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Peridynamic simulation of dynamic fracture in functionally graded materials subjected to impact load

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Abstract

Dynamic crack propagation assessment in functionally graded materials (FGMs) with micro-cracks is accomplished using bond-based Peridynamics (PD). The dynamic fracture behaviour of various FGMs’ material models is studied in Kalthoff–Winkler experiment. Dynamic crack growth predictions and associated material damage of the specimen under dynamic loading conditions are considered. The effect of micro-cracks near macro-crack tips on the toughening mechanism is evaluated in terms of crack propagation velocities. Stochastically pre-located micro-cracks are modelled to obtain the toughening effect in the material. In addition, the velocities and time required for fracture are compared in different FGM cases. It is frankly found that if a crack propagates in the harder region of the specimen, velocities decrease and toughness increase in contrast to the softer region. Furthermore, micro-cracks around a macro-crack decelerate the crack propagation and enhance toughening mechanism in FGM body depending on gradation of material properties.

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References

  1. Madenci E, Özütok A (2017) Variational approximate and mixed-finite element solution for static analysis of laminated composite plates. Solid State Phenom 267:35–39. https://doi.org/10.4028/www.scientific.net/SSP.267.35

    Article  Google Scholar 

  2. Özütok A, Madenci E (2017) Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method. Int J Mech Sci 130:234–243. https://doi.org/10.1016/j.ijmecsci.2017.06.013

    Article  Google Scholar 

  3. Madenci E, Özütok A (2020) Variational approximate for high order bending analysis of laminated composite plates. Struct Eng Mech 73:97–108. https://doi.org/10.12989/sem.2020.73.1.097

  4. Madenci E (2021) Free vibration and static analyses of metal-ceramic FG beams via high-order variational MFEM. Steel Compos Struct 39:493–509. https://doi.org/10.12989/scs.2021.39.5.493

  5. Madenci E, Gülcü Ş (2020) Optimization of flexure stiffness of FGM beams via artificial neural networks by mixed FEM. Struct Eng Mech 75:633–642. https://doi.org/10.12989/sem.2020.75.5.633

  6. Madenci E (2019) A refined functional and mixed formulation to static analyses of fgm beams. Struct Eng Mech 69:427–437. https://doi.org/10.12989/sem.2019.69.4.427

  7. Madenci E, Özkılıç YO (2021) Free vibration analysis of open-cell FG porous beams: Analytical, numerical and ANN approaches. Steel Compos Struct 40:157–173. https://doi.org/10.12989/scs.2021.40.2.157

  8. Delale F, Erdogan F (1983) The crack problem for a nonhomogeneous plane. J Appl Mech Trans ASME 50:609–614. https://doi.org/10.1115/1.3167098

    Article  MATH  Google Scholar 

  9. Eischen JW (1987) Fracture of nonhomogeneous materials. Int J Fract 34:3–22. https://doi.org/10.1007/BF00042121

    Article  Google Scholar 

  10. Jin ZH, Batra RC (1996) Some basic fracture mechanics concepts in functionally graded materials. J Mech Phys Solids 44:1221–1235. https://doi.org/10.1016/0022-5096(96)00041-5

    Article  Google Scholar 

  11. Marur PR, Tippur HV (2000) Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. Int J Solids Struct 37:5353–5370. https://doi.org/10.1016/S0020-7683(99)00207-3

    Article  MATH  Google Scholar 

  12. Itou S (2010) Dynamic stress intensity factors for two parallel interface cracks between a nonhomogeneous bonding layer and two dissimilar elastic half-planes subject to an impact load. Int J Solids Struct 47:2155–2163. https://doi.org/10.1016/j.ijsolstr.2010.04.020

    Article  MATH  Google Scholar 

  13. Guo LC, Wu LZ, Ma L, Zeng T (2004) Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface - Part I: Static problem. Int J Fract 127:21–38. https://doi.org/10.1023/b:frac.0000035049.26772.2d

    Article  MATH  Google Scholar 

  14. Ma L, Wu LZ, Guo LC (2005) On the moving Griffith crack in a nonhomogeneous orthotropic strip. Int J Fract 136:187–205. https://doi.org/10.1007/s10704-005-6023-z

    Article  MATH  Google Scholar 

  15. Xia CH, Ma L (2007) Dynamic behavior of a finite crack in functionally graded materials subjected to plane incident time-harmonic stress wave. Compos Struct 77:10–17. https://doi.org/10.1016/j.compstruct.2005.05.012

    Article  Google Scholar 

  16. Kidane A, Chalivendra VB, Shukla A, Chona R (2010) Mixed-mode dynamic crack propagation in graded materials under thermo-mechanical loading. Eng Fract Mech 77:2864–2880. https://doi.org/10.1016/j.engfracmech.2010.07.004

    Article  Google Scholar 

  17. Cheng Z, Zhong Z (2007) Analysis of a moving crack in a functionally graded strip between two homogeneous layers. Int J Mech Sci 49:1038–1046. https://doi.org/10.1016/j.ijmecsci.2007.01.003

    Article  Google Scholar 

  18. Cheng Z, Gao D, Zhong Z (2010) Crack propagating in functionally graded coating with arbitrarily distributed material properties bonded to homogeneous substrate. Acta Mech Solid Sin 23:437–446. https://doi.org/10.1016/S0894-9166(10)60046-8

    Article  Google Scholar 

  19. Lee KH (2009) Analysis of a transiently propagating crack in functionally graded materials under mode I and II. Int J Eng Sci 47:852–865. https://doi.org/10.1016/j.ijengsci.2009.05.004

    Article  MathSciNet  MATH  Google Scholar 

  20. Matbuly MS (2009) Multiple crack propagation along the interface of a non-homogeneous composite subjected to anti-plane shear loading. Meccanica 44:547–554. https://doi.org/10.1007/s11012-009-9190-6

    Article  MathSciNet  MATH  Google Scholar 

  21. Hassan SF, Siddiqui O, Ahmed MF, Al Nawwah AI (2019) Development of gradient concentrated single-phase fine Mg-Zn particles and effect on structure and mechanical properties. J Eng Mater Technol Trans ASME. https://doi.org/10.1115/1.4041865

    Article  Google Scholar 

  22. Jin X, Wu L, Guo L et al (2009) Experimental investigation of the mixed-mode crack propagation in ZrO2/NiCr functionally graded materials. Eng Fract Mech 76:1800–1810. https://doi.org/10.1016/j.engfracmech.2009.04.003

    Article  Google Scholar 

  23. Abanto-Bueno J, Lambros J (2006) An experimental study of mixed mode crack initiation and growth in functionally graded materials. Exp Mech 46:179–196. https://doi.org/10.1007/s11340-006-6416-6

    Article  Google Scholar 

  24. Jain N, Shukla A (2006) Mixed mode dynamic fracture in particulate reinforced functionally graded materials. Exp Mech 46:137–154. https://doi.org/10.1007/s11340-006-5867-0

    Article  Google Scholar 

  25. Kirugulige MS, Tippur HV (2006) Mixed-mode dynamic crack growth in functionally graded glass-filled epoxy. Exp Mech 46:269–281. https://doi.org/10.1007/s11340-006-5863-4

    Article  Google Scholar 

  26. Rousseau CE, Tippur HV (2001) Dynamic fracture of compositionally graded materials with cracks along the elastic gradient: experiments and analysis. Mech Mater 33:403–421. https://doi.org/10.1016/S0167-6636(01)00065-5

    Article  Google Scholar 

  27. Toktaş SE, Dag S (2020) Oblique surface cracking and crack closure in an orthotropic medium under contact loading. Theor Appl Fract Mech 109:102729. https://doi.org/10.1016/j.tafmec.2020.102729

    Article  Google Scholar 

  28. Shukla A, Jain N, Chona R (2007) A review of dynamic fracture studies in functionally graded materials. Strain 43:76–95. https://doi.org/10.1111/j.1475-1305.2007.00323.x

    Article  Google Scholar 

  29. Lorentz E (2008) A mixed interface finite element for cohesive zone models. Comput Methods Appl Mech Eng 198:302–317. https://doi.org/10.1016/j.cma.2008.08.006

    Article  MATH  Google Scholar 

  30. Unosson M, Olovsson L, Simonsson K (2006) Failure modelling in finite element analyses: element erosion with crack-tip enhancement. Finite Elem Anal Des 42:283–297. https://doi.org/10.1016/j.finel.2005.07.001

    Article  Google Scholar 

  31. Lancaster IM, Khalid HA, Kougioumtzoglou IA (2013) Extended FEM modelling of crack propagation using the semi-circular bending test. Constr Build Mater 48:270–277. https://doi.org/10.1016/j.conbuildmat.2013.06.046

    Article  Google Scholar 

  32. Zhou X, Wang Y, Qian Q (2016) Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended non-ordinary state-based peridynamics. Eur J Mech A/Solids 60:277–299. https://doi.org/10.1016/j.euromechsol.2016.08.009

    Article  MathSciNet  MATH  Google Scholar 

  33. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1%3c131::AID-NME726%3e3.0.CO;2-J

    Article  MathSciNet  MATH  Google Scholar 

  34. Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58:1873–1905. https://doi.org/10.1002/nme.941

    Article  MATH  Google Scholar 

  35. Zhuang Z, Bin CB (2011) Development of X-FEM methodology and study on mixed-mode crack propagation. Acta Mech Sin Xuebao 27:406–415. https://doi.org/10.1007/s10409-011-0436-x

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang H, Liu Z, Xu D et al (2016) Extended finite element method analysis for shielding and amplification effect of a main crack interacted with a group of nearby parallel microcracks. Int J Damage Mech 25:4–25. https://doi.org/10.1177/1056789514565933

    Article  Google Scholar 

  37. Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H (2010) A simple and robust three-dimensional cracking-particle method without enrichment. Comput Methods Appl Mech Eng 199:2437–2455. https://doi.org/10.1016/j.cma.2010.03.031

    Article  MATH  Google Scholar 

  38. Kosteski L, Barrios D’Ambra R, Iturrioz I (2012) Crack propagation in elastic solids using the truss-like discrete element method. Int J Fract 174:139–161. https://doi.org/10.1007/s10704-012-9684-4

    Article  Google Scholar 

  39. Braun M, Fernández-Sáez J (2014) A new 2D discrete model applied to dynamic crack propagation in brittle materials. Int J Solids Struct 51:3787–3797. https://doi.org/10.1016/j.ijsolstr.2014.07.014

    Article  Google Scholar 

  40. Kim J-H, Paulino GH (2004) Simulation of crack propagation in functionally graded materials under mixed-mode and non-proportional loading. Int J Mech Mater Des 1:63–94. https://doi.org/10.1023/b:mamd.0000035457.78797.c5

    Article  Google Scholar 

  41. Kirugulige M, Tippur HV (2008) Mixed-mode dynamic crack growth in a functionally graded particulate composite: Experimental measurements and finite element simulations. J Appl Mech Trans ASME 0511021–05110214

  42. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209. https://doi.org/10.1016/S0022-5096(99)00029-0

    Article  MathSciNet  MATH  Google Scholar 

  43. Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535. https://doi.org/10.1016/j.compstruc.2004.11.026

    Article  Google Scholar 

  44. Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93:13–37. https://doi.org/10.1007/s10659-008-9163-3

    Article  MathSciNet  MATH  Google Scholar 

  45. Oterkus E, Madenci E, Weckner O et al (2012) Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Compos Struct 94:839–850. https://doi.org/10.1016/j.compstruct.2011.07.019

    Article  Google Scholar 

  46. Liu S, Fang G, Liang J, Lv D (2020) A coupling model of XFEM/peridynamics for 2D dynamic crack propagation and branching problems. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2020.102573

    Article  Google Scholar 

  47. De Meo D, Russo L, Oterkus E (2017) Modeling of the onset, propagation, and interaction of multiple cracks generated from corrosion pits by using peridynamics. J Eng Mater Technol Trans ASME 139:1–9. https://doi.org/10.1115/1.4036443

    Article  Google Scholar 

  48. Bang DJ, Madenci E (2017) Peridynamic modeling of hyperelastic membrane deformation. J Eng Mater Technol Trans ASME. https://doi.org/10.1115/1.4035875

    Article  Google Scholar 

  49. De Meo D, Zhu N, Oterkus E (2016) Peridynamic modeling of granular fracture in polycrystalline materials. J Eng Mater Technol Trans ASME. https://doi.org/10.1115/1.4033634

    Article  Google Scholar 

  50. Diyaroglu C, Oterkus E, Madenci E et al (2016) Peridynamic modeling of composite laminates under explosive loading. Compos Struct 144:14–23. https://doi.org/10.1016/j.compstruct.2016.02.018

    Article  Google Scholar 

  51. Javili A, McBride AT, Steinmann P (2021) A geometrically exact formulation of peridynamics. Theor Appl Fract Mech 111:102850. https://doi.org/10.1016/j.tafmec.2020.102850

    Article  Google Scholar 

  52. Nguyen CT, Oterkus S (2021) Ordinary state-based peridynamics for geometrically nonlinear analysis of plates. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2020.102877

    Article  MATH  Google Scholar 

  53. Nguyen CT, Oterkus S, Oterkus E (2021) A physics-guided machine learning model for two-dimensional structures based on ordinary state-based peridynamics. Theor Appl Fract Mech 112:102872. https://doi.org/10.1016/j.tafmec.2020.102872

    Article  Google Scholar 

  54. Dai MJ, Tanaka S, Oterkus S, Oterkus E (2020) Mixed-mode stress intensity factors evaluation of flat shells under in-plane loading employing ordinary state-based peridynamics. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2020.102841

    Article  Google Scholar 

  55. Shou Y, Zhou X, Berto F (2019) 3D numerical simulation of initiation, propagation and coalescence of cracks using the extended non-ordinary state-based peridynamics. Theor Appl Fract Mech 101:254–268. https://doi.org/10.1016/j.tafmec.2019.03.006

    Article  Google Scholar 

  56. Karpenko O, Oterkus S, Oterkus E (2021) Peridynamic Investigation of the effect of porosity on fatigue nucleation for additively manufactured titanium alloy Ti6Al4V. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2021.102925

    Article  Google Scholar 

  57. Bang DJ, Ince A, Oterkus E, Oterkus S (2021) Crack growth modeling and simulation of a peridynamic fatigue model based on numerical and analytical solution approaches. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2021.103026

    Article  Google Scholar 

  58. Yang Z, Oterkus E, Oterkus S (2021) Peridynamic modelling of higher order functionally graded plates. Math Mech Solids. https://doi.org/10.1177/10812865211004671

    Article  MathSciNet  MATH  Google Scholar 

  59. Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, New York

    Book  MATH  Google Scholar 

  60. Javili A, Morasata R, Oterkus E, Oterkus S (2019) Peridynamics review. Math Mech Solids 24:3714–3739. https://doi.org/10.1177/1081286518803411

    Article  MathSciNet  MATH  Google Scholar 

  61. Rubinstein AA (1985) Macrocrack interaction with semi-infinite microcrack array. Int J Fract 27:113–119. https://doi.org/10.1007/BF00040390

    Article  Google Scholar 

  62. Lrf ROSE (1986) Effective fracture toughness of microcracked materials. J Am Ceram Soc 69:212–214. https://doi.org/10.1111/j.1151-2916.1986.tb07409.x

    Article  Google Scholar 

  63. Brencich A, Carpinteri A (1998) Stress field interaction and strain energy distribution between a stationary main crack and its process zone. Eng Fract Mech 59:797–814. https://doi.org/10.1016/S0013-7944(97)00158-6

    Article  Google Scholar 

  64. Petrova V, Schmauder S (2011) Thermal fracture of a functionally graded/homogeneous bimaterial with system of cracks. Theor Appl Fract Mech 55:148–157. https://doi.org/10.1016/j.tafmec.2011.04.005

    Article  Google Scholar 

  65. Singh IV, Mishra BK, Bhattacharya S (2011) XFEM simulation of cracks, holes and inclusions in functionally graded materials. Int J Mech Mater Des 7:199–218. https://doi.org/10.1007/s10999-011-9159-1

    Article  Google Scholar 

  66. Vazic B, Wang H, Diyaroglu C et al (2017) Dynamic propagation of a macrocrack interacting with parallel small cracks. AIMS Mater Sci 4:118–136. https://doi.org/10.3934/matersci.2017.1.118

    Article  Google Scholar 

  67. Basoglu MF, Zerin Z, Kefal A, Oterkus E (2019) A computational model of peridynamic theory for deflecting behavior of crack propagation with micro-cracks. Comput Mater Sci 162:33–46. https://doi.org/10.1016/j.commatsci.2019.02.032

    Article  Google Scholar 

  68. Cheng Z, Zhang G, Wang Y, Bobaru F (2015) A peridynamic model for dynamic fracture in functionally graded materials. Compos Struct 133:529–546. https://doi.org/10.1016/j.compstruct.2015.07.047

    Article  Google Scholar 

  69. Cheng Z, Liu Y, Zhao J et al (2018) Numerical simulation of crack propagation and branching in functionally graded materials using peridynamic modeling. Eng Fract Mech 191:13–32. https://doi.org/10.1016/j.engfracmech.2018.01.016

    Article  Google Scholar 

  70. Cheng ZQ, Sui ZB, Yin H et al (2019) Studies of dynamic fracture in functionally graded materials using peridynamic modeling with composite weighted bond. Theor Appl Fract Mech 103:102242. https://doi.org/10.1016/j.tafmec.2019.102242

    Article  Google Scholar 

  71. Pathrikar A, Tiwari SB, Arayil P, Roy D (2021) Thermomechanics of damage in brittle solids: a peridynamics model. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2020.102880

    Article  Google Scholar 

  72. Xia W, Oterkus E, Oterkus S (2021) Ordinary state-based peridynamic homogenization of periodic micro-structured materials. Theor Appl Fract Mech 113:102960. https://doi.org/10.1016/j.tafmec.2021.102960

    Article  Google Scholar 

  73. Ghajari M, Iannucci L, Curtis P (2014) A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Methods Appl Mech Eng 276:431–452. https://doi.org/10.1016/j.cma.2014.04.002

    Article  MathSciNet  MATH  Google Scholar 

  74. Ozdemir M, Kefal A, Imachi M et al (2020) Dynamic fracture analysis of functionally graded materials using ordinary state-based peridynamics. Compos Struct 244:112296. https://doi.org/10.1016/j.compstruct.2020.112296

    Article  Google Scholar 

  75. He D, Huang D, Jiang D (2021) Modeling and studies of fracture in functionally graded materials under thermal shock loading using peridynamics. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2020.102852

    Article  Google Scholar 

  76. Candaş A, Oterkus E, İmrak CE (2021) Dynamic crack propagation and its interaction with micro-cracks in an impact problem. J Eng Mater Technol 143:1–10. https://doi.org/10.1115/1.4047746

    Article  Google Scholar 

  77. Kalthoff JF, Winkler S (1987) Failure mode transition at high rates of loading. Impact Load Dyn Behav Mater 1:185–195

    Google Scholar 

  78. Kalthoff JF (2000) Modes of dynamic shear failure in solids. Int J Fract 101:1–31. https://doi.org/10.1023/a:1007647800529

    Article  Google Scholar 

  79. Silling SA (2003) Dynamic fracture modeling with a meshfree peridynamic code. Comput Fluid Solid Mech 2003:641–644

    Google Scholar 

  80. Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Methods Eng 108:1451–1476. https://doi.org/10.1002/nme.5257

    Article  MathSciNet  Google Scholar 

  81. Ren H, Zhuang X, Rabczuk T (2017) Dual-horizon peridynamics: a stable solution to varying horizons. Comput Methods Appl Mech Eng 318:762–782. https://doi.org/10.1016/j.cma.2016.12.031

    Article  MathSciNet  MATH  Google Scholar 

  82. Amani J, Oterkus E, Areias P et al (2016) A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87:83–94. https://doi.org/10.1016/j.ijimpeng.2015.06.019

    Article  Google Scholar 

  83. Gu X, Zhang Q, Xia X (2017) Voronoi-based peridynamics and cracking analysis with adaptive refinement. Int J Numer Methods Eng 112:2087–2109. https://doi.org/10.1002/nme.5596

    Article  MathSciNet  Google Scholar 

  84. Guo JS, Gao WC (2019) Study of the Kalthoff–Winkler experiment using an ordinary state-based peridynamic model under low velocity impact. Adv Mech Eng 11:168781401985256. https://doi.org/10.1177/1687814019852561

    Article  Google Scholar 

  85. Trask N, You H, Yu Y, Parks ML (2019) An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics. Comput Methods Appl Mech Eng 343:151–165. https://doi.org/10.1016/j.cma.2018.08.016

    Article  MathSciNet  MATH  Google Scholar 

  86. Wang H, Xu Y, Huang D (2019) A non-ordinary state-based peridynamic formulation for thermo-visco-plastic deformation and impact fracture. Int J Mech Sci 159:336–344. https://doi.org/10.1016/j.ijmecsci.2019.06.008

    Article  Google Scholar 

  87. Silling SA, Epton M, Weckner O et al (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184. https://doi.org/10.1007/s10659-007-9125-1

    Article  MathSciNet  MATH  Google Scholar 

  88. Zhang Z, Paulino GH, Celes W (2008) Cohesive modeling of dynamic crack growth in homogeneous and functionally graded materials. AIP Conf Proc 973:562–567. https://doi.org/10.1063/1.2896840

    Article  Google Scholar 

  89. Stukowski A (2010) Visualization and analysis of atomistic simulation data with OVITO-the open visualization tool. Model Simul Mater Sci Eng. https://doi.org/10.1088/0965-0393/18/1/015012

    Article  Google Scholar 

  90. Kalthoff JF (1988) Shadow optical analysis of dynamic shear fracture. Opt Eng. https://doi.org/10.1117/12.7976772

    Article  Google Scholar 

  91. Kalthoff JF (2003) Failure methodology of mode-II loaded cracks. Strength Fract Complex 1:121–138

    MATH  Google Scholar 

  92. Loehnert S, Belytschko T (2007) Crack shielding and amplification due to multiple microcracks interacting with a macrocrack. Int J Fract 145:1–8. https://doi.org/10.1007/s10704-007-9094-1

    Article  MATH  Google Scholar 

  93. Bleyer J, Roux-Langlois C, Molinari JF (2017) Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms. Int J Fract 204:79–100. https://doi.org/10.1007/s10704-016-0163-1

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Istanbul Technical University, Beyoğlu, Istanbul, 34437, Turkey

    Adem Candaş & C. Erdem İmrak

  2. Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, G4 0LZ, UK

    Erkan Oterkus

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AC: visualisation, methodology, software, writing—original draft preparation. EO: software, methodology, reviewing and editing, supervision. CEİ: methodology, writing—reviewing and editing, supervision.

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Candaş, A., Oterkus, E. & İmrak, C.E. Peridynamic simulation of dynamic fracture in functionally graded materials subjected to impact load. Engineering with Computers 39, 253–267 (2023). https://doi.org/10.1007/s00366-021-01540-2

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