A Finite Element Formulation

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. 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), Italy2 Istituto di Fotonica e Nanotecnologie CNR-FBK, via alla Cascata, 56/C 38100 Povo (Trento), Italy

3 Istituto 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 Biots 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 ReissnerMindlin 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 the

computational 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; ReissnerMindlin 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.


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E. SERRA AND M. BONALDI

and NEMS devices, thermoelastic damping establishes an absolute lower bound on internal friction

and, in many cases, is the dominant component of damping at room temperature (300 K). For

this reason, the study of thermoelastic damping in flexures is an active area of current theoretical

research

[1, 2

]. In precision measurements, thermoelastic damping acts as a source of mechanical

thermal 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 the

thermoelastic damping in beams and he gave an analytical approximation for the loss angle of

those 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. They

calculated the loss angle of microbeams and found results close to those obtained with Zeners

model. Another simple formula based on the extension of the Lifshitz and Roukes approach was

obtained by Wong 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 damping

was 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 more

complex geometries or boundary conditions are considered. In fact, the loss angles of structural

modes, which vary spatially across the plate width, cannot be predicted using a simple beam or

plate 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 relaxation

times 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 problem

involved in beam resonators has been addressed using a combination of finite element method

and eigenvalue formulation [11]. This approach allows the evaluation of the eigenfrequencies andtheir associated quality factor of resonance, even in case of complex geometries and boundary

conditions. 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 fully

described. The purpose of this paper is to present a new finite element formulation for computing

the loss angle of a vibrating structure using either planar or full three-dimensional elements. We

derive a weak form for linearly coupled thermoelasticity boundary value problem (BVP), where

an infinite velocity of the thermal field and small harmonic temperature variations with respect to

the equilibrium temperature are assumed. From the weak form a dissipation function is recognized

and used for evaluating the work lost due to thermoelastic effect. We show that the dissipation

function is substantially the same of the dissipation function introduced by M. A. Biot in its early

variational 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 the

M. A. Biot variational principle. In the finite element formulation two elements are considered.

The first is a new 8-node plane ReissnerMindlin element which can be used in modeling thin andmoderately thick structures like coatings or the interface layer between two bonded materials. The

second thermoelastic element is a 20-node element and can be used to model thick structures. For

the 8-node element we use the well-known ReissnerMindlin assumptions and the temperature

field along the element thickness is taken into account by properly modifying the element shape

functions. To validate the procedure we compute the thermoelastic damping of several vibrational

modes 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


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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 constitutive

equations used in the finite element formulation. We introduce here the dissipation function used

to evaluate the thermoelastic damping. In Section 3 we present the weak formulation of coupled

thermoelasticity BVP and show the connections with the well-known Biots variational principle.Section 4 introduces the thermoelastic elements with their constitutive relations. The general Finite

Element framework is detailed in Section 5, where the numerical procedure used to compute the

thermoelastic loss angle is also illustrated. In Section 6 analytical results are compared with the

numerical ones for the dissipation of a thin beam. The paper ends with concluding remarks in

Section 7.

2. MATHEMATICAL FRAMEWORK FOR THE THERMOELASTIC DAMPING

We now review the BVP for linearly coupled thermoelasticity in case of small vibrations and small

temperature changes with respect to the equilibrium temperature T0. Straindisplacement relation,balance equations and constitutive equations are needed to formulate the general thermoelastic

problem. We also recall the definition of thermoelastic damping of a body.

2.1. Basic equations for linear coupled thermoelasticity in solids

2.1.1. Straindisplacement equation. When considering small strains and displacements, the kine-

matics equations consist of the well-known straindisplacement 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 and

the 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 of

the 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 by

the values of a certain set of independent variables: the kinematic variables (strains i j , . . .) and

the 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


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to the strain tensor i j , in the same way the entropy per unit volume s is thermodynamically

conjugated to the temperature shift = TT0:

i j =

*F

*i j, s

=*F

*(4)

where F(i j ,) is Helmholtzs free energy. If we retain only the quadratic terms in the Taylor

series expansion in the vicinity of the equilibrium state (i j =0,=0), we obtain

F(i j ,)=1

2Ci j k l i j kl i j i j +

cE

2T02 (5)

where

Ci j k l =*

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 j k l kl i j (7)

s = *F*

=i j i j +cE

T0 (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 Fouriers

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 j k l = i j kl +(i kj l +il j k)i j = (3+2)i jki j = i j

(10)

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

conductivity.

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 of

motion to obtain the normal modes of vibration and their corresponding eigenfrequencies n . In

general, the eigenfrequencies are complex and their real part gives the oscillation frequency while


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the imaginary part gives the attenuation of the vibration. The loss angle at the frequency n is

then obtained as:

=2 Im(n)

Re(n) (11)

The loss angle can be also evaluated as the work lost per radiant normalized to the maximum

elastic energy stored in body during a cycle. In this case

= W2Wm

(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 (GouyStodola

theorem):

W=

cycle

V

,i ki j ,j

T0dVdt (13)

as can be easily shown by combining Equations (2), (9) and after integrating over the bodys

volume [12, 15, 16]. If we define the dissipation function D as

D= 12

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

(12)*kk

*t(16)

where cE,E, is the volumetric specific heat per unit volume in the absence of deformation,

Youngs modulus of the body and the Poisson ratio. Small temperature variations are assumed and

the 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 temperature

field and conversely the temperature field affects the strain field or the stress field. This is known

as two-way coupling. In many cases the thermoelastic damping is evaluated by considering the

strain or the stress trace in isothermal condition. This approximation is called one-way coupling

because the heat equation is solved with a source term (strain or stress trace) which is independent

from the temperature field [17].


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3. FINITE ELEMENTS FOR THERMOELASTIC DAMPING ANALYSIS

We now overview the weak form for thermoelasticity and the approach of M. A. Biot, showing

its 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 of

the thermoelastic BVP. Consider a solid at thermodynamic equilibrium, described by the state

variables (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) can

be 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:V

qi,i dV=

V

qi ,i dV+

A

qi ni dA (19)

then the integral equation (18) may be rewritten as:V

T0s dV

V

qi ,i dV+

A

qi ni 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 straindisplacement

equation (1): V

i j,j ui dV=

V

i j i j dV+

A

i j nj 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.


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Then integral equation (21) may be rewritten as:

V

i j i j dV+

A

i j n j ui dA+

V

( fi ui )ui dV=0 (23)

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

V

T0s dV

V

qi ,i dV+

V

i j i j dV =

A

qi ni dA

+

A

i j nj 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+V cE dV+V ,i ki j ,j dV+

V

i j Ci j k l kl dV

V

i j i j d V+

V

ui ui dV

=

A

qi ni dA+

A

i j nj ui dA+

V

fi ui dV (25)

Appropriate boundary conditions must be specified to guarantee that the thermoelastic BVP is

well-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 Auqi ni

= Q on A

Q,

= on A

(26)

The thermoelastic damping is due to irreversible heat flux inside the body. In the variational

equation (25) we identify the term linked to this dissipation mechanism and consider it as the

variation 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 Biots principle

Our dissipation function is closely related to the dissipation function in Biots variational principle.

According to Biot[12, 13

]a variational principle can be obtained in the framework of irreversible

thermodynamics by considering (ui ,Hi ) as state variables, where Hi is related to the entropy of the

system 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 thermoelastic

potential and the dissipation function D, whose variation is:

(+D)=

A

Hi dA+

A

i j nj ui dA+

V

( fi ui )ui dV (28)


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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 variationW, 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

V

k1i j ( Hj Hi)dV (29)

The dissipation function is related to the irreversible entropy produced inside the body due to the

volume deformations. In fact we haveDdt=

T0

2

V

k1i j Hj Hi dVdt (30)

and taking into account the definition of Hi , Hj and the symmetry of the conductivity tensorki j =kj i (Onsagers reciprocity relations), we obtain on a cycle:

Ddt=1

2T0

V

,i ki j ,j dVdt (31)

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

D=DT0 (32)This relation gives a deeper insight into the dissipation function D, which we use in the following

to evaluate the thermoelastic loss angle. We point out that a finite element formulation based on

Biots 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 solid

bodies. 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 ReissnerMindlin plate element

We seek a coupled-field element which takes into account the in-plane and out-of-plane mechanical

deformations and the thermal balance but at the same time we want to preserve a small number

of DOFs. For this reason we derive the deformation along the plate thickness due to bending

and shear stresses by following a given plate theory. A new planar 8-node serendipity coupled-

field element is developed on the basis of the ReissnerMindlin theory. The basic hypothesis of

ReissnerMindlin theory are:

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

u1 =x3x1 (x1,x2), u2 =x3x2 (x1,x2), u3 =u3(x1,x2) (33)where x1 ,x2 are the rotations while x3 is the element local coordinate along the plate

thickness. This implies that the normal remains straight after deformation, but it is not


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necessarily perpendicular to the deflected mid-surface. The transverse displacement u3 does

not vary along the thickness.

(ii) the stress along the plates thickness is negligible (33 =0).(iii) a shear correction factor m

=56

is introduced to account for the parabolic distribution of

the shear stress along the thickness.

A full three-dimensional description of the thermoelastic damping requires a proper handling of the

thermal balance in the element. As shown in Equation (16), the heat flux due to the thermoelastic

coupling represents the source term in the heat equation and is related to the time derivative of

the 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 constitutive

equation (7):

out-of-plane 33 =

+2x3*x1*x1

+ *x2*x2

(35)

Combining the above relations and using the definitions of Lame constants, the heat generated

by the thermoelastic coupling for small harmonic oscillations with respect to the equilibrium

temperature 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 plates neutral plane and changes sign when going from the upper to the

lower surface.

Now we discuss the linear approximation of the temperature shift following the approach

developed 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 temperature

profile can be determined solving heat equation (16) that becomes

*2

*x 23 =jcE

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)


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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)=ET0

cE(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 be

written as:

(x1,x2,x3)=ET0

cE(1)

*x1*x1

+ *x2*x2

x3

1 1

cos(h/2)

+1

6

2x 33cos(h/2)

+o(x 33)

(40)

If we consider only the linear part in order to maintain the equations system of small size, only

one coefficient is needed to compute the temperature shift. If we would add the cubic term, an

additional 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 functions

are the well-known quadratic serendipity functions that satisfy completeness and compatibility

requirements. These functions are well suited in modeling thin or moderately thick structures as

described 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 in

Figure 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 middle

plane u3 inside each element are obtained by interpolating the nodal values with shape functions:

x1 =8

i=1N(i)(i)x1 , x2 =

8i=1

N(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)


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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 plates 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 =

8i=1

N(i)x(i)2 , x3 =

h

23 (46)

where x(i)1 ,x

(i)2 are the cartesian coordinate of the i th node. Here we assume a linear dependence

of through-the-thickness coordinate x3 and h is the plates thickness which does not involve any

nodal coordinate.


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4.1.2. Straindisplacement and constitutive equations. Let us first derive the straindisplacement

relation. 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 ={11,22,212} are the bending components and s ={223,231} 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

*x2x2

*u3

*x1x1

(49)

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

33=

+2

(11+

22) (50)

Thus, the straindisplacement 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)


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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

{}= CB{e}{}TN{e} (55)

where elastic matrix can be written as:

C =

Cb 0

0 Cs

=

+2

0

+2 00 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 {}={bs}={{112212}{2331}}. The entropy constitutive law(8) may be written in matrix form as:

s ={}TB{e}+cE

T0N{e} (58)

Fouriers law in matrix notation is

{q}=K {}=KB{e} (59)


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where

B =J1

3*N(1)

*1 3

*N(8)

*1

3*N(1)

*2 3

*N(8)

*2

N(1) N(8)

(60)

and J1 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 straindisplacement and constitutive relations are formally the same as it is

described in the above section. In deriving the straindisplacement and constitutive matrices the

six component of the stress and the strain tensor must be considered. In this case the unknowns

vectors 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 . The

variational equation (24) becomes:

2

V

{u}T{u}dV+j

V

T0 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. The

solution in time is given by as the real part of these phasors. In order to derive the algebraic system

of equations we observe that each volume integral on the left-hand side of (61) is the sum of the

volume integral over the elements of the mesh. Each integral must be computed by substituting

the 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)


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where

M, =e

VeNT N dV (63)

The shape function matrix N has size 3n 1 while the mass matrix M, has size 3n3n where n isthe number of nodes into volume V which is discretized by the 8-node or by 20-node thermoelastic

elements. 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

V

T0 sdV=j{g}TD,{g}+{g}TD,{g} (64)

where

D,=

T0e

VeN

T

{}

T

{B

}dV (65)

and

D, =

e

Ve

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 nn. The global temperature shift is defined as{g}={(1) . . . (n)}T. N has size n 1.

The third term becomes

V

{

}T

K

{

}dV

={g

}T

K,

{g

}(67)

where:

K, =

e

Ve

BT KB 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 CB dV (70)

and

K, =

e

Ve

BT {}N dV (71)


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The stiffness matrix K, has size 3n3n. The stiffness matrix K, has size 3nn and representsthe effects of the thermal field in the equation of motion.

Volume integrals are then evaluated in the elements natural coordinates. The matrix system

becomes:

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 on

displacements 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 the

two-way coupling due to thermoelasticity. These matrices are related according to the equation:

D, = T0KT, (73)

5.1. Evaluation of the thermoelastic loss angleWe evaluate the thermoelastic loss on the basis of the dissipation function defined in Equation (14),

which in vector notation becomes:

D= 12

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= 12

e

Ve

{||}TK{||}cos2(t+)dV (76)

According to Equation (15), to evaluate the thermoelastic damping we integrate the dissipation

function over a cycle:

cycle

Ddt=2/

0

Ddt= 14

2/0

e

Ve

{||}TK{||}dVdt

+1

4

2/

0

e

Ve{|

|}T

K

{|

|}cos(2t

+2)dVdt (77)

where we have used the cosine bisection formula. In Equation (77) we observe that the second

term vanishes and after some algebras it changes into:cycle

Ddt= 2

e

Ve

({eRe}TBT KB{eRe}+{eIm}TBT KB{eIm})dV (78)


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The temperature shift module may be written as: ||=(

gRe)

2 +(gIm)2. Hence, according toEquation (15) the loss angle is given by:

()= {

g

Re}TK,

{

g

Re}+{

g

Im}TK,

{

g

Im}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 the

performances of the thermoelastic elements developed in this work. As test bench we consider

an isotropic thin cantilever silicon beam L =57mm long and w=10mm wide and with thicknessh =92m. 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: clampedfree and clampedclamped.The isothermal resonant frequencies are given by the well-known solution of the eigenvalue problem

for bending vibrations of beams:

n =a2nh

L2

E

12(80)

where an ={1.875,4.694,7.855, . . .} for clampedfree while an ={4.730,7.853,10.996, . . .} in theclampedclamped case. The eigenfrequencies are summarized in Tables II and III for the first five

bending modes. The expected loss angle is computed by the relation [1]:

n =E2T0

Cp 6

2 6

3

sinh

+sin

cosh +cos (81)where =hn/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 structuralthermal harmonicanalysis, forcing one mode at a time. Each modal shape is driven at its resonances by a proper

set of harmonic forces: the clampedfree 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

Youngs modulus, E 162.4 GPaPoisson modulus, 0.28

Density, 2330 kg m3Thermal expansion coefficient, 2.54106 K1Thermal conductivity, 145 W m1K1Heat capacity per unit volume, cE/ 711 J kg

1K1


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Table II. Analytical (ANA) and numerical (FE) modal frequencies of the first five normalbending modes for the thin clampedfree 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 clampedclamped 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 beams width, whiledifferent discretization levels are used along the beams length. We show here planar discretization levels:(3,10),(3,30),(3,60). The first index is the number of division along the beams while the second is the

number of division along the beams length.

clampedclamped 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 a

commercial grid generator, performs the global matrix assembly and solves the complex equation

system according to the formulation described in the previous sections. The loss angle is then

evaluated 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 convergence

is guaranteed with a number of 60 elements along the beams length.


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10 20 30 40 50 60

3.0x107

3.5x107

4.0x107

4.5x107

5.0x107

5.5x107

Convergence of the loss angle (f=38 Hz)

_LR

number of division along a plate's axis

_FE

_

FE

Figure 5. Convergence analysis. The loss angle of the first bending mode of the clampedfree 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 beams length.

1E-7

1E-7

1E-6

1E-5

1E-4

38 Hz

240 Hz

671 Hz

1315 Hz2174 Hz

_FE= _ANA

_

FE

1E-6

1E-5

1E-4

_

FE

_ANA(a)

Clamped-free beam Clamped- clamped beam

(b)

_ANA=_FE

3243 Hz

2171 Hz

1313 Hz

670 Hz

243 Hz

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

_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 clampedfree beam (a) and for the clampedclamped 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 7

respectively. Here the loss angles for the first five bending modes are compared with the theoretical

values given by Equation (81). An agreement within 5% is obtained for the majority of the modes

and 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 computational

efforts than three-dimensional 20-node elements.


DOI: 10.1002/nme


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1E-7

1E-7

1E-6

1E-5

1E-4

1E-6

1E-6

1E-5

1E-4

(b)(a)

3335 Hz

2226 Hz

1343 Hz

684 Hz

248 Hz

Clamped-clamped beam

_

FE

_

2219 Hz1338Hz

681 Hz

243 Hz

39 Hz

_FE= _ANA

_

FE

_

Clamped-free beam

_FE= _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 clampedfree beam (a) and for the clampedclamped 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. Starting

from a variational form of thermoelasticity, the thermoelastic damping is calculated from the

irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain

variations. We introduced a dissipation function D, which can be integrated over an oscillation

period to evaluate the dissipated energy. On this basis we developed a finite element framework for

the computation of the thermoelastic damping involving plates and/or massive bodies, with two

elements for solving problems involving either thin or thick solid bodies. For the thin structures

we proposed a plane ReissnerMindlin element with extended capabilities to model through-the-thickness thermal effects. In this element out-of-plane mechanical deformation follows the

ReissnerMindlin assumptions while heat flux exchanges through-the-thickness are modeled by a

linear interpolation of the a temperature shift . We showed that this element allows an accurate

evaluation of the thermoelastic damping in a thin beam with small computational efforts. Thick

bodies 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 some

microresonators [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).


DOI: 10.1002/nme


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DOI: 10.1002/nme

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A Finite Element Formulation

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