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Some remarks on the viscous approximation of crack growth
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Local minimality and crack prediction in quasi-static Griffith fracture evolution
1. | Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road,Worcester, MA 01609, United States |
References:
[1] |
L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital.(B), 3 (1989), 857-881. |
[2] |
L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics, in "Homogenization andAapplications to Material Sciences" (Nice, 1995), 1-2, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 9 (1995), 1-22. |
[3] |
L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation, Arch.Rational Mech. Anal., 139 (1997), 201-238.
doi: 10.1007/s002050050051. |
[4] |
L. Ambrosio, E. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, TheClarendon Press, Oxford University Press, New York, 2000. |
[5] |
G. Bellettini, A. Coscia and G. Dal Maso, Compactness and lower semicontinuity properties in $SBD(\Omega)$, Math.Z., 228 (1998), 337-351.
doi: 10.1007/PL00004617. |
[6] |
A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in brittle materials, Arch. Ration. Mech. Anal., 188 (2008), 309-349.
doi: 10.1007/s00205-007-0080-6. |
[7] |
G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rat. Mech. Anal., 176 (2005), 165-225.
doi: 10.1007/s00205-004-0351-4. |
[8] |
G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 257-290. |
[9] |
G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rat. Mech. Anal., 162 (2002), 101-135.
doi: 10.1007/s002050100187. |
[10] |
G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1800.
doi: 10.1142/S0218202502002331. |
[11] |
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. |
[12] |
G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), 1465-1500.
doi: 10.1002/cpa.3039. |
[13] |
G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342.
doi: 10.1016/S0022-5096(98)00034-9. |
[14] |
A. Griffith, The phenomena of rupture and flow insolids, Phil. Trans. Roy. Soc. London, CCXXI-A (1920), 163-198. |
[15] |
D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci., 18 (2008), 1529-1569.
doi: 10.1142/S0218202508003121. |
[16] |
C. J. Larsen, Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl. Math., 63 (2010), 630-654. |
[17] |
C. J. Larsen, M. Ortiz and C. L. Richardson, Fracture paths from front kinetics: relaxation and rate independence, Arch. Ration. Mech. Anal., 193 (2009), 539-583.
doi: 10.1007/s00205-009-0216-y. |
[18] |
M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925.
doi: 10.1142/S0218202508003236. |
show all references
References:
[1] |
L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital.(B), 3 (1989), 857-881. |
[2] |
L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics, in "Homogenization andAapplications to Material Sciences" (Nice, 1995), 1-2, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 9 (1995), 1-22. |
[3] |
L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation, Arch.Rational Mech. Anal., 139 (1997), 201-238.
doi: 10.1007/s002050050051. |
[4] |
L. Ambrosio, E. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, TheClarendon Press, Oxford University Press, New York, 2000. |
[5] |
G. Bellettini, A. Coscia and G. Dal Maso, Compactness and lower semicontinuity properties in $SBD(\Omega)$, Math.Z., 228 (1998), 337-351.
doi: 10.1007/PL00004617. |
[6] |
A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in brittle materials, Arch. Ration. Mech. Anal., 188 (2008), 309-349.
doi: 10.1007/s00205-007-0080-6. |
[7] |
G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rat. Mech. Anal., 176 (2005), 165-225.
doi: 10.1007/s00205-004-0351-4. |
[8] |
G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 257-290. |
[9] |
G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rat. Mech. Anal., 162 (2002), 101-135.
doi: 10.1007/s002050100187. |
[10] |
G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1800.
doi: 10.1142/S0218202502002331. |
[11] |
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. |
[12] |
G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), 1465-1500.
doi: 10.1002/cpa.3039. |
[13] |
G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342.
doi: 10.1016/S0022-5096(98)00034-9. |
[14] |
A. Griffith, The phenomena of rupture and flow insolids, Phil. Trans. Roy. Soc. London, CCXXI-A (1920), 163-198. |
[15] |
D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci., 18 (2008), 1529-1569.
doi: 10.1142/S0218202508003121. |
[16] |
C. J. Larsen, Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl. Math., 63 (2010), 630-654. |
[17] |
C. J. Larsen, M. Ortiz and C. L. Richardson, Fracture paths from front kinetics: relaxation and rate independence, Arch. Ration. Mech. Anal., 193 (2009), 539-583.
doi: 10.1007/s00205-009-0216-y. |
[18] |
M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925.
doi: 10.1142/S0218202508003236. |
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