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

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].
Citation: Gianni Dal Maso, Giuliano Lazzaroni. Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem. Discrete and Continuous Dynamical Systems, 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), 405-425. doi: 10.1016/0362-546X(94)90180-5.

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

[3]

J. M. Ball, Some open problems in elasticity, in "Geometry, Mechanics, and Dynamics" (eds. P. Newton, P. Holmes and A. Weinstein), 3-59, Springer, New York, 2002.

[4]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3.

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal., 167 (2003), 211-233. doi: 10.1007/s00205-002-0240-7.

[6]

P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.

[7]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 97 (1987), 171-188. doi: 10.1007/BF00250807.

[8]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4.

[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), 257-290.

[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), 101-135. doi: 10.1007/s002050100187.

[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), 199-210.

[12]

H. Federer, "Geometric Measure Theory," Die Grundlehren der Mathematischen Wissenschaften, 153, Springer-Verlag, New York, 1969.

[13]

G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), 1465-1500. doi: 10.1002/cpa.3039.

[14]

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.

[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), 55-91. doi: 10.1515/CRELLE.2006.044.

[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), 321-336. doi: 10.1017/S0308210500004571.

[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), 1019-1041. doi: 10.1017/S0308210507000121.

[18]

A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163-198.

[19]

G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data, Ann. Mat. Pura Appl. (4), 190 (2011), 165-194. doi: 10.1007/s10231-010-0145-2.

[20]

A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations" (eds. C. M. Dafermos and E. Feireisl), Vol. II, 461-559, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, 2005.

[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), 565-584. doi: 10.1098/rspa.1972.0026.

[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), 567-583. doi: 10.1098/rspa.1972.0096.

show all references

References:
[1]

L. Ambrosio, On the lower semicontinuity of quasiconvex integrals in $SBV(\Omega,\R^k)$, Nonlinear Anal., 23 (1994), 405-425. doi: 10.1016/0362-546X(94)90180-5.

[2]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

[3]

J. M. Ball, Some open problems in elasticity, in "Geometry, Mechanics, and Dynamics" (eds. P. Newton, P. Holmes and A. Weinstein), 3-59, Springer, New York, 2002.

[4]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3.

[5]

A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal., 167 (2003), 211-233. doi: 10.1007/s00205-002-0240-7.

[6]

P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.

[7]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 97 (1987), 171-188. doi: 10.1007/BF00250807.

[8]

G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4.

[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), 257-290.

[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), 101-135. doi: 10.1007/s002050100187.

[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), 199-210.

[12]

H. Federer, "Geometric Measure Theory," Die Grundlehren der Mathematischen Wissenschaften, 153, Springer-Verlag, New York, 1969.

[13]

G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), 1465-1500. doi: 10.1002/cpa.3039.

[14]

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.

[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), 55-91. doi: 10.1515/CRELLE.2006.044.

[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), 321-336. doi: 10.1017/S0308210500004571.

[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), 1019-1041. doi: 10.1017/S0308210507000121.

[18]

A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London Ser. A, 221 (1920), 163-198.

[19]

G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data, Ann. Mat. Pura Appl. (4), 190 (2011), 165-194. doi: 10.1007/s10231-010-0145-2.

[20]

A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations" (eds. C. M. Dafermos and E. Feireisl), Vol. II, 461-559, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, 2005.

[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), 565-584. doi: 10.1098/rspa.1972.0026.

[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), 567-583. doi: 10.1098/rspa.1972.0096.

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