Fractional order generalised thermoelasticity to an infinite body with a cylindrical cavity and variable material properties

Abstract

This paper is concerned with the determination of thermoelastic displacement, stress and temperature produced in an infinite isotropic elastic body having a cylindrical cavity in the context of the new consideration of heat conduction with fractional order generalised thermoelasticity with Lord–Shulman model (LS model) and Green–Naghdi model with energy dissipation (GN-III model). Here, the elastic parameters and the thermal conductivity are temperature dependent. The boundary of the cavity is subjected to time-dependent thermal and mechanical shocks. The fractional order generalised coupled thermoelasticity theories for the problem are combined into a unified formulation introducing the unified parameters. The governing equations of generalised thermoelasticity theory are obtained in the Laplace transform domain and are solved in that domain by finding out the roots by using the Laguerre’s method. The inversion of the transform solution is carried out numerically by applying a method based on the Fourier series expansion technique. The numerical estimates for thermophysical quantities (displacement, temperature and stress) are obtained for copper-like material for weak, normal and strong conductivity and have been depicted graphically to estimate the effects of the fractional order parameter. The comparison of the results for different theories have been presented and the effects of temperature-dependent parameters are also discussed.

Keywords:

• generalised thermoelasticity
• Lord–Shulman model
• Green–Naghdi model
• cylindrical cavity
• fractional order

Acknowledgements

We are grateful to Prof S.C. Bose of the Department of Applied Mathematics, University of Calcutta, for his valuable suggestions and guidance in preparation of the paper. We also express our sincere thanks to the reviewers for their valuable suggestions for the improvement of the paper.