Modeling of memory-dependent derivative in a functionally graded plate

ABSTRACT

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of heat transport in a functionally graded thick plate in the context of Taylor's series expansion involving memory-dependent derivative for the dual-phase-lag (DPL) heat conduction law, which is defined in an integral form of a common derivative with a kernel function on a slipping interval. The medium is considered as a thick plate, both the surfaces of which is taken to be traction free and the lower surface is subjected to different time-dependent thermal loadings (thermal shock, periodically varying thermal loading and ramp-type heating) while the upper surface is kept at zero temperature. Laplace transform technique is employed to find out the analytical solutions and the inversion of Laplace transform is carried out using a method based on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusion about the new theory is constructed due to the effect of nonhomogeneity. Excellent predictive capability is demonstrated due to the presence of memory-dependent derivative and nonhomogeneity also.

Keywords:

• Memory-dependent derivative
• functionally graded material
• dual-phase-lag thermoelastic model
• Eigen value approach
• Laplace transform
• time delay

Acknowledgements

We are grateful to the reviewers for their valuable suggestions which help us to present the paper in the modified form.

Disclosure statement

No potential conflict of interest was reported by the authors.