A non-Fourier and couple stress-based model for thermoelastic dissipation in circular microplates according to complex frequency approach

Abstract

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).

Appendix

In this section, the method of computing the couple stress-based isothermal frequency ${\omega }_{0}$ is explained. For this purpose, parameter $\lambda$ is introduced as follows:

$${\lambda }^{4}=\frac{\rho h{\omega }_{0}^{2}}{D\left(1+\kappa \right)}$$

(52)

By inserting relation above into Eq. (43), this equation takes the following form:

$${\nabla }^{4}W-{\lambda }^{4}W=0$$

(53)

The equation above can be decomposed as follows:

$${\nabla }^{2}W+{\lambda }^{2}W=0\,\, OR\,\, {\nabla }^{2}W-{\lambda }^{2}W=0$$

(54)

Note that ${\nabla }^{2}=\left({\partial }^{2}/\partial {r}^{2}\right)+\left(1/r\right)\left(\partial /\partial r\right)$. Accordingly, the solution of equations above are of the Bessel functions type. Given the finite value of deflection at the center of circular plate, the general solution of Eq. (53) can be expressed as:

$$W\left(r\right)={B}_{1}{J}_{0}\left(\lambda r\right)+{B}_{2}{I}_{0}(\lambda r)$$

(55)

in which ${J}_{0}$ and ${I}_{0}$ denote the Bessel and modified Bessel functions of the first kind of order zero, respectively. Coefficients ${B}_{1}$ and ${B}_{2}$ are also integration constants. In what follows, the frequency ${\omega }_{0}$ is extracted for the clamped (C) and simply-supported (SS) boundary conditions.

• (1) Clamped boundary conditions: According to Eqs. (28a) and (28b), one can write:

  $$W(a)=\frac{dW}{dr}(a)=0$$

  (56)

With the help of the properties of Bessel functions, the above two boundary conditions lead to the system of equations below:

$$\left[\begin{array}{cc}{J}_{0}\left(\lambda a\right)& {I}_{0}(\lambda a)\\ -\lambda {J}_{1}(\lambda a)& \lambda {I}_{1}(\lambda a)\end{array}\right]\left\{\begin{array}{c}{B}_{1}\\ {B}_{2}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$$

(57)

To attain nontrivial solution, determinant of coefficient matrix must be equal to zero. Therefore, one can derive the characteristic equation of circular plates with clamped edge conditions as follows:

$${J}_{0}\left(\lambda a\right){I}_{1}\left(\lambda a\right)+{J}_{1}\left(\lambda a\right){I}_{0}\left(\lambda a\right)=0$$

(58)

By solving equation above and inserting the result in Eq. (52), the couple stress-based isothermal frequency ${\omega }_{0}$ for clamped circular plates is obtained.

2) Simply-supported boundary conditions: On the basis of Eqs. (28a) and (28b), the boundary conditions can be written as:

$$W(a)=\left(1+\kappa \right)\frac{{\partial }^{2}w}{\partial {r}^{2}}(a)+\left(\nu -\kappa \right)\frac{1}{r}\frac{\partial w}{\partial r}(a)=0$$

(59)

By substituting Eq. (55) into relations above and applying the properties of Bessel functions, these relations yield the following system of equations:

$$\left[\begin{array}{cc}{J}_{0}\left(\lambda a\right)& {I}_{0}(\lambda a)\\ -\left(1+\kappa \right){\lambda }^{2}{J}_{0}\left(\lambda a\right)+\frac{1-\nu +2\kappa }{a}\lambda {J}_{1}\left(\lambda a\right)& \left(1+\kappa \right){\lambda }^{2}{I}_{0}\left(\lambda a\right)-\frac{1-\nu +2\kappa }{a}\lambda {I}_{1}\left(\lambda a\right)\end{array}\right]\left\{\begin{array}{c}{B}_{1}\\ {B}_{2}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$$

(60)

Similar to previous section, to achieve nontrivial solution, determinant of coefficient matrix must be set to zero. So, the characteristic equation of circular plates with simply-supported edge conditions is expressed by:

$$2\left(1+\kappa \right)\lambda {J}_{0}\left(\lambda a\right){I}_{0}\left(\lambda a\right)-\left(\frac{1-\nu +2\kappa }{a}\right)\left[{J}_{0}\left(\lambda a\right){I}_{1}\left(\lambda a\right)+{J}_{1}\left(\lambda a\right){I}_{0}\left(\lambda a\right)\right]=0$$

(61)

By finding the root of equation above and substituting it in Eq. (52), the couple stress-based isothermal frequency ${\omega }_{0}$ for simply-supported circular plates can be specified.