f stress intensity factrs ( KII / K I ) btained frm hybrid phtelasticity, FEM and empirical apprximatin fr the plate with inclined crack. Stress intensity factr KII KI Angle Hybrid FEM [19] Equatin [21] hybrid FEM (iii) f Figs. 8(a)-(e) are the same sharpened fringes (iv) f Figs. 7(a)-(e). Als, the fringes f phtelastic experiment (ii) f Figs. 8(a)-(e) are the same dark-field fringes (i) f Figs. 4(a)-(e). As shwn in Figs. 8(a)-(e), all the recnstructed and sharpened fringes analyzed by hybrid FEM are quite cmparable t actual fringes btained frm the phtelastic experiment. Table 3 shws the cmparisn f nrmalized Mde I stress intensity factr, KI / σ0 π a, btained frm hybrid phtelasticity, FEM and empirical apprximatin fr the plate with inclined crack. Tward a relatinship with mixed-mde stress intensity factr, Table 4 shws cmparisn f the rati f stress intensity factrs ( KI / K II ) btained frm hybrid phtelasticity, FEM and empirical apprximatin fr the plate with inclined crack. 4. Discussin and cnclusins In this study, we used stress cmpnents analyzed by finite element methd with their respective crdinates n the straight lines far away frm the tip f inclined crack in a finite width plate t calculate ischrmatic data. Using these ischrmatic data and cnfrmal mapping prcedure, a hybrid methd was emplyed t recnstruct phtelastic fringes. These recnstructed fringes were cmpared t the actual fringes btained by phtelastic experiment. Als sharpened fringes btained by hybrid FEM methd were cmpared t thse f experimental fringes. All the recnstructed fringes by hybrid FEM methd are quite cmparable t experimental nes. Frm Tables 3 and 4 and Figs. 8(a)-(e) presented abve, we can see that the results f mixed mde stress intensity factrs calculated by three kinds f different methd are well agreed with each. Cnsidering the experimental and calculated errrs, the technique presented in this paper is effective and reliable. Here we utilized FEM t btain the ischrmatic data and crdinate infrmatin f given pints alng the straight lines far away frm the crack tip. This methd can then be directly applied t btain mixed mde stress intensity factrs frm the experimentally measured ischrmatic data alng the straight lines. It may be cnvenient t use phtelastic phase-shifting methd t btain ischrmatic data alng a straight line [7, 22]. The use f hybrid methd has a ptential future and the results attained in this study can be used fr bench mark test in theretical simulatin and experiment. Acknwledgment This research was supprted by Basic Science Research Prgram thrugh the Natinal Research Fundatin f Krea (NRF) funded by the Ministry f Educatin, Science and Technlgy (Grant number: ). References [1] J. W. Dally and W. F. Riley, Experimental stress analysis, 3rd Editin, McGraw-Hill, Inc. New Yrk, USA (1991). [2] A. S. Kbayshi, Handbk n experimental mechanics, Secnd Revised Editin, Sciety fr Experimental Mechanics, VCH Publishers, Inc., New Yrk, USA (1993). [3] E. Cllett, Plarized light: Fundamentals and Applicatins, Marcel Dekker, Inc., New Yrk, USA (1993). [4] G. D. Gerhardt, A hybrid/finite element apprach fr stress analysis f ntched anistrpic materials, ASME Jurnal f Applied Mechanics, 51 (1984) [5] T. H. Baek and T. J. Rudlphi, A hybrid stress measurement using nly x-displacement by phase shifting methd with furier transfrm (PSM/FT) in laser speckle interfermetry and least squares methd, Internatinal Jurnal f Precisin Engineering and Manufacturing, 11 (1) (2010) [6] T. H. Baek, Phtelastic stress analysis by use f hybrid technique and fringe phase shifting methd, Jurnal f Experimental Mechanics, 21 (1) (2006) [7] T. H. Baek, Measurement f stress distributin arund a circular hle in a plate under bending mment using phaseshifting methd with reflective plariscpe arrangement, Jurnal f Slid Mechanics and Material Engineering, 2 (4) (2008) [8] T. H. Baek, T. J. Chung and H. Panganivan, Full-field stress determinatin arund circular discntinuity in a tensileladed plate using x-displacements nly, Jurnal f Slid Mechanics and Material Engineering, 2 (6) (2008) [9] T. H. Baek, H. Panganiban and T. J. Chung, A hybrid phtelastic stress analysis arund hles in tensile-laded plates using ischrmatic data and finite element methd, Lecture Series n Cmputer and Cmputatinal Sciences, 8, Brill

9 T. H. Baek et al. / Jurnal f Mechanical Science and Technlgy 25 (10) (2011) 2535~ Academic Publishers, The Netherlands (2007) [10] T. H. Baek, M. S. Kim, J. Rhee and R. E. Rwlands, Hybrid stress analysis f perfrated tensile plates using multiplied and sharpened phtelastic data and cmplex-variable techniques, JSME Internatinal Jurnal, Series A: Slid Mechanics and Material Engineering, 43 (4) (2000) [11] O. S. Lee, J. C. Park and G. H. Kim, Dynamic mixed mde crack prpagatin behavir f structural bnded jints, KSME Internatinal Jurnal, 14 (7) (2000) [12] J. S. Hawng, J. H. Nam, K. H. Kim, O. S. Kwn, G. Kwn and S. H. Park, A study n the develpment f phtelastic experimental hybrid methd fr clr ischrmatics, Jurnal f Mechanical Science and Technlgy, 24 (6) (2010) [13] J. S. Hawng, C. H. Lin, S. T. Lin, J. Rhee and R. E. Rwlands, A hybrid methd t determine individual stresses in rthtrpic cmpsites using nly measured ischrmatic data, Jurnal f Cmpsite Material, 29 (18) (1995) [14] G. N. Savin, Stress cncentratin arund hles, Pergamn Press, New Yrk, USA (1961). [15] R. J. Sanfrd, Applicatin f the least squares methd t the phtelastic analysis, Experimental Mechanics, 20 (6) (1980) [16] T. H. Baek and J. C. Lee, Develpment f image prcessing technique fr phtelastic fringe analysis, Transactins f Krean Sciety fr Mechanical Engineers, 18 (10) (1994) [17] T. H. Baek, Digital image prcessing technique fr phtelastic ischrmatic fringe sharpening, Jurnal f the Krean Sciety fr Precisin Engineering, 10 (3) (1993) [18] T. H. Baek and C. P. Burger, Accuracy imprvement technique fr measuring stress intensity factr in phtelastic experiment, KSME Internatinal Jurnal, 5 (1) (1991) [19] ABAQUS analysis user s manual, ABAQUS Inc., Pr- vidence, RI 02909, USA. [20] L. Chen, S. Jin, B. H. Lee, M. S. Kim and T. H. Baek, Analysis f phtelastic stress field arund inclined crack tip by using hybrid technique, Transactins f Krean Sciety fr Mechanical Engineers A, 34 (9) (2010) [21] T. L. Andersn, Fracture mechanics, 3 rd Ed., CRC Press Taylr & Francis Grup, USA (2005). [22] T. H. Baek, H. Panganiban, C. T. Lee and T. J. Chung, Hybrid full-field stress analysis arund hles in tensileladed plates by Phase-shifting phtelasticity, Key Engineering Materials, (2007) Tae Hyun Baek received a B.S degree in Mechanical Engineering frm Hanyang University in Then he received his M.S degree in 1984 and Ph. D degree in 1986 frm Iwa State University in USA, respectively. Dr. Baek is currently a prfessr f Mechanical and Autmtive Engineering at Kunsan Natinal University, Jenbuk, Krea. Prf. Baek s research interests include experimental stress analysis, finite element methd and numerical analysis, etc. Lei Chen received a B.S degree in Mechanical Engineering frm Qingda Ocean University in Then he received his M.S degree frm Ocean University f China in Mr. Chen is currently a Ph.D. candidate at the graduate schl f Kunsan Natinal University, in Kunsan city, Krea. Mr. Chen s research interests include stress analysis, experimental mechanics and tplgy ptimizatin, etc.