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Atomistic modelling of fracture
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Buze, Maciej (2019) Atomistic modelling of fracture. PhD thesis, University of Warwick.
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WRAP_Theses_Buze_2019.pdf - Submitted Version - Requires a PDF viewer. Download (3009Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3492993~S15
Abstract
This thesis is devoted to the mathematical analysis of atomistic modelling of fracture in a crystalline solid. In particular, we focus on a single Mode III crack defect in an infinite two-dimensional square lattice under anti-plane displacements and nearest neighbour interactions, show that the associated lattice equilibration problem is well-defined over a suitable function space and discuss different regimes of the key parameter known as the (rescaled) stress intensity factor k _ 0, which in continuum fracture mechanics characterises the strength of the stress singularity at the crack tip and more broadly acts as a loading parameter on the crack.
In the first part of the work, we focus on the small-loading regime with k sufficiently small and, under the assumption that interactions across the crack are disregarded, prove existence, local uniqueness and stability of atomistic solutions and further establish their qualitatively sharp far-field decay estimates.
The latter result requires establishing existence and decay estimates for the corresponding lattice Green’s function in the anti-plane crack geometry, which constitutes the main technical result of the thesis.
In the final part, we go beyond the small-loading regime and focus on capturing crack propagation in a quasi-static analysis aided by bifurcation theory. We provide evidence that k is a natural bifurcation parameter and that the resulting bifurcation diagram is a periodic “snaking curve”. Subsequently we investigate cell size effects in a finite-cell approximation to the infinite problem by proving sharp convergence rates and obtaining a superconvergence result for critical values of k. This enables us to capture the phenomenon of lattice trapping and how it is significantly influenced by the computational domain size.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QD Chemistry |
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Library of Congress Subject Headings (LCSH): | Fracture mechanics -- Mathematical models, Crystalline polymers -- Fracture, Atomic structure, Stress corrosion, Crystallography, Mathematical, Green's functions | ||||
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: | Ortner, Christoph ; Hudson, Thomas | ||||
Format of File: | |||||
Extent: | v, 132 leaves : illustrations | ||||
Language: | eng |
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