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Crack growth with non-interpenetration: A simplified proof for the pure Neumann problem
1. | SISSA, via Bonomea 265, 34136 Trieste, Italy |
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. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).
|
[3] |
J. M. Ball, Some open problems in elasticity,, in, (2002), 3.
|
[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. |
[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. |
[6] |
P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).
|
[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. |
[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. |
[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.
|
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.
|
[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. |
[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. |
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. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).
|
[3] |
J. M. Ball, Some open problems in elasticity,, in, (2002), 3.
|
[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. |
[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. |
[6] |
P. G. Ciarlet, "Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).
|
[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. |
[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. |
[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.
|
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.
|
[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. |
[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. |
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