### Аннотация

We study the asymptotic behavior of solutions to the linear problem of elasticity in a domain Ω with paraboloidal exit at infinity. Properties of solutions and the condition of the existence of solutions depend on a parameter γ ∈ [0,1] characterizing the velocity of extending the paraboloid (a cylinder and a cone correspond to the cases γ = 0 γ = 1 respectively). Asymptotic formulas are deduced for displacement fields generating forces and moments "applied at infinity." The Saint-Venant principle is verified for "oblong" bodies such as paraboloids, cylinders, and narrow cones. The following question turns out to be a key one: What rigid displacements belong to the energy space obtained by completion of C
_{0}
^{∞}
(Ω̄)
^{3}
by the energy norm? The dimension d
_{γ}
of the lineal R
_{γ}
of rigid energy displacements is computed (in this case, d
_{0}
= 6, d
_{1}
= 0, and the function γ → d
_{γ}
has jumps at the points γ = 1/4, 1/2, 3/4). We also clarify the reasons why it is necessary to distinguish the notions "energy solution" and "solution with finite energy." We also discuss the phenomenon of a boundary layer that appears near the endpoints of spindle-like rods and is described by energy solutions in paraboloids. As is shown, in order to have the well-posed formulation of the boundary conditions in one-dimensional models of such rods, it is necessary to use the weakened Saint-Venant principle, i.e., replace R
_{0}
with R
_{γ}
: for γ > 1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem.

Язык оригинала | английский |
---|---|

Страницы (с-по) | 717-752 |

Число страниц | 36 |

Журнал | Journal of Mathematical Sciences |

Том | 98 |

Номер выпуска | 6 |

DOI | |

Состояние | Опубликовано - 1 янв 2000 |

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### Предметные области Scopus

- Теория вероятности и статистика
- Математика (все)
- Прикладная математика

### Цитировать

*Journal of Mathematical Sciences*,

*98*(6), 717-752. https://doi.org/10.1007/BF02355387