thermoelastic vibrations in micro-/nano-scale beam resonators with voids

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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 2964–2977

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Thermoelastic vibrations in micro-/nano-scale beam resonatorswith voids

J.N. Sharma, D. Grover n

Department of Mathematics, National Institute of Technology, Hamirpur 177005, India

a r t i c l e i n f o

Article history:

Received 1 December 2010

Received in revised form

6 January 2011

Accepted 11 January 2011

Handling Editor: S. Ilankodamping in (micro-electromechanical systems) MEMS/(nano-electromechanical sys-

Available online 19 February 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.01.012

esponding author. Tel.: +91 9882544618.

ail addresses: [email protected] (J.N. Sharma),

a b s t r a c t

In this paper the closed form expressions for the transverse vibrations of a homogenous

isotropic, thermoelastic thin beam with voids, based on Euler–Bernoulli theory have

been derived. The effects of voids, relaxation times, thermomechanical coupling, surface

conditions and beam dimensions on energy dissipation induced by thermoelastic

tems) NEMS resonators are investigated for beams under clamped and simply

supported conditions. Analytical expressions for deflection, temperature change,

frequency shifts and thermoelastic damping in the beam have been derived. Some

numerical results with the help of MATLAB programming software in case of magne-

sium like material have also been presented. The computer simulated results in respect

of damping factor and frequency shift have been presented graphically.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The theory of elastic materials with voids is concerned with the elastic materials consisting of a distribution of smallpores (voids) which contains nothing of mechanical or energetic significance and is one of the most recent generalizationsof the classical theory of elasticity. The nonlinear theory of elastic materials with voids was proposed by Nunziato andCowin [1] and the linearized version was deduced by Cowin and Nunziato [2] where voids have been included as anadditional kinematics’ variable. Iesan [3] developed a linear theory of thermoelastic material with voids and alsoestablished the theorems concerning the uniqueness of solution, the reciprocity relation and variational characterization ofsolution. Sharma and Kaur [4] studied the plane harmonic waves in generalized thermoelastic materials with voids andSharma et al. [5] carried out an exact free vibration analysis of simply supported, homogenous isotropic, cylindrical panelin the three-dimensional generalized thermoelasticity with voids.

Many decades ago Zener [6] explained the mechanism of thermoelastic damping and derived an analytical solution torelate the energy dissipation (i.e. quality factor) and the material properties of a thin beam structure. However, somemathematical and physical simplifications were assumed in this derivation. Lifshitz and Roukes [7] reported a higherprecision solution, although the direct application of solution was still limited to beam systems. Yi and Martin [8] usedfinite element analysis to study eigenvalue solution of TED in a Beam Resonator. Lapage and Golinval [9] presented a2-D finite element modal to compute the frequency shifts in a small set of doubly clamped single-crystal siliconmicro-resonators and validated the results with those of Lipshitz and Roukes [7] in case of one-dimension. Prabhakaret al. [10] used Galerkin technique to calculate the frequency shifts due to thermoelastic coupling in micro-mechanical and

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[email protected] (D. Grover).

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J.N. Sharma, D. Grover / Journal of Sound and Vibration 330 (2011) 2964–2977 2965

nano-mechanical resonators by considering two-dimensional (2-D) heat conduction. Guo and Rogerson [11] and Sunet al. [12] presented two-dimensional (2-D) analysis of frequency shifts by considering heat conduction along the beamthickness and beam span whereas former assumed cubic and later took sinusoidal temperature gradients across thethickness of the beam prior to the solution of coupled equation for flexural vibrations. Sun and Saka [13] investigated thethermoelastic damping of the vibrations in arbitrary direction in a coupled thermoelastic circular plate. Sharma andSharma [14] studied damping and phase velocities of surface wave modes of generalized thermoelastic circular plateresonators. In order to compute the magnitude of TED in plates [15,16], ring [17]. Sun and Tohmyoh [18] studiedthermoelastic damping of the axisymmetric vibration in circular beams.

Micro- or nano-electromechanical structures have received great interests in recent years due to their extremely highresonance frequencies, small size, and large-scale integration-capability. Modeling and simulation of thermoelasticdamping is a recurrent interest in the community of micro-mechanics and micro-engineering, mainly motivated by therecent advancement of MEMS and NEMS technologies. Thermoelastic damping is recognized as a significant lossmechanism at room temperature in micro-scale thin beam resonators. Honsten et al. [19] predicted that the internalfriction in 50 nm scale silicon based MEMS structures is strong due to thermoelastic damping. Vig et al. [20] proposed amicro-scale resonator based sensor array to be used as Infrared (IR) sensors. Keeping in view the applications of sensors(resonators) in detecting Infrared (IR) imaging, chemical and biological agent sensing and design/construction of precisionthermometers, the present paper is devoted to study frequency shifts and damping due to thermal variations inhomogenous isotropic thermoelastic beam with voids in the context of Lord and Shulman [21] model of non-classical(generalized) thermoelasticity.

2. Basic equations

We consider a homogenous isotropic thermoelastic solid with voids in Cartesian coordinate systems ox1x2x3 initiallyundeformed and at uniform temperature T0 with initial volume fraction f0. The basic governing equations of motion,balance of equilibrated force and heat conduction in the context of generalized (non-Fourier) thermoelasticity fordisplacement vector u

!ðx1,x2,x3,tÞ ¼ ðu1,u2,u3Þ and temperature change T(x1,x2,x3,t), in the absence of body forces, external

loads, extrinsic equilibrated body force and heat sources, are given by [4]

@s11

@x1þ@s12

@x2þ@s13

@x3¼ r @

2u1

@t2(1)

@s21

@x1þ@s22

@x2þ@s23

@x3¼ r @

2u2

@t2(2)

@s31

@x1þ@s32

@x2þ@s33

@x3¼ r @

2u3

@t2(3)

rw €f ¼ ar2f�brU u!�x1f�x2

_fþmT (4)

Kr2T�rCeð_Tþt0

€T Þ ¼ bT0 rU_u!þt0rU

€u!

� �þmT0

_fþt0€f

� �(5)

where

sij ¼ ldijekkþ2meijþ bf�bTð Þdij, i,j,k¼ 1,2,3 (6)

Here l, m are the Lam�e’s parameters; K is the thermal conductivity; r and Ce are the density and specific heat at constantstrain, respectively; b=(3l+2m)aT, aT is the linear thermal expansion; a, b, x1, x2, m and w are the material constants dueto the presence of voids and t0 is the thermal relaxation time.

3. Modeling of beam structures

We consider small flexural deflection of a thin thermoelastic beam with voids and having dimensions of lengthL(0rx1rL), width a(0rx2ra) and thickness h �h=2rx3rh=2

� �: We take x1-axis along the axis of beam, x2-axis along

width and x3-axis along the thickness. In equilibrium the beam is unstrained, unstressed and also kept at uniformtemperature T0 and volume fraction f0. The beam undergoes bending vibrations of small amplitude about the x1-axis suchthat the deflection is consistent with the linear Euler–Bernoulli theory. That is, any plane cross-section initiallyperpendicular to axis of beam remains, plane and perpendicular to the neutral surface during bending. Thus thedisplacements are given by

u1 ¼�x3@u3

@x1, u2 ¼ 0, u3ðx1,x2,x3,tÞ ¼ u3ðx1,tÞ (7)

J.N. Sharma, D. Grover / Journal of Sound and Vibration 330 (2011) 2964–29772966

where t is the time. Then, the flexural moment of cross-section is given as follows:

Mðx1,tÞ ¼

Z h=2

�h=2as11x3 dx3 ¼ ðlþ2mÞIu3,11�bMfþbMT (8)

where I¼ ah3=12 is moment of inertia of the cross-section and

Mf ¼ a

Z h=2

�h=2fx3 dx3 and MT ¼ a

Z h=2

�h=2Tx3 dx3

are the moment of beam due to the presence of voids and thermal effects, respectively.Now the equation of transverse motion of a beam is

@2M

@x21

þrA@2u3

@t2¼Dp (9)

where r is the density; A=ah is the area of cross-section and Dp=p1�p2. Here p1and p2 being the pressures on the upperand lower surface of the beam. Using Eq. (8) in Eq. (9), we obtain

ðlþ2mÞI @4u3

@x41

�b@2Mf

@x21

þb@2MT

@x21

þrA@2u3

@t2¼Dp (10)

The equation of balance of equilibrated force is given by

@2f@x2

1

þ@2f@x2

3

�rwa€fþ

bx3

a@2u3

@x21

�x1

a ðfþx_fÞþ

m

a T ¼ 0 (11)

The heat conduction equation in the context of generalized theory (LS) with voids is given by

@2T

@x21

þ@2T

@x23

�rCe

Kð _Tþt0

€T Þ�mT0

Kð _fþt0

€fÞþbT0x3

K

@2

@x21

ð _u3þt0 €u3Þ ¼ 0 (12)

Hence system of Eqs. (10)–(12) governs the transverse vibrations in a thermoelastic beam with voids.

4. Solution along thickness direction

In order to solve system of Eqs. (10)–(12), we take the solution of time harmonic vibrations of the beam as

½u3ðx1,tÞ,fðx1,x3,tÞ,Tðx1,x3,tÞ� ¼ ½Uðx1Þ,Fðx1,x3Þ,Yðx1,x3Þ�expðiotÞ (13)

Using Eq. (13) in system of Eqs. (10)–(12), we obtain

ðlþ2mÞI d4U

dx41

�bd2Mf0

dx21

þbd2MT0

dx21

�rAo2U ¼ 0 (14)

@2F@x2

1

¼�@2F@x2

3

þx1x0�rwo2

a

!F�

bx3

a@2U

@x21

�m

aY (15)

@2Y@x2

1

¼�@2Y@x2

3

þiot0rCe

KYþ

iomT0t0

KF�

iobT0t0x3

K

@2U

@x21

(16)

where we have taken Dp=0 and

Mf0¼ a

Z h=2

�h=2Fx3 dx3, MT0

¼ a

Z h=2

�h=2Yx3 dx3, x0 ¼ 1þ iox, x¼

x2

x1, t0 ¼ 1þ iot0 (17)

In case there is no flow of heat and change of volume fraction across the upper and lower surfaces of the beam, then we have

@Y@x3¼ 0¼

@F@x3

at x3 ¼ 7h

2(18)

In case Eqs. (15) and (16) are uncoupled to each other in respect of temperature (Y) and volume fraction (F) so thatm=0 and the variation of temperature and volume fraction field are steady in the plane perpendicular to thicknessdirection and hence the trial solution of resulting system of equations satisfying the boundary conditions (18) is written as

Fðx1,x3Þ ¼b

ðx1x0�rwo2Þx3�

sinqx3

qcosðqh=2Þ

� �d2U

dx21

Yðx1,x3Þ ¼bT0

rCex3�

sinpx3

pcosðph=2Þ

� �d2U

dx21

(19)

It is mentioned here that the values of p and q are still subjected to modifications.


Differentiating solutions (19) w.r.t. x3 twice and then substituting for @2F=@x23 and @2Y=@x2

3 in Eqs. (15)–(16), we obtain

d2Fdx2

1

¼�q�b

x1x0�rwo2

!sinqx3

qcosðqh=2Þþ

mbT0

arCe

� �x3�

sinpx3

pcosðph=2Þ

� �" #d2U

dx21

d2Ydx2

1

¼iot0mT0b

K x1x0�rwo2� � x3�

sinqx3

qcosðqh=2Þ

� ��

p�bT0

rCe

� �sinpx3

pcosðph=2Þ

24

35d2U

dx21

(20)

where

p� ¼ p2þiot0rCe

K, q� ¼ q2þ

ðx1x0�rwo2Þ

aNow

Mf0¼ a

Z h=2

�h=2Fx3 dx3 and MT0

¼ a

Z h=2

�h=2Yx3 dx3

with the help of Eq. (20) provide us

d2Mf0

dx21

¼ a

Z h=2

�h=2

d2Fdx2

1

x3 dx3 ¼ Ibq�

ðx1x0�rwo2ÞgðoÞ�mbT0

arCeð1þ f ðoÞÞ

" #d2U

dx21

(21)

d2MT0

dx21

¼ a

Z h=2

�h=2

d2Ydx2

1

x3 dx3 ¼ IbT0p�

rCef ðoÞþ iot0mT0b

ðx1x0�rwo2ÞKð1þgðoÞÞ

" #d2U

dx21

(22)

where

f ðoÞ ¼ 24

p3h3

ph

2�tan

ph

2

� , gðoÞ ¼ 24

q3h3

qh

2�tan

qh

2

� (23)

Upon using Eqs. (21)–(22) in Eq. (14), we obtain

ðlþ2mÞI d4U

dx41

þ bFðoÞþbGðoÞ �

Id2U

dx21

�rAo2U ¼ 0 (24)

where

GðoÞ ¼ mbT0

arCeð1þ f ðoÞÞ� bq�

ðx1x0�rwo2ÞgðoÞ

FðoÞ ¼ bT0p�

rCef ðoÞþ iot0mT0b

ðx1x0�rwo2ÞKð1þgðoÞÞ (25)

Also Eq. (17) with the help of Eq. (19) implies that

d2Mf0

dx21

¼bI

ðx1x0�rwo2Þ1þgðoÞ½ �

d2

dx21

d2U

dx21

!(26)

d2MT0

dx21

¼IbT0

rCe1þ f ðoÞ � d2

dx21

d2U

dx21

!(27)

The comparison of Eqs. (21)–(22) with Eqs. (26) and (27) provides us

GðoÞffi� b

ðx1x0�rwo2Þ1þgðoÞ½ �

d2

dx21

(28)

FðoÞffi bT0

rCe1þ f ðoÞ � d2

dx21

(29)

Using Eqs. (28)–(29) in Eq. (24), we obtain

Dod4U

dx41

�rAo2U ¼ 0 (30)

where

Do ¼ ðlþ2mÞI 1þeT ð1þ f ðoÞÞ�ef x0�rwo2

x1

� ��1

ð1þgðoÞÞ" #

,

ef ¼b2

x1ðlþ2mÞ, eT ¼

b2T0

rCeðlþ2mÞ


Here ef, eT are the elasto-voids and thermomechanical coupling constants of the beam, respectively. Usually, the thermalgradients in the plane of cross-section along the thickness direction of the beam are much larger than those along theperpendicular to it [13], so that @2Y=@x2

1 � 0. On the same analogy we assume that the gradient of volume-fraction field isnegligible small along perpendicular to the thickness direction of the beam and hence we may take @2F=@x2

1 � 0. Underthese assumptions, we have

p2 ¼�iorCet0

K1þ

mb

bðrwo2�x1x0Þ

" #

q2 ¼rwo2�x1x0

a

!1þ

mbT0

brCe

� (31)

Thus the solution given by Eq. (19) now represent the solution of coupled Eqs. (15) and (16) with modified values of p

and q given by Eq. (31). Thus, Eq. (19) and Eq. (30) constitute a complete set of the governing equations for thehomogenous isotropic thin thermoelastic beam with voids (TEV), when there is no pressure difference at the surfaceoccurs. In addition, these equations can also be supplemented with appropriate initial and boundary conditions of therelevant problem to be modeled.

5. Application

We consider the case of a micro- and nano-beam whose edges are either clamped or simply supported, so that we havethe following two sets of boundary conditions:

Set I: For clamped beam, we have

U ¼ 0,dU

dx1¼ 0 at x1 ¼ 0,L (32)

Set II: For simply supported beam, we have

U ¼ 0,d2U

dx21

¼ 0 at x1 ¼ 0,L (33)

As U=U(x1), so Eq. (30) provide us

d4

dx41

�Z4

" #U ¼ 0, Z4 ¼

rAo2

Do(34)

The solution of Eq. (34) can be written as

Uðx1Þ ¼ A1 sinZx1þA2 cosZx1þA3 sinhZx1þA4 coshZx1, Z2 ¼o

ffiffiffiffiffiffiffirA

Do

s(35)

Using solution (35) in Eqs. (32)–(33), we obtain the following characteristics:Equations which govern the vibrations of the beam

Set I : cosZLcoshZL¼ 1 (36)

Set II : sinZLsinhZL¼ 0 (37)

The corresponding characteristic roots of Eqs. (36)–(37) are obtained as

Set I : Z¼ 2kpL

, k 2 I

Set II : Z¼ kpL

, k 2 I (38)

The expressions for the deflection U(x1) are obtained as

Set I : Uðx1Þ ¼X1k ¼ 1

Ak sin2kp

Lx1�sinh

2kpL

x1

�

Set II : Uðx1Þ ¼X1k ¼ 1

Ak sinkpL

x1 (39)


Thus the deflection u3(x1, t) of the beam can be written from Eq. (13) with the help of Eq. (39) as

Set I : u3ðx1,tÞ ¼X1k ¼ 1

Ak sin2kp

Lx1�sinh

2kpL

x1

� eiot , ok ¼

4k2p2

L2

ffiffiffiffiffiffiffiDo

rA

s

Set II : u3ðx1,tÞ ¼X1k ¼ 1

Ak sinkpL

x1eiot , ok ¼k2p2

L2


rA

s(40)

The volume fraction and temperature distributions in the beam are given bySet I:

fðx1,x3,tÞ ¼�b

x1x0�rwo2� � x3�

sinqx3

qcosðqh=2Þ

� �X1k ¼ 1

Ak4k2p2

L2sin

2kpL

x1þsinh2kp

Lx1

� eiot

Tðx1,x3,tÞ ¼ �bT0

rCex3�

sinpx3

pcosðph=2Þ

� �X1k ¼ 1

Ak4k2p2

L2sin

2kpL

x1þsinh2kp

Lx1

� eiot (41)

Set II:

fðx1,x3,tÞ ¼ �b

x1x0�rwo2� � x3�

sinqx3

qcosðqh=2Þ

� �X1k ¼ 1

Akk2p2

L2sin

kpL

x1eiot

Tðx1,x3,tÞ ¼�bT0

rCex3�

sinpx3

pcosðph=2Þ

� � X1k ¼ 1

Akk2p2

L2sin

kpL

x1eiot (42)

Clearly, solution (40)–(42) are consistent with the physical situation of the problem including boundary conditions.

6. Frequency shift and damping

Noting that for most of the materials x1 is large so that x�11 is nearly zero and hence the vibration frequency of the beam

with voids in the presence of thermoelastic coupling and thermal relaxation time is given by

ok ¼ gk


rh

s¼o0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ef 1þ

rwx1

o2

� �1þgðoÞ½ �þeT 1þ f ðoÞ

�s(43)

where

o0 ¼gk hD�

2, D� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffilþ2m

3r

s, gk ¼

4k2p2

L2 for Set I

k2p2

L2 for Set II

8<: (44)

For most of the materials ef51, eT51 ðef ¼ 0:1613, eT ¼ 0:00182 at T0 ¼ 25 3CÞ for magnesium, we can replace f(o) withf(o0) and g(o) with g(o0) and expand Eq. (43) upto first order to obtain

ok ¼o0 1�ef2

1þrwx1

o20

� �ð1þgðo0ÞÞþ

eT

2ð1þ f ðo0ÞÞ

� (45)

Clearly the quantity p and q given by Eq. (32) are complex, therefore using Euler theorem and replacing ok with o0,we obtain

p¼ffiffiffi2p

p0 cosy1þy2

2

� �þ isin

y1þy2

2

� ��

q¼ffiffiffi2p

q0 cosy3

2

� �þ isin

y3

2

� �� (46)

where p0, q0, y1, y2 and y3 are defined in Appendix.For convenience, we set

k¼ p�0h, k¼ q�0h (47)

Using Eqs. (A.1)–(A.4) and Eq. (47) in Eq. (45) and simplifying, we obtain

ok ¼oRkþ ioI

k (48)

where

oRk ¼o0 1�

ef2

1þrwx1

o20

� �H1þ

eT

2H2

�

oIk ¼�

o0

2�ef 1þ

rwx1

o20

� �H3þeT H4

� (49)

where H1, H2, H3, H4, k0, T * , ku and T�

are defined in Appendix and o0 is given by Eq. (44).


The thermoelastic damping arises from thermal currents generated due to contraction and expansion of elasticstructures. At the low-frequency range ðtR5o�1Þ, the vibrations are isothermal and a small amount of energy is dissipated,where tR is the characteristic time required for equalization of temperature and ois the vibration frequency. At high-frequency ðtRbo�1Þ adiabatic conditions prevail with low-energy dissipation similar to the low-frequency domain. In case(tREo�1), the stress and strain are out of phase and a minimum value, called Debye peak, of internal friction occurs andalso due to the presence of voids. Thus, for thin choice of the value of characteristic time, we take relaxation times (due tovoids and thermal effects) as tR=t0=o�1=x2. Eqs. (46) become

p¼ffiffiffi2p

p00 cosy10þy20

2

� �þ isin

y10þy20

2

� �� and q¼

ffiffiffi2p

q00 cosy30

2

� �þ isin

y30

2

� �� (50)

where p00, q00, y10, y20 and y30 are defined in Eq. (A.7).The thermoelastic damping (TED) and frequency shift are given by

Q�1LS ¼ 2

onkI

onkR

¼ ef 1þ

rwx1

o20

� �HLS

3 �eT HLS4

(51)

OSLS ¼

onkR �o0

o0

¼ � ef2 1þ

rwx1

o20

� �HLS

1 þeT

2HLS

2

(52)

where HLSi ði¼ 1,2,3,4Þ can be written from Hi(i=1,2,3,4) defined in Eqs. (A.5)–(A.7) by setting t0 ¼o�1

0 ¼ x2 and

^k¼ q00h, ^ku¼ffiffiffi2p ^kcos y30=2

� �,^T�

¼ tan y30=2� �

,

~T�¼ tan

y10þy20

2

� �, ~k ¼ p00h, ~ku¼

ffiffiffi2p

~k cosy10þy20

2

� �

Here we have used in that ef51 and eT51 for most of the materials.In the absence of relaxation time, thermal (t0-0) for coupled theory of thermoelasticity, we have

p¼ffiffiffi2p

p��0 cosy10�ðp=2Þ

2

� �þ i sin

y10�ðp=2Þ

2

� �� , q¼

ffiffiffi2p

q00 cosy30

2

� �þ isin

y30

2

� �� (53)

where p��0 , q00, y10, y002 and y30 are defined in Eq. (A.8). Consequently, the TED and frequency shift for coupled thermoelastic

beam are given by

Q�1CT ¼ ef 1þ

rwx1

o20

� �HCT

3 �eT HCT4

(54)

OSCT ¼ �

ef2

1þrwx1

o20

� �HCT

1 þeT

2HCT

2

(55)

where HCTi ði¼ 1,2,3,4Þ can be written from Hi(i=1, 2, 3, 4) defined in Eqs. (A.5)–(A.7) by setting t0=0 and tR ¼ x2 ¼o�1

0 and

^k¼ q00h, ^ku¼ffiffiffi2p

^kcosy30

2

� �,

^T�

¼ tany30

2

� �

~~k ¼ p��0 h, ~~ku¼ffiffiffi2p

~~k cosy0

10�ðp=2Þ

2

!, ~T

�¼ tan

y010�ðp=2Þ

2

!

From Eq. (47) and Eq. (43) the thickness of the beam with fixed aspect ratios AR ¼ L=h are obtained as

Set I : hkLS ¼

w3k2A2Rffiffiffi

2p

r010p2k2D�

, hkCT ¼

w3k2A2R

r010p2k2D�

(56)

Set II : hkLS ¼

2ffiffiffi2p

w3k2A2R

r010p2k2D�

, hkCT ¼

4w3k2A2R

r010p2k2D�

(57)

where w3 ¼ K3=rCe is the thermal diffusivity along the thickness direction. Clearly hnkCT ¼

ffiffiffi2p

hnkLS , which shows that the

presence of thermal relaxation time results in decreasing of thickness and hence the value of critical thickness. Moreover,thickness increases with increasing thermal diffusivity (w3) along the axis of symmetry in both Fourier and non-Fourierbeams in the presence of voids.

7. Numerical results and discussion

This section is devoted to the discussion of the dependency of damping factor (Q�1) and frequency shift (OS) on thebeam dimensions, boundary conditions, vibration modes, environmental temperature, thermal relaxation time and voidsfor magnesium like material MEMS devices. The mechanical and thermal properties of magnesium like material are givenin Table 1. The values of damping factor (Q�1) and frequency shift of first two vibration modes have been computed from


Eqs. (51)–(52) and Eqs. (54)–(55) in the presence and the absence of thermal relaxation time viz. generalized (non-Fourier)and coupled (Fourier) beams with voids, respectively. Here Q�1

LS , Q�1CT and OS

LS, OSCT represent the damping factor and

frequency shift in respective thermoelastic theories in the presence of voids. The numerical computations have beencarried out with the help of MATLAB software programming for magnesium like material beams. The dimensions of thebeam and parameters of resonance frequencies of fundamental mode (thickness-shear mode) have been taken in theprescribe limits [13] for micro-scale beam resonators. The computer simulated results have been presented graphically inFigs. 1–13 for clamped and simply supported beams.

Figs. 1–3 represent the variations of damping factor of first two modes in case of clamped beam with fixed aspect ratioAR=50 and varying thickness (h), with fixed length L=500 mm and varying thickness (h) and fixed thickness h=10 mm andvarying length (L), respectively, in the context of generalized and coupled theories of thermoelasticity with voids. It isobserved that the damping factor of vibration modes first increases and then decreases in the considered range ofthickness (h). Thus, there exists a critical thickness denoted by ðhk

LSÞc and ðhk

CTÞc for which maximum value of damping

factor occurs. The value of damping factor of vibration modes is observed to have greater value in case of generalized thanthat for coupled thermoelasticity with voids.

Figs. 4–6 represent the variations of damping factor of first two modes in case of simply supported beam with fixedaspect ratio AR=50 and varying thickness (h), with fixed length L=500 mm and varying thickness (h) and fixed thicknessh=10 mm and varying length (L), respectively, in the context of generalized and coupled theories of thermoelasticity withvoids. Moreover, it is observed that the behavior of damping factor remains unaltered as discussed for Figs. 1–3.

Figs. 7 and 8 represent the vibrations of frequency shift in case of clamped and simply supported beams, respectively,with fixed aspect ratio AR=50 and varying thickness (h) in the context of generalized and coupled theories ofthermoelasticity with voids. It is observed that the frequency shift increases rapidly with increasing thickness to attainits maximum value and it becomes stable for large values of thickness afterwards for considered vibration modes. It is alsoobserved that the frequency shift of coupled thermoelastic beam with voids approximately equals to that of generalizedthermoelastic beam with voids. Moreover, it is observed that the behavior of frequency shift remains unaltered for beamsof fixed length and fixed thickness in case of both clamped and simply supported conditions and it remains the same asthat of beam with fixed aspect ratio except having minor difference in magnitude.

Table 1Physical data of magnesium [4].

Quantity Unit Magnesium

r kg m�3 1.74 �103

l N m�2 2.17� 1010

m N m�2 1.639�1010

b N m�2 deg.�1 2.68�106

K W m�1 deg.�1 170

Ce J kg�1 deg.�1 1040

a N 3.688 �10�5

m N m�2 deg.�1 2.0 �106

b N m�2 1.13849 �1010

w m�2 1.753 �10�15

x1=x2 N m�2 1.475�1010

T01C 25

t0 S 0.09�10�10

Fig. 1. Variation of damping of few modes with thickness (h) in a clamped beam of fixed aspect ratio (AR=50).

Fig. 3. Variation of damping of few modes with length (L)in a clamped beam of fixed thickness (h=10 mm).

Fig. 4. Variation of damping of few modes with thickness (h) in a simply supported beam of fixed aspect ratio (AR=50).

Fig. 2. Variation of damping of few modes with thickness (h) in a clamped beam of fixed length (L=500 mm).


Figs. 9 and 10 represent the variations of damping factor of first two modes in case of clamped beam and simplysupported with fixed aspect ratio AR=50 and varying thickness (h), respectively, in the context of generalized theory ofthermoelasticity with voids (TEV) and in the absence of voids (TE). It is observed that the damping factor of vibrationmodes first increases and then decreases in the considered range of thickness (h) and the magnitude of damping increasesdue to the presence of voids in comparison of damping in thermoelastic beam for both modes of clamped and simply

Fig. 6. Variation of damping of few modes with length (L) in a simply supported beam of fixed thickness (h=10 mm).

Fig. 7. Variation of frequency shift of few modes with thickness (h) in a clamped beam of fixed aspect ratio (AR=50).

Fig. 5. Variation of damping of few modes with thickness (h) in a simply supported beam of fixed length (L=500 mm).


supported beams. Thus, there exists a critical thickness denoted by hkLS

� �cand hk

LSWV

� �cfor which maximum value of

damping factor occurs for TEV beam and TE beam and also noticed that the critical thickness of TEV beam has greater valuethan critical thickness of TE beam.

Figs. 11–13 represent the variations of damping factor of first two modes in case of clamped beam with fixed aspectratio AR=2 and varying thickness (h), with fixed length L=100 nm and varying thickness (h) and fixed thickness h=5 nmand varying length (L), respectively, in the context of generalized and coupled theories of thermoelasticity with voids. It is

Fig. 9. Variation of damping of few modes with thickness (h) in a TEV and TE clamped beam of fixed aspect ratio (AR=50).

Fig. 10. Variation of damping of few modes with thickness (h) in a TEV and TE simply supported beam of fixed aspect ratio (AR=50).

Fig. 8. Variation of frequency shift of few modes with thickness (h) in a simply supported beam of fixed aspect ratio (AR=50).


observed that the damping factor of vibration modes first increases and then decreases in the considered range ofthickness (h). Thus, there exists a critical thickness denoted by hk

LS

� �cand hk

CT

� �cfor which maximum value of damping

factor occurs. The value of damping factor of vibration modes is observed to have greater value in case of generalized thanthat for coupled thermoelasticity with voids.

Fig. 12. Variation of damping of few modes with thickness (h) in a clamped beam of fixed length (L=100 nm).

Fig. 13. Variation of damping of few modes with length (L) in a clamped beam of fixed thickness (h=5 nm).

Fig. 11. Variation of damping of few modes with thickness (h) in a clamped beam of fixed aspect ratio (AR=2).



8. Conclusions

It is concluded that voids and thermal relaxation time contribute in increasing the magnitude of peak value of dampingin addition to decreasing the critical thickness, which is consistent as the derived analytical results. It is observed that thecritical thickness of the beam under resonant conditions decreases with increasing values of modes in both Fourier (CT)and non-Fourier (LS) beam resonators. It is observed that thickness of the beam increases with increasing values of thermaldiffusivity, however it decreases with increase in temperature in both LS and CT beams. For beams with fixed aspect ratioand fixed length, the critical size is the critical thickness. It is observed that critical thickness of simply supported beam islarger than that of clamped one for the same mode of vibration. Also under the same boundary conditions, the criticalthickness decreases for the higher modes.

Appendix A

The quantities p0, q0, yi(i=1, 2, 3) in Eq. (46) are defined as

p0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirCeo0r1r2

2K

r, r2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þt0o2

0

q, y2 ¼ tan�1 �

1

t0o0

� �(A.1)

r1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

0þS20

q, y1 ¼ tan�1ðS0=R0Þ (A.2)

R0 ¼bðrwo2

0�x1Þ2þmbðrwo2

0�x1Þþo20x

22

b ðrwo20�x1Þ

2þo2

0x22

h i

S0 ¼o0x2ðmbþðb�1Þðrwo2

0�x1ÞÞ

b½ðrwo20�x1Þ

2þo2

0x22�

(A.3)

q0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2a1þ

mbT0

brCe

� �r3

s, r3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrwo2

0�x1Þ2þo2

0x22

q, y3 ¼ tan�1 �

o0x2

rwo20�x1

!(A.4)

The quantities Hi(i=1, 2, 3, 4) in Eq. (49) are given by

H1 ¼ 1þ6

k2cosy3�

6ffiffiffi2p

cosð3y3=2Þ

k3

sinkuþtanð3y3=2Þsinh kuT�

� �cos kuþcosh kuT

��

0@

1A

H2 ¼ 1þ6

k2cos y1þy2ð Þ�

6ffiffiffi2p

cosðð3ðy1þy2ÞÞ=2Þ

k3

sinkuþtanðð3ðy1þy2ÞÞ=2ÞsinhðkuT�Þcoskuþcosh kuT�ð Þ

� �( )

H3 ¼ �6

k2siny3�

6ffiffiffi2p

cosð3y3=2Þ

k3

sinh kuT�

� ��tanð3y3=2Þsinku

cos kuþcoshðkuT�Þ

0@

1A

8<:

9=;

H4 ¼ �6

k2sinðy1þy2Þ�

6ffiffiffi2p

cosðð3ðy1þy2ÞÞ=2Þ

k3

sinhðkuT�Þ�tanðð3ðy1þy2ÞÞ=2ÞsinkucoskuþcoshðkuT�Þ

� �( )(A.5)

ku¼ffiffiffi2p

kcosy1þy2

2

� �, T� ¼ tan

y1þy2

2

� �, ku¼

ffiffiffi2p

kcosy3

2

� �, T

�¼ tan

y3

2

� �(A.6)

p00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirCeo0r10ffiffiffi

2p

K

s, r20 ¼

ffiffiffi2p

, y20 ¼�p4

r10 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

00þS200

q, y10 ¼ tan�1 S00

R00

� �

R00 ¼bðrwo2

0�x1Þ2þmbðrwo2

0�x1Þþ1

b ðrwo20�x1Þ

2þ1

h i , S00 ¼mbþðb�1Þðrwo2

0�x1Þ

b½ðrwo20�x1Þ

2þ1�

q00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2a 1þmbT0

brCe

� �r30

s, r30 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrwo2

0�x1Þ2þ1

q, y30 ¼ tan�1 �1

rwo20�x1

!(A.7)


p��0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirCeo0r0

10ffiffiffi2p

K

s, q00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2a 1þmbT0

brCe

� �r30

s

r002 ¼ 1, y00

2 ¼�p2

, r010 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR�002þS�002

q, y0

10 ¼ tan�1 S�00

R�00

� �,

R�00 ¼bðrwo2

0�x1Þ2þmbðrwo2

0�x1Þþ1

b ðrwo20�x1Þ

2þ1

h i , S�00 ¼mbþðb�1Þðrwo2

0�x1Þ

b ðrwo20�x1Þ

2þ1

h i

r30 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrwo2

0�x1Þ2þ1

q, y30 ¼ tan�1 �1

rwo20�x1

!(A.8)

References

[1] J.W. Nunziato, S.C. Cowin, A nonlinear theory of elastic materials with voids, Archive for Rational Mechanics and Analysis 72 (1979) 175–201.[2] S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, Journal of Elasticity 13 (1983) 125–147.[3] D. Iesan, A theory of thermoelastic material with voids, Acta Mechanica 60 (1986) 67–89.[4] J.N. Sharma, D. Kaur, Generalized thermoelastic plane harmonic waves in materials with voids, International Journal of Applied Mechanics and

Engineering 13 (4) (2008) 1019–1043.[5] P.K. Sharma, D. Kaur, J.N. Sharma, Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids, International Journal of Solids

and Structure 45 (2008) 5049–5058.[6] C. Zener, Internal friction in solids, I: theory of internal friction in reeds, Physical Review 52 (1937) 230–235.[7] R. Lifshitz, M.L. Roukes, Thermoelastic damping in micro- and nanomechanical systems, Physical Review B 61 (2000) 5600–5609.[8] Y.B. Yi, M.A. Martin, Eigen-value solution of thermoelastic damping in a beam resonator using a finite element analysis, Journal of Vibration and

Acoustics 129 (2007) 478–483.[9] S. Lapage, J. Golinval, Finite element modeling of thermoelastic damping in filleted micro-beams, Proceedings of IEEE EuroSimE, London, April 2007,

pp. 1–7.[10] S. Prabhakar, M.P. Paidousis, S. Vengallatore, Analysis of frequency shifts due to thermoelastic coupling in flexural-mode micromechanical and

nano-mechanical resonators, Journal of Sound and Vibration 323 (2009) 385–396.[11] F.L. Guo, G.A. Rogerson, Thermoelastic coupling effect on a micro-machined beam resonator, Mechanics Research Communications 30 (2003)

513–518.[12] Y.X. Sun, D.N. Fang, A.K. Soh, Thermoelastic damping in micro-beam resonators, International Journal of Solids and Structure 43 (2006) 3213–3229.[13] Y. Sun, M. Saka, Thermoelastic damping in micro-scale circular plate resonators, Journal of Sound and Vibration 329 (2010) 328–337.[14] J.N. Sharma, R. Sharma, Damping in micro-scale generalized thermoelastic circular plate resonators, Ultrasonics 51 (2011) 352–358.[15] Ali H. Nayfeh, Mohammad I. Younis, Modeling and simulations of thermoelastic damping in microplates, Journal of Micromechanics and

Microengineering 14 (2004) 1711–1717.[16] H.N. Arafat, A.H. Nayfeh, W. Farris, Natural frequencies of heated annular and circular plates, International Journal of Solids and Structure 41 (2004)

3031–3051.[17] S.J. Wong, C.H.J. Fox, S. McWilliam, Thermoelastic damping of the in-plane vibration of thin silicon rings, Journal of Sound and Vibration 293 (2006)

266–285.[18] Y.X. Sun, H. Tohmyoh, Thermoelastic damping of the axisymmetric vibration of circular plate resonators, Journal of Sound and Vibration 319 (2009)

392–405.[19] B.H. Houston, D.M. Photiadis, J.F. Vignola, M.H. Marcus, X. Liu, D. Czaplewski, L. Sekaric, J. Butler, P. Pehrsson, J.A. Bucaro, Loss due to transverse

thermoelastic currents in microscale resonators, Materials Science and Engineering A 370 (2004) 407–411.[20] J.R. Vig, R.L. Filler, Y. Kim, Uncooled IR imaging array based on quartz resonators, IEEE Journal of Microelectromechanical Systems 5 (1996) 131–137.[21] H.W. Lord, Y. Shulman, The generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15 (1967) 299–309.

thermoelastic vibrations in micro-/nano-scale beam resonators with voids

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