Relaxation in Optimization Theory and Variational Calculus

Roubíček, Tomáš

doi:10.1515/9783110811919

Cited by 269 publications

(297 citation statements)

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“…For more sophisticated ways of relaxing energy functions and the corresponding mechanical interpretations, see [7,8]. For a comparison of upper and lower bounds to the free energy in order to estimate the quality of the convexification bound, see [9][10][11].…”

Section: Micromechanical Modelmentioning

confidence: 99%

Experimental verification of a micromechanical model for polycrystalline shape memory alloys in dependence of martensite orientation distributions

Hackl

Heinen

Schmahl

et al. 2008

Materials Science and Engineering: A

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We make use of a micromechanical model for polycrystalline shape memory alloys, whose main focus is the orientation distribution of the martensitic low symmetry variant. By energy minimization, the internal reorientation of martensite can be predicted. Hysteresis effects are included via the hypothesis that changes in the orientation distribution are connected to energy dissipation. From these considerations, we obtain evolution equations for the orientation distribution in terms of the thermomechanical driving forces. Comparing our model to results from synchrotron diffraction experiments, good agreement is found between experimentally observed and analytically predicted orientations of austenite and stressinduced martensite.

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Section: Micromechanical Modelmentioning

confidence: 99%

Experimental verification of a micromechanical model for polycrystalline shape memory alloys in dependence of martensite orientation distributions

Hackl

Heinen

Schmahl

et al. 2008

Materials Science and Engineering: A

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“…x ∈Ω the collection {ν x } x∈Ω is the so-called Young measure on (Ω, σ) ( [43], see also [5,36,39,41,42]). …”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

“…They can be considered as generalization of Young measures, see for example DiPerna's and Majda's treatment of concentrations [9], Alibert's and Bouchitté's approach [2] or Fonseca's method described in [13]. An overview can be found in [36,40]. In many cases we are interested in oscillation/concentration effects generated by sequences of gradients.…”

Section: Introductionmentioning

confidence: 99%

Oscillations and concentrations in sequences of gradients

Kałamajska

Kružík

2007

ESAIM: COCV

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Abstract.We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, {∇u k }, bounded in L p (Ω; R m×n ) if p > 1 and Ω ⊂ R n is a bounded domain with the extension property in W 1,p . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.Mathematics Subject Classification. 49J45, 35B05.

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“…A generalization of this result was formulated by Schonbek [31] (cf. also [27]): if 1 ≤ p < +∞: for every sequence {y k } k∈N bounded in L p (Ω; R d ) there exists its subsequence (denoted by the same indices) and a Young measure ν = {ν x } x∈Ω ∈ Y (Ω; R d ) such that…”

Section: Discussionmentioning

confidence: 99%

Computational modeling of magnetic hysteresis with thermal effects

Kružík

Valdman

2018

Mathematics and Computers in Simulation

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We study computational behavior of a mesoscopic model describing temperature/external magnetic field-driven evolution of magnetization. Due to nonconvex anisotropy energy describing magnetic properties of a body, magnetization can develop fast spatial oscillations creating complicated microstructures. These microstructures are encoded in Young measures, their first moments then identify macroscopic magnetization. Our model assumes that changes of magnetization can contribute to dissipation and, consequently, to variations of the body temperature affecting the length of magnetization vectors. In the ferromagnetic state, minima of the anisotropic energy density depend on temperature and they tend to zero as we approach the so-called Curie temperature. This brings the specimen to a paramagnetic state. Such a thermo-magnetic model is fully discretized and tested on two-dimensional examples. Computational results qualitatively agree with experimental observations. The own MATLAB code used in our simulations is available for download.

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Relaxation in Optimization Theory and Variational Calculus

Cited by 269 publications

References 0 publications

Experimental verification of a micromechanical model for polycrystalline shape memory alloys in dependence of martensite orientation distributions

Experimental verification of a micromechanical model for polycrystalline shape memory alloys in dependence of martensite orientation distributions

Oscillations and concentrations in sequences of gradients

Computational modeling of magnetic hysteresis with thermal effects

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