Browsing by Author "Ortiz, Michael"
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- ItemA class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Ortiz, MichaelThis 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 elliptic regularizations of the original evolutionary problem. We find that the $Gamma$-limits of interest are highly degenerate and provide 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.
- ItemMicrostructures in Solids: From Quantum Models to Continua(Zürich : EMS Publ. House, 2010) Ortiz, MichaelThe mathematical theory of solids was studied from the modern perspective of materials with microcstructures. The discussed topics ranged from experimental findings, via numerical simulations and mathematical modeling to the analysis of models with microstructures. A special emphasis was given to theories providing rigorous insight into and justification of the limit passage between different scales.
- ItemMini-Workshop: Analysis and Computation of Microstructures in Finite Plasticity(Zürich : EMS Publ. House, 2005) Conti, Sergio; Ortiz, MichaelPlastic material behaviour is typically the result of the interaction of complex substructures on a microscopic scale. Common models of finite plasticity are based on macroscopic, phenomenological approaches and do not take into account any microstructural information. The miniworkshop focuses on the application of methods from the calculus of variations to models for microstructures in plasticity. In particular, the investigation of the relaxation of the underlying functional, corresponding to quasiconvexification of the energy density, allows us to gain interesting microscopic as well as macroscopic information.
- ItemVariational Methods for the Modelling of Inelastic Solids(Zürich : EMS Publ. House, 2018) Garroni, Adriana; Hackl, Klaus; Ortiz, MichaelThis workshop brought together two communities working on the same topic from different perspectives. It strengthened the exchange of ideas between experts from both mathematics and mechanics working on a wide range of questions related to the understanding and the prediction of processes in solids. Common tools in the analysis include the development of models within the broad framework of continuum mechanics, calculus of variations, nonlinear partial differential equations, nonlinear functional analysis, Gamma convergence, dimension reduction, homogenization, discretization methods and numerical simulations. The applications of these theories include but are not limited to nonlinear models in plasticity, microscopic theories at different scales, the role of pattern forming processes, effective theories, and effects in singular structures like blisters or folding patterns in thin sheets, passage from atomistic or discrete models to continuum models, interaction of scales and passage from the consideration of one specific time step to the continuous evolution of the system, including the evolution of appropriate measures of the internal structure of the system.