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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. [ABSTRACT FROM AUTHOR]

The purpose of current research is to establish a theoretical size-dependent framework for estimating the amount of thermoelastic damping (TED) in miniaturized rings with circular cross section via the nonlocal dual-phase-lag (NDPL) generalized thermoelasticity theory, as one of the most exhaustive non-Fourier heat conduction models. The method used to achieve this goal is based on the definition of TED in the energy dissipation (ED) approach. To do so, after deriving heat equation in the context of NDPL model, the distribution of temperature all over the volume of the ring is specified. Then, the relations of dissipated thermal energy and strain energy in the ring are obtained. Lastly, by employing ED approach, a formula comprising the nonclassical parameters of NDPL model is rendered to determine TED value in small-scaled toroidal rings. By presenting various numerical examples, the dependence of TED on influential parameters including nonclassical thermal constants in NDPL model, ring geometry, vibrational mode number and ring material is surveyed. The Findings enlighten that the inclusion of nonlocal parameter into the model can have conspicuous impacts on TED value, especially in smaller rings or higher vibrational mode numbers. The results also reveal that among the investigated materials (i.e. silicon, copper, silver, and lead), rings made of lead and silicon exhibit the maximum and minimum TED values, respectively. [ABSTRACT FROM AUTHOR]

Precise estimation of the amount of thermoelastic damping (TED) in small-sized resonators is one of the crucial issues in their optimal design. According to several experimental, numerical and theoretical findings, the behavior of structures with such small dimensions in structural and thermal domains does not follow classical elasticity theory and the Fourier law of heat conduction. The objective of this work is to develop a nonclassical formulation for computing TED in asymmetric vibrations of circular nanoplates according to nonlocal theory (NT) and Moore-Gibson-Thompson (MGT) heat transfer model. To do so, first, the coupled equations of motion and heat are derived in the purview of NT and MGT model. Then, the real and imaginary parts of the system frequency are extracted from the frequency equation affected by thermoelastic coupling. Finally, by means of the existing definition for TED, a mathematical expression including nonlocal parameter of NT and nonclassical constants of MGT model is obtained to make an estimate of TED value. In the numerical examples section, to ensure the credibility of the provided formulation, the results obtained through this model are compared with those reported in the literature in the context of simpler models. Then, by presenting various graphical data, the type and extent of the role of different factors in TED changes are appraised. The obtained results indicate that employing NT and MGT model can substantially impact the amount of TED compared to the estimations of the classical formulation. Furthermore, the comparison of results reveals that the magnitude of TED in asymmetric vibration modes surpasses that in symmetric ones, and consequently incorporating asymmetric vibration modes is crucial for thorough analysis of TED in circular nanoplates. Thus, the formulation presented in this article can aid in achieving an optimal design for such plates, particularly in terms of minimizing energy loss. [ABSTRACT FROM AUTHOR]

It has been proven that mechanical elements display size-dependent behavior in structural and thermal fields at microscales. It has also been found that thermoelastic damping (TED) is one of the dominant reasons in confining the quality factor (Q-factor) of such elements. This paper aims to develop a novel formulation for evaluating TED in microbeams by accounting for the size effect on the mechanical and thermal areas via the nonclassical theory of modified strain gradient (MSG) and the non-Fourier heat conduction model of Moore-Gibson-Thompson (MGT). In the first step, the heat equation for beams is derived within the framework of MGT model. Through this equation, the function of temperature fluctuation can be obtained. Then, the constitutive relations of the beam according to MSG theory (MSGT) are extracted. By using the temperature distribution and nonclassical constitutive relations obtained, the maximum amounts of potential and wasted thermal energies during one cycle of beam vibration are calculated. Finally, by placing the value of these energies in the existing relationship for computing the value of TED, an explicit expression for TED is presented. With the aim of clarifying the sensitivity of TED value to the characteristic parameters of MSGT and MGT model, a variety of numerical data are provided. According to the obtained outcomes, the inclusion of size effect in the structural and thermal equations can cause a remarkable difference compared to the classical model. The dependency of TED on some factors like beam thickness and aspect ratio, vibration mode number and material of the beam is also investigated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study the forced flexural vibrations caused by time-harmonic concentrated loads on a transversely isotropic thin rectangular plate (TRP). A mathematical model has been formed using a new modified Green–Naghdi (GN) III theory of thermoelasticity by introducing higher-order memory-dependent derivatives (MDD). The memory-dependent derivatives represent the memory effect (i.e., the instantaneous rate of change depends on the previous state). This study not only considers the size effect but also studies the memory, mechanical, and thermal-field effects. With the help of the double finite Fourier-transform technique (DFFT), the expressions for lateral deflection, thermoelastic damping, temperature distribution, frequency shift, and thermal moment, have been found in the transformed domain for simply supported (SS) TRP. Also, we have graphically demonstrated the effect of the MDD kernel function on the lateral deflection, thermoelastic damping, temperature distribution, frequency shift, and thermal moment. [ABSTRACT FROM AUTHOR]

The micro disk resonator gyroscope is a micro-mechanical device with potential for navigation-grade applications, where the performance is significantly influenced by the quality factor, which is determined by various energy dissipation mechanisms within the micro resonant structure. To enhance the quality factor, these gyroscopes are typically enclosed in high-vacuum packaging. This paper investigates a wafer-level high-vacuum-packaged (<0.1 Pa) cobweb-like disk resonator gyroscope, presenting a systematic and comprehensive theoretical analysis of the energy dissipation mechanisms, including air damping, thermoelastic damping, anchor loss, and other factors. Air damping is analyzed using both a continuous fluid model and an energy transfer model. The analysis results are validated through quality factor testing on batch samples and temperature characteristic testing on individual samples. The theoretical results obtained using the energy transfer model closely match the experimental measurements, with a maximum error in the temperature coefficient of less than 2%. The findings indicate that air damping and thermoelastic damping are the predominant energy dissipation mechanisms in the cobweb-like disk resonant gyroscope under high-vacuum conditions. Consequently, optimizing the resonator to minimize thermoelastic and air damping is crucial for designing high-performance gyroscopes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The thermoelastic coupling between strain and temperature fields arising out from temperature gradient, while bending of an oscillating structure, leads to irreversible heat conduction, which causes energy dissipation in such vibrating systems. The estimation of thermoelastic damping (TED) in micro- and nano-mechanical resonators is a significant aspect of acquiring a high quality factor (Q-factor). Yang et al. [Couple stress based strain gradient theory for elasticity. Int J Solids Struct. 2002;39(10):2731–2743.] developed the modified couple stress theory (MCST) that involves only one material length scale parameter. The present work attempts to investigate TED in micro/nano-beam resonators utilizing MCST and the recently proposed Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory. Adopting the frequency approach method, we derive the size-dependent expression of the inverse quality factor for evaluating TED in rectangular micro/nano-beam resonators. Furthermore, the impact of the material length scale parameter on TED associated with MCST for silicon micro-beam resonator is investigated in detail by comparing the present results with those predicted by classical continuum theory (CCT). The obtained results are also compared with the existing results for the Lord–Shulman (LS) and Green–Naghdi (GN-III) thermoelasticity theories. Moreover, the effects of other important parameters like beam thickness, beam length, aspect ratio, phase-lag on TED in the present context are highlighted. [ABSTRACT FROM AUTHOR]

This paper introduces a size-dependent model for evaluating thermoelastic damping (TED) in small-scaled rectangular plates, incorporating three-dimensional (3D) heat transfer. Utilizing the modified couple stress theory (MCST) and Moore–Gibson–Thompson (MGT) heat equation, we enhance the thermomechanical analysis accuracy in micro/nano-structures. The model employs MCST to derive size-dependent constitutive relations for rectangular plates, coupled with the MGT model for formulating the 3D heat conduction equation. This approach facilitates the analysis of the 3D temperature field and aids in defining TED using the energy loss method. Consequently, an analytical expression is developed to predict 3D TED in rectangular plate resonators, integrating characteristic MCST length and nonclassical MGT parameters. Comparative analyses with existing studies and a series of simulated numerical results are presented. These simulations primarily focus on contrasting the 3D model with conventional 1D models and examining the effects of implementing MCST and MGT models. Findings reveal that the proposed formulation significantly alters outcomes for very small and relatively thick plates compared to simpler models. This advancement in modeling provides a more precise understanding of TED in micro/nano-plate structures, offering vital insights for their optimal design and application in advanced technological fields. [ABSTRACT FROM AUTHOR]

This paper assesses thermoelastic damping (TED) in circular nanoplates by incorporation of the small-scale effect into structural and thermal domains. The nonlocal elasticity theory and dual-phase-lag (DPL) heat conduction model are exploited for achieving the size-dependent coupled thermoelastic equations. By choosing time-harmonic and asymmetric form for deflection and temperature change, and solving the size-dependent thermoelastic eigenvalue problem, the damped natural frequency of circular nanoplate is extracted. On the basis of the complex frequency approach, an analytical relation, for the description of TED in circular nanoplates, is derived. For different boundary conditions and vibration modes, a comparison study is performed between the size-dependent results and those provided by classical continuum mechanics and heat conduction theories. Outcomes show a discrepancy between results acquired by the size effect on the structure and heat conduction. It is elucidated that how nonlocal and DPL characteristic parameters can represent the size-dependent phenomenon and can affect the magnitude of TED. A comprehensive parametric study is conducted to specify the influence of boundary constraints, vibration mode and the type of material on the amount of energy loss from TED. [ABSTRACT FROM AUTHOR]

The current work investigates the transverse vibration of a piezothermoelastic (PTE) nanobeam in the frame of dual-phase-lag thermoelasticity theory. Closed-form analytical expression for the thermoelastic damping (TED) in terms of quality factor for a homogeneous transversely isotropic PTE beam is derived by using Euler–Bernoulli beam theory and complex frequency approach. The size effect of the nanostructured beam is tackled by applying modified couple stress theory (MCST). Detailed analysis on damping of vibration owing to thermal fluctuations and electric potential in the present context under three sets of boundary conditions is attempted to investigate the influences of two-phase-lag parameters, piezoelectric parameter, thermal effect and size-dependent behaviour on energy dissipation caused by TED in PTE beam resonators. Analytical results are illustrated with the help of graphical plots on numerical findings for lead zirconate titanate (PZT-5A) PTE material. The investigation brings out some significant key findings and observations in view of the present heat conduction model. [ABSTRACT FROM AUTHOR]

AbstractThe present study examines the vibrations in micro- and nano-piezothermoelastic beams with voids. In this work, the modified couple stress theory is employed to generate closed-form formulas for deflection, volume fraction, electric potential and temperature distribution. The effects of voids, modified couple stress, beam dimensions and electric potential on energy dissipation (thermoelastic damping) are examined for beams under various boundary conditions. We have established analytical formulas for frequency shift, attenuation and thermoelastic damping. MATLAB software is used to get numerical results and exhibit them visually. [ABSTRACT FROM AUTHOR]

This paper addresses the exploration of thermoelastic damping (TED) in a microbeam resonator, utilizing the Quintanilla generalized thermoelasticity theory. The study emphasized analyzing the influence of beam height on the microbeam resonators TED and also conducted a thorough assessment of the frequency shift and normalized attenuation in microbeam resonators. These outcomes are then compared with different materials, like silicon (Si) and lead (Pb). The current undertaking demonstrates its utility in tackling challenges associated with science and engineering. [ABSTRACT FROM AUTHOR]

Functionally graded (FG) bi-layer structures have become one of the most promising candidates for micro-devices, which are widely used as high-efficient micro-resonators due to their excellent thermo-mechanical properties. In addition, the design of high performance micro-resonators requires sufficiently accurate analysis of their thermoelastic damping (TED). Nevertheless, the classical analysis model of TED fail on the micro-structures owing to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. To address this issue, present study focuses on investigating the size-dependent TED model of FG bi-layered microbeam resonators for TED analysis by combining the nonlocal dual-phase-lag heat conduction model and the modified coupled stress theory. It is assumed that the FG bi-layered microbeam resonators consist of double FG surfaces. The corresponding governing equation are formulated, and the analytical solution is solved by complex frequency method. The obtained TED model is theoretically verified, and then, 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. This article provides a theoretical analysis model of the TED in FG bi-layered microbeam resonators, which has practical significance in the design of high quality factor devices. [ABSTRACT FROM AUTHOR]

The micro disk resonator gyroscope is a micro-mechanical device with potential for navigation-grade applications, where the performance is significantly influenced by the quality factor, which is determined by various energy dissipation mechanisms within the micro resonant structure. To enhance the quality factor, these gyroscopes are typically enclosed in high-vacuum packaging. This paper investigates a wafer-level high-vacuum-packaged (

AbstractThe extraordinary properties of graphene nanoplatelets (GPLs) have attracted great attention in the thermomechanical behaviors. Additionally, thermoelastic damping (TED), as a dominant intrinsic dissipation mechanism, is a major challenge in optimizing high-performance micro/nano-resonators. However, the classical TED models fail to explain the thermomechanical behavior considering the influences of the small-scale effect and the memory-dependent effect. To fill these gaps, the present study aims to investigate TED analysis of functionally graded (FG) microplate resonators reinforced with GPLs in the frame of the modified strain gradient theory and the fractional dual-phase-lag heat conduction model. Four patterns of GPLs distribution namely the UD, FG-O, FG-X, and FG-A pattern distributions are taken into account, and the effective material properties of the plate-type nanocomposite are evaluated based on the Halpin–Tsai model. The energy equation and the motion equation in the Kirchhoff microplate model are solved, and then, the closed-form analytical expression of TED is obtained by complex frequency technique. A detailed parametric study has been conducted to discuss the influence of the material length-scale parameter, the fractional-order parameter, and the total weight fraction of GPLs on the TED. The results demonstrated that the energy dissipation of FG microplate resonators reinforced with GPLs is determined by the small-scale effect, the memory-dependent effect, and the total weight fraction of GPLs. This article provides a theoretical approach to predict TED in the design of FG microplate resonators reinforced with GPLs with high performance. [ABSTRACT FROM AUTHOR]

The present article pertains to the assessment of thermoelastic damping (TED) in rectangular nanoplates by incorporation of size effect within the constitutive and heat conduction relations. With the purpose of establishing size-dependent coupled thermoelastic equations, nonlocal elasticity theory and dual-phase-lag heat conduction model are utilized. By considering time-harmonic vibrations, nonclassical frequency equation affected by thermoelastic coupling is derived. By solving this equation and extracting the real and imaginary parts of damped natural frequency via some mathematical operations, an explicit expression for description of TED in circular nanoplates is obtained. With the goal of better understanding the small-scale effect on TED, a comparison is made between the results predicted by the size-dependent formulation and those determined according to classical continuum mechanics and heat conduction theories for two kinds of boundary constraints. Considerable difference between classical and nonclassical results, especially at smaller sizes, implies the necessity of incorporating small-scale effect into both structural and thermal areas. [ABSTRACT FROM AUTHOR]

Purpose: The present investigation is devoted to providing two/three-dimensional (2D/3D) models for estimating the amount of thermoelastic damping (TED) in circular cross-sectional micro/nanorings by capturing the effects of size on thermal domain via dual-phase-lag (DPL) heat conduction model. Methods: To achieve the goal of the article, first of all, the equation of heat conduction derived in the framework of DPL model is solved. In this way, for 2D and 3D models of heat propagation, the temperature field in the ring is obtained in the form of infinite series. Next, by exploiting the relation of quality factor in entropy generation (EG) approach, a formulation including the two phase lag parameters of DPL model is extracted to anticipate TED value in small-sized rings with circular cross section. Results: By comparing the results of this investigation with those of studies in the literature that are based on simpler heat conduction models, a validation study is accomplished. An intensive numerical study is also performed to discern the influence of some of the most significant factors such as phase lag parameters of DPL model, vibration mode, the dimensions and ring material on TED. Conclusion: The findings reveal the noticeable effect of phase lag parameters of DPL model on the magnitude of TED in miniaturized circular cross-sectional rings, especially in smaller dimensions and higher vibration modes. [ABSTRACT FROM AUTHOR]

Functionally graded sandwich micro/nano-structures have attracted great attention due to the capability to resist high noise and thermal stress in a non-isothermal environment. Additionally, the design of high quality-factor micro/nano-resonators requires accurate estimation of their thermoelastic damping. However, the classical thermoelastic damping models fail at the micro/nano-scale due to the influences of the size-dependent effects related to heat transfer and elastic deformation. This work aims to investigate the influences of the size-dependent effects on the thermoelastic damping of functionally graded sandwich micro-beam resonators by combining the nonlocal dual-phase-lag heat conduction model and the nonlocal elasticity model. It is assumed that the functionally graded sandwich micro-beam resonators consist of a ceramic core and functionally graded surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of thermoelastic damping is obtained by complex frequency method. Numerical results are analyzed for the effects of the thermal nonlocal parameter, the elastic nonlocal parameter, the power-law index, and the vibration modes on the thermoelastic damping of functionally graded sandwich micro-beam resonators. The results show that the thermoelastic damping of functionally graded sandwich micro-beam resonators can be adjusted by the suitably modified parameters, which strongly depends on the double nonlocal effects and the power-law index. [ABSTRACT FROM AUTHOR]

To investigate the size-dependent behavior of micro-beams with thermoelastic damping (TED), a new size-dependent thermoelastic damping model based on modified gradient elasticity is proposed in this study. The governing equations and boundary conditions are derived based on the modified gradient elasticity theory and the single-phase-lag thermal relaxation model. By utilizing the complex frequency method, the exact expression of thermoelastic damping is obtained from the model. The results indicate that there is a significant size effect of thermoelastic damping in micro-beams, and the critical thickness of the beam is related to the internal length scale. Different thermal parameters have varying degrees of influence on thermoelastic damping. Compared with other size-dependent thermoelastic damping models, the proposed thermoelastic damping model in this paper is characterized by its clear physical interpretation and simple form. Some theoretical guidance for the designing of high-quality micro/nano-electromechanical resonators can be provided by the new model. [ABSTRACT FROM AUTHOR]

Based on the modified Moore–Gibson–Thompson (MGT) model, transversely isotropic visco-thermoelastic material is investigated for frequency shift and thermoelastic damping. The Green–Naghdi (GN) III theory of thermoelasticity with two temperatures is used to express the equations that govern heat conduction in deformable bodies based on the difference between conductive and dynamic temperature acceleration. A mathematical model for a simply supported scale beam is formed in a closed form using Euler Bernoulli (EB) beam theory. We have figured out the lateral deflection, conductive temperature, frequency shift, and thermoelastic damping. To calculate the numerical values of various physical quantities, a MATLAB program has been developed. Graphical representations of the memory-dependent derivative's influence have been made. [ABSTRACT FROM AUTHOR]

Thermoelastic dissipation or thermoelastic damping (TED) can restrict the quality factor of micro/nanoring resonators seriously. This paper employs the non-Fourier model of Guyer–Krumhansl (GK model) to render a size-dependent formulation and analytical solution for approximating the amount of TED in micro/nanorings with circular cross-section by inclusion of nonlocal and single-phase-lagging effects. To fulfill this objective, the equation of heat conduction in the ring is first established according to GK model. Then, by placing the temperature distribution obtained from the heat conduction equation in the TED relation defined on the basis of thermal energy approach, an expression in the form of infinite series is given for TED, which includes non-classical parameters of GK model. Finally, after checking the validity of the model through a comparative study, several simulation results are prepared to emphasize on the influence of different factors such as non-classical parameters of GK model, geometry of ring, vibrational mode and ambient temperature on TED value. Numerical examples reveal that the mentioned factors along with the two- or three-dimensional heat transfer (2D or 3D) model have major influences on TED variations. [ABSTRACT FROM AUTHOR]

As a main source of energy dissipation, thermoelastic damping (TED) cannot be ignored in designing resonators with high-quality factor. However, due to the size-dependent effect and the thermal relaxation, the TED models formulated in the classical theory are no longer able to accurately estimate the energy dissipation in the micro-/nano-resonators. To fill theses gaps, the present work aims at developing a new TED model for micro-plate resonators based on the modified couple stress theory incorporating memory-dependent derivative heat conduction model. The corresponding governing equations are derived, and the analytical solution for the TED is obtained. In order to discover the variation law of TED of micro-plate resonators, the numerical results corresponding to different materials and different theories are illustrated and compared. In calculation, the effects of the material parameters, the boundary conditions, the length scale parameters and the length on the peak value as well as the critical thickness of TED are further studied. It is expected that this novel model may provide a theoretical basis for designing high-performance micro-plate resonators under complex working conditions. [ABSTRACT FROM AUTHOR]

This research tries to render an unconventional model for thermoelastic dissipation or thermoelastic damping (TED) in circular microplates by accommodating small-scale effect into both structure and heat transfer fields. To accomplish this purpose, the modified couple stress theory (MCST) and Guyer−Krumhansl (GK) heat conduction model are utilized for providing the coupled thermoelastic equations of motion and heat conduction. The equation of heat conduction is then solved to acquire the closed-form of temperature profile in the circular microplate. By placing the extracted temperature profile in the equation of motion, the size-dependent frequency equation influenced by thermoelastic coupling is established. By conducting some mathematical manipulations, the real and imaginary parts of damped frequency are obtained. In the next stage, with the help of the description of TED based upon the complex frequency (CF) approach, an explicit single-term relation consisting of structural and thermal scale parameters is derived for making a size-dependent estimation of TED value in circular microplates. For evaluating the precision and veracity of the proposed model, the results obtained through the presented solution are compared with the ones available from the literature. In addition, by way of several examples, the pivotal role of length scale parameter of MCST and thermal nonlocal parameter of GK model in the magnitude of TED is assessed. Various numerical results are also given to place emphasis on the impact of some parameters such as boundary conditions, geometrical features, material and ambient temperature on TED value. The formulation and results provided in this study can be used as a benchmark for optimal design of microelectromechanical systems (MEMS). [ABSTRACT FROM AUTHOR]

It has been confirmed that structures with micro dimensions display size-dependent thermomechanical behaviors. Moreover, according to the findings of empirical and theoretical researches, thermoelastic damping (TED) has been recognized as one of inescapable causes of energy dissipation in microstructures. The current article is an effort to provide a novel size-dependent framework for approximating the amount of TED in microring resonators with rectangular cross section. To include size effect into structural and thermal constitutive relations, the modified couple stress theory (MCST) and the Moore–Gibson–Thompson (MGT) heat equation are utilized, respectively. By solving the coupled heat equation in the purview of MGT model, the fluctuation temperature throughout the ring is determined. By employing the obtained temperature distribution and constitutive relations of MCST, the peak values of strain and wasted thermal energies during one cycle of vibration are computed. Based on the description of TED in the energy dissipation (ED) method, a mathematical expression containing the scale parameters of MCST and MGT model is derived for estimating TED value. To ensure the correctness and veracity of the established solution, a comparative study is carried out on the basis of the data released by other researchers for more plain models. A section is also designated for an all-out study to ascertain the association between TED spectrum and some influential factors like scale parameters of MCST and MGT model, vibration mode number, one-dimensional (1D) and two-dimensional (2D) heat conduction, geometry and material. The extracted data enlighten that the impact of applying MCST and MGT model on TED has a close relationship with the vibration mode number of the ring. [ABSTRACT FROM AUTHOR]

The In this paper, we study the forced flexural vibrations caused by time-harmonic concentrated loads on a transversely isotropic thin rectangular plate (TRP) with a memory-dependent derivative (MDD).This study not only considered the size effect but also studied the memory, mechanical, and thermal field effects. If only one of them is studied, one-sided conclusions may result. we develop a closed-form mathematical model for the thin plate based on the Green–Naghdi (GN) III theory and non-local generalized Kirchhoff's love plate theory of thermoelasticity using MDD. With the double finite Fourier transform technique, expressions for lateral deflection, thermoelastic damping, temperature distribution, frequency shift, and thermal moment, have been found in the transformed domain for simply supported (SS) TRP. We have demonstrated the effectiveness of the MDD kernel function on the resultant quantities. [ABSTRACT FROM AUTHOR]

Thermoelastic damping (TED) has been discerned as a definite source of intrinsic energy loss in miniaturized mechanical elements. The size-dependent structural and thermal behavior of these small-sized structures has been proven through experimental observations. As a first attempt, this article exploits nonlocal strain gradient theory (NSGT) and nonlocal dual-phase-lag (NDPL) heat conduction model simultaneously to acquire a mathematical formulation and analytical solution for TED in nanobeams that can accommodate size effect into both structural and heat transfer fields. For this purpose, the coupled equations of motion and heat conduction are firstly extracted via NSGT and NDPL model. Next, by deriving the distribution of temperature from heat conduction equation and substituting it in the motion equation, the unconventional thermoelastic frequency equation is established. By deriving the real and imaginary parts of the frequency from this equation and employing the definition of quality factor, an explicit solution is given for approximating TED value. The veracity of the proposed model is checked by comparing it with the solutions reported in the literature for specific and simpler cases. A diverse set of numerical results is then presented to appraise the influence of some factors like structural and thermal nonlocal parameters, strain gradient length scale parameter, geometrical parameters, mode number and material on the amount of TED. According to the results, use of NDPL model yields a smaller value for TED than DPL model, but prediction of NSGT about the magnitude of TED, in addition to the relative amounts of its two scale parameters, strongly depend on other factors such as aspect ratio, vibration mode and material type. [ABSTRACT FROM AUTHOR]

In this paper, the steel internal damping due to both the thermoelastic and the magnetoelastic phenomena has been investigated through a formulation based on thermodynamical potential joints with a hysteretic damping model. With the aim of focusing on the temperature transient in the solid, a first configuration has been considered, which is characterized by a steel rod with an imposed alternating pure shear strain in which only the thermoelastic contribution was studied. The magnetoelastic contribution was then introduced in a further configuration, in which a steel rod in free motion was subjected to torsion on its ends in the presence of a constant magnetic field. A quantitative assessment of the influence of the magnetoelastic dissipation in steel has been computed according to the Sablik-Jiles model by giving a comparison between the thermoelastic and the prevailing magnetoelastic damping coefficients. [ABSTRACT FROM AUTHOR]

Thermoelastic damping has emerged as a critical issue in the modeling and design of micro and nanomechanical systems. Therefore, an extensive research interest is being devoted towards the reduction in thermoelastic damping for micro and nanomechanical systems. The present study intends to examine thermoelastic damping in nanobeam resonators using the modified couple stress theory and Eringen's nonlocal elasticity theory, thus so-called modified nonlocal couple stress theory within the context of the recently proposed Moore-Gibson-Thompson thermoelasticity theory. In order to observe size effects, the size-dependent coupled thermoelastic equations are derived by combining modified couple stress theory and nonlocal elasticity theory in the frame of Moore-Gibson-Thompson thermoelasticity. The coupled governing equations for thermoelastic damping of nanobeam resonators are solved analytically in terms of the inverse quality factor. The size-dependent thermoelastic damping of nanobeams is presented graphically with the help of numerical results predicted by modified nonlocal couple stress theory. The results obtained under the combined effects of modified couple stress theory and nonlocal theory in the present context are further compared to those of the classical, modified couple stress, and nonlocal elasticity theories as special cases of the modified nonlocal couple stress theory. It is observed that thermoelastic damping weakens at the submicron scale under Eringen's nonlocal elasticity theory, whereas it becomes greater at the submicron scale under modified couple stress theory. However, at the submicron scale, the modeling of nanobeam resonators using modified nonlocal couple stress theory reveals a smaller amount of thermoelastic damping than modified couple stress, nonlocal, and classical theories. In addition, as compared to the Green-Naghdi thermoelasticity theory of type III (GN-III), the Moore-Gibson-Thompson model predicts a higher thermoelastic damping value, while agreeing with the results predicted by the Lord-Shulman (LS) model under modified nonlocal couple stress theory. Some important points highlighted here are believed to be useful in the design of mechanical resonators at micron and submicron scales. [ABSTRACT FROM AUTHOR]

• An analytical model is developed to evaluate thermoelastic damping in laminated microring. • More items are needed to converge when values of ∆ E for the layers are quite different. • The temperature gradient in the circumferential direction has little effect on the TED. • Increasing thickness of metallic coating will increase TED of laminated microring resonators. Predicting thermoelastic damping (TED) is crucial in the design of micro-resonators with high quality factors. Until now, some analytical models have been developed to evaluate the TED effect on microrings which are widely used in many micro-resonators. However, most previous works are limited to microrings with single homogeneous material. This paper develops an analytical model for investigating the TED in the laminated microrings. The temperature fields along the thickness and circumferential directions in the laminated microring are presented using the generalized orthogonal expansion technique. The analytical model for the TED is obtained by computing the dissipated energy and the maximum elastic energy in each layer. The present analytical model for evaluating the TED is validated by comparing with the previously published result and the finite element method (FEM). The effects of the thickness and the property of the metallic layer on the TED characteristics of two- and three-layered microrings are analyzed. The present model can be used to optimize the design of the microring resonators with high quality factors. [ABSTRACT FROM AUTHOR]

• Two generalized models of thermoelastic damping (TED) in micro/nano-rings are derived. • Temperature varies with the nonlocal single-phase-lag (SPL) effect of heat conduction. • Frequency-shift ratio for micro/nano-rings under the nonlocal-SPL effect is investigated • TED at high frequencies is affected by the nonlocal-SPL effect. • Equivalent relaxation times for rings under the nonlocal-SPL effect are discussed. Estimating thermoelastic damping (TED) and frequency shift with sufficient accuracy is of importance for the quality factor prediction of micro/nano-resonators operating in certain special cases containing the ultra-high frequency excitation and the ultra-instantaneous thermal heating etc. At the design stage before the engineering application of micro/nano-resonators, the mathematical modelling of the frequency-shift ratio and TED gets challenging when considering the non-Fourier effect and size-dependent effect in the thermal field. In this paper, one/two-dimensional (1D/2D) analytical formulations of frequency-shift ratio and TED for micro/nano-ring resonators are firstly developed by adopting the nonlocal single-phase-lag (SPL) model. Basing on the present proposed models, the nonlocal thermal-relaxation time τ N is firstly introduced in terms of the nonlocal thermal-length l Q and the thermal diffusion coefficient χ. In simulation cases, the typical semiconductor material silicon is selected, the SPL time τ Q is computed basing on the second sound, and l Q refers to the mean-free path of energy carriers. The convergence performance of the proposed models is analyzed. Meanwhile, the results of existed TED models for rings are presented for comparison. The impacts of the nonlocal-SPL effect on the TED spectrum and fluctuation temperature are examined emphatically. And the peak phenomenon of TED spectrum characterized by the peak damping, the equivalent relaxation time associated with the peak frequency, and the frequency-shift ratio are explored. Results demonstrate that the thermo-dynamical behaviors of micro/nano-ring resonators are conjunctively determined by the amount relationship of τ Q , τ N , and the thermal relaxation time. Especially, when the radial thickness of the ring operating at ultra-high frequencies is comparable to l Q , the nonlocal-SPL effect of heat conduction should be considered to estimate the frequency-shift ratio and TED. [ABSTRACT FROM AUTHOR]

In this paper, thermoelastic dissipation or thermoelastic damping (TED) in micro/nanorings with rectangular cross-section is examined by accounting for small-scale effect on both structural and thermal areas. The modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat conduction model are exploited for incorporating size effect within constitutive relations and heat conduction equation. By employing simple harmonic form for asymmetric vibrations of the miniaturized ring and solving the heat conduction equation, for one-dimensional (1D) and two-dimensional (2D) cases of heat transfer, the solution of temperature distribution in the ring is extracted in the form of infinite series. By employing the definition of TED on the basis of entropy generation approach, an analytical relation in the series form containing structural and thermal scale parameters is established to estimate TED value. To appraise the precision and validity of the developed solution, a comparison study is performed by utilizing the outcomes of researches published in the literature. An exhaustive parametric study is then conducted to ascertain the role of structural and thermal scale parameters in the magnitude of TED. The influence of some key parameters such as vibration mode, geometrical properties, directions of heat conduction (1D and 2D model) and material on TED is also addressed. [ABSTRACT FROM AUTHOR]

Based on the modified Moore–Gibson–Thompson (MGT) model, transversely isotropic visco-thermoelastic material is investigated for frequency shift and thermoelastic damping. The Green–Naghdi (GN) III theory of thermoelasticity with two temperatures is used to express the equations that govern heat conduction in deformable bodies based on the difference between conductive and dynamic temperature acceleration. A mathematical model for a simply supported scale beam is formed in a closed form using Euler Bernoulli (EB) beam theory. We have figured out the lateral deflection, conductive temperature, frequency shift, and thermoelastic damping. To calculate the numerical values of various physical quantities, a MATLAB program has been developed. Graphical representations of the memory-dependent derivative’s influence have been made.

The analysis of free vibrations of a thermoelastic hollow sphere with voids is investigated under dual-phase-lag generalized thermoelasticity and Erigen's nonlocal elasticity. By implementing the stress-free thermal boundary conditions, the frequency relation is obtained for possible modes in a compact form. Numerical iteration technique with the help of MATLAB is used for generating numerical data to determine free vibrations. The analytical results are presented graphically with the help of numerical computations and simulations. At the end, the frequency shift, and the thermoelastic damping with and without voids are presented graphically for various cases of interest and analyzed the results. [ABSTRACT FROM AUTHOR]

The thermoelastic damping in a micro-beam of rectangular cross-section is studied using the Euler–Bernoulli beam theory incorporating Lord and Shulman's theory of generalized thermoelasticity with memory-dependent heat conduction. By employing the mode analysis method, explicit formulae for the thermoelastic damping, frequency shift and attenuation are derived. The kernel function, the relaxation time, and the memory time are selected to analyze their effects on thermoelastic damping. The dependence of the inverse quality factor Q − 1 on the boundary conditions, vibration modes, time delay, aspect ratios, and relaxation time of small-scale mechanical resonators is shown for the memory-dependent heat conduction with ideal, linear, and nonlinear kernels. Numerical results of the Q − 1 factor are calculated and compared with those based on the Lord–Shulman generalized thermoelasticity and the Lifshitz–Roukes model. The obtained numerical results show that the memory-dependent Lord–Shulman model is more accurate in describing thermoelastic damping than Lifshitz–Roukes' model and the classical Lord–Shulman model. [ABSTRACT FROM AUTHOR]

With the extensive applications of the micro/nano-electro-mechanical systems, a great deal of attention has been paid to explore the influences of the arising size-dependent effect and surface effect on their performance. Along with these effects, the nonlocal elasticity, strain gradient theory, and surface elasticity theory were successively proposed. Moreover, it is inevitable for MEMS/NEMS to work in a variable temperature environment, correspondingly, the thermal-induced stress or deformation becomes another issue that needs serious concern. Thermoelastic damping (TED), as a main energy dissipation source, is a major challenge in designing higher-quality factor resonators. However, considering the thermoelastic coupling effect, the aforementioned theories are not capable enough to depict the thermoelastic performance of micro/nano-resonators. To amend this deficiency, based on nonlocal strain gradient theory, a new model for assessing the TED of nanobeam resonators by incorporating the thermal relaxation effect and the surface effect is developed. The corresponding governing equations for the Euler–Bernoulli microbeam resonator model are formulated, and then, solved by means of a complex frequency method. The influences of the nonlocal parameter, strain gradient coefficients, aspect ratio, surface stress and surface elasticity on the TED are examined and discussed in detail. [ABSTRACT FROM AUTHOR]

High-Q micro-beam resonators have been widely applied in communication fields such as narrowband filters, but thermoelastic damping restricts the further improvement of quality factor. Traditional method such as overall size optimization used for thermoelastic quality factor (QTED) improvement will cause the resonant frequency of the resonator change greatly, which will not be allowed in narrowband filters. In this article, three slots are introduced into a beam resonator to greatly reduce the thermoelastic damping by rationally arranging the length ratio of the central slot and the side slot, while the resonant frequency is basically kept unchanged. The feasibility is verified by parametric finite element simulation method. The results indicate that without changing the overall size of the micro-beam resonator, the resonant frequency remains unchanged and the QTED of the resonator can be doubled over twice both in the isothermal region and adiabatic region by placing side slots of 10% and a central slot of 30%. The proposed thermoelastic damping suppression method shows potential applicability for MEMS resonators in high-order mode of vibration. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Polymers are attractive materials for the fabrication of micro-electro-mechanical system (MEMS) microphones because of their low Young’s moduli, generally lower processing temperatures than silicon, and simple fabrication; such microphones are sensitive and economical. However, an earlier polymer diaphragm revealed lower sensitivity than expected. In addition, the viscoelastic properties of polymer diaphragm have been considered to degrade the sensing performance due to a high-level mechanical noise. Here, we investigated the possibility of the polymer material as a MEMS microphone in terms of stiffness and damping. Stiffness was analyzed and experimentally verified. The obtained polymer diaphragm damping coefficient was compared with the coefficient of a silicon MEMS microphone to explore whether performance was degraded by material damping. All samples were fabricated using a thin film transfer method. SU-8 was mainly used but Polymethyl methacrylate (PMMA) was also selected for comparing the material damping effect. The measured residual stress of an SU-8 diaphragm was 18 MPa; the measured flexibility was similar to the state-of-art of a silicon-based MEMS microphone, and it was proven that there is room for further improvement through process optimization. The damping effect induced by viscoelasticity was very small, compared with the effect of air damping; indeed, the intrinsic damping effect of the polymer on device performance was negligible. [2021-0213] [ABSTRACT FROM AUTHOR]

This paper aims to inspect the nonlocal Erigen's elasticity model for free vibrations of thermoelastic diffusive sphere in the reference of dual-phase-lag (DPL) model of generalized thermoelasticity. The effect of non-locality and different models of generalized thermoelasticity, i.e. dual-phase-lag (DPL), Lord–Shulman (LS) models and coupled thermoelasticity (CTE) in the presence and absence of diffusion, has been taken under investigation. Due to time harmonic vibrations, the constitutive relations and governing equations are transformed into ordinary differential equations. By applying the stress free thermal boundary conditions, the frequency equation has been taken under investigation for the possible modes of vibrations in compact form. To discover free vibration analysis from frequency equations, the numerical functional iteration technique has been applied for generating numerical data with the assistance of MATLAB software tools. The obtained analytical results have been presented graphically from numerical computations and simulations. The effect of elastic nonlocality on the frequency shift and thermoelastic damping is represented graphically in some cases of interest and analyzed the obtained numerical results. [ABSTRACT FROM AUTHOR]

• This paper effectively calculates TED of hemi-ellipsoidal shell with variable thickness. • Hemi-ellipsoidal shell with b < a has a smaller TED compared with hemispherical shell. • Reduction of R 0 can decrease TED of hemi-ellipsoidal shell with variable thickness. • Increasing the thickness of the rim part of hemi-ellipsoidal shell can reduce TED. Thermoelastic damping (TED) is a fundamental dissipation mechanism that inevitably exists in shell resonators with high quality factors. Based on the thermal energy method, this paper demonstrates an effective method for the TED characterization of the hemi-ellipsoidal shells with variable thickness which manifest lower TED compared with the hemispherical shells. The equation of motion of the hemi-ellipsoidal shell under clamped-free boundary conditions is established by Hamilton's principle and the assumed mode method, and the natural frequencies and mode shape functions of the hemi-ellipsoidal shell with variable thickness are obtained by solving the eigenvalue problem. The temperature field is acquired by solving the heat conduction equation along the radial direction, and an analytical model for the TED of the hemi-ellipsoidal shell with variable thickness is presented by calculating the maximum elastic potential energy and the work lost per cycle of vibration due to irreversible heat conduction. Analysis on TED at the vibration patterns of meridional wave number m = 1 and the circumferential wave number n = 2 or 3 where the shell resonators typically operate is carried out. The analytically calculated TED results are compared with those of the finite element method (FEM) to verify the feasibility and correctness of the present method. The influences of the geometrical parameters on the TED characteristics of the hemi-ellipsoidal shells with variable thickness are analyzed in detail. A meaningful discovery is that compared with the hemispherical shell, the hemi-ellipsoidal shell with variable thickness has a smaller TED when its semiminor axis is shorter than the semimajor axis, which is particularly significant for optimizing the design of the shell resonators with high quality factors. [ABSTRACT FROM AUTHOR]

• Consider circular cross-section micro/nano-rings undergoing out-of-plane vibration. • Develop generalized thermoelastic damping (TED) models for rings. • Consider size-dependence effects in thermal and mechanical fields. • MCS size-dependence effect affects stored energy and enhances quality factor. • NDPL effect impacts fluctuation temperature and TED. Micro/nano-ring or ring-based resonators are popular core components of sensors and actuators. Accurately predicting the quality factor (Q) based on thermoelastic damping (TED) is intensively crucial for designing high-performance micro/nano-ring resonators. In this work, adopting the modified-couple-stress (MCS) and nonlocal-dual-phase-lagging (NDPL) models, analytical TED expressions are developed for circular cross-section micro/nano-rings undergoing out-of-plane vibration for the first time. Multiple effects are considered in the modeling procedure, which are size-dependence effects in the mechanical and thermal fields, the non-Fourier effect of heat conduction, and three-dimensional temperature gradients in the circumferential, radial, and tangential directions. Therefore, the proposed TED models are more comprehensive compared to existing TED models. The coefficients (C C and C M) associated with the out-of-plane vibration mode are examined from an energy-ratio perspective. Moreover, the investigation is focused on the impacts of the MCS and NDPL non-Fourier effects on the fluctuation temperature and TED spectra. The results indicate as C C and C M approach one, the bending energy emerges as the predominant component of the stored energy. Notably, the NDPL non-Fourier effect significantly affects the fluctuation temperature within the ring operated in high-order modes. In special, the distributions of fluctuation temperature in the circumferential direction exhibit periodic patterns that are closely correlated with the modal order. Furthermore, TED can be suppressed by the MCS size-dependence effect. Meanwhile, the behavior of TED at high frequencies is simultaneously determined by the dimension of heat conduction, phase-lagging times, and nonlocal thermal-length parameter. [Display omitted] [ABSTRACT FROM AUTHOR]

• A model for TED in symmetrical three-layer piezoelectric microbeam is presented. • The peak TED value tends to decrease slightly with the increment of the dimension. • TED is inversely proportional to the electromechanical coupling coefficient. • Reducing the piezoelectric material to isotropy results in an increase in TED value. • The residual polarization intensity affects TED by thermal expansion coefficients. Thermoelastic damping (TED) plays a crucial role in the design of multilayer micro-resonators, especially in the investigation of coupled multi-physical fields. An analytical model for TED in a symmetrical three-layer piezoelectric microresonator was proposed with electric-mechanical-thermal coupling. The three-layer microbeam consists of piezoelectric ceramic layers covering the upper and lower surfaces of a silicon-based layer, and the beam undergoes forced vibration due to electrical excitation. This model can be verified by comparing experimental data and simplifying to previous models. It is observed that with an increase in the proportion of the piezoelectric layers, there are unimodal-bimodal-unimodal variations. Two-dimensional heat conduction is considered in this model, which makes the peak damping value decrease slightly with the increases of the thickness, length and mode. The presence of piezoelectric layers and a high electromechanical coupling coefficient have inhibitory effects on TED. The simplification of the piezoelectric material into isotropic and a reduced residual polarization intensity all induce an increase in thermal stress modulus, which further amplifies differences in Zener modulus among different materials, thereby accelerating double peak formation and enhancing the contribution of the piezoelectric layers to TED. What's more, increasing the beam thickness, proportion of piezoelectric layers, and decreasing excitation voltage amplitude all contribute to decreasing the beam displacement amplitude. [ABSTRACT FROM AUTHOR]

• An effective method is developed for dynamic modelling of hemispherical shells with fillets. • Effects of fillets on thermoelastic quality factors of hemispherical shells are deeply explored. • The lower fillet has larger influence on the natural frequencies than the upper fillet does. • For the lower fillet, there exists size interval where hemispherical shell has higher energy losses. In engineering applications, hemispherical shell resonators are typically machined with fillets to reduce stress concentration and enhance structure strength. The fillets will inevitably affect the dynamic properties and the mechanical quality factor of hemispherical shell resonators, which has been seldom investigated before. In this paper, an effective analytical method is developed to explore the free vibration and thermoelastic damping (TED) characteristics of the hemispherical shell with fillets. The fillets are characterized by the variation in the thickness of the hemispherical shell during modelling. The first-order shear deformation theory (FSDT) is used to describe the theoretical formulas of the hemispherical shell with fillets. By employing the unified Jacobi polynomials and Fourier series as the assumed mode shape functions, the equation of motion of the structure is established by Hamilton's principle and the assumed mode method. The analytical model for thermoelastic quality factor (Q TED) which is determined by TED is obtained by computing the dissipated energy and the maximum elastic potential energy of the hemispherical shell with fillets. The validity and accuracy of the present method are confirmed by comparing the present solutions with the published results and those obtained from the finite element method (FEM). The influences of fillets on the vibration behaviors and Q TED characteristics of the hemispherical shells are analyzed in detail. The present model can be used to optimize the design of the fillets of the hemispherical shell resonators with high quality factors. [ABSTRACT FROM AUTHOR]

The present work investigates the vibration of a piezothermoelastic nanobeam in the frame of Moore–Gibson–Thompson (MGT) thermoelasticity theory. Closed form analytical expressions for the thermoelastic damping in terms of quality factor and frequency shift for a homogeneous transversely isotropic piezothermoelastic (PTE) beam is derived by using Euler–Bernoulli beam theory and complex frequency approach. Detailed analysis of damping of vibration owing to thermal fluctuations and electric potential in the present context under three sets of boundary conditions is attempted to investigate the influence of modes of vibration, electric potential, and beam length on energy dissipation caused by thermoelastic damping in PTE beam resonators. Analytical results are illustrated with the help of graphical plots on numerical findings in the case of the nanobeam resonator of Lead Zirconate Titanate (PZT-5A) material. The investigation brings out some significant key findings and observations in view of MGT heat conduction model. • The present work investigates the vibration of a piezothermoelastic nanobeam under MGT thermoelasticity theory. • Closed-form analytical expression for TED and frequency shift is derived for a homogeneous transversely isotropic PTE beam by using Euler–Bernoulli beam theory and complex frequency approach. • Detailed analysis of damping of vibration under three sets of boundary conditions is attempted to investigate the influence of various factors on energy dissipation caused by TED. • Analytical results are illustrated with the help of graphical plots by considering Lead Zirconate Titanate (PZT-5A) material. [ABSTRACT FROM AUTHOR]