Mathematical models of two-dimensional sheets and foundations
<p>The ubiquity of thin elastic materials in biology and engineering has motivated the development of many reduced models to describe their elastic deformation. This thesis is concerned with the validity of some common reduced models to problems motivated by recent experiments, as well as the...
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| Format: | Thesis |
| Language: | English |
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2021
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| _version_ | 1797089504486490112 |
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| author | Chandler, TGJ |
| author2 | Vella, D |
| author_facet | Vella, D Chandler, TGJ |
| author_sort | Chandler, TGJ |
| collection | OXFORD |
| description | <p>The ubiquity of thin elastic materials in biology and engineering has motivated the development of many reduced models to describe their elastic deformation. This thesis is concerned with the validity of some common reduced models to problems motivated by recent experiments, as well as the derivation of appropriate alternatives.</p>
<p>The first problem is motivated by the indentation of suspended elastic sheets, as is often done to characterize ultra-thin solids (including truly two-dimensional materials like graphene). While this is a convenient means of measuring properties such as the stretching modulus of these materials, experiments on ostensibly similar systems have reported material properties that differ by more than an order of magnitude. We demonstrate that such reported differences may arise from the inappropriate use of asymptotic results as well as commonly neglected effects, namely the indenter geometry and non-Hookean material behaviour. In particular, we present a modelling study of this indentation process assuming linear elasticity and implement a model that accounts for large strains and plate rotations.</p>
<p>The second problem comes from the deformation of a thin elastic foundation. A linear force--deflection relationship (known as Winkler's mattress model) is often used as a simplified model to understand how a thin elastic layer deforms when subject to a distributed normal load. For an incompressible material, however, the model predicts infinite resistance to deformation and, hence, breaks down. We derive a model that describes the deformation of thin elastic layers in response to pressure and shear loading, and that holds for both incompressible and compressible materials alike. We find that the applicability of Winkler's model in not determined by the value of the Poisson's ratio alone, but by a compressibility parameter that combines the Poisson's ratio with a measure of the layer’s slenderness. We illustrate the application of our combined foundation model to three example problems.</p>
<p>Finally, we consider the macroscopic elastic properties of thin cellular structures. Internal cellular pressure (turgor) is known to provide structural rigidity to non-woody plants and has been used to induce Gaussian curvature in inflatable shells. However, the route through which pressure induces this macroscopic rigidity remains unclear. We calculate the macroscopic bulk, stretching, and bending moduli of a two-dimensional single-cell thick structure with variable internal pressure in a simple model. Using these effective moduli, we then consider problems purely at a macroscale, considering the implications of our results for bryophytes (e.g. mosses) and soft robotics.</p> |
| first_indexed | 2024-03-07T03:05:03Z |
| format | Thesis |
| id | oxford-uuid:b2368224-dc2d-4240-8b21-db5c9e370198 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:05:03Z |
| publishDate | 2021 |
| record_format | dspace |
| spelling | oxford-uuid:b2368224-dc2d-4240-8b21-db5c9e3701982022-03-27T04:10:08ZMathematical models of two-dimensional sheets and foundationsThesishttp://purl.org/coar/resource_type/c_db06uuid:b2368224-dc2d-4240-8b21-db5c9e370198Applied mathematicsEnglishHyrax Deposit2021Chandler, TGJVella, D<p>The ubiquity of thin elastic materials in biology and engineering has motivated the development of many reduced models to describe their elastic deformation. This thesis is concerned with the validity of some common reduced models to problems motivated by recent experiments, as well as the derivation of appropriate alternatives.</p> <p>The first problem is motivated by the indentation of suspended elastic sheets, as is often done to characterize ultra-thin solids (including truly two-dimensional materials like graphene). While this is a convenient means of measuring properties such as the stretching modulus of these materials, experiments on ostensibly similar systems have reported material properties that differ by more than an order of magnitude. We demonstrate that such reported differences may arise from the inappropriate use of asymptotic results as well as commonly neglected effects, namely the indenter geometry and non-Hookean material behaviour. In particular, we present a modelling study of this indentation process assuming linear elasticity and implement a model that accounts for large strains and plate rotations.</p> <p>The second problem comes from the deformation of a thin elastic foundation. A linear force--deflection relationship (known as Winkler's mattress model) is often used as a simplified model to understand how a thin elastic layer deforms when subject to a distributed normal load. For an incompressible material, however, the model predicts infinite resistance to deformation and, hence, breaks down. We derive a model that describes the deformation of thin elastic layers in response to pressure and shear loading, and that holds for both incompressible and compressible materials alike. We find that the applicability of Winkler's model in not determined by the value of the Poisson's ratio alone, but by a compressibility parameter that combines the Poisson's ratio with a measure of the layer’s slenderness. We illustrate the application of our combined foundation model to three example problems.</p> <p>Finally, we consider the macroscopic elastic properties of thin cellular structures. Internal cellular pressure (turgor) is known to provide structural rigidity to non-woody plants and has been used to induce Gaussian curvature in inflatable shells. However, the route through which pressure induces this macroscopic rigidity remains unclear. We calculate the macroscopic bulk, stretching, and bending moduli of a two-dimensional single-cell thick structure with variable internal pressure in a simple model. Using these effective moduli, we then consider problems purely at a macroscale, considering the implications of our results for bryophytes (e.g. mosses) and soft robotics.</p> |
| spellingShingle | Applied mathematics Chandler, TGJ Mathematical models of two-dimensional sheets and foundations |
| title | Mathematical models of two-dimensional sheets and foundations |
| title_full | Mathematical models of two-dimensional sheets and foundations |
| title_fullStr | Mathematical models of two-dimensional sheets and foundations |
| title_full_unstemmed | Mathematical models of two-dimensional sheets and foundations |
| title_short | Mathematical models of two-dimensional sheets and foundations |
| title_sort | mathematical models of two dimensional sheets and foundations |
| topic | Applied mathematics |
| work_keys_str_mv | AT chandlertgj mathematicalmodelsoftwodimensionalsheetsandfoundations |