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Phase-field approximation for a class of cohesive fracture energies with an activation threshold

  • Antonin Chambolle and Vito Crismale ORCID logo EMAIL logo
Published/Copyright: January 8, 2020

Abstract

We study the Γ-limit of Ambrosio–Tortorelli-type functionals D ε ( u , v ) , whose dependence on the symmetrised gradient e ( u ) is different in 𝔸 u and in e ( u ) - 𝔸 u , for a -elliptic symmetric operator 𝔸 , in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano [Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity, SIAM J. Math. Anal. 46 2014, 4, 2936–2955]. In particular, we prove that G(S)BD functions with bounded 𝔸 -variation are (S)BD.


Communicated by Gianni Dal Maso


Award Identifier / Grant number: 793018

Award Identifier / Grant number: ANR-11-LABX-0056-LMH

Funding statement: Vito Crismale has been supported by the Marie Skłodowska-Curie Standard European Fellowship No. 793018 and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

References

[1] R. Alessi, M. Ambati, T. Gerasimov, S. Vidoli and L. De Lorenzis, Comparison of phase-field models of fracture coupled with plasticity, Advances in Computational Plasticity, Comput. Methods Appl. Sci. 46, Springer, Cham (2018), 1–21. 10.1007/978-3-319-60885-3_1Search in Google Scholar

[2] R. Alessi, J.-J. Marigo, C. Maurini and S. Vidoli, Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples, Int. J. Mech. Sci. 149 (2018), 559–576. 10.1016/j.ijmecsci.2017.05.047Search in Google Scholar

[3] R. Alicandro, A. Braides and J. Shah, Free-discontinuity problems via functionals involving the L 1 -norm of the gradient and their approximations, Interfaces Free Bound. 1 (1999), no. 1, 17–37. 10.4171/IFB/2Search in Google Scholar

[4] R. Alicandro and M. Focardi, Variational approximation of free-discontinuity energies with linear growth, Commun. Contemp. Math. 4 (2002), no. 4, 685–723. 10.1142/S0219199702000816Search in Google Scholar

[5] S. Almi, Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening, ESAIM Control Optim. Calc. Var. 23 (2017), no. 3, 791–826. 10.1051/cocv/2016014Search in Google Scholar

[6] M. Ambati, T. Gerasimov and L. De Lorenzis, A review on phase-field models of brittle fracture and a new fast hybrid formulation, Comput. Mech. 55 (2015), no. 2, 383–405. 10.1007/s00466-014-1109-ySearch in Google Scholar

[7] L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990), no. 4, 291–322. 10.1007/BF00376024Search in Google Scholar

[8] L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation, Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238. 10.1007/s002050050051Search in Google Scholar

[9] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University Press, New York, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar

[10] L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math. 43 (1990), no. 8, 999–1036. 10.1002/cpa.3160430805Search in Google Scholar

[11] A. Arroyo-Rabasa, G. De Philippis, J. Hirsch and F. Rindler, Dimensional estimates and rectifiability for measures satisfying linear PDE constraints, Geom. Funct. Anal. 29 (2019), no. 3, 639–658. 10.1007/s00039-019-00497-1Search in Google Scholar

[12] J.-F. Babadjian, Traces of functions of bounded deformation, Indiana Univ. Math. J. 64 (2015), no. 4, 1271–1290. 10.1512/iumj.2015.64.5601Search in Google Scholar

[13] M. Barchiesi, G. Lazzaroni and C. I. Zeppieri, A bridging mechanism in the homogenization of brittle composites with soft inclusions, SIAM J. Math. Anal. 48 (2016), no. 2, 1178–1209. 10.1137/15M1007343Search in Google Scholar

[14] G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics. Vol. 7, Academic Press, New York (1962), 55–129. 10.1016/S0065-2156(08)70121-2Search in Google Scholar

[15] G. Bellettini, A. Coscia and G. Dal Maso, Compactness and lower semicontinuity properties in SBD ( Ω ) , Math. Z. 228 (1998), no. 2, 337–351. 10.1007/PL00004617Search in Google Scholar

[16] D. Breit, L. Diening and F. Gmeineder, Traces of functions of bounded 𝔸 -variation and variational problems with linear growth, preprint (2017), https://arxiv.org/abs/1707.06804v1. Search in Google Scholar

[17] B. Bourdin, Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound. 9 (2007), no. 3, 411–430. 10.4171/IFB/171Search in Google Scholar

[18] B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids 48 (2000), no. 4, 797–826. 10.1016/S0022-5096(99)00028-9Search in Google Scholar

[19] S. Burke, C. Ortner and E. Süli, An adaptive finite element approximation of a generalized Ambrosio–Tortorelli functional, Math. Models Methods Appl. Sci. 23 (2013), no. 9, 1663–1697. 10.1142/S021820251350019XSearch in Google Scholar

[20] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser. 207, Longman Scientific & Technical, Harlow, 1989. Search in Google Scholar

[21] M. Caroccia and N. Van Goethem, Damage-driven fracture with low-order potentials: Asymptotic behavior, existence and applications, ESAIM Math. Model. Numer. Anal. 53 (2019), no. 4, 1305–1350. 10.1051/m2an/2019024Search in Google Scholar

[22] A. Chambolle, An approximation result for special functions with bounded deformation, J. Math. Pures Appl. (9) 83 (2004), no. 7, 929–954. 10.1016/j.matpur.2004.02.004Search in Google Scholar

[23] A. Chambolle, Addendum to: An approximation result for special functions with bounded deformation [J. Math. Pures Appl. (9) 83 (2004), no. 7, 929–954], J. Math. Pures Appl. (9) 84 (2005), 137–145. 10.1016/j.matpur.2004.11.001Search in Google Scholar

[24] A. Chambolle, S. Conti and G. Francfort, Korn–Poincaré inequalities for functions with a small jump set, Indiana Univ. Math. J. 65 (2016), no. 4, 1373–1399. 10.1512/iumj.2016.65.5852Search in Google Scholar

[25] A. Chambolle, S. Conti and G. A. Francfort, Approximation of a brittle fracture energy with a constraint of non-interpenetration, Arch. Ration. Mech. Anal. 228 (2018), no. 3, 867–889. 10.1007/s00205-017-1207-zSearch in Google Scholar

[26] A. Chambolle and V. Crismale, Compactness and lower semicontinuity in GSBD, preprint (2018), https://arxiv.org/abs/1802.03302; to appear in J. Eur. Math. Soc. (JEMS). 10.4171/JEMS/1021Search in Google Scholar

[27] A. Chambolle and V. Crismale, Existence of strong solutions to the Dirichlet problem for the Griffith energy, Calc. Var. Partial Differential Equations 58 (2019), no. 4, 10.1007/s00526-019-1571-7. 10.1007/s00526-019-1571-7Search in Google Scholar

[28] A. Chambolle and V. Crismale, A density result in GSBD p with applications to the approximation of brittle fracture energies, Arch. Ration. Mech. Anal. 232 (2019), no. 3, 1329–1378. 10.1007/s00205-018-01344-7Search in Google Scholar

[29] S. Conti, M. Focardi and F. Iurlano, Phase field approximation of cohesive fracture models, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 4, 1033–1067. 10.1016/j.anihpc.2015.02.001Search in Google Scholar

[30] S. Conti, M. Focardi and F. Iurlano, Approximation of fracture energies with p-growth via piecewise affine finite elements, ESAIM Control Optim. Calc. Var. 25 (2019), Article No. 34. 10.1051/cocv/2018021Search in Google Scholar

[31] S. Conti, M. Focardi and F. Iurlano, Existence of strong minimizers for the Griffith static fracture model in dimension two, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, 455–474. 10.1016/j.anihpc.2018.06.003Search in Google Scholar

[32] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal. Real World Appl. 38 (1999), no. 5, 585–604. 10.1016/S0362-546X(98)00132-1Search in Google Scholar

[33] V. Crismale, Density in SBD and approximation of fracture energies, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 3, 533–542. 10.4171/RLM/859Search in Google Scholar

[34] V. Crismale, On the approximation of SBD functions and some applications, SIAM J. Math. Anal. 51 (2019), no. 6, 5011–5048. 10.1137/18M119522XSearch in Google Scholar

[35] S. Dain, Generalized Korn’s inequality and conformal Killing vectors, Calc. Var. Partial Differential Equations 25 (2006), no. 4, 535–540. 10.1007/s00526-005-0371-4Search in Google Scholar

[36] G. Dal Maso, An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser, Boston, 1993. 10.1007/978-1-4612-0327-8Search in Google Scholar

[37] G. Dal Maso, Generalised functions of bounded deformation, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1943–1997. 10.4171/JEMS/410Search in Google Scholar

[38] G. Dal Maso, G. Orlando and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: The antiplane case, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Article No. 45. 10.1007/s00526-016-0981-zSearch in Google Scholar

[39] G. de Philippis, N. Fusco and A. Pratelli, On the approximation of SBV functions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 2, 369–413. 10.4171/RLM/768Search in Google Scholar

[40] G. De Philippis and F. Rindler, On the structure of 𝒜 -free measures and applications, Ann. of Math. (2) 184 (2016), no. 3, 1017–1039. 10.4007/annals.2016.184.3.10Search in Google Scholar

[41] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. Search in Google Scholar

[42] M. Focardi, On the variational approximation of free-discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci. 11 (2001), no. 4, 663–684. 10.1142/S0218202501001045Search in Google Scholar

[43] M. Focardi and F. Iurlano, Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity, SIAM J. Math. Anal. 46 (2014), no. 4, 2936–2955. 10.1137/130947180Search in Google Scholar

[44] I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 24 (1997), no. 3, 463–499. Search in Google Scholar

[45] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 , SIAM J. Math. Anal. 23 (1992), no. 5, 1081–1098. 10.1137/0523060Search in Google Scholar

[46] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), no. 8, 1319–1342. 10.1016/S0022-5096(98)00034-9Search in Google Scholar

[47] M. Friedrich, A piecewise Korn inequality in SBD and applications to embedding and density results, SIAM J. Math. Anal. 50 (2018), no. 4, 3842–3918. 10.1137/17M1129982Search in Google Scholar

[48] M. Fuchs and S. Repin, Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 385 (2010), no. 41, 224–233, 237. 10.1007/s10958-011-0554-9Search in Google Scholar

[49] A. Giacomini, Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 129–172. 10.1007/s00526-004-0269-6Search in Google Scholar

[50] F. Gmeineder and B. Raiţă, Embeddings for 𝔸 -weakly differentiable functions on domains, J. Funct. Anal. 277 (2019), no. 12, Article ID 108278. 10.1016/j.jfa.2019.108278Search in Google Scholar

[51] F. Gmeineder and B. Raita, On critical L p -differentiability of BD-maps, Rev. Mat. Iberoam. 35 (2019), no. 7, 2071–2078. 10.4171/rmi/1111Search in Google Scholar

[52] A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. Lond. Ser. A 221 (1920), 163–198. 10.1098/rsta.1921.0006Search in Google Scholar

[53] J.W̃. Hutchinson, A Course on Nonlinear Fracture Mechanics, Department of Solid Mechanics, Technical University of Denmark, Lyngby, 1989. Search in Google Scholar

[54] F. Iurlano, Fracture and plastic models as Γ-limits of damage models under different regimes, Adv. Calc. Var. 6 (2013), no. 2, 165–189. 10.1515/acv-2011-0011Search in Google Scholar

[55] F. Iurlano, A density result for GSBD and its application to the approximation of brittle fracture energies, Calc. Var. Partial Differential Equations 51 (2014), no. 1–2, 315–342. 10.1007/s00526-013-0676-7Search in Google Scholar

[56] D. Mumford and J. Shah, Boundary detection by minimizing functionals, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (San Francisco 1985), IEEE Press, Piscataway (1985), 22–26. Search in Google Scholar

[57] M. Negri, A non-local approximation of free discontinuity problems in SBV and SBD, Calc. Var. Partial Differential Equations 25 (2006), no. 1, 33–62. 10.1007/s00526-005-0356-3Search in Google Scholar

[58] J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237–248. 10.1051/m2an/1981150302371Search in Google Scholar

[59] W. F. Osgood, Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, Ann. of Math. (2) 3 (1901/02), no. 1–4, 25–34. 10.2307/1967630Search in Google Scholar

[60] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Appl. Austral. Nat. Univ. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. Search in Google Scholar

[61] D. Spector and J. Van Schaftingen, Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 3, 413–436. 10.4171/RLM/854Search in Google Scholar

[62] R. Temam, Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1983. Search in Google Scholar

[63] J. Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 877–921. 10.4171/JEMS/380Search in Google Scholar

Received: 2019-03-09
Revised: 2019-10-01
Accepted: 2019-12-10
Published Online: 2020-01-08
Published in Print: 2021-10-01

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