Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-28T22:53:12.927Z Has data issue: false hasContentIssue false
  • English
  • Français

Brittle fracture in linearly elastic plates

Part of: Existence theories Equilibrium (steady-state) problems Thin bodies, structures Fracture and damage

Published online by Cambridge University Press:  19 November 2021

Stefano Almi Open the ORCID record for Stefano Almi [Opens in a new window]  and
Emanuele Tasso Open the ORCID record for Emanuele Tasso [Opens in a new window]
Stefano Almi
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria (stefano.almi@univie.ac.at)
Emanuele Tasso
Affiliation:
Technische Universität Dresden, Faculty of Mathematics, 01062 Dresden, Germany (emanuele.tasso@tu-dresden.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work we derive by $\Gamma$-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

Keywords

Dimension reductionΓ-convergencebrittle fracturefree discontinuity problems

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 74R10: Brittle fracture 74G65: Energy minimization 74K15: Membranes
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 1 , February 2023 , pp. 68 - 103
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abels, H., Mora, M. G. and Müller, S.. The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity. Calc. Var. Partial Differ. Equ. 41 (2011), 241–259.CrossRefGoogle Scholar
Almi, S., Belz, S., Micheletti, S. and Perotto, S.. A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity. Math. Models Methods Appl. Sci. 31 (2021), 37–81.CrossRefGoogle Scholar
Almi, S. and Tasso, E., A new proof of compactness in $G(S)BD$, Preprint, (2021).CrossRefGoogle Scholar
Ambrosio, L., Coscia, A. and Dal Maso, G.. Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139 (1997), 201–238.CrossRefGoogle Scholar
Ambrosio, L., Fusco, N. and Pallara, D., Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.Google Scholar
Anzellotti, G., Baldo, S. and Percivale, D.. Dimension reduction in variational problems, asymptotic development in $\Gamma$-convergence and thin structures in elasticity. Asymptotic Anal. 9 (1994), 61–100.CrossRefGoogle Scholar
Babadjian, J.-F.. Quasistatic evolution of a brittle thin film. Calc. Var. Partial Differ. Equ. 26 (2006), 69–118.CrossRefGoogle Scholar
Babadjian, J.-F. and Henao, D.. Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination. Interfaces Free Bound. 18 (2016), 545–578.CrossRefGoogle Scholar
Bourdin, B., Francfort, G. A. and Marigo, J.-J.. The variational approach to fracture. J. Elasticity 91 (2008), 5–148.CrossRefGoogle Scholar
Braides, A. and Fonseca, I.. Brittle thin films. Appl. Math. Optim. 44 (2001), 299–323.CrossRefGoogle Scholar
Cagnetti, F., Chambolle, A. and Scardia, L., Korn and poincaré-korn inequalities for functions with small jump set, Preprint cvgmt.sns.it/paper/4636/, (2020).Google Scholar
Chambolle, A.. An approximation result for special functions with bounded deformation. J. Math. Pures Appl. (9) 83 (2004), 929–954.CrossRefGoogle Scholar
Chambolle, A.. Addendum to: “An approximation result for special functions with bounded deformation”. J. Math. Pures Appl. (9) 84 (2005), 137–145. [J. Math. Pures Appl. (9) 83 (2004), no. 7, 929–954; mr2074682].CrossRefGoogle Scholar
Chambolle, A. and Crismale, V.. A density result in $GSBD^{p}$ with applications to the approximation of brittle fracture energies. Arch. Ration. Mech. Anal. 232 (2019), 1329–1378.CrossRefGoogle Scholar
Chambolle, A. and Crismale, V.. Compactness and lower semicontinuity in $GSBD$. J. Eur. Math. Soc. (JEMS) 23 (2021), 701–719.CrossRefGoogle Scholar
Ciarlet, P., Mathematial Elasticity; Volume II: Theory of Plates, vol. 27 of Studies in Mathematics and its Applications, Elsevier, Amsterdam, 1997.Google Scholar
Dal Maso, G.. Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS) 15 (2013), 1943–1997.CrossRefGoogle Scholar
Davoli, E.. Linearized plastic plate models as $\Gamma$-limits of 3D finite elastoplasticity. ESAIM Control Optim. Calc. Var. 20 (2014), 725–747.CrossRefGoogle Scholar
Davoli, E.. Quasistatic evolution models for thin plates arising as low energy $\Gamma$-limits of finite plasticity. Math. Models Methods Appl. Sci. 24 (2014), 2085–2153.CrossRefGoogle Scholar
Davoli, E. and Mora, M. G.. A quasistatic evolution model for perfectly plastic plates derived by $\Gamma$-convergence. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), 615–660.CrossRefGoogle Scholar
Federer, H., Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.Google Scholar
Freddi, L., Paroni, R. and Zanini, C.. Dimension reduction of a crack evolution problem in a linearly elastic plate. Asymptot. Anal. 70 (2010), 101–123.Google Scholar
Friedrich, M. and Kružík, M.. Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity. Arch. Ration. Mech. Anal. 238 (2020), 489–540.CrossRefGoogle Scholar
Friesecke, G., James, R. D., Mora, M. G. and Müller, S., Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris, 336 (2003), pp. 697–702.Google Scholar
Friesecke, G., James, R. D. and Müller, S.. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (2002), 1461–1506.CrossRefGoogle Scholar
Friesecke, G., James, R. D. and Müller, S.. A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180 (2006), 183–236.CrossRefGoogle Scholar
Griffith, A. A.. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. London. Seri. A, Cont. Papers Math. Phys. Char. 221 (1921), 163–198.Google Scholar
Hornung, P., Neukamm, S. and Velčić, I.. Derivation of a homogenized nonlinear plate theory from 3d elasticity. Calc. Var. Partial Differ. Equ. 51 (2014), 677–699.CrossRefGoogle Scholar
Hornung, P. and Velčić, I.. Derivation of a homogenized von-Kármán shell theory from 3D elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 1039–1070.CrossRefGoogle Scholar
Iurlano, F.. A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Equ. 51 (2014), 315–342.CrossRefGoogle Scholar
Kholmatov, S. and Piovano, P.. A unified model for stress-driven rearrangement instabilities. Arch. Ration. Mech. Anal. 238 (2020), 415–488.CrossRefGoogle ScholarPubMed
León Baldelli, A. A., Babadjian, J.-F., Bourdin, B., Henao, D. and Maurini, C.. A variational model for fracture and debonding of thin films under in-plane loadings. J. Mech. Phys. Solids 70 (2014), 320–348.CrossRefGoogle Scholar
Lewicka, M., Mora, M. G. and Pakzad, M. R.. Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), 253–295.Google Scholar
Lewicka, M., Mora, M. G. and Pakzad, M. R.. The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. Arch. Ration. Mech. Anal. 200 (2011), 1023–1050.CrossRefGoogle Scholar
Maggiani, G. B. and Mora, M. G.. Quasistatic evolution of perfectly plastic shallow shells: a rigorous variational derivation. Ann. Mat. Pura Appl. (4) 197 (2018), 775–815.CrossRefGoogle Scholar
Mielke, A. and Roubíček, T., Rate-independent systems, Vol. 193 of Applied Mathematical Sciences, Springer, New York, 2015. Theory and application.CrossRefGoogle Scholar
Mielke, A., Roubíček, T. and Stefanelli, U.. $\Gamma$-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008), 387–416.CrossRefGoogle Scholar
Neukamm, S. and Velčić, I.. Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity. Math. Models Methods Appl. Sci. 23 (2013), 2701–2748.CrossRefGoogle Scholar
Velčić, I.. On the derivation of homogenized bending plate model. Calc. Var. Partial Differ. Equ. 53 (2015), 561–586.CrossRefGoogle Scholar