Model For Thermoelastic Damping Research Articles

Three-dimensional analysis of thermoelastic damping in couple stress-based rectangular plates with nonlocal dual-phase-lag heat conduction

Since thermoelastic damping (TED) plays a decisive role in the functioning of micro/nanoresonators, accurate assessment of its magnitude in these tiny systems is imperative. Successfully simulating the thermomechanical behavior of various mechanical elements requires consideration of heat transfer in all three principal directions. Furthermore, there is no denying the impact of size on the mechanical and thermal domains. In light of these reasons, the current study strives to create a three-dimensional (3D) size-dependent TED model for rectangular micro/nanoplates via the use of the modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat transfer model concurrently for the first time. To accomplish this, by applying the Galerkin method to the 3D NDPL-based heat equation, the relation of temperature field is derived. Moreover, the coupled thermoelastic constitutive relations are extracted by including the couple stress effect. Lastly, by exploiting the obtained relations in the energy dissipation approach, an analytical TED formula in the form of infinite trigonometric series is attained. To appraise the rightness of derived formulation, the method of model reduction is applied. Moreover, an elaborate convergence analysis is implemented to realize the number of sufficient terms of the provided solution that result in well-grounded outcomes. A rigorous simulation study is also accomplished on the role of 3D heat conduction, scale parameters of MCST and NDPL model, boundary constraints, geometry and material in the variations of TED. Findings imply the definite impact of heat conduction dimension on TED value in rectangular plates with a large ratio of thickness to length and width.

Nonlocal dual-phase-lag thermoelastic damping analysis in functionally graded sandwich microbeam resonators utilizing the modified coupled stress theory

Functionally graded (FG) sandwich structures stand as one of the most representative composite structures owing to the thermal resistance and the energy absorption property in non-unfiorm thermal environment. Additionally, accurately estimating thermoelastic damping (TED) is of great importance for the design of high-performance micro/nano-resonators. Nevertheless, the classical TED models fail on the micro/nano-scale structures due to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. The nonlocal heat conduction model and modified coupled stress theory are responsible for the size-dependent effects. To address this issue, present study aims to conduct the size-dependent TED model of FG sandwich microbeam resonators for TED analysis by incorporating the nonlocal dual-phase-lag (NDPL) heat conduction model and the modified coupled stress theory (MCST). It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are formulated, and then, the analytical solution is solved by complex frequency method. Exact and closed-form expressions for TED can be obtained through the complex frequency method. The results are validated, and the parameter effects of the nonlocal thermal parameter, the material length-scale parameter, the power-law index and the vibration modes on the TED are analyzed. The results show that the energy dissipation can be reduced by the size-dependent effects resulting in improving the quality factor of microstructures. It is expected that these results may provide a theoretical basis for predicating TED in the design of FG sandwich micro/nano-resonators with high quality factor in extreme heat transfer environment.

Scale-Dependent Generalized Thermoelastic Damping in Vibrations of Small-Sized Rectangular Plate Resonators by Considering Three-Dimensional Heat Conduction

Given that thermoelastic damping (TED) is one of the main causes of energy dissipation in miniaturized structures, calculation of its exact amount is of particular importance in the design of such elements. Considering three-dimensional (3D) heat transfer along with the scale effect on both mechanical and thermal areas is one of the decisive factors in the more rigorous modeling of TED in small-sized resonators. This paper exploits the modified couple stress theory (MCST) and dual-phase-lag (DPL) heat conduction model to establish an analytical TED relation for rectangular micro/nanoplate resonators with 3D heat conduction. At the start, the non-Fourier heat equation corresponding to 3D DPL model is derived. Then, the solution of temperature distribution is determined by employing infinite trigonometric series. Moreover, the scale-dependent frequency of the system is presented in the context of MCST. Eventually, with the help of entropy generation (EG) method, an analytical TED expression considering 3D heat conduction is established. The precision of the presented solution is examined by comparing it with a simpler model available in the literature. By choosing two common types of boundary conditions, i.e. fully-clamped (CCCC) and fully-simply (SSSS) supported plates, the influence of various factors on TED changes is comprehensively investigated in simulation section. According to the outcomes, in plates with a lower ratio of length and width to thickness, the dimension considered for heat conduction model has a noticeable impact on the variations of TED.

Thermoelastic damping analysis of functionally graded sandwich microbeam resonators incorporating nonlocal and surface effects

Functionally graded (FG) sandwich microstructures/nanostructures stand as one of the most promising composite structures due to the capability to resist thermal shock and high noise in extreme thermal environments. Meanwhile, accurately prediction quality-factor based on thermoelastic damping (TED) is of great significance for the design of high-performance microresonators /nanoresonators. However, the classical TED models fail to explain the size-dependent thermo-mechanical behavior. The nonlocal elasticity and surface elasticity are responsible for the size-dependent phenomenon. In this article, a modified TED model is proposed to estimate the impact of the size-dependent effect on the TED of FG sandwich microbeam resonators in unsteady heat transfer processes by combining the nonlocal elasticity model, surface elasticity model, and dual-phase-lag (DPL) heat conduction model. It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of TED is obtained by the complex frequency method. The influences of the factors including the nonlocal parameter, the surface effect, the power-law index, and the vibration modes on the TED are analyzed. These results provide a reasonable theoretical analysis to predicate TED in the design of FG sandwich microresonators/nanoresonators with high performance.

Thermoelastic dissipation of circular-cross-sectional ring including nonlocal and dual-phase-lagging effects

In the high-frequency region, thermoelastic damping (TED) is essential to ensure the accuracy of micro- or nano-precision sensors or actuators. The size-effect of molecules should be considered in the design and analysis since the atomic volume is not negligible in the nanoscale model. Moreover, a toroidal ring is easy to manufacture, so there is a high possibility that it will be developed in the near future. And thus, the 3-dimensional (3D) TED model of a circular cross-sectional ring is investigated by adopting dual-phase-lagging (DPL) and modified couple stress (MCS) theories based on nonlocal elasticity. Additionally, the size-dependent influences are specified by length-scale parameters in the elastic and thermal fields. Then the TED results are predicted as a quality factor (Q) with the peaks of the spectra and are compared separately by the causes of physical quantity. Moreover, the phenomenon is analyzed in terms of how the properties and length-scale parameters correlate with each other. Finally, it is confirmed that nonlocal elasticity should be considered when the size of the resonator decreases to the scale of the energy-carrying mean-free path. In this regard, this work provides an idea for the analysis of DPL TED of the toroidal ring, including nanoscale effects based on nonlocal elasticity and MCS theories.

Thermoelastic damping in micro-scale T-, U- and Z-shaped frame structures using wave propagation approach

Studying the thermo-mechanical coupling behavior and establishing the thermoelastic damping (TED) model are essential for improving the quality factor of the microelectromechanical resonator. In this work, an analytical TED model for planar frame resonators is first developed using the wave propagation approach. The method is illustrated for typical T-, U- and Z-shaped frame structures and can be extended to more complex frame structures. The frame is viewed as a set of beam segments interconnected by rigid joints, each following the Euler-Bernoulli beam theory. Based on the wave propagation approach, the wave transmission and reflection properties at discontinuities are determined taking into account the flexural and longitudinal motions. The displacement fields of the frame structure are established for both free and forced vibrations. The coupled temperature field of the frame is obtained by solving for the temperature variation arising from the flexural and longitudinal vibrations separately. Finally, the expression of TED is achieved by its energy definition. The validity of the present TED model is confirmed by comparison with finite element (FE) simulations. Numerical results indicate that the TED behavior of frame structures is too complicated to be predicted by previous TED models of simple structures. The often-overlooked longitudinal motion will cause a dramatic TED reduction around specific frequencies. The overall TED is determined by all segments together, and the influence of a segment is reflected by its energy share. Moreover, the effects of different boundary conditions and geometric parameters on TED are evaluated. Practical conclusions to guide the design of high Q-factor resonators are drawn based on the TED behavior of frame structures as summarized from the numerical study.

Thermoelastic damping in rectangular micro/nanoplate resonators by considering three-dimensional heat conduction and modified couple stress theory

Thermoelastic damping (TED) has been confirmed as an intrinsic energy loss and determines the upper limit of the quality factor of micro/nanoelectromechanical system (M/NEMS) resonators. A framework for determining the TED in the full clamped (C-C-C-C), clamped-free-clamped-free (C-F-C-F), and cantilever (C-F-F-F) rectangular microplate resonators operating in the fundamental mode by considering three-dimensional heat conduction in conjunction with modified couple stress theory (MCST) is proposed in this article. Then by using Rayleigh’s method, three analytical models expressed by Fourier series for TED are derived, respectively. The difference between the present TED model and the previous one-dimensional (1-D) MCST model and 3-D classical model are comprehensively investigated. The comparison results indicate that the size effect on TED decreases to be ignorable as the thickness increases to h/l ≥ 50. In the case of h/l ≤ 50, the size effect cannot be ignored. The corresponding critical thicknesses for TED peaks are independent of the size effect. TED behaviors vary significantly under different structural support types.

Thermoelastic damping analysis model of transversely isotropic micro/nano-resonators based on dual-phase-lag heat conduction model and surface effect

Thermoelastic damping (TED) is one of the key factors for lowering the quality factor (Q-factor) of micro/nano-resonators. However, due to a complex small-scale effect and governing equations of non-homogeneous isotropic materials, the existing TED models usually focus on homogeneous isotropic micro/nano-resonators. In this paper, a closed-form TED model is derived to estimate the influence of the small-scale effect on TED of transversely isotropic micro/nano-resonators. The surface effect and the dual-phase-lag model are included to distinguish the influence of mechanical and thermal small-scale effects on TED, respectively. The obtained TED model is theoretically verified. The results indicate that TED values are underestimated if the classical TED model is employed. Moreover, a critical thickness can be determined by the frequency shift curve is proposed. The small-scale effect results in higher TED values within the critical thickness. However, since the small-scale effect has a weak influence on TED, it can be neglected when the resonator thickness is higher than the critical thickness. Additionally, the surface effect plays a dominant role in improving the TED of nano-resonators. In this paper, a more reasonable theoretical approach for estimating TED in transversely isotropic micro/nano-resonators is provided.

Nonlocal dual-phase-lag thermoelastic dissipation of size-dependent micro/nano-ring resonators

The modeling of thermoelastic damping or dissipation (TED) is significant but challenging for the quality-factor estimation of the size-dependent micro/nano-resonator operating in some extreme situations, such as the ultra-instantaneous thermal heating and ultra-high frequency excitation. Considering these special situations, this study intends to explore the more generalized TED mechanism for the micro/nano-ring resonator with the size-dependent effect and non-Fourier heat conduction effect. In this work, one/two-dimensional (1D/2D) TED models of rings are firstly developed by adopting the nonlocal dual-phase-lag (DPL) model and the modified-couple-stress (MCS) model. In detail, the size-dependent effects both in the elastic and thermal fields are respectively characterized by the length-scale parameters lQ and lM, while the effect of non-Fourier heat conduction is characterized by DPL times τQ and τT. In simulations, two typical materials involving silicon and gold popularly manufactured in micro/nano-devices are selected, and the results obtained by previous models are shown for comparison. Firstly, the fluctuation temperature and TED spectra under the MCS size-dependent and nonlocal-DPL effects are investigated. Secondly, the peak phenomenon of TED spectra is analyzed, then the equivalent relaxation times for the 1D case are discussed according to the peak-frequency calculation. Results demonstrate that the MCS size-dependent effect can slightly improve the quality factor of the ring by weakening TED, but its existence should be recognized. Meanwhile, thermal behaviors of solid resonators at micro/nano-scales are jointly affected by the amount relationship of τ1, τQ, τT, and τl. When the characteristic dimension of the resonator is in the level of lQ referring to the mean-free path of energy carries, the nonlocal size-dependent effect is necessary to be considered.

Thermoelastic damping analysis in micro-beam resonators considering nonlocal strain gradient based on dual-phase-lag model

Thermoelastic damping (TED), as a main source of intrinsic energy dissipation, is crucial to the design of the micro/nano-devices and -systems with higher quality factor (Q-factor). However, the classical analysis model of TED fails in micro/nano-scale due to the influence of small-scale effect. The present article focuses on investigating the influence of scale effect on the TED of micro-beam resonators by considering stress nonlocal and higher-order strain gradient effects. Firstly, the governing differential equations are formulated by employing the nonlocal strain gradient theory (NSG) in conjunction with the dual-phase-lag (DPL) heat conduction model. According to the assumption of vibration mode and the boundary conditions, the size-dependent Q-factor expression of TED is derived by the complex frequency method. Finally, the influences of various parameters on the TED of micro-beam resonators, such as nonlocal parameter, length scale coefficient, slenderness ratio and material type, are discussed in detail. And then, the analysis results are compared with the TED of classical thermal-mechanical model. This article provides a novel theoretical analysis model of the TED in micro/nano-meter scale field, which has practical significance in the design of high-efficiency devices and systems.

Thermoelastic damping in rectangular microplate/nanoplate resonators based on modified nonlocal strain gradient theory and nonlocal heat conductive law

Size effect is a significant factor for accurate estimation of the thermoelastic damping (TED) in microplate/nanoplate resonators. The actual TED in microstructures/nanostructures is comprehensively affected by various size-dependent mechanisms in the mechanical and thermal field and studying only one of them may lead to one-sided conclusions. In this article, the governing equations of coupled thermoelasticity in a transversely vibrating rectangular plate are derived by the modified nonlocal strain gradient theory and the nonlocal heat conductive law (GK model). Then utilizing the energy approach, a size-dependent TED model for rectangular plates is presented in the form of infinite series. The constitutive boundary conditions are considered to close the nonlocal strain gradient problem in differential form. The silicon material with typical length scale parameters and relaxation time is selected in simulation. The influences of nonlocal elasticity, strain gradient elasticity and nonlocal heat conduction on TED in rectangular plates are investigated under different frequencies, thicknesses and boundary conditions. The simulation results show that the TED expression converges rapidly and the three size-dependent parameters have distinct influences on TED value, critical thickness and critical frequency. By taking advantage of the size effects reasonably, the quality of resonators can be improved.

A generalized methodology for thermoelastic damping in axisymmetric vibration of circular plate resonators covered by multiple partial coatings

Microresonators used for microelectromechanical systems (MEMS) are becoming more and more complex in terms of structural geometry and mode shapes. Accompanying this increase in complexity is the challenge to develop an accurate analytical method to evaluate the thermoelastic damping (TED). This paper firstly presents a generalized methodology for TED in axisymmetric vibration of circular plate microresonators covered by multiple partial coatings. The frequency and mode shape function of plates are solved by the Kirchhoff–Love plate theory. The thermoelastic temperature fields are acquired by solving the one-way coupled heat conduction along the thickness direction with Green’s function method, accordingly obtaining the effects of partial coating on the microplate. The present model can reduce to that of the fully covered bilayer plates, and is validated by finite element model (FEM). This methodology produces an explicit model for TED in the form of one infinite series with rapid converge, and a simple model is given by retaining the first term. The accuracy and practicality of the simple model are investigated. The results suggest that SiO2 coating with thermally perfect interfaces can reduce the TED at the fundamental mode. However, SiO2 coating significantly increases TED values with thermally imperfect interfaces. Three distinctive peaks of TED spectra in Si/DLC/SiO2 plates are observed. This work can be utilized to optimize the TED in the composite circular microplates and gain more insights into the intrinsic dissipation.

Thermoelastic Damping in Partially Covered Bilayer Microbeam Resonators with Two-Dimensional Heat Conduction

Thermoelastic damping (TED) has been proved as an intrinsic mechanism of energy dissipation in the microelectromechanical systems (MEMS) resonators. However, the previous TED models developed for the fully covered bilayer beam resonators cannot be used for the partially covered cases. This paper firstly derives an analytical TED model for partially covered bilayer microbeams. The bilayer beam performs small-amplitude vibration in pure bending mode, and the mode shape is achieved from the dynamic of Euler-Bernoulli beam. To obtain the coupled temperature field, Green's functions are utilized to solve the heat conduction along thickness and length directions within the framework of Fourier's law. The expression for TED is derived in the form of an infinite series. The present TED model can reduce to that of a fully covered bilayer beam, and matches well with the finite element method (FEM). The behaviors of TED spectrum are investigated comprehensively. Two comparable Debye peaks are noticed at approximately two corresponding critical frequencies. The partial coating greatly reduces the peak values at high critical frequency, but causes an additional TED peak at low critical frequency. The TED peaks of the coating slightly increase as the length increases. The effects of the length and position of the metal coating on the TED at the fundamental frequency are significant. To reduce TED, the metal coating should be located away from the substrate clamped end. This paper provides a developing methodology for controlling TED.

Study of thermoelastic damping in fully clamped bilayered rectangular microplate resonators based on three-dimensional heat conduction

Under the condition that microresonators work at room temperature or vaccum, thermoelastic damping is one of the main mechanisms of energy dissipation. Thermoelastic damping caused by the internal consumption of thermoelastic materials has always prevented the improvement of the quality of microresonators. In this paper, the theoretical model of thermoelastic damping in fully clamped bilayered plate microresonators based on the theory of three-dimensional heat conduction is first established and then verified to be equivalent to the previous single-layer model or not through the formula derivation. Analysis on thermoelastic damping at the first-order frequency where microresonators usually work is carried out afterwards. The differences of thermoelastic damping in the present three-dimensional model with different materials are investigated, including the convergence speed and the value of thermoelastic damping with different thicknesses. Then, with different lengths, widths, and thicknesses, but the same combination of materials, the thermoelastic damping is investigated in the present model. Furthermore, the present bilayered model is compared with the single-layer model to investigate their equivalent relationship. Finally, the present three-dimensional model is compared with the one-dimensional model and FEM models to investigate its feasibility.

Thermoelastic damping in bilayer microbeam resonators with two-dimensional heat conduction

Accurate modeling of thermoelastic damping (TED) is a significant and challenging task in the design of multilayer micro-resonator devices. Several analytical models were proposed to predict TED in bilayer and trilayer microbeams. However, previous models theoretically only considered one-dimensional (1-D) heat conduction in the thickness direction. 1-D model cannot capture the effects of length-to-thickness ratio, structural boundary conditions and flexural-mode order, which is only suitable for the long slender beam. To address these issues, this paper presents an analytical TED model based on the two-dimensional (2-D) heat conduction in the thickness and length directions in a bilayer microbeam with a rectangular cross-section. The expression for TED in the form of two infinite series is more explicit and simpler than others. And two key factors that determine the difference between the 2-D model and the 1-D model are first summarized in this paper. The simulation results of the present model match better with those of the finite element method (FEM) than the previous 1-D model. The shape of TED spectrum is affected by the ratio of the thermal diffusivity of materials. Two peaks occur in the case of ratio of thermal diffusivity higher than 100. However, two peaks will reduce to one as the ratio decreases to 1. The convergence rate of the present model is weaker than the 1-D model.

Single-phase-lag thermoelastic damping models for rectangular cross-sectional micro- and nano-ring resonators

Employing the non-Fourier theory of single-phase-lag (SPL) model, the analytical thermoelastic damping (TED) models for rectangular cross-sectional micro- and nano-ring resonators with heat conduction along the radial thickness direction and the circumferential direction are derived in this paper. Specially, for only considering one-dimensional (1D) heat conduction along the radial thickness direction, the 1D-SPL TED model of the ring derived is consistent with that of the beam reported previously. The dependence of TED behaviors on the single-phase-lag time, geometry dimensions, the modal order, the material type, and the equilibrium temperature is explored. The TED spectra with single- or multiple-peak phenomenon under the effect of single-phase-lag time are discussed, which can be completely characterized by the present models in the series form. The results reveal that the dimensions of heat conduction and the effect of single-phase-lag time have significant influences on TED behaviors of micro- and nano-ring resonators.

Thermoelastic damping in flexural vibration of bilayered microbeams with circular cross-section

Predicting of thermoelastic damping (TED) is crucial in the design of micro-resonators with composite structures. Several analytical models were developed to evaluate TED in bilayered and three-layered microbeams in the past. However, the previous models focus on the microbeams with rectangular cross-section. This paper aims to study the TED in a bilayered microbeam with circular cross-section. The temperature field is approximated by using sine series and Bessel series in the circular cross-section. An analytical full model for TED in flexural vibration of bilayered microbeam is presented in the form of an infinite series. A simple model is also developed by retaining only the first term. The simulation results of the present model have a good agreement with those of the finite element method (FEM). The results indicate that well-resolved two peaks of TED are typically observed in the bilayered microbeams in which the value of k2/C2 is much less or higher than that of k1/C1. The present model is not a rapidly converging infinite series for most of the bilayered microbeams. The present model is more suitable for the long slender beams.

Analysis of thermoelastic damping in the clamped-free microbeam with linearly tapered circular cross-section

An accurate estimation of thermoelastic damping (TED) is of significance and importance in the design for the microbeam resonators with variable cross-section operated in vacuum. TED determines the up limit of quality factor (Q-factor) used for evaluating the microresonator performance. However, in the past, the previous works focus on the TED modelling of uniform microbeam resonators with rectangular cross-section. There is lack of works aiming at the TED modelling of non-uniform microbeam resonators with circular cross-section. In addition, the previous TED model developed by Zener for uniformly circular cross-section beams is not suitable for calculating TED values of the non-uniform beams. Therefore, the TED model for clamped –free microbeam resonators with linearly tapered circular cross-section is meaningful to be derived in this work. The radius and length dependences of TED are investigated. The results of finite element method (FEM) model are calculated for purpose of comparison. Results indicate that for large aspect ratio (l/r0), TED values of linearly tapered microbeams with circular cross-section are larger than those of uniform microbeams, while for small aspect ratio (l/r0), TED values of linearly tapered microbeams are lower than those of uniform microbeams. Moreover, the maximum values of TED called Debye peaks in linearly tapered microbeams are lower than those of uniform microbeams.

Thermoelastic damping in circular cross-section micro/nanobeam resonators with single-phase-lag time

Estimating thermoelastic damping (TED) is a crucial and significant procedure for the design of the high quality-factor micro/nanobeam resonators operated in vacuum. However, most of the previous TED models were derived by employing the classical thermoelasticity theory established on the Fourier heat conduction. These existing models all focused on beam resonators with rectangular cross-section. In this work, employing the generalized thermoelasticity theory of single-phase-lag model, an analytical TED model is derived for circular cross-section micro/nanobeams considering the non-Fourier heat conduction. The influences of non-Fourier effect on TED behaviors and the temperature field of the circular cross-section beam are examined. The single- and multiple-peak phenomena of TED spectrum under the non-Fourier effect are studied. The results indicate that TED in circular cross-section beams is significantly dependent on the equilibrium temperature and the ratio of single-phase-lag time to thermal relaxation time. In addition, the temperature distribution in the beam exhibits distinct differences due to the non-Fourier effect.

Thermoelastic damping in bilayered microbar resonators with circular cross-section

It is always a challenge to determine the Thermoelastic damping (TED) in bilayered microbars precisely. In this paper, a model for TED in the bilayered and cantilevered microbar was proposed, in which the total damping was derived by calculating the energy evanished in each layer. The distribution of temperature in the bilayered microbar with a thermodynamically ideal boundary receiving a time-harmonic force is obtained. An infinite summation for the computing of TED in the bilayered slender microbars under axial loading is presented, and the convergence rate of it is discussed. There are little differences between the results computed by our model and that by finite element method (FEM).