Abstract
In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → t g(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time t i > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional \(\epsilon(\nu, \Gamma)\,:=\int_{\Omega} f(x,\nabla v)\,{\rm d}x+k{\mathcal{H}}^{1} (\Gamma),\) where \(\Omega \subseteq {\mathbb{R}}^{2}\) , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with \({\mathcal{H}}^{1} (\Gamma)\) small enough are such that \(\epsilon(u,\emptyset) < \epsilon(u_{\Gamma},\Gamma)\) .
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References
Ambrosio L. (1989) A compactness theorem for a new class of functions of bounded variations. Boll. Un. Mat. Ital. 3-B: 857–881
Ambrosio L. (1990) Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111: 291–322
Ambrosio L. (1995) A new proof of the SBV compactness theorem. Calc. Var. Partial Differ. Equ. 3: 127–137
Ambrosio L., Fusco N., Pallara D. (2000) Functions of Bounded Variations and Free Discontinuity Problems. Clarendon, Oxford
Ambrosio L. Caselles V., Masnou S., Morel J.-M. (2001) Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. (JEMS) 3: 39–92
Astala K. (1994) Area distortion of quasiconformal mappings. Acta Math. 103(1): 37–60
Bonnetier E., Chambolle A. (2002) Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62:1093–1121
Boyarski B. (1955) Homeomorphic solutions of Beltrami systems (Russian). Dokl. Akad. Nauk SSSR (N.S) 102: 661–664
Boyarski B. (1957) Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients (Russian). Mat. Sb. N.S. 43(85): 451–503
Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, 1973
Caffarelli L., Peral I. (1998) On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51: 1–21
Chambolle A. (2003) A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167: 211–233
Dal Maso G., Morel J.-M., Solimini S. (1992) A variational method in image segmentation: existence and approximation results. Acta Math. 168(1–2): 89–151
Dal Maso G., Toader R. (2002) A model for the quasistatic growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162: 101–135
Dal Maso G., Toader R. (2002) A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Methods Appl. Sci. 12: 1773–1799
David G. (2005) Singular Sets of Minimizers for the Mumford–Shah Functional. Progress in Mathematics. Birkhaüser, Basel
De Giorgi E., Carriero M., Leaci A. (1989) Existence theorem for a minimum problem with free discontinuity set. Arch Ration. Mech. Anal. 108: 195–218
Ebobisse F., Ponsiglione M. (2004) A duality approach for variational problems in domains with cracks. J. Convex Anal. 11: 17–40
Francfort G.A., Larsen C.J. (2003) Existence and convergence for quasistatic evolution in brittle fracture. Commun. Pure Appl. Math. 56: 1465–1500
Francfort G.A., Marigo J.-J. (1998) Revisiting brittle fractures as an energy minimization problem. J. Mech. Phys. Solids 46: 1319–1342
Grisvard P. (1985) Elliptic Problems in Nonsmooth Domains. Pitman, Boston
Leonetti F., Nesi V. (1997) Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton. J. Math. Pure Appl. 76: 109–124
Li Y., Nirenberg L. (2003) Estimates for elliptic systems from composite material. Dedicated to the memory of Jürgen K. Moser. Commun. Pure Appl. Math. 56: 892–925
Li Y., Vogelius M. (2000) Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal. 153: 91–151
Maddalena F., Solimini S. (2001) Lower semicontinuity properties of functionals with free discontinuities, Arch. Ration. Mech. Anal 159(4): 273–294
Meyers N.G. (1963) An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17: 189–206
Mielke, A.: Analysis of energetic models for rate-independent materials. In: Proceedings of the International Congress of Mathematicians,Vol. III (Beijing, 2002), pp. 817–828, Higher Ed. Press, Beijing, 2002
Morel J.-M., Solimini S. (1995) Variational Methods in Image Segmentation. Birkhaüser, Boston
Mumford D., Shah J. (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42: 577–685
Rockafellar R.T. (1970) Convex Analysis. Princeton Mathematical Series, No. 28 Princeton University Press, Princeton
Solimini S. (1997) Simplified excision techniques for free discontinuity problems in several variables. J. Funct. Anal. 151: 1–34
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Chambolle, A., Giacomini, A. & Ponsiglione, M. Crack Initiation in Brittle Materials. Arch Rational Mech Anal 188, 309–349 (2008). https://doi.org/10.1007/s00205-007-0080-6
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DOI: https://doi.org/10.1007/s00205-007-0080-6