8.2 Literature Review
297
Berry [36] obtained experimentally Q factor as a function of frequency. Duwel
et al. [8] presented experimental data illustrating the importance of TED in the resonance MEMS sensors. They also investigated the influence of various materials on
TED behaviour. Zhang and Turner [37] developed a theory for TED for micro- and
submicro-longitudinally vibrating beams. Vengallatore [38] studied TED in laminated composite micromechanical beam resonators. Prabhakar and Vengallatore
[39] constructed an exact theory useful for TED computation in asymmetric twolayer micromechanical beam resonators. Yi [40] investigated the effect of geometric
parameters on TED in the MEMS resonators. Analytical methods to study TED in
nanobeams are presented in reference [41].
On the other hand, most of the models described in the literature use EulerBernoulli’s beam theory which does not apply to a small ratio of beam length to its
thickness. Massalas and Kalpakidis [42] investigated coupled thermoelastic vibrations of simply supported beams. They presented analytical solutions for the coupled
problem of thermoelasticity of beams made from homogeneous and isotropic materials applicable for the Euler-Bernoulli and Timoshenko beam models. Shieh [43]
studied TED of a simply supported Timoshenko beam with a circular cross section.
He used 2D heat transfer equations and obtained damping coefficients. In reference
[44], the authors proposed an analytical solution for the resonator quality factor under
thermoelastic damping in beams based on the Timoshenko theory using the Lifshitz
and Roukes method [3] for the Euler-Bernoulli beams.
There are known attempts to solve the problems of thermoelastic damping numerically. Prevost and Tao [45] employed finite element method (FEM) to solve a coupled thermoelastic problem. The authors used the second-order heat transfer PDE
instead of the Fourier equation. Prabhakar and Vengallatore [46] matched 1D EulerBernoulli beam theory and 2D heat transfer PDE and obtained the quality factor
using the introduced theory of series. Silver and Peterson [47] investigated elastic
thermodynamic damping for the beam using FEM and also employed the method
of perturbation. Sun et al. [48] obtained TED of microbeam resonators using the
method of finite Fourier transform and modal analysis. Lepage [49] applied FEM
to estimate the quality factor of TED of the Euler-Bernoulli beams. He employed
the cubic approximation of temperature variation along with beam thickness. Guo
et al. [50] used 2D FEM to study MEMS resonators. De and Aluru [51] employed
a combined method of finite and boundary elements to validate the results of their
TED-related modified theory.
In MEMS structures made of metals and polymers in which the smallest length of
elements was in the micro-range, and the size-dependent phenomena were observed
experimentally in many systems [52–56]. Effects of the dimensional dependence
can be divided into two categories. The first one is implied by the boundary effect
coupled with a molecular layer located on the specimen surfaces. Brezny and Green
[57] and Onck et al. [58] observed that the effective Young modulus and resistance
against compression of some materials essentially decreased while decreasing the
size of the specimen. The second one known as the Kosser effect, or micropolar
effect [59], occurs due to violation of the classical mechanical statements. Owing to
297
Berry [36] obtained experimentally Q factor as a function of frequency. Duwel
et al. [8] presented experimental data illustrating the importance of TED in the resonance MEMS sensors. They also investigated the influence of various materials on
TED behaviour. Zhang and Turner [37] developed a theory for TED for micro- and
submicro-longitudinally vibrating beams. Vengallatore [38] studied TED in laminated composite micromechanical beam resonators. Prabhakar and Vengallatore
[39] constructed an exact theory useful for TED computation in asymmetric twolayer micromechanical beam resonators. Yi [40] investigated the effect of geometric
parameters on TED in the MEMS resonators. Analytical methods to study TED in
nanobeams are presented in reference [41].
On the other hand, most of the models described in the literature use EulerBernoulli’s beam theory which does not apply to a small ratio of beam length to its
thickness. Massalas and Kalpakidis [42] investigated coupled thermoelastic vibrations of simply supported beams. They presented analytical solutions for the coupled
problem of thermoelasticity of beams made from homogeneous and isotropic materials applicable for the Euler-Bernoulli and Timoshenko beam models. Shieh [43]
studied TED of a simply supported Timoshenko beam with a circular cross section.
He used 2D heat transfer equations and obtained damping coefficients. In reference
[44], the authors proposed an analytical solution for the resonator quality factor under
thermoelastic damping in beams based on the Timoshenko theory using the Lifshitz
and Roukes method [3] for the Euler-Bernoulli beams.
There are known attempts to solve the problems of thermoelastic damping numerically. Prevost and Tao [45] employed finite element method (FEM) to solve a coupled thermoelastic problem. The authors used the second-order heat transfer PDE
instead of the Fourier equation. Prabhakar and Vengallatore [46] matched 1D EulerBernoulli beam theory and 2D heat transfer PDE and obtained the quality factor
using the introduced theory of series. Silver and Peterson [47] investigated elastic
thermodynamic damping for the beam using FEM and also employed the method
of perturbation. Sun et al. [48] obtained TED of microbeam resonators using the
method of finite Fourier transform and modal analysis. Lepage [49] applied FEM
to estimate the quality factor of TED of the Euler-Bernoulli beams. He employed
the cubic approximation of temperature variation along with beam thickness. Guo
et al. [50] used 2D FEM to study MEMS resonators. De and Aluru [51] employed
a combined method of finite and boundary elements to validate the results of their
TED-related modified theory.
In MEMS structures made of metals and polymers in which the smallest length of
elements was in the micro-range, and the size-dependent phenomena were observed
experimentally in many systems [52–56]. Effects of the dimensional dependence
can be divided into two categories. The first one is implied by the boundary effect
coupled with a molecular layer located on the specimen surfaces. Brezny and Green
[57] and Onck et al. [58] observed that the effective Young modulus and resistance
against compression of some materials essentially decreased while decreasing the
size of the specimen. The second one known as the Kosser effect, or micropolar
effect [59], occurs due to violation of the classical mechanical statements. Owing to