Fracture models as \Gamma-limits of damage models

Previous Article
On the backgrounds of the theory of m-Hessian equations
CPAA Home
This Issue
Next Article
A global attractor for a fluid--plate interaction model

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

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

Gianni Dal Maso ^1, and Flaviana Iurlano ^2,

1.	SISSA, via Bonomea 265, 34136 Trieste
2.	SISSA, Via Bonomea 265, 34136 Trieste, Italy

Received June 2011 Revised June 2012 Published November 2012

Abstract
Related Papers

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.

Keywords: Damage problems, free discontinuity problems., $\Gamma$-convergence, fracture mechanics.

Mathematics Subject Classification: Primary: 74A45, 49J45; Secondary: 35A3.

Citation: Gianni Dal Maso, Flaviana Iurlano. Fracture models as $\Gamma$-limits of damage models. Communications on Pure and 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), 1-46. doi: 10.1007/s002050050111.
[2]	F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773. doi: 10.2307/1990893.
[3]	L. Ambrosio, L. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, Oxford, 2000.
[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.
[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), 999-1036. doi: 10.1002/cpa.3160430805.
[6]	L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital., 6-B (1992), 105-123.
[7]	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.
[8]	G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation," Pitman Res. Notes Math. Ser., 203, Longman, Harlow, 1989.
[9]	J.-M. Coron, The continuity of the rearrangement in $W^{1,p}(R)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 57-85.
[10]	G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal., 38 (1999), 585-604. doi: 10.1016/S0362-546X(98)00132-1.
[11]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.
[12]	L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992.
[13]	H. Federer, "Geometric Measure Theory," Springer-Verlag, 1969.
[14]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics 80, Birkhäuser, Basel, 1984.
[15]	K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math., 18 (1976), 215-235.
[16]	F. Iurlano, Fracture and plastic models as $\Gamma$-limits of damage models under different regimes,, Adv. Calc. Var., ().
[17]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.

show all references

References:

[1]	G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with line-tension effect, Arch. Rational Mech. Anal., 144 (1998), 1-46. doi: 10.1007/s002050050111.
[2]	F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773. doi: 10.2307/1990893.
[3]	L. Ambrosio, L. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, Oxford, 2000.
[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.
[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), 999-1036. doi: 10.1002/cpa.3160430805.
[6]	L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital., 6-B (1992), 105-123.
[7]	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.
[8]	G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation," Pitman Res. Notes Math. Ser., 203, Longman, Harlow, 1989.
[9]	J.-M. Coron, The continuity of the rearrangement in $W^{1,p}(R)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 57-85.
[10]	G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal., 38 (1999), 585-604. doi: 10.1016/S0362-546X(98)00132-1.
[11]	G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.
[12]	L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992.
[13]	H. Federer, "Geometric Measure Theory," Springer-Verlag, 1969.
[14]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics 80, Birkhäuser, Basel, 1984.
[15]	K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math., 18 (1976), 215-235.
[16]	F. Iurlano, Fracture and plastic models as $\Gamma$-limits of damage models under different regimes,, Adv. Calc. Var., ().
[17]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.

[1]	Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355
[2]	Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems and Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267
[3]	Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030
[4]	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 and Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139
[5]	Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control and Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018
[6]	Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems and Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215
[7]	Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[8]	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 and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[9]	Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013
[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 and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[11]	Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347
[12]	Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[13]	Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
[14]	Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[15]	Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
[16]	Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems and Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033
[17]	Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[18]	Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[19]	Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679
[20]	Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Gianni Dal Maso Flaviana Iurlano
on Google Scholar
Gianni Dal Maso Flaviana Iurlano

[Back to Top]