# American Institute of Mathematical Sciences

2013, 6(1): 121-129. doi: 10.3934/dcdss.2013.6.121

## 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

Received  April 2011 Revised  July 2011 Published  October 2012

The mathematical analysis developed for energy minimizing fracture evolutions has been difficult to extend to locally minimizing evolutions. The reasons for this difficulty are not obvious, and our goal in this paper is to describe in some detail what precisely the issues are and why the previous analysis in fact cannot be extended to the most natural models based on local minimality. We also indicate how the previous methods can be modified for the analysis of models based on a recent definition of stability that is a bit stronger than local minimality.
Citation: Christopher J. Larsen. Local minimality and crack prediction in quasi-static Griffith fracture evolution. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 121-129. doi: 10.3934/dcdss.2013.6.121
##### References:
 [1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation,, Boll. Un. Mat. Ital.(B), 3 (1989), 857. [2] L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics,, in, 9 (1995), 1. [3] L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation,, Arch.Rational Mech. Anal., 139 (1997), 201. doi: 10.1007/s002050050051. [4] L. Ambrosio, E. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). [5] G. Bellettini, A. Coscia and G. Dal Maso, Compactness and lower semicontinuity properties in $SBD(\Omega)$,, Math.Z., 228 (1998), 337. doi: 10.1007/PL00004617. [6] A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in brittle materials,, Arch. Ration. Mech. Anal., 188 (2008), 309. 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. 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. [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. 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. doi: 10.1142/S0218202502002331. [11] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (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. 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. 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. [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. doi: 10.1142/S0218202508003121. [16] C. J. Larsen, Epsilon-stable quasi-static brittle fracture evolution,, Comm. Pure Appl. Math., 63 (2010), 630. [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. 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. doi: 10.1142/S0218202508003236.

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##### References:
 [1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation,, Boll. Un. Mat. Ital.(B), 3 (1989), 857. [2] L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics,, in, 9 (1995), 1. [3] L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation,, Arch.Rational Mech. Anal., 139 (1997), 201. doi: 10.1007/s002050050051. [4] L. Ambrosio, E. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). [5] G. Bellettini, A. Coscia and G. Dal Maso, Compactness and lower semicontinuity properties in $SBD(\Omega)$,, Math.Z., 228 (1998), 337. doi: 10.1007/PL00004617. [6] A. Chambolle, A. Giacomini and M. Ponsiglione, Crack initiation in brittle materials,, Arch. Ration. Mech. Anal., 188 (2008), 309. 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. 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. [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. 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. doi: 10.1142/S0218202502002331. [11] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (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. 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. 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. [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. doi: 10.1142/S0218202508003121. [16] C. J. Larsen, Epsilon-stable quasi-static brittle fracture evolution,, Comm. Pure Appl. Math., 63 (2010), 630. [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. 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. doi: 10.1142/S0218202508003236.
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