American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Kružík and T. Roubíček, On the measures of DiPerna and Majda, Math. Bohem., 122 (1997), 383-399.  Google Scholar

[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.  Google Scholar

[8]

M. Kružík and J. Zimmer, Vanishing regularisation for gradient flows via $\Gamma$-limit,, in preparation., ().   Google Scholar

[9]

M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces, ESAIM Control Optim. Calc. Var., 16 (2010), 1-22.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Kružík and T. Roubíček, On the measures of DiPerna and Majda, Math. Bohem., 122 (1997), 383-399.  Google Scholar

[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.  Google Scholar

[8]

M. Kružík and J. Zimmer, Vanishing regularisation for gradient flows via $\Gamma$-limit,, in preparation., ().   Google Scholar

[9]

M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces, ESAIM Control Optim. Calc. Var., 16 (2010), 1-22.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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