Distinctive features of thermoelastic dissipation in microbeam resonators at nanoscale

In this work, thermoelastic damping in microbeam resonators is evaluated using the generalized thermoelasticity theory based on the dual-phase-lagging thermal conduction model with relaxation between temperature increment and thermal expansion. An explicit formula of thermoelastic damping has been derived. Influences of various affecting factors on thermoelastic damping, such as the beam height, aspect ratio, and relaxation time between temperature increment and thermal expansion, are examined. Numerical results show that the thermoelastic damping, obtained by the generalized thermoelasticity theory in the present study, exhibits distinctive features at nanoscale. This work reveals that non-Fourier thermal conduction and relaxation between temperature increment and thermal expansion may play a nonnegligible role at nanometer scale.     

Effect of phase-lag on thermoelastic vibration of Timoshenko beam

Abstract

An explicit formula of coupled three-phase-lag (TPL) thermoelasticity theory under the Timoshenko beam is constructed for microbeam resonators. The constructed mathematical model is based on the modified couple stress theory which implies a prediction of size-dependent effects in microbeam resonators. By using Hamilton’s principle, governing equations for motion and boundary conditions are derived. The thermal moment and thermal deflection of microbeam resonators are studied analytically and numerically. A comparison of the results between modified coupled stress theory and classical theory is executed for TPL, GN-II, and Lord–Shulman (LS) models. Also, a comparison of the results between TPL, GN-II, and LS models for modified coupled stress theory is done. Besides, the result is presented for silicon microbeam for different aspect ratios and phase-lags. It demonstrated the result corresponding to the behavior of thermoelastic frequencies of microbeam resonators.     

Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Abstract

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.     

Enlarging quality factor in microbeam resonators by topology optimization

The fundamental characteristic of mechanical resonators can be evaluated by resonance eigenfrequency and energy dissipation (or inverse quality factor). Reduced length scales are necessary for achieving high resonance eigenfrequency, which can afford fast response and extraordinary sensitivity to external forces. However, smaller scales will result in higher energy dissipation (or smaller quality factor) in the eigenfrequency of mechanical resonators. A topology optimization process is presented for enlarging quality factor of a microbeam resonator made of linear, isotropic and homogeneous thermoelastic material. The beam’s width is supposed to be considerable so that the plane strain assumption can be satisfied and the design domain is the cross section along the beam’s thickness. By combining the equation of motion and the heat transfer equation, the damping problem of thermoelastic coupling can be solved by a finite-element method. A topology optimization method is used to enlarge quality factor in microbeam resonators. It is found by clamped-clamped and clamped-free microbeam resonators that significant improvements of quality factors can be achieved through this optimization process.     

Thermoelastic analysis of functionally graded porous beam

We present a thermoelastic analysis of functionally graded porous beams under in-plane thermal loading which is applied as uniform temperature distribution over the entire beam. The pore distributions are modeled by a power law and assumed to vary smoothly across the thickness of beam. We consider both saturated and unsaturated pores filled with fluid. The governing equations of porous beam are derived using the variational formulation based on the Timoshenko beam theory. We study the influence of pore’s material properties comparing the results to solutions of homogeneous beams.     

Large-Deflection Effect on Thermoelastic Dissipation of Microbeam Resonators

In real applications, beam resonators in MEMS/NEMS often vibrate beyond the linear regime. The present paper aims to study the effect of large-deflection on thermoelastic dissipation of doubly-clamped microbeam resonators. Detailed formulas are derived for quality (Q-) factor due to thermoelastic dissipation which depends on the amplitude of vibration deflection. Under adiabatic or isothermal surface thermal conditions, the nonlinear effect of large-deflection on thermoelastic dissipation is demonstrated with a comparison to the results based on linearized small deflection vibration. Our results show that thermoelastic dissipation is reduced monotonically with increasing amplitude of vibration deflection under adiabatic surface condition, while thermoelastic dissipation is increased monotonically with increasing amplitude under isothermal surface condition. Under both adiabatic and isothermal surface conditions, the large-deflection effect on thermoelastic dissipation becomes more significant for higher vibration frequencies than lower ones. For the first time to the best of our knowledge, these results reveal that large deflection has a significant effect on thermoelastic dissipation of microbeam resonators and surface thermal condition plays an important role in the large-deflection effect.     

Generalized Thermoelastic Vibration of an Axially Moving Clamped Microbeam Subjected to Ramp-Type Thermal Loading

In this article, the generalized thermoelastic problem of an axially moving microbeam subjected to ramp-type heating is studied. Based on the heat conduction equation containing the thermoelastic coupling term and the classical Euler–Bernoulli thin beam theory, the thermoelastic coupling differential equation of motion of the microbeam is established. The generalized thermoelasticity theory with dual-phase-lags (DPLs) model is used to solve this problem. An analytical technique is used to calculate the vibration of deflections and temperature. The effects of the PLs, the transport speed and the ramp-time parameters on the lateral vibration, temperature, displacement, stress, bending moment, and strain energy density of the microbeam are discussed. Some comparisons have been shown graphically to estimate the effects of the dimensionless speed as well as the time on all the studied fields. The through-the-thickness distributions of all fields are also investigated.     

Effects of phase-lag on thermoelastic damping in micromechanical resonators

The present article attempts to investigate thermoelastic damping (TED) of a microbeam resonator by employing the three-phase-lag (TPL) thermoelasticity theory proposed by Roychoudhuri. An explicit formula of TED has been derived and the effects of the beam height, the phase-lag parameters on TED of the microbeam resonator have been studied. Effects of beam height and phase-lags on TED have been shown with the numerical results. A comparison of the results with the corresponding results of the theory of thermoelasticity of type GN-III is also presented. It has been observed that GN-III predicts a high-quality factor of the resonator’s sensitivity as compared to TPL model.     

Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation

Buckling and post-buckling thermomechanical deformations of a functionally graded material (FGM) Timoshenko beam resting on a two-parameter non-linear elastic foundation and subjected to only a temperature rise have been numerically investigated with the shooting method. The material properties are assumed to vary only in the thickness direction according to a power law function. Through-the-thickness temperature distribution is determined by numerically solving the one-dimensional heat conduction equation. Geometric non-linearities in the strain-displacement relations and the non-linear traction-displacement relations at the interface between the beam and the foundation are considered. For clamped-clamped and immovable simply supported beams, critical values of the ratio of temperatures of the top and the bottom surfaces of the beam for transitions in buckling modes to occur are determined. Post-buckled equilibrium paths and configurations of the heated FGM beam are illustrated for different values of the elastic foundation stiffness parameters, exponent in the power law variation of material properties and the slenderness ratio. Results for the Timoshenko beam are compared with those of the corresponding homogeneous Euler–Bernoulli beam available in the literature.     

Generalized differential quadrature nonlinear buckling analysis of smart SMA/FG laminated beam resting on nonlinear elastic medium under thermal loading

The nonlinear thermal buckling analysis of functionally graded (FG) beam integrated with shape memory alloy (SMA) layer(s), with different lay-up configurations and supported on a nonlinear elastic foundation, has been investigated. The FG layer is graded through the beam thickness direction and thermomechanical properties are assumed to be temperature dependent. The Brinson one-dimensional constitutive law are used to model the characteristics of SMA. The von Kármán strain–displacement fields with the Timoshenko beam theory are applied to the Hamilton’s principle to derive the set of nonlinear equilibrium equations. Generalized differential quadrature method along with direct iterative scheme is utilized to discretize and solve the nonlinear equilibrium equations. The accuracy of proposed model is compared and validated with previous research in literature. The detailed parametric study has been performed to investigate the influence of geometrical, material, and some other key parameters on the nonlinear thermal buckling solutions. The results show that selecting the proper lay-up is of great importance because the type of SMA/FG lay-up can considerably affect the nonlinear buckling solutions. Moreover, adequate application of SMA layers in a proper lay-up configuration significantly postpones the thermal buckling temperature of the beam.     

Effects of Thermoelastic Damping on The Operation of The Ring Shape Anchored Contour Mode Disk Resonators

Thermoelastic damping is one of the most important energy loss mechanisms in MEMS resonators especially when the resonator is miniaturized to achieve higher frequencies. Based on thermal energy produced as the result of the resonator expansion and contraction, by considering the thermoelastic coupled equation, this article presents a solution to thermoelastic damping for previously demonstrated ring shape anchored contour mode disk RF MEMS resonator. This research proves that the thermoelastic damping has a negligible effect on the quality factor of the resonator. In addition the results mention that this effect becomes stronger when the surface to volume ratio is decreased for achieving higher frequency resonators. Obtained results reveal that the resonator could be utilized for ultra-low far-from-carrier phase noise oscillators.     

Thermoelastic damping of graphene nanobeams by considering the size effects of nanostructure and heat conduction

Thermoelastic damping of nanobeams by considering the size effects of nanostructure and heat conduction is studied herein. The size effect of nanostructure is investigated based on Euler–Bernoulli beam assumptions in the framework of nonlocal strain gradient elasticity, and the size dependence of heat conduction is taken into account by incorporating phase-lagging and nonlocal effects. Closed-form solutions of thermoelastic damping and quality factor characterized by thermoelastic coupling are derived. Graphene nanoribbon is chosen as a nanobeam. The effects of relaxation time, aspect ratio, elastic modulus, thermal expansion, and thermal conductivity on quality factor of graphene nanobeams are discussed in detail.     

Thermoelastic Damping and Frequency Shift in Micro/Nanoscale Anisotropic Beams

The governing equations of flexural vibrations in a transversely isotropic thermoelastic beam are derived in closed form based on Euler–Bernoulli theory. The out-of- plane vibrations have been studied under different beam dimensions and boundary conditions. The analytical expressions for thermoelastic damping and frequency shift of vibrations are obtained. The damping and frequency shift of beam vibrations significantly depend on thermal relaxation time and surface conditions at resonance. The expressions for displacement and temperature fields in the beam resonator are obtained. Some numerical results with help of MATLAB software have been computed and presented graphically for silicon material beams.     

Thermoelastic Damping in Nanomechanical Resonators Based on Two-Temperature Generalized Thermoelasticity Theory

This work is concerned with the study of the thermoelastic damping of nanobeam resonators in the context of the two-temperature generalized thermoelasticity theory. An explicit formula of thermoelastic damping has been derived. Influences of the beam height, the relaxation time parameter, the two-temperature parameter and the isothermal value of frequency have been studied with some comparisons between the Biot model and Lord–Shulman model (L–S). Numerical results show that the values of thermal relaxation parameter and the two-temperature parameter have a strong influence on thermoelastic damping in nanoscales.     

Evaluation of thermoelastic damping in micromechanical resonators with axial pretension: An analytical model accounting for two-dimensional thermal conduction

It has been reported that application of tensile axial stress can simultaneously increase quality factor and resonant frequency for micromechanical resonators. In this study, we formulate an analytical model for evaluating thermoelastic damping in micromechanical resonators based on the thermal energy method, in which thermal conductions in both thickness direction and axial direction are considered. An explicit expression for thermoelastic damping in the form of infinite series has been obtained. The proposed analytical model is further validated by finite element analysis. Results of the present study demonstrate that the 2D model needs to be adopted in order to accurately evaluate thermoelastic damping of micromechanical resonators with axial pretension. In addition, the 2D model proposed in the present study eliminates the inherent inconsistency entailed in the 1D model.     

Coupled Thermoelasticity of Functionally Graded Beams

This paper presents the finite element solution of an Euler–Bernoulli beam with functionally graded material (FGM) subjected to lateral thermal shock loads. The FGM beam is assumed to be graded across the thickness. The material properties across the thickness direction follow the volume fraction of the constitutive materials in power law form. The solution is obtained under coupled thermoelastic assumption. The equation of motion and the conventional coupled energy equation are simultaneously solved to obtain the transverse deflection and temperature distribution in the beam. The governing partial differential equations of the problem are solved simultaneously using the Galerkin finite element method with the C ¹-continuous shape function leading to fast convergence of the solution. Results are presented for different power law indexes and coupling coefficients for simply supported boundary conditions. The results are verified with those reported in the literature.     

Axisymmetric and One-Dimensional Thermoelastic Fields of Radially Graded Bodies

In this paper, both Young's modulus and Poisson's ratio along with thermal expansion coefficient are allowed to vary across the radius in a solid ring and a curved beam. Effects of non-constant Poisson's ratio on the thermoelastic field in these graded axisymmetric and one-dimensional problems are studied. A governing differential equation in terms of stress function is obtained for general axisymmetric and one-dimensional problems. Two linearly independent solutions in terms of hypergeometric functions are then attained to calculate the stresses and the strains. Using Green's function method, a form of a solution for the stress functions in terms of integral equations for a curved beam and a solid ring are obtained. Specifically, closed form solutions for the stress functions, when Young's modulus and Poisson's ratio are expressed as power law functions across the radius, are calculated. The results show that the effect of varying Poisson's ratio upon the thermal stresses is considerable for the solid ring. In addition, a non-constant Poisson's ratio has significant influences on the thermal strain field in solid rings. The effect of varying Poisson's ratio upon the thermal stresses is negligible for the curved beam. However, non-constant Poisson's ratios have substantial effects on the thermal strain field in curved beams. Finally, the effects of varying Poisson's ratio on the thermal stresses in thick solid rings and curved beams are also investigated.     

Thermoelastic Buckling Analysis of Functionally Graded Circular Plates Integrated with Piezoelectric Layers

Thermal buckling of circular plates made of functionally graded materials with surface-bounded piezoelectric layers are studied. The material properties of the FG plates are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituent materials. The general thermoelastic nonlinear equilibrium and linear stability equations for the piezoelectric FG plate are derived using the variational formulations. Buckling temperatures are derived for solid circular plates under uniform temperature rise, nonlinear and linear temperature variation through the thickness for immovable clamped edge of boundary conditions. The effects of piezo-to-host thickness ratio, applied actuator voltage, boundary condition, and power law index of functionally graded plates on the buckling temperature of plate are investigated. The results are verified with the data in literature.     

THERMOELASTIC ANALYSIS OF THICK-WALLED FINITE-LENGTH CYLINDERS OF FUNCTIONALLY GRADED MATERIALS

ABSTRACT

A semianalytical thermoelasticity solution for thick-walled finite-length cylinders made of functionally graded (FG) materials is presented. The governing partial differential equations are reduced to ordinary differential equations using Fourier expansion series in the axial coordinate. The radial domain is divided into some virtual subdomains in which the power-law distribution is used for the thermomechanical properties of the constituent components. Imposing the necessary continuity conditions between adjacent subdomains, together with the global boundary conditions, a set of linear algebraic equations are obtained. Solution of the linear algebraic equations yields the thermoelastic responses for each subdomain as exponential functions of the radial coordinate. Some results for the stress, strain, and displacement components through the thickness and along the length are presented due to uniform internal pressure and thermal loading. Based on the results, the gradation of the constitutive components is a significant parameter in the thermomechanical responses of FG cylinders.