Modified DPL Green–Naghdi theory for thermoelastic vibration of temperature-dependent nanobeams

Abstract

A unified multi-dual-phase-lag thermoelasticity theory is presented to study the vibration of a temperature-dependent nanobeam subjected to a ramp-type heating. The nonlocal thermoelasticity theory based on Euler-Bernoulli hypothesis is applied. Laplace transform domain is adopted to solve the governing partial differential equations using the state space approach. Numerical computations are carried out using the inverse of Laplace transforms. The present heat conduction and constitutive equations are covering at least five models of the generalized thermoelasticity. Comparison between the classical thermoelasticity (CTE), the Lord–Shulman (L–S), the Green–Lindsay (G–L), and the simple and refined-phase-lag models are made. The effects of nonlocal, ramp-type heating, and temperature-dependent parameters on all quantities have been discussed and presented graphically. It is found that the ramp-type heating parameter has significant effects on all quantities. However, the thermoelastic deflection, axial displacement, dilatation, and bending moment have strong dependencies on the nonlocal and temperature-dependent parameters.