Rate-independent processes with linear growth energies and time-dependent boundary conditions

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June 2012, 5(3): 591-604. doi: 10.3934/dcdss.2012.5.591

Rate-independent processes with linear growth energies and time-dependent boundary conditions

Martin Kružík ^1, and Johannes Zimmer ^2,

1.	Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic
2.	Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

Received August 2010 Revised January 2011 Published October 2011

Abstract
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A rate-independent evolution problem is considered for which the stored energy density depends on the gradient of the displacement. The stored energy density does not have to be quasiconvex and is assumed to exhibit linear growth at infinity; no further assumptions are made on the behaviour at infinity. We analyse an evolutionary process with positively $1$-homogeneous dissipation and time-dependent Dirichlet boundary conditions.

Keywords: time-dependent boundary conditions, rate-independent evolution., oscillations, Concentrations.

Mathematics Subject Classification: Primary: 74C15; Secondary: 49J45, 74G6.

Citation: Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591

References:

[1]	I. V. Chenchiah, M. O. Rieger and J. Zimmer, Gradient flows in asymmetric metric spaces, Nonlinear Anal., 71 (2009), 5820-5834. doi: 10.1016/j.na.2009.05.006.
[2]	S. Conti and M. Ortiz, Dislocation microstructures and the effective behavior of single crystals, Arch. Ration. Mech. Anal., 176 (2005), 103-147. doi: 10.1007/s00205-004-0353-2.
[3]	G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4.
[4]	R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: 10.1007/BF01214424.
[5]	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. doi: 10.1515/CRELLE.2006.044.
[6]	M. Kružík and T. Roubíček, On the measures of DiPerna and Majda, Math. Bohem., 122 (1997), 383-399.
[7]	M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA Journal of Applied Mathematics, 76 (2011), 193-216. doi: 10.1093/imamat/hxq058.
[8]	M. Kružík and J. Zimmer, Vanishing regularisation for gradient flows via $\Gamma$-limit,, in preparation., ().
[9]	M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces, ESAIM Control Optim. Calc. Var., 16 (2010), 1-22.
[10]	A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009), 221-248. doi: 10.1007/s00332-008-9033-y.
[11]	A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597 (electronic). doi: 10.1137/S1540345903422860.
[12]	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. doi: 10.1007/s002050200194.
[13]	M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47 (1999), 397-462. doi: 10.1016/S0022-5096(97)00096-3.
[14]	T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus,'' de Gruyter Series in Nonlinear Analysis and Applications, 4, Walter de Gruyter & Co., Berlin, 1997.
[15]	J. Souček, Spaces of functions on domain $\Omega $, whose $k$-th derivatives are measures defined on $\bar \Omega $, Časopis Pĕst. Mat., 97 (1972), 10-46, 94.
[16]	D. W. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233, Springer-Verlag, Berlin-New York, 1979.

show all references

References:

[1]	I. V. Chenchiah, M. O. Rieger and J. Zimmer, Gradient flows in asymmetric metric spaces, Nonlinear Anal., 71 (2009), 5820-5834. doi: 10.1016/j.na.2009.05.006.
[2]	S. Conti and M. Ortiz, Dislocation microstructures and the effective behavior of single crystals, Arch. Ration. Mech. Anal., 176 (2005), 103-147. doi: 10.1007/s00205-004-0353-2.
[3]	G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4.
[4]	R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: 10.1007/BF01214424.
[5]	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. doi: 10.1515/CRELLE.2006.044.
[6]	M. Kružík and T. Roubíček, On the measures of DiPerna and Majda, Math. Bohem., 122 (1997), 383-399.
[7]	M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA Journal of Applied Mathematics, 76 (2011), 193-216. doi: 10.1093/imamat/hxq058.
[8]	M. Kružík and J. Zimmer, Vanishing regularisation for gradient flows via $\Gamma$-limit,, in preparation., ().
[9]	M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces, ESAIM Control Optim. Calc. Var., 16 (2010), 1-22.
[10]	A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009), 221-248. doi: 10.1007/s00332-008-9033-y.
[11]	A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597 (electronic). doi: 10.1137/S1540345903422860.
[12]	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. doi: 10.1007/s002050200194.
[13]	M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids, 47 (1999), 397-462. doi: 10.1016/S0022-5096(97)00096-3.
[14]	T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus,'' de Gruyter Series in Nonlinear Analysis and Applications, 4, Walter de Gruyter & Co., Berlin, 1997.
[15]	J. Souček, Spaces of functions on domain $\Omega $, whose $k$-th derivatives are measures defined on $\bar \Omega $, Časopis Pĕst. Mat., 97 (1972), 10-46, 94.
[16]	D. W. Stroock and S. R. S. Varadhan, "Multidimensional Diffusion Processes,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233, Springer-Verlag, Berlin-New York, 1979.

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