Analysis of an Inelastic Contact Problem for the Damped Wave Equation
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915288127111168 |
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| author | Muha, Boris Trifunović, Srđan |
| author_facet | Muha, Boris Trifunović, Srđan |
| contents | In this paper, we examine the dynamic behavior of a viscoelastic string oscillating above a rigid obstacle in a one-dimensional setting, accounting for inelastic contact between the string and the obstacle. We construct a global-in-time weak solution to this problem by using an approximation method that incorporates a penalizing repulsive force of the form $\frac1\varepsilonχ_{\{η<0\}} (\partial_tη)^-$. The weak solution exhibits well-controlled energy dissipation, occurring only during contact on a set of zero measure and exclusively when the string moves downward. Furthermore, the velocity is shown to vanish after contact in a specific weak sense. This model serves as a simplified framework for studying contact problems in fluid-structure interaction contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis of an Inelastic Contact Problem for the Damped Wave Equation Muha, Boris Trifunović, Srđan Analysis of PDEs 74M15 (Primary) 74K05, 35L05 (Secondary) In this paper, we examine the dynamic behavior of a viscoelastic string oscillating above a rigid obstacle in a one-dimensional setting, accounting for inelastic contact between the string and the obstacle. We construct a global-in-time weak solution to this problem by using an approximation method that incorporates a penalizing repulsive force of the form $\frac1\varepsilonχ_{\{η<0\}} (\partial_tη)^-$. The weak solution exhibits well-controlled energy dissipation, occurring only during contact on a set of zero measure and exclusively when the string moves downward. Furthermore, the velocity is shown to vanish after contact in a specific weak sense. This model serves as a simplified framework for studying contact problems in fluid-structure interaction contexts. |
| title | Analysis of an Inelastic Contact Problem for the Damped Wave Equation |
| topic | Analysis of PDEs 74M15 (Primary) 74K05, 35L05 (Secondary) |
| url | https://arxiv.org/abs/2412.06185 |