\(\Gamma\)-convergence approximation of fracture and cavitation in nonlinear elasticity. (English) Zbl 1457.35081
Summary: Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by D. Henao and C. Mora-Corral [Arch. Ration. Mech. Anal. 197, No. 2, 619–655 (2010; Zbl 1248.74006)]. The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of \(\Gamma\)-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica-Mortola approximation of the perimeter and the Ambrosio-Tortorelli approximation of the Mumford-Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving.
MSC:
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74B20 | Nonlinear elasticity |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74R10 | Brittle fracture |
Citations:
Zbl 1248.74006References:
| [1] | Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York, 2000 · Zbl 0957.49001 |
| [2] | Mumford D., Shah J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577-685 (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503 |
| [3] | De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108(3), 195-218 (1989) · Zbl 0682.49002 |
| [4] | Griffith A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A 221, 163-198 (1921) · Zbl 1454.74137 · doi:10.1098/rsta.1921.0006 |
| [5] | Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids46(8), 1319-1342 (1998) · Zbl 0966.74060 |
| [6] | Ball J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. Ser. A 306, 557-611 (1982) · Zbl 0513.73020 · doi:10.1098/rsta.1982.0095 |
| [7] | Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131(1), 1-66 (1995) · Zbl 0836.73025 |
| [8] | Henao, D., Mora-Corral, C.: Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Rational Mech. Anal197, 619-655 (2010) · Zbl 1248.74006 |
| [9] | Henao, D., Mora-Corral, C.: Fracture surfaces and the regularity of inverses for BV deformations. Arch. Rational Mech. Anal. 201(2), 575-629 (2011) · Zbl 1262.74007 |
| [10] | Henao D., Mora-Corral C.: Lusin’s condition and the distributional determinant for deformations with finite energy. Adv. Calc. Var. 5(4), 355-409 (2012) · Zbl 1252.49016 · doi:10.1515/acv.2011.016 |
| [11] | Modica, L., Mortola, S.: Un esempio di Γ−-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285-299 (1977) · Zbl 0356.49008 |
| [12] | Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98(2), 123-142 (1987) · Zbl 0616.76004 · doi:10.1007/BF00251230 |
| [13] | Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43(8), 999-1036 (1990) · Zbl 0722.49020 |
| [14] | Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6(1), 105-123 (1992) · Zbl 0776.49029 |
| [15] | Braides, A.: Approximation of free-discontinuity problems. In: Lecture Notes in Mathematics, Vol. 1694. Springer, Berlin, 1998 · Zbl 0909.49001 |
| [16] | Bourdin, B., Chambolle, A.: Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85(4), 609-646 (2000) · Zbl 0961.65062 |
| [17] | Giacomini, A.: Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22(2), 129-172 (2005) · Zbl 1068.35189 |
| [18] | Bourdin B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interf. Free Bound. 9(3), 411-430 (2007) · Zbl 1130.74040 · doi:10.4171/IFB/171 |
| [19] | Burke, S., Ortner, C., Süli, E.: An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numer. Anal. 48(3), 980-1012 (2010) · Zbl 1305.74080 |
| [20] | Bourdin, B., Francfort, G.A., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids48(4), 797-826 (2000) · Zbl 0995.74057 |
| [21] | Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1-3), 5-148 (2008) · Zbl 1176.74018 |
| [22] | Burke, S.: A numerical analysis of the minimisation of the Ambrosio-Tortorelli functional, with applications in brittle fracture. Ph.D. thesis, University of Oxford, 2010 |
| [23] | Chambolle, A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl. (9) 83(7), 929-954 (2004) · Zbl 1084.49038 |
| [24] | Focardi, M.: On the variational approximation of free-discontinuity problems in the vectorial case. Math. Models Methods Appl. Sci. 11(4), 663-684 (2001) · Zbl 1010.49010 |
| [25] | Williams, M., Schapery, R.: Spherical flaw instability in hydrostatic tension. Int. J. Fract. Mech. 1(1), 64-71 (1965) · Zbl 0368.73040 |
| [26] | Rice, J.R., Tracey, D.M.: On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids17, 201-217 (1969) · Zbl 0837.49011 |
| [27] | Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2-15 (1977) · Zbl 1252.49016 |
| [28] | Goods, S.H., Brown, L.M.: The nucleation of cavities by plastic deformation. Acta Metall. 27, 1-15 (1979) · Zbl 0616.76004 |
| [29] | Tvergaard, V.: Material failure by void growth and coalescence. In: Advances in Applied Mechanics, pp. 83-151. Academic Press, San Diego, 1990 · Zbl 0728.73058 |
| [30] | Gent, A.N., Wang, C.: Fracture mechanics and cavitation in rubber-like solids. J. Mater. Sci. 26(12), 3392-3395 (1991) · Zbl 0916.49002 |
| [31] | Petrinic, N., Curiel Sosa, J.L., Siviour, C.R., Elliott, B.C.F.: Improved predictive modelling of strain localisation and ductile fracture in a Ti-6Al-4V alloy subjected to impact loading. J. Phys. IV134, 147-155 (2006) |
| [32] | Henao, D., Mora-Corral, C., Xu, X.: A numerical study of void coalescence and fracture in nonlinear elasticity (2014, in preparation) · Zbl 1425.74080 |
| [33] | 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) · Zbl 1149.49017 |
| [34] | Mora-Corral, C.: Approximation by piecewise affine homeomorphisms of Sobolev homeomorphisms that are smooth outside a point. Houston J. Math. 35(2), 515-539 (2009) · Zbl 1182.57019 |
| [35] | Bellido, J.C., Mora-Corral, C.: Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms. Houston J. Math. 37(2), 449-500 (2011) · Zbl 1228.57009 |
| [36] | Iwaniec T., Kovalev L.V., Onninen J.: Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Rational Mech. Anal. 201(3), 1047-1067 (2011) · Zbl 1260.46023 · doi:10.1007/s00205-011-0404-4 |
| [37] | Daneri, S., Pratelli, A.: Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms. Ann. Inst. H. Poincaré Anal. Non Linéaire31(3), 567-589 (2014) · Zbl 1348.37071 |
| [38] | Mora-Corral C., Pratelli A.: Approximation of piecewise affine homeomorphisms by diffeomorphisms. J. Geom. Anal. 24(3), 1398-1424 (2014) · Zbl 1300.41014 · doi:10.1007/s12220-012-9378-1 |
| [39] | Cortesani G.: Strong approximation of GSBV functions by piecewise smooth functions. Ann. Univ. Ferrara Sez. VII (N.S.) 43, 27-49 (1997) · Zbl 0916.49002 |
| [40] | Cortesani, G., Toader, R.: A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38(5), 585-604 (1999) · Zbl 0939.49024 |
| [41] | Federer, H.: Geometric measure theory. Springer, New York, 1969 · Zbl 0176.00801 |
| [42] | Ziemer, W.P.: Weakly differentiable functions. Springer, New York, 1989 · Zbl 0692.46022 |
| [43] | Conti, S., De Lellis, C.: Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2(3), 521-549 (2003) · Zbl 1114.74004 |
| [44] | Sivaloganathan, J., Spector, S.J.: On the existence of minimizers with prescribed singular points in nonlinear elasticity. J. Elasticity59(1-3), 83-113 (2000) · Zbl 0987.74016 |
| [45] | Ambrosio, L.: On the lower semicontinuity of quasiconvex integrals in SBV(Ω, Rk). Nonlinear Anal. 23(3), 405-425 (1994) · Zbl 0817.49017 |
| [46] | Ambrosio, L.: A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B (7) 3(4), 857-881 (1989) · Zbl 0767.49001 |
| [47] | Ambrosio, L.: Variational problems in SBV and image segmentation. Acta Appl. Math. 17(1), 1-40 (1989) · Zbl 0697.49004 |
| [48] | Giaquinta, M., Modica, G., Soucek, J.: Cartesian Currents in the Calculus of Variations. I. Springer, Berlin, 1998 · Zbl 0914.49001 |
| [49] | Ambrosio L.: A new proof of the SBV compactness theorem. Calc. Var. Partial Differ. Equ. 3(1), 127-137 (1995) · Zbl 0837.49011 · doi:10.1007/BF01190895 |
| [50] | Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337-403 (1977) · Zbl 0368.73040 · doi:10.1007/BF00279992 |
| [51] | Müller S., Tang Q., Yan B.S.: On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré Anal. Non Linéaire 11(2), 217-243 (1994) · Zbl 0863.49002 |
| [52] | Dacorogna, B.: Direct methods in the calculus of variations. In: Applied Mathematical Sciences, Vol. 78, 2nd edn. Springer, New York, 2008 · Zbl 1140.49001 |
| [53] | Ball J.M., Currie J.C., Olver P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41(2), 135-174 (1981) · Zbl 0459.35020 · doi:10.1016/0022-1236(81)90085-9 |
| [54] | Braides, A.: A handbook of Γ-convergence. Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 3, pp. 101-213 (Eds. M. Chipot and P. Quittner) North-Holland, Amsterdam, 2006 · Zbl 1195.35002 |
| [55] | Alvarado, R., Brigham, D., Maz’ya, V., Mitrea, M., Ziadé, E.: On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik boundary point principle. J. Math. Sci. (N. Y.)176(3), 281-360 (2011) · Zbl 1290.35046 |
| [56] | de Pascale, L.: The Morse-Sard theorem in Sobolev spaces. Indiana Univ. Math. J.50(3), 1371-1386 (2001) · Zbl 1027.58008 |
| [57] | Alberti, G.: Variational models for phase transitions, an approach via Γ-convergence. In: Calculus of Variations and Partial Differential Equations (Pisa, 1996), pp. 95-114. Springer, Berlin, 2000 · Zbl 0957.35017 |
| [58] | Henao D., Serfaty S.: Energy estimates and cavity interaction for a critical-exponent cavitation model. Comm. Pure Appl. Math. 66, 1028-1101 (2013) · Zbl 1388.74082 · doi:10.1002/cpa.21396 |
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