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Relaxace v mechanice kontinua tuhé fáze
Pathó, Gabriel ; Kružík, Martin (advisor) ; Zeman, Jan (referee)
This work deals with the modelling of shape-memory alloys, in particular with the steady-state model of martensitic thin films. After the introductory motivation the crystallographic structure of the materials is described followed by the introduction of the link between the lattice and continuum model. The next parts of the work focus on the possible solutions of the given 3D variational problem (quasiconvexification, Young measures) and on derivation of thin film theories with the aid of different tools (regularization,-convergence). The last part takes over an approximation of an obtained model and sketches numerical experiments on a Ni-Mn-Ga alloy.
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Mathematical modelling of thin films of martensitic materials
Pathó, Gabriel ; Kružík, Martin (advisor) ; Kalamajska, Agnieszka (referee) ; Šilhavý, Miroslav (referee)
The aim of the thesis is the mathematical and computer modelling of thin films of martensitic materials. We derive a thermodynamic thin-film model on the meso-scale that is capable of capturing the evolutionary process of the shape-memory effect through a two-step procedure. First, we apply dimension reduction techniques in a microscopic bulk model, then enlarge gauge by neglecting microscopic interfacial effects. Computer modelling of thin films is conducted for the static case that accounts for a modified Hadamard jump condition which allows for austenite--martensite interfaces that do not exist in the bulk. Further, we characterize $L^p$-Young measures generated by invertible matrices, that have possibly positive determinant as well. The gradient case is covered for mappings the gradients and inverted gradients of which belong to $L^\infty$, a non-trivial problem is the manipulation with boundary conditions on generating sequences, as standard cut-off methods are inapplicable due to the determinant constraint. Lastly, we present new results concerning weak lower semicontinuity of integral functionals along (asymptotically) $\mathcal{A}$-free sequences that are possibly negative and non-coercive. Powered by TCPDF (www.tcpdf.org)
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Mathematical modelling of thin films of martensitic materials
Pathó, Gabriel ; Kružík, Martin (advisor) ; Kalamajska, Agnieszka (referee) ; Šilhavý, Miroslav (referee)
The aim of the thesis is the mathematical and computer modelling of thin films of martensitic materials. We derive a thermodynamic thin-film model on the meso-scale that is capable of capturing the evolutionary process of the shape-memory effect through a two-step procedure. First, we apply dimension reduction techniques in a microscopic bulk model, then enlarge gauge by neglecting microscopic interfacial effects. Computer modelling of thin films is conducted for the static case that accounts for a modified Hadamard jump condition which allows for austenite--martensite interfaces that do not exist in the bulk. Further, we characterize $L^p$-Young measures generated by invertible matrices, that have possibly positive determinant as well. The gradient case is covered for mappings the gradients and inverted gradients of which belong to $L^\infty$, a non-trivial problem is the manipulation with boundary conditions on generating sequences, as standard cut-off methods are inapplicable due to the determinant constraint. Lastly, we present new results concerning weak lower semicontinuity of integral functionals along (asymptotically) $\mathcal{A}$-free sequences that are possibly negative and non-coercive. Powered by TCPDF (www.tcpdf.org)
|
|
|
Relaxace v mechanice kontinua tuhé fáze
Pathó, Gabriel ; Kružík, Martin (advisor) ; Zeman, Jan (referee)
This work deals with the modelling of shape-memory alloys, in particular with the steady-state model of martensitic thin films. After the introductory motivation the crystallographic structure of the materials is described followed by the introduction of the link between the lattice and continuum model. The next parts of the work focus on the possible solutions of the given 3D variational problem (quasiconvexification, Young measures) and on derivation of thin film theories with the aid of different tools (regularization,-convergence). The last part takes over an approximation of an obtained model and sketches numerical experiments on a Ni-Mn-Ga alloy.
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