References
- Y. Shao, Y. Zhang, X. Xu, Z. Zhou, W. Li and B. Liu, “Effect of crack pattern on the residual strength of ceramics after quenching,” J. Am. Ceram. Soc., vol. 94, no. 9, pp. 2804–2807, 2011. DOI: https://doi.org/10.1111/j.1551-2916.2011.04728.x.
- C. P. Jiang, et al., “A study of the mechanism of formation and numerical simulations of crack patterns in ceramics subjected to thermal shock,” Acta Mater., vol. 60, no. 11, pp. 4540–4550, 2012. DOI: https://doi.org/10.1016/j.actamat.2012.05.020.
- Y. F. Fu, Y. L. Wong, C. A. Tang and C. S. Poon, “Thermal induced stress and associated cracking in cement-based composite at elevated temperatures - Part I: Thermal cracking around single inclusion,” Cem. Concr. Compos., vol. 26, no. 2, pp. 99–111, 2004. DOI: https://doi.org/10.1016/S0958-9465(03)00086-6.
- J. F. Geyer and S. Nemat-Nasser, “Experimental investigation of thermally induced interacting cracks in brittle solids,” Int. J. Solids Struct., vol. 18, no. 4, pp. 349–356, 1982. DOI: https://doi.org/10.1016/0020-7683(82)90059-2.
- O. Ronsin, F. Heslot and B. Perrin, “Experimental study of quasistatic brittle crack propagation,” Phys. Rev. Lett., vol. 75, no. 12, pp. 2352–2355, 1995. DOI: https://doi.org/10.1103/PhysRevLett.75.2352.
- B. Kilic and E. Madenci, “Prediction of crack paths in a quenched glass plate by using peridynamic theory,” Int. J. Fract., vol. 156, no. 2, pp. 165–177, 2009. DOI: https://doi.org/10.1007/s10704-009-9355-2.
- S. A. Silling, “Reformulation of elasticity theory for discontinuities and long-range forces,” J. Mech. Phys. Solids, vol. 48, no. 1, pp. 175–209, 2000. DOI: https://doi.org/10.1016/S0022-5096(99)00029-0.
- H. A. Bahr, H. J. Weiss, H. G. Maschke and F. Meissner, “Multiple crack propagation in a strip caused by thermal shock,” Theor. Appl. Fract. Mech., vol. 10, no. 3, pp. 219–226, 1988. DOI: https://doi.org/10.1016/0167-8442(88)90014-6.
- M. Duflot, “The extended finite element method in thermoelastic fracture mechanics,” Int. J. Numer. Meth. Engng., vol. 74, no. 5, pp. 827–847, 2008. DOI: https://doi.org/10.1002/nme.2197.
- A. Zamani and M. R. Eslami, “Implementation of the extended finite element method for dynamic thermoelastic fracture initiation,” Int. J. Solids Struct., vol. 47, no. 10, pp. 1392–1404, 2010. DOI: https://doi.org/10.1016/j.ijsolstr.2010.01.024.
- T. Menouillard and T. Belytschko, “Analysis and computations of oscillating crack propagation in a heated strip,” Int. J. Fract., vol. 167, no. 1, pp. 57–70, 2011. DOI: https://doi.org/10.1007/s10704-010-9519-0.
- H. Pathak, “Crack interaction study in functionally graded materials (FGMs) using XFEM under thermal and mechanical loading environment,” Mech. Adv. Mater. Struct., vol. 27, no. 11, pp. 903–926, 2020. DOI: https://doi.org/10.1080/15376494.2018.1501834.
- G. A. Francfort and J. J. Marigo, “Revisiting brittle fracture as an energy minimization problem,” J. Mech. Phys. Solids, vol. 46, no. 8, pp. 1319–1342, 1998. DOI: https://doi.org/10.1016/S0022-5096(98)00034-9.
- B. Bourdin, G. A. Francfort and J. J. Marigo, “Numerical experiments in revisited brittle fracture,” J. Mech. Phys. Solids, vol. 48, no. 4, pp. 797–826, 2000. DOI: https://doi.org/10.1016/S0022-5096(99)00028-9.
- H. Amor, J. J. Marigo and C. Maurini, “Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments,” J. Mech. Phys. Solids, vol. 57, no. 8, pp. 1209–1229, 2009. DOI: https://doi.org/10.1016/j.jmps.2009.04.011.
- C. Miehe, M. Hofacker and F. Welschinger, “A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits,” Comput. Methods Appl. Mech. Eng., vol. 199, no. 45–48, pp. 2765–2778, 2010. DOI: https://doi.org/10.1016/j.cma.2010.04.011.
- M. Ambati, T. Gerasimov and L. De Lorenzis, “A review on phase-field models of brittle fracture and a new fast hybrid formulation,” Comput. Mech., vol. 55, no. 2, pp. 383–405, 2015. DOI: https://doi.org/10.1007/s00466-014-1109-y.
- M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. R. Hughes and C. M. Landis, “A phase-field description of dynamic brittle fracture,” Comput. Methods Appl. Mech. Eng., vol. 217–220, pp. 77–95, 2012. DOI: https://doi.org/10.1016/j.cma.2012.01.008.
- M. Hofacker and C. Miehe, “Continuum phase field modeling of dynamic fracture: Variational principles and staggered FE implementation,” Int. J. Fract., vol. 178, no. 1–2, pp. 113–129, 2012. DOI: https://doi.org/10.1007/s10704-012-9753-8.
- A. Schlüter, A. Willenbücher, C. Kuhn and R. Müller, “Phase field approximation of dynamic brittle fracture,” Comput. Mech., vol. 54, no. 5, pp. 1141–1161, 2014. DOI: https://doi.org/10.1007/s00466-014-1045-x.
- M. Klinsmann, D. Rosato, M. Kamlah and R. M. McMeeking, “Modeling crack growth during Li insertion in storage particles using a fracture phase field approach,” J. Mech. Phys. Solids, vol. 92, pp. 313–344, 2016. DOI: https://doi.org/10.1016/j.jmps.2016.04.004.
- J. Y. Wu and V. P. Nguyen, “A length scale insensitive phase-field damage model for brittle fracture,” J. Mech. Phys. Solids, vol. 119, pp. 20–42, 2018. DOI: https://doi.org/10.1016/j.jmps.2018.06.006.
- J. Y. Wu, Y. Huang and V. P. Nguyen, “On the BFGS monolithic algorithm for the unified phase field damage theory,” Comput. Methods Appl. Mech. Eng., vol. 360, pp. 112704, 2020. DOI: https://doi.org/10.1016/j.cma.2019.112704.
- T. K. Mandal, V. P. Nguyen, J. Y. Wu, C. Nguyen-Thanh and A. de Vaucorbeil, “Fracture of thermo-elastic solids: Phase-field modeling and new results with an efficient monolithic solver,” Comput. Methods Appl. Mech. Eng., vol. 376, pp. 113648, 2021. DOI: https://doi.org/10.1016/j.cma.2020.113648.
- E. M. Pañeda, A. Golahmar and C. F. Niordson, “A phase field formulation for hydrogen assisted cracking,” Comput. Methods Appl. Mech. Eng., vol. 342, pp. 742–761, 2018. DOI: https://doi.org/10.1016/j.cma.2018.07.021.
- R. U. Patil, B. K. Mishra and I. V. Singh, “An adaptive multiscale phase field method for brittle fracture,” Comput. Methods Appl. Mech. Eng., vol. 329, pp. 254–288, 2018. DOI: https://doi.org/10.1016/j.cma.2017.09.021.
- Hirshikesh, S. Natarajan and R. K. Annabattula, “A FEniCS implementation of the phase field method for quasi-static brittle fracture,” Front. Struct. Civ. Eng., vol. 13, no. 2, pp. 380–396, 2019. DOI: https://doi.org/10.1007/s11709-018-0471-9.
- Hirshikesh, S. Natarajan, R. K. Annabattula and E. M. Pañeda, “Phase field modelling of crack propagation in functionally graded materials,” Compos. Part B., vol. 169, pp. 239–248, 2019. DOI: https://doi.org/10.1016/j.compositesb.2019.04.003.
- S. Mohanty, P. Y. Kumbhar, N. Swaminathan and R. K. Annabattula, “A phase-field model for crack growth in electro-mechanically coupled functionally graded piezo ceramics,” Smart Mater. Struct., vol. 29, no. 4, pp. 045005, 2020. DOI: https://doi.org/10.1088/1361-665X/ab7145.
- P. Raghu, A. Rajagopal and J. N. Reddy, “Thermodynamically consistent variational approach for modeling brittle fracture in thick plates by a hybrid phase field model,” J. Appl. Mech., vol. 87, no. 2, 021002, 2020.
- P. Raghu, A. Rajagopal, S. K. Jalan and J. N. Reddy, “Modeling of brittle fracture in thick plates subjected to transient dynamic loads using a hybrid phase field model,” Meccanica, vol. 2020, pp. 1–18, 2020.
- P. Kasirajan, S. Bhattacharya, A. Rajagopal and J. N. Reddy, “Phase field modeling of fracture in Quasi-Brittle materials using natural neighbor Galerkin method,” Comput. Methods Appl. Mech. Eng., vol. 366, pp. 113019, 2020. DOI: https://doi.org/10.1016/j.cma.2020.113019.
- B. Bourdin, J. J. Marigo, C. Maurini and P. Sicsic, “Morphogenesis and propagation of complex cracks induced by thermal shocks,” Phys. Rev. Lett., vol. 112, no. 1, pp. 014301, 2014. DOI: https://doi.org/10.1103/PhysRevLett.112.014301.
- C. Miehe, L. M. Schänzel and H. Ulmer, “Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids,” Comput. Methods Appl. Mech. Eng., vol. 294, pp. 449–485, 2015. DOI: https://doi.org/10.1016/j.cma.2014.11.016.
- C. Miehe, M. Hofacker, L. M. Schänzel and F. Aldakheel, “Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic–plastic solids,” Comput. Methods Appl. Mech. Eng., vol. 294, pp. 486–522, 2015. DOI: https://doi.org/10.1016/j.cma.2014.11.017.
- H. Badnava, M. A. Msekh, E. Etemadi and T. Rabczuk, “An h-adaptive thermo-mechanical phase field model for fracture,” Finite Elem. Anal. Des., vol. 138, no. C, pp. 31–47, 2018. DOI: https://doi.org/10.1016/j.finel.2017.09.003.
- D. Chu, X. Li and Z. Liu, “Study the dynamic crack path in brittle material under thermal shock loading by phase field modeling,” Int. J. Fract., vol. 208, no. 1–2, pp. 115–130, 2017. DOI: https://doi.org/10.1007/s10704-017-0220-4.
- T. Wang, X. Ye, Z. Liu, X. Liu, D. Chu and Z. Zhuang, “A phase-field model of thermo-elastic coupled brittle fracture with explicit time integration,” Comput. Mech., vol. 65, no. 5, pp. 1305–1321, 2020. DOI: https://doi.org/10.1007/s00466-020-01820-6.
- P. L. Hans and L. Anders, Solving PDEs in Python. New York, NY: Springer, 2017.
- C. Geuzaine and J. F. Remacle, “Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities,” Int. J. Numer. Meth. Engng., vol. 79, no. 11, pp. 1309–1331, 2009. DOI: https://doi.org/10.1002/nme.2579.
- J. D. R, “Optimal spacing and penetration of cracks in a shrinking slab,” Phys. Rev. E., vol. 71, no. 5, pp. 056117, 2005.