### Abstract

Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton-Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh-Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov-Galerkin method.

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

Pages (from-to) | 51-72 |

Number of pages | 22 |

Journal | International Journal of Non-Linear Mechanics |

Volume | 74 |

DOIs | |

Publication status | Published - 1 Sep 2015 |

Externally published | Yes |

### Fingerprint

### Keywords

- General solutions
- Non-classical theory
- Non-linear boundary value problems
- Shallow multi-layer shells

### ASJC Scopus subject areas

- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics

### Cite this

*International Journal of Non-Linear Mechanics*,

*74*, 51-72. https://doi.org/10.1016/j.ijnonlinmec.2015.03.011

**On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections.** / Kirichenko, V. F.; Awrejcewicz, J.; Kirichenko, A. V.; Krysko, A. V.; Krysko, V. A.

Research output: Contribution to journal › Article

*International Journal of Non-Linear Mechanics*, vol. 74, pp. 51-72. https://doi.org/10.1016/j.ijnonlinmec.2015.03.011

}

TY - JOUR

T1 - On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections

AU - Kirichenko, V. F.

AU - Awrejcewicz, J.

AU - Kirichenko, A. V.

AU - Krysko, A. V.

AU - Krysko, V. A.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton-Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh-Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov-Galerkin method.

AB - Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton-Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh-Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov-Galerkin method.

KW - General solutions

KW - Non-classical theory

KW - Non-linear boundary value problems

KW - Shallow multi-layer shells

UR - http://www.scopus.com/inward/record.url?scp=84928732897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928732897&partnerID=8YFLogxK

U2 - 10.1016/j.ijnonlinmec.2015.03.011

DO - 10.1016/j.ijnonlinmec.2015.03.011

M3 - Article

AN - SCOPUS:84928732897

VL - 74

SP - 51

EP - 72

JO - International Journal of Non-Linear Mechanics

JF - International Journal of Non-Linear Mechanics

SN - 0020-7462

ER -