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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)
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Mathematical and computational modeling of shape-memory alloys
Benešová, Barbora ; Roubíček, Tomáš (advisor) ; Krejčí, Pavel (referee) ; Stefanelli, Ulisse (referee)
This dissertation thesis is concerned with developing a mesoscopic model for sin- gle crystalline shape-memory alloys including thermo-dynamically consistent thermo- mechanical coupling - here the term "mesoscopic" refers to the ability of the model to capture fine spatial oscillations of the deformation gradient by means of gradient Young measures. Existence of solutions to the devised model is proved in a "phase-field-like approach" by a scale transition from a microscopic model that features a term related to the interfacial energy; this scale transition from a physically relevant model justifies the mesoscopic relaxation. Further, existence of solutions is also proved by backward- Euler time discretization which forms a conceptual numerical algorithm. Based on this conceptual algorithm a computer implementation of the model has been developed and further optimized in the rate-independent isothermal setting; some calculations using this implementation are also presented. Finally, refinements s of the analysis in the convex case as well as a limit of the phase-field-like approach in this case are exposed, too.
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