Archive of Applied Mechanics 72 (2002) 470 – 482 Ó Springer-Verlag 2002 DOI 10.1007/s00419-002-0215-z
Wave propagation in transversely isotropic plates in generalized thermoelasticity K. L. Verma, N. Hasebe
470 Summary In this paper, the boundary value problem in generalized thermoelasticity concerning the propagation of plane harmonic waves in a thin, flat, infinite homogeneous, transversely isotropic plate of finite width is solved. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained. The limiting and special cases of the frequency equations have also been discussed. Finally, a numerical solution of the frequency equations for a NaF crystal is carried out, and the dispersion curves for the lowest six modes of the symmetric and antisymmetric vibrations are represented graphically at different values of thermal relaxation time. Keywords Thermoelasticity, Frequency equation, NaF crystal, Thermal relaxation time, Vibration, Harmonic wave
1 Introduction The dynamical theory of generalized thermoelasticity proposed in [1] has aroused much interest in recent years. This theory is the generalization of the classical coupled thermoelasticity, [2], which includes the time needed for acceleration of the heat flow. The new theory, which has been named the ‘Generalized Theory of Thermoelasticity’ and received much attention, [2–6], eliminates the paradox of an infinite velocity of propagation and admits finite speed for the propagation of thermoelastic disturbances. Extensive theoretical efforts have been made so far to model thermoelastic waves in solids. The propagation of generalized thermoelastic waves in plates of isotropic media has considered by [7–11]. In [12], the propagation of thermoelastic waves in infinite plates has studied in the context of generalized thermoelasticity and linear theory of thermoelasticity without energy dissipation. The generalized theory of thermoelasticity has been extended to heat conducting anisotropic elastic solids by [13, 14]. Propagation of plane harmonic waves in a homogeneous anisotropic generalized thermoelastic solid was discussed [15]. It was found that four dispersive wave modes are possible namely, three quasi-elastic wave modes (E) and one quasi-thermal wave mode (T), which in coupled thermoelasticity is diffusive but now becomes wave-like with the finite velocity of propagation. In [16], the propagation of thermoelastic waves was studied in bilaminated periodic waveguides in the context of the generalized theory of thermoelasticity. The same authors studied the propagation of harmonic waves in a laminated composite consisting of an arbitrary number of layered anisotropic plates, [17]. The present paper is a continuation of our previous work [12]. Its aim is to study in generalized thermoelasticity the propagation of plane harmonic waves in a thin, flat, infinite Received 17 July 2000; accepted for publication 26 March 2002 K. L. Verma (&) Department of Mathematics, Government Post Graduate College, Hamirpur, (H.P.) 177005 India e-mail:
[email protected] N. Hasebe Department of Civil Engineering, Nogoya Institute of Technology, Gokiso-Cho, Showa-Ku, Nagoya 466, Japan
homogeneous, transversely isotropic plate of given thickness, [14]. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained, and the limiting and special cases of the frequency equations discussed. Finally, a numerical solution of the frequency equations for a NaF crystal is carried out, and the dispersion curves for the lowest six modes of the symmetric and antisymmetric vibrations are represented graphically at different values of thermal relaxation time.
2 Preliminary formulations We consider an infinite, homogeneous, transversely isotropic, thermally conducting elastic plate at uniform temperature T0 in the undisturbed state having thickness 2d. Let the faces of the plate be the planes z ¼ d, referred to a rectangular set of cartesian axes Oðx; y; zÞ. We choose x-axis in the direction of the propagation of waves so that all particles on a line parallel to y-axis are equally displaced. Therefore, all the field quantities will be independent of y-coordinate. The motion is supposed to take place in the dimensions ðx; zÞ. Here, u; w are the displacements of a point in the x; z directions, respectively. In linear generalized theory of thermoelasticity, the governing field equations for the temperature Tðx; z; tÞ and the displacement vector uðx; z; tÞ ¼ fu; 0; wg in the absence of body forces and heat sources are given by
c11 u;xx þ c44 u;zz þ ðc13 þ 12 c44 Þw;xz q€ u ¼ b1 T;x ;
ð1Þ
€ ¼ b3 T;z ; ðc13 þ 12 c44 Þu;xz þ 12 c44 w;xx þ c33 w;zz qw € ;z Þ ; K1 T;xx þ K3 T;zz qCe T_ þ s0 T€ ¼ T0 ½b1 ðu_ ;x þ s0 u€;x Þ þ b3 ðw_ ;z þ s0 w
ð2Þ ð3Þ
where
b1 ¼ ðc11 þ c12 Þa1 þ c13 a3 ;
b3 ¼ 2c11 a1 þ c13 a3 ;
ð4Þ
cij are the elastic parameters, q is the density of the medium, Ce and s0 are the specific heat at constant strain and the thermal relaxation time, respectively; K1 , K3 and a1 , a3 are, respectively, the coefficients of thermal conductivities and linear thermal expansions along and perpendicular to the plane of symmetry. The comma notation is used for spatial derivatives and a superposed dot denotes time differentiation. We define the following dimensionless quantities:
x ¼
v1 x; k1
T¼
T ; T0
K3 K ¼ ; K1
v1 z; k1
t ¼
v21 t; k1
v21 s0 ; k1
c1 ¼
c33 ; c11
z ¼ s 0 ¼
b b ¼ 3 ; b1
e1 ¼
b21 T0 ; qCe v21
u ¼
c2 ¼
c44 ; 2c11
f2 ¼
where
1=2 c11 v1 ¼ q is the velocity of compressional waves and
k1 ¼
K1 qCe
v31 q u; k1 b1 T0
qc2 c11
c3 ¼
w ¼
v31 q w ; k1 b1 T0
c13 þ 12 c44 ; c11 ð5Þ
471
is the thermal diffusivity in the x-direction. Here, e1 is the thermoelastic coupling constant and s 0 is the thermal relaxation constant. Introducing quantities (5) in Eqs. (1)–(3), after suppressing the asterisk and using a superposed dot for time differentiation, we obtain
472
u;xx þc2 u;zz þc3 w;xz € u ¼ T;x ;
ð6Þ
€ ¼ bT;z ; c3 u;xz þc2 w;xx þc1 w;zz w
ð7Þ
€ ;z Þ : T;xx þKT;zz ðT_ þ s0 T€Þ ¼ e1 ½ðu_ ;x þs0 u€;x Þ þ bðw_ ;z þs0 w
ð8Þ
The stresses and temperature gradient relevant to our problem in the plate are
szz ¼ ðc3 c2 Þu;x þ c1 w;z b1 T ;
ð9Þ
szx ¼ b1 T0 c2 ðu;z þ w;x Þ ;
ð10Þ
@T : @z
T;z ¼
ð11Þ
For a plane harmonic wave traveling in the x-direction, the solutions u, w, and T of Eqs. (6)–(8) take the form
u ¼ f ðzÞ exp½inðx ctÞ ;
ð12Þ
w ¼ gðzÞ exp½inðx ctÞ ;
ð13Þ
T ¼ hðzÞ exp½inðx ctÞ ;
ð14Þ
where c ¼ x=n and p n are ffiffiffiffiffiffi the phase velocity and the wave number, respectively, x is the circular frequency, and i ¼ 1. Substituting for u, w and T from Eqs. (12)–(14) into (6)–(8), we get
ðc2 D2 n2 þ n2 c2 Þf þ inc3 Dg inh ¼ 0; inc3 Df þ ðc1 D2 c2 n2 þ n2 c2 Þg bDh ¼ 0;
ð15Þ
in e1 sc f þ e1 bsn c Dg þ ðKD n þ sn c Þh ¼ 0 ; 3
2 2
2
2
2
2 2
where
D¼
d ; dz
s ¼ s0 þ
i : nc
The solution to Eq. (15) is
f ðzÞ ¼ gðzÞ ¼
3 X j¼1 3 X
½Rj chðnsj zÞ Sj shðnsj zÞ ; mj ½Sj chðnsj zÞ Rj shðnsj zÞ ;
j¼1 3 X
hðzÞ ¼ n
ð16Þ
lj ½Rj chðnsj zÞ Sj chðnsj zÞ ;
j¼1
where
mj ¼
½bðc2 s2j þ c2 1Þ þ c3 sj ; i½s2 ðc3 b c1 Þ þ c2 c2
j
lj ¼
½ðc1 s2j þ c2 1Þ ic3 sj mj
i
ð17Þ :
Here, Rj , Sj ð j ¼ 1; 2; 3Þ are arbitrary constants, and s1 , s2 , and s3 are the roots of the equation
s6 þ A1 s4 þ A2 s2 þ A3 ¼ 0 ;
ð18Þ
where
A1 ¼ Kc2 c2 c2 þ c1 c2 1 sc2 þ Kc1 1 c2 Kc23 Kc1 c2 =c2 bsc2 e1 ;
A2 ¼ K c2 c2 þ c1 1 sc2 e1 sb2 c2 ð1 c2 Þ c23 ð1 sc2 Þ þ c2 c2 c2 1 sc2 þKc1 c2 =e1 sc2 ð2c3 b c1 Þ ; A3 ¼ Kc1 c2 = ðc2 c2 Þ½ð1 sc2 Þð1 c2 Þ e1 sc2 :
473
The displacements and temperature to the plate are thus 3 X
u¼
½Rj chðnsj zÞ Sj shðnsj zÞ exp½inðx ctÞ ;
j¼1
w¼
3 X
mj ½Sj chnsj z Sj shðnsj zÞ exp½inðx ctÞ ;
ð19Þ
j¼1
T¼n
3 X
lj ½Rj chðnsj zÞ Sj shðnsj zÞ exp½inðx ctÞ :
j¼1
3 Boundary conditions The boundary conditions are that stresses and temperature gradient on the surfaces of the plate should vanish. Hence, we demand for all x and t, szz ¼ sxz ¼ T;z ¼ 0 on z ¼ d :
ð20Þ
Substituting expressions (19) for the displacement components and temperature into Eqs. (9)–(11), we obtain the thermal stresses and the temperature gradient. Using boundary conditions (20) in the resulting thermal stresses and temperature gradient yield six equations involving the arbitrary constants R1 , R2 , R3 , S1 , S2 and S3 . 3 X ðiF c1 mj sj lj Þ½Rj shðnsj dÞ Sj shðnsj dÞ ¼ 0; j¼1 3 X ðimj sj Þ½Sj chðnsj dÞ Rj shðnsj dÞ ¼ 0; j¼1 3 X ðlj sj ÞðSj chðnsj dÞ Rj shðnsj dÞ ¼ 0; j¼1 3 X
ð21Þ ðiF c1 mj sj blj Þ½Rj chðnsj dÞ þ Sj shðnsj dÞ ¼ 0;
j¼1 3 X
ðimj sj Þ½Sj chðnsj dÞ þ Rj shðnsj dÞ ¼ 0;
j¼1 3 X
ðlj sj Þ½Sj chðnsj dÞ þ Rj shðnsj dÞ ¼ 0 ;
j¼1
where F ¼ c3 c2 :
4 Frequency equation In order for the six boundary conditions to be satisfied simultaneously, the determinant of the coefficients of the arbitrary constants in Eqs. (21) must vanish. This gives an equation for the frequency of the plate oscillations. The frequency equation is found to factorize into two factors, each of which yields the equations D1 G1 chðns1 dÞshðns2 dÞshðns3 dÞ D2 G2 shðns1 dÞchðns2 dÞshðns3 dÞ þ D3 G3 shðns1 dÞshðns2 dÞchðns3 dÞ ¼ 0; 474
D1 G1 shðns1 dÞchðns2 dÞchðns3 dÞ D2 G2 chðns1 dÞshðns2 dÞchðns3 dÞ
ð22Þ
þ D3 G3 chðns1 dÞchðns2 dÞshðns3 dÞ ¼ 0 ; where
Dj ¼ iF c1 mj sj blj ; G1 ¼ Y2 Z3 Y3 Z2 ; Yj ¼ imj sj ;
ð23Þ
G2 ¼ Y1 Z3 Y3 Z1 ;
Zj ¼ lj sj ;
G3 ¼ Y1 Z2 Y2 Z ;
j ¼ 1; 2; 3 ;
ð24Þ ð25Þ
with mj and lj given in Eqs. (17). These are the period equations which correspond to the symmetric and antisymmetric motions of the plate with respect to the medial plane z ¼ 0. It can be shown that (22a) corresponds to the symmetric motion, and (22b) corresponds to the antisymmetric motion. If we take
c11 ¼ c33 ¼ k þ 2l; a1 ¼ a3 ¼ at ;
c44 ¼ l;
K1 ¼ K3 ¼ K ;
ð26Þ
b1 ¼ b3 ¼ ð3k þ 2lÞat ;
ð27Þ
the above equations reduce to the corresponding forms for an isotropic body with Lame’s parameters k, l, thermal conductivity K and the coefficient of linear thermal expansion at . The discussion of transcendental equations (22), in general, is difficult; therefore, we consider the results for some limiting cases.
5 Limiting cases 5.1 Symmetric modes For waves long as compared with the thickness 2d of the plate, nd is small and, consequently, nds1 , nds2 and nds3 may be assumed small as long as c is finite. Then, if the hyperbolic functions in (22a) are replaced by unity or linear terms, we then obtain ðs22 s23 Þðs21 s23 Þðs21 s22 Þ½F11 H2 A3 F33 H3 F22 H1 A3 F11 H3 A2 F22 H3 A1 ¼ 0 ;
ð28Þ
where
F11 ¼ ðc3 b c1 Þc2 bðc2 c3 Þ c22 b2 ;
ð29Þ 2
F22 ¼ ðc3 b c1 Þðc2 1Þc2 c2 c3 bð1 þ c2 c2 Þ ðc2 1Þc2 b ;
ð30Þ
F33 ¼ ðc2 1Þ2 b½ðc3 b c1 Þ b 2c3 bðc2 1Þ ½ðc2 1Þb þ c3 ðc2 c2 Þ ;
ð31Þ
H1 ¼ c2 bðc3 b c1 Þ ;
ð32Þ
H2 ¼ ðc3 b c1 Þ½ðbðc2 1Þ þ c2 ð1 bÞ ;
ð33Þ
H3 ¼ bðc2 1Þ½ðc3 b c1 Þ ðc2 c2 Þ ðc3 c2 Þðc2 c2 Þ :
ð34Þ
Hence, Eq. (28) is either
ðs22 s23 Þðs21 s23 Þðs21 s22 Þ ¼ 0 ;
ð35aÞ
or
F11 H2 A3 F33 H3 F22 H1 A3 F11 H3 A2 F22 H3 A1 ¼ 0 :
ð35bÞ
If s22 ¼ s23 ; s21 ¼ s23 ; s21 ¼ s22 , the form of the original solution assumed, Eqs. (19) cannot satisfy the boundary conditions. Hence, Eq. (35b) holds. On using the isotropic relations (26, 27), expression (35b) reduces to
2 c2 2 ½1 c2 ðs þ e1 sÞ ¼ 4½ðc2 s 1Þðc2 1Þ e1 c2 s : c2
ð36Þ
This equation gives the phase velocity of long compressional or plate waves cp in generalized theory of thermoelasticity. When the strain and thermal fields are uncoupled from each other, the coupling constant e1 is identically zero, and Eq. (36) reduces to
b2 c ¼ 4b 1 2 ; a 2
2
ð37Þ
which agrees with [18]. For very short waves and c such that s1 ; s2 and s3 are real, nd is large and we obtain the approximation
ðs1 s2 Þðs2 s3 Þðs3 s1 Þ½ðs1 þ s2 þ s3 ÞðF11 H3 A2 F11 H2 A3 þ F22 H1 A3 þ F22 H3 A1 þ F33 H3 Þ þ s1 s2 s3 fðs1 s2 þ s2 s3 þ s3 s1 ÞðF11 H3 þ F22 H2 F33 H1 Þ þ ðF11 H1 A3 þ F22 H1 A2 þ F22 H2 A2 þ F22 H3 þ F33 H2 Þg ¼ 0 : Evidently ðs1 s2 Þðs2 s3 Þðs3 s1 Þ is a nonvanishing factor. Therefore, from the last equation we obtain
ðs1 þ s2 þ s3 ÞðF11 H3 A2 F11 H2 A3 þ F22 H1 A3 þ F22 H3 A1 þ F33 H3 Þ þ s1 s2 s3 fðs1 s2 þ s2 s3 þ s3 s1 ÞðF11 H3 þ F22 H2 F33 H1 Þ þ ðF11 H1 A3 þ F22 H1 A2 þ F22 H2 A2 þ F22 H3 þ F33 H2 Þg ¼ 0 ;
ð38Þ
where s1 ; s2 and s3 are roots of Eq. (18). Equation (38) can be identified as the phase velocity equation for Rayleigh waves in transversely isotropic half-space, which has been discussed in detail in [19] and [20]. On using the isotropic relations (26, 27), expression (38) becomes
ð1 þ s23 Þ2 fs21 þ s21 þ s1 s2 þ c2 1g þ 4s1 s2 s3 ðs1 þ s2 Þ ¼ 0 :
ð39aÞ
Equation (39a) can be identified as the phase velocity equation for Rayleigh waves in isotropic half-space. This is in agreement with the corresponding result of Ref. [19]. When the strain and thermal fields are uncoupled from each other, the coupling constant e1 is identically zero, and Eq. (39a) reduces to
4 c2 c2 : 2 ¼ 16 1 c2 1 c2 c2
This is in agreement with the corresponding result in [19].
ð39bÞ
475
5.2 Antisymmetric modes For waves long compared with thickness of the plate, s1 ; s2 and s3 are real and then we may replace the hyperbolic functions by the approximation of the order OðxÞ3 . After some algebraic transformation and reductions, we obtain ðs21 s22 Þðs22 s23 Þðs23 s21 Þ½F33 H1 F22 H2 F11 H3 476
c2 ðF11 H1 A3 þ F33 H2 þ F22 H1 A2 þ F33 H1 A1 þ F22 H3 Þ ¼ 0 : 3
ð40Þ
where c ¼ nd. Hence either
ðs21 s22 Þðs22 s23 Þðs23 s21 Þ ¼ 0 ;
ð41Þ
or
F33 H1 F22 H2 F11 H3
c2 ðF11 H1 A3 þ F33 H2 þ F22 H1 A2 þ F33 H1 A1 þ F22 H3 Þ ¼ 0 : 3 ð42Þ
Equation (41) cannot satisfy the boundary conditions. Hence, Eq. (42) holds and it represent the dispersion equation of long flexural waves. It can be seen that in generalized thermoelasticity the phase velocity decreases and tends to zero as the wavelength increases. Using the isotropic relations (26) and (27), Eq. (42) reduces to the isotropic case, [16]. For waves short compared with the thickness of the plate, that is at nd ! 1 and c such that s1 ; s2 , and s3 are real, Eq. (22a) reduces to Rayleigh equation (39) and the propagation degenerates to Rayleigh waves associated with both surfaces of the plate.
6 Thermoelastic surface waves 6.1 Wave propagates in an arbitrary direction In order to have a surface wave, the s2i ði ¼ 1; 2; 3Þ, the roots of Eq. (18) must be either negative (so the square roots are pure imaginary) or a complex number; this ensures that superposition of partial waves has the property of exponential decay. There are two cases: (i) s2i , i ¼ 1; 2; 3, all are negative; and (ii) s21 is negative and s22 ¼ s2 3 , are complex conjugates ( stands for complex conjugate). For case (i), as d ! 1, ftanhðnsj dÞg1 ! ð1Þ, from Eqs. (22), we have
D1 G1 D2 G2 þ D3 G3 ¼ 0 :
ð43Þ
For case (ii), as d ! 1, ftanhðns1 dÞg1 ! 1 and if s22 ¼ p þ iq, s23 ¼ p iq, ðq > 0Þ then ftanhðns2 dÞg1 ! 1 and ftanhðns3 dÞg1 ! ð1Þ, and so we have from Eqs. (22),
D1 G1 D2 G2 D3 G3 ¼ 0 :
ð44Þ
Although Eqs. (43) and (44) are functions of the wavenumber n and frequency x, thermoelastic surface wave velocity can be obtained by solving these equations for heat conducting plate in the context of generalized thermoelasticity.
6.2 Wave propagation in principal direction (say x direction) Similar to the situation described in Sec. 6.1, we have two cases: (i) s2i , i ¼ 2; 3 are negative; and (ii) s21 is negative and s22 ¼ s2 3 , are complex conjugates. Equations (22) become Eq. (43), which can be simplified to find the thermoelastic surface waves.
7 Special cases
477
7.1 Classical case This case corresponds to the situation when the strain and temperature fields are not coupled with each other. In this case, the thermomechanical coupling constant e1 is identically zero. Equation (18) reduces to s2 K þ c2 s 1 ¼ 0 ;
ð45aÞ
and
c1 c2 s4 þ ðc23 c22 þ c2 c2 þ c1 c2 c1 Þs2 þ ðc2 c2 Þð1 c2 Þ ¼ 0 :
ð45bÞ
Fig. 1. Dispersion curves of NaF crystal plate for (a) first antisymmetric mode, (b) second antisymmetric mode, (c) third antisymmetric mode, (d) fourth antisymmetric mode, (e) fifth antisymmetric mode, (f) sixth antisymmetric mode
Equation (45a) gives us
i 1 þ c2 s0 þ xn ; s2 ¼ K
478
ð45cÞ
which defines the speed and the attenuation constant for the thermal wave. Clearly this is influenced by the thermal relaxation time s0 . Equation (45b) (nondimensional) is exactly the same equation which has been obtained and discussed in [21]. It gives two period equations for the symmetric and antisymmetric modes, respectively, for a free homogeneous transversely isotropic plate of thickness 2d. If we define
V2 ¼
x2 ; n2
then after some algebraic manipulations Eqs. (43) and (44) with (45b) reduce to
½c1 ðc3 c2 Þ2 c1 V 2 2 ðc2 V 2 Þ c1 c2 V 4 ð1 V 2 Þ ¼ 0 :
ð46Þ
This equation is the same as one obtained in [22], representing phase velocity equation for Rayleigh waves in a transversely isotropic half-space. It reduces to the equation giving the velocity of Rayleigh waves in the isotropic case. It was proved in [22] that this equation has only one real value for V 2 in the range 0 < V 2 < c2 .
7.2 Isotropic case For thermoelastic isotropic material in the context of generalized thermoelasticity, Eqs. (43) and (44) reduce to
Fig. 2. Dispersion curves of NaF crystal plate for (a) first symmetric mode, (b) second symmetric mode, (c) third symmetric mode, (d) fourth symmetric mode, (e) fifth symmetric mode, (f) sixth symmetric mode
ð1 s23 Þ2 fs21 þ s25 þ s1 s5 þ 1 f2 g 4s1 s3 s5 ðs1 þ s5 Þ ¼ 0 :
ð47Þ
Equation (47) is the same as that obtained and discussed by [11,18], and Eq. (46) reduces to
4 f2 f2 ð1 f2 Þ : 2 ¼ 16 1 c2 c2
ð48Þ
This reveals that the elastic waves will be nondispersive in this case, which is in agreement with [22] in the nondimensional term.
7.3 Coupled themoelasticity This case corresponds to lack of thermal relaxation time, i.e. s0 ¼ 0 and, hence s¼
i : x
Fig. 3. Attenuation coefficients of NaF crystal plate versus wavenumber for (a) first antisymmetric mode, (b) second antisymmetric mode, (c) third antisymmetric mode, (d) fourth antisymmetric mode, (e) fifth antisymmetric mode, (f) sixth antisymmetric mode
479
In case of isotropic material, proceeding on the same lines, we again arrived at frequency equation of the form (47). This is in agreement with the corresponding results obtained in [11, 23, 24]. If we use the condition x
