Abstract
Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over an SBD type space. The corresponding functional can in turn be approximated in the sense of \({\Gamma}\)-convergence by a sequence of functionals involving a phase field as well as the displacement field. We show that a similar approximation persists if additionally imposing a non-interpenetration constraint in the minimization, namely that only nonnegative normal jumps should be permissible.
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Ambrosio L., Coscia A., Dal Maso G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139(3), 201–238 (1997)
Ambrosio L., Tortorelli V.M.: Approximation of functionals depending on jumps by elliptic functionals via \({\Gamma}\)-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York 2000
Amor H., Marigo J.J., Maurini C.: Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J. Mech, Phys. Solids 57(8), 1209–1229 (2009)
Bellettini G., Coscia A., Dal Maso G.: Compactness and lower semicontinuity properties in \({SBD(\Omega)}\). Mathematische Zeitschrift 228, 337–351 (1998)
Bourdin B., Chambolle A.: Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85(4), 609–646 (2000)
Bourdin B., Francfort G.A., Marigo J.J.: The variational approach to fracture. J. Elasticity 91(1,2,3), 5–148 (2008)
Braides A., Chambolle A., Solci M.: A relaxation result for energies defined on pairs set-function and applications. ESAIM Control Optim. Calc. Var. 13(4), 717–734 (2007)
Carriero M., Leaci A.: Existence theorem for a Dirichlet problem with free discontinuity set. Nonlinear Anal. 15(7), 661–677 (1990)
Chambolle A.: An approximation result for special functions with bounded variations. J. Math Pures Appl. 83, 929–954 (2004)
Chambolle, A.: Addendum to “An approximation result for special functions with bounded deformation” [J. Math. Pures Appl. (9) 83 (7) (2004) 929–954]: the N-dimensional case, J. Math Pures Appl. 84, 137–145 2005
Chambolle A., Conti S., Francfort G.A.: Korn-Poincaré inequalities for functions with a small jump set. Indiana Univ. Math. J. 65(4), 1373–1399 (2016)
Chambolle, A., Crismale, V.: A density result in GSBD p with applications to the approximation of brittle fracture energies, preprint arXiv:1708.03281, 2017
Chambolle, A., Conti, S., Iurlano, F.: Approximation of functions with small jump sets and existence of strong minimizers of Griffith’s energy, preprint arXiv:1710.01929 2017
Conti S., Focardi M., Iurlano F.: Existence of minimizers for the 2d stationary Griffith fracture model. C. R. Math. Acad. Sci. Paris 354, 1055–1059 (2016)
Conti S., Focardi M., Iurlano F.: Integral representation for functionals defined on SBD p in dimension two. Arch. Ration. Mech. Anal. 223(3), 1337–1374 (2017)
Conti, S., Focardi, M., Iurlano, F.: Approximation of fracture energies with p-growth via piecewise affine finite elements, preprint arXiv:1706.01735, 2017
Dal Maso G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. 15(5), 1943–1997 (2013)
Dal Maso G., Lazzaroni G.: Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 257–290 (2010)
Dal Maso G., Lazzaroni G.: Crack growth with non-interpenetration: a simplified proof for the pure Neumann problem. Discrete Contin. Dyn. Syst. 31, 1219–1231 (2011)
De Giorgi E., Carriero M., Leaci A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108(3), 195–218 (1989)
Del Piero G.: Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials. Meccanica 24(3), 150–162 (1989)
Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)
Freddi F., Royer-Carfagni G.: Regularized variational theories of fracture: a unified approach. J. Mech. Phys. Solids 58, 1154–1174 (2010)
Freddi F., Royer-Carfagni G.: Variational fracture mechanics to model compressive splitting of masonry-like materials. Ann. Solid Struct. Mech. 2, 57–67 (2011)
Friedrich, M., Solombrino, F.: Quasistatic crack growth in 2d-linearized elasticity, to appear in Annales Institut H Poincaré Anal. Non Linéaire
Giacomini A., Ponsiglione M.: Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. Roy. Soc. Edinburgh Sect. A 138(5), 1019–1041 (2008)
Griffith, A.A.: 1920, The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. A CCXXI(A), 163–198 1920
Iurlano F.: A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Equ. 51(1,2), 315–342 (2014)
Lancioni G., Royer-Carfagni G.: The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris. J. Elast. 95(1,2), 1–30 (2009)
Ortiz M.: A constitutive theory for the inelastic behavior of concrete. Mech. Mater. 1(4), 67–93 (1985)
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Chambolle, A., Conti, S. & Francfort, G.A. Approximation of a Brittle Fracture Energy with a Constraint of Non-interpenetration. Arch Rational Mech Anal 228, 867–889 (2018). https://doi.org/10.1007/s00205-017-1207-z
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DOI: https://doi.org/10.1007/s00205-017-1207-z