a finite element formulation for thermoelastic … · strain–displacement relation, balance...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2008)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2502

A finite element formulation for thermoelastic damping analysis

Enrico Serra1,∗,† and Michele Bonaldi2,3

1Fondazione Bruno Kessler, FBK-irst MicroTechnologies Laboratory, via Sommarive 18,I-38100 Povo (Trento), Italy

2Istituto di Fotonica e Nanotecnologie CNR-FBK, via alla Cascata, 56/C 38100 Povo (Trento), Italy3Istituto Nazionale di Fisica Nucleare, via Sommarive 14, I-38100 Povo (Trento), Italy

SUMMARY

We present a finite element formulation based on a weak form of the boundary value problem for fullycoupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropydue to the thermal fluxes that have originated from the volumetric strain variations. Within our weakformulation we define a dissipation function that can be integrated over an oscillation period to evaluate thethermoelastic damping. We show the physical meaning of this dissipation function in the frameworkof the well-known Biot’s variational principle of thermoelasticity. The coupled finite element equationsare derived by considering harmonic small variations of displacement and temperature with respect to thethermodynamic equilibrium state. In the finite element formulation two elements are considered: the firstis a new 8-node thermoelastic element based on the Reissner–Mindlin plate theory, which can be used formodeling thin or moderately thick structures, while the second is a standard three-dimensional 20-nodeiso-parametric thermoelastic element, which is suitable to model massive structures. For the 8-nodeelement the dissipation along the plate thickness has been taken into account by introducing a through-the-thickness dependence of the temperature shape function. With this assumption the unknowns and thecomputational effort are minimized. Comparisons with analytical results for thin beams are shown toillustrate the performances of those coupled-field elements. Copyright q 2008 John Wiley & Sons, Ltd.

Received 9 May 2008; Revised 7 October 2008; Accepted 9 October 2008

KEY WORDS: finite elements; thermoelastic damping; Reissner–Mindlin plate theory

1. INTRODUCTION

Thermoelastic damping is an internal friction mechanism that originates from stress inhomo-geneities which in turn generate heat fluxes that increase the entropy of a vibrating solid. In MEMS

∗Correspondence to: Enrico Serra, Fondazione Bruno Kessler, FBK-irst MicroTechnologies Laboratory, via Sommarive18, I-38100 Povo (Trento), Italy.

†E-mail: [email protected]

Contract/grant sponsor: European Community; contract/grant number: RII3-CT-2004-506222Contract/grant sponsor: Provincia Autonoma di Trento

Copyright q 2008 John Wiley & Sons, Ltd.

E. SERRA AND M. BONALDI

and NEMS devices, thermoelastic damping establishes an absolute lower bound on internal frictionand, in many cases, is the dominant component of damping at room temperature (300K). Forthis reason, the study of thermoelastic damping in flexures is an active area of current theoreticalresearch [1, 2]. In precision measurements, thermoelastic damping acts as a source of mechanicalthermal noise and contributes to limit the sensitivity of gravitational wave detectors [3, 4].

The thermoelastic damping was studied first by Zener [5–7]. In particular, he considered thethermoelastic damping in beams and he gave an analytical approximation for the loss angle ofthose structures. More recently, Lifshitz and Roukes [1] obtained a closed form of the thermoelasticdamping problem in a thin beam of rectangular cross section under flexural vibrations. Theycalculated the loss angle of microbeams and found results close to those obtained with Zener’smodel. Another simple formula based on the extension of the Lifshitz and Roukes approach wasobtained byWong et al. [8] for the computation of thermoelastic damping of MEMS ring resonators.Nayfeh and Younis [9] developed an independent approach for the evaluation of the loss angle ofmicroplates: they considered the biharmonic differential plate equation for small transverse (out-of-plane) displacement of the thin plate together with the heat equation. The thermoelastic dampingwas then computed by a perturbative analysis starting from the normal modes of the plate.

The ability to predict thermoelastic damping of all these analytical approaches fails when morecomplex geometries or boundary conditions are considered. In fact, the loss angles of structuralmodes, which vary spatially across the plate width, cannot be predicted using a simple beam orplate model. In such cases, a numerical procedure may overcome this limitations.

Recently some efforts have been done in developing finite elements for thermoelastic analysis.The finite element equations corresponding to the generalized thermoelasticity with two relaxationtimes were derived and solved directly in time-domain [10] to investigate second sound effect ofheat conduction in solids subject to thermal shock loading. The thermoelastic damping probleminvolved in beam resonators has been addressed using a combination of finite element methodand eigenvalue formulation [11]. This approach allows the evaluation of the eigenfrequencies andtheir associated quality factor of resonance, even in case of complex geometries and boundaryconditions. However, a complete framework for the finite element computation of the thermoelasticdamping involving plates and/or massive bodies is still lacking and the physical context is not fullydescribed. The purpose of this paper is to present a new finite element formulation for computingthe loss angle of a vibrating structure using either planar or full three-dimensional elements. Wederive a weak form for linearly coupled thermoelasticity boundary value problem (BVP), wherean infinite velocity of the thermal field and small harmonic temperature variations with respect tothe equilibrium temperature are assumed. From the weak form a dissipation function is recognizedand used for evaluating the work lost due to thermoelastic effect. We show that the dissipationfunction is substantially the same of the dissipation function introduced by M. A. Biot in its earlyvariational principle [12, 13]. On the other hand our choice of variables leads to a finite elementscheme which requires less DOFs with respect to a finite element implementation based on theM. A. Biot variational principle. In the finite element formulation two elements are considered.The first is a new 8-node plane Reissner–Mindlin element which can be used in modeling thin andmoderately thick structures like coatings or the interface layer between two bonded materials. Thesecond thermoelastic element is a 20-node element and can be used to model thick structures. Forthe 8-node element we use the well-known Reissner–Mindlin assumptions and the temperaturefield along the element thickness is taken into account by properly modifying the element shapefunctions. To validate the procedure we compute the thermoelastic damping of several vibrationalmodes of a thin beam and show the comparison with the analytical values obtained by Lifshitz

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2008)DOI: 10.1002/nme

A FINITE ELEMENT FORMULATION FOR THERMOELASTIC DAMPING

and Roukes [1]. The outline of the paper is as follows. Section 2 gives an overview of the basicequations for thermoelasticity, presenting a brief summary of the balance and of the constitutiveequations used in the finite element formulation. We introduce here the dissipation function usedto evaluate the thermoelastic damping. In Section 3 we present the weak formulation of coupledthermoelasticity BVP and show the connections with the well-known Biot’s variational principle.Section 4 introduces the thermoelastic elements with their constitutive relations. The general FiniteElement framework is detailed in Section 5, where the numerical procedure used to compute thethermoelastic loss angle is also illustrated. In Section 6 analytical results are compared with thenumerical ones for the dissipation of a thin beam. The paper ends with concluding remarks inSection 7.

2. MATHEMATICAL FRAMEWORK FOR THE THERMOELASTIC DAMPING

We now review the BVP for linearly coupled thermoelasticity in case of small vibrations and smalltemperature changes with respect to the equilibrium temperature T0. Strain–displacement relation,balance equations and constitutive equations are needed to formulate the general thermoelasticproblem. We also recall the definition of thermoelastic damping of a body.

2.1. Basic equations for linear coupled thermoelasticity in solids

2.1.1. Strain–displacement equation. When considering small strains and displacements, the kine-matics equations consist of the well-known strain–displacement relation:

�i j = ui, j +u j,i

2(1)

2.1.2. Balance equations. Any state of the body must satisfy the balance equation of entropy andthe equation of motion.

The local balance equation of entropy is

T0s=−qi,i (2)

where s is the entropy rate per unit volume, T0 the equilibrium temperature, qi,i the divergence ofthe thermal flux. We assumed small temperature changes close to the equilibrium temperature T0to obtain a linear equation. No internal heat sources are considered in Equation (2).

The equation of motion in local form is

�i j, j + fi =�ui (3)

where �i j, j is the derivative of the stress tensor, fi the volume forces per unit volume, � the densityand ui the displacement vector.

2.1.3. Constitutive equations. In thermodynamics, the state of a solid is entirely determined bythe values of a certain set of independent variables: the kinematic variables (strains �i j , . . .) andthe temperature. Starting from the first and the second law of thermodynamics and, for instance,following the approach in [14], it is possible to show that the Cauchy stress tensor �i j is conjugated



to the strain tensor �i j , in the same way the entropy per unit volume s is thermodynamicallyconjugated to the temperature shift �=T −T0:

�i j = �F��i j

, s=−�F��

(4)

where F(�i j ,�) is Helmholtz’s free energy. If we retain only the quadratic terms in the Taylorseries expansion in the vicinity of the equilibrium state (�i j =0,�=0), we obtain

F(�i j ,�)= 1

2Ci jkl�i j �kl −�i j �i j�+ cE

2T0�2 (5)

where

Ci jkl = �2F(0,0)

��i j��kl, �i j =−�2F(0,0)

��i j��, cE =T0

�2F(0,0)

��2(6)

In deriving (5) we assume that the free energy, entropy and stress vanish in the equilibriumstate: F(0,0)=0,s(0,0)=0,�i j (0,0)=0. From (4) and (5) the two constitutive equations can beobtained

�i j = �F��i j

=Ci jkl�kl −�i j� (7)

s = −�F��

=�i j �i j +cET0

� (8)

where cE is the specific heat for the unit volume in the absence of deformation.The constitutive relation between the heat flux and the temperature gradient is given by Fourier’s

law of heat conduction:

qi =−ki j�, j (9)

where ki j is the thermal conductivity tensor.In the following we limit our study to the case of isotropic body, where:

Ci jkl = ��i j�kl +�(�ik� j l+�il� jk)

�i j = (3�+2�)�i j

ki j = �i j

(10)

Here, is the coefficient of thermal expansion, �,� are the Lame constants and is the thermalconductivity.

2.2. Thermoelastic damping

According to the approach developed by Liftshitz and Roukes [1], the effect of thermoelasticcoupling on the vibrations of a thin beam can be evaluated by solving the coupled equation ofmotion to obtain the normal modes of vibration and their corresponding eigenfrequencies �n . Ingeneral, the eigenfrequencies are complex and their real part gives the oscillation frequency while



the imaginary part gives the attenuation of the vibration. The loss angle � at the frequency �n isthen obtained as:

�=2

∣∣∣∣Im(�n)

Re(�n)

∣∣∣∣ (11)

The loss angle � can be also evaluated as the work lost per radiant normalized to the maximumelastic energy stored in body during a cycle. In this case

�= �W

2 Wm(12)

where �W is the total work lost per cycle and Wm is the maximum strain energy during that cycle.The work lost per cycle �W is related to the irreversible entropy produced per cycle (Gouy–Stodolatheorem):

�W =∮cycle

∫V

�,i ki j�, j

T0dVdt (13)

as can be easily shown by combining Equations (2), (9) and after integrating over the body’svolume [12, 15, 16]. If we define the dissipation function D as

D= 1

2

∫V

�,i ki j�, j dV (14)

the loss angle � may be written as:

�=∮Ddt

T0Wm(15)

2.3. Two-way and one-way thermoelastic coupling

From Equation (2), using the entropy constitutive equation (8) and definitions (10) for an isotropicbody, we derive the classical linear heat differential equation that can be found in many textbooks:

�,i i =cE��

�t+ ET0

(1−2�)

��kk�t

(16)

where cE , E,� is the volumetric specific heat per unit volume in the absence of deformation,Young’s modulus of the body and the Poisson ratio. Small temperature variations are assumed andthe term related to thermoelastic damping contains time variation of the strain trace �kk .

Equivalently [17] the source term can be written in terms of the stress trace �kk and the heatequation becomes:

�,i i =cE��

�t+T0

��kk�t

(17)

These equations take into account that the strain field or the stress field affects the temperaturefield and conversely the temperature field affects the strain field or the stress field. This is knownas two-way coupling. In many cases the thermoelastic damping is evaluated by considering thestrain or the stress trace in isothermal condition. This approximation is called one-way couplingbecause the heat equation is solved with a source term (strain or stress trace) which is independentfrom the temperature field [17].Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2008)

DOI: 10.1002/nme


3. FINITE ELEMENTS FOR THERMOELASTIC DAMPING ANALYSIS

We now overview the weak form for thermoelasticity and the approach of M. A. Biot, showingits connections with the dissipation function D.

3.1. Derivation of the weak form for linear coupled thermoelasticity

We derive the matrix equations for the finite element implementation by using a weak form ofthe thermoelastic BVP. Consider a solid at thermodynamic equilibrium, described by the statevariables (ui ,�) and identified with the closed domain V in Figure 1. Consider now a variation(�ui ,��) of the state variables around their equilibrium value. The local balance equation (2) canbe multiplied by �� and integrated over the region V of the body, to obtain:∫

V(T0s+qi,i)��dV =0 (18)

On the second term of this equation we apply the divergence theorem:∫Vqi,i��dV =−

∫Vqi��,i dV +

∫Aqini��dA (19)

then the integral equation (18) may be rewritten as:∫VT0s��dV −

∫Vqi��,i dV +

∫Aqini��dA=0 (20)

The same procedure is applied to the balance equation (3), by multiplying by �ui and integratingover the region V : ∫

V(�i j, j + fi −�ui )�ui dV =0 (21)

On the first term of this equation we apply the divergence theorem and the strain–displacementequation (1): ∫

V�i j, j�ui dV =−

∫V

�i j��i j dV +∫A�i j n j�ui dA (22)

Figure 1. Body V in equilibrium with mechanical and thermal mixed boundary conditions: (a) mechanical:the body is subject to a pressure on A� and to an imposed displacement in Au . Small displacementsvariations �ui are also represented and (b) thermal: the body is subject to a heat flux on Aq and to a

temperature shift A�. Small temperature shift variations �� are also represented.



Then integral equation (21) may be rewritten as:

−∫V

�i j��i jdV +∫A�i j n j�uidA+

∫V( fi −�ui )�uidV =0 (23)

Subtracting Equation (23) from Equation (20) we obtain the weak form of the thermoelastic BVP:

∫VT0s��dV −

∫Vqi��,i dV +

∫V

�i j��i j dV = −∫Aqini��dA

+∫A�i j n j�ui dA+

∫V( fi −�ui )�ui dV (24)

We note that the second term in the left-hand side of (24) is linked to the thermoelastic dissipation.By substituting the constitutive equations (7), (8) and (9) we obtain:

T0

∫V

�i j �i j��dV +∫VcE ��dV +

∫V

��,i ki j�, j dV

+∫V

��i jCi jkl�kl dV −∫V

��i j�i j�dV +∫V

�ui�ui dV

=−∫Aqini��dA+

∫A�i j n j�ui dA+

∫Vfi�ui dV (25)

Appropriate boundary conditions must be specified to guarantee that the thermoelastic BVP iswell-posed and a unique solution can be obtained [18]. In Figure 1 we show the mixed mechanicalboundary conditions on A= A�∪Au and the mixed thermal boundary conditions on A= A�∪Aq :

�i j n j = � on A�, ui = u on Au

qini = Q on AQ, �= � on A�

(26)

The thermoelastic damping is due to irreversible heat flux inside the body. In the variationalequation (25) we identify the term linked to this dissipation mechanism and consider it as thevariation of a dissipation function D:

�D=∫V

��,i ki j�, j dV (27)

The loss angle � may be evaluated according to Equation (15).

3.2. Connections with Biot’s principle

Our dissipation function is closely related to the dissipation function in Biot’s variational principle.According to Biot [12, 13] a variational principle can be obtained in the framework of irreversiblethermodynamics by considering (ui ,Hi ) as state variables, where Hi is related to the entropy of thesystem and to the heat flux according to equations: s=−Hi,i ,qi =T0 Hi . Consider small variations(�ui ,�Hi ) of the state variables in the solid V . Biot introduces two invariants, the thermoelasticpotential � and the dissipation function D∗, whose variation is:

�(�+D∗)=−∫A��Hi dA+

∫A�i j n j�ui dA+

∫V( fi −�ui )�ui dV (28)



The thermoelastic potential is �=W+P, where W is the isothermal strain elastic energy and Pis the heat potential. Further details can be found in [14]. The isothermal elastic energy variation�W, the heat potential variation �P and the dissipation function variation �D∗ can be expressed as:

�W=∫V

�i j��i j dV+∫V

��i j��i i dV , �P=cET0

∫V

��dV , �D∗=T02

∫Vk−1i j �(H j Hi )dV (29)

The dissipation function is related to the irreversible entropy produced inside the body due to thevolume deformations. In fact we have∮

D∗dt=∮

T02

∫Vk−1i j H j Hi dVdt (30)

and taking into account the definition of Hi , H j and the symmetry of the conductivity tensorki j =k j i (Onsager’s reciprocity relations), we obtain on a cycle:∮

D∗dt=∮

1

2T0

∫V

�,i ki j�, j dV dt (31)

Comparing the definition of the dissipation function (14) it is easy to find that:

D=D∗T0 (32)

This relation gives a deeper insight into the dissipation function D, which we use in the followingto evaluate the thermoelastic loss angle. We point out that a finite element formulation based onBiot’s variational principle is also possible but it would require the use of vector variable Hiinstead of the scalar �, increasing the size of the algebraic system of equation.

4. THERMOELASTIC ELEMENTS

In the following we define two elements for solving problems involving either thin or thick solidbodies. For the thin structures we develop a new thermoelastic element satisfying the Reissner–Mindlin plate theory with a through-the-thickness linear approximation of the temperature field.For thick bodies, we use a classical quadratic iso-parametric element.

4.1. 8-Node thermoelastic Reissner–Mindlin plate element

We seek a coupled-field element which takes into account the in-plane and out-of-plane mechanicaldeformations and the thermal balance but at the same time we want to preserve a small numberof DOFs. For this reason we derive the deformation along the plate thickness due to bendingand shear stresses by following a given plate theory. A new planar 8-node serendipity coupled-field element is developed on the basis of the Reissner–Mindlin theory. The basic hypothesis ofReissner–Mindlin theory are:

(i) the displacements in an external coordinate system must satisfy the relations:

u1=−x3�x1 (x1, x2), u2=−x3�x2 (x1, x2), u3=u3(x1, x2) (33)

where �x1 ,�x2 are the rotations while x3 is the element local coordinate along the platethickness. This implies that the normal remains straight after deformation, but it is not



necessarily perpendicular to the deflected mid-surface. The transverse displacement u3 doesnot vary along the thickness.

(ii) the stress along the plate’s thickness is negligible (�33=0).(iii) a shear correction factor m= 5

6 is introduced to account for the parabolic distribution ofthe shear stress along the thickness.

A full three-dimensional description of the thermoelastic damping requires a proper handling of thethermal balance in the element. As shown in Equation (16), the heat flux due to the thermoelasticcoupling represents the source term in the heat equation and is related to the time derivative ofthe trace of the strain tensor �kk = �11+�22+�33. According to the hypothesis (i), we can write thein-plane component of the trace as

�in-plane≡ �11+�22=−x3

(��x1

�x1+ ��x2

�x2

)(34)

while the out-of-plane can be derived considering hypothesis (i), (ii) and the constitutiveequation (7):

�out-of-plane≡ �33= �

�+2�x3

(��x1

�x1+ ��x2

�x2

)(35)

Combining the above relations and using the definitions of Lame constants, the heat generatedby the thermoelastic coupling for small harmonic oscillations with respect to the equilibriumtemperature T0, may be written as

qThEl =−j�ET0(1−�)

x3

(��x1

�x1+ ��x2

�x2

)(36)

where � is the angular frequency.Hence, the heat flux generated inside the element due to thermoelastic damping depends linearly

on the distance from the plate’s neutral plane and changes sign when going from the upper to thelower surface.

Now we discuss the linear approximation of the temperature shift following the approachdeveloped by Lifshitz and Roukes [1], which is here extended to thin plates. We consider anisotropic body and we analyze only temperature changes along the plate thickness. The temperatureprofile can be determined solving heat equation (16) that becomes

�2��x23

= j�cE

(�− ET0

cE (1−�)

(��x1

�x1+ ��x2

�x2

)x3

)(37)

and whose solution is

�− ET0cE (1−�)

(��x1

�x1+ ��x2

�x2

)x3= A sin(�x3)+B cos(�x3) (38)



where �= (1+ j)�cE/. Applying the boundary conditions (no heat flow across the boundaries ofthe plate), we derive the constants A and B and find the solution:

�(x1, x2, x3)= ET0cE (1−�)

(��x1

�x1+ ��x2

�x2

)(x3− sin(�x3)

�cos(�h/2)

)(39)

where ��x1/�x1+��x2/�x2 is the sum of the middle-plane curvatures of the plate as it is shownin Figure 3 and h is the plate thickness. Using a Taylor series expansion, Equation (39) may bewritten as:

�(x1, x2, x3)= ET0cE (1−�)

(��x1

�x1+ ��x2

�x2

)(x3

(1− 1

�cos(�h/2)

)+1

6

�2x33cos(h�/2)

+o(x33)

)(40)

If we consider only the linear part in order to maintain the equations system of small size, onlyone coefficient is needed to compute the temperature shift. If we would add the cubic term, anadditional DOF in the temperature shift should also be considered. For thin plates the error madewith the first order approximation is small as it will be shown in Section 6. Hence we can write,the temperature shift as:

�(x1, x2, x3)= �(x1, x2)2

hx3= �(x1, x2)�3 (41)

To summarize, this element has four nodal DOFs (�x1 ,�x2 ,u3,�), and the in-plane functionsare the well-known quadratic serendipity functions that satisfy completeness and compatibilityrequirements. These functions are well suited in modeling thin or moderately thick structures asdescribed in Reference [19]. We assume also a linear dependence between natural coordinate �3and the cartesian coordinate x3 in the coordinate transformation. The element is represented inFigure 2(a) where the transformation from the natural space to the physical space is also shown.

4.1.1. Shape functions. The rotations along the axis x1, x2 and the displacement of the middleplane u3 inside each element are obtained by interpolating the nodal values with shape functions:

�x1 =8∑

i=1N (i)�(i)

x1 , �x2 =8∑

i=1N (i)�(i)

x2 , u3=8∑

i=1N (i)u(i)

3 (42)

The temperature shift � is expressed as:

�=�38∑

i=1N (i)�(i) (43)

The displacement vector {u} in the element and the temperature shift may be written using matrixnotation and Equations (42) and (43), respectively:

{u} =N�{�e} (44)

� =N�{�e} (45)



Figure 2. (a) The 8-node thermoelastic element with its node connectivity and geometry. T1is the transformation from the natural coordinate to cartesian coordinate and (b) the 20-nodethermoelastic element with its node connectivity and geometry. T2 is the transformation from

the natural coordinate to cartesian coordinate.

Figure 3. Profile of the temperature shift � along the thickness. The dashed line is the real part of thetemperature shift along the plate’s thickness according to Equation (39). The continuous line is the linear

approximation used to describe the through-the-thickness temperature shift of our plate element.

where {�e}={�(1)x1 �(1)

x2 u(1)3 . . . �(8)

x1 �(8)x2 u

(8)3 }T and {�e}={�(1) . . . �(8)}T. Natural coordinates and

cartesian coordinate are related by the transformation:

x1=8∑

i=1N (i)x (i)

1 , x2=8∑

i=1N (i)x (i)

2 , x3= h

2�3 (46)

where x (i)1 , x (i)

2 are the cartesian coordinate of the i th node. Here we assume a linear dependenceof through-the-thickness coordinate x3 and h is the plate’s thickness which does not involve anynodal coordinate.



4.1.2. Strain–displacement and constitutive equations. Let us first derive the strain–displacementrelation. Following the theory of plates [20], the strain tensor of a plate can be represented by fivecomponents and can be decoupled into bending and shear component:

{�}={

�b

�s

}(47)

where �b={�11,�22,2�12} are the bending components and �s={2�23,2�31} are the shear compo-nents. By using Equation (33) the bending strain tensor becomes:

{�b}=−x3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

��x1

�x1��x2

�x2��x1

�x2+ ��x2

�x1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(48)

while the shear strain tensor:

{�s}=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

�u3�x2

−�x2

�u3�x1

−�x1

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(49)

From assumption (ii) the following relation between strain components can be derived:

�33= −�

�+2�(�11+�22) (50)

Thus, the strain–displacement relation may be written as follows:

{�}=B�{�e} (51)

where

B� =[Bb

�

Bs�

](52)

with

Bb� =−x3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

�N (1)

�x10 0 · · · �N (8)

�x10 0

0�N (1)

�x20 · · · 0

�N (8)

�x20

�N (1)

�x2

�N (1)

�x10 · · · �N (8)

�x2

�N (8)

�x10

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)



and

Bs� =

⎡⎢⎢⎢⎣

0 −N (1) �N (1)

�x2· · · 0 −N (8) �N (8)

�x2

−N (1) 0�N (1)

�x1· · · −N (8) 0

�N (8)

�x1

⎤⎥⎥⎥⎦ (54)

The stress constitutive equation (7) for the plate is

{�}=C�B�{�e}−{�}TN�{�e} (55)

where elastic matrix can be written as:

C� =[Cb

� 0

0 Cs�

]=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎡⎢⎢⎢⎣

�+2� � 0

� �+2� 0

0 0 �

⎤⎥⎥⎥⎦ 0

0

[� 0

0 �

]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(56)

and

{�}=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3�+2�)

(3�+2�)

0

0

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(57)

with �=2��/(�+2�). In the first term of Equation (55) the stress tensor is decoupled into itsbending and shear component {�}={�b�s}={{�11�22�12}{�23�31}}. The entropy constitutive law(8) may be written in matrix form as:

s={�}TB�{�e}+ cET0

N�{�e} (58)

Fourier’s law in matrix notation is

{q}=−K� {∇�}=−K�B�{�e} (59)



where

B� =J−1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�3�N (1)

��1· · · �3

�N (8)

��1

�3�N (1)

��2· · · �3

�N (8)

��2

N (1) · · · N (8)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(60)

and J−1 is the Jacobian matrix of the coordinate transformation.

4.2. 20-Node thermoelastic element

A three-dimensional element is needed to describe thick structures. We chose a hexahedral elementof the serendipity family containing only exterior nodes (20-node element shown in Figure 2(b)).This element has four nodal DOFs (ui ,�) with i =1, . . .,3 and a midside node along its edges.In this case the strain–displacement and constitutive relations are formally the same as it isdescribed in the above section. In deriving the strain–displacement and constitutive matrices thesix component of the stress and the strain tensor must be considered. In this case the unknownsvectors should also be changed according to the definition of the 20-node shape function as{�e}={u(1)

1 u(1)2 u(1)

3 . . .u(20)1 u(20)

2 u(20)3 }T and {�e}={�(1) . . .�(20)}T.

5. u−�-BASED FINITE ELEMENT FORMULATION

To get the numerical solution, we must rewrite the variational equation (24) in matrix notationand in the frequency domain. From now on we consider only the case of null body forces fi . Thevariational equation (24) becomes:

−�2∫V

� �{u}T{u}dV + j�∫VT0�� sdV −

∫V

�{∇�}T{q}dV +∫V

�{�}T{�}dV

=−∫Aq

��{Q}dA+∫A�

�{u}T {�}dA (61)

where � is the angular frequency. We point out that all unknowns are intended as phasors. Thesolution in time is given by as the real part of these phasors. In order to derive the algebraic systemof equations we observe that each volume integral on the left-hand side of (61) is the sum of thevolume integral over the elements of the mesh. Each integral must be computed by substitutingthe shape functions for the 8-node or 20-node thermoelastic element and the constitutive relations.Each term of (61) may be derived separately, hence:

first term becomes

−�2∫V

� �{u}T{u}dV =−�2{��g}TM�,�{�g} (62)



where

M�,� =∑e

∫V e

�NT�N� dV (63)

The shape function matrixN� has size 3n×1 while the mass matrixM�,� has size 3n×3n where n isthe number of nodes into volume V which is discretized by the 8-node or by 20-node thermoelasticelements. The global displacement vector is defined as: {�g}={�(1)

x1 �(1)x1 u

(1)3 . . .�(n)

x1 �(n)x1 u

(n)3 }T or

{�g}={u(1)1 u(1)

2 u(1)3 . . .u(n)

1 u(n)2 u(n)

3 }T for the 8-node or the 20-node element, respectively.The second term becomes after substituting the entropy constitutive equation (58):

j�∫VT0��sdV = j�{��g}TD�,�{�g}+{��g}TD�,�{�g} (64)

where

D�,� =T0∑e

∫Ve

NT� {�}T{B�}dV (65)

and

D�,� =∑e

∫V e

cENT�N� dV (66)

The damping matrix D�,� has size n×3n and represents the effects of the strain field in the heatequation while the damping matrix D�,� has size n×n. The global temperature shift is defined as{�g}={�(1) . . . �(n)}T. N� has size n×1.

The third term becomes∫V

�{∇�}TK�{∇�}dV ={��g}TK�,�{�g} (67)

where:

K�,� =∑e

∫V e

BT�K�B� dV (68)

The stiffness matrix K�,� has size n×n.The fourth term becomes after substituting the stress constitutive equation (55):∫

V�{�}T{�}dV ={��g}TK�,�{��g}+{��g}TK�,�{�g} (69)

where:

K�,� =∑e

∫Ve

BT�C�B� dV (70)

and

K�,� =∑e

∫V e

BT� {�}N�dV (71)



The stiffness matrix K�,� has size 3n×3n. The stiffness matrix K�,� has size 3n×n and representsthe effects of the thermal field in the equation of motion.

Volume integrals are then evaluated in the element’s natural coordinates. The matrix systembecomes:

−�2

[M�,� 0

0 0

]{{�g}{�g}

}+j�

[0 0

D�,� D�,�

]{{�g}{�g}

}+[K�,� −K�,�

0 K�,�

]{{�g}{�g}

}={{ f�}

{0}

}(72)

The uniqueness of the solution must be guaranteed by a proper choice of boundary conditions ondisplacements and thermal shifts as well as by the specification of thermal and mechanical loads.The off-diagonal terms in the global damping and stiffness matrices represent the effect of thetwo-way coupling due to thermoelasticity. These matrices are related according to the equation:

D�,� =T0KT�,� (73)

5.1. Evaluation of the thermoelastic loss angle

We evaluate the thermoelastic loss on the basis of the dissipation function defined in Equation (14),which in vector notation becomes:

D= 1

2

∑e

∫Ve

{∇�}TK�{∇�}dV (74)

In the time domain, we may write the temperature gradient vector as follows:

{∇�}={|∇�|}cos(�t+�) (75)

which is the real part of the temperature gradient phasor obtained from the solution of the system(72). Substituting (75) into (74) we obtain:

D= 1

2

∑e

∫Ve

{|∇�|}TK�{|∇�|}cos2(�t+�)dV (76)

According to Equation (15), to evaluate the thermoelastic damping we integrate the dissipationfunction over a cycle:

∮cycle

Ddt=∫ 2 /�

0Ddt = 1

4

∫ 2 /�

0

∑e

∫V e

{|∇�|}TK�{|∇�|}dVdt

+1

4

∫ 2 /�

0

∑e

∫Ve

{|∇�|}TK�{|∇�|}cos(2�t+2�)dVdt (77)

where we have used the cosine bisection formula. In Equation (77) we observe that the secondterm vanishes and after some algebras it changes into:∮

cycleDdt=

2�

∑e

∫Ve

({�eRe}TBT�K�B�{�eRe}+{�eIm}TBT

�K�B�{�eIm})dV (78)



The temperature shift module may be written as: |�|=√

(�gRe)2+(�gIm)2. Hence, according to

Equation (15) the loss angle is given by:

�(�)= {�gRe}TK�,�{�gRe}+{�gIm}TK�,�{�gIm}�T0{�gRe}TK�,�{�gRe}

(79)

6. ANALYTICAL AND NUMERICAL RESULTS FOR THIN BEAMS

In this section we present the results obtained in a series of numerical tests to evaluate theperformances of the thermoelastic elements developed in this work. As test bench we consideran isotropic thin cantilever silicon beam L=57mm long and w=10mm wide and with thicknessh=92�m. Material properties of silicon are reported in Table I. The loss angle of this beam wasmeasured at a number of frequencies [22] and was found to be in agreement with the theoreticalvalues obtained by solving the two-way coupled equations for thermoelastic damping [1].

We consider two sets of boundary conditions for the beam: clamped–free and clamped–clamped.The isothermal resonant frequencies are given by the well-known solution of the eigenvalue problemfor bending vibrations of beams:

�n =a2nh

L2

√E

12�(80)

where an ={1.875,4.694,7.855, . . .} for clamped–free while an ={4.730,7.853,10.996, . . .} in theclamped–clamped case. The eigenfrequencies are summarized in Tables II and III for the first fivebending modes. The expected loss angle is computed by the relation [1]:

�n = E2T0Cp

(6

�2− 6

�3sinh�+sin�

cosh�+cos�

)(81)

where �=h√

�n/2�, � is the thermal diffusivity and Cp is the volumetric specific heat at constantpressure.

The thermoelastic damping analysis of the beam was implemented in a finite element solver(APFEM) developed using Mathematica [23]. We performed a structural–thermal harmonicanalysis, forcing one mode at a time. Each modal shape is driven at its resonances by a properset of harmonic forces: the clamped–free beam is driven by a force applied on its tip while the

Table I. Material properties of Silicon (data from [21]) at 300K.Material properties

Young’s modulus, E 162.4 GPaPoisson modulus, � 0.28Density, � 2330 kgm−3

Thermal expansion coefficient, 2.54×10−6 K−1

Thermal conductivity, 145 Wm−1K−1

Heat capacity per unit volume, cE/� 711 Jkg−1K−1



Table II. Analytical (ANA) and numerical (FE) modal frequencies of the first five normalbending modes for the thin clamped–free silicon beam.

Frequency (Hz) Frequency (Hz) Frequency (Hz)an (ANA) (FE 8-node element) (FE 20-node-element)

1.87 38.18 38.562 38.7924.69 239.31 241.45 242.957.85 670.15 676.93 681.2310.99 1313.15 1329.6 1338.314.14 2170.74 2203.5 2218.7

The numerical frequency is referred to planar discretization level (3, 60).

Table III. Analytical (ANA) and numerical (FE) modal frequencies of the first five normalbending modes for the thin clamped–clamped silicon beam.

Frequency (Hz) Frequency (Hz) Frequency (Hz)an (ANA) (FE 8-node element) (FE 20-node-element)

4.730 243.00 246.97 248.127.853 669.81 680.42 684.0810.996 1313.26 1335.4 1343.414.137 2170.69 2211.3 2226.117.2788 3242.72 3309.3 3334.6

The numerical frequency is referred to planar discretization level (3, 60).

Figure 4. The beam is described by a single layer of 8-node plane elements or by four layers of 20-nodethree-dimensional elements. A fixed number of three elements are placed along the beam’s width, whiledifferent discretization levels are used along the beam’s length. We show here planar discretization levels:(3,10), (3,30), (3,60). The first index is the number of division along the beam’s while the second is the

number of division along the beam’s length.

clamped–clamped beam is driven by a set of forces applied where the modal shape has itsmaximum values. The solver reads geometry and boundary conditions data from the output of acommercial grid generator, performs the global matrix assembly and solves the complex equationsystem according to the formulation described in the previous sections. The loss angle is thenevaluated on the solution according to Equation (79). Meshes with different number of elements(Figure 4) are used to test the convergence of the result. As shown in Figure 5, the convergenceis guaranteed with a number of 60 elements along the beam’s length.



10 20 30 40 50 60

3.0x10–7

3.5x10–7

4.0x10–7

4.5x10–7

5.0x10–7

5.5x10–7

Convergence of the loss angle (f=38 Hz)

φ_LR

number of division along a plate's axis

φ_FE

φ_F

E

Figure 5. Convergence analysis. The loss angle of the first bending mode of the clamped–free beamis evaluated by an FE analysis based on the 8-node plane element. The mesh is gradually refined by

increasing the number of elements along the beam’s length.

1E-71E-7

1E-6

1E-5

1E-4

38 Hz

240 Hz

671 Hz

1315 Hz2174 Hz

φ_FE= φ_ANA

φ_F

E

1E-6

1E-5

1E-4

φ_F

E

φ_ANA(a)

Clamped-free beam Clamped- clamped beam

(b)

φ_ANA=φ_FE

3243 Hz2171 Hz

1313 Hz

670 Hz

243 Hz

1E-6 1E-5 1E-4

φ_ANA

1E-6 1E-5 1E-4

Figure 6. 8-node thermoelastic element. Comparison of the loss angle computed with finite element �FEand the theoretical values �ANA for the clamped–free beam (a) and for the clamped–clamped beam (b).The continuous line represents the points where �FE=�ANA. The frequency values close to the data refer

to the modes listed in Tables II and III.

The results obtained with the 8-node thermoelastic element and the 20-node thermoelasticelement for the two different boundary conditions are summarized in Figure 6 and in Figure 7respectively. Here the loss angles for the first five bending modes are compared with the theoreticalvalues given by Equation (81). An agreement within 5% is obtained for the majority of the modesand for the two boundary conditions, confirming a good match with the theory. We remark that 8-node thermoelastic plane elements give accurate results while requiring much lower computationalefforts than three-dimensional 20-node elements.



1E-71E-7

1E-6

1E-5

1E-4

1E-6

1E-6

1E-5

1E-4

(b)(a)

3335 Hz2226 Hz

1343 Hz

684 Hz

248 Hz

Clamped-clamped beam

φ_F

E

φ_ΑΝΑ

2219 Hz1338Hz

681 Hz

243 Hz

39 Hz

φ_FE= φ_ANA

φ_F

E

φ_ΑΝΑ

Clamped-free beam

φ_FE= φ_ANA

1E-5 1E-41E-6 1E-5 1E-4

Figure 7. 20-node thermoelastic element. Comparison of the loss angle computed with finite element �FEand the theoretical values �ANA for the clamped–free beam (a) and for the clamped–clamped beam (b).The continuous line represents the points where �FE=�ANA. The frequency values close to the data refer

to the modes listed in Tables II and III.

7. CONCLUSIONS

This paper presented a new finite elements formulation with thermoelastic capabilities. Startingfrom a variational form of thermoelasticity, the thermoelastic damping is calculated from theirreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strainvariations. We introduced a dissipation function D, which can be integrated over an oscillationperiod to evaluate the dissipated energy. On this basis we developed a finite element framework forthe computation of the thermoelastic damping involving plates and/or massive bodies, with twoelements for solving problems involving either thin or thick solid bodies. For the thin structureswe proposed a plane Reissner–Mindlin element with extended capabilities to model through-the-thickness thermal effects. In this element out-of-plane mechanical deformation follows theReissner–Mindlin assumptions while heat flux exchanges through-the-thickness are modeled by alinear interpolation of the a temperature shift �. We showed that this element allows an accurateevaluation of the thermoelastic damping in a thin beam with small computational efforts. Thickbodies may be also modeled by a standard quadratic iso-parametric element.

Our finite element framework will have immediate application in the evaluation of the thermoe-lastic damping in multi-layered structures made by bonding of silicon wafers [24]. It could alsobe used to model laminated composites of layered thin films, adopted in the manufacture of somemicroresonators [25] and in the production of high-reflectivity mirrors [4]. Future extensions ofthis work could include the treatment of a finite value of the thermal wave speed.

ACKNOWLEDGEMENTS

The authors are grateful to Ana Maria Alonso Rodriguez for the helpful discussions and suggestionsabout the finite element formulation for thermoelasticity. This work was supported partially by EuropeanCommunity (project ILIAS, c.n. RII3-CT-2004-506222) and by Provincia Autonoma di Trento (projectQL-Readout).



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