A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 494-516.

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as a weighted dissipation-energy functional with a weight decaying with a rate 1/ε. The corresponding Euler-Lagrange equation leads to an elliptic regularization of the original evolutionary problem. The Γ-limit of these functionals for ε→0 is highly degenerate and provides limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.

DOI : 10.1051/cocv:2007064
Classification : 49J40, 49M20, 49S05, 74N10
Mots-clés : weighted energy-dissipation functional, incremental minimization problems, relaxation of evolutionary problems, rate-independent processes, energetic solutions 
@article{COCV_2008__14_3_494_0,
     author = {Ortiz, Michael and Mielke, Alexander},
     title = {A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {494--516},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {3},
     year = {2008},
     doi = {10.1051/cocv:2007064},
     mrnumber = {2434063},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2007064/}
}
TY  - JOUR
AU  - Ortiz, Michael
AU  - Mielke, Alexander
TI  - A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
SP  - 494
EP  - 516
VL  - 14
IS  - 3
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2007064/
DO  - 10.1051/cocv:2007064
LA  - en
ID  - COCV_2008__14_3_494_0
ER  - 
%0 Journal Article
%A Ortiz, Michael
%A Mielke, Alexander
%T A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 494-516
%V 14
%N 3
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2007064/
%R 10.1051/cocv:2007064
%G en
%F COCV_2008__14_3_494_0
Ortiz, Michael; Mielke, Alexander. A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 494-516. doi : 10.1051/cocv:2007064. https://www.numdam.org/articles/10.1051/cocv:2007064/

[1] J. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag (1984). | MR | Zbl

[2] S. Aubry and M. Ortiz, The mechanics of deformation-induced subgrain-dislocation structures in metallic crystals at large strains. Proc. Royal Soc. London, Ser. A 459 (2003) 3131-3158. | MR | Zbl

[3] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | MR | Zbl

[4] D. Brandon, I. Fonseca and P. Swart, Oscillations in a dynamical model of phase transitions. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 59-81. | MR | Zbl

[5] H. Brézis and I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. C. R. Acad. Sci. Paris 282 (1976) 971-974 and 1197-1198. | Zbl

[6] C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity. Proc. Royal Soc. London, Ser. A 458 (2002) 299-317. | MR | Zbl

[7] F.H. Clarke, Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990). | MR | Zbl

[8] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations. Comm. Partial Diff. Eq. 15 (1990) 737-756. | MR | Zbl

[9] S. Conti and M. Ortiz, Dislocation microstructures and the effective behavior of single crystals. Arch. Rational Mech. Anal. 176 (2005) 103-147. | MR | Zbl

[10] S. Conti and F. Theil, Single-slip elastoplastic microstructures. Arch. Rational Mech. Anal. 178 (2005) 125-148. | MR | Zbl

[11] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989). | MR | Zbl

[12] G. Dal Maso, An introduction to Γ-convergence. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl

[13] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165-225. | MR | Zbl

[14] I. Fonseca, D. Brandon and P. Swart, Dynamics and oscillatory microstructure in a model of displacive phase transformations, in Progress in partial differential equations: the Metz surveys 3, Longman Sci. Tech., Harlow (1994) 130-144. | MR | Zbl

[15] 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. | MR | Zbl

[16] N. Ghoussoub and L. Tzou, A variational principle for gradient flows. Math. Ann. 330 (2004) 519-549. | MR | Zbl

[17] A. Giacomini and M. Ponsiglione, A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications. Arch. Rational Mech. Anal. 180 (2006) 399-447. | MR | Zbl

[18] M.E. Gurtin, Variational principles in the linear theory of viscoelasticity. Arch. Rational Mech. Anal. 3 (1963) 179-191. | MR | Zbl

[19] M.E. Gurtin, Variational principles for linear initial-value problems. Quart. Applied Math. 22 (1964) 252-256. | Zbl

[20] K. Hackl and U. Hoppe, On the calculation of microstructures for inelastic materials using relaxed energies, in IUTAM Symposium on Computational Mechanics of Solids at Large Strains, C. Miehe Ed., Kluwer (2003) 77-86. | MR | Zbl

[21] R. Jordan, D. Kinderlehrer and F. Otto, Free energy and the Fokker-Planck equation. Physica D 107 (1997) 265-271. | MR | Zbl

[22] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17. | MR | Zbl

[23] R. Jordan, D. Kinderlehrer and F. Otto, Dynamics of the Fokker-Planck equation. Phase Transit. 69 (1999) 271-288.

[24] M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi. Meccanica 40 (2005) 389-418. | MR | Zbl

[25] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Springer-Verlag, New York (1972). | MR | Zbl

[26] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. PDEs 22 (2005) 73-99. | MR

[27] A. Mielke, Flow properties for Young-measure solutions of semilinear hyperbolic problems. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 85-123. | MR | Zbl

[28] A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 5095-5127. | MR | Zbl

[29] 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. | MR | Zbl

[30] A. Mielke and S. Müller, Lower semicontinuity and existence of minimizers for a functional in elastoplasticity. Z. angew. Math. Mech. 86 (2006) 233-250. | MR | Zbl

[31] A. Mielke and R. Rossi, Existence and uniqueness results for a class of rate-independent hysteresis problems. Math. Models Methods Appl. Sci. 17 (2007) 81-123. | MR | Zbl

[32] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity. ESAIM: M2AN (submitted). WIAS Preprint 1169.

[33] 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 (1999) 117-129.

[34] A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA Nonlinear Differ. Equ. Appl. 11 (2004) 151-189. | MR | Zbl

[35] A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162 (2002) 137-177. (Essential Science Indicator: Emerging Research Front, August 2006.) | MR | Zbl

[36] A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Part. Diff. Equ. (2007) Online first. DOI: 10.1007/s00526-007-0119-4 | MR

[37] M. Ortiz and E. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47 (1999) 397-462. | MR | Zbl

[38] M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Engrg. 171 (1999) 419-444. | MR | Zbl

[39] M. Ortiz, E. Repetto and L. Stainier, A theory of subgrain dislocation structures. J. Mech. Physics Solids 48 (2000) 2077-2114. | MR | Zbl

[40] T. Roubíček, Nonlinear Partial Differential Equations with Applications. Birkhäuser Verlag, Basel (2005). | MR | Zbl

[41] S.M. Sivakumar and M. Ortiz, Microstructure evolution in the equal channel angular extrusion process. Comput. Methods Appl. Mech. Engrg. 193 (2004) 5177-5194. | MR | Zbl

[42] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York (1988). | MR | Zbl

[43] F. Theil, Young-measure solutions for a viscoelastically damped wave equation with nonmonotone stress-strain relation. Arch. Rational Mech. Anal. 144 (1998) 47-78. | MR | Zbl

[44] F. Theil, Relaxation of rate-independent evolution problems. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 463-481. | MR | Zbl

[45] Q. Yang, L. Stainier and M. Ortiz, A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. J. Mech. Phys. Solids 54 (2006) 401-424. | MR | Zbl

Cité par Sources :