American Institute of Mathematical Sciences

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

Local minimality and crack prediction in quasi-static Griffith fracture evolution

Christopher J. Larsen 1,

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.
Keywords: rate independent., quasi-static evolution, Fracture mechanics.
Mathematics Subject Classification: Primary: 74R10; Secondary: 49Q20 74G65 74H5.
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.   Google Scholar

[2]

L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics,, in, 9 (1995), 1.   Google Scholar

[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.  Google Scholar

[4]

L. Ambrosio, E. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Griffith, The phenomena of rupture and flow insolids,, Phil. Trans. Roy. Soc. London, CCXXI-A (1920), 163.   Google Scholar

[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.  Google Scholar

[16]

C. J. Larsen, Epsilon-stable quasi-static brittle fracture evolution,, Comm. Pure Appl. Math., 63 (2010), 630.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics,, in, 9 (1995), 1.   Google Scholar

[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.  Google Scholar

[4]

L. Ambrosio, E. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Griffith, The phenomena of rupture and flow insolids,, Phil. Trans. Roy. Soc. London, CCXXI-A (1920), 163.   Google Scholar

[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.  Google Scholar

[16]

C. J. Larsen, Epsilon-stable quasi-static brittle fracture evolution,, Comm. Pure Appl. Math., 63 (2010), 630.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Przemysław Górka. Quasi-static evolution of polyhedral crystals. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 309-320. doi: 10.3934/dcdsb.2008.9.309
[2]

Alice Fiaschi. Young-measure quasi-static damage evolution: The nonconvex and the brittle cases. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 17-42. doi: 10.3934/dcdss.2013.6.17
[3]

Dorothee Knees, Andreas Schröder. Computational aspects of quasi-static crack propagation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 63-99. doi: 10.3934/dcdss.2013.6.63
[4]

Irina F. Sivergina, Michael P. Polis. About global null controllability of a quasi-static thermoelastic contact system. Conference Publications, 2005, 2005 (Special) : 816-823. doi: 10.3934/proc.2005.2005.816
[5]

Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079
[6]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076
[7]

Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i
[8]

M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495
[9]

T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215
[10]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
[11]

K. R. Rajagopal. The thermo-mechanics of rate-type fluids. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1133-1145. doi: 10.3934/dcdss.2012.5.1133
[12]

Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[13]

Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585
[14]

Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070
[15]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
[16]

Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331
[17]

Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i
[18]

Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447
[19]

Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256
[20]

Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences