In many engineering applications, one is interested in structures with elastic properties which undergo deformation due to external loads. This interest has motivated a deep study of the equations underlying the deformation and how to solve them during the two last centuries. In this thesis, we propose a method that allows us to construct exact solutions to these equations. We prove that the partial differential equation governing the deformation is equivalent to the well-studied Cauchy-Riemann equation on the unit disk. Furthermore, we prove sufficient conditions for when the exact solutions to the Cauchy-Riemann equation on the unit disk can be used to construct solutions to the physical problem. We end the thesis by outlining a method for solving the elasticity equation in a general simply connected domain with a known conformal mapping to the unit disk. This method simplifies the formulas of Kolosov and Muskhelishvili, which are constructed by complex potentials in a similar, but more indirect way. It also allows us to obtain solutions in domains where standard numerical methods, such as e.g. the finite element method, proves difficult or even impossible to apply.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:umu-196386 |
| Date | January 2022 |
| Creators | Granath, Andreas |
| Publisher | Umeå universitet, Institutionen för fysik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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