Study on Crack Growth Resulting from Spacing and Alignment of Two Circular Holes: A Phase Field Approach

Sayahdin Alfat

doi:10.24423/cames.2024.1620

Sayahdin Alfat Physics Education Department, Halu Oleo University

DOI: http://dx.doi.org/10.24423/cames.2024.1620

Abstract

This study investigates the effects of spacing and alignment between two circular holes on crack growth simulation. Key aspects analyzed include: (a) crack growth behavior, (b) von Mises stress distribution, and (c) energy profiles, all through variations in the spacing and alignment of the holes. The material is assumed to be homogeneous and isotropic, with the following non-dimensional properties: Young’s modulus E = 70, Poisson's ratio ν = 0.35, which correspond to the real values E = 70 GPa, ν = 0.35, and ɣ = 2800 J · m^–2. Additionally, the body force is neglected (f(x; t) = 0). The numerical method used in this research is the adaptive finite element method, which is considered highly robust for solving the phase field model for crack growth. Notable findings include: (a) spacing between the two holes did not significantly alter the crack path, while alignment differences had a significant impact; (b) during the cracking process, the highest stress occurs at the crack tip and the lowest at the crack center; and (c) the time for cracking in materials with two holes varies with spacing and alignment, and elastic and surface energy curves help predict total damage.

Keywords

phase field model, crack path, stress distribution, spacing and alignment of two holes,

References

1. L.P. Borrego, J.M. Ferreira, J.M. Pinho da Cruz, J.M. Costa, Evaluation of overload effects on fatigue crack growth and closure, Engineering Fracture Mechanics, 70(11): 1379–1397, 2003, doi: 10.1016/S0013-7944(02)00119-4.
2. S. Alfat, New Frameworks of PFM for Thermal Fracturing in The Linear Thermoelasticity Solids Based on a Microforce Balance Approach, 19 December 2023, PREPRINT (Ver. 1) available at Research Square, doi: 10.21203/rs.3.rs-3776383/v1.
3. S. Alfat, M. Kimura, A.M. Maulana, Phase field models for thermal fracturing and their variational structures, Materials, 15(7): 2571, 2022, doi: 10.3390/ma15072571.
4. M.K. Hubbert, D.G. Willis, Mechanics of hydraulic fracturing, Transactions of the AIME, 210(01): 153–168, 1957, doi: 10.2118/686-G.
5. T. Takaishi, Phase field crack growth model with hydrogen embrittlement, [in:] Mathematical Analysis of Continuum Mechanics and Industrial Applications: Proceedings of the International Conference CoMFoS15, pp. 27–34, Springer, Singapore, 2017.
6. E. Martinez-Paneda, A. Golahmar, C.F. Niordson, A phase field formulation for hydrogen assisted cracking, Computer Methods in Applied Mechanics and Engineering, 342: 742–761, 2018, doi: 10.1016/j.cma.2018.07.021.
7. M. Kimura, T. Takaishi, S. Alfat, T. Nakano, Y. Tanaka, Irreversible phase field models for crack growth in industrial applications: thermal stress, viscoelasticity, hydrogen embrittlement, SN Applied Sciences, 3(9): 781, 2021, doi: 10.1007/s42452-021-04593-6.
8. D. Taylor, Geometrical effects in fatigue: A unifying theoretical model, International Journal of Fatigue, 21(5): 413–420, 1999, doi: 10.1016/S0142-1123(99)00007-9.
9. S. Alfat, M. Kimura, M.Z. Firihu, R. Rahmat, Numerical investigation of shape domain effect to its elasticity and surface energy using adaptive finite element method, [in:] Metallurgy and Advanced Material Technology for Sustainable Development (ISMM2017), 2018, 1964(1): 020011, doi: 10.1063/1.5038293.
10. H. Haeri, V. Sarfarazi, The effect of micro pore on the characteristics of crack tip plastic zone in concrete, Computers and Concrete, 17(1): 107–127, 2016, doi: 10.12989/cac.2016.17.1.107.
11. K.S. Churn, D.N. Yoon, Pore formation and its effect on mechanical properties in W–Ni–Fe heavy alloy, Powder Metallurgy, 22(4): 175–178, 1979.
12. X.S. Wang, J.H. Fan, B.S. Wu, Y. Li, Effects of distance and alignment holes on fatigue crack behaviors of cast magnesium alloys, Advanced Materials Research, 33: 13–18, 2008, doi: 10.4028/www.scientific.net/AMR.33-37.13.
13. X.S. Wang, C.H. Tan, J. Ma, X.D. Zhu, Q.Y. Wang, Influence of multi-holes on fatigue behaviors of cast magnesium alloys based on in-situ scanning electron microscope technology, Materials, 11(9): 1700, 2018, doi: 10.3390/ma11091700.
14. Y.M. Chen, Numerical solutions of three dimensional dynamic crack problems and simulation of dynamic fracture phenomena by a “non-standard” finite difference method, Engineering Fracture Mechanics, 10(4): 699–708, 1978, doi: 10.1016/0013-7944(78)90028-0.
15. A. Dorogoy, Finite difference method for solving crack problems in a functionally graded material, Simulation, 95(10): 941–953, 2019, doi: 10.1177/003754971880289.
16. M. Kimura, T. Takaishi, A phase field approach to mathematical modeling of crack propagation, [in:] R. Nishii et al. [Eds.], A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, Vol. 5, pp. 161–170, Springer, Tokyo, 2014, doi: 10.1007/978-4-431-55060-0_13.
17. B. Bourdin, G.A. Francfort, J.J. Marigo, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids, 48(4): 797–826, 2000, doi: 10.1016/S0022-5096(99)00028-9.
18. G. Zi, T. Belytschko, New crack-tip elements for XFEM and applications to cohesive cracks, International Journal for Numerical Methods in Engineering, 57(15): 2221–2240, 2003, doi: 10.1002/nme.849.
19. N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46(1): 131–150, 1999, doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J.
20. G.L. Golewski, P. Golewski, T. Sadowski, Numerical modelling crack propagation under Mode II fracture in plain concretes containing siliceous fly-ash additive using XFEM method, Computational Materials Science, 62: 75–78, 2012, doi: 10.1016/j.commatsci.2012.05.009.
21. H. Kim, M.P. Wagoner, W.G. Buttlar, Simulation of fracture behavior in asphalt concrete using a heterogeneous cohesive zone discrete element model, Journal of Materials in Civil Engineering, 20(8): 552–563, 2008, doi: 10.1061/(ASCE)0899-1561(2008)20:8(552).
22. L.U.C. Scholtès, F.V. Donzé, Modelling progressive failure in fractured rock masses using a 3D discrete element method, International Journal of Rock Mechanics and Mining Sciences, 52: 18–30, 2012, doi: 10.1016/j.ijrmms.2012.02.009.
23. Y. Mi, M. Aliabadi, Three-dimensional crack growth simulation using BEM, Computers & Structures, 52(5): 871–878, 1994, doi: 10.1016/0045-7949(94)90072-8.
24. C.S. Chen, E. Pan, B. Amadei, Fracture mechanics analysis of cracked discs of anisotropic rock using the boundary element method, International Journal of Rock Mechanics and Mining Sciences, 35(2): 195–218, 1998.
25. S. Wang, H. Liu, Modeling brittle–ductile failure transition with meshfree method, International Journal of Impact Engineering, 37(7), 783–791, 2010, doi: 10.1016/j.ijimpeng.2010.01.006.
26. B.N. Rao, S. Rahman, An efficient meshless method for fracture analysis of cracks, Computational Mechanics, 26(4): 398–408, 2000, doi: 10.1007/s004660000189.
27. S. Li, W.K. Liu, Meshfree and particle methods and their applications, Applied Mechanics Review, 55(1): 1–34, 2002, doi: 10.1115/1.1431547.
28. B. Giovanardi, A. Scotti, L. Formaggia, A hybrid XFEM–Phase field (Xfield) method for crack propagation in brittle elastic materials, Computer Methods in Applied Mechanics and Engineering, 320: 396–420, 2017, doi: 10.1016/j.cma.2017.03.039.
29. C. Miehe, M. Hofacker, F. Welschinger, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering, 199(45–48): 2765–2778, 2010, doi: 10.1016/j.cma.2010.04.011.
30. A. Karma, D.A. Kessler, H. Levine, Phase-field model of mode III dynamic fracture, Physical Review Letters, 87(4): 045501, 2001, doi: 10.1103/PhysRevLett.87.045501.
31. G.A. Francfort, J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, 46(8): 1319–1342, 1998, doi: 10.1016/S0022-5096(98)00034-9.
32. L. Ambrosio, V.M. Tortorelli, On the approximation of free discontinuity problems, Bollettino della Unione Matematica Italiana, 6: 105–123, 1992.
33. F. Hecht, New development in FreeFem++, Journal of Numerical Mathematics, 20(3–4): 251–266, 2012, doi: 10.1515/jnum-2012-0013.
34. T.T. Nguyen, J. Yvonnet, Q.-Z. Zhu, M. Bornert, C. Chateau, A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure, Engineering Fracture Mechanics, 139: 18–39, 2015, doi: 10.1016/j.engfracmech.2015.03.045.
35. P. Chakraborty, P. Sabharwall, M.C. Carroll, A phase-field approach to model multiaxial and microstructure dependent fracture in nuclear grade graphite, Journal of Nuclear Materials, 475: 200–208, 2016, doi: 10.1016/j.jnucmat.2016.04.006.
36. J.G. Londono, L. Berger-Vergiat, H. Waisman, An equivalent stress-gradient regularization model for coupled damage-viscoelasticity, Computer Methods in Applied Mechanics and Engineering, 322: 137–166, 2017, doi: 10.1016/j.cma.2017.04.010.
37. J. Ahrens, B. Geveci, C. Law, ParaView: An End-User Tool for Large-Data Visualization, Technical Report, LA-UR-03-1560, Los Alamos National Laboratory, 2005.
38. D.P.H. Hasselman, Elastic energy at fracture and surface energy as design criteria for thermal shock, Journal of the American Ceramic Society, 46(11): 535–540, 1963.
39. E. Orowan, Energy criteria of fracture, Welding Journal Research Supplement, 34: 157–160, 1955.