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A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity

Stefano Almi

Corresponding author.

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

E-mail Address: [email protected]

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Sandro Belz

Department of Mathematics, Technical University Munich, Boltzmannstr. 3, 85748 Garching (Munich), Germany

E-mail Address: [email protected]

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Stefano Micheletti

MOX, Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

E-mail Address: [email protected]

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and

Simona Perotto

MOX, Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218202521500020Cited by:2

Abstract

In this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the $Γ$ -limit of the functional as the thickness tends to zero.

The numerical discretization is tackled by first approximating the fracture through a phase field, following an Ambrosio–Tortorelli like approach, and then resorting to an alternating minimization procedure, where the irreversibility of the crack propagation is rigorously imposed via an inequality constraint. The minimization is enriched with an anisotropic mesh adaptation driven by an a posteriori error estimator, which allows us to sharply track the whole crack path by optimizing the shape, the size, and the orientation of the mesh elements.

Finally, the overall algorithm is successfully assessed on two Riemannian settings and proves not to bias the crack propagation.

Communicated by G. Dal Maso

Keywords:

AMSC: 49M25, 65K15, 65N50, 74G65, 74K25, 74R10, 74S05

References

1. S. Almi, Irreversibility and alternate minimization in phase field fracture: A viscosity approach, Z. Angew. Math. Phys. 71 (2020) 128. Crossref, ISI, Google Scholar
2. S. Almi and S. Belz, Consistent finite-dimensional approximation of phase-field models of fracture, Ann. Mat. Pura Appl. (4), 198 (2019) 1191–1225. Crossref, ISI, Google Scholar
3. S. Almi, S. Belz and M. Negri, Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics, ESAIM Math. Model. Numer. Anal. 53 (2019) 659–699. Crossref, ISI, Google Scholar
4. S. Almi and M. Negri, Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture, Arch. Ration. Mech. Anal. 236 (2020) 189–252. Crossref, ISI, Google Scholar
5. L. Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal. 139 (1997) 201–238. Crossref, ISI, Google Scholar
6. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs (The Clarendon Press, 2000). Google Scholar
7. L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $Γ$ -convergence, Comm. Pure Appl. Math. 43 (1990) 999–1036. Crossref, ISI, Google Scholar
8. L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7) 6 (1992) 105–123. Google Scholar
9. M. Artina, M. Fornasier, S. Micheletti and S. Perotto. Anisotropic adaptive meshes for brittle fractures: Parameter sensitivity, in Numerical Mathematics and Advanced Applications—ENUMATH 2013, Lecture Notes in Computational Science and Engineering, Vol. 103 (Springer, 2015), pp. 293–301. Crossref, Google Scholar
10. M. Artina, M. Fornasier, S. Micheletti and S. Perotto, Anisotropic mesh adaptation for crack detection in brittle materials, SIAM J. Sci. Comput. 37 (2015) B633–B659. Crossref, ISI, Google Scholar
11. M. Artina, M. Fornasier, S. Micheletti and S. Perotto, The benefits of anisotropic mesh adaptation for brittle fractures under plane-strain conditions, in New Challenges in Grid Generation and Adaptivity for Scientific Computing, SEMA SIMAI Springer Series, Vol. 5 (Springer, 2015), pp. 43–67. Crossref, Google Scholar
12. J.-F. Babadjian, Quasistatic evolution of a brittle thin film, Calc. Var. Partial Dif. 26 (2006) 69–118. Crossref, ISI, Google Scholar
13. J.-F. Babadjian, Lower semicontinuity of quasi-convex bulk energies in $SBV$ and integral representation in dimension reduction, SIAM J. Math. Anal. 39 (2008) 1921–1950. Crossref, ISI, Google Scholar
14. J.-F. Babadjian and D. Henao, Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination, Interfaces Free Bound. 18 (2016) 545–578. Crossref, ISI, Google Scholar
15. S. Belz and K. Bredies, Approximation of the Mumford–Shah functional by functions of bounded variation, submitted (2019), arXiv:1903.02349 [math.AP], DOI: 10.1142/S0219530520500190. Google Scholar
16. B. Bourdin, Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound. 9 (2007) 411–430. Crossref, ISI, Google Scholar
17. B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids 48 (2000) 797–826. Crossref, ISI, Google Scholar
18. A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics, Vol. 1694 (Springer-Verlag, 1998). Crossref, Google Scholar
19. A. Braides and I. Fonseca, Brittle thin films, Appl. Math. Optim. 44 (2001) 299–323. Crossref, ISI, Google Scholar
20. S. Burke, C. Ortner and E. Süli, An adaptive finite element approximation of a variational model of brittle fracture, SIAM J. Numer. Anal. 48 (2010) 980–1012. Crossref, ISI, Google Scholar
21. S. Burke, C. Ortner and E. Süli, Adaptive finite element approximation of the Francfort–Marigo model of brittle fracture, in Approximation and Computation, Springer Optimization and its Applications, Vol. 42 (Springer, 2011), pp. 297–310. Google Scholar
22. A. Chambolle and V. Crismale, A density result in $G S B D^{p}$ with applications to the approximation of brittle fracture energies, Arch. Ration. Mech. Anal. 232 (2019) 1329–1378. Crossref, ISI, Google Scholar
23. P. Ciarlet, Mathematial Elasticity Volume II: Theory of Plates, Studies in Mathematics and its Applications, Vol. 27 (Elsevier, 1997). Google Scholar
24. P. Ciarlet, Mathematial Elasticity Volume III: Theory of Shells, Studies in Mathematics and its Applications, Vol. 29 (Elsevier, 2000). Google Scholar
25. P. G. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973) 17–31. Crossref, Google Scholar
26. P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numr. 9 (1975) 77–84. Google Scholar
27. G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Anal. 38 (1999) 585–604. Crossref, ISI, Google Scholar
28. G. Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and their Applications, Vol. 8 (Birkhäuser, 1993). Crossref, Google Scholar
29. G. Dal Maso, Generalized functions of bounded deformation, J. Eur. Math. Soc. 15 (2013) 1943–1997. Crossref, ISI, Google Scholar
30. G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal. 176 (2005) 165–225. Crossref, ISI, Google Scholar
31. G. Dal Maso and F. Iurlano, Fracture models as $Γ$ -limits of damage models, Commun. Pure Appl. Anal. 12 (2013) 1657–1686. Crossref, ISI, Google Scholar
32. G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002) 101–135. Crossref, ISI, Google Scholar
33. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics (CRC Press, 1992). Google Scholar
34. H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 (Springer-Verlag, 1969). Google Scholar
35. M. Focardi, On the variational approximation of free-discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci. 11 (2001) 663–684. Link, ISI, Google Scholar
36. A. Föppl, Vorlesungen über Technische Mechanik, Vol. 5 (B.G. Teubner, 1907). Google Scholar
37. L. Formaggia, S. Micheletti and S. Perotto, Anisotropic mesh adaption with application to CFD problems, eds. H. MangF. RammerstorferJ. Eberhardsteiner, Proc. WCCM V, Fifth World Congress on Computational Mechanics, 2002, pp. 1481–1493. Google Scholar
38. L. Formaggia, S. Micheletti, and S. Perotto. Anisotropic mesh adaption in computational fluid dynamics: Application to the advection-diffusion-reaction and the Stokes problems, Appl. Numer. Math. 51 (2004) 511–533. Crossref, ISI, Google Scholar
39. L. Formaggia and S. Perotto, New anisotropic a priori error estimates, Numer. Math. 89 (2001) 641–667. Crossref, ISI, Google Scholar
40. L. Formaggia and S. Perotto, Anisotropic error estimates for elliptic problems, Numer. Math. 94 (2003) 67–92. Crossref, ISI, Google Scholar
41. G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342. Crossref, ISI, Google Scholar
42. G. Friesecke, R. D. James, M. G. Mora and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris 336 (2003) 697–702. Crossref, ISI, Google Scholar
43. G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002) 1461–1506. Crossref, ISI, Google Scholar
44. G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal. 180 (2006) 183–236. Crossref, ISI, Google Scholar
45. K. Genevey, Justification of 2D linear shell models by the use of $Γ$ -convergence theory, in Plates and Shells (Québec, QC, 1996), CRM Proceedings and Lecture Notes, Vol. 21 (Amer. Math. Soc., 1999), pp. 185–197. Crossref, Google Scholar
46. P.-L. George and H. Borouchaki, Delaunay Triangulation and Meshing, Application to finite elements (Hermès, 1998). Translated from the 1997 French original by the authors, P. J. Frey and S. A. Canann. Google Scholar
47. A. Giacomini, Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations 22 (2005) 129–172. Crossref, ISI, Google Scholar
48. F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012) 251–265. Crossref, ISI, Google Scholar
49. F. Iurlano, Fracture and plastic models as $Γ$ -limits of damage models under different regimes, Adv. Calc. Var. 6 (2013) 165–189. Crossref, ISI, Google Scholar
50. N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, Vol. 8 (Society for Industrial and Applied Mathematics (SIAM), 1988). Crossref, Google Scholar
51. G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. Reine Angew. Math. 40 (1850) 51–88. Crossref, Google Scholar
52. D. Knees and M. Negri, Convergence of alternate minimization schemes for phase-field fracture and damage, Math. Models Methods Appl. Sci. 27 (2017) 1743–1794. Link, ISI, Google Scholar
53. D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci. 23 (2013) 565–616. Link, ISI, Google Scholar
54. A. E. H. Love, On the Equilibrium of a Thin Elastic Spherical Bowl, Proc. Lond. Math. Soc. 20 (1888/89) 89–102. Crossref, Google Scholar
55. S. Micheletti and S. Perotto, Output functional control for nonlinear equations driven by anisotropic mesh adaption: The Navier-Stokes equations, SIAM J. Sci. Comput. 30 (2008) 2817–2854. Crossref, ISI, Google Scholar
56. S. Micheletti and S. Perotto, The effect of anisotropic mesh adaptation on PDE-constrained optimal control problems, SIAM J. Control Optim. 49 (2011) 1793–1828. Crossref, ISI, Google Scholar
57. M. Negri, A unilateral $L^{2}$ -gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement, Adv. Calc. Var. 12 (2019) 1–29. Crossref, ISI, Google Scholar
58. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990) 483–493. Crossref, ISI, Google Scholar
59. G. Strang and G. J. Fix, An Analysis of the Finite Element Method, 2nd edn. (Cambridge Press, 2008). Google Scholar
60. R. Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal. 33 (1999) 695–713. Crossref, ISI, Google Scholar
61. T. von Kármán, Festigkeitsprobleme im maschinenbau, in Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwedungen, Vol. 4 (B.G. Teubner, 1910), pp. 314–385. Google Scholar
62. A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. 106 (2006) 25–57. Crossref, ISI, Google Scholar
63. O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Meth. Eng. 24 (1987) 337–357. Crossref, ISI, Google Scholar
64. O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates ii: Error estimates and adaptivity, Int. J. Numer. Methods Eng. 33 (1992) 1365–1382. Crossref, ISI, Google Scholar
65. M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S (2013). ISI, Google Scholar

Published: 22 December 2020

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Received 9 April 2020

Accepted 23 October 2020

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A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity

Abstract

References

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