Analysis of a dislocation model for earthquakes
Authors:
Jing Liu, Xin Yang Lu and Noel J. Walkington
Journal:
Quart. Appl. Math. 74 (2016), 765-786
MSC (2010):
Primary 86A17, 74S05, 49J45
DOI:
https://doi.org/10.1090/qam/1445
Published electronically:
July 21, 2016
MathSciNet review:
3539031
Full-text PDF Free Access
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Abstract: Approximation of problems in linear elasticity having small shear modulus in a thin region is considered. Problems of this type arise when modeling ground motion due to earthquakes where rupture occurs in a thin fault. It is shown that, under appropriate scaling, solutions of these problems can be approximated by solutions of a limit problem where the fault region is represented by a surface. In a numerical context this eliminates the need to resolve the large deformations in the fault; a numerical example is presented to illustrate efficacy of this strategy.
References
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- A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn. 15 (2003), no. 4, 351–382. MR 1999280, DOI 10.1007/s00161-003-0120-x
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- R. E. Showalter, Hilbert space methods for partial differential equations, Monographs and Studies in Mathematics, Vol. 1, Pitman, London-San Francisco, Calif.-Melbourne, 1977. MR 0477394
- R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252, DOI 10.1090/surv/049
- Augusto Visintin, On the homogenization of visco-elastic processes, IMA J. Appl. Math. 77 (2012), no. 6, 869–886. MR 2999142, DOI 10.1093/imamat/hxs055
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References
- Amit Acharya, New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale, J. Mech. Phys. Solids 58 (2010), no. 5, 766–778. MR 2642309, DOI 10.1016/j.jmps.2010.02.001
- Amit Acharya and Luc Tartar, On an equation from the theory of field dislocation mechanics, Boll. Unione Mat. Ital. (9) 4 (2011), no. 3, 409–444. MR 2906769
- Amit Acharya and Xiaohan Zhang, From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics, Chin. Ann. Math. Ser. B 36 (2015), no. 5, 645–658. MR 3377868, DOI 10.1007/s11401-015-0970-0
- Giovanni Alberti, Guy Bouchitté, and Gianni Dal Maso, The calibration method for the Mumford-Shah functional and free-discontinuity problems, Calc. Var. Partial Differential Equations 16 (2003), no. 3, 299–333. MR 2001706, DOI 10.1007/s005260100152
- L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics, Homogenization and applications to material sciences (Nice, 1995) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 9, Gakkōtosho, Tokyo, 1995, pp. 1–22. MR 1473974
- Luigi Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 191–246 (Italian, with English and Italian summaries). MR 1387558
- Pedro Areias and Ted Belytschko, Analysis of three-dimensional crack initiation and propagation using the extended finite element method, International Journal for Numerical Methods in Engineering 63 (2005), no. 5, 760–788.
- B. Bourdin, G. A. Francfort, and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids 48 (2000), no. 4, 797–826. MR 1745759, DOI 10.1016/S0022-5096(99)00028-9
- Blaise Bourdin, Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound. 9 (2007), no. 3, 411–430. MR 2341850, DOI 10.4171/IFB/171
- Blaise Bourdin, Gilles A. Francfort, and Jean-Jacques Marigo, The variational approach to fracture, J. Elasticity 91 (2008), no. 1-3, 5–148. MR 2390547, DOI 10.1007/s10659-007-9107-3
- Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
- Gianni Dal Maso, Antonio DeSimone, and Maria Giovanna Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal. 180 (2006), no. 2, 237–291. MR 2210910, DOI 10.1007/s00205-005-0407-0
- Gianni Dal Maso, Gilles A. Francfort, and Rodica Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), no. 2, 165–225. MR 2186036, DOI 10.1007/s00205-004-0351-4
- Gianni Dal Maso and Rodica Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002), no. 2, 101–135. MR 1897378, DOI 10.1007/s002050100187
- Ennio De Giorgi, New problems on minimizing movements, Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 81–98. MR 1260440
- John Dolbow and Ted Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Meth. Eng 46 (1999), no. 1, 131–150.
- G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), no. 8, 1319–1342. MR 1633984, DOI 10.1016/S0022-5096(98)00034-9
- Gilles Francfort and Alexander Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math. 595 (2006), 55–91. MR 2244798, DOI 10.1515/CRELLE.2006.044
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- Morton E. Gurtin, Eliot Fried, and Lallit Anand, The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010. MR 2884384, DOI 10.1017/CBO9780511762956
- J.P. Hirth and J. Lothe, Theory of dislocations, Krieger Publishing Company, 1982.
- H Horii and Siavouche Nemat-Nasser, Brittle failure in compression: splitting, faulting and brittle-ductile transition, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 319 (1986), no. 1549, 337–374.
- A. Mesgarnejad, B. Bourdin, and M. M. Khonsari, Validation simulations for the variational approach to fracture, Comput. Methods Appl. Mech. Engrg. 290 (2015), 420–437. MR 3340163, DOI 10.1016/j.cma.2014.10.052
- A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn. 15 (2003), no. 4, 351–382. MR 1999280, DOI 10.1007/s00161-003-0120-x
- Alexander Mielke and Florian Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl. 11 (2004), no. 2, 151–189. MR 2210284, DOI 10.1007/s00030-003-1052-7
- R. E. Showalter, Hilbert space methods for partial differential equations, Pitman, London-San Francisco, Calif.-Melbourne, 1977. Monographs and Studies in Mathematics, Vol. 1. MR 0477394
- R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252
- Augusto Visintin, On the homogenization of visco-elastic processes, IMA J. Appl. Math. 77 (2012), no. 6, 869–886. MR 2999142, DOI 10.1093/imamat/hxs055
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI 10.1016/j.jmps.2015.07.004
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Additional Information
Jing Liu
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
jingliu1@andrew.cmu.edu
Xin Yang Lu
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Canada
MR Author ID:
974881
Email:
xinyang.lu@mcgill.ca
Noel J. Walkington
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
noelw@andrew.cmu.edu
Received by editor(s):
April 21, 2016
Published electronically:
July 21, 2016
Additional Notes:
The third author was supported in part by National Science Foundation grants DMS–1418991 and DMREF–1434734. This work was also supported by the NSF through the Center for Nonlinear Analysis.
Article copyright:
© Copyright 2016
Brown University
