INFLUENCE OF THERMO-PIEZOELECTRIC FIELD IN A CIRCULAR

BAR SUBJECTED TO THERMAL LOADING DUE TO LASER PULSE

R. Selvamani

Department of mathematics, Karunya University, Coimbatore, Tamil Nadu, India

e-mail: selvam1729@gmail.com

Abstract. The influence of thermo-piezoelectric field in a circular bar subjected to thermal

loading due to non-Gaussian laser pulse is discussed using linear theory of thermoelasticity.

The equations of motion of the rod are formulated using the constitutive equations of a thermo-

piezoelectric material. Displacement potential functions are introduced to uncouple the

equations of motion, heat conduction and charge equation. The frequency equation of the

system is developed under the assumption of stress free, thermal and electrically shorted

boundary conditions. The numerical calculations are carried out for the material PZT-5A and

the computed stress, strain and temperature distribution are plotted as the dispersion curves and

their physical characteristics are discussed for longitudinal and flexural (symmetric and

antisymmetric) modes.

1. Introduction

Laser-induced vibrations of micro-beam resonators have attracted considerable attention

recently due to their many important technological applications in Micro-Electro-Mechanical

Systems (MEMS) and Nano- Electro-Mechanical Systems (NMES). Very rapid thermal

processes, under the action of a thermal shock (e.g., ultra-short laser pulse) are interesting from

the standpoint of thermoelasticity, since they require an analysis of the coupled temperature and

deformation fields. This means that the temperature shock induces very rapid movements in the

structure elements, thus causing the rise of very significant inertial forces, and thereby, the rise

of vibrations.

In recent years, polymers piezoelectric materials have been used in numerous fields taking

advantage of the flexible characteristics of these polymers. Some of the applications of these

polymers include Audio device-microphones, high frequency speakers, tone generators and

acoustic modems; Pressure switches – position switches, accelerometers, impact detectors, flow

meters and load cells; Actuators- electronic fans and high shutters. Since piezoelectric polymers

allow their use in a multitude of compositions and geometrical shapes for a large variety of

applications from transducers in acoustics, ultrasonic’s and hydrophone applications to

resonators in band pass filters, power supplies, delay lines, medical scans and some industrial

non-destructive testing instruments.

The coupling between the thermal/electric/elastic fields in piezo electric materials

provides a mechanism for sensing thermomechanical disturbances from measurements of

induced electric potentials, and for altering structural responses via applied electric fields. One

of the applications of the piezo thermoelastic material is to detect the responses of a structure

by measuring the electric charge, sensing or to reduce excessive responses by applying

additional electric forces or thermal forces actuating. If sensing and actuating can be integrated

smartly, a so-called intelligent structure can be designed. The piezoelectric materials are also

often used as resonators whose frequencies need to be precisely controlled. The coupling

Materials Physics and Mechanics 27 (2016) 1-8 Received: July 11, 2015

© 2016, Institute of Problems of Mechanical Engineering

between the thermoelastic and pyroelectric effects, it is important to qualify the effect of heat

dissipation on the propagation of wave at low and high frequencies.

The governing equations of a thermo-piezoelectric plate was first proposed by

Mindlin [1]. The physical laws for the thermo-piezoelectric materials have been discussed by

Nowacki [2]. Chandrasekhariah [3] presented the generalized theory of thermo-piezoelectricity

by taking into account the finite speed of propagation of thermal disturbance. Yang and Batra

[4] studied the effect of heat conduction on shift in the frequencies of a freely vibrating linear

piezoelectric body with the help of perturbation methods. Sharma and Pal [5] discussed the

propagation of Lamb waves in a transversely isotropic piezothermoelastic plate.

Tang and Xu [6] derived the general dynamic equations, which include mechanical,

thermal and electric effects, based on the anisotropic composite laminated plate theory. They

also obtained analytical dynamical solutions for the case of general force acting on a simply

supported piezo thermoelastic laminated plate and harmonic responses to temperature variation

and electric field have been examined as a special case. Tauchert [7] applied thermo-

piezoelectricity theory to composite plate. The generalized theory of thermoelasticity was

developed by Lord and Shulman [8] involving one relaxation time for isotropic homogeneous

media, which is called the first generalization to the coupled theory of elasticity. These

equations determine the finite speeds of propagation of heat and displacement distributions, the

corresponding equations for an isotropic case were obtained by Dhaliwal and Sheried [9]. The

second generalization to the coupled theory of elasticity is what is known as the theory of

thermoelasticity, with two relaxation times or the theory of temperature – dependent

thermoelasticity. Longitudinal vibrations of a circular cylindrical shell coupled with a thermal

field were also obtained independently by Suhubi [10]. Dispersion analysis of generalized

magneto-thermoelastic waves in a transversely isotropic cylindrical panel is analyzed by

Ponnusamy and Selvamani [11]. Later, extensional waves in a transversely isotropic solid bar

immersed in an inviscid fluid calculated using Chebyshev polynomials were discussed by

Selvamani and Ponnusamy [12].

This theory contains two constants that act as relaxation times and modify not only the

heat equations, but also all the equations of the coupled theory. Wang and Xu [13] have studied

the stress wave induced by nanoseconds, picoseconds, and femtoseconds laser pulses in a semi-

infinite solid. The Precursor in Laser-Generated Ultrasound Waveforms in Metals were

discussed by McDonald [14].The so-called ultra-short lasers are those with pulse duration

ranging from nanoseconds to femtoseconds in general. In the case of ultra-short-pulsed laser

heating, the high-intensity energy flux and ultra-short duration laser beam, have introduced

situations where very large thermal gradients or an ultra-high heating speed may exist on the

boundaries Sun et al. [15].

In this paper, the influence of piezo thermoelastic field in a circular bar subjected to

thermal loading due to laser pulse is studied. The frequency equations are derived for stress

free, temperature and electrically shorted boundary conditions. The numerical calculations are

carried out for the material PZT-5A and the computed stress, strain and temperature distribution

are plotted as the dispersion curves and their physical characteristics are discussed for

longitudinal and flexural (symmetric and antisymmetric) modes.

2. Formulation of the problem

We consider homogeneous transversely isotropic, thermally and electrically conducting

piezoelectric bar of finite length with uniform temperature 0T in the undistributed state initially.

This piezoelectric rod is subjected to laser pulse with Non-Gaussian temporal profile discussed

in Sun et al. [15]. 2.1. The Non-Gaussian Laser pulse model. The piezo thermoelastic rod is heated

uniformly by a laser beam of Non-Gaussian form defined by Sun et al. [15]

2 R. Selvamani

0

2( ) p

t

t

p

LI t e

t

, (1)

where 2pt ps is the characterstic time of the laser pulse, the total energy carried by laser

pulse per unit area of the laser beam is denoted by 0L (laser intensity). The conduction of heat

transfer in the rod can be modeled with an energy source ( , )Q z t as

/2/2( )

0

2

1( , ) p

z h tz hI t

ta

p

R LRQ z t e te

t

, (2)

where is the absorption depth of heating energy and aR is the surface reflexivity, when we

take the laser pulse lie on the surface of the rod when 0z , we can get the energy source in

the following form

20

2( , ) p

h t

ta

p

R LQ z t te

t

. (3)

Fig. 1. Geometry of the bar.

The equations of motion, heat conduction and electric potentials in the absence of body force

are

1 1

, , , ,rr r r rz z rr r ttr r u , 1 1

, , , ,2r r z z r ttr r u ,

1 1

, , , ,rz r z zz z rz z ttr r u . (4)

The heat conduction equation is

1 2

1 , , , 3 , , 1 3 3 , ,rr r zz v t o rr zz z tK T r T r T K T c T T e e e p Q . (5)

The electric charge equation is

1 1

0zr

D DrD

r r r r

. (6)

The elastic, the piezoelectric, and dielectric matrices of the 6mm crystal class, the thermo-

piezoelectric relations are

, 12 11 13 1 31rr zz zc e c e c e T e E ,

, 66r rc e , 1

44 15z zc e r e E ,

44 152rz rz rc e e E ,

15 11r rz rD e e E , 1

15 11zD e e r E , 31 33 33 3z rr zz zD e e e e e E p T , (7)

where , , , , ,rr zz r z rz are the stress components, , , , , ,rr zz r z rze e e e e e are the strain

components, T is the temperature change about the equilibrium temperature

11 12 13 33 44, , , , ,oT c c c c c and 66 11 12 2c c c are the five elastic constants, 1 3, and 1 3,K K

11 12 13 1 31rr rr zz zc e c e c e T e E

13 13 33 3 33zz rr zz zc e c e c e T e E

3Influence of thermo-piezoelectric field in a circular bar subjected to thermal loading...

respectively thermal expansion coefficients and thermal conductivities along and perpendicular

to the symmetry, is the mass density, vc is the specific heat capacity , 3p is the pyroelectric

effect.

The strain ije are related to the displacements are given by

,rr r re u , 1

,re r u u

, ,zz z ze u (8a)

1

, ,r r re u r u u

, 1

, ,z z ze u r u

, , ,rz z r r ze u u (8b)

The comma in the subscripts denotes the partial differentiation with respect to the variables.

Substituting the Eqs.(8), (7) in the Eqs. (4)-(6), results in the following three-dimensional

equations of motion, heat electric conductions are obtained as follows:

1 2 2 2

11 , , 11 66 , 66 , 44 , 44 13 ,

1

66 12 , 31 15 , 1 , , ,

rr r r r r r r zz z rz

r rz r r tt

c u r u r u r c c u r c u c u c c u

r c c u e e T u

(9a)

1 2 1 2 2

12 66 , 66 11 , 66 , , 11 , 44 ,

1 1

44 13 , 31 15 , 1 , , ,

r r r rr r zz

z z z tt

r c c u r c c u c u r u r u r c u c u

r c c u e e r T u

(9b)

1 2 1

44 , , , 44 13 , , 44 13 , 33 , 33 , 3 , ,z rr z r z r z z r rz z zz zz z z ttc u r u r u r c c u u c c u c u e T u ,(9c)

1 2 1 1

1 , , , 3 , , 1 , , 3 , 3 ,( )rr r zz v t o r r r z z zK T r T r T K T c T T u r u r u u p Qt

, (9d)

1 2 1 1

15 , , , 31 15 , , , 33 , 33 ,

1 2

11 , , , 3 , 0.

z rr z r z r zr r z z z zz zz

rr r z

e u r u r u e e u r u r u e u

r r p T

(9e)

3. Solutions of the Field Equation

To obtain the propagation of harmonic waves in thermo-piezoelectric circular bar, we assume

the solutions of the displacement components to be expressed in terms of derivatives and we

seek the solution of the Eqs. (9) in the form

1

, ,, , ,i kz t

r ru r z t r e

, 1

, ,, , ,i kz t

ru r z t r e

,

, , ,i kz t

z

iu r z t We

a

,

, , ,

i kz tr z tV iVe

,

,, , ,i kz t

r rr z tE E e

, 1

,, , ,i kz t

r z tE r E e

,

,, , ,i kz t

z zr z tE E e

(10)

where 1i , k is the wave number, is the angular frequency, , , ,r W r , ,r

and ,E r are the displacement potentials and ,V r , is the electric potentials, ,T r is

the temperature change and a is the geometrical parameter of the bar.

By introducing the dimensionless quantities such as x r a , ka , 2 2 2

44a c ,

11 11 44c c c , 13 13 44c c c , 33 33 44c c c , 0 22

3 0

ha

p

R Le

T t

, 66 66 44c c c , 1 3 , pt t

1 2 2

44 3 ok c T a , 2

44 3v od c c T , 3 44 3 333p p c e and substituting Eq. (10) in Eqs. (9),

we obtain

2 2 211 13 31 151 0c c W e e V T ,

2 2 2 2 2 213 33 151 0c c W e V T ,

2 2 2 2 231 15 15 33 113

0e e e W p T V ,

4 R. Selvamani

2 2 21 3 3

(1 ) 0W d e ik ik T p V , (11)

and 2 2 266 0c , (12)

where 2 2

2 1 2

2 2x x

x x

The Eq. (11) can be written as

2 2 211 13 31 15

2 2 2 2 2 213 33 15

2 2 21 3 3

2 2 2 2 231 15 15 33 113

1

1, , , 0

(1 )

c c e e

c c eW T V

d e ik ik p

e e e p

(13)

Evaluating the determinant given in Eq. (13), we obtain a partial differential equation of the

form

8 6 4 2 , , , 0A B C D E W T V , (14)

where

1 11 18A ik c g , 2

2 211 1 11 15 19 2 17 1 18 3 18B c g ik g e g ik g g g g ,

2 211 11 15 1 1 11 152 9 5 8 1 9 2 17

211 33 15 152 3 33

2 215 1 15 15 14 4 14 3 15 3 4 3 16 23

2

2

,

C c g g g e g ik g g ik g e g

g p g g e e

g g g e g g g ik e g p e g g ik g

2 2 2 211 11 15 12 5 8 7 1 2 9 5 8

22

114 4 11 3 123 3

2 2 2334 13 2 3 3 3 4 11 2 3 16 33 3

2

,

D c g g g g g g g g e g ik

g g g p g g p

g g g p g g g g g g p g g g

2 2 2

1 2 5 1 8 1 7 4 6 4 7 4 3 8E g g g g g g g g g g g g g g ,

in which 2 2

1g , 2 2

332g c , 31 153g e e , 134 1g c , 2

23 3 335 3

g d ik p ,

336 3(1 )g d p e , 337 3

g p ,2

38 3g d p ik ,

2 215 1 3 1511g de ik ik e

2 21 33 3 11 119 (1 )g ik e ik d ,

211 15 110 82g e g ik ,

2 233 312 3 3

(1 )g e d p ik p , 11 1513 3g p e , 1 15 314g i k e k ,

11 115 3g i p k ,

2 2316g d ik ,

2117 11g ik g ,

2

11 1518g e .

Factorizing the relation given in Eq. (14) into biquadratic equation for 2

ia , 1,2,3,4i , the

solutions for the symmetric modes are obtained as

4

1

cosii n

i

axA J n

, 4

1

cosii i n

i

axW a A J n

,

4

1

cosii i n

i

axT b A J n

, 4

1

cosii i n

i

axV c A J n

. (15)

5Influence of thermo-piezoelectric field in a circular bar subjected to thermal loading...

Here 2

0ia , 1,2,3,4i are the roots of the algebraic equation

8 6 4 2

0A a B a C a D a E . (16)

The solutions corresponding to the root 2

0ia is not considered here, since 0nJ is zero,

except for 0n . The Bessel function nJ is used when the roots 2

ia , 1,2,3,4i are real or

complex and the modified Bessel function nI is used when the roots 2,ia 1,2,3,4i are

imaginary.

The constants ,i ia b and ic defined in the Eq. (15) can be calculated from the equations

3 43ia p g L g L , 2 2

113 1 43i i ib g a p g c a g L ,

2

114 1 4i ic a g c g g L , (17)

where 2 21 3iL d ik a ik .

Solving the Eq.(12), the solution to the symmetric mode is obtained as

5 5 sinnA J ax n , (18)

where 2 2 2

5a . If 2

5 0a , the Bessel function nJ is replaced by the modified

Bessel function nI .

4. Boundary conditions and Frequency equations

In this problem, the free vibration of transversely isotropic thermo-piezoelectric solid bar of

circular cross-section is considered. The stress free, temperature and electrically shorted can be

written as

, , , , 0,0,0,0 ,rr r rz V T r a . (19)

Substituting the solutions given in the Eqs. (16) and (18) in the boundary condition in the Eq.

(19), we obtain a system of five linear algebraic equation as follows:

0A X , (20)

where A is a 5 5 matrix of unknown wave amplitudes, and X is an 5 1 column vector

of the unknown amplitude coefficients 1, 2, 3 4, 5,A A A A A . The solution of Eq. (20) is nontrivial

when the determinant of the coefficient of the wave amplitudes X vanishes, that is

0A . (21)

Eq. (21) is the frequency equation of the coupled system consisting of a transversely isotropic

thermo-piezoelectric solid circular bar.

22

66 11 131 112i i i i ii n n i i i nn n J ax ax J ax a J axa c x c c a b c

, 1,2,3,4i ,

5 5 56615 112 n nn n ax ax axa c J J ,

2 12 1 ,i i i in nn ax J ax n J axa 1,2,3,4,i 5 5

2

25 5 5 1 ,2 1 2n nax axa ax n n J ax J

153 1i i ii i i n nJ ax ax J axa a e b n , 1,2,3,4i , 535 n axa n J ,

6 R. Selvamani

15 114 1i i ii i i n nn ax ax axa e a b J J , 1,2,3,4i , 5

1545 n axa e nJ ,

5 1i i ii i n nnJ ax ax J axa c , 1,2,3,4i , 55 0a .

5. Numerical results and discussion The free vibration of transversely isotropic thermo-piezoelectric solid bar of circular cross-

section is considered. The frequency equation is numerically solved for the material PZT-5A

and the material properties of PZT-5A is given below: 10 2

11 13.9 10c Nm , 10 2

12 7.78 10c Nm , 10 2

13 7.54 10c Nm , 10 2

33 11.3 10c Nm ,

10 2

44 2.56 10c Nm , 2

31 6.98e Cm , 2

33 13.8e Cm , 2

15 13.4e Cm ,

10 2 1 2

11 60.0 10 C N m , 10 2 1 2

33 54.7 10 C N m , 27750Kgm ,6 1 2

1 1.52 10 NK m , 6 1 2

3 1.53 10 NK m , 1 1420eC J kg K , 1 1

1 3 1.5K K Wm K .

6 1 2

3 1.53 10 NK m , 0 298T K , 6 1 23 452 10p CK m .

In this problem, there are two kinds of basic independent modes of wave propagation have been

considered, namely, the longitudinal and flexural modes. By choosing respectively n=0 and

n=1, we can obtain the non-dimensional frequencies of longitudinal and flexural modes of

vibrations.

5.1. Dispersion curves. The results of longitudinal and flexural modes (symmetric and

antisymmetric) are plotted in the form of dispersion curves. The notation used in the figures,

namely LM, FLS and FLAS respectively denote the longitudinal mode, flexural symmetric

mode and, flexural anti symmetric mode. The dispersion curves are drawn for stress distribution

versus the different position of the thermo-piezoelectric bar for longitudinal and flexural

(symmetric and antisymmetric) modes are respectively shown in Figs. 2 and 3. From the Figs. 2

and 3, it is clear that the stress distributions have the same trend in both piezoelectric and

thermo-piezoelectric solid bar with respect to its different positions. But there is a small

deviation in the lower limit of the position in Fig. 3 due to the influence of laser pulse.

Fig. 2. Stress distribution verses different

position of the piezoelectric bar.

Fig. 3. Stress distribution verses different

position of the thermo-piezoelectric bar.

Fig. 4. Strain distribution verses different

position of the piezoelectric bar.

Fig. 5. Strain distribution verses different

position of the thermo-piezoelectric bar.

7Influence of thermo-piezoelectric field in a circular bar subjected to thermal loading...

A comparison is made among the longitudinal and flexural symmetric and antisymmetric

mode of strain distribution of piezoelectric and thermo-piezoelectric vibrations are respectively

shown in the Figs. 3 and 4 respectively. From the Figs. 3 and 4, it is observed that, both the

piezoelectric and thermo-piezoelectric bar the strain energy experiences some cross over and

peak values for a particular position after that, it starts decreases. The inclusion of thermal

energy due to laser pulse is oscillating the strain energy in Fig. 4.

A dispersion curve is drawn to compare the temperature distribution of longitudinal and

flexural symmetric and antisymmetric modes of vibration for a piezoelectric and thermo-

piezoelectric bar is shown respectively in the Figs. 5 and 6. From the Figs. 5 and 6, it is observed

that the temperature distribution are decreases with respect to the position of bar.

Fig. 6. Temperature distribution verses

different position of the piezoelectric bar.

Fig. 7. Temperature distribution verses

different position of the thermo-piezoelectric

bar.

6. Conclusion

The influence of thermo-piezoelectric field in a circular bar subjected to thermal loading due to

non-Gaussian laser pulse is discussed using linear theory of thermoelasticity. The frequency

equation of the system is developed under the assumption of stress free, thermal and electrically

shorted boundary conditions. The numerical calculations are carried out for the material

PZT-5A and the computed stress, strain and temperature distribution are plotted as the

dispersion curves and their physical characteristics are discussed for longitudinal and flexural

(symmetric and antisymmetric) modes of piezoelectric and thermo-piezoelectric bar. From the

result, it is clear that the thermal load due to non-Gaussian laser pulse has the significant effect

on the thermal and mechanical interactions.

References

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[2] W. Nowacki // Journal of Thermal Stresses 1 (1978) 171.

[3] D.S. Chandrasekhariah // Journal of Thermal Stresses 7 (1984) 293.

[4] J.S. Yang, R.C. Batra // Journal of Thermal Stresses 18 (1995) 247.

[5] J.N. Sharma, M. Pal // Journal of Sound and Vibration 270 (2004) 587.

[6] Y.X. Tang, K. Xu // Journal of Thermal Stresses 18 (1995) 87.

[7] T.R. Tauchert // Journal of Thermal Stresses 15 (1992) 25.

[8] H.W. Lord, Y. Shulman // Journal of Mechanical Physics Solids 5 (1967) 299.

[9] R.S. Dhaliwal, H.H. Sherief // Quarterly Journal of Applied Mathematics 8(1) (1990) 1.

[10] E.S. Suhubi // Journal of Mechanical Physics Solids 12 (1964) 69.

[11] P. Ponnusamy, R. Selvamani // Journal of Thermal Stresses 35 (2012) 1119.

[12] R. Selvamani, P. Ponnusamy // Materials Physics and Mechanics 16 (2013) 82.

[13] X. Wang, X. Xu // Applied Physics A: Materials Science & Processing 73(1) (2001) 107.

[14] F.A. McDonald // Applied Physics Letters 56(3) (1990) 230.

[15] Y. Sun, D. Fang, M. Saka, A.K. Soh // International Journal of Solids and Structures 45

(2008) 1993.

8 R. Selvamani