Two-dimensional fracture analysis of FGM under mechanical loading



This paper extends the concept to isotropic functionally graded materials and addresses fracture problems under mechanical loading. The mode-I and mixed mode stress intensity factors (SIFs) are determinate by combination between the finite element method and the displacement extrapolation technique (DET). The variation continue of the elastics properties is incorporated at the centroid of each finite element, using the Ansys Parametric Design Language (APDL). In this work, two examples are analysed to check for the robustness of the present approach, the FGM disk with a central inclined crack subjected to concentrated couple forces and the three-point bending specimen with crack parallel to material gradation. The numerical results obtained by present technique are discussed by comparison with other published results.


Displacement Extrapolation ; Functionally Graded Material ; Stress Intensity Factors ; Mechanical Loading

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- S.K. Chan, I.S. Tuba, W.K. Wilson, On the finite element method in linear fracture mechanics. Eng. Fract. Mech. 2(1970)1–17. doi:10.1016/0013-7944(70)90026-3

- C.F. Shih, H.G. De Lorenzi, M.D. German, Crack extension modeling with singular quadratic isoparametric elements. Int. J. Fract. 12(1976) 647–51. doi:10.1007/BF00034654

- D.M. Parks, A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int. J. Fract. 10(1974) 487–502. doi:10.1007/BF00155252

- H.G. De Lorenzi, Energy release rate calculations by the finite element method. Eng. Fract. Mech. 21 (1985) 129–43. doi:10.1016/0013-7944(85)90060-8

- B. Moran, C.F. Shih, A general treatment of crack tip contour integrals. Int. J. Fract. 35(1987) 295–310. doi:10.1007/BF00276359

- P. Gu, M. Dao, R. Asaro, A simplified method for calculating the crack-tip field of functionally graded materials using the domain integral. J. Appl. Mech. 66(1999) 101–8. doi:10.1115/1.2789135

- A. Anlas, M.H. Santare, J. Lambros, Numerical calculation of stress intensity factors in functionally graded materials. Int. J. Fract. 104(2) (2000) 131–143. doi:10.1023/A:1007652711735

- J. Yau , S. Wang, H. Corten, A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J. Appl. Mech. 47(2) (1980) 335–41. doi:10.1115/1.3153665

- J. Dolbow, M. Gosz, On the computation of mixed-mode stress intensity factors in functionally graded materials. Int. J. Solids. Struct. 39 (2002) 2557–74. doi:10.1016/S0020-7683(02)00114-2

- X.W. Gao, Ch. Zhang, J. Sladek, V. Sladek, Fracture analysis of functionally graded materials by a BEM. Compos. Sci. Technol. 68 (2008) 1209–1215. doi:10.1016/j.compscitech.2007.08.029

- B.N. Rao, S. Rahman, Mesh-free analysis of cracks in isotropic functionally graded materials. Eng. Fract. Mech. 70 (2003) 1–27. doi:10.1016/S0013-7944(02)00038-3

- J.H. Kim, H. Glaucio, G.H. Paulino, Finite element evaluation of mixed mode stress intensity factors in functionally graded materials. Int. J. Numer. Methods. Eng. 53(8) (2002) 1903–1935. doi:10.1002/nme.364

- S. Dag, B. Yildirim, Computation of Thermal Fracture Parameters for Inclined Cracks in Functionally Graded Materials Using Jk-Integral. J. Therm. Stresses. 32 (2009) 530–556. doi:10.1080/01495730802637480

- B. Yildirim, S. Yilmaz, S. Kadioglu, Delamination of Compressively Stressed Orthotropic Functionally Graded Material Coatings under Thermal Loading. J. Appl. Mech, 75(5) (2008) 051106. doi:10.1115/1.2936239

- I. Eshraghi, N. Soltani, Stress intensity factor calculation for internal circumferential cracks in functionally graded cylinders using the weight function approach. Eng. Fract. Mech. 134 (2015) 1–19. doi:10.1016/j.engfracmech.2014.12.007

- N. Benamara, A. Boulenouar, M. Aminallah, Strain Energy Density Prediction of Mixed-Mode Crack Propagation in Functionally Graded Materials. Period. Polytech. Mech. Eng. 61(1) (2017) 60–67. doi:10.3311/PPme.9682

- J.H. Kim, G.H. Paulino, Mixed-mode J-integral formulation and implementation using graded finite elements for fracture analysis of nonhomogeneous orthotropic materials. Mech. Mater. 35(1) (2003) 107–128. doi:10.1016/S0167-6636(02)00159-X

- Y. Ait Ferhat, A. Boulenouar, N. Benamara, L. Benabou. Generalized displacement correlation method for mechanical and thermal fracture of FGM. Int. J. Comput. Mater. Sci. Surf. Eng. accepted 2020. doi:10.1142/S2047684120500049

- ANSYS, Inc. Programmer’s Manual for Mechanical APDL. Release 12.1, 2009.

- A. Boulenouar, N. Benseddiq, M. Mazari, N. Benamara, FE model for linear elastic mixed mode loading: estimation of SIFs and crack propagation. J. Theor. Appl. Mech. 52(2) (2014) 373–383

- A. Boulenouar, A. Benouis, N. Benseddiq, Numerical modelling of crack propagation in cement PMMA: Comparison of different criteria. Mat. Res. 19 (4) (2016) 846–855. doi:10.1590/1980-5373-MR-2015-0784

- A. Benouis, A. Boulenouar, B. Serier, Finite element analysis of the behaviour of a crack in the orthopedic cement. J. Theor. Appl. Mech. 54(1) (2016) 277–284. doi:10.15632/jtam-pl.54.1.277

- M. Souiyah, A. Muchtar, A.K. Ariffin, A. Malek, M.I. Fadhel, B. Abu Zneid, Finite Element Model of Crack Growth under Mixed Mode Loading. Int. J. Mater. Eng. 2(5) (2012) 67–74. doi:10.5923/j.ijme.20120205.02

- N. Benamara, A. Boulenouar, M. Aminallah, N. Benseddiq, On the mixed-mode crack propagation in FGMs plates: comparison of different criteria. Struct. Eng. Mech. 615(3) (2017) 371–379. doi:10.12989/sem.2017.61.3.371

- M. Chafi, A. Boulenouar, A Numerical Modelling of Mixed Mode Crack Initiation and Growth in Functionally Graded Materials. Mat. Res. 615(3) 2019. doi:10.1590/1980-5373-mr-2018-0701

- R.S. Barsoum, On the use of isoparametric finite element in linear fracture mechanics. Int. J. Numer. Methods. Eng. 10 (1974) 25–37. doi:10.1002/nme.1620100103

- J.H. Kim, H. Glaucio, G.H. Paulino, The interaction integral for fracture of orthotropic functionally graded materials: evaluation of stress intensity factors. Int. J. Solids. Struct. 40(15) (2003) 3967–4001. doi:10.1016/S0020-7683(03)00176-8

- A. Boulenouar, N. Benseddiq, M. Mazari, Strain energy density prediction of crack propagation for 2D linear elastic materials. Theor. Appl. Frac. Mech. 67–68 (2013) 29–37. doi:10.1016/j.tafmec.2013.11.001

- C. Ki-Hyun, Y. Won-Ho, Fracture mechanics analysis on the bonded repair of a skin/stiffener with an inclined central crack. Comp. Struc. 55(3) (2002) 269–276. doi:10.1016/S0263-8223(01)00163-5


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