We re-examine the Love equation, which forms the first historical attempt at improving on the classical wave equation to encompass for dispersion of longitudinal waves in rods. Dispersion is introduced by accounting for lateral inertia through the Love hypothesis. Our aim is to provide a rigorous justification of the Love hypothesis, which may be generalized to other contexts. We show that the procedure by which the Love equation is traditionally derived is misleading: indeed, proper variational dealing of the Love hypothesis in a two -modal kinematics (the Mindlin-Herrmann system) leads to the Bishop-Love equation instead. The latter is not asymptotically equivalent to the Love equation, which is in fact a long wave low frequency approximation of the Pochhammer-Chree solution. However, the Love hypothesis may still be retrieved from the Mindlin-Herrmann system, by a slow -time perturbation process. In so doing, the linear KdV equation is retrieved. Besides, consistent approximation demands that a correction term be added to the classical Love hypothesis. Surprisingly, in the case of isotropic linear elasticity, this correction term produces no effect in the correction term of the Lagrangian, so that, to first order, the same Bishop-Love equation is the Euler-Lagrange equation corresponding to a family of Love -like hypotheses, all being different by the correction term. Besides, ill-posedness coming from non-standard (namely non static) natural boundary conditions is now amended.
Revisiting the Love hypothesis for introducing dispersion of longitudinal waves in elastic rods / Nobili, A.; Saccomandi, G.. - In: EUROPEAN JOURNAL OF MECHANICS. A, SOLIDS. - ISSN 0997-7538. - 105:(2024), pp. 105257-105257. [10.1016/j.euromechsol.2024.105257]
Revisiting the Love hypothesis for introducing dispersion of longitudinal waves in elastic rods
Nobili A.;
2024
Abstract
We re-examine the Love equation, which forms the first historical attempt at improving on the classical wave equation to encompass for dispersion of longitudinal waves in rods. Dispersion is introduced by accounting for lateral inertia through the Love hypothesis. Our aim is to provide a rigorous justification of the Love hypothesis, which may be generalized to other contexts. We show that the procedure by which the Love equation is traditionally derived is misleading: indeed, proper variational dealing of the Love hypothesis in a two -modal kinematics (the Mindlin-Herrmann system) leads to the Bishop-Love equation instead. The latter is not asymptotically equivalent to the Love equation, which is in fact a long wave low frequency approximation of the Pochhammer-Chree solution. However, the Love hypothesis may still be retrieved from the Mindlin-Herrmann system, by a slow -time perturbation process. In so doing, the linear KdV equation is retrieved. Besides, consistent approximation demands that a correction term be added to the classical Love hypothesis. Surprisingly, in the case of isotropic linear elasticity, this correction term produces no effect in the correction term of the Lagrangian, so that, to first order, the same Bishop-Love equation is the Euler-Lagrange equation corresponding to a family of Love -like hypotheses, all being different by the correction term. Besides, ill-posedness coming from non-standard (namely non static) natural boundary conditions is now amended.File | Dimensione | Formato | |
---|---|---|---|
refinedLove.pdf
Open access
Tipologia:
Versione originale dell'autore proposta per la pubblicazione
Dimensione
454.6 kB
Formato
Adobe PDF
|
454.6 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris