July  2013, 12(4): 1657-1686. doi: 10.3934/cpaa.2013.12.1657

Fracture models as $\Gamma$-limits of damage models

1. 

SISSA, via Bonomea 265, 34136 Trieste

2. 

SISSA, Via Bonomea 265, 34136 Trieste, Italy

Received  June 2011 Revised  June 2012 Published  November 2012

We analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals $\Gamma$-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.
Citation: Gianni Dal Maso, Flaviana Iurlano. Fracture models as $\Gamma$-limits of damage models. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1657-1686. doi: 10.3934/cpaa.2013.12.1657
References:
[1]

G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with line-tension effect,, Arch. Rational Mech. Anal., 144 (1998), 1. doi: 10.1007/s002050050111. Google Scholar

[2]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.2307/1990893. Google Scholar

[3]

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

[4]

L. Ambrosio, A. Lemenant and G. Royer-Carfagni, A variational model for plastic slip and its regularization via $\Gamma$-convergence,, J. Elasticity, (): 10659. doi: 10.1007/s10659-012-9390-5. Google Scholar

[5]

L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence,, Comm. Pure Appl. Math., 43 (1990), 999. doi: 10.1002/cpa.3160430805. Google Scholar

[6]

L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems,, Boll. Un. Mat. Ital., 6-B (1992), 105. Google Scholar

[7]

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

[8]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation,", Pitman Res. Notes Math. Ser., 203 (1989). Google Scholar

[9]

J.-M. Coron, The continuity of the rearrangement in $W^{1,p}(R)$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 57. Google Scholar

[10]

G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies,, Nonlinear Anal., 38 (1999), 585. doi: 10.1016/S0362-546X(98)00132-1. Google Scholar

[11]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkh\, (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar

[12]

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

[13]

H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969). Google Scholar

[14]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monographs in Mathematics \textbf{80}, 80 (1984). Google Scholar

[15]

K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality,, Manuscripta Math., 18 (1976), 215. Google Scholar

[16]

F. Iurlano, Fracture and plastic models as $\Gamma$-limits of damage models under different regimes,, Adv. Calc. Var., (). Google Scholar

[17]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. Google Scholar

show all references

References:
[1]

G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with line-tension effect,, Arch. Rational Mech. Anal., 144 (1998), 1. doi: 10.1007/s002050050111. Google Scholar

[2]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.2307/1990893. Google Scholar

[3]

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

[4]

L. Ambrosio, A. Lemenant and G. Royer-Carfagni, A variational model for plastic slip and its regularization via $\Gamma$-convergence,, J. Elasticity, (): 10659. doi: 10.1007/s10659-012-9390-5. Google Scholar

[5]

L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence,, Comm. Pure Appl. Math., 43 (1990), 999. doi: 10.1002/cpa.3160430805. Google Scholar

[6]

L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems,, Boll. Un. Mat. Ital., 6-B (1992), 105. Google Scholar

[7]

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

[8]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation,", Pitman Res. Notes Math. Ser., 203 (1989). Google Scholar

[9]

J.-M. Coron, The continuity of the rearrangement in $W^{1,p}(R)$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 57. Google Scholar

[10]

G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies,, Nonlinear Anal., 38 (1999), 585. doi: 10.1016/S0362-546X(98)00132-1. Google Scholar

[11]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkh\, (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar

[12]

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

[13]

H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969). Google Scholar

[14]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monographs in Mathematics \textbf{80}, 80 (1984). Google Scholar

[15]

K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality,, Manuscripta Math., 18 (1976), 215. Google Scholar

[16]

F. Iurlano, Fracture and plastic models as $\Gamma$-limits of damage models under different regimes,, Adv. Calc. Var., (). Google Scholar

[17]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. Google Scholar

[1]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267

[2]

Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030

[3]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291

[4]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[5]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

[6]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215

[7]

Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018

[8]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013

[9]

Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347

[10]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[11]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006

[12]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025

[13]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386

[14]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017

[15]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679

[16]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787

[17]

Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

[18]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53

[19]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

[20]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]