### Выдержка

A mathematical model of flexible physically non-linear micro-shells is presented in this paper, taking into account the coupling of temperature and deformation fields. The geometric non-linearity is introduced by means of the von Kármán shell theory and the shells are assumed to be shallow. The Kirchhoff-Love hypothesis is employed, whereas the physical non-linearity is yielded by the theory of plastic deformations. The coupling of fields is governed by the variational Biot principle. The derived partial differential equations are reduced to ordinary differential equations by means of both the finite difference method of the second order and the Faedo-Galerkin method. The Cauchy problem is solved with methods of the Runge-Kutta type, i.e. the Runge-Kutta methods of the 4th (RK4) and the 2nd (RK2) order, the Runge-Kutta-Fehlberg method of the 4th order (rkf45), the Cash-Karp method of the 4th order (RKCK), the Runge-Kutta-Dormand-Prince (RKDP) method of the 8th order (rk8pd), the implicit 2nd-order (rk2imp) and 4th-order (rk4imp) methods. Each of the employed approaches is investigated with respect to time and spatial coordinates. Analysis of stability and nature (type) of vibrations is carried out with the help of the Largest Lyapunov Exponent (LLE) using the Wolf, Rosenstein and Kantz methods as well as the modified method of neural networks. The existence of a solution of the Faedo-Galerkin method for geometrically non-linear problems of thermoelasticity is formulated and proved. A priori estimates of the convergence of the Faedo-Galerkin method are reported. Examples of calculation of vibrations and loss of stability of square shells are illustrated and discussed.

Язык оригинала | Английский |
---|---|

Страницы (с-по) | 635-654 |

Число страниц | 20 |

Журнал | Chaos, Solitons and Fractals |

Том | 104 |

DOI | |

Состояние | Опубликовано - 1 ноя 2017 |

### Отпечаток

### ASJC Scopus subject areas

- Mathematics(all)

### Цитировать

*Chaos, Solitons and Fractals*,

*104*, 635-654. https://doi.org/10.1016/j.chaos.2017.09.008

**Mathematical modelling of physically/geometrically non-linear micro-shells with account of coupling of temperature and deformation fields.** / Awrejcewicz, J.; Krysko, V. A.; Sopenko, A. A.; Zhigalov, M. V.; Kirichenko, A. V.; Krysko, A. V.

Результат исследований: Материалы для журнала › Статья

*Chaos, Solitons and Fractals*, том. 104, стр. 635-654. https://doi.org/10.1016/j.chaos.2017.09.008

}

TY - JOUR

T1 - Mathematical modelling of physically/geometrically non-linear micro-shells with account of coupling of temperature and deformation fields

AU - Awrejcewicz, J.

AU - Krysko, V. A.

AU - Sopenko, A. A.

AU - Zhigalov, M. V.

AU - Kirichenko, A. V.

AU - Krysko, A. V.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - A mathematical model of flexible physically non-linear micro-shells is presented in this paper, taking into account the coupling of temperature and deformation fields. The geometric non-linearity is introduced by means of the von Kármán shell theory and the shells are assumed to be shallow. The Kirchhoff-Love hypothesis is employed, whereas the physical non-linearity is yielded by the theory of plastic deformations. The coupling of fields is governed by the variational Biot principle. The derived partial differential equations are reduced to ordinary differential equations by means of both the finite difference method of the second order and the Faedo-Galerkin method. The Cauchy problem is solved with methods of the Runge-Kutta type, i.e. the Runge-Kutta methods of the 4th (RK4) and the 2nd (RK2) order, the Runge-Kutta-Fehlberg method of the 4th order (rkf45), the Cash-Karp method of the 4th order (RKCK), the Runge-Kutta-Dormand-Prince (RKDP) method of the 8th order (rk8pd), the implicit 2nd-order (rk2imp) and 4th-order (rk4imp) methods. Each of the employed approaches is investigated with respect to time and spatial coordinates. Analysis of stability and nature (type) of vibrations is carried out with the help of the Largest Lyapunov Exponent (LLE) using the Wolf, Rosenstein and Kantz methods as well as the modified method of neural networks. The existence of a solution of the Faedo-Galerkin method for geometrically non-linear problems of thermoelasticity is formulated and proved. A priori estimates of the convergence of the Faedo-Galerkin method are reported. Examples of calculation of vibrations and loss of stability of square shells are illustrated and discussed.

AB - A mathematical model of flexible physically non-linear micro-shells is presented in this paper, taking into account the coupling of temperature and deformation fields. The geometric non-linearity is introduced by means of the von Kármán shell theory and the shells are assumed to be shallow. The Kirchhoff-Love hypothesis is employed, whereas the physical non-linearity is yielded by the theory of plastic deformations. The coupling of fields is governed by the variational Biot principle. The derived partial differential equations are reduced to ordinary differential equations by means of both the finite difference method of the second order and the Faedo-Galerkin method. The Cauchy problem is solved with methods of the Runge-Kutta type, i.e. the Runge-Kutta methods of the 4th (RK4) and the 2nd (RK2) order, the Runge-Kutta-Fehlberg method of the 4th order (rkf45), the Cash-Karp method of the 4th order (RKCK), the Runge-Kutta-Dormand-Prince (RKDP) method of the 8th order (rk8pd), the implicit 2nd-order (rk2imp) and 4th-order (rk4imp) methods. Each of the employed approaches is investigated with respect to time and spatial coordinates. Analysis of stability and nature (type) of vibrations is carried out with the help of the Largest Lyapunov Exponent (LLE) using the Wolf, Rosenstein and Kantz methods as well as the modified method of neural networks. The existence of a solution of the Faedo-Galerkin method for geometrically non-linear problems of thermoelasticity is formulated and proved. A priori estimates of the convergence of the Faedo-Galerkin method are reported. Examples of calculation of vibrations and loss of stability of square shells are illustrated and discussed.

KW - A priori estimates

KW - Coupled thermo-elasticity

KW - Faedo-Galerkin method

KW - Finite difference method

KW - Lyapunov exponents

KW - Micro-shells

KW - Runge-Kutta type methods

KW - Theorem of solution existence

UR - http://www.scopus.com/inward/record.url?scp=85029889603&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029889603&partnerID=8YFLogxK

U2 - 10.1016/j.chaos.2017.09.008

DO - 10.1016/j.chaos.2017.09.008

M3 - Article

AN - SCOPUS:85029889603

VL - 104

SP - 635

EP - 654

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

SN - 0960-0779

ER -