Abstract

A high order theory for linear thermoelasticity and heat conductivity of shells has been developed. The proposed theory is based on expansion of the 3-D equations of theory of thermoelasticity and heat conductivity into Fourier series in terms of Legendre polynomials. The first physical quantities that describe thermodynamic state have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby all equations of elasticity and heat conductivity including generalized Hooke's and Fourier's laws have been transformed to the corresponding equations for coefficients of the polynomial expansion. Then in the same way as in the 3D theories system of differential equations in terms of displacements and boundary conditions for Fourier coefficients has been obtained. First approximation theory is considered in more detail. The obtained equations for the first approximation theory are compared with the corresponding equations for Timoshenko's and Kirchhoff-Love's theories. Special case of plates and cylindrical shell is also considered, and corresponding equations in displacements are presented.

Highlights

  • The development of microelectromechanical and nanoelectromechanical technologies extends the field of application of the classical or nonclassical theories of plates and shells towards the new thin-walled structures

  • In the case if only the first two terms of the Legendre polynomials series are considered in the expansion (20) we have the first approximation shell theory which is usually referred to as Vekua’s shell theory

  • Differential operators that appear in the equations of thermoelasticity (41) and heat conductivity (42) for shells of arbitrary geometry are presented in the Appendix Section

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Summary

Introduction

The development of microelectromechanical and nanoelectromechanical technologies extends the field of application of the classical or nonclassical theories of plates and shells towards the new thin-walled structures. An approach based on expansion of the equations of thermoelasticity and heat conductivity into Fourier series in terms of Legendre polynomials has been developed and applied to high order theory of arbitrary geometry shells. For that purpose we expand functions that describe thermodynamic state of elastic body into Fourier series in terms of Legendre polynomials with respect to thickness and find corresponding relations of thermoelasticity and heat conductivity for Fourier coefficients of those. We consider that xα = (x1, x2) are curvilinear coordinates associated with main curvatures of the middle surface of the shell In this case 3-D equations (4)–(8) can be simplified taking into account that Lame coefficients and their derivatives have the form. Order of the system of differential equations depends on assumption regarding thickness distribution of the stressstrain parameters of the shell

Vekua’s Shell Equations
Timoshenko’s Shell Equations
Kirchhoff-Love’s Shell Equations
Comparative Study and Applications
Conclusions
A A u1k A2

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