×

BV solutions constructed using the epsilon-neighborhood method. (English) Zbl 1338.49025

Summary: We study a certain class of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order \(\varepsilon\) and then taking the limit \(\varepsilon\to 0\). We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke et al.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J20 Existence theories for optimal control problems involving partial differential equations
49M30 Other numerical methods in calculus of variations (MSC2010)
49M25 Discrete approximations in optimal control
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] G. Alberti and A. DeSimone, Quasistatic evolution of sessile drops and contact angle hysteresis. Arch. Ration. Mech. Anal.202 (2011) 295-348. · Zbl 1276.76016 · doi:10.1007/s00205-011-0427-x
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Clarendon Press (2000). · Zbl 0957.49001
[3] 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. · Zbl 1156.74308 · doi:10.3934/nhm.2008.3.567
[4] 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. · Zbl 1219.35305 · doi:10.1007/s00205-008-0117-5
[5] G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling. Cal. Var. Partial Differ. Equ.40 (2008) 125-181. · Zbl 1311.74024 · doi:10.1007/s00526-010-0336-0
[6] G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire27 (2010) 257-290. · Zbl 1188.35205 · doi:10.1016/j.anihpc.2009.09.006
[7] 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
[8] G. Francfort and C.J. Larsen, Existence and convergence for quasistatic evolution in brittle fracture. Comm. Pure Appl. Math.56 (2003) 1465-1500. · Zbl 1068.74056 · doi:10.1002/cpa.3039
[9] G. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids46 (1998) 1319-1342. · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[10] 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 · doi:10.1515/CRELLE.2006.044
[11] C.J. Larsen, Epsilon-stable quasistatic brittle fracture evolution. Comm. Pure Appl. Math.63 (2010) 630-654. · Zbl 1423.74835
[12] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ.22 (2005) 73-99. · Zbl 1161.74387 · doi:10.1007/s00526-004-0267-8
[13] A. Mielke, Finite Elastoplasticity, Lie Groups and Geodesics on SL(d), In Geometry, Dynamics, and Mechanics. Edited by P. Newton, A. Weinstein and P. Holmes. Springer-Verlag (2003) 61-90.
[14] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Cont. Mech. Thermodyn.15 (2003) 351-382. · Zbl 1068.74522 · doi:10.1007/s00161-003-0120-x
[15] A. Mielke, Evolution of Rate-Independent Systems. Handb. Differ. Equ. Evol. Equ. Elsevier B. V. 2 (2005) 461-559. · Zbl 1120.47062
[16] A. Mielke, A Mathematical Framework for Generalized Standard Materials in the Rate-independent Case, in Multifield problems in Fluid and Solid Mechanics. In Ser. Lect. Notes Appl. Comput. Mechanics. Springer (2006).
[17] A. Mielke, Modeling and Analysis of Rate-independent Processes. Lipschitz Lectures. University of Bonn (2007).
[18] A. Mielke, Differential, Energetic and Metric Formulations for Rate-independent Processes. Lect. Notes of C.I.M.E. Summer School on Nonlinear PDEs and Applications. Cetraro (2008). · Zbl 1251.35003
[19] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst.2 (2010) 585-615. · Zbl 1170.49036
[20] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems. ESAIM: COCV18 (2012) 36-80. · Zbl 1250.49041 · doi:10.1051/cocv/2010054
[21] A. Mielke, R. Rossi and G. Savaré, Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems. To appear in J. Eur. Math. Soc. (2016). · Zbl 1357.35007
[22] A. Mielke and F. Theil, A Mathematical Model for Rate-Independent Phase Transformations with Hysteresis. In Models of Continuum Mechanics in Analysis and Engineering. Shaker Ver. Aachen (1999).
[23] A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA Nonlin. Differ. Equ. Appl.11 (2004) 151-189. · Zbl 1061.35182 · doi:10.1007/s00030-003-1052-7
[24] A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal.162 (2002) 137-177. · Zbl 1012.74054 · doi:10.1007/s002050200194
[25] M.N. Minh, Weak solutions to rate-independent systems: Existence and Regularity. Ph.D. thesis (2012).
[26] S. Müller, Variational Models for Microstructure and Phase Transitions, In Calculus of Variations and Geometric Evolution Problems, Cetraro. Springer, Berline (1999) 85-210. · Zbl 0968.74050
[27] I.P. Natanson, Theory of Functions of a Real Variable. Frederick Ungar, New York (1965).
[28] M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation. Adv. Calc. Var.3 (2010) 149-212. · Zbl 1193.49055 · doi:10.1515/acv.2010.008
[29] F. Schmid and A. Mielke, Vortex pinning in super-conductivity as a rate-independent process. Eur. J. Appl. Math. (2005). · Zbl 1118.82053
[30] U. Stefanelli, A variational characterization of rate-independent evolution. Math. Nach.282 (2009) 1492-1512. · Zbl 1217.34104 · doi:10.1002/mana.200810803
[31] R. Rossi and G. Savaré, A characterization of energetic and BV solutions to one-dimensional rate-independent systems. Discrete Contin. Dyn. Syst. Ser. S.6 (2013) 167-191. · Zbl 1270.34027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.