SHEN Yongxing1,2* (沈泳星), MOLLAALI Mostafa1,2, LI Yihuan1,2 (李毅环),MA Weixin1,2 (马维馨), JIANG Jiahao3 (蒋家皓) . Implementation Details for the Phase Field Approaches to Fracture[J]. Journal of Shanghai Jiaotong University(Science), 2018 , 23(1) : 166 -174 . DOI: 10.1007/s12204-018-1922-0

[1] LANDAU L D, LIFSHITZ E M. Statistical physics [M].Oxford: Pergamon Press, 1980. [2] ARANSON I, KALATSKY V, VINOKUR V. Continuum field description of crack propagation [J]. Physical Review Letters, 2000, 85(1): 118. [3] KARMA A, KESSLER D A, LEVINE H. Phase-field model of mode iii dynamic fracture [J]. Physical Review Letters, 2001, 87(4): 045501. [4] HENRY H, LEVINE H. Dynamic instabilities of fracture under biaxial strain using a phase field model [J].Physical Review Letters, 2004, 93(10): 105504. [5] AMBATI M, GERASIMOV T, DE LORENZIS L. A review on phase-field models of brittle fracture and a new fast hybrid formulation [J]. Computational Mechanics,2015, 55(2): 383-405. [6] FRANCFORT G A, MARIGO J J. Revisiting brittle fracture as an energy minimization problem [J].Journal of the Mechanics and Physics of Solids, 1998,46(8): 1319-1342. [7] BOURDIN B, FRANCFORT G A, MARIGO J J. Numerical experiments in revisited brittle fracture [J].Journal of the Mechanics and Physics of Solids, 2000,48(4): 797-826. [8] BOURDIN B. Numerical implementation of the variational formulation for quasi-static brittle fracture [J].Interfaces and Free Boundaries, 2007, 9(3): 411-430. [9] BOURDIN B, MARIGO J J, MAURINI C, et al. Morphogenesis and propagation of complex cracks induced by thermal shocks [J]. Physical Review Letters, 2014,112: 014301. [10] TANN′ EE, LI T, BOURDINB, et al. Crack nucleation in variational phase-field models of brittle fracture [J].Journal of the Mechanics and Physics of Solids, 2018,110: 80-99. [11] BORDEN M J, HUGHES T J, LANDIS C M, et al. A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework [J]. Computer Methods in Applied Mechanics and Engineering, 2014, 273: 100-118. [12] BORDEN M J. Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture [D].Austin: University of Texas at Austin, 2012. [13] SARGADO J M, KEILEGAVLEN E, BERRE I, et al.High-accuracy phasefield models for brittle fracture based on a new family of degradation functions [J].Journal of the Mechanics and Physics of Solids,2017. https://doi.org/10.1016/j.jmps.2017.10.015(published online). [14] AMOR H, MARIGO J J, MAURINI C. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments [J]. Journal of the Mechanics and Physics of Solids, 2009, 57: 1209-1229. [15] MIEHE C, WELSCHINGER F, HOFACKER M.Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations[J]. International Journal for Numerical Methods in Engineering, 2010, 83: 1273-1311. [16] GERASIMOV T, DE LORENZIS L. A line search assisted monolithic approach for phase-field computing of brittle fracture [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 276-303. [17] WHEELER M, WICK T, WOLLNER W. An augmented-Lagrangian method for the phasefield approach for pressurized fractures [J]. Computer Methods in Applied Mechanics and Engineering, 2014, 271: 69-85. [18] ZIAEI-RAD V, SHEN Y. Massive parallelization of the phase field formulation for crack propagation with time adaptivity [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 312, 224-253.