A Theoretical Analysis of Thermoelastic Damping Model in Laminated Trilayered Circular Plate Resonators

Thermoelastic damping of the axisymmetric vibration of laminated circular plate resonators is discussed in this paper. Based on the classical laminated plate theory assumptions, the governing equations of coupled thermoelastic problems are established for axisymmetric out-of-plane vibration of trilayered circular plate. The analytical expression for thermoelastic damping is obtained and the accuracy is verified through comparison with finite element analysis results. Then some simplifications are made on the theoretical model.


Introduction
Miniaturized resonators are widely used as sensors and modulators in micro-and nanoelectromechanical systems (MEMS/NEMS) [1] [2].For resonators, it is desired to design and construct systems with loss of energy as little as possible [3].Various energy dissipation mechanisms exist in MEMS and NEMS [4]- [6].While, thermoelastic damping is an intrinsic dissipation mechanism and it will not be affected by the changes of environment.Therefore, structural damping is essentially dominated by thermoelastic damping (TED), and it is more important to study the effect of thermoelastic damping on the mechanical behavior of MEMS.
Zener [7] firstly developed the thermoelastic damping theory by studying the transverse vibration of homogeneous and isotropic thin beam.Thermoelastic damping arises from thermal currents generated by compression/decompression in elastic media.The bending of the reed causes dilations of opposite signs to exist on the upper and lower halves.One side is compressed and heated, and the other side is stretched and cooled.Thus, in the presence of finite thermal expansion, a transverse temperature gradient is produced.The temperature gradient generates local heat currents, which cause increase of the entropy of the reed, lead to energy dissipation.
Many researchers have studied the thermoelastic damping effect in resonators made of single materials [8]- [10].However, in many applications, it is necessary to coat a resonator with thin metallic layers to improve the optical reflectivity or enhance the electrical conductivity [11] [12].Examples include the probe of scanning force microscopy which is Au-coated Si microcantilevers, and SiC microreonators coated by Al [13].In such cases, the resonators must be treated as laminated composites of layered thin films.Up to date, a little of work relative to the thermoelastic damping in laminated composite resonators has been reported.Bishop and Kinra [14] first developed analysis of thermoelastic damping in laminated rectangular plate.Later on, Vengallatore and his co-author studied thermoelastic damping in bilayered [15] and symmetric, three-layered [11] beam resonators.All the analysis started from the definition that the magnitude of thermoelastic damping is the ratio of the energy dissipated per cycle of vibration to the maximum elastic energy stored in the body.
This paper deals with thermoelastic damping effects on the out-of-plane vibration of laminated circular plate resonators, which are common elements in many sensors and resonators [16].Different from the above work, the expression of thermoelastic damping is obtained from the complex vibration frequency.

Formulation of the Problem
Consider a laminated circular plate symmetrically composed of three layers with thermally perfect interfaces.The cylindrical coordinate system ( ) is applied to study the vibration of the trilayered circular plate.The neutral surface is put on the ( ) , r θ coordinate plane, and the z-axis normal to the neutral surface.The structure and the coordinate system are illustrated in Figure 1.The two outer layers are made of the same materials and have an identical thickness.The radius of the resonator is a .The thicknesses of the inner and outer layers are 1 2z and ( ) z z − , respectively.Hence, the volume fraction of the outer layer is . For axisymmetric circular plate, the displacement and the temperature are independent of θ , so we define ( ) v r z t and ( ) , , w r z t to be the displacement components along the radial, circumferential and axial directions, respectively, and ( ) , , T r z t the temperature field.In equilibrium, the resonator is assumed to be unstrained, unstressed and keeps at environmental temperature 0 T everywhere.Assuming small strains and displacements, the relationship between the displacement components can be given by ,  , ,   , , , , where t is time.The strain components are given by We assume plane stresses for the laminated circular plate, which leads to ( ) ( ) Thus the cubical dilation j e for each layer is obtained as where the subscript j ranges from 1 to 3, with the convention that 1 j = denotes the substrate layer (the inner layer) and 2,3 j = denotes the two covering layers (the outer layers); j υ and Tj α denote the Poisson's ratio and the coefficient of thermal expansion, respectively; and is the temperature increment of the resonator as a function of ( ) , , r z t . 2  ∇ is the Laplace operator, and is expressed as Now, the thermal conduction equation containing the thermoelastic coupling term for each layer has the following term: where C and j κ are the Young's modulus, the specific heat at constant volume and the thermal conductivity , respectively.
Typically, the second item of * j s can be ignored.For silicon at 300 K, the ratio of the first item to the second item of * j s is 1776, and the ratios for Au, Al, Cu and Ag are about 30.Although the ratios for metals are not very large, still we can replace * j s by j j vj s C ρ = without introducing large error to the final results.In Section 4, we will first compare the results of the two cases: the heat conduction equation with * j s (case I) and that with j s (case II).
Noting that thermal gradients in the plane of the cross-section along the plate thickness direction are much larger than gradients along the radial direction, we can ignore the term 2 j ϕ ∇ in Equation ( 6) [2].Hence, Equation (6) reduces to ( ) Note that the second term of the right side of Equation ( 7) is an odd function of z , and the operator 2 2 z ∂ ∂ on the left side is symmetric in z , it follows that the temperature field j ϕ is an odd function of z , and [14] Thus, it is sufficient to consider only the upper half of the plate, 0 z > while solving the thermal conduction Equation (7).
The stress components of each layer are given by the constitutive equation as Now, the moments of flexure in the substrate layer are ( ) ( ) where ( ) And the moments of flexure in the outer layers are ( ) where ( ) ( ) Hence the resultant moments of flexure of the plate are as follows: The equation of transverse motion for an axisymmetric circular plate is where ( ) , and 1 ρ and 2 ρ are the density of the substrate and covering layers, respec- tively.Substitute Equations ( 9)-( 13) into Equation ( 14), we can get the differential equation of the out of plane vibration of the laminated circular plate: In summary, the governing equations of this problem are composed of Equations ( 7) and (15).

Solution of the Governing Equations
To calculate the effect of thermoelastic coupling on the vibrations of a circular plate, we solve the coupled thermoelastic Equations ( 7) and ( 15) for the case of harmonic vibrations.We set where ( ) w r is the mode shape.In general, the frequency Ω is complex, the real part

( )
Re Ω giving the new eigenfrequencies of the plate in the presence of thermoelastic coupling effect, and the imaginary part

( )
Im Ω giving the attenuation of the vibration.
Substituting Equation ( 16) into Equation ( 15) yield the following equation: It is assumed that the upper and lower surfaces are adiabatic and the interfaces are thermally perfect.Taking into account the property of j ϕ related to z , the boundary and interface conditions are Following the standard integral transform techniques presented by Özisik [17], the solution of the boundary value problem, in terms of Equation ( 16), was found to be: where ( ) ) , , cos cos cos sin , sin sin The eigenvalues n γ is obtained from the transcendental equation sin sin cos cos cos sin 0 0 cos sin From Equations ( 16) and (20), it is obtained ( ) ( ) Substitution of Equations ( 18) and ( 27) into Equation ( 16) yields, ( ) ( ) where ( ) For a laminated circular plate with the boundary fully clamped, the boundary conditions are The solution of Equation ( 28) in combination with Equation (30) gives the value of frequency considering thermoelastic coupling effect ( ) where ( ) is the frequency when the thermoelastic coupling effect is ignored, and { } 10.21,39.78,89.10,m q =  .Because the difference between Ω and 0 Ω is small, we can replace ( ) f Ω in Equation ( 31) by ( ) 0 f Ω and expand Equation (31) into a series up to the first order.That is ( ) So, we can easily extract the real and imaginary parts of Ω , which are shown as ( ) ( ) Thus, we arrive at an expression of thermoelastic damping in a symmetric trilayered circular plate, which is given by

Results and Discussions
In this section, we will present some numerical results for the relationship between 1 Q − and the radius of the plate and make some discussions about this model.From [11] [18], we can get the thermodynamic properties of the used materials under room temperature (300 K), which are listed in Table 1.For abbreviation, we will only consider the fundamental vibration mode, namely, the 1 m = mode.First, the influence of the reduction of * j s to j s is checked.The heat conduction equation with * j s is named case I and that with j s is named case II. Figure 2 shows the themoelastic damping, 1 Q − , against the Table 1.Mechanical and thermal properties of the materials (300 K).

Material
( ) To verify the accuracy of the model proposed in the paper, we also simulate this problem through the commercial software COMSOL Multiphysics.In the calculation, the resonator is composed of Au/Si/Au, the thicknesses of the Si and Au layers are 20 and 1.11 µm, respectively.We check the slice of the normal strain along the z direction of the circulate plate with the radius of 400 µm, as shown in Figure 3, and find that the strain component z ε is nearly zero in the substrate layer of Si, while it varies significantly in the Au layer.
According to this result, the heat conduction equation should be modified into ( ) ( ) , for the substrate 1 2 , for the covering layer Now, the governing equations are composed of Equations ( 7) and (43), and the expression of 1 Q − can be obtained through repeating the derivation in Section 3, except for replacing 1 . This is defined as Model 2. Correspondingly, the model composed of Equations ( 7) and ( 15) is defined as Model 1.
Figure 4 shows the calculation results based on Model 1, Model 2 and the finite element method (FEM), where the abscissa is the plate radius and the ordinate is the thermoelastic damping.
It is shown in Figure 4 that the results can be classified into three regions: (1) 500 μm a ≤ , the results of Model 1 and Model 2 are nearer to each other; (2) 500 1200 μm a < ≤ , the results of Model 2 and FEM analysis are nearer to each other; (3) 1200 μm a > , the results of the three methods are almost the same.In addition, the variation of  Comparing Figure 4, Figure 5, it is shown that the three regions in Figure 4 are related to the value of 1 a z .In the case of small value of 1 a z , the results of Model 1 and Model 2 are nearer to each other.On the other hand, in the case of large value of 1 a z , the results of Model 2 and FEM analysis are nearer to each other.In summarize, the results of both models are in good agreement with FEM results, and those of Model 2 agree better than those of Model 1.To further illustrate the influence of the normal strain along the z direction, the distribution of z ε at the center of the plate along the z direction is calculated for plates with different radius.
The variation tendency of z ε on z is the same for plates with different radius.As an example, the results for the plate with the radius of 200 and 400 µm are shown in Figure 6.It can be found that the normal strain in Si layer is quite small, however, the strain in Au layer is much larger, and it increases suddenly at the interface.At the interface, the ratio of the strain of Si layer, Si ε , to that of Au layer, Au ε , increases a little as the radius of  the plate increases.The value is about 0.38.So, we can ignore the normal strain z ε in Si layer and obtain good results, as shown in Figure 4.

Conclusion
As MEMS and NEMS technologies evolve, there is an increasing use of sophisticated structural geometries and advanced materials, especially laminated composites of layered thin film.A detailed theory is presented for thermoelastic damping in trilayered circular plate resonators.The analytical expression for thermoelastic damping in these structures is obtained.Then the accuracy of the model is verified through comparison with finite element analysis results and some simplifications are made on the theory.In the heat conduction equation, the coefficient of ratio of temperature obtained from the plane stress assumption is composed of two items.It is found that the equation can be simplified by ignoring the item related to mechanical deformation.In addition, the strain along the thickness direction of Silicon can be ignored.Finally, the model is accurate for thin plate structures.However, the Mindlin model should be used for thick plate to derive the expression of thermoelastic damping.

Figure 1 .
Figure 1.Schematic illustration of the trilayered circular plate and the coordinate system.

Figure 2 .
Figure 2. Comparison of the two cases: the heat conduction with * j s , and the heat conduction with j s .

Figure 4 .
Figure 4. Comparison of the analytical solution and FEM analysis.

Figure 6 .
Figure 6.Distribution of z ε at the center of the plate along the z direction for the plates with radius of 200 and 400 µm.