Abstract
This thesis studies two-dimensional nonlinear elastic shell models. These models are systems of fully nonlinear partial differential equations whose solutions predict the deformation of the middle surface of a thin elastic shell under the impact of applied forces. This is the case in many practical situations, such as in construction or software design. However, existence and uniqueness results for these models are not yet fully satisfactory. This is the main purpose of this thesis.We begin with the nonlinear shell model of W.T. Koiter. For this model, the existence of a minimizer remains an open question, one of the difficulties being that Koiter’s energy is not coercive, which prevents us from applying the classical method of calculus of variations. We will tackle this problem by modifying Koiter’s energy in an appropriate sense under some additional assumptions on the geometry of the shell and/or on the set of admissible deformations so that the new energies are coercive and weakly lower semicontinuous and, thus, the existence results are ensured.
Finally, we study P. Destuynder’s nonlinear shell model of Budiansky-Sanders’ type. Destuynder proved the existence of a minimizer of this model by considering two cases involving additional assumptions on the geometry of the shell and/or the applied forces. However, no result has been obtained regarding the uniqueness of the minimizer. We will partially solve this question for a class of shells whose middle surfaces are a portion of a cylinder under the impact of special applied forces. We will also generalize one of the two existence theorems by Destuynder to general shells, that is, regardless of the geometry of their middle surfaces, and to a larger set of applied forces. Moreover, we establish a uniqueness result when the applied forces are small enough.
| Date of Award | 9 Sept 2025 |
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| Original language | English |
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| Supervisor | Moritz Andreas REINTJES (Supervisor) & Cristinel MARDARE (External Co-Supervisor) |