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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.23253 |
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| _version_ | 1866913062317981696 |
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| author | Kiselev, Oleg |
| author_facet | Kiselev, Oleg |
| contents | We construct a local matched-asymptotic description of time-harmonic elastic fields generated by Rayleigh waves near cuspidal elements of a traction-free surface. The free surface is represented locally by a cusp graph with exponent $0<α<1$, or equivalently by a vanishing-width horn $b(s)=B s^m$, $m=1/α>1$. A cuspidal gorge is a zero-opening re-entrant notch: its leading field is the Williams crack-tip field, and the stresses behave as $r^{-1/2}$. The cusp exponent affects the gorge through lower-order corrections and through the stress cut-off produced by rounding the bottom. In contrast, a cuspidal ridge behaves as an elastic horn with vanishing width. The leading admissible free-tip field is asymptotically rigid (bounded stress), distinct from the high-energy branch supported by a finite tip truncation, where stresses grow as $σ\sim ρ^{-m}$. Finite-element calculations for the local static Lamé problems support these predictions: the free-tip ridge test confirms the absence of crack-like growth, the truncated ridge recovers the high-energy law, and the gorge stress slope is found to be close to $-1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Matched asymptotics of Rayleigh-wave fields near cuspidal ridges and gorges Kiselev, Oleg Mathematical Physics 35C20 We construct a local matched-asymptotic description of time-harmonic elastic fields generated by Rayleigh waves near cuspidal elements of a traction-free surface. The free surface is represented locally by a cusp graph with exponent $0<α<1$, or equivalently by a vanishing-width horn $b(s)=B s^m$, $m=1/α>1$. A cuspidal gorge is a zero-opening re-entrant notch: its leading field is the Williams crack-tip field, and the stresses behave as $r^{-1/2}$. The cusp exponent affects the gorge through lower-order corrections and through the stress cut-off produced by rounding the bottom. In contrast, a cuspidal ridge behaves as an elastic horn with vanishing width. The leading admissible free-tip field is asymptotically rigid (bounded stress), distinct from the high-energy branch supported by a finite tip truncation, where stresses grow as $σ\sim ρ^{-m}$. Finite-element calculations for the local static Lamé problems support these predictions: the free-tip ridge test confirms the absence of crack-like growth, the truncated ridge recovers the high-energy law, and the gorge stress slope is found to be close to $-1/2$. |
| title | Matched asymptotics of Rayleigh-wave fields near cuspidal ridges and gorges |
| topic | Mathematical Physics 35C20 |
| url | https://arxiv.org/abs/2604.23253 |