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December  2011, 31(4): 1219-1231. doi: 10.3934/dcds.2011.31.1219

Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem

Gianni Dal Maso 1, and  Giuliano Lazzaroni 1,

1. 

SISSA, via Bonomea 265, 34136 Trieste, Italy

Received  December 2009 Revised  December 2010 Published  September 2011

We present a recent existence result concerning the quasistatic evolution of cracks in hyperelastic brittle materials, in the framework of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in [9].
Keywords: brittle fracture, polyconvexity, Variational models, finite elasticity, non-interpenetration., energy minimization, crack propagation, free-discontinuity problems, quasistatic evolution, rate-independent processes, Griffith's criterion.
Mathematics Subject Classification: 35R35, 74R10, 74B20, 49J45, 49Q20, 35A3.
Citation: Gianni Dal Maso, Giuliano Lazzaroni. Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1219-1231. doi: 10.3934/dcds.2011.31.1219
References:
[1]

L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in $SBV(\Omega,\R^k)$,, Nonlinear Anal., 23 (1994), 405.  doi: 10.1016/0362-546X(94)90180-5.  Google Scholar

[2]

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

[3]

J. M. Ball, Some open problems in elasticity,, in, (2002), 3.   Google Scholar

[4]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, J. Elasticity, 91 (2008), 5.  doi: 10.1007/s10659-007-9107-3.  Google Scholar

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications,, Arch. Ration. Mech. Anal., 167 (2003), 211.  doi: 10.1007/s00205-002-0240-7.  Google Scholar

[6]

P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[7]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Ration. Mech. Anal., 97 (1987), 171.  doi: 10.1007/BF00250807.  Google Scholar

[8]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[9]

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

[10]

G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results,, Arch. Ration. Mech. Anal., 162 (2002), 101.  doi: 10.1007/s002050100187.  Google Scholar

[11]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 199.   Google Scholar

[12]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der Mathematischen Wissenschaften, 153 (1969).   Google Scholar

[13]

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

[14]

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

[15]

G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies,, J. Reine Angew. Math., 595 (2006), 55.  doi: 10.1515/CRELLE.2006.044.  Google Scholar

[16]

N. Fusco, C. Leone, R. March and A. Verde, A lower semi-continuity result for polyconvex functionals in SBV,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 321.  doi: 10.1017/S0308210500004571.  Google Scholar

[17]

A. Giacomini and M. Ponsiglione, Non interpenetration of matter for $SBV$-deformations of hyperelastic brittle materials,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1019.  doi: 10.1017/S0308210507000121.  Google Scholar

[18]

A. A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163.   Google Scholar

[19]

G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data,, Ann. Mat. Pura Appl. (4), 190 (2011), 165.  doi: 10.1007/s10231-010-0145-2.  Google Scholar

[20]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.   Google Scholar

[21]

R. W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids,, Proc. Roy. Soc. London A, 326 (1972), 565.  doi: 10.1098/rspa.1972.0026.  Google Scholar

[22]

R. W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids,, Proc. Roy. Soc. London A, 328 (1972), 567.  doi: 10.1098/rspa.1972.0096.  Google Scholar

show all references

References:
[1]

L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in $SBV(\Omega,\R^k)$,, Nonlinear Anal., 23 (1994), 405.  doi: 10.1016/0362-546X(94)90180-5.  Google Scholar

[2]

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

[3]

J. M. Ball, Some open problems in elasticity,, in, (2002), 3.   Google Scholar

[4]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, J. Elasticity, 91 (2008), 5.  doi: 10.1007/s10659-007-9107-3.  Google Scholar

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications,, Arch. Ration. Mech. Anal., 167 (2003), 211.  doi: 10.1007/s00205-002-0240-7.  Google Scholar

[6]

P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[7]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Ration. Mech. Anal., 97 (1987), 171.  doi: 10.1007/BF00250807.  Google Scholar

[8]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[9]

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

[10]

G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results,, Arch. Ration. Mech. Anal., 162 (2002), 101.  doi: 10.1007/s002050100187.  Google Scholar

[11]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 199.   Google Scholar

[12]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der Mathematischen Wissenschaften, 153 (1969).   Google Scholar

[13]

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

[14]

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

[15]

G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies,, J. Reine Angew. Math., 595 (2006), 55.  doi: 10.1515/CRELLE.2006.044.  Google Scholar

[16]

N. Fusco, C. Leone, R. March and A. Verde, A lower semi-continuity result for polyconvex functionals in SBV,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 321.  doi: 10.1017/S0308210500004571.  Google Scholar

[17]

A. Giacomini and M. Ponsiglione, Non interpenetration of matter for $SBV$-deformations of hyperelastic brittle materials,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1019.  doi: 10.1017/S0308210507000121.  Google Scholar

[18]

A. A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163.   Google Scholar

[19]

G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data,, Ann. Mat. Pura Appl. (4), 190 (2011), 165.  doi: 10.1007/s10231-010-0145-2.  Google Scholar

[20]

A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.   Google Scholar

[21]

R. W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids,, Proc. Roy. Soc. London A, 326 (1972), 565.  doi: 10.1098/rspa.1972.0026.  Google Scholar

[22]

R. W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubberlike solids,, Proc. Roy. Soc. London A, 328 (1972), 567.  doi: 10.1098/rspa.1972.0096.  Google Scholar

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