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BV solutions and viscosity approximations of rate-independentsystems∗

Published online by Cambridge University Press:  23 December 2010

Alexander Mielke ,
Riccarda Rossi  and
Giuseppe Savaré
Alexander Mielke
Affiliation:
Weierstraß-Institut, Mohrenstraße 39, 10117 Berlin, Germany. mielke@wias-berlin.de Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany
Riccarda Rossi
Affiliation:
Dipartimento di Matematica, Università di Brescia, via Valotti 9, 25133 Brescia, Italy; riccarda.rossi@ing.unibs.it
Giuseppe Savaré
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, 27100 Pavia, Italy; giuseppe.savare@unipv.it
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Abstract

In the nonconvex case, solutions of rate-independent systems may develop jumps as afunction of time. To model such jumps, we adopt the philosophy that rate-independenceshould be considered as limit of systems with smaller and smaller viscosity. For thefinite-dimensional case we study the vanishing-viscosity limit of doubly nonlinearequations given in terms of a differentiable energy functional and a dissipation potentialthat is a viscous regularization of a given rate-independent dissipation potential. Theresulting definition of “BV solutions” involves, in a nontrivial way, both therate-independent and the viscous dissipation potential, which play crucial roles in thedescription of the associated jump trajectories. We shall prove general convergenceresults for the time-continuous and for the time-discretized viscous approximations andestablish various properties of the limiting BV solutions. In particular, we shall providea careful description of the jumps and compare the new notion of solutions with therelated concepts of energetic and local solutions to rate-independent systems.

Keywords

Doubly nonlineardifferential inclusionsgeneralized gradient flowsviscous regularizationvanishing-viscosity limitvanishing-viscosity contact potentialparameterized solutions

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 36 - 80
Copyright
© EDP Sciences, SMAI, 2010

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References

Ambrosio, L., Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191–246. Google Scholar
Ambrosio, L. and Dal Maso, G., A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108 (1990) 691–702. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000).
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, 2nd edn., Birkhäuser Verlag, Basel (2008).
Auricchio, F., Mielke, A. and Stefanelli, U., A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Meth. Appl. Sci. 18 (2008) 125–164. Google Scholar
Bouchitté, G., Mielke, A. and Roubíček, T., A complete-damage problem at small strains. Z. Angew. Math. Phys. 60 (2009) 205–236. Google Scholar
Buliga, M., de Saxcé, G. and Vallée, C., Existence and construction of bipotentials for graphs of multivalued laws. J. Convex Anal. 15 (2008) 87–104. Google Scholar
Colli, P., On some doubly nonlinear evolution equations in Banach spaces. Japan J. Indust. Appl. Math. 9 (1992) 181–203. Google Scholar
Colli, P. and Visintin, A., On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15 (1990) 737–756. Google Scholar
Dal Maso, G. and Toader, R., A model for 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., A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Meth. Appl. Sci. 12 (2002) 1773–1799. Google Scholar
Dal Maso, G. and Zanini, C., Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. R. Soc. Edinb., Sect. A, Math. 137 (2007) 253–279. Google Scholar
Dal Maso, G., Francfort, G. and Toader, R., Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176 (2005) 165–225. Google Scholar
Dal Maso, G., DeSimone, A. and Mora, M.G., Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237–291. Google Scholar
Dal Maso, G., DeSimone, A., Mora, M.G. and Morini, M., Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media 3 (2008) 567–614. Google Scholar
Dal Maso, G., DeSimone, A., Mora, M.G. and Morini, M., A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189 (2008) 469–544. Google Scholar
G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for cam-clay plasiticity : a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differential Equations (to appear).
Efendiev, M. and Mielke, A., On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Analysis 13 (2006) 151–167. Google Scholar
A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy. Ann. Inst. Henri Poincaré, Anal. Non Linéaire (to appear).
Francfort, G. and Garroni, A., A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182 (2006) 125–152. Google Scholar
Francfort, G. and Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55–91. Google Scholar
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms. II : Advanced theory and bundle methods, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 306. Springer-Verlag, Berlin (1993).
Knees, D., Mielke, A. and Zanini, C., On the inviscid limit of a model for crack propagation. Math. Models Meth. Appl. Sci. 18 (2008) 1529–1569. Google Scholar
Knees, D., Zanini, C. and Mielke, A., Crack propagation in polyconvex materials. Physica D 239 (2010) 1470–1484. Google Scholar
Kočvara, M., Mielke, A. and Roubíček, T., A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423–447. Google Scholar
P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in Nonlinear differential equations (Chvalatice, 1998), Res. Notes Math. 404, Chapman & Hall/CRC, Boca Raton, FL (1999) 47–110.
Krejčí, P., and Liero, M., Rate independent Kurzweil processes. Appl. Math. 54 (2009) 117–145. Google Scholar
Larsen, C.J., Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math. 63 (2010) 630–654. Google Scholar
Mainik, A. and Mielke, A., Existence results for energetic models for rate-independent systems. Calc. Var. PDEs 22 (2005) 73–99. Google Scholar
Mainik, A. and Mielke, A., Global existence for rate-independent gradient plasticity at finite strain. J. Nonlin. Sci. 19 (2009) 221–248. Google Scholar
Mielke, A., Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351–382. Google Scholar
A. Mielke, Evolution in rate-independent systems (Chap. 6), in Handbook of differential equations, evolutionary equations 2, C. Dafermos and E. Feireisl Eds., Elsevier B.V., Amsterdam (2005) 461–559.
A. Mielke, Differential, energetic and metric formulations for rate-independent processes. Lecture Notes, Summer School Cetraro (in press).
Mielke, A. and Roubčíek, T., A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1 (2003) 571–597. Google Scholar
Mielke, A. and Roubčíek, T., Rate-independent damage processes in nonlinear elasticity. M 3 ! AS Math. Models Meth. Appl. Sci. 16 (2006) 177–209. Google Scholar
A. Mielke and T. Roubčíek, Rate-Independent Systems : Theory and Application. (In preparation).
A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R. Balean and R. Farwig Eds., Shaker-Verlag, Aachen (1999) 117–129.
Mielke, A. and Theil, F., On rate-independent hysteresis models. NoDEA 11 (2004) 151–189. Google Scholar
Mielke, A. and Timofte, A., An energetic material model for time-dependent ferroelectric behavior : existence and uniqueness. Math. Meth. Appl. Sci. 29 (2006) 1393–1410. Google Scholar
A. Mielke and S. Zelik, On the vanishing viscosity limit in parabolic systems with rate-independent dissipation terms. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (submitted).
Mielke, A., Theil, F. and Levitas, V.I., A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137–177. Google Scholar
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
A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations. (In preparation).
Negri, M. and Ortner, C., Quasi-static crack propagation by Griffith’s criterion. Math. Models Meth. Appl. Sci. 18 (2008) 1895–1925. Google Scholar
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970).
Rossi, R. and Savaré, G., Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM : COCV 12 (2006) 564–614. Google Scholar
Rossi, R., Mielke, A. and Savaré, G., A metric approach to a class of doubly nonlinear evolution equations and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008) 97–169. Google Scholar
Roubčíek, T., Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32 (2009) 825–862. Google Scholar
Stefanelli, U., A variational characterization of rate-independent evolution. Math. Nachr. 282 (2009) 1492–1512. Google Scholar
Thomas, M. and Mielke, A., Damage of nonlinearly elastic materials at small strains – Existence and regularity results. Zeits. Angew. Math. Mech. 90 (2009) 88–112. Google Scholar
Toader, R. and Zanini, C., An artificial viscosity approach to quasistatic crack growth. Boll. Unione Mat. Ital. 2 (2009) 1–35. Google Scholar
A. Visintin, Differential models of hysteresis, Applied Mathematical Sciences 111. Springer-Verlag, Berlin (1994).