VALIDATION OF CRACK INTERACTION LIMIT MODEL FOR PARALLEL EDGE CRACKS USING TWO-DIMENSIONAL FINITE ELEMENT ANALYSIS

Transcription

1 nternational Journal of Automotive and Mechanical Engineering (JAME) SSN: (Print); SSN: (Online); Volume 7, pp , January-June 0 Universiti Malaysia Pahang DO: VALDATON OF CRACK NTERACTON LMT MODEL FOR PARALLEL EDGE CRACKS USNG TWO-DMENSONAL FNTE ELEMENT ANALYSS R. Daud, A.K. Ariffin and S. Abdullah Department of Mechanical and Materials Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 4600, Bangi, Selangor, Malaysia ruslizam@unimap.edu.my Phone: ; Fax: ABSTRACT Shielding interaction effects of two parallel edge in finite thickness plates subjected to remote tension load is analyzed using a developed finite element analysis program. n the present study, the crack interaction limit is evaluated based on the fitness of service (FFS) code, and focus is given to the weak crack interaction region as the crack interval exceeds the length of (b > a). Crack interaction factors are evaluated based on stress intensity factors (SFs) for Mode SFs using a displacement extrapolation technique. Parametric studies involved a wide range of crack-to-width (0.05 a/w 0.5) and crack interval ratios (b/a > ). For validation, crack interaction factors are compared with single edge crack SFs as a state of zero interaction. Within the considered range of parameters, the proposed numerical evaluation used to predict the crack interaction factor reduces the error of existing analytical solution from.9% to 0.97% at higher a/w. n reference to FFS codes, the small discrepancy in the prediction of the crack interaction factor validates the reliability of the numerical model to predict crack interaction limits under shielding interaction effects. n conclusion, the numerical model gave a successful prediction in estimating the crack interaction limit, which can be used as a reference for the shielding orientation of other. Keywords: Crack interaction limit; interacting ; shielding effects; fitness for service. NTRODUCTON The presence of an edge cracking strip in aging critical engineering structures, e.g., aerospace structures and pressure vessel components, always places the structural integrity at risk of failure. n structural safety assessments, the presence of isolated single edge may be easier to conduct. For multiple surface, the damage tolerance of a single crack is no longer applicable owing to the existence of crack interaction behavior between the (i.e., shielding and amplification). The pattern of evolution of multiple crack interaction is different and becomes increasingly complex when two or more are present in close proximity. Table shows the past and most recent solution models for evaluating stress-shielding parameters in parallel. The list of analytical models and approaches in Table that are used to develop the model, can be categorized into energy release rate criterion, stress intensity criterion, stress field traction, and the combination of many approaches. These models have contributed to the advancement of stress shielding assessment. 99

2 Validation of crack interaction limit model for parallel edge using two-dimensional finite element analysis Parallel multiple Parallel and collinear Table. Analytical elastic shielding interaction models. Models and approaches used References Simultaneous Singular (Ratwani and Gupta, 974) ntegral based on Disturbance Stress Coefficient of nteraction (Yokobori et al., 979) Parallel-Collinear Parallel Asymptotic Approximation (Chang, 98) Row of periodic Fredholm ntegral Equations (Chen, 987) Parallel and collinear Kachanov Method by Superposition-Average Traction (Kachanov, 987, 99, 00; Kachanov and Laures, 988; Kachanov and Montagut, 986; Kachanov et al., 990) Offset parallel Edge function methods (Dwyer, 997) Periodic array Hypersingular ntegral (Choi, 997) Equation Shedding arrays of Energy release rate (Parker, 999) edge Micro- Actual Displacement Discontinuity (Gorbatikh and Kachanov, 000) Collinear and parallel Mushkelishvili and Burger s (Han et al., 00) interface vector Distributed parallel Rotated Staggered Grid (Orlowsky et al., 00) Parallel and collinear Modified Kachanov Method (Gorbatikh et al., 007; Li et al., 008) nclined and offset parallel and collinear ntegral Equation Method (Chen, 007; Saha and Ganguly, 005) Offset parallel Parallel Complex Stress Function (Sankar and Lesser, 006) Method Muskhelishvili and (Li et al., 008) Laurent s series Schmidt Method (Yang, 009) Parallel symmetric Parallel surface Singular ntegral Method (Feng and Jin, 009) Periodic edge Fourier ntegral (Yildirim et al., 0) Parallel edge Fourier and Cauchy ntegral (Chen and Wang, 0) Safety assessment of the presence of multiple is commonly referred to recommended FFS codes. However, the theoretical and practical examination of the recommended FFS codes, such as ASME Boiler and Pressure Vessel Code Section X (ASME, 998, 004), AP 579 (ASME, 007), British Standard PD6495 (BS, 99) and BS790 (BS, 997, 005), Nuclear Electric CEGB R6 (R6, 006), and JSME Fitness-for-Service Code (JSME, 000), have found that the exception of crack interaction in FFS combination rules has resulted in over-estimated and unrealistic 994

3 Daud et al. /nternational Journal of Automotive and Mechanical Engineering 7(0) fracture and failure predictions, particularly for closely spaced. Burdekin (98) presented elastic and plastic crack interaction solutions based on crack open displacement and the J-integral. The formulation by ida (98) established a new single coalesced crack, but crack interaction is excluded. Moreover, based on the ida solution, Leek and Howard (994a) proposed an estimation of a single coalesced crack and a function of crack interaction factor (Leek and Howard, 994b, 996) corresponding to stress intensity factors (SFs) at crack tips obtained by finite element (FE) analysis. An FE alternating method has been developed by O donoghue et al. (984) to solve the problem of interacting multiple in a finite solid. Moussa et al. (999) used the finite element method (FEM) to calculate the J-integral and used it to analyze the interaction of two identical parallel non-coplanar surface subjected to remote tension and pure bending loads in a D finite body. The application of the body force method (BFM) by Kamaya (00) justified the direction of coalesced with a proposed new formulation. Owing to the limitations of BFM, FEM and the virtual crack extension method is employed by Kamaya (008a, 008b) to investigate the formation of a single coalesced crack as a result of crack growth under fatigue loading. n creep loading conditions, Xuan et al. (009) introduced a creep interaction factor to address the interaction between as a reaction to the stress field under loading. Kamaya et al. (00) used an S-version FEM to analyze the crack growth of surface. n relation to the FFS code limitations, crack interaction problems are more concentrated on elastic crack interaction with crack propagation, rather than on elastic crack interaction without crack propagation. n the past, many techniques have focused on the strong interaction region, particularly on the combination rule of parallel to coplanar (shielding effect), and the assumption of two as a single crack for coplanar (amplification effect). Most proposed solutions have considered mainly the strong interaction region and none of them question how crack interaction will behave within the weak interaction region, especially when the interaction approaches the crack interaction limit (CL) (Zulkifli et al., 0). To date, there are few investigations with emphasis on investigating the shielding effects in the weak interaction region as the interaction approaches CL. This paper presents a numerical approach for modeling multiple edge strip that experience weak elastic crack interaction in a finite plate (Kamal et al., 0; Domínguez Almaraz et al., 00). The crack interaction is limited to crack interaction without crack propagation, where the shielding effect is dominated in promoting fracture and failure. The aim of this paper is to study the effect of the relative position of two parallel edge, subjected to a variation of crack interval b/ aand crack-to-width ratio a/ W. The numerical analysis was performed using D linear FE analysis using developed APDL codes in ANSYS. The obtained value of the SFs and elastic crack interaction factor are compared with corresponding FFS rules and numerical data from literature. For the present investigation of elastic CL, the BS codes (99, 997, 005) statements and the works of Jiang et al. (990, 99, 99), and Leek and Howard (994a, 994b, 996) are taken as the interaction limit reference. t is used for determining the onset of weak interaction and where the weak interaction is diminished, focusing specifically on the effect of shielding for parallel edge in a finite body. For reference, the elastic crack interaction (EC) of all parallel may be redefined to diminish approximately at b a with the assumption that the crack lengths 995

4 Validation of crack interaction limit model for parallel edge using two-dimensional finite element analysis ai ( i,... n) are equal length. The new analytical EC factor EC to address the CL problem is proposed and it written as Eq. () ( a / W, b / a ),( i 0,,,... n ) () EC EC CL i i and needs to comply with FFS codes in the condition of ( a / W, b / a ) FFS (ASME,BS,JSME) EC i i ( a / W, b / a ) FFS ( b.7mm, b ( a a ) /, b 0mm) EC i i () CRACK NTERACTON LMT DETERMNATON The new model of CL is based on the formulation of the state of no interaction or a single cracked body to the state of having weak interaction, until the interaction becomes stronger as the crack interval decreases. The elastic CL is based on the analytical works of Jiang et al. (990, 99, 99), and Leek and Howard (994a, 994b, 996) as reference. The SF calculation is limited to a linear elastic problem with a homogeneous, isotropic material near the crack region. The SF determination is based on the creation of a singular element at the crack tip based on a quadratic isoparametric finite element. A quarter-point singular element or 8-node collapsed quadrilateral element developed by Henshell and Shaw (975) is used; this has distinct advantages in terms of time, meshing, and re-meshing over the other quarter-point element (Banks Sills, 00). The singularity is obtained by shifting the mid-side node ¼ points close to the crack tip. To calculate the SF, we assumed the elements to be in rigid body motion and constant strain modes. The accuracy of this special element has been addressed by Murakami (976), where the crack tip nodal point is enclosed by a number of special elements and in analysis, the size, number, and compatibility of special elements really affect the accuracy. By assuming the crack interaction will be diminished at b a i and therefore, it is about equivalent to the SF of a single crack in the state of zero crack interaction factor, the analytical single crack Mode SF proposed by Brown and Strawley (966) is used as the SF reference K, expressed as Eq. (). K 4 a a a a a W W W W () Theoretically, to validate the model, the Mode SF of the CL model must be close to the value of K CL K or K K, but it cannot be equal, i.e., CL K CL K. Therefore, based on Tada et al. (00), the new CL written as: K function is K a C C C C C C W W W W W 4 n a a a a a CL i 4 5 n (4) where the constants,... n C C are assumed to be constant over the crack surface. Shown in Figure (a) are two straight parallel edge in a finite plate and the two 996

5 Daud et al. /nternational Journal of Automotive and Mechanical Engineering 7(0) are under uniform far-field tension P ( ) normal to crack faces, and where the are of equal lengths: a and a. n this analysis, the ratios of ( a/ W ) and ( b/ a ) are used to evaluate the in the weak interaction region, which is assumed to be in the range of b/ a. Therefore, the following cases are considered, where ( b/ a ) =.5,.0,.5,.0 and ( a/ W ) = 0.05, 0., 0.5, 0.0, 0.5, 0.0, 0.5, 0.40, 0.45, Figure (a) (c) also shows the two-dimensional configuration of the singular element. Based on Arakere et al. (008), the meshing scheme is generated using ANSYS, as shown in Figure (d). The mesh encompassing the interacting is divided into two zones: a global mesh, and the local mesh zone. Both zones are meshed with the 8-node isoparametric quadrilateral element that is used to build up the entire element for the two-dimensional plate, as illustrated in Figure (b). The SF calculation is limited to the linear elastic problem with a homogeneous, isotropic material near the crack region. The studies are conducted in a pure Mode loading condition with the specified material, Aluminium Alloy 7475 T75 solid plate with constant thickness, homogenous isotropic continuum material, linear elastic behavior, small strain and displacements, and crack surfaces are smooth with the crack surfaces are almost contacting each other ( c c 0). P (σ ) (a) h Global mesh (b) 7 8 η ELEMENTS (d) ξ JUN 0 :06:4 c Front free surface a Ct b Sub-model mesh Ct Local mesh H Back free surface (c) y Ct Sub-model mesh Ct ly / c a h Global mesh Crack tip v 5 (u,v) 4 θ = const x ly / u lx /4 lx /4 W P (σ ) Y SC PSTRAN L ZF X Figure. (a) Two parallel edge in finite body, (b) 8-node quadrilateral element, (c) Barsoum singular element, and (d) complete meshing. 997

6 Validation of crack interaction limit model for parallel edge using two-dimensional finite element analysis The SF for CL K CL, is evaluated at the both crack tips of the edge strip, as shown in Figure (a). The reduction of K comparative to CL K is denoted as K CL and is expressed as K CL K K CL. The SF for Mode K, is determined using the displacement extrapolation method using written APDL macro code in ANSYS, and is expressed as Eq. (5). / y K CL E / ( )( ) / l v v v v / (5) where E=Young s Modulus, 4 for plane stress, 4 / for plane strain, l y is the length of element, v and u are displacements in a local Cartesian coordinate system, and υ is Poisson s ratio. The elastic crack interaction factor for Mode SF is expressed as Eq. (6). K / K (6) CL 0 where denotes the elastic interaction factor for Mode and K o is the SF for the plain specimen. RESULTS AND DSCUSSON Figure shows the relationship between elastic crack interaction factor and the crack-to-width ratio a/ W for the crack interval ratios ( b/ a ) and ( b/ a.5). The elastic crack interaction factor is based on the SF at the crack tips K and is 998 CL normalized by K 0, which is the SF for the plain specimen. n the weak interaction range, ( b/ a ) for ( b/ a ), strong interaction appears at a/ W 0.5 and it rapidly decreases when a/ W approaches a/ W Therefore, the intensity of depends on the variation of a/ W ; it becomes larger as a/ W increases. Figure also shows a very small discrepancy (0.05%) between the intensity of for crack tips Ct and Ct. This indicates that under the assumption of a homogenous material with equal length, the intensity of can be assumed equal. According to FFS codes by ASME (004, 007), BS (997, 005), and JSME (000, 008), parallel can be assumed a single coplanar crack under certain conditions of crack interval. n reduction of the a/ W ratio from 0.5 to 0.05, the present numerical model supports the FFS code by the indication of the intersection point at 0. and 0.07 (see Figure ) and 0.05 (see Figure ) as the crack interval decreases from b/ a to b/ a.5. The prediction by Jiang et al. (990) of F ( K ) shows no intersection point for a single crack of f ( K )(Brown and Strawley, 966), and shows no possible sign for single coplanar crack agreement, as outlined by the FFS codes. Again, the present ( C CL ) and CL ( C ) show a good trend line prediction of elastic crack interaction intensity and comply with the recommended FFS codes. n

7 Daud et al. /nternational Journal of Automotive and Mechanical Engineering 7(0) Crack interaction factor, γ b/a =.0 Brown & Strawley -single edge crack (966) Present (Ct) Present (Ct) Jiang et al. (990) ntersection point Crack to width ratio, a/w (a) Crack interaction factor, γ b/a =.5 Brown & Strawley-single edge crack (966) Present (Ct) Present (Ct) Jiang et al. (990).6.4 ntersection point Crack to width ratio, a/w (b) Figure. Elastic crack interaction factor: (a) b/ a, and (b) b/ a

8 Validation of crack interaction limit model for parallel edge using two-dimensional finite element analysis Crack interaction factor, γ b/a =.0 Brown & Strawley-single edge crack(966) Present (Ct) Present (Ct) Jiang et al. (990) ntersection point Crack to width ratio, a/w (a) Crack interaction factor, γ b/a =.5 Brown & Strawley-single edge crack(966) Present (Ct) Present (Ct) Jiang et al. (990) Crack to width ratio, a/w (b) Figure. Elastic crack interaction factor: (a) b/ a.0, and (b) b/ a

9 Daud et al. /nternational Journal of Automotive and Mechanical Engineering 7(0) n order to validate the numerical model, the comparison is made with single crack SF reference K (Brown and Strawley, 966) Eq. () expressed as f ( K ). Good agreement was obtained, which provides confidence on the FE modeling and CL interaction analysis of shielding effects. When compared with the condition of CL claimed by Jiang et al. (990), denoted as Fn( K ), the present numerical model shows a significant improvement in the reduction of errors as a/ W increases, as shown in Table. Table. Crack interaction factor at ( b/ a.0). a/w Analytical Analytical Present f ( K ) ( ) ( ) n CL ( CL ) CONCLUSON An alternative solution for the elastic CL for multiple parallel and equal edge strip in a finite continuum body has been presented based on FE analysis. The close agreement with well-known analytical solutions for single edge in a finite body validates the proposed solution for further applications for CL analysis for closer crack distance, which involves higher crack interaction. Meanwhile, the compliance with recommended ASME codes, BS codes, and JSME FFS codes provides additional evidence corroborating the present CL prediction. Finally, it can be concluded that the CL for equal and parallel edge is depicted best at b/ a because the crack interaction factor is approximately the normalized SF value of a single edge crack with zero crack interaction. ACKNOWLEDGMENTS The authors wish to thank Dr. Mostafa Jamshidian and Prof. Dr. Masanori Kikuchi for crack interaction mechanic consultation. This research is supported by Fundamental Research Grant Scheme (FRGS), Ministry of Higher Education, Malaysia. All authors declare that there are no conflicts of interest that may arise as a result of this research. REFERENCES Arakere, N.K., Knudsen, E.C., Wells, D., McGill, P. and Swanson, G.R Determination of mixed-mode stress intensity factors, fracture toughness and 00

10 Validation of crack interaction limit model for parallel edge using two-dimensional finite element analysis crack turning angle for anisotropic foam material. nternational Journal of Solids and Structures, 46: ASME ASME Boiler and Pressure Vessel Code, Section X. New York, USA. ASME ASME Boiler and Pressure Vessel Code, Section X. New York, USA. ASME / FFS- Fitness-for-service, Section 9, American Society of Mechanical Engineers. New York, USA. Banks-Sills, L. 00. Update: Application of the finite element method to linear elastic fracture mechanics. Applied Mechanics Reviews, 6: -7. Brown, W.F. and Strawley, J.E ASTM STP 40. BS. 99. British Standard nstitute, PD 649, Section 8. Guidance on methods for assessing the acceptability of flaws in fusion welded structures. BS British Standard nstitute, BS790, Section 8. Guidance on methods for assessing the acceptability of flaws in structures. BS British Standard nstitute, BS790. Guidance on methods for assessing the acceptability of flaws in metallic structures. Burdekin, F.M. 98. The role of fracture mechanics in the safety analysis of pressure vessels. nternational Journal of Mechanical Sciences, 4(4): Chang, R. 98. On crack-crack interaction and coalescence in fatigue. Engineering Fracture Mechanics, 6(5): Chen, Y.Z Plane problem of two rows of periodic crack. Theoretical and Applied Fracture Mechanics, 7: Chen, Y.Z ntegral equation methods for multiple crack problems and related topics. ASME Journal of Applied Mechanics, 60: Chen, Y.Z. and Wang, Z.X. 0. Solution of multiple crack problem in a finite plate using coupled integral equations. nternational Journal of Solids and Structures, 49: Choi, H.J Mode interaction of a periodic array of parallel in a functionally graded nonhomogeneous plane. KSME nternational Journal, (): Domínguez Almaraz, G.M., Guzmán Tapia, M., Tapia Silva, E.E. and Cadenas Calderón, E. 00. Fatigue life prediction based on macroscopic plastic zone on fracture surface of AS-SAE 08 steel. nternational Journal of Automotive and Mechanical Engineering, : 9-7. Dwyer, J.F Edge function analysis of crack interaction in anisotropic materials. Engineering Fracture Mechanics, 56(): -48. Feng, Y. and Jin, Z Thermal fracture of functionaly graded plate with parallel surface. Acta Mechanica Solida Sinica, (5): Gorbatikh, L. and Kachanov, M A simple technique for constructing the full stress and displacement fields in elastic plates with multiple. Engineering Fracture Mechanics, 66: 5-6. Gorbatikh, L., Lomov, S. and Verpoest, On stress intensity factors of multiple at small distances in -D problems. nternational Journal of Fracture, 4: Han, X., Ellyin, F. and Xia, Z. 00. nteraction among interface, multiple and dislocations. nternational Journal of Solids and Structures, 9: Henshell, R.D. and Shaw, K.G Crack tip finite elements are unnecessary. nternational Journal for Numerical Methods in Engineering, 9(): ida, K. 98. Shapes and coalescence of surface fatigue. Paper presented at the nternational Symposium on Fracture Mechanics, Beijing, China. 00

11 Daud et al. /nternational Journal of Automotive and Mechanical Engineering 7(0) Jiang, Z.D., Petit, J. and Bezine, G Fatigue propagation of two parallel. Engineering Fracture Mechanics, 7(5): Jiang, Z.D., Petit, J., and Bezine, G. 99. Stress intensity factors of two parallel D surface. Engineering Fracture Mechanics, 40(): Jiang, Z.D., Petit, J. and Bezine, G. 99. An investigation of stress intensity factors for two unequal parallel in a finite width plate. Engineering Fracture Mechanics, 4(): 9-8. Jiang, Z.D., Zeghloul, A., Bezine, G. and Petit, J Stress intensity factor of parallel in a finite width sheet. Engineering Fracture Mechanics, 5(6): JSME JSME Fitness-for-Service Code S NA-000. JSME JSME Fitness-for-Service Code S NA-008. Kachanov, M Elastic solids with many : a simple method of analysis. nternational Journal of Solid Structures, (): -4. Kachanov, M. 99. Elastic solids with many and related problems. Advances in Applied Mechanics, 0: Kachanov, M. 00. On the problems of the crack interactions and crack coalescence. nternational Journal of Fracture, 0: Kachanov, M. and Laures, J.P Strong three dimensional interaction of several arbitrarily located penny-shaped. nternational Journal of Fracture, 7: R6-R68. Kachanov, M. and Montagut, E nteraction of a crack with certain microcrack arrays. Engineering Fracture Mechanics, 5(5/6): Kachanov, M., Montagut, E.L.E. and Laures, J.P Mechanics of crack-microcrack interactions. Mechanics of Materials, 0: Kamal, M., Rahman, M.M. and Rahman, A.G.A. 0. Fatigue life evaluation of suspension knuckle using multi body simulation technique. Journal of Mechanical Engineering and Sciences, : Kamaya, M. 00. A crack growth evaluation method for interacting multiple. JSME nternational Journal, 46(): 5-. Kamaya, M. 008a. Growth evaluation of multiple interacting surface. Part : Growth evaluation of parallel. Engineering Fracture Mechanics, 75: Kamaya, M. 008b. Growth evaluation of multiple interacting surface. Part : Experiments and simulation of coalesced crack. Engineering Fracture Mechanics, 75: Kamaya, M., Miyokawa, E. and Kikuchi, M. 00. Growth prediction of two interacting surface of dissimilar sizes. Engineering Fracture Mechanics, 77: 0-. Leek, T.H. and Howard,.C. 994a. Estimating the elastic interaction factor of two coplanar surface under Mode load. nternational Journal of Pressure Vessels and Piping, 60: 07-. Leek, T.H. and Howard,.C. 994b. Rules for the assessment of interacting surface under Mode load. nternational Journal of Pressure Vessels and Piping, 60: -9. Leek, T.H. and Howard,.C An examination of methods of assessing interacting surface by comparison with experimental data. nternational Journal of Pressure Vessels and Piping, 68:

12 Validation of crack interaction limit model for parallel edge using two-dimensional finite element analysis Li, D.F., Li, C.F., Shu, S.Q., Wang, Z.X. and Lu, J A fast and accurate analysis of the interacting in linear elastic solids. nternational Journal of Fracture, 5: Moussa, W.A., Bell, R. and Tan, C.L The interaction of two parallel noncoplanar identical surface under tension and bending. nternational Journal of Pressure Vessels and Piping, 76: Murakami, Y A simple procedure for the accurate determination of stress intensity factors by finite element method. Engineering Fracture Mechanics, 8: Murakami, Y. and Nasser, S.N. 98. nteracting dissimilar semi-elliptical surface flaws under tension and bending. Engineering Fracture Mechanics, 6(): O'donoghue, P.E., Nishioka, T. and Atluri, S.N Multiple surface in pressure vessel. Engineering Fracture Mechanics, 0(): Orlowsky, B., Saenger, E.H., Gueguen, Y. and Saphiro, S.A. 00. Effects of parallel distributions on effective elastic properties- A numerical study. nternational Journal of Fracture, 4: Parker, A.P Stability of arrays of multiple edge. Engineering Fracture Mechanics, 6: R Nuclear Electric: Assessment of the integrity of structures containing defects, Revision 4. Gloucester, British Energy Generation Ltd. Ratwani, M. and Gupta, G.D nteraction between parallel in layered composites. nternational Journal of Solids and Structures, 0: Saha, T.K. and Ganguly, S nteraction of penny-shaped with an eliptic crack under shear loading. nternational Journal of Fracture, : Sankar, R. and Lesser, A.J Generic overlapping in polymers: modeling of interaction. nternational Journal of Fracture, 4: Tada, H., Paris, P. C. and rwin, G.R. 00. The stress analysis of handbook. New York: ASME Press. Xuan, F.Z., Si, J. and Tu, S.T Evaluation of C* integral for interacting in plates under tension. Engineering Fracture Mechanics, 76: 9-0. Yang, Y.H Multiple parallel symmetric mode in a functionally graded material plane. Journal of Solid Mechanics and Materials Engineering, (5): Yildirim, B., Kutlu, O. and Kadioglu, S. 0. Periodic crack problem for a functionally graded half-plane an analytic solution. nternational Journal of Solids and Structures, 48: Yokobori, T., Sawaki, Y. and Nakanishi, S Criterion for brittle fracture of notched or cracked specimens based on combined micro-and macro crack mechanics-. Engineering Fracture Mechanics, : 5-4. Zulkifli, A., Ariffin, A.K. and Rahman, M.M. 0. Probabilistic finite element analysis on vertebra lumbar spine under hyperextension loading. :