Characterization of the Vibration and Strain Energy Density of a Nanobeam under Two-Temperature Generalized Thermoelasticity with Fractional-Order Strain Theory

Hamzah Abdulrahman Alharthi

doi:10.3390/mca26040078

Abstract

In this work, fractional-order strain theory was applied to construct a novel model that introduces a thermal analysis of a thermoelastic, isotropic, and homogeneous nanobeam. Under supported conditions of fixed aspect ratios, a two-temperature generalized thermoelasticity theory based on one relaxation time was used. The governing differential equations were solved using the Laplace transform, and their inversions were found by applying the Tzou technique. The numerical solutions and results for a thermoelastic rectangular silicon nitride nanobeam were validated and supported in the case of ramp-type heating. Graphs were used to present the numerical results. The two-temperature model parameter, beam size, ramp-type heat, and beam thickness all have a substantial influence on all of the investigated functions. Moreover, the parameter of the ramp-type heat might be beneficial for controlling the damping of nanobeam energy.

Highlights

IntroductionCattaneo’s rule of heat conduction was developed to change the classic Fourier law by including the heat flow and its time scale shift
The mixed heat thermoelasticity principle is composed of two different partial differential equations: the energy conservation equation and the mechanical equation, both of which are based on the Fourier fundamental theorem [1]
This work introduces thermal analysis for a thermoelastic, isotropic, and homogenous nanobeam within the theory of fractional-order strain in the context of the generalized two-temperature thermoelasticity model with one relaxation time theory

Summary

Introduction

Cattaneo’s rule of heat conduction was developed to change the classic Fourier law by including the heat flow and its time scale shift. Sun and Saka formulated the thermoelastic damping vibrations of circular plate resonators using out-of-plane microplates [9] They included a component in their thermoelastic damping formula that is not included in Lifshitz and Roukes’s formula and is based on Poisson’s ratio [10]. Youssef tested, using two different temperatures, the concept of universal twotemperature thermoelasticity He developed principles of dynamic and conductive temperatures, where the difference between them is related to the material’s heat supply [16]. This work introduces thermal analysis for a thermoelastic, isotropic, and homogenous nanobeam within the theory of fractional-order strain in the context of the generalized two-temperature thermoelasticity model with one relaxation time theory. The numerical results for a thermoelastic rectangular nanobeam of silicon nitride were validated in the case of ramp-type heating and simple support

Basic Equations

Problem Formulation

Problem Formulation in the Laplace Transform Domain

The Function of the Strain Energy Density

Numerical Results and Discussion

Conclusions