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

Abstract

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.