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$$\Gamma$$-convergence for high order phase field fracture: continuum and isogeometric formulations. (English) Zbl 1439.74027
Summary: We consider high order phase field functionals introduced in [M. J. Borden et al., Comput. Methods Appl. Mech. Eng. 273, 100–118 (2014; Zbl 1296.74098)] and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy. We take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in Sobolev spaces and in spaces of tensor product $$B$$-splines. In the latter convergence holds when the mesh size vanishes faster than the internal length of the phase-field model. On the theoretical level, this condition is natural since the size of the phase field layer, around the crack, itself scales like the internal length; on the numerical level, it should be satisfied by local $$h$$-refinement.
Technically, convergence holds in the sense of $$\Gamma$$-convergence, with respect to the strong topology of $$L^1$$, while the sharp crack energy is defined in $$GSBD^2$$. The constraint on the phase field to take values in $$[0,1]$$ is taken into account both in the Sobolev setting and in the iso-geometric setting; in the latter, it requires a special treatment since the projection operator on the space of tensor product $$B$$-splines is not Lagrangian (i.e., interpolatory).
##### MSC:
 74A45 Theories of fracture and damage 74R10 Brittle fracture 74S22 Isogeometric methods applied to problems in solid mechanics 49J45 Methods involving semicontinuity and convergence; relaxation
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##### References:
 [1] Bourdin, B.; Francfort, G. A.; Marigo, J.-J., Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, 48, 4, 797-826 (2000) · Zbl 0995.74057 [2] Ambrosio, L.; Tortorelli, V. M., Approximation of functionals depending on jumps by elliptic functionals via $$\Gamma$$-convergence, Comm. Pure Appl. Math., 43, 8, 999-1036 (1990) · Zbl 0722.49020 [3] March, R., Visual reconstruction with discontinuities using variational methods, Vis. Comput., 10, 30-38 (1992) [4] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42, 5, 577-685 (1989) · Zbl 0691.49036 [5] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83, 10, 1273-1311 (2010) · Zbl 1202.74014 [6] Kuhn, C.; Müller, R., A continuum phase field model for fracture, Eng. Fract. Mech., 77, 18, 3625-3634 (2010) [7] Borden, M. J.; Hughes, T. J.R.; Landis, C. M.; Verhoosel, C. V., A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework, Comput. Methods Appl. Mech. Engrg., 273, 100-118 (2014) · Zbl 1296.74098 [8] Sicsic, P.; Marigo, J.-J.; Maurini, C., Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling, J. Mech. Phys. Solids, 63, 256-284 (2014) · Zbl 1303.74008 [9] Li, B.; Peco, C.; Millán, D.; Arias, I.; Arroyo, M., Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy, Internat. J. Numer. Methods Engrg., 102, 3-4, 711-727 (2015) · Zbl 1352.74290 [10] Mikelić, A.; Wheeler, M. F.; Wick, T., A quasi-static phase-field approach to pressurized fractures, Nonlinearity, 28, 5, 1371-1399 (2015) · Zbl 1316.35287 [11] Kiendl, J.; Ambati, M.; De Lorenzis, L.; Gomez, H.; Reali, A., Phase-field description of brittle fracture in plates and shells, Comput. Methods Appl. Mech. Engrg., 312, 374-394 (2016) [12] Bourdin, B.; Francfort, G. A.; Marigo, J.-J., The variational approach to fracture, J. Elasticity, 91, 5-148 (2008) · Zbl 1176.74018 [13] Ambati, M.; Gerasimov, T.; De Lorenzis, L., A review on phase-field models of brittle fracture and a new fast hybrid formulation, Comput. Mech., 55, 2, 383-405 (2015) · Zbl 1398.74270 [14] Burger, M.; Esposito, T.; Zeppieri, C. I., Second-order edge-penalization in the Ambrosio-Tortorelli functional, Multiscale Model. Simul., 13, 4, 1354-1389 (2015) · Zbl 1332.49016 [15] Larsen, C. J.; Ortner, C.; Süli, E., Existence of solutions to a regularized model of dynamic fracture, Math. Models Methods Appl. Sci., 20, 7, 1021-1048 (2010) · Zbl 1425.74418 [16] Negri, M., Quasi-static evolutions in brittle fracture generated by gradient flows: sharp crack and phase-field approaches, (Innovative Numerical Approaches for Multi-Physics and Multi-Scale Problems. Innovative Numerical Approaches for Multi-Physics and Multi-Scale Problems, Lecture Notes in Applied and Computational Mechanics, vol. 81 (2016), Springer), 197-216 [17] Knees, D.; Negri, M., Convergence of alternate minimization schemes for phase field fracture and damage, Math. Models Methods Appl. Sci., 27, 9, 1743-1794 (2017) · Zbl 1376.49038 [18] Almi, S.; Negri, M., Analysis of staggered evolutions for nonlinear energies in phase field fracture, Arch. Ration. Mech. Anal. (2019) [19] Dal Maso, G., Generalised functions of bounded deformation, J. Eur. Math. Soc., 15, 5, 1943-1997 (2013) · Zbl 1271.49029 [20] Dal Maso, G., An Introduction to $$\Gamma$$-Convergence (1993), Birkhäuser: Birkhäuser Boston · Zbl 0816.49001 [21] Braides, A., Approximation of Free-Discontinuity Problems (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0909.49001 [22] Iurlano, F., A density result for $$G S B D$$ and its application to the approximation of brittle fracture energies, Calc. Var. Partial Differential Equation, 51, 315-342 (2014) · Zbl 1297.49080 [23] Chambolle, A.; Crismale, V., A density result in $$G S B D^p$$ with applications to the approximation of brittle fracture energies, Arch. Ration. Mech. Anal., 232, 3, 1329-1378 (2019) · Zbl 1411.74050 [24] Bellettini, G.; Coscia, A., Discrete approximation of a free discontinuity problem, Numer. Funct. Anal. Optim., 15, 3-4, 201-224 (1994) · Zbl 0806.49002 [25] Chambolle, A., An approximation result for special functions with bounded deformation, J. Math. Pures Appl. (9), 83, 7, 929-954 (2004) · Zbl 1084.49038 [26] Chambolle, A.; Conti, S.; Francfort, G. A., Approximation of a brittle fracture energy with a constraint of non-interpenetration, Arch. Ration. Mech. Anal., 228, 3, 867-889 (2018) · Zbl 1391.35366 [27] Fonseca, I.; Mantegazza, C., Second order singular perturbation models for phase transitions, SIAM J. Math. Anal., 31, 5, 1121-1143 (2000) · Zbl 0958.49007 [28] Bach, A., Anisotropic free-discontinuity functionals as the $$\Gamma$$-limit of second-order elliptic functionals, ESAIM Control Optim. Calc. Var., 24, 3, 1107-1140 (2018) · Zbl 1412.49032 [29] Fonseca, I.; Müller, S., Quasi-convex integrands and lower semicontinuity in $$L^1$$, SIAM J. Math. Anal., 23, 5, 1081-1098 (1992) · Zbl 0764.49012 [30] Bazilevs, Y.; Beirão da Veiga, L.; Cottrell, J. A.; Hughes, T. J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for $$h$$-refined meshes, Math. Models Methods Appl. Sci., 16, 7, 1031-1090 (2006) · Zbl 1103.65113 [31] Beirão da Veiga, L.; Buffa, A.; Sangalli, G.; Vázquez, R., An introduction to the numerical analysis of isogeometric methods, (Numerical Simulation in Physics and Engineering. Numerical Simulation in Physics and Engineering, SEMA SIMAI Springer Ser., vol. 9 (2016), Springer), 3-69 · Zbl 1351.65086 [32] Schumaker, L. L., Spline Functions: Basic Theory (2007), Cambridge Mathematical Library. Cambridge University Press: Cambridge Mathematical Library. Cambridge University Press Cambridge · Zbl 1123.41008 [33] Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S., Anisotropic mesh adaptation for crack detection in brittle materials, SIAM J. Sci. Comput., 37, 4, B633-B659 (2015) · Zbl 1325.74134 [34] 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 [35] 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 [36] Hennig, P.; Müller, S.; Kästner, M., Bézier extraction and adaptive refinement of truncated hierarchical NURBS, Comput. Methods Appl. Mech. Engrg., 305, 316-339 (2016) · Zbl 1425.65031 [37] Hesch, C.; Schuß, S.; Dittmann, M.; Franke, M.; Weinberg, K., Isogeometric analysis and hierarchical refinement for higher-order phase-field models, Comput. Methods Appl. Mech. Engrg., 303, 185-207 (2016) · Zbl 1425.74464 [38] Paul, K.; Zimmermann, C.; Mandadapu, K. K.; Hughes, T. J.R.; Landis, C. M.; Sauer, R. A., An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS, Comput. Mech. (2019) [39] Ciarlet, P. G., The finite element method for elliptic problems, (Studies in Mathematics and Its Applications, vol. 4 (1978), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam) · Zbl 0383.65058 [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
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