Effect of hyperbolic heat conduction on the linear and nonlinear vibration of CNT reinforced size-dependent functionally graded microbeams

Abstract

As a first attempt, the combined application of the differential quadrature method (DQM) and the Newton–Raphson method is used to solve the hyperbolic (non-Fourier) heat conduction equations to obtain temperature, displacements and nonlinear frequency in the functionally graded (FG) nanocomposite Timoshenko microbeam. To do so, we need to follow two steps: (1): solving the hyperbolic heat conduction to obtain the temperature in the spatial and temporal domains by using DQM and Newton–Raphson method; (2): implementation of the obtained temperature in thermoelastic equations of microbeam to obtain displacements and frequency at each time step by direct iterative method. The material length scale parameter is introduced in the non-classical Timoshenko beam model, to interpret the size effect in microstructures. The material properties of the FG nanocomposite beam are estimated using the Eshelby-Mori-Tanaka approach and carbon nanotubes (CNTs) are randomly distributed within the composite. The nonlinear governing equations and boundary conditions are derived using the Hamilton principle and von Kármán geometric nonlinearity. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the beams. All material properties such as Young modulus (E), heat capacity (Cp), relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of temperature and CNT volume fraction. The effects of temperature change, thermal conductivity, CNTs volume fraction, length to span ratio, heat wave speed, heat flux, and end support conditions on the nonlinear vibration of the beam are discussed in detail. Unlike all previous publications, the present results show that increasing thickness-to-length scale ratio (h/l) will increase the frequency.