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Relaxation of integral functionals depending on the symmetrised gradient
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Kosiba, Kamil (2019) Relaxation of integral functionals depending on the symmetrised gradient. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3517145~S1
Abstract
In this thesis we aim to advance the variational theory of integral functionals depending on the symmetrised gradient. New contributions to this theory are contained in chapters 3, 4 and 5, where we study relaxations of integral functionals of the form: F : u 7→ Z Ω f x, 1 2 ∇u(x) + ∇u(x) T d x, u : Ω ⊂ R d → R d under various ‘shape’ constraints imposed on the integrand f. Functionals of this form arise naturally in the mathematical theory of solid mechanics. In Chapter 3 we investigate the linear growth case, that is we additionally assume that f satisfies bounds: m|A| ≤ f(A) ≤ M(1 + |A|) for all symmetric matrices A ∈ R d×d sym and some constants 0 < m ≤ M. Sometimes this growth is called linear isotropic. In Chapter 4 we deal with the case of mixed growth, that is we assume that the inequality m (tr A) 2 + | dev A| ≤ f(A) ≤ M 1 + (tr A) 2 + | dev A| holds for all symmetric matrices A ∈ R d×d sym and some constants 0 < m ≤ M. In Chapter 5 we look at the special case of mixed-growth functionals, the Hencky’s plasticity functional and its inhomogeneous generalisation. The main result of this chapter is the proof of lower semicontinuity of the aforementioned inhomogeneous functional in a sufficiently weak topology. This result relies on the theory of Young measures, which we briefly recall. We also discuss new developments in this theory and state open problems.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Integral equations, Relaxation methods (Mathematics), Plasticity, Symmetry (Mathematics) | ||||
Official Date: | September 2019 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Rindler, Filip ; Theil, Florian | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | 91 leaves | ||||
Language: | eng |
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