### Abstract

In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component Γ_{N}of the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lamé equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ_{0}, thereby for the spatial elasticity problem, we confirm the Babuška paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.

Original language | English |
---|---|

Pages (from-to) | 399-428 |

Number of pages | 30 |

Journal | Journal of Mathematical Sciences (United States) |

Volume | 210 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Oct 2015 |

### Fingerprint

### Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics

### Cite this

}

*Journal of Mathematical Sciences (United States)*, vol. 210, no. 4, pp. 399-428. https://doi.org/10.1007/s10958-015-2573-4

**Approximation of Thin Three-Dimensional Plates with Smooth Lateral Surface by Polygonal Plates.** / Nazarov, S. A.; Chechkin, G. A.

Research output › › peer-review

TY - JOUR

T1 - Approximation of Thin Three-Dimensional Plates with Smooth Lateral Surface by Polygonal Plates

AU - Nazarov, S. A.

AU - Chechkin, G. A.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component ΓNof the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lamé equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ0, thereby for the spatial elasticity problem, we confirm the Babuška paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.

AB - In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component ΓNof the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lamé equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ0, thereby for the spatial elasticity problem, we confirm the Babuška paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.

UR - http://www.scopus.com/inward/record.url?scp=84944698977&partnerID=8YFLogxK

U2 - 10.1007/s10958-015-2573-4

DO - 10.1007/s10958-015-2573-4

M3 - Article

VL - 210

SP - 399

EP - 428

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 4

ER -