Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-20T06:49:48.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics∗

Published online by Cambridge University Press:  04 August 2014

Matteo Negri
Matteo Negri*
Affiliation:
Department of Mathematics, University of Pavia, Via A. Ferrata 1, 27100 Pavia, Italy. [email protected]
Get access

Abstract

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

Keywords

Quasi-static evolutionsphase-field
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 983 - 1008
Copyright
© EDP Sciences, SMAI, 2014

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

Ambrosio, L., Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191–246. Google Scholar
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lect. Math. ETH Zürich. Birkhäuser Verlag, Basel (2005).
Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105–123. Google Scholar
J.-F. Babadjian and V. Millot, Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements (2012).
Bourdin, B., Francfort, G.A. and Marigo, J.-J., Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797–826. Google Scholar
Bourdin, B., Francfort, G.A. and Marigo, J.-J., The variational approach to fracture. J. Elasticity 91 (2008) 5–148. Google Scholar
A. Braides, Γ -convergence for beginners. Oxford University Press, Oxford (2002).
A. Braides, Local Minimization, Variational Evolution and Γ-convergence, vol. 2094 of Lect. Notes Math. Springer, Berlin (2013).
Chambolle, A., A density result in two-dimensional linearized elasticity and applications. Arch. Ration. Mech. Anal. 167 (2003) 211–233. Google Scholar
Chambolle, A., Francfort, G.A. and Marigo, J.-J., Revisiting Energy Release Rates in Brittle Fracture. J. Nonlinear Sci. 20 (2010) 395–424. Google Scholar
G. Dal Maso, An introduction to Γ -convergence. Birkhäuser, Boston (1993).
Dal Maso, G. and Toader, R., A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Methods Appl. Sci. 12 (2002) 1773–1799. Google Scholar
Dal Maso, G. and Toader, R., A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101–135. Google Scholar
Dal Maso, G. and Toader, R., On a notion of unilateral slope for the Mumford-Shah functional. NoDEA Nonlin. Differ. Equ. Appl. 13 (2007) 713–734. Google Scholar
E. De Giorgi, New problems on minimizing movements. In Boundary value problems for partial differential equations and applications. RMA Res. Notes Appl. Math. Masson, Paris (1993) 81–98.
Efendiev, M.A. and Mielke, A., On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13 (2006) 151–167. Google Scholar
Francfort, G.A. and Larsen, C.J., Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56 (2003) 1465–1500. Google Scholar
Francfort, G.A. and Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. Google Scholar
Giacomini, A., Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22 (2005) 129–172. Google Scholar
Knees, D., Mielke, A. and Zanini, C., On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18 (2008) 1529–1569. Google Scholar
D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model (2013).
Larsen, C.J., Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math. 63 (2010) 630–654. Google Scholar
Larsen, C.J., Ortner, C. and Süli, E., Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20 (2010) 1021–1048. Google Scholar
A. Mielke, Evolution of rate-independent systems, volume Evolutionary equations. Handb. Differ. Equ. Elsevier, Amsterdam (2005) 461–559.
A. Mielke, Differential, energetic, and metric formulations for rate-independent processes. In Nonlinear PDE’s and applications, vol. 2028 of Lecture Notes in Math. Springer, Heidelberg (2011) 87–170.
Mielke, A., Rossi, R. and Savaré, G., Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. 25 (2009) 585–615. Google Scholar
Mielke, A., Rossi, R., and Savaré, G., BV solutions and viscosity approximations of rate-independent systems. ESAIM: COCV 18 (2012) 36–80. Google Scholar
Mielke, A., Rossi, R. and Savaré, G., Variational convergence of gradient flows and rate-independent evolutions in metric spaces. Milan J. Math. 80 (2012) 381–410. Google Scholar
Mielke, A., T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008) 387–416. Google Scholar
Negri, M., From phase-field to sharp cracks: convergence of quasi-static evolutions in a special setting. Appl. Math. Lett. 26 (2013) 219–224. Google Scholar
Negri, M. and Ortner, C., Quasi-static propagation of brittle fracture by Griffith’s criterion. Math. Models Methods Appl. Sci. 18 (2008) 1895–1925. Google Scholar
Ortner, C., Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal. 38 (2006) 1214–1234. Google Scholar
Rossi, R. and Savaré, G., Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: COCV 12 (2006) 564–614. Google Scholar
Sandier, E. and Serfaty, S., Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Comm. Pure Appl. Math. 57 (2004) 1627–1672. Google Scholar
Serfaty, S., Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31 (2011) 1427–1451. Google Scholar
Sicsic, P. and Marigo, J.-J., From gradient damage laws to Griffith’s theory of crack propagation. J. Elasticity 113 (2013) 55–74. Google Scholar
Stefanelli, U., A variational characterization of rate-independent evolution. Math. Nach. 282 (2009) 1492–1512. Google Scholar
Toader, R. and Zanini, C., An artificial viscosity approach to quasi-static crack growth. Boll. Unione Mat. Ital. 2 (2009) 1–35. Google Scholar