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BV solutions and viscosity approximations of rate-independent systems. (English) Zbl 1250.49041

Summary: In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case, we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of “BV solutions” involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.
Reviewer: Th. M. Rassias

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
58E99 Variational problems in infinite-dimensional spaces
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References:

[1] L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.19 (1995) 191-246. · Zbl 0957.49029
[2] L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives. Proc. Am. Math. Soc.108 (1990) 691-702. · Zbl 0685.49027
[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000). · Zbl 0957.49001
[4] 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). · Zbl 1145.35001
[5] F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Meth. Appl. Sci.18 (2008) 125-164. Zbl1151.74319 · Zbl 1151.74319
[6] G. Bouchitté, A. Mielke and T. Roubíček, A complete-damage problem at small strains. Z. Angew. Math. Phys.60 (2009) 205-236. · Zbl 1238.74005
[7] M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws. J. Convex Anal.15 (2008) 87-104. · Zbl 1133.49018
[8] P. Colli, On some doubly nonlinear evolution equations in Banach spaces. Japan J. Indust. Appl. Math.9 (1992) 181-203. · Zbl 0757.34051
[9] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ.15 (1990) 737-756. · Zbl 0707.34053
[10] G. Dal Maso and R. Toader, A model for quasi-static growth of brittle fractures : existence and approximation results. Arch. Ration. Mech. Anal.162 (2002) 101-135. Zbl1042.74002 · Zbl 1042.74002
[11] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Meth. Appl. Sci.12 (2002) 1773-1799. · Zbl 1205.74149
[12] G. Dal Maso and C. Zanini, Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. R. Soc. Edinb., Sect. A, Math.137 (2007) 253-279. · Zbl 1116.74004
[13] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal.176 (2005) 165-225. · Zbl 1064.74150
[14] G. Dal Maso, A. DeSimone and M.G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal.180 (2006) 237-291. · Zbl 1093.74007
[15] G. Dal Maso, A. DeSimone, M.G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media3 (2008) 567-614. Zbl1156.74308 · Zbl 1156.74308
[16] G. Dal Maso, A. DeSimone, M.G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal.189 (2008) 469-544. Zbl1219.35305 · Zbl 1219.35305
[17] 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). · Zbl 1311.74024
[18] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Analysis13 (2006) 151-167. · Zbl 1109.74040
[19] A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy. Ann. Inst. Henri Poincaré, Anal. Non Linéaire (to appear). · Zbl 1167.74005
[20] G. Francfort and A. Garroni, A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal.182 (2006) 125-152. · Zbl 1098.74006
[21] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math.595 (2006) 55-91. · Zbl 1101.74015
[22] 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). Zbl0795.49002 · Zbl 0795.49002
[23] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation. Math. Models Meth. Appl. Sci.18 (2008) 1529-1569. · Zbl 1151.49014
[24] D. Knees, C. Zanini and A. Mielke, Crack propagation in polyconvex materials. Physica D239 (2010) 1470-1484. · Zbl 1201.49013
[25] M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem. Math. Mech. Solids11 (2006) 423-447. · Zbl 1133.74038
[26] 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. · Zbl 0949.47053
[27] P. Krejčí, and M. Liero, Rate independent Kurzweil processes. Appl. Math.54 (2009) 117-145. · Zbl 1212.49007
[28] C.J. Larsen, Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math.63 (2010) 630-654. · Zbl 1423.74835
[29] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. PDEs22 (2005) 73-99. · Zbl 1161.74387
[30] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain. J. Nonlin. Sci.19 (2009) 221-248. Zbl1173.49013 · Zbl 1173.49013
[31] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn.15 (2003) 351-382. · Zbl 1068.74522
[32] A. Mielke, Evolution in rate-independent systems (Chap. 6), in Handbook of differential equations, evolutionary equations2, C. Dafermos and E. Feireisl Eds., Elsevier B.V., Amsterdam (2005) 461-559.
[33] A. Mielke, Differential, energetic and metric formulations for rate-independent processes. Lecture Notes, Summer School Cetraro (in press). · Zbl 1251.35003
[34] A. Mielke and T. Roubčíek, A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul.1 (2003) 571-597. Zbl1183.74207 · Zbl 1183.74207
[35] A. Mielke and T. Roubčíek, Rate-independent damage processes in nonlinear elasticity. M3 ! AS Math. Models Meth. Appl. Sci.16 (2006) 177-209. · Zbl 1094.35068
[36] A. Mielke and T. Roubčíek, Rate-Independent Systems : Theory and Application. (In preparation).
[37] 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.
[38] A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA11 (2004) 151-189. · Zbl 1061.35182
[39] A. Mielke and A. Timofte, An energetic material model for time-dependent ferroelectric behavior : existence and uniqueness. Math. Meth. Appl. Sci.29 (2006) 1393-1410. · Zbl 1096.74017
[40] 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). · Zbl 1295.35036
[41] A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal.162 (2002) 137-177. · Zbl 1012.74054
[42] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst.25 (2009) 585-615. · Zbl 1170.49036
[43] A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations. (In preparation). · Zbl 1270.35289
[44] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith’s criterion. Math. Models Meth. Appl. Sci.18 (2008) 1895-1925. · Zbl 1155.74035
[45] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). · Zbl 0193.18401
[46] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM : COCV12 (2006) 564-614. · Zbl 1116.34048
[47] R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci.7 (2008) 97-169. · Zbl 1183.35164
[48] T. Roubčíek, Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci.32 (2009) 825-862. · Zbl 1194.35226
[49] U. Stefanelli, A variational characterization of rate-independent evolution. Math. Nachr.282 (2009) 1492-1512. Zbl1217.34104 · Zbl 1217.34104
[50] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strains - Existence and regularity results. Zeits. Angew. Math. Mech.90 (2009) 88-112. · Zbl 1191.35159
[51] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth. Boll. Unione Mat. Ital.2 (2009) 1-35. Zbl1180.35521 · Zbl 1180.35521
[52] A. Visintin, Differential models of hysteresis, Applied Mathematical Sciences111. Springer-Verlag, Berlin (1994). Zbl0820.35004 · Zbl 0820.35004
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