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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

L. Ambrosio, L. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, Oxford, 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), 999-1036. 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), 105-123.  Google Scholar

[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.  Google Scholar

[8]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation," Pitman Res. Notes Math. Ser., 203, Longman, Harlow, 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), 57-85.  Google Scholar

[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.  Google Scholar

[11]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Basel, 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, CRC Press, Boca Raton, 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 80, Birkhäuser, Basel, 1984.  Google Scholar

[15]

K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math., 18 (1976), 215-235.  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), 353-372.  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), 1-46. 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), 683-773. 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, Oxford, 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), 999-1036. 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), 105-123.  Google Scholar

[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.  Google Scholar

[8]

G. Buttazzo, "Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation," Pitman Res. Notes Math. Ser., 203, Longman, Harlow, 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), 57-85.  Google Scholar

[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.  Google Scholar

[11]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Basel, 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, CRC Press, Boca Raton, 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 80, Birkhäuser, Basel, 1984.  Google Scholar

[15]

K. Hilden, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math., 18 (1976), 215-235.  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), 353-372.  Google Scholar

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