The large deformation of nonlinearly elastic rings in a two-dimensional compressible flow |
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Authors: | DE CRISTOFORIS MASSIMO LANZA |
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Affiliation: | Dipartimento di Matematica Pura ed Applicata, Università di Padova Via Belzoni 7, 35131 Padova, Italy |
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Abstract: | The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((0, 0), 0, P0) in the solution-parameter space of ((0, 0),0, P0) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((0, 0), 0, P0) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P0 and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((0, 0), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((0, 0),0). |
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