Reflection of SV waves from the free surface of an elastic solid in generalized thermoelastic diffusion

ARTICLE IN PRESS

JOURNAL OFSOUND ANDVIBRATION

Journal of Sound and Vibration 291 (2006) 764–778

0022-460X/$ -

doi:10.1016/j.

�Tel.: +9 11

E-mail add

www.elsevier.com/locate/jsvi

Reflection of SV waves from the free surface of an elastic solidin generalized thermoelastic diffusion

Baljeet Singh�

Department of Mathematics, Government College, Sector-11, Chandigarh - 160 011, India

Received 1 November 2004; received in revised form 14 March 2005; accepted 25 June 2005

Available online 24 August 2005

Abstract

The governing equations for two-dimensional generalized thermoelastic diffusion in an elastic solid aresolved. There exist three compressional waves and a shear vertical (SV) wave. The reflection phenomena ofSV wave from the free surface of an elastic solid with generalized thermoelastic diffusion is considered. Theclosed-form expressions for the reflection coefficients for various reflected waves are obtained. Thesereflection coefficients are found to depend upon the angle of incidence of SV wave, thermoelastic diffusionparameter and other material constants. The numerical values of modulus of the reflection coefficients arepresented graphically for different thermal and diffusion parameters.r 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Duhamel [1] and Neumann [2] introduced the theory of uncoupled thermoelasticity. There aretwo shortcomings of this theory. First, the fact that the mechanical state of the elastic body has noeffect on the temperature, is not in accordance with true physical experiments. Second, the heatequation being parabolic predicts an infinite speed of propagation for the temperature, which isnot physically admissible.Biot [3] developed the coupled theory of thermoelasticity which eliminates the first defect, but

shares the second defect of uncoupled theory. In the classical theory of thermoelasticity, when an

see front matter r 2005 Elsevier Ltd. All rights reserved.

jsv.2005.06.035

723203140; fax: +911 662245675.

ress: [email protected].

www.elsevier.com/locate/jsvi

ARTICLE IN PRESS

B. Singh / Journal of Sound and Vibration 291 (2006) 764–778 765

elastic solid is subjected to a thermal disturbance, the effect is felt at a location far from thesource, instantaneously. This implies that the thermal wave propagates with infinite speed, aphysically impossible result. In contrast to conventional thermoelasticity, non-classical theoriescame into existence during the last part of 20th century. These theories, referred to as generalizedthermoelasticity, were introduced in the literature in an attempt to eliminate the shortcomings ofthe classical dynamical thermoelasticity. For example, Lord and Shulman [4], by incorporating aflux-rate term into Fourier’s law of heat conduction, formulated a generalized theory whichinvolves a hyperbolic heat transport equation admitting finite speed for thermal signals. Greenand Lindsay [5], by including temperature rate among the constitutive variables, developed atemperature-rate-dependent thermoelasticity that does not violate the classical Fourier law of heatconduction, when body under consideration has center of symmetry and this theory also predictsa finite speed for heat propagation. Chandresekharaiah [6] referred to this wavelike thermaldisturbance as ‘‘second sound’’. The Lord and Shulman theory of generalized thermoelasticitywas further extended by Sherief [7] and Dhaliwal and Sherief [8] to include the anisotropic case. Asurvey article of representative theories in the range of generalized thermoelasticity is due toHetnarski and Ignaczak [9].Sinha and Sinha [10] and Sinha and Elsibai [11,12] studied the reflection of thermoelastic waves

from the free surface of a solid half-space and at the interface of two semi-infinite media in weldedcontact, in the context of generalized thermoelasticity. Abd-Alla and Al-Dawy [13] studied thereflection phenomena of SV waves in a generalized thermoelastic medium. Recently, Sharma et al.[14] investigated the problem of thermoelastic wave reflection from the insulated and isothermalstress-free as well as rigidly fixed boundaries of a solid half-space in the context of differenttheories of generalized thermoelasticity.Diffusion may be defined as the random walk, of an ensemble of particles, from regions of high

concentration to regions of lower concentration. The study of this phenomenon is of greatconcern due to its many geophysical and industrial applications. In integrated circuit fabrication,diffusion is used to introduce ‘‘dopants’’ in controlled amounts into the semiconductor substrate.In particular, diffusion is used to form the base and emitter in bipolar transistors, form integratedresistors, form the source/drain regions in metal oxide semiconductor (MOS) transistors and dopepoly-silicon gates in MOS transistors. The phenomenon of diffusion is used to improve theconditions of oil extractions (seeking ways of more efficiently recovering oil from oil deposits).These days, oil companies are interested in the process of thermoelastic diffusion for moreefficient extraction of oil from oil deposits.Using the coupled thermoelastic model, Nowacki [15–17] developed the theory of thermoelastic

diffusion. Using Lord–Shulman (L-S) model, Sherief et al. [18] generalized the theory ofthermoelastic diffusion, which allows the finite speeds of propagation of waves. The present studyis motivated by the importance of thermoelastic diffusion process in the field of oil extraction.Also, the development of generalized theory of thermoelastic diffusion by Sherief et al. [18]provide a chance to study the wave propagation in such an interesting media. The paper isorganized as follows: in Section 2, the wave propagation in an isotropic, homogeneous model ofelastic solid with generalized thermoelastic diffusion is studied for Lord–Shulman (L-S) as well asfor Green–Lindsay (G-L) model. The governing equations for L-S and G-L models are solved inx–z plane to show the existence of three compressional waves and a SV wave. In Section 3, theclosed-form expressions for reflection coefficients are obtained for the incidence of SV wave at a

ARTICLE IN PRESS

B. Singh / Journal of Sound and Vibration 291 (2006) 764–778766

thermally insulated free surface. In the last section, a numerical example is given to discuss thedependence of reflection coefficients upon thermal relaxation times, diffusion relaxation times,angle of incidence of SV wave and other thermal and diffusion parameters. This dependence isshown graphically.

2. Governing equations and solution

Following, Lord and Shulman [4], Green and Lindsay [5] and Sherief et al. [18], the governingequations for an isotropic, homogeneous elastic solid with generalized thermoelastic diffusionwith reference temperature T0 in the absence of body forces are:

(i)
the constitutive equations
sij ¼ 2meij þ dij½lekk � b1ðYþ t1 _YÞ � b2ðC þ t1 _CÞ�, (1)

rT0S ¼ rcEðYþ a _YÞ þ b1T0ekk þ aT0ðC þ b _CÞ, (2)

P ¼ �b2ekk þ bðC þ t1 _CÞ � aðYþ t1 _YÞ. (3)

Here a ¼ b ¼ t1 ¼ t1 ¼ 0 for L-S model and a ¼ t0; b ¼ t0 for G-L model.
(ii) the equation of motion
mui;jj þ ðlþ mÞuj;ij � b1ðYþ t1 _YÞ;i � b2ðC þ t1 _CÞ;i ¼ r €ui, (4)

(iii)
the equation of heat conduction
rcEð _Yþ t0 €YÞ þ b1T0ð_eþ Ot0 €eÞ þ aT0ð _C þ g €CÞ ¼ KY;ii, (5)

(iv)
the equation of mass diffusion
Db2e;ii þDaðYþ t1 _YÞ;ii þ ð _C þ Ot0 €CÞ �DbðC þ t1 _CÞ;ii ¼ 0, (6)

where b1 ¼ ð3lþ 2mÞat and b2 ¼ ð3lþ 2mÞac, l;m are Lame’s constants, at is the coefficient oflinear thermal expansion and ac is the coefficient of linear diffusion expansion. Y ¼T � T0;T0 is the temperature of the medium in its natural state assumed to be such thatjY=T0j51. sij are the components of the stress tensor, ui are the components of thedisplacement vector, r is the density assumed independent of time, eij are the components ofthe strain tensor, e ¼ ekk, T is the absolute temperature, S is the entropy per unit mass, P isthe chemical potential per unit mass, C is the concentration, cE is the specific heat at constantstrain, K is the coefficient of thermal conductivity, D is thermoelastic diffusion constant. t0 isthe thermal relaxation time, which will ensure that the heat conduction equation, satisfied bythe temperature Y will predict finite speeds of heat propagation. t is the diffusion relaxationtime, which will ensure that the equation, satisfied by the concentration C will also predictfinite speeds of propagation of matter from one medium to the other. The constants a and bare the measures of thermoelastic diffusion effects and diffusive effects, respectively. Thesuperposed dots denote derivative with respect to time.

ARTICLE IN PRESS


For the L-S model, t1 ¼ 0; t ¼ 0; O ¼ 1; g ¼ t0. The governing equations in L-S model aresame as given by Sherief et al. [18].

1

For G-L model, t140; t140; O ¼ 0; g ¼ t0. The thermal relaxation times t0 and t1 satisfy theinequality t1Xt0X0 for G-L model. The diffusion relaxation times t0 and t1 also satisfy theinequality t1Xt0X0 for G-L model.For two-dimensional motion in x–z plane, Eqs. (4)–(6) are written as

ðlþ 2mÞu1;11 þ ðmþ lÞu3;13 þ mu1;33 � b1t1yY;1 � b2t

1cC;1 ¼ r €u1, (7)

mu3;11 þ ðmþ lÞu1;13 þ ðlþ 2mÞu3;33 � b1t1yY;3 � b2t

1cC;3 ¼ r €u3, (8)

Kr2Y ¼ rcEt0y _Yþ b1T0t0e _eþ aT0t0c _C, (9)

Db2r2eþDat1yr

2Y�Dbt1cr2C þ t0f _C ¼ 0, (10)

where

t1y ¼ 1þ t1qqt; t0y ¼ 1þ t0

qqt; t1c ¼ 1þ t1

qqt,

t0c ¼ 1þ gqqt; t0e ¼ 1þ Ot0

qqt; t0f ¼ 1þ Ot0

qqt; r2 ¼

q2

qx2þ

q2

qz2.

The displacement components u1 and u3 may be written in terms of potential functions f and c as

u1 ¼qfqx�

qcqz; u3 ¼

qfqzþqcqx

. (11)

Using Eq. (11) into Eqs. (7)–(10), we obtain

mr2c ¼ rq2cqt2

, (12)

ðlþ 2mÞr2f� b1t1yY� b2t

1cC ¼ r

q2fqt2

, (13)

Kr2Y ¼ rcEt0yqYqtþ b1T0t0e

qqtr2fþ aT0t0c

qC

qt, (14)

Db2r4fþDat1yr

2Y�Dbt1cr2C þ t0f

qC

qt¼ 0. (15)

Eq. (12) is uncoupled, whereas Eqs. (13)–(15) are coupled in f, Y and C. From Eqs. (12)–(15), wesee that while the P-wave is affected due to the presence of thermal and diffusion fields, the SVremains unaffected. The solution of Eq. (12) corresponds to the propagation of SV wave withvelocity v0 ¼

ffiffiffiffiffiffiffiffim=r

p.

Solutions of Eqs. (13)–(15) are now sought in the form of the harmonic travelling wave

ff;Y;Cg ¼ ff0;Y0;C0geikðx sin yþz cos y�vtÞ, (16)

where v is the phase speed, k is the wavenumber and ðsin y; cos yÞ denotes the projection of thewave normal onto x–z plane.

ARTICLE IN PRESS


The homogeneous system of equations in f0;Y0; and C0, obtained by inserting Eq. (16) intoEqs. (13)–(15), admits non-trivial solutions and enables to conclude that x satisfies the cubicequation

x3 þ Lx2 þMxþN ¼ 0, (17)

where

x ¼ rv2,

L ¼ �ð�þ ��1�2�3 þ d1 þ d2 þ lþ 2mÞ,

M ¼ ðlþ 2mÞðd1 þ d2 þ ��1�2�3Þ þ d1d2 þ d2�� 2ð�1 þ �3Þ � �2,

N ¼ �d1d2ðlþ 2mÞ þ �2d1,

d1 ¼K

cEty; d2 ¼

rDbt11c

tf

,

� ¼b21T0tet11yrcEty

; �1 ¼ �a

b1b2; �2 ¼

rDb22t11c

tf

; �3 ¼ �atc

b1b2tet11c

,

ty ¼ t0 þio; tc ¼ gþ

io; te ¼ Ot0 þ

io; tf ¼ Ot0 þ

io,

t11y ¼ 1� iot1; t11c ¼ 1� iot1.

Eq. (17) is cubic in x. The roots of this equation give three values of x. Each value of x correspondsto a wave if v2 is real and positive. Hence, three positive values of v will be the velocities ofpropagation of three possible waves. Cardan’s method is used to solve Eq. (17). Using Cardan’smethod, Eq. (17) is written as

L3 þ 3HLþ G ¼ 0, (18)

where

L ¼ xþL

3; H ¼

3M � L2

9; G ¼

27N � 9LM þ 2L3

27. (19)

For all the three roots of Eq. (18) to be real, D0ð¼ G2 þ 4H3Þ should be negative. Assuming the D0

to be negative, we obtain the three roots of Eq. (18) as

Ln ¼ 2ffiffiffiffiffiffiffiffiffi�Hp

cosfþ 2pðn� 1Þ

3

� �ðn ¼ 1; 2; 3Þ, (20)

where

f ¼ tan�1ffiffiffiffiffiffiffiffijD0jp

�G

� �. (21)

ARTICLE IN PRESS


Hence,

vn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLn �

L

3

� ��r

sðn ¼ 1; 2; 3Þ (22)

are velocities of propagation of the three possible coupled dilatational waves. The waves withvelocities v1; v2 and v3 correspond to P wave, mass diffusion (MD) wave and thermal (T) wave,respectively. This fact may be verified, when we solve Eq. (17), using a computer program ofCardan’s method. If we neglect thermal effects, i.e. for � ¼ 0; d1 ¼ 0, the cubic equation (17)reduces to a quadratic equation whose roots are as

2rv2 ¼ ½d2 þ ðlþ 2mÞ� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½d2 � ðlþ 2mÞ�2 þ 4�2

q, (23)

where the positive and negative signs correspond to P wave and MD wave, respectively.Moreover, the MD wave exists if b22obðlþ 2mÞ. Similarly, if we neglect the diffusion effects, i.e.for �1 ¼ �2 ¼ d2 ¼ 0, Eq. (17) reduces to a quadratic equation whose roots are as

2rv2 ¼ ½fd1 þ ðlþ 2mÞg þ �� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½fd1 � ðlþ 2mÞg � ��2 þ 4�d1

q, (24)

where the positive and negative signs correspond to P wave and T waves, respectively. The T waveexists, if d140, which is true. These two-dimensional roots are in agreement with the non-dimensional roots obtained by Abd-Alla and Al-Dawy [13] for Lord and Shulman theory. Inabsence of thermoelastic diffusion effects, Eq. (17) corresponds to P wave with velocityv1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlþ 2mÞ=r

p.

3. Reflection coefficients

In previous section, it has been discussed that there exists three compressional waves and a SVwave in an isotropic elastic solid with generalized thermoelastic diffusion. Any incident wave atthe interface of two elastic solid bodies, in general, produce compressional and distortional wavesin both media (see for example, Refs. [19,20]). Let us consider an incident SV wave (Fig. 1). Theboundary conditions at the free surface z ¼ 0 are satisfied, if the incident SV wave gives rise to areflected SV and three reflected compressional waves (i.e. P, MD and T waves). The surface z ¼ 0is assumed to be traction free and thermally insulated so that there is no variation of temperatureand concentration on it. Therefore, the boundary conditions on z ¼ 0 are written as

szz ¼ 0; szx ¼ 0;qYqz¼ 0;

qC

qz¼ 0; on z ¼ 0. (25)

The appropriate displacement potentials f and c, temperature Y and concentration C are takenin the form

c ¼ B0 exp½ik0ðx sin y0 þ z cos y0Þ � iotÞ� þ B1 exp½ik0ðx sin y0 � z cos y0Þ � iotÞ�, (26)

f ¼ A1 exp½ik1ðx sin y1 � z cos y1Þ � iotÞ� þ A2 exp½ik2ðx sin y2 � z cos y2Þ � iotÞ�

þ A3 exp½ik3ðx sin y3 � z cos y3Þ � iot�, ð27Þ

ARTICLE IN PRESS

z

x

z > 0 ( Vaccum)

θ0θ0

θ1θ2

θ3

B0B1

A1

A2

A3

Fig. 1. Schematic diagram for the problem.


Y ¼ z1A1 exp½ik1ðx sin y1 � z cos y1Þ � iotÞ� þ z2A2 exp½ik2ðx sin y2 � z cos y2Þ � iotÞ�

þ z3A3 exp½ik3ðx sin y3 � z cos y3Þ � iot�, ð28Þ

C ¼ Z1A1 exp½ik1ðx sin y1 � z cos y1Þ � iotÞ� þ Z2A2 exp½ik2ðx sin y2 � z cos y2Þ � iotÞ�

þ Z3A3 exp½ik3ðx sin y3 � z cos y3Þ � iot�, ð29Þ

where the wave normal of the incident SV wave makes angle y0 with the positive direction of thez-axis, and those of reflected P, MD and T waves make y1; y2 and y3 with the same direction, and

zi ¼ k2i Giðrv2i � l� 2mÞ; Zi ¼ k2

i Hiðrv2i � l� 2mÞ ði ¼ 1; 2; 3Þ (30)

and

Gi ¼�rv2i ð�1�2 � d2 þ rv2i Þ

d1�2 þ rv2i ½�ðd2 � rv2i Þ � �2 � ��2ð�1 þ �3Þ�, (31)

Hi ¼�2½rv2i ð��1 þ 1Þ � d1�

d1�2 þ rv2i ½�ðd2 � rv2i Þ � �2 � ��2ð�1 þ �3Þ�. (32)

The ratios of the amplitudes of the reflected waves to the amplitude of the incident wave, namelyB1=B0; A1=B0; A2=B0 and A3=B0 give the reflection coefficients for reflected SV, reflected P,reflected MD and reflected T waves, respectively. The wavenumber k0; k1; k2; k3 and the anglesy0; y1; y2; y3 are connected by the relation

k0 sin y0 ¼ k1 sin y1 ¼ k2 sin y2 ¼ k3 sin y3 (33)

at surface z ¼ 0. Relation (33) may also be written in order to satisfy the boundary conditions (25) as

sin y0v0¼

sin y1v1¼

sin y2v2¼

sin y3v3

, (34)

ARTICLE IN PRESS


where v0 ¼ffiffiffiffiffiffiffiffim=r

pis the velocity of SV wave and vi; ði ¼ 1; 2; 3Þ are the velocities of three sets of

reflected compressional waves. Using the potentials given by Eqs. (26)–(29) in boundary conditions(25), the reflection coefficients may be expressed as

B1

B0¼

D1

D;

A1

B0¼

D2

D;

A2

B0¼

D3

D;

A3

B0¼

D4

D, (35)

where

D ¼x43x31 � x33x41

x42x31 � x32x41�

x33ðx21 � x11Þ � x31ðx23 � x13Þ

x32ðx21 � x11Þ � x31ðx22 � x12Þ,

D1 ¼x43x31 � x33x41

x42x31 � x32x41�

x33ðx21 þ x11Þ � x31ðx23 þ x13Þ

x32ðx21 þ x11Þ � x31ðx22 þ x12Þ,

D2 ¼�2ðx43x32 � x42x33Þ

x42x32,

D3 ¼2ðx43x31 � x41x33Þ

x41x31,

D4 ¼�2ðx42x31 � x41x32Þ

x41x31,

x1i ¼ �½lþ 2m cos2 yi þ t11y ðzi=k2

i Þb1 þ t11c ðZi=k2i Þb2�ðki=k0Þ

2

m sin 2y0,

x2i ¼sin 2yiðki=k0Þ

2

cos 2y0,

x3i ¼ cos yiðzi=k2i Þðki=k0Þ

3,

x4i ¼ cos yiðZi=k2i Þðki=k0Þ

3.

In the absence of thermoelastic diffusion, these reflection coefficients reduce to

B1

B0¼

sin 2y1 sin 2y0 � ðv1=v0Þ2 cos 2y0

sin 2y1 sin 2y0 þ ðv1=v0Þ2 cos 2y0

, (36)

A1

B0¼

ðv1=v0Þ sin 4y0sin 2y1 sin 2y0 þ ðv1=v0Þ

2 cos 2y0, (37)

ARTICLE IN PRESS


which are the same as those given by Ben-Menahem and Singh [20], if y1, y0, v1 and v0 are replacedby e, f, a and b, respectively. It may also be mentioned that the MD and T waves will disappear inabsence of thermoelastic diffusion.

4. Numerical results

For computational work, the following material constants at T0 ¼ 27 �C are considered for anelastic solid with generalized thermoelastic diffusion

l ¼ 5:775� 1011 dyn=cm2; m ¼ 2:646� 1011 dyn=cm2; r ¼ 2:7 g=cm3,

cE ¼ 2:361 cal=g �C; K ¼ 0:492 cal=cm s �C; at ¼ 0:05 cm3=g,

ac ¼ 0:06 cm3=g; o ¼ 10 s�1; a ¼ 0:005 cm2=s2 �C; b ¼ 0:05 cm5=g s,

D ¼ 0:5 g s=cm3.

The numerical values of reflection coefficients of various reflected waves are computed for angle ofincidence varying from 1� to 45� for L-S (Lord–Shulman) and G-L (Green–Lindsay) models whent1 ¼ t0 ¼ 0:05 s and t1 ¼ t0 ¼ 0:04 s. These numerical values of reflection coefficients are showngraphically in Figs. 2–5 where solid and dotted lines correspond to L-S and G-L models, respectively.Fig. 2 shows the reflection coefficients of reflected SV waves for L-S and G-L models. The

reflection coefficient for each model, first increases and then decreases to its minima, thereafter, itattains its value one at y0 ¼ 45�. Figs. 3–5 show the variations for reflected P, reflected MD and

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s L-S Model

G-L Model

Reflected SV waves

Fig. 2. Variations of reflection coefficients of SV waves with the angle of incidence for L-S and G-L models.

ARTICLE IN PRESS


reflected T, respectively. Comparing the solid and dotted lines in these figures, the effects ofsecond thermal relaxation time and second diffusion relaxation time are observed on the reflectioncoefficients. The further numerical study is restricted to L-S model only. The numerical values of

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

L-S Model

G-L Model

Reflected P waves

Fig. 3. Variations of reflection coefficients of P waves with the angle of incidence for L-S and G-L models.

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

G-L Model

L-S Mode l

Reflected MD waves

Fig. 4. Variations of reflection coefficients of MD waves with the angle of incidence for L-S and G-L models.

ARTICLE IN PRESS

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

Reflected T waves

L-S Model

G-L Model

Fig. 5. Variations of reflection coefficients of T waves with the angle of incidence for L-S and G-L models.

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

Reflected SV waves

S1

S2

S3

Fig. 6. Variations of reflection coefficients of SV waves with the angle of incidence for sets S1, S2 and S3 of L-S model.


reflection coefficients are computed for three different sets of thermal and diffusion relaxationtimes, namely S1ðt0 ¼ 0:2; t0 ¼ 0:1Þ; S2ðt0 ¼ 0:02; t0 ¼ 0:01Þ and S3ðt0 ¼ 0:002; t0 ¼ 0:001Þ.These coefficients are shown graphically in Figs. 6–9. The reflection coefficients of various

ARTICLE IN PRESS

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

S3

S2

S1

Reflected P waves

Fig. 7. Variations of reflection coefficients of P waves with the angle of incidence for sets S1, S2 and S3 of L-S model.

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

S3

S2

S1

Reflected MD waves

Fig. 8. Variations of reflection coefficients of MD waves with the angle of incidence for sets S1, S2 and S3 of L-S

model.


ARTICLE IN PRESS

Angle of Incidence

Ref

lect

ion

Coe

ffic

eint

s

S3

S2

S1

Reflected T waves

Fig. 9. Variations of reflection coefficients of T waves with the angle of incidence for sets S1, S2 and S3 of L-S model.

D

Vel

ocity G-L

L-S

L-S

G-LThermal waves

Mass Diffusion waves

Fig. 10. Variations of velocities of MD and T waves with thermoelastic diffusion constant D for L-S and G-L models.


reflected waves are affected considerably by thermal as well as diffusion relaxation times. Oncomparing the curves for sets S1, S2 and S3, in Figs. 6–9, it may me remarked that the rate ofchange of reflection coefficients decreases as we take values of thermal relaxation time t0 anddiffusion relaxation time t0 less than those given in set S3.

ARTICLE IN PRESS


The variations of the velocities of MD and T waves with D are also shown graphically in Fig. 10for L-S and G-L models. The effects of second thermal relaxation time and second diffusionrelaxation time are noted on velocities of MD and T waves. The velocities P and SV waves aresame at each value of D and are not affected by second thermal and diffusion relaxation times.Therefore, the graphs of these two waves are not included in Fig. 10.

5. Conclusions

From theory and numerical computation, the following points are concluded.

1.
The theory of generalized thermoelastic diffusion is extended in the frame of G-L model. Thesolutions of governing equations lead to the existence of one shear and three compressionalwaves travelling with distinct speeds for two-dimensional motion in a solid with thermoelasticdiffusion.
2.
The reflection of SV wave is studied. The reflection coefficients of various waves are expressedin closed-form and computed numerically for both L-S and G-L models. The reflectioncoefficients are also computed for different values of thermal and diffusion relaxation times forL-S model only.
3.
The velocities as well as reflection coefficients of various plane waves depend on variousthermal and diffusion parameters.
The present model of elastic solid with thermoelastic diffusion becomes more realistic due to theexistence of these new compressional waves. The present theoretical results may provideinteresting information for experimental scientists/researchers/seismologists working on subjectsof wave propagation.

Acknowledgements

This work was partially completed while author was visiting the University of Salford underBilateral Exchange Programme of Indian National Science Academy and Royal Society in year2003. The award of Visiting Fellowship from Royal Society, London is very gratefullyacknowledged.

References

[1] J. Duhamel, Some memoire sur les phenomenes thermo-mechanique, Journal de l’ Ecole Polytechnique 15 (1885).

[2] F. Neumann, Vorlesungen Uber die Theorie der Elasticitat, Brestau, Meyer, 1885.

[3] M. Biot, Thermoelasticity and irreversible thermo-dynamics, Journal of Applied Physics 27 (1956) 249–253.

[4] H. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of

Solids 15 (1967) 299–309.

[5] A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elasticity 2 (1972) 1–7.

ARTICLE IN PRESS


[6] D.S. Chandresekharaiah, Thermoelasticity with second sound: a review, Applied Mechanical Review 39 (1986)

355–376.

[7] H.H. Sherief, On Generalized Thermoelasticity, PhD Thesis, University of Calgary, Canada, 1980.

[8] R.S. Dhaliwal, H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quarterly of Applied

Mathematics 33 (1980) 1–8.

[9] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, Journal of Thermal Stresses 22 (1999) 451–476.

[10] A.N. Sinha, S.B. Sinha, Reflection of thermoelastic waves at a solid half-space with thermal relaxation, Journal of

Physics of the Earth 22 (1974) 237–244.

[11] S.B. Sinha, K.A. Elsibai, Reflection of thermoelastic waves at a solid half-space with two thermal relaxation times,

Journal of Thermal Stresses 19 (1996) 763–777.

[12] S.B. Sinha, K.A. Elsibai, Reflection and refraction of thermoelastic waves at an interface of two semi-infinite

media with two thermal relaxation times, Journal of Thermal Stresses 20 (1997) 129–146.

[13] A.N. Abd-Alla, A.A.S. Al-Dawy, The reflection phenomena of SV waves in a generalized thermoelastic medium,

International Journal of Mathematics and Mathematical Sciences 23 (2000) 529–546.

[14] J.N. Sharma, V. Kumar, D. Chand, Reflection of generalized thermoelastic waves from the boundary of a

half-space, Journal of Thermal Stresses 26 (2003) 925–942.

[15] W. Nowacki, Dynamical problems of thermoelastic diffusion in solids I, Bulletin de l’ Academie Polonaise des

Sciences Serie des Sciences Techniques 22 (1974) 55–64.

[16] W. Nowacki, Dynamical problems of thermoelastic diffusion in solids II, Bulletin de l’ Academie Polonaise des


[17] W. Nowacki, Dynamical problems of thermoelastic diffusion in solids III, Bulletin de l’ Academie Polonaise des


[18] H.H. Sherief, F. Hamza, H. Saleh, The theory of generalized thermoelastic diffusion, International Journal of

Engineering Science 42 (2004) 591–608.

[19] W.B. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York, 1957, p. 76.

[20] A. Ben-Menahem, S.J. Singh, Seismic Waves and Sources, Springer, New York, 1981, pp. 89–95.

Documents

Reflection of SV waves from the free surface of an elastic solid in generalized thermoelastic diffusion