| [1] | 钱学森. 物理力学讲义[M]. 上海: 上海交通大学出版社, 2007. |
| [2] | BECKER R. How metals fail [R]. Metal Fracture-Lawrence Livermore Nation Laboratory, 2002: 13–30. |
| [3] | FAN R, FISH J. The rs-method for material failure simulations [J]. International Journal for Numerical Methods in Engineering, 2008, 73(11): 1607–1623. doi: 10.1002/nme.2134 |
| [4] | HALLQUIST J O. LS-DYNA theoretical manual [M]. USA: Livemore Software Technology Corporation, 1998. |
| [5] | XU X P, NEEDLEMAN A. Numerical simulations of fast crack growth in brittle solids [J]. Journal of the Mechanics and Physics of Solids, 1994, 42(9): 1397–1434. doi: 10.1016/0022-5096(94)90003-5 |
| [6] | CAMACHO G T, ORTIZ M. Computational modelling of impact damage in brittle materials [J]. International Journal of Solids and Structures, 1996, 33(20/21/22): 2899–2938. |
| [7] | BELYTSCHKO T, BLACK T. Elastic crack growth in finite elements with minimal remeshing [J]. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620. doi: 10.1002/(sici)1097-0207(19990620)45:5<601::aid-nme598>3.0.co;2-s |
| [8] | MOËS N, DOLBOW J, BELYTSCHKO T. A finite element method for crack growth without remeshing [J]. International Journal for Numerical Methods in Engineering, 1999, 46(1): 131–150. doi: 10.1002/(SICI)1097-0207(19990910)46:13.0.CO;2-J |
| [9] | 庄茁, 成斌斌. 发展基于CB壳单元的扩展有限元模拟三维任意扩展裂纹 [J]. 工程力学, 2012, 29(6): 12–21. doi: 10.6052/j.issn.1000-4750.2010.08.0616 ZHUANG Z, CHENG B B. Development of X-FEM on CB shell element for simulating 3D arbitrary crack growth [J]. Engineering Mechanics, 2012, 29(6): 12–21. doi: 10.6052/j.issn.1000-4750.2010.08.0616 |
| [10] | PROVATAS N, ELDER K. Book-phase-field methods in materials science and engineering [M]. Wiley-VCH Press, 2010. |
| [11] | SONG J H, WANG H, BELYTSCHKO T. A comparative study on finite element methods for dynamic fracture [J]. Computational Mechanics, 2008, 42(2): 239–250. doi: 10.1007/s00466-007-0210-x |
| [12] | BORDEN M J, VERHOOSEL C V, SCOTT M A, et al. A phase-field description of dynamic brittle fracture [J]. Computer Methods in Applied Mechanics and Engineering, 2012, 217(220): 77–95. |
| [13] | RAMULU M, KOBAYASHI A S. Mechanics of crack curving and branching-a dynamic fracture analysis [J]. International Journal of Fracture, 1985, 27(3/4): 187–201. |
| [14] | KALTHOFF J F, WINKLER S. In failure mode transition at high rates of shear loading [C]//Impact Loading and Dynamic Behavioavior of Materials. FRG, 1987: 185–195. |
| [15] | 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. doi: 10.1016/S0022-5096(99)00028-9 |
| [16] | GIACOMINI A, PONSIGLIONE M. A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications [J]. Archive for Rational Mechanics and Analysis, 2006, 180(3): 399–447. doi: 10.1007/s00205-005-0392-3 |
| [17] | GRAVOUIL A, MOËS N, BELYTSCHKO T. Non-planar 3D crack growth by the extended finite element and level sets-Part II: level set update [J]. International Journal for Numerical Methods in Engineering, 2002, 53(11): 2569–2586. doi: 10.1002/nme.v53:11 |
| [18] | SUKUMAR N, CHOPP D L, BÉCHET E, et al. Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method [J]. International Journal for Numerical Methods in Engineering, 2008, 76(5): 727–748. doi: 10.1002/nme.v76:5 |
| [19] | MIEHE C, SCHÄNZEL L M, ULMER H. 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 [J]. Computer Methods in Applied Mechanics and Engineering, 2015, 294(1): 449–485. |
| [20] | HAKIM V, KARMA A. Laws of crack motion and phase-field models of fracture [J]. Journal of the Mechanics and Physics of Solids, 2009, 57(2): 342–368. doi: 10.1016/j.jmps.2008.10.012 |
| [21] | PONS A J, KARMA A. Helical crack-front instability in mixed-mode fracture [J]. Nature, 2010, 464(7285): 85–89. doi: 10.1038/nature08862 |
| [22] | 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. doi: 10.1016/S0022-5096(98)00034-9 |
| [23] | SPATSCHEK R, BRENER E, KARMA A. Phase field modeling of crack propagation [J]. Philosophical Magazine, 2011, 91(1): 75–95. doi: 10.1080/14786431003773015 |
| [24] | GRIFFITH A A. The phenomena of rupture and flow in solid [J]. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1920, 221: 163–198. |
| [25] | Bourdin B, Francfort G A, Marigo J J. The variational approach to fracture[M]. Springer Science+Business Media B. V., 2008. |
| [26] | 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(8): 1209–1229. doi: 10.1016/j.jmps.2009.04.011 |
| [27] | 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(10): 1273–1311. doi: 10.1002/nme.v83:10 |
| [28] | MIEHE C, HOFACKER M, WELSCHINGER F. A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits [J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199: 2765–2778. doi: 10.1016/j.cma.2010.04.011 |
| [29] | MESGARNEJAD A, BOURDIN B, KHONSARI M M. Validation simulations for the variational approach to fracture [J]. Computer Methods in Applied Mechanics and Engineering, 2015, 290(1): 420–437. |
| [30] | BOURDIN B, LARSEN C J, RICHARDSON C L. A time-discrete model for dynamic fracture based on crack regularization [J]. International Journal of Fracture, 2011, 168(2): 133–143. doi: 10.1007/s10704-010-9562-x |
| [31] | LARSEN C J, ORTNER C, SÜLI E. Existence of solutions to a regularized model of dynamic fracture [J]. Mathematical Models and Methods in Applied Sciences, 2010, 20(7): 1021–1048. doi: 10.1142/S0218202510004520 |
| [32] | HOFACKER M, MIEHE C. Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation [J]. International Journal of Fracture, 2012, 178(1): 113–129. |
| [33] | HOFACKER M, MIEHE C. A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns [J]. International Journal for Numerical Methods in Engineering, 2013, 93(3): 276–301. doi: 10.1002/nme.v93.3 |
| [34] | SCHLÜTER A, WILLENBÜCHER A, KUHN C, et al. Phase field approximation of dynamic brittle fracture [J]. Computational Mechanics, 2014, 54(5): 1141–1161. doi: 10.1007/s00466-014-1045-x |
| [35] | MIEHE C, MAUTHE S. Phase field modeling of fracture in multi-physics problems. Part III. crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 304(1): 619–655. |
| [36] | MIEHE C, HOFACKER M, SCHÄNZEL L M, et al. 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 [J]. Computer Methods in Applied Mechanics and Engineering, 2015, 294(1): 486–522. |
| [37] | WILSON Z A, BORDEN M J, LANDIS C M. A phase-field model for fracture in piezoelectric ceramics [J]. International Journal of Fracture, 2013, 183(2): 135–153. doi: 10.1007/s10704-013-9881-9 |
| [38] | ABDOLLAHI A, ARIAS I. Phase-field modeling of fracture in ferroelectric materials [J]. Archives of Computational Methods in Engineering, 2015, 22(2): 153–181. doi: 10.1007/s11831-014-9118-8 |
| [39] | MIEHE C, SCHÄNZEL L M. Phase field modeling of fracture in rubbery polymers. Part I: finite elasticity coupled with brittle failure [J]. Journal of the Mechanics and Physics of Solids, 2014, 64: 93–113. |
| [40] | ZIAEI-RAD V, SHEN L, JIANG J, et al. Identifying the crack path for the phase field approach to fracture with non-maximum suppression [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 312(1): 304–321. |
| [41] | SANTILLÁN D, MOSQUERA J C, CUETO-FELGUEROSO L. Phase-field model for brittle fracture. validation with experimental results and extension to dam engineering problems [J]. Engineering Fracture Mechanics, 2017, 178: 109–125. doi: 10.1016/j.engfracmech.2017.04.020 |
| [42] | DUDA F P, CIARBONETTI A, SÁNCHEZ P J, et al. A phase-field/gradient damage model for brittle fracture in elastic-plastic solids [J]. International Journal of Plasticity, 2015, 65: 269–296. doi: 10.1016/j.ijplas.2014.09.005 |
| [43] | AMBATI M, GERASIMOV T, LORENZIS L D. Phase-field modeling of ductile fracture [J]. Computational Mechanics, 2015, 55(5): 1017–1040. doi: 10.1007/s00466-015-1151-4 |
| [44] | AMBATI M, KRUSE R, LORENZIS L D. A phase-field model for ductile fracture at finite strains and its experimental verification [J]. Computational Mechanics, 2016, 57: 149–167. doi: 10.1007/s00466-015-1225-3 |
| [45] | MCAULIFFE C, WAISMAN H. A coupled phase field shear band model for ductile-brittle transition in notched plate impacts [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 305: 173–195. doi: 10.1016/j.cma.2016.02.018 |
| [46] | 柳占立, 李想, 初东阳, 等. 多物理场耦合断裂的相场方法模拟及工程应用[C]//第十二届全国爆炸力学会议. 桐乡, 2018. |