In the setting of planar linearized elasticity, we study a fracture model depending on the crack opening. Assuming that the crack path is known a priori and sufficiently smooth, we prove that the energy release rate is well defined. Then, we consider the problem of quasi-static evolution for our model. Thanks to a vanishing viscosity approach, we show the existence of such an evolution satisfying a weak Griffith’s criterion.
Mots-clés : Variational models, free-discontinuity problems, energy release rate, energy derivative, cohesive fracture, crack propagation, quasi-static evolution, local minimizers, Griffith’s criterion, vanishing viscosity
@article{COCV_2017__23_3_791_0,
author = {Almi, Stefano},
title = {Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {791--826},
publisher = {EDP-Sciences},
volume = {23},
number = {3},
year = {2017},
doi = {10.1051/cocv/2016014},
mrnumber = {3660449},
zbl = {1373.49011},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2016014/}
}
TY - JOUR AU - Almi, Stefano TI - Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 791 EP - 826 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016014/ DO - 10.1051/cocv/2016014 LA - en ID - COCV_2017__23_3_791_0 ER -
%0 Journal Article %A Almi, Stefano %T Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 791-826 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016014/ %R 10.1051/cocv/2016014 %G en %F COCV_2017__23_3_791_0
Almi, Stefano. Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 791-826. doi : 10.1051/cocv/2016014. http://archive.numdam.org/articles/10.1051/cocv/2016014/
L Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2nd edition (2008). | MR | Zbl
, and , A bridging mechanism in the homogenisation of brittle composites with soft inclusions. SIAM J. Math. Anal. 48 (2016) 1178–1209. | DOI | MR | Zbl
G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture. In Vol. 7, Advances in Applied Mechanics. Academic Press, New York (1962) 55–129. | MR
B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture. Springer, New York (2008). Reprinted with a foreword by Roger Fosdick, from J. Elasticity 91 (2008) [MR2390547]. | MR | Zbl
, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path. Math. Models Methods Appl. Sci. 18 (2008) 1027–1071. | DOI | MR | Zbl
and , Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach. ESAIM: COCV 17 (2011) 1–27. | Numdam | MR | Zbl
and , Fracture models as gamma-limits of damage models. Commun. Pure Appl. Anal. 12 (2013) 1657–1686. | DOI | MR | Zbl
and , A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Methods Appl. Sci. 12 (2002) 1773–1799. | DOI | MR | Zbl
and , Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. R. Soc. Edinb., Sect. A 137 (2007) 253–279. | DOI | MR | Zbl
and , Homogenization of fiber reinforced brittle materials: the intermediate case. Adv. Calc. Var. 3 (2010) 345–370. | MR | Zbl
and , On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13 (2006) 151–167. | MR | Zbl
and , Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity. SIAM J. Math. Anal. 46 (2014) 2936–2955. | DOI | MR | Zbl
and , Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. | DOI | MR | Zbl
M. Giaquinta and S. Hildebrandt, Calculus of variations. I. The Lagrangian formalism. Vol. 310 of Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1996). | MR | Zbl
, The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond., Series A 221 (1921) 163–198. | DOI | Zbl
M.E. Gurtin, E. Fried and L. Anand, The mechanics and thermodynamics of continua. Cambridge University Press, Cambridge (2010). | MR
, Fracture and plastic models as gamma-limits of damage models under different regimes. Adv. Calc. Var. 6 (2013) 165–189. | DOI | MR | Zbl
and , Energy release rate for cracks in finite-strain elasticity. Math. Methods Appl. Sci. 31 (2008) 501–528. | DOI | MR | Zbl
, and , On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18 (2008) 1529–1569. | DOI | MR | Zbl
, and , Crack growth in polyconvex materials. Phys. D 239 (2010) 1470–1484. | DOI | MR | Zbl
and , Energy release rate and stress intensity factor in antiplane elasticity. J. Math. Pures Appl. 95 (2011) 565–584. | DOI | MR | Zbl
and , A model for crack propagation based on viscous approximation. Math. Models Methods Appl. Sci. 21 (2011) 2019–2047. | DOI | MR | Zbl
A. Mielke, Evolution of rate-independent systems. In Vol. II of Evolutionary equations of Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam (2005) 461–559. | MR | Zbl
, and , Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. 25 (2009) 585–615. | DOI | MR | Zbl
, and , BV solutions and viscosity approximations of rate-independent systems. ESAIM: COCV 18 (2012) 36–80. | Numdam | MR | Zbl
and , Quasi-static crack propagation by Griffith’s criterion. Math. Models Methods Appl. Sci. 18 (2008) 1895–1925. | DOI | MR | Zbl
, A variational characterization of rate-independent evolution. Math. Nachr. 282 (2009) 1492–1512. | DOI | MR | Zbl
and , An artificial viscosity approach to quasistatic crack growth. Boll. Unione Mat. Ital. 2 (2009) 1–35. | MR | Zbl
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