[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.