Conti, S. and Ortiz, M. (2008) Minimum principles for the trajectories of systems governed by rate problems. Journal of the Mechanics and Physics of Solids, 56 (5). pp. 1885-1904. ISSN 0022-5096. doi:10.1016/j.jmps.2007.11.006. https://resolver.caltech.edu/CaltechAUTHORS:20171117-152209057
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Abstract
Recently, Mielke and Ortiz [2007. A class of minimum principles for characterizing the trajectories of dissipative systems, ESAIM Control Optim. Calc. Var., in press] have proposed a variational reformulation of evolutionary problems that characterizes entire trajectories of a system as minimizers of certain energy–dissipation functionals. In this paper we present two examples of energy–dissipation functionals for which relaxations and optimal scalings can be rigorously derived. The first example concerns a one-dimensional bar characterized by a quadratic dissipation function and a bistable energy density; the second example concerns the coarsening kinetics of island growth in thin films exhibiting a preferred slope. In both cases, we present closed-form relaxations in the local limit of the problem and optimal scaling relations for the nonlocal problems. The relaxations rigorously characterize macroscopic properties of complex microstructural evolution by means of well-posed effective problems. The scaling relations rigorously characterize some average properties of the coarsening kinetics of the systems and lead to predictions on the growth exponents.
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| Additional Information: | © 2007 Elsevier. Received 16 August 2007, Revised 3 November 2007, Accepted 14 November 2007, Available online 28 November 2007. We thank Francesco Maggi for interesting discussions. The support of the Department of Energy through Caltech's ASCI ASAP Center for the Simulation of the Dynamic Response of Materials and of the Deutsche Forschungsgemeinschaft through the Forschergruppe 797 Analysis and computation of microstructure in finite plasticity is gratefully acknowledged. | |||||||||
| Group: | GALCIT | |||||||||
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| Subject Keywords: | Minimum principlesRate problemsDissipative systemsMetastabilityRelaxation | |||||||||
| Issue or Number: | 5 | |||||||||
| DOI: | 10.1016/j.jmps.2007.11.006 | |||||||||
| Record Number: | CaltechAUTHORS:20171117-152209057 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20171117-152209057 | |||||||||
| Official Citation: | S. Conti, M. Ortiz, Minimum principles for the trajectories of systems governed by rate problems, In Journal of the Mechanics and Physics of Solids, Volume 56, Issue 5, 2008, Pages 1885-1904, ISSN 0022-5096, https://doi.org/10.1016/j.jmps.2007.11.006. (http://www.sciencedirect.com/science/article/pii/S0022509607002220) | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 83310 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | Lydia Suarez | |||||||||
| Deposited On: | 21 Nov 2017 00:18 | |||||||||
| Last Modified: | 15 Nov 2021 19:57 |
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