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 C^{1} 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 ℱ_{n} 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 evolutions, phase-field

@article{COCV_2014__20_4_983_0, author = {Negri, Matteo}, title = {Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {983--1008}, publisher = {EDP-Sciences}, volume = {20}, number = {4}, year = {2014}, doi = {10.1051/cocv/2014004}, mrnumber = {3264231}, zbl = {1301.49017}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014004/} }

TY - JOUR AU - Negri, Matteo TI - Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 983 EP - 1008 VL - 20 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014004/ DO - 10.1051/cocv/2014004 LA - en ID - COCV_2014__20_4_983_0 ER -

%0 Journal Article %A Negri, Matteo %T Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 983-1008 %V 20 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014004/ %R 10.1051/cocv/2014004 %G en %F COCV_2014__20_4_983_0

Negri, Matteo. Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 983-1008. doi : 10.1051/cocv/2014004. http://archive.numdam.org/articles/10.1051/cocv/2014004/

[1] Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR | Zbl

,[2] Gradient flows in metric spaces and in the space of probability measures. Lect. Math. ETH Zürich. Birkhäuser Verlag, Basel (2005). | MR | Zbl

, and ,[3] On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6 (1992) 105-123. | MR | Zbl

and ,[4] Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements (2012).

and ,[5] Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797-826. | MR | Zbl

, and ,[6] The variational approach to fracture. J. Elasticity 91 (2008) 5-148. | MR | Zbl

, and ,[7] Γ-convergence for beginners. Oxford University Press, Oxford (2002). | MR | Zbl

,[8] Local Minimization, Variational Evolution and Γ-convergence, vol. 2094 of Lect. Notes Math. Springer, Berlin (2013).

,[9] A density result in two-dimensional linearized elasticity and applications. Arch. Ration. Mech. Anal. 167 (2003) 211-233. | MR | Zbl

,[10] Revisiting Energy Release Rates in Brittle Fracture. J. Nonlinear Sci. 20 (2010) 395-424. | MR | Zbl

, and ,[11] An introduction to Γ-convergence. Birkhäuser, Boston (1993). | MR | Zbl

,[12] A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Methods Appl. Sci. 12 (2002) 1773-1799. | MR | Zbl

and ,[13] A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101-135. | MR | Zbl

and ,[14] On a notion of unilateral slope for the Mumford-Shah functional. NoDEA Nonlin. Differ. Equ. Appl. 13 (2007) 713-734. | MR | Zbl

and ,[15] 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. | MR | Zbl

,[16] On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13 (2006) 151-167. | MR | Zbl

and ,[17] Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56 (2003) 1465-1500. | MR | Zbl

and ,[18] Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319-1342. | MR | Zbl

and ,[19] Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22 (2005) 129-172. | MR | Zbl

,[20] On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18 (2008) 1529-1569. | MR | Zbl

, and ,[21] A vanishing viscosity approach to a rate-independent damage model (2013). | MR | Zbl

, and ,[22] Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math. 63 (2010) 630-654. | MR

,[23] Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20 (2010) 1021-1048. | MR

, and ,[24] Evolution of rate-independent systems, volume Evolutionary equations. Handb. Differ. Equ. Elsevier, Amsterdam (2005) 461-559. | MR | Zbl

,[25] 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. | MR | Zbl

,[26] Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. 25 (2009) 585-615. | MR | Zbl

, and ,[27] BV solutions and viscosity approximations of rate-independent systems. ESAIM: COCV 18 (2012) 36-80. | Numdam | MR | Zbl

, , and ,[28] Variational convergence of gradient flows and rate-independent evolutions in metric spaces. Milan J. Math. 80 (2012) 381-410. | MR | Zbl

, and ,[29] Γ-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008) 387-416. | MR

, and ,[30] From phase-field to sharp cracks: convergence of quasi-static evolutions in a special setting. Appl. Math. Lett. 26 (2013) 219-224. | MR

,[31] Quasi-static propagation of brittle fracture by Griffith's criterion. Math. Models Methods Appl. Sci. 18 (2008) 1895-1925. | MR | Zbl

and ,[32] Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal. 38 (2006) 1214-1234. | MR | Zbl

,[33] Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: COCV 12 (2006) 564-614. | Numdam | MR | Zbl

and ,[34] Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Comm. Pure Appl. Math. 57 (2004) 1627-1672. | MR | Zbl

and ,[35] Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31 (2011) 1427-1451. | MR | Zbl

,[36] From gradient damage laws to Griffith's theory of crack propagation. J. Elasticity 113 (2013) 55-74. | MR | Zbl

and ,[37] A variational characterization of rate-independent evolution. Math. Nach. 282 (2009) 1492-1512. | MR | Zbl

,[38] An artificial viscosity approach to quasi-static crack growth. Boll. Unione Mat. Ital. 2 (2009) 1-35. | MR | Zbl

and ,*Cited by Sources: *