- Email: [email protected]
Thermoelastic plane waves without energy dissipation
Mechanics R©seareh Communications, VoL 23, No. 5, pp. 549-555, 1996 Copyright 0 1996 El~viec ~ Ltd Printed in the USA. All d ~ r-'=,e~ed 0093-6413/96 $12.00 + .00
Pergamon
PU S~93-6413(96)00056-0
THERMOELASTIC PLANE WAVES WITHOUT ENERGY DISSIPATION
D. S. Chandrasekharaiah Department of Mathematics, Bangalore University, Central College Campus, Bangalore 560 001, India.
(Received 10 May 1996; acceptedfor print 24 July 1996)
1. Introduction Thermoelasticity theories that predict a finite speed for the propagation of thermal signals have aroused much interest in the last three decades. Unlike the conventional thermoelasticity theory [1] based on a parabolic heat equation, which predicts an infinite speed for the propagation of heat, these theories involve a hyperbolic heat equation and are referred to as generalized thermoelasticity theories. Among these generalized thermoelasticity theories, the theory
proposed by Lord and Shulman [2] and the one
developed by Green and Lindsay [3] have been subjected to a large number of investigations. For a review of the relevant literature, see [4, 5]. Recently, Green and Naghdi [6] formulated a new generalized thermoelasticity theory by including the "thermal-displacement gradient" among the independent constitutive variables.
An important feature of this theory, which is not present in other
thermoelasticity theories, is that this theory does not accomodate dissipation of thermal
549
550
D.S. CHANDRASEKHARAIAH
energy. The discussion presented in [6] includes the derivation of a complete system of field equations of the linearized version of this theory for homogeneous and isotropic materials in terms of displacement and temperature fields and a proof of the uniqueness of solution of the corresponding initial, mixed boundary value problem. The uniqueness of solution of an initial boundary value problem formulated in terms of stress and energy flux has been proved in [7]. In this paper we study plane harmonic waves in an unbounded linear thermoelastic body by employing the theory of thermoelasticity without energy dissipation, presented in [6] (which we will henceforth refer to as the GN theory). We find that, as in the conventional theory and in the generalized theories of Lord and Shulman (LS theory) and Green and Lindsay (GL theory), the shear waves are not influenced by the thermal field and there exist two coupled dilatational waves influenced by both the elastic field and the thermal field. Interestingly, we find that the qualitative behaviour of dilatational waves in the GN theory
is substantially different from that in the conventional as well as LS and GL
theories. Whereas these waves
experience both dispersion and attenuation in the
conventional as well as LS and GL theories, there occurs neither dispersion nor attenuation in the GN theory.
Furthermore, whereas it is practically not possible to construct exact
expressions for the speeds of propagation in the conventional as well as LS and GL theories, the derivation of exact expressions, in closed form, is a trivial task in the GN theory. Thermoelastic plane harmonic waves have been studied, among others, by Deresiewicz [8] in the context of the conventional theory, by Puri [9] in the context of the LS theory, and by Agarwal [10] in the context of the GL theory.
At the appropriate stages of our
discussion, we compare our results with those obtained by these authors.
2. Governin~ eauations
In the context of the GN theory, the equation of motion and the equation of heat conduction for a linear, homogeneous and isotropic thermoelastic solid, in the absence of body forces and heat sources, are as follows [6]: ~V2u + (A+ ~ t ) V d i v u -
7VO = p u
(1)
THERMOELASTICWAVES WITHOUTDISSIPATION
c0 + 7T0divu = z'V20
551
(2)
In these equations, u is the displacement vector, 0 is the temperature-change above the uniform reference temperature To, p is the mass density, c is the specific heat,
). and
are the Lame' constants, Y= ( 3~. + 2~ ) ~ *, ~ * being the coefficient of volume expansion, and z
• .
is a material constant characteristic of the theory. Also, an over dot denotes the
partial derivative with respect to the time variable t. Further, the direct vector and tensor notation [11] is employed.
Some of the symbols and the notation used here are slightly
different from those in [6]. It is convenient to have the governing equations (1) and (2) rewritten in non-dimensional form. To this end, we consider the following transformations: 1
v
x'=-/x,
u'-
t'=~t,
1 X+2~t
l
0'=
~'T0 u,
0
To°
(3)
Here, x is the position vector of the field point, l is a standard length and v is a standard speed. Introducing (3) into (1) and (2) and suppressing primes, we obtain the following governing equations: Cs2V2u CT2V20--
+ ( Cp 2 - Cs2)Vdivu 0+
- Cp 2 Ve=
u
(4)
edivu
(5)
Here, Cp2-
~+~1 pv 2 '
~1
2 CS
= -p-v 2'
~* C T 2 _ CV 2 '
T2T0 2~)
e = C (k +
(6)
Equations (4) and (5) serve as a coupled hyperbolic system of partial differential equations for the non-dimensional displacement vector u and non-dimensional temperature e. We note that Cp and C s represent the non-dimensional speeds of purely elastic dilatational and shear waves respectively, and
CT
represents the non-dimensional speed of purely
thermal waves. Also, e is the thermoelastic coupling parameter [4].
552
D.S. CHANDRASEKHARAIAH
3. Plane waves
For examining free plane harmonic waves, we seek the solutions of the field equations (4) and (5) in the form u = a e x p [ i ( c o t - 11n.x)];
0= b e x p [ i ( c o t - 1 1 n . x ) ]
(7)
where a and b are constants not both zero, co is a positive real number, 11 is generally a complex number and n is the unit vector along the direction of propagation. For the waves to be physically realistic, we should have Re ~ > 0 and Im ~ < 0. Then, for these waves, c°/2n is the frequency 2~/Re q is the wave length, V = a)/Req is the speed of propagation and Im TI is the decay (attenuation) coefficient. All the quantities being considered here are non-dimensional. Substituting expressions (7) into the field equations (4) and (5), we obtain the following system of linear algebraic equations for a and b:
(¢0 2- C s 2 ~ 2 ) a - [(Cp 2- C s 2 ) ~ 2 a . n ) (CT2q 2- co2)b+ ie~co2(a.n)=
Cp2i~b]n
=0
0
(8) (9)
For puly shear waves, we have a . n = 0, and equations (8) and (9) become (co2_ C s 2 T i 2 ) a +
Cp2i~bn= 0
(CT2~ 2- c02)b= 0
(10) (11)
Taking the scalar product with n of equation (10), we find that b = 0; then (11) is trivially satisfied. Also, since a and b do not vanish together, (10) yields
¢02- CS2~ 2= 0
(12)
which is the secular equation for pure shear waves. This equation shows that pure shear waves propagate with (dimensionless) speed
Cs and that the thermal field has no influence
on these waves. This result agrees with that observed in the conventional as well as LS and GL theories [8, 9, 10]. For purely dilatational waves, we have a • n = a, where a is the magnitude of a, and equations (8) and (9) become, on taking the scalar product with n of equation (8),
THERMOELASTICWAVESWITHOUTDISSIPATION
553
( ( 0 2 Cp2~2)a+ Cp2i~b= 0 (CT2TI 2- ( 0 2 ) b + ie(02~a= 0
(13) (14)
From these equations it is evident that, in the case of dilatational waves, the thermal field and the elastic field are coupled together in general, as in the conventional as well as LS and GL theories [8, 9, 10]. Eliminating the constants a and b from these equations, we obtain the following secular equation for the dilatational
waves in the GN
theory: ((02- Cp2~2)((02-
CT2~ 2 ) -
ECp2~2(02 = 0
(15)
If the coupling between the elastic and thermal fields is negligibly small (that is, if -- 0 ), then equation (15) factorizes into the following two equations:
(02_ Cp2.q2 = O;
(02_ CT21~2
--
(16a,b)
0
The first of these equations corresponds to a purely elastic dilatational influenced by the thermal field) propagating with (dimensionless) speed
wave (not
Cp. The
second
equation corresponds to a purely thermal dilatational wave (not influenced by the elastic field) propagating with (dimensionless) speed
CT.
If the coupling between the elastic and thermal fields is not negligibly small (that is, if e ~ 0), then the secular equation (15) may be rewritten as a biquadratic equation for ~ as given below: 2
CT2Cp2
4
. CT2Cp2 - 0
(17)
For a given (0, this equation yields four roots of the form _+ ~1' + 112' for 11. Of these four roots, only two roots yield positive values for Re ~. Hence, there occur two distinct dilatational waves, both influenced by elastic as well as thermal fields. This result agrees with that in the conventional as well as LS and GL theories [8, 9, 10]. In these theories it has also been observed that one of the dilatational waves is a quasi-elastic wave and the other is a quasi-thermal wave [8, 9, 10]. The same is true in the GN theory also; see the last paragraph of this section. The roots of equation (17) that are associated with the waves under consideration are given by
554
D.S. CHANDRASEKHARAIAH
)]a -
CO
V '
a=1,2
(18)
{2
g
=
+(l+e)Cp2
+(_I)~+~A
v2
(19)
-
= V12 -
(20)
V22
(21)
From the above expressions, we immediately note that
V1 and V2 and therefore
111 and "112 a r e purely real. Consequently, it follows that, in the GN theory, neither of the two dilatational waves experiences any attenuation. This is not the case in the conventional as well as LS and GL theories; in
all
these
theories the
roots of the corresponding
secular equations are complex numbers and consequently the waves do experience exponential attenuation [8, 9, 10]. Since T11and ~2 a r e (now) purely real, we find from (18) that V1 and V2 are indeed the (dimensionless) speeds of the waves that occur. We note that (19) are exact expressions for the speeds; from these expressions we immediately observe that both of the speeds are determined completely by the material constants and are independent that,
of co. This means
in the GN theory, the dilatational waves propagate with constant speeds without
experiencing any dispersion. Again, this is not the case in the conventional as well as LS and GL theories; in all these theories the speeds of propagation do depend on the frequency and consequently the waves do experience dispersion [8, 9, 10]. Furthermore, unlike in the present analysis, the extraction of exact expressions for the speeds of propagation in the conventional and LS and GL theories is a formidable task; in these theories
only
approximate expressions for the speeds, valid for small or large frequencies or for small values of the thermoelastic coupling parameter, are available
[8, 9, 10].
Returning to expressions (19) - (21), we find (from these expressions) that (i) VI > (Cp, C T) > V 2 (ii) If Cp > C T ,
then VI > Cp > C T > V 2 and V1 tends to Cp, V 2 t e n d s t o C T a s
tends to zero. (iii) I f C T > Cp, then V1 > C r > Cp > V 2 and VI tends to C 7. V 2 t e n d s t o C p a s e tends to zero.
THERMOELASTICWAVES WITHOUTDISSIPATION
555
Accordingly, it follows that V I is the (dimensionless) speed of the faster of the two thermoelastic dilatational waves that occur in the GN theory and V2 is the (dimensionless) In materials for which Cp > C T , the faster wave is a
speed of the slower wave.
predominantly elastic wave (quasi-elastic wave or the e-wave) and the slower wave is a predominantly thermal wave (quasi-thermal wave or the 0-wave). which CT> Cp, the faster wave is the
In
materials
for
0-wave and the slower wave is the e-wave. *
We note from (6) that
Cp > C T or C T > Cp according as ~¢ < c ( k + 2 ~ t ) / p , or
> c (k + 2~t)/p. The value of ~* has not yet been experimentally determined for any material.
Acknowledgement: The author thanks the reviewer for his kind observations and suggestions.
References
1. M.A. Biot, J. Appl. Phys., 27, 240 (1956). 2. H . W . Lord and Y. Shulman, J. Mech. Phys. Solids 15, 299 (1967). 3. A. E. Green and K. A. Lindsay, J. Elasticity 2, 1 (1972). 4. D. S. Chandrasekharaiah, Appl. Mech. Rev., 39, 355 (1986). 5. D. D. Joseph and L. Preziosi, Rev. Mod. Phys., 61, 41 (1989) and 62, 375 (1990). 6. A. E. Green and P. M. Naghdi, J. Elasticity 31, 189 (1993). 7. D. S. Chandrasekharaiah, J. Thermal Stresses 19, 267 (1996). 8. H. Deresiewicz, J. Acous. Soc. Am., 29, 204 (1957). 9. P. Puri, Int. J. Eng. Sci., 11,735 (1973) and 13, 339 (1975). 10. V. K. Agarwal, Acta Mech., 31, 185 (1979). 11. D. S. Chandrasekharaiah and L. Debnath, Continuum Mechanics, chaps. 1-4, Academic Press, New York (1994).
Pergamon
PU S~93-6413(96)00056-0
THERMOELASTIC PLANE WAVES WITHOUT ENERGY DISSIPATION
D. S. Chandrasekharaiah Department of Mathematics, Bangalore University, Central College Campus, Bangalore 560 001, India.
(Received 10 May 1996; acceptedfor print 24 July 1996)
1. Introduction Thermoelasticity theories that predict a finite speed for the propagation of thermal signals have aroused much interest in the last three decades. Unlike the conventional thermoelasticity theory [1] based on a parabolic heat equation, which predicts an infinite speed for the propagation of heat, these theories involve a hyperbolic heat equation and are referred to as generalized thermoelasticity theories. Among these generalized thermoelasticity theories, the theory
proposed by Lord and Shulman [2] and the one
developed by Green and Lindsay [3] have been subjected to a large number of investigations. For a review of the relevant literature, see [4, 5]. Recently, Green and Naghdi [6] formulated a new generalized thermoelasticity theory by including the "thermal-displacement gradient" among the independent constitutive variables.
An important feature of this theory, which is not present in other
thermoelasticity theories, is that this theory does not accomodate dissipation of thermal
549
550
D.S. CHANDRASEKHARAIAH
energy. The discussion presented in [6] includes the derivation of a complete system of field equations of the linearized version of this theory for homogeneous and isotropic materials in terms of displacement and temperature fields and a proof of the uniqueness of solution of the corresponding initial, mixed boundary value problem. The uniqueness of solution of an initial boundary value problem formulated in terms of stress and energy flux has been proved in [7]. In this paper we study plane harmonic waves in an unbounded linear thermoelastic body by employing the theory of thermoelasticity without energy dissipation, presented in [6] (which we will henceforth refer to as the GN theory). We find that, as in the conventional theory and in the generalized theories of Lord and Shulman (LS theory) and Green and Lindsay (GL theory), the shear waves are not influenced by the thermal field and there exist two coupled dilatational waves influenced by both the elastic field and the thermal field. Interestingly, we find that the qualitative behaviour of dilatational waves in the GN theory
is substantially different from that in the conventional as well as LS and GL
theories. Whereas these waves
experience both dispersion and attenuation in the
conventional as well as LS and GL theories, there occurs neither dispersion nor attenuation in the GN theory.
Furthermore, whereas it is practically not possible to construct exact
expressions for the speeds of propagation in the conventional as well as LS and GL theories, the derivation of exact expressions, in closed form, is a trivial task in the GN theory. Thermoelastic plane harmonic waves have been studied, among others, by Deresiewicz [8] in the context of the conventional theory, by Puri [9] in the context of the LS theory, and by Agarwal [10] in the context of the GL theory.
At the appropriate stages of our
discussion, we compare our results with those obtained by these authors.
2. Governin~ eauations
In the context of the GN theory, the equation of motion and the equation of heat conduction for a linear, homogeneous and isotropic thermoelastic solid, in the absence of body forces and heat sources, are as follows [6]: ~V2u + (A+ ~ t ) V d i v u -
7VO = p u
(1)
THERMOELASTICWAVES WITHOUTDISSIPATION
c0 + 7T0divu = z'V20
551
(2)
In these equations, u is the displacement vector, 0 is the temperature-change above the uniform reference temperature To, p is the mass density, c is the specific heat,
). and
are the Lame' constants, Y= ( 3~. + 2~ ) ~ *, ~ * being the coefficient of volume expansion, and z
• .
is a material constant characteristic of the theory. Also, an over dot denotes the
partial derivative with respect to the time variable t. Further, the direct vector and tensor notation [11] is employed.
Some of the symbols and the notation used here are slightly
different from those in [6]. It is convenient to have the governing equations (1) and (2) rewritten in non-dimensional form. To this end, we consider the following transformations: 1
v
x'=-/x,
u'-
t'=~t,
1 X+2~t
l
0'=
~'T0 u,
0
To°
(3)
Here, x is the position vector of the field point, l is a standard length and v is a standard speed. Introducing (3) into (1) and (2) and suppressing primes, we obtain the following governing equations: Cs2V2u CT2V20--
+ ( Cp 2 - Cs2)Vdivu 0+
- Cp 2 Ve=
u
(4)
edivu
(5)
Here, Cp2-
~+~1 pv 2 '
~1
2 CS
= -p-v 2'
~* C T 2 _ CV 2 '
T2T0 2~)
e = C (k +
(6)
Equations (4) and (5) serve as a coupled hyperbolic system of partial differential equations for the non-dimensional displacement vector u and non-dimensional temperature e. We note that Cp and C s represent the non-dimensional speeds of purely elastic dilatational and shear waves respectively, and
CT
represents the non-dimensional speed of purely
thermal waves. Also, e is the thermoelastic coupling parameter [4].
552
D.S. CHANDRASEKHARAIAH
3. Plane waves
For examining free plane harmonic waves, we seek the solutions of the field equations (4) and (5) in the form u = a e x p [ i ( c o t - 11n.x)];
0= b e x p [ i ( c o t - 1 1 n . x ) ]
(7)
where a and b are constants not both zero, co is a positive real number, 11 is generally a complex number and n is the unit vector along the direction of propagation. For the waves to be physically realistic, we should have Re ~ > 0 and Im ~ < 0. Then, for these waves, c°/2n is the frequency 2~/Re q is the wave length, V = a)/Req is the speed of propagation and Im TI is the decay (attenuation) coefficient. All the quantities being considered here are non-dimensional. Substituting expressions (7) into the field equations (4) and (5), we obtain the following system of linear algebraic equations for a and b:
(¢0 2- C s 2 ~ 2 ) a - [(Cp 2- C s 2 ) ~ 2 a . n ) (CT2q 2- co2)b+ ie~co2(a.n)=
Cp2i~b]n
=0
0
(8) (9)
For puly shear waves, we have a . n = 0, and equations (8) and (9) become (co2_ C s 2 T i 2 ) a +
Cp2i~bn= 0
(CT2~ 2- c02)b= 0
(10) (11)
Taking the scalar product with n of equation (10), we find that b = 0; then (11) is trivially satisfied. Also, since a and b do not vanish together, (10) yields
¢02- CS2~ 2= 0
(12)
which is the secular equation for pure shear waves. This equation shows that pure shear waves propagate with (dimensionless) speed
Cs and that the thermal field has no influence
on these waves. This result agrees with that observed in the conventional as well as LS and GL theories [8, 9, 10]. For purely dilatational waves, we have a • n = a, where a is the magnitude of a, and equations (8) and (9) become, on taking the scalar product with n of equation (8),
THERMOELASTICWAVESWITHOUTDISSIPATION
553
( ( 0 2 Cp2~2)a+ Cp2i~b= 0 (CT2TI 2- ( 0 2 ) b + ie(02~a= 0
(13) (14)
From these equations it is evident that, in the case of dilatational waves, the thermal field and the elastic field are coupled together in general, as in the conventional as well as LS and GL theories [8, 9, 10]. Eliminating the constants a and b from these equations, we obtain the following secular equation for the dilatational
waves in the GN
theory: ((02- Cp2~2)((02-
CT2~ 2 ) -
ECp2~2(02 = 0
(15)
If the coupling between the elastic and thermal fields is negligibly small (that is, if -- 0 ), then equation (15) factorizes into the following two equations:
(02_ Cp2.q2 = O;
(02_ CT21~2
--
(16a,b)
0
The first of these equations corresponds to a purely elastic dilatational influenced by the thermal field) propagating with (dimensionless) speed
wave (not
Cp. The
second
equation corresponds to a purely thermal dilatational wave (not influenced by the elastic field) propagating with (dimensionless) speed
CT.
If the coupling between the elastic and thermal fields is not negligibly small (that is, if e ~ 0), then the secular equation (15) may be rewritten as a biquadratic equation for ~ as given below: 2
CT2Cp2
4
. CT2Cp2 - 0
(17)
For a given (0, this equation yields four roots of the form _+ ~1' + 112' for 11. Of these four roots, only two roots yield positive values for Re ~. Hence, there occur two distinct dilatational waves, both influenced by elastic as well as thermal fields. This result agrees with that in the conventional as well as LS and GL theories [8, 9, 10]. In these theories it has also been observed that one of the dilatational waves is a quasi-elastic wave and the other is a quasi-thermal wave [8, 9, 10]. The same is true in the GN theory also; see the last paragraph of this section. The roots of equation (17) that are associated with the waves under consideration are given by
554
D.S. CHANDRASEKHARAIAH
)]a -
CO
V '
a=1,2
(18)
{2
g
=
+(l+e)Cp2
+(_I)~+~A
v2
(19)
-
= V12 -
(20)
V22
(21)
From the above expressions, we immediately note that
V1 and V2 and therefore
111 and "112 a r e purely real. Consequently, it follows that, in the GN theory, neither of the two dilatational waves experiences any attenuation. This is not the case in the conventional as well as LS and GL theories; in
all
these
theories the
roots of the corresponding
secular equations are complex numbers and consequently the waves do experience exponential attenuation [8, 9, 10]. Since T11and ~2 a r e (now) purely real, we find from (18) that V1 and V2 are indeed the (dimensionless) speeds of the waves that occur. We note that (19) are exact expressions for the speeds; from these expressions we immediately observe that both of the speeds are determined completely by the material constants and are independent that,
of co. This means
in the GN theory, the dilatational waves propagate with constant speeds without
experiencing any dispersion. Again, this is not the case in the conventional as well as LS and GL theories; in all these theories the speeds of propagation do depend on the frequency and consequently the waves do experience dispersion [8, 9, 10]. Furthermore, unlike in the present analysis, the extraction of exact expressions for the speeds of propagation in the conventional and LS and GL theories is a formidable task; in these theories
only
approximate expressions for the speeds, valid for small or large frequencies or for small values of the thermoelastic coupling parameter, are available
[8, 9, 10].
Returning to expressions (19) - (21), we find (from these expressions) that (i) VI > (Cp, C T) > V 2 (ii) If Cp > C T ,
then VI > Cp > C T > V 2 and V1 tends to Cp, V 2 t e n d s t o C T a s
tends to zero. (iii) I f C T > Cp, then V1 > C r > Cp > V 2 and VI tends to C 7. V 2 t e n d s t o C p a s e tends to zero.
THERMOELASTICWAVES WITHOUTDISSIPATION
555
Accordingly, it follows that V I is the (dimensionless) speed of the faster of the two thermoelastic dilatational waves that occur in the GN theory and V2 is the (dimensionless) In materials for which Cp > C T , the faster wave is a
speed of the slower wave.
predominantly elastic wave (quasi-elastic wave or the e-wave) and the slower wave is a predominantly thermal wave (quasi-thermal wave or the 0-wave). which CT> Cp, the faster wave is the
In
materials
for
0-wave and the slower wave is the e-wave. *
We note from (6) that
Cp > C T or C T > Cp according as ~¢ < c ( k + 2 ~ t ) / p , or
> c (k + 2~t)/p. The value of ~* has not yet been experimentally determined for any material.
Acknowledgement: The author thanks the reviewer for his kind observations and suggestions.
References
1. M.A. Biot, J. Appl. Phys., 27, 240 (1956). 2. H . W . Lord and Y. Shulman, J. Mech. Phys. Solids 15, 299 (1967). 3. A. E. Green and K. A. Lindsay, J. Elasticity 2, 1 (1972). 4. D. S. Chandrasekharaiah, Appl. Mech. Rev., 39, 355 (1986). 5. D. D. Joseph and L. Preziosi, Rev. Mod. Phys., 61, 41 (1989) and 62, 375 (1990). 6. A. E. Green and P. M. Naghdi, J. Elasticity 31, 189 (1993). 7. D. S. Chandrasekharaiah, J. Thermal Stresses 19, 267 (1996). 8. H. Deresiewicz, J. Acous. Soc. Am., 29, 204 (1957). 9. P. Puri, Int. J. Eng. Sci., 11,735 (1973) and 13, 339 (1975). 10. V. K. Agarwal, Acta Mech., 31, 185 (1979). 11. D. S. Chandrasekharaiah and L. Debnath, Continuum Mechanics, chaps. 1-4, Academic Press, New York (1994).