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 & 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

[1]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304

[2]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178

[3]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

[4]

Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363

[5]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[6]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360

[7]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[8]

Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035

[9]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[10]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015

[11]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[12]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

[13]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018

[14]

Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, 2021, 20 (2) : 559-582. doi: 10.3934/cpaa.2020281

[15]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233

[16]

Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230

[17]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096

[18]

Junkee Jeon. Finite horizon portfolio selection problems with stochastic borrowing constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 733-763. doi: 10.3934/jimo.2019132

[19]

Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133

[20]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]