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Free vibration of microstretch thermoelastic plate with one relaxation time
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Theoretical and Applied Fracture Mechanics 48 (2007) 238–257 www.elsevier.com/locate/tafmec
Free vibration of microstretch thermoelastic plate with one relaxation time R. Kumar
a,*
, G. Partap
b
a
b
Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India Department of Applied Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, Punjab, India Available online 20 September 2007
Abstract The propagation of waves in microstretch thermoelastic homogeneous isotropic plate subjected to stress free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skew-symmetric wave mode propagation have been obtained. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical computations are found to be in close agreement. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Microstretch; Secular equations; Phase velocity; Attenuation
1. Introduction The theory of micropolar elasticity introduced and developed in [1,2] has aroused much interest in recent years because of its possible utility in investigating deformation properties of solids for which the classical (Hookean) theory is inadequate. This theory is believed to be particularly useful in investigating materials consisting of barlike molecules which exhibit microrotational effects and which can support body and surface couples. Furthermore, the work in [3] considered the micropolar elastic model to be more realistic than classical elastic model in studying earth science problems. Considered in [4] is a wave-type heat equation by postulating a new law of heat conduction (the Maxwell– Cattaneo equation) to replace the classical Fourier law. Because the heat equation of this theory is of the wave-type, it automatically ensures finite speeds of propagation for heat and elastic waves. The remaining
*
Corresponding author. E-mail addresses: rajneesh_kuk@rediffmail.com (R. Kumar), [email protected] (G. Partap).
0167-8442/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2007.08.003
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
239
governing equations for this theory, namely, the equations of motions and constitutive relations, remain the same as those for the coupled and the uncoupled theories. Microstretch [5] theory is a generalization of the micropolar theory, for such a material, a homogeneous stretch microdeformation is added to every particle i.e. besides the translation and rigid rotation, each particle can have an independently breathe-like degree of freedom. Such a generalized media can catch more detailed information about the microdeformation inside a material point. The material points of microstretch solids can stretch and contract independently of their translations and rotations. The microstretch continua are used to characterize composite materials and various porous media. The linear theory of micropolar thermoelasticity was developed by extended the theory micropolar continua to include thermal effects in the works of [6–8]. Derived also are the basic equations of linear theory of micropolar coupled thermoelasticity. Formulated in [9] a theory of micropolar elasticity which includes the heat-flux among the constitutive. The work in [10] extended the theory of microstretch elastic solids to include the heat conduction. For the detailed study in microstretch thermoelastic solid, one can refer the book of [11]. Investigated in [12] is the inclusion problem of microstretch. Constructed in [13] is the fundamental solution of the system of equations of steady oscillations in the theory of microstretch elastic solids. It was investigated in [14] that the stress concentration effects in microstretch elastic bodies. It was shown in [15] that the reflection of plane waves in a heat flux depends on the microstretch thermoelastic solid half spaces. Discussed in [16] is the reflection and transmission of waves from a plane interface between two microstretch elastic solid half spaces. Studied in [17] is the wave propagation in a generalized thermo-microstretch elastic solid and in [18] the wave propagation through cylindrical bore contained in a microstretch elastic medium. The present investigation is concerned to study the propagation of waves in an infinite homogeneous isotropic microstretch generaliized thermoelastic plate of finite thickness. More general dispersion equations of microstretch thermoelastic Lamb type waves are derived and discussed. The secular equations for different conditions of solutions have been deduced from the present one. Numerical solutions of the dispersion equations and attenuation coefficients for symmetric and skew symmetric modes are presented graphically. 2. Basic equations The equations of motion and the constitutive relations in a homogeneous isotropic microstretch generalized thermoelastic solid in the absence of body forces, body couples, stretch force and heat sources are given in [4,11]. 2
o~ u ðk þ 2l þ KÞrðr ~ uÞ ðl þ KÞr r ~ u þ Kr ~ / mrT þ k0 r/ ¼ q 2 ; ot 2~ * / o ða þ b þ cÞrðr ~ /Þ cr ðr ~ /Þ þ Kr ~ u 2K / ¼ qj 2 ; ot 2 qj o / a0 r2 / þ m1 T k1 / k0 r ~ u¼ 0 ; 2 ot2 oT o2 T o o2 o o2 þ s0 2 þ mT 0 þ s0 2 ðr ~ þ s0 2 / ; K r2 T ¼ qC uÞ þ m1 T 0 ot ot ot ot ot ot tij ¼ kur;r dij þ lðui;j þ uj;i Þ þ Kðuj;i eijr /r Þ mT dij þ k0 dij / ; mij ¼ a/r;r dij þ b/i;j þ c/j;i þ
b0 emji /;m ;
ki ¼
a0 /;i
þ b0 eijm /j;m :
ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ
where k, l, a, b, c, K, a0, k0, k1 are material constants, q is the density, j is the microinertia, j0 is the microinertia of microelements, tij and mij are the components of force stress and couple stress tensors respectively. ~ u ¼ ðu1 ; u2 ; u3 Þ is the displacement vector, ~ / ¼ ð/1 ; /2 ; /3 Þ is the microrotation vector and /* is the scalar point microstretch function, ki is the component of microstretch tensor, T is the temperature change, T0 is uniform temperature, m ¼ ð3k þ 2l þ KÞat1 , m1 ¼ ð3k þ 2l þ KÞat2 , at1 and at2 are the coefficients of linear thermal expansion and K* is the coefficient of thermal conductivity, C* is specific heat at constant strain, dij is Kronecker delta. The comma notation denotes spatial derivatives.
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3. Formulation of the problem and its solution We consider an infinite homogeneous isotropic thermally conducting microstretch elastic plate of thickness 2d initially undisturbed and at uniform temperature T0. The plate is axisymmetric with the z-axis as the axis of the symmetry. The origin of the co-ordinate system (x, y,z) is taken as the middle surface of the plate and z-axis normal to it along the thickness. The surface z = ±d is subjected to different boundary conditions. The x–yplane is chosen to coincide with the middle surface and z-axis normal to it along the thickness as illustrated in Fig. 1. For two dimensional problem, we take ~ u ¼ ðu1 ; 0; u3 Þ and ~ / ¼ ð0; /2 ; 0Þ:
ð7Þ
Define the non-dimensional quantities x0 ¼
x x x z qx c1 qx c1 qc2 qc2 T ; z0 ¼ ; u01 ¼ u1 ; u03 ¼ u3 ; t0 ¼ x t; /02 ¼ 1 /2 ; /0 ¼ 1 / ; T 0 ¼ ; c1 c1 T0 mT 0 mT 0 mT 0 mT 0
s00 ¼ x s0 ; t0ij ¼ c22 ¼ 2
1 x mij c1 h k þ 2l þ K ; tij ; m0ij ¼ ; h0 ¼ ; c21 ¼ mT 0 x q c1 mT 0
lþK c 2a0 K k1 k0 c2 c2 c2 ; c23 ¼ ; c24 ¼ ; p ¼ 2 ; p1 ¼ 2 ; p0 ¼ 2 ; d2 ¼ 22 ; d21 ¼ 32 ; d22 ¼ 42 ; q qj qc1 qj0 qc1 qc1 c1 c1 c1
d ¼
Kc21 qc41 m2 T 0 qC c21 m1 x ki 0 ; x ; k ; d ¼ ; 2¼ ¼ ; m ¼ ¼ : 1 i cx qC ðk þ 2l þ KÞ K m c1 mT 0 a0 x2 ð8Þ
where x* is the characteristic frequency of the medium. With the help of non-dimensional quantities defined by Eq. (8), Eqs. (1)–(4) can be recast into following dimensionless form after suppressing the primes 2 o u1 o2 u3 o/ o/2 oT o2 u1 2 2 2 ¼ 2 ; p þ r u þ p ð9Þ ð1 d Þ þ d 1 0 ox ox2 oxoz ox oz ot 2 o u1 o2 u3 o/ o/ oT o2 u3 ¼ 2 ; þ 2 þ d2 r 2 u3 þ p 0 þp 2 ð1 d2 Þ ð10Þ oz oxoz oz oz ox ot 2 ou3 1 o2 /2 2 ou1 2 r /2 þ d ; ð11Þ 2d /2 ¼ 2 oz ox d1 ot2 ou3 1 o2 / ou1 2 þ ; ð12Þ r / þ md1 T p1 d1 / p0 d1 ¼ 2 ox oz d2 ot2
Fig. 1. Geometry of the problem.
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
oT o2 T o o2 o o2 ou1 ou3 þ s0 2 ¼ m 2 þ s0 2 / þ 2 þ s0 2 þ rT : ot ot ot ot ot ot ox oz 2
241
ð13Þ
where
r2 ¼
o2 o2 þ ; ox2 oz2
Introducing the potential functions / and w through the relations u1 ¼
o/ ow þ ; ox oz
u3 ¼
o/ ow ; oz ox
ð14Þ
in Eqs. (9)–(13), we obtain o2 r2 2 / þ p0 / T ¼ 0; ot r2 w
ð15Þ
p/2 1 o2 w 2 2 ¼ 0; d ot d2 2
2
r2 /2 þ d r2 w 2d /2
ð16Þ 1 o2 /2 ¼ 0; d21 ot2
r2 / p1 d1 / p0 d1 r2 / þ md1 T
1 o2 / ¼ 0; d22 ot2
€ þ m 2 r T ðT_ þ s0 T€ Þ ¼2 r2 ð/_ þ s0 /Þ 2
ð17Þ
ð18Þ
o/ o2 / þ s0 2 : ot ot
ð19Þ
Assume the solutions of Eqs. (15)–(19) of the form ð/; w; /2 ; T ; / Þ ¼ ½f ðzÞ; gðzÞ; wðzÞ; hðzÞ; gðzÞeinðxctÞ ;
ð20Þ
where x is the circular frequency and n is the wave number. Using Eq. (20) in Eqs. (15)–(19) and solving the resulting differential equations, the expressions for /, w, /2, T and /* are obtained as / ¼ ðA1 cos m1 z þ A2 cos m2 z þ A3 cos m3 z þ B1 sin m1 z þ B2 sin m2 z þ B3 sin m3 zÞeinðxctÞ ; w ¼ ðA4 cos m4 z þ B4 sin m4 z þ A5 cos m5 z þ B5 sin m5 zÞe /2 ¼
inðxctÞ
;
d2 2 ½ðb m24 ÞðA4 cos m4 z þ B4 sin m4 zÞ þ ðb2 m25 ÞðA5 cos m5 z þ B5 sin m5 zÞeinðxctÞ ; p
T ¼ ½S 1 ðA1 cos m1 z þ B1 sin m1 zÞ þ S 2 ðA2 cos m2 z þ B2 sin m2 zÞ þ S 3 ðA3 cos m3 z þ B3 sin m3 zÞeinðxctÞ ;
ð21Þ ð22Þ ð23Þ ð24Þ
/ ¼ ½V 1 ðA1 cos m1 z þ B1 sin m1 zÞ þ V 2 ðA2 cos m2 z þ B2 sin m2 zÞ þ V 3 ðA3 cos m3 z þ B3 sin m3 zÞeinðxctÞ : ð25Þ
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where m2i ¼ n2 ðc2 a2i 1Þ;
i ¼ 1; 2; 3; 4; 5: 2 2 2 2 2 c 2 a ¼ n ðc 1Þ; b ¼ n 1 ; d2 k 0 ¼ s0 þ ix1 ; k 1 ¼ ix1 ; 1 1 d ðp 2d2 Þ a24 þ a25 ¼ 2 þ 2 þ ; d d1 x2 d2 1 1 2d a24 a25 ¼ 2 ð 2 2 Þ; d d1 x X 1 p d p 2 d a21 ¼ 1 þ k 0 þ 2 1 21 þ 0 21 ix 2 k 1 k 0 ; x x d2 X 1 þ k d ix 2 k 0 k 1 0 a21 a22 ¼ k 0 þ 12 ½ð1 þ k 0 Þp1 p20 k 0 þ ix 2 k 0 k 1 ðmp0 p1 Þ ix 2 k 0 k 1mðm p0 Þ ; 2 x d2 d22 ! 1 p d ik 0 k 1 2 d1m2 ; a21 a22 a23 ¼ k 0 2 1 21 x x d2 d1 ðm p0 Þm2i mn2 ðc2 þ pm0 1Þ h i ; i ¼ 1; 2; 3 Vi ¼ d m2i n2 c2 d12 x12 ðmp0 p1 Þ 1 2 h i d ½m2i n2 ðc2 1Þ m2i n2 ðc2 d12 x12 ðmp0 p1 Þ 1 þ d1 p0 ðm p0 Þm2i mn2 ðc2 þ pm0 1Þ h2 i Si ¼ d ixk 1 m2i n2 ðc2 d12 x12 ðmp0 p1 Þ 1Þ 2
ð26Þ With the help of Eqs. (20) and (21), the displacement components u1 and u3 are obtained: u1 ¼ ½inðA1 cos m1 z þ A2 cos m2 z þ A3 cos m3 z þ B1 sin m1 z þ B2 sin m2 z þ B3 sin m3 zÞ þ m4 ðB4 cos m4 z A4 sin m4 zÞ þ m5 ðB5 cos m5 z A5 sin m5 zÞeinðxctÞ ;
ð27Þ
u3 ¼ ½ðm1 A1 sin m1 z þ m1 B1 cos m1 z m2 A2 sin m2 z þ m2 B2 cos m2 z m3 A3 sin m3 z þ m3 B3 cos m3 zÞ inðA4 cos m4 z þ B4 sin m4 z þ A5 cos m5 z þ B5 sin m5 zÞeinðxctÞ :
ð28Þ
3.1. Boundary conditions Consider the following mechanical and thermal boundary conditions at the plate surfaces z = ± d. 3.2. Mechanical conditions The non-dimensional mechanical boundary conditions at z = ± d are given as follows t33 ¼ 0;
t31 ¼ 0;
m32 ¼ 0;
k;3 ¼ 0:
ð29Þ
where ou3 ou1 þk mT þ k0 / ; oz ox ou1 ou3 þl K/2 ; t31 ¼ ðl þ KÞ oz ox o/ o/ ; m32 ¼ c 2 þ b0 oz ox o/ o/ k;3 ¼ a0 b0 2 : oz ox t33 ¼ ðk þ 2l þ KÞ
ð30Þ
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243
3.3. Thermal conditions The thermal boundary conditions at z = ±d are given by oT þ hT ¼ 0: oz
ð31Þ
where h is the surface heat transfer coefficient. Here h ! 0 corresponds to thermal insulated boundaries and h ! 1 refers to isothermal one. 4. Derivation of the secular equations Using the boundary conditions (29) and (31) on the surfaces z = ±d of the plate and with the help of Eqs. (21)–(25), there results a system of ten simultaneous equations P ðA1 c1 þ A2 c2 þ A3 c3 þ B1 s1 þ B2 s2 þ B3 s3 Þ þ QfðA4 m4 s4 þ A5 m5 s5 Þ þ ðB4 m4 c4 þ B5 m5 c5 Þg ¼ 0;
ð32aÞ
P ðA1 c1 þ A2 c2 þ A3 c3 B1 s1 B2 s2 B3 s3 Þ þ QfðA4 m4 s4 þ A5 m5 s5 Þ þ ðB4 m4 c4 þ B5 m5 c5 Þg ¼ 0;
ð32bÞ
QfðA1 m1 s1 þ A2 m2 s2 þ A3 m3 s3 Þ þ ðB1 m1 c1 þ B2 m2 c2 þ B3 m3 c3 Þg þ P ðA4 c4 þ A5 c5 þ B4 s4 þ B5 s5 Þ ¼ 0; ð32cÞ QfðA1 m1 s1 þ A2 m2 s2 þ A3 m3 s3 Þ þ ðB1 m1 c1 þ B2 m2 c2 þ B3 m3 c3 Þg þ P ðA4 c4 þ A5 c5 B4 s4 B5 s5 Þ ¼ 0; ð32dÞ RðV 1 A1 c1 þ V 2 A2 c2 þ V 3 A3 c3 þ V 1 B1 s1 þ V 2 B2 s2 þ V 3 B3 s3 Þ þ SðA4 f4 s4 m4 A5 f5 s5 m5 þ B4 f4 c4 m4 þ B5 f5 c5 m5 Þ ¼ 0;
ð32eÞ
RðV 1 A1 c1 þ V 2 A2 c2 þ V 3 A3 c3 V 1 B1 s1 V 2 B2 s2 V 3 B3 s3 Þ þ SðA4 f4 s4 m4 þ A5 f5 s5 m5 þ B4 f4 c4 m4 þ B5 f5 c5 m5 ¼ 0
ð32fÞ
U ½V 1 m1 s1 A1 V 2 m2 s2 A2 V 3 m3 s3 A3 þ V 1 m1 c1 B1 þ V 2 m2 c2 B2 þ V 3 m3 c3 B3 þ V ½f4 c4 A4 þ f5 c5 A5 þ f4 s4 B4 þ f5 s5 B5 ¼ 0;
ð32gÞ
U ½V 1 m1 s1 A1 þ V 2 m2 s2 A2 þ V 3 m3 s3 A3 þ V 1 m1 c1 B1 þ V 2 m2 c2 B2 þ V 3 m3 c3 B3 þ V ½f4 c4 A4 þ f5 c5 A5 f4 s4 B4 f5 s5 B5 ¼ 0;
ð32hÞ
S 1 ½ðm1 s1 þ hc1 ÞA1 þ ðm1 c1 þ hs1 ÞB1 þ S 2 ½ðm2 s2 þ hc2 ÞA2 þ ðm2 c2 þ hs2 ÞB2 þ S 3 ½ðm3 s3 þ hc3 ÞA3 þ ðm3 c3 þ hs3 ÞB3 ¼ 0;
ð32iÞ
S 1 ½ðm1 s1 þ hc1 ÞA1 þ ðm1 c1 hs1 ÞB1 þ S 2 ½ðm2 s2 þ hc2 ÞA2 þ ðm2 c2 hs2 ÞB2 þ S 3 ½ðm3 s3 þ hc3 ÞA3 þ ðm3 c3 hs3 ÞB3 ¼ 0:
ð32jÞ
where pn2 p P ¼ b n þ 2 ; Q ¼ 2in 1 2 ; f i ¼ b2 m2i ; 2d d 2 cd b0 ind2 R ¼ inb0 ; S ¼ ; U ¼ a0 ; V ¼ ; p p si ¼ sin mi z; ci ¼ cos mi z ; i ¼ 1; 2; 3; 4; 5: 2
2
i ¼ 4; 5;
The system of Eqs. (32a)–(32j) has a non-trivial solution if the determinant of the coefficients of amplitudes [A1, A2, A3, A4, A5, B1, B2, B3, B4, B5]T vanishes. This after lengthy algebraic reductions and manipulations leads to the secular equations for the plate with force stress free and couple stress free thermally insulated and isothermal boundaries, there results
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R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
T1 T4
m1 ðV 1 S 3 V 3 S 1 Þ T 2 m1 ðV 1 S 2 V 2 S 1 Þ T 3 RV ðf5 f4 Þ m1 S 1 ðV 3 V 2 Þ þ þ m2 ðV 2 S 3 V 3 S 2 Þ T 4 m3 ðV 2 S 3 V 3 S 2 Þ T 4 SU ðm5 f5 T 4 m4 f4 T 5 Þ m2 m3 ðV 2 S 3 V 3 S 2 Þ ( ) T 2T 3 m2 S 2 ðV 3 V 1 Þ T 1 T 3 m3 S 3 ðV 2 V 1 Þ T 1 T 2 þ m1 S 1 ðV 3 V 2 Þ T 24 T 5 m1 S 1 ðV 3 V 2 Þ T 24 T 5 T 24 T 5 ( " # m4 m5 TT 54 Q V f5 f4 ðS 3 S 2 Þ T1 m1 ðS 3 S 1 Þ T 2 m1 ðS 2 S 1 Þ T 3 þ þ U ðV S V S Þ T P m2 ðS 3 S 2 Þ T 4 m3 ðS 3 S 2 Þ T 4 2 3 3 2 4 m5 f5 m4 f4 TT 54 R þ S
¼
"
T1 T4
#) m1 V 2 ðV 1 S 3 V 3 S 1 Þ T 2 m1 V 3 ðV 1 S 2 V 2 S 1 Þ T 3 þ m2 V 1 ðV 2 S 3 V 3 S 2 Þ T 4 m3 V 1 ðV 2 S 3 V 3 S 2 Þ T 4
Q2 m1 m4 m5 ððV 1 V 2 ÞðS 2 S 3 Þ ðV 2 V 3 ÞðS 1 S 2 ÞÞðf5 f4 Þ Q2 RV ðm5 f4 T 4 m4 f5 T 5 Þ 2 P SU ðm5 f5 T 4 m4 f4 T 5 Þ 2 P ðV 2 S 3 V 3 S 2 Þ m5 f5 m4 f4 TT 54 V 1 ðS 3 S 2 Þ ðV 2 S 3 V 3 S 2 Þ
( ) T1 m1 V 2 ðS 3 S 1 Þ T 2 m1 V 3 ðS 2 S 1 Þ T 3 þ m2 V 1 ðS 3 S 2 Þ T 4 m3 V 1 ðS 3 S 2 Þ T 4 T4
ð33Þ
for stress free insulated boundaries (h ! 0) of the plate.
m1 S 2 ðV 3 V 1 Þ T 2 m1 S 3 ðV 2 V 1 Þ T 3 RV ðm5 f4 T 4 m4 f5 T 5 Þ m1 ðV 2 S 3 V 3 S 2 Þ þ þ m2 S 1 ðV 3 V 2 Þ T 4 m3 S 1 ðV 3 V 2 Þ T 4 SU m4 m5 ðf5 f4 Þ m2 m3 S 1 ðV 3 V 2 Þ ( ) T 2T 3 m2 ðV 1 S 3 V 3 S 1 Þ T 1 T 3 m3 ðV 1 S 2 V 2 S 1 Þ T 1 T 2 P ðm4 T 5 m5 T 4 Þ þ þ m1 ðV 2 S 3 V 3 S 2 Þ T 4 m1 ðV 2 S 3 V 3 S 2 Þ T 4 Q m4 m5 ðf5 f4 Þ T4 ( " # U m1 f5 f4 ðS 3 S 2 Þ T 2T 3 m2 ðS 3 S 1 Þ T 1 T 3 m3 ðS 2 S 1 Þ T 1 T 2 þ V m2 m3 S 1 ðV 3 V 2 Þ m1 ðS 3 S 2 Þ T 4 m1 ðS 3 S 2 Þ T 4 T4 " #) R m1 V 1 ðV 2 S 3 V 3 S 2 Þ T 2 T 3 m2 V 2 ðV 1 S 3 V 3 S 1 Þ T 1 T 3 m3 V 3 ðV 1 S 2 V 2 S 1 Þ T 1 T 2 þ þ S m2 m3 S 1 ðV 3 V 2 Þ m1 V 1 ðV 2 S 3 V 3 S 2 Þ T 4 m1 V 1 ðV 2 S 3 V 3 S 2 Þ T 4 T4 ) V 2 ðS 3 S 1 Þ 1 mm45 ff45 TT 54 f5 P 2 ( T 1T 3 m1 V 1 ðS 3 S 2 Þ T 2 T 3 m3 V 3 ðS 2 S 1 Þ T 1 T 2 ¼ m2 V 2 ðS 3 S 1 Þ T 24 m2 V 2 ðS 3 S 1 Þ T 24 T 24 Q2 m3 m4 S 1 ðV 3 V 2 Þðf5 f4 Þ
P 2 RV S 2 ðV 3 V 1 Þ T 1T 2T 3 S 1 ðV 3 V 2 Þ S 3 ðV 2 V 1 Þ þ 2 1 ð34Þ S 2 ðV 3 V 1 Þ S 2 ðV 3 V 1 Þ Q SU S 1 ðV 3 V 2 Þm2 m3 m4 m5 T 24 T 5
T1 T4
for stress free isothermal boundaries (h ! 1)of the plate. Where T i ¼ tan mi d;
i ¼ 1; 2; 3; 4; 5:
Here the superscript +1 refers to skew symmetric and 1 refers to symmetric modes. Eqs. (33) and (34) are the most general dispersion relations involving wave number and phase velocity of various modes of propagation in a microstretch generalized thermoelastic plates under the considered situations. These equations can be recognized as modified Rayleigh-Lamb equations which respectively govern the symmetric and anti symmetric modes of wave propagation for force stress and couple stress free, thermally insulated and isothermal
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245
microstretch thermoelastic plate. Refer to such waves as microstretch thermoelastic plate waves rather than Lamb waves whose properties were derived in[19] for isotropic elastic solids in elastokinetics. 4.1. Particular cases 4.1.1. Microstretch coupled thermoelastic plate In case of coupled theory of thermoelasticity (CT), the thermal relaxation times vanish i.e s0 = 0 so that k0 = k1 = ix1. In this case, the secular equations are given by the Eqs. (33) and (34) with changed values of characteristic roots. 4.1.2. Micropolar thermoelastic plate In the absence of microstretch effect, let R = V = U = V1 = V3 = 0, V2 = 1 and S2 = 0, S i ¼ 2 2 ix1 k 1 1 ðmi a Þ; i ¼ 1; 3 and consequently, the secular Eqs. (33) and (34) reduce to tan m1 d m1 ðm21 a2 Þ tan m3 d ¼ tan m4 d m3 ðm23 a2 Þ tan m4 d
2 4n2 1 2dp2 m1 m4 m5 ðm23 m21 Þðm24 m25 Þ 2 ; 2 2 2 2 pn2 2 2 2 2 b n þ d2 ðm3 a Þ m5 ðb m5 Þ m4 ðb m4 Þ TT 54 2 2
m ðb2 m2 Þ
b2 n2 þ pnd2 ðb2 m25 Þðm23 m21 Þ 1 m4 ðb2 m42 Þ 5 tan m1 d m1 ðm23 a2 Þ tan m3 d 5 ¼ 2 2 2 tan m4 d m3 ðm1 a Þ tan m4 d 2 p 2 2 2 4n 1 2d2 m3 m4 ðm5 m4 Þðm1 a2 Þ
T5 T4
ð35Þ
T 1T 3 T 24
;
ð36Þ 4.1.3. Thermoelastic plate In the absence of micropolarity effect (K = p = 0), there prevails ! 2 c m24 ¼ b2 ; m25 ¼ n2 2 1 ; d1 and consequently, the secular Eqs. (35) and (36) for insulated and isothermal boundaries reduce to tan m1 d m1 ðm21 a2 Þ tan m3 d 4n2 m1 bðm23 m21 Þ ; ¼ 2 2 2 2 tan m4 d m3 ðm3 a Þ tan m4 d ðb n2 Þ ðm23 a2 Þ 2 tan m1 d m1 ðm23 a2 Þ tan m3 d ðb2 n2 Þ ðm23 m21 Þ : ¼ 2 2 tan m4 d m3 ðm1 a2 Þ tan m4 d 4n m3 bðm21 a2 Þ
ð37Þ
ð38Þ
5. Discussion of the secular equation 5.1. Regions of the secular equation In order to explore various regions of the secular equations, here consider the Eq. (33) as an example for the purpose of discussion. Depending upon whether m1, m2, m3, m4, m5, b being real, purely imaginary or complex, the frequency Eq. (33) is correspondingly altered as follows: Region I. When the characteristic roots are of the type, a2 = a 0 2, b2 = b 0 2, m2k ¼ a2k , k = 1, 2, 3, 4, 5 so that a = ia 0 , b = ib 0 , mk = iak, i = 1, 2, 3, 4, 5 are purely imaginary or complex numbers. This ensures that the superposition of partial waves has the property of exponential decay. The secular Eq. (33) becomes
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tanh a1 d a1 ðV 01 S 03 V 03 S 01 Þ tanh a2 d a1 ðV 01 S 02 V 02 S 01 Þ tanh a3 d þ tanh a4 d a2 ðV 02 S 03 V 03 Sv2 Þ tanh a4 d a3 ðV 02 S 03 V 03 S 02 Þ tanh a4 d 0 0 RV ðf5 f4 Þ a1 S 01 ðV 03 V 02 Þ þ SU ða f 0 tanh a4 d a4 f40 tanh a5 dÞ a2 a3 ðV 02 S 03 V 03 S 02 Þ # " # (" 5 5 tanh a2 d tanh a3 d a2 S 02 ðV 03 V 01 Þ tanh a1 d tanh a3 d a1 S 01 ðV 03 V 02 Þ ðtanh a4 dÞ2 tanh a5 d ðtanh a4 dÞ2 tanh a5 d " # ) a5 d ða4 a5 ðtanh Þ Þ a3 S 03 ðV 02 V 01 Þ tanh a1 d tanh a2 d Q tanh a4 d þ þ a1 S 01 ðV 03 V 02 Þ ðtanh a4 dÞ2 tanh a5 d P tanh a5 d a5 f50 a4 f40 tanh a4 d ( " # 0 0 0 0 0 0 V f5 f4 ðS 3 S 2 Þ tanh a1 d a1 ðS 3 S 1 Þ tanh a2 d a1 ðS 02 S 01 Þ tanh a3 d þ U ðV 0 S 0 V 0 S 0 Þ tanh a4 d a2 ðS 03 S 02 Þ tanh a4 d a3 ðS 03 S 02 Þ tanh a4 d " 2 3 3 2 #) R tanh a1 d a1 V 02 ðV 01 S 03 V 03 S 01 Þ tanh a2 d a1 V 03 ðV 01 S 02 V 02 S 01 Þ tanh a3 d þ þ S tanh a4 d a2 V 01 ðV 02 S 03 V 03 S 02 Þ tanh a4 d a3 V 01 ðV 02 S 03 V 03 S 02 Þ tanh a4 d Q2 RV ða5 f40 tanh a4 d a4 f50 tanh a5 dÞ V 01 ðS 03 S 02 Þ P 2 SU ða5 f50 tanh a4 d a4 f40 tanh a5 dÞ ðV 02 S 03 V 03 S 02 Þ ( ) tanh a1 d a1 V 02 ðS 03 S 01 Þ tanh a2 d a1 V 03 ðS 02 S 01 Þ tanh a3 d þ tanh a4 d a2 V 01 ðS 03 S 02 Þ tanh a4 d a3 V 01 ðS 03 S 02 Þ tanh a4 d
¼
Q2 a1 a4 a5 ððV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 ÞÞða24 a25 Þ ; a5 d P 2 ðV 02 S 03 V 03 S 02 Þ a5 f50 a4 f40 tanh tanh a4 d
ð39Þ
where V 0i , S 0i , f 0 can be obtained from the Eq. (26) for the above cases. Region II. In case, the characteristic roots a2 = a 0 2, m2k ¼ a2k , k = 1, 2, 3 and m2k ¼ m2k for k = 4, 5 are real, the frequency equation can be obtained from Eq. (33) as tanh a1 d a1 ðV 01 S 03 V 03 S 01 Þ tanh a2 d a1 ðV 01 S 02 V 02 S 01 Þ tanh a3 d þ tan m4 d a2 ðV 02 S 03 V 03 S 02 Þ tan m4 d a3 ðV 02 S 03 V 03 S 02 Þ tan m4 d RV ðm24 m25 Þ a1 S 01 ðV 03 V 02 Þ þ SU ðm f tan m4 d m4 f4 tan m5 dÞ a2 a3 ðV 02 S 03 V 03 S 02 Þ # " # (" 5 5 tanh a2 d tanh a3 d a2 S 02 ðV 03 V 01 Þ tanh a1 d tanh a3 d a3 S 03 ðV 02 V 01 Þ þ 0 0 0 a1 S 1 ðV 3 V 2 Þ ðtan m4 dÞ2 tan m5 d a1 S 01 ðV 03 V 02 Þ ðtan m4 dÞ2 tan m5 d m5 d ( " " # ) m4 m5 tan d tan m 4 tanh a1 d tanh a2 d Q V f5 f4 ðS 03 S 02 Þ tanh a1 d þ U ðV 0 S 00 V 0 S 0 Þ tan m d 2 P 4 ðtan m4 dÞ tan m5 d m5 d 2 3 3 2 m5 f5 m4 f4 tan tan m4 d " # a1 ðS 03 S 01 Þ tanh a2 d a1 ðS 02 S 01 Þ tanh a3 d R tanh a1 d þ þ a2 ðS 03 S 02 Þ tan m4 d a3 ðS 03 S 02 Þ tan m4 d S tan m4 d #) 0 0 0 0 0 0 0 0 0 0 a1 V 2 ðV 1 S 3 V 3 S 1 Þ tanh a2 d a1 V 3 ðV 1 S 2 V 2 S 1 Þ tanh a3 d þ a2 V 01 ðV 02 S 03 V 03 S 02 Þ tan m4 d a3 V 01 ðV 02 S 03 V 03 S 02 Þ tan m4 d ( Q2 RV ðm5 f4 tan m4 d m4 f5 tan m5 dÞ V 01 ðS 03 S 02 Þ tanh a1 d ¼ 2 tan m4 d P SU ðm5 f5 tan m4 d m4 f4 tan m5 dÞ ðV 02 S 03 V 03 S 02 Þ ) 0 0 0 0 0 0 a1 V 2 ðS 3 S 1 Þ tanh a2 d a1 V 3 ðS 2 S 1 Þ tanh a3 d þ 0 0 0 a2 V 1 ðS 3 S 2 Þ tan m4 d a3 V 01 ðS 03 S 02 Þ tan m4 d
Q2 a1 m4 m5 ððV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 ÞÞðm24 m25 Þ
m5 d P 2 ðV 02 S 03 V 03 S 02 Þðm5 f5 m4 f4 ðtan Þ Þ tan m4 d
:
ð40Þ
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Region III. In this case, the characteristic roots are given by m2k , k = 1, 2, 3, 4, 5 and the secular equation is given by Eq. (33). Similar regions can be characterized for secular Eq. (34). 6. Thin plate results Consider the case when transverse wavelength with respect to thickness of the plate is quite large, so that nd 6 1. Regions I and II yield the results of interest in this case. In Region I, the symmetric case has no roots. In Region II, skew symmetric case has no roots. The skew symmetric case and symmetric case, upon retaining the first two terms in the expansion of hyperbolic tangents, the secular Eqs. (39) and (40) reduce to 2 G3 þ d3 GG4 G d 2 G RV 1 þ SU b02 d 2 1 a2 a2 d 2 ð1 a24 d 2 Þ 1 a25 d 2 3G 5 4 3 3 3 !) d 2 ða24 þa25 Þ ( 2 R G2 d3 G2 3 P 1 V d 2 G1 02 2 02 2 2 2 b ða4 þ a5 b Þ a4 a5 þ þ Q a5 b02 d 2 1 U 3 G S Ga4 V 01 3
Q2 RV ¼ 2 P SU
h i 2 d 2 ða24 þa25 b02 Þ a24 d 2 2 3 d2 1 Q2 a4 1 3 02 2 G2 G2 2 02 2 b d b d 3 P 1 G 1 3
ð41Þ
3
RV G V 01 a21 ða22 S 02 a23 S 03 Þ þ V 02 a22 ða23 S 03 a21 S 01 Þ þ V 03 a23 ða21 S 01 a22 S 02 Þ SU 3
Q 1 V 2 2 R G ½m m b2 ðm24 þ m25 b2 Þ þ 0 G5 P m4 ðm4 þ m5 Þ U 4 4 5 SV 1 ¼
Q2 a21 a22 a23 G Q2 RV G þ 2 P2 P SU 5
ð42Þ
where G ¼ ðV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 Þ; G ¼ a21 ðV 02 S 03 V 03 S 02 Þ a22 ðV 01 S 03 V 03 S 01 Þ þ a23 ðV 01 S 02 V 02 S 01 Þ; G1 ¼ a21 ðS 03 S 02 Þ a22 ðS 03 S 01 Þ þ a23 ðS 03 S 01 Þ; G2 ¼ V 01 ðV 02 S 03 V 03 S 02 Þ V 02 ðV 01 S 03 V 03 S 01 Þ þ V 03 ðV 01 S 02 V 02 S 01 Þ; G2 ¼ a21 V 01 ðV 02 S 03 V 03 S 02 Þ a22 V 02 ðV 01 S 03 V 03 S 01 Þ þ a23 V 03 ðV 01 S 02 V 02 S 01 Þ; G3 ¼ V 01 ðS 03 S 02 Þ V 02 ðS 03 S 01 Þ þ V 03 ðS 02 S 01 Þ; G3 ¼ a21 V 01 ðS 03 S 02 Þ a22 V 02 ðS 03 S 01 Þ þ a23 V 03 ðS 02 S 01 Þ; a22 a23 2 0 0 a2 a2 d ÞS 1 ðV 3 V 02 Þ ða21 þ a23 1 3 d 2 ÞS 02 ðV 03 V 01 Þ 3 3 2 2 aa þ ða21 þ a22 1 2 d 2 ÞS 03 ðV 02 V 01 Þ 3 0 0 2 2 G4 ¼ a2 a3 ðS 3 S 2 Þ a21 a23 ðS 03 S 01 Þ þ a21 a22 ðS 02 S 01 Þ; G4 ¼ ða22 þ a23
G5 ¼ a22 a23 V 01 ðV 02 S 03 V 03 S 02 Þ a21 a23 V 02 ðV 01 S 03 V 03 S 01 Þ þ a21 a22 V 03 ðV 01 S 02 V 02 S 01 Þ; G5 ¼ a22 a23 V 01 ðS 03 S 02 Þ a21 a23 V 02 ðS 03 S 01 Þ þ a21 a22 V 03 ðS 02 S 01 Þ; In the absence of microstretch effect, there prevails R = V = U = V1 = V3 = 0, V2 = 1 and S2 = 0, 2 2 G 2 S i ¼ ix1 k 1 1 ðmi a Þ; i ¼ 1; 3, G ¼ a and consequently, the secular Eqs. (41) and (42) reduce to
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02 2 2 2 pn2 2 d2 2 b d p a24 d 2 2 2 a 1 ¼ 4n Þ 1 1 a 1 ; 4 3 3 3 2d2 d2 2 4n2 1 2dp2 a21 a23 ða23 þ a21 a02 Þ ¼ : 2 2 b02 n2 þ pnd2
ðb02 þ n2
ð43Þ
ð44Þ
If we ignore the micropolarity effect i.e. (K = p = 0), then Eqs. (43) and (44) reduce to ðb02 þ n2 Þ2 4n2 b02 ¼ ðb02 þ n2 Þ2 ða23 þ a21 a02 Þ ¼
4n2 a21 a23 ðb02 þ n2 Þ
2
a2 d 2 4 2 04 2 nb d ; 3 3
:
ð45Þ ð46Þ
In general, here the wave modes depend upon the micropolar parameter. 7. Waves of short wavelength Some information on the asymptotic behavior is obtainable when the transverse wavelength with respect to the thickness of the plate is quite small, so that nd 1. Then the characteristic roots a, b, mi, i = 1, 2, 3, 4 ,5 lie in Region I and secular Eqs. (33) and (34) reduce to ðb02 a24 a25 a4 a5 Þ a2 a3 ðV 02 S 03 V 03 S 02 Þ a1 a3 ðV 01 S 03 V 03 S 01 Þ þ a1 a2 ðV 01 S 02 V 02 S 01 Þ RV ða4 þ a5 Þ a1 S 01 ðV 03 V 02 Þ a2 S 02 ðV 03 V 01 Þ þ a3 S 03 ðV 02 V 01 Þ þ SU
QV 2 2 a4 a5 b02 a24 þ a25 b02 a3 a2 ðS 03 S 02 Þ a1 a3 ðS 03 S 01 Þ þ a1 a2 ðS 02 S 01 Þ þ PU QR a2 a3 V 01 ðV 02 S 03 V 03 S 02 Þ a1 a3 V 02 ðV 01 S 03 V 03 S 01 Þ þ a1 a2 V 03 ðV 01 S 02 V 02 S 01 Þ þ PS Q2 ða4 a5 b02 Þ RV a3 a2 V 01 ðS 03 S 02 Þ a1 a3 V 02 ðS 03 S 01 Þ þ a1 a2 V 03 ðS 02 S 01 Þ ¼ 2 SU P Q2 ð47Þ 2 a1 a2 a3 a4 a5 ða4 þ a5 Þ ðV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 Þ P a2 a3 S 01 ðV 03 V 02 Þ a1 a3 S 02 ðV 03 V 01 Þ þ a1 a2 S 03 ðV 02 V 01 Þ RV ðb2 a4 a5 Þ a1 ðV 02 S 03 V 03 S 02 Þ a2 ðV 01 S 03 V 03 S 01 Þ þ a3 ðV 01 S 02 V 02 S 01 Þ SU a4 a5 ða4 þ a5 Þ
02 02 PU a24 a25 b a24 þ a25 b þ a1 ðS 03 S 02 Þ a2 ðS 03 S 01 Þ þ a3 ðS 02 S 01 Þ QV a4 a5 ða4 þ a5 Þ 2 2
02 02 PR a4 a5 b a24 þ a25 b a1 V 01 ðV 02 S 03 V 03 S 02 Þ a2 V 02 ðV 01 S 03 V 03 S 01 Þ þ a3 V 03 ðV 01 S 02 V 02 S 01 Þ þ QS a4 a5 ða4 þ a5 Þ P 2 ðb02 a24 a25 a4 a5 Þ P 2 RV G3 : ð48Þ þ ¼ a2 V 02 ðS 03 S 01 Þ a3 V 03 ðS 02 S 01 Þ a1 V 01 ðS 03 S 02 Þ Q2 a4 a5 ða4 þ a5 Þ Q2 a4 a5 SU þ
The Eqs. (47) and (48) are respectively the Rayleigh surface wave equations for a force stress and couple stress thermally insulated and isothermal, microstretch generalized thermoelastic half space. In the absence of micro2 2 stretch effect, we have V1 = V3 = 0, V2 = 1 and S2 = 0, S i ¼ ix1 k 1 1 ðmi a Þ, i = 1,3; R = V = U = 0 so that the Eqs. (47) and (48) reduce to
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2 p 4n2 1 2 a1 a3 a4 a5 ða1 þ a3 Þða4 þ a5 Þ 2d 2 p ¼ b2 n2 1 2 ða21 þ a23 þ a1 a3 a02 Þðb02 a24 a25 a4 a5 Þ; 2d 2 2 p p 2 2 2 02 4n 1 2 a4 a5 ða þ a1 a3 Þða4 þ a5 Þ ¼ b n 1 2 ða1 þ a3 Þðb02 a24 a25 a4 a5 Þ: 2d 2d
249
ð49Þ ð50Þ
The Eqs. (49) and (50) are respectively the Rayleigh surface wave equations for a force stress and couple stress free thermally insulated and isothermal, micropolar generalized thermoelastic solid half space. If the micropolarity effect is neglected i.e. (p = 0), the Eqs. (49) and (50) reduce to 2
4n2 a1 a3 a4 a5 ða1 þ a3 Þða4 þ a5 Þ ¼ ðb02 þ n2 Þ ða21 þ a23 þ a1 a3 a02 Þðb02 a24 a25 a4 a5 Þ; 2
02
02
2 2
02
4n a4 a5 ða þ a1 a3 Þða4 þ a5 Þ ¼ ðb þ n Þ ða1 þ a3 Þðb
a24
a25
a4 a5 Þ:
ð51Þ ð52Þ
for thermally insulated and isothermal stress free thermoelastic plates. The Eqs. (51) and (52) are merely Rayleigh surface wave equations. The Rayleigh results enter here since at such wave lengths the finite thickness plate appears as a semi-infinite medium and hence vibrational energy is transmitted mainly along the surface of the plate. 8. Lame modes A special class of exact solutions, called theLamemodes, but evidently first identified in [19] can be obtained by considering the special case b2 ¼ n2 1 dp2 , the characteristic roots for this case are in Region II and the frequency Eq. (33) reduces to Symmetric modes: tan m4d = 1, ) m4 ¼ np , n = 1, 3, 5, . . . 2d Anti symmetric modes: tan m4d = 0, ) m4 ¼ np , n = 0, 2, 4, . . . 2d Here, the frequency is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4b2 d 2 þ n2 p2 ð1 dp2 Þ qffiffiffiffiffiffiffiffiffiffiffiffi x¼ : 2da4 1 dp2
ð53Þ
It is obvious that these modes depend upon the micropolar parameter (K or p) and thickness of the plate. However in the absence of stretch and micropolarity effect and consequently the frequency is given by ffiffi which agrees with the result in [20]. As these modes don’t depend upon the thermomechanical coux ¼ pnpd 2d pling and thermal relaxation time. However, it is obvious that these modes depend upon the micropolar and stretch parameters in addition to thickness of the plate. 9. Amplitudes of dilatation, microrotation, microstretch and temperature In this section, the amplitudes of dilatation, microrotation, microstretch and temperature distribution for symmetric and skew symmetric modes of plate waves have been computed for stress free microstretch thermoelastic plate. Upon using Eqs. 27,28 and (32a)–(32j), there results ðeÞsy ¼ fðn2 þ m21 Þ cos m1 z þ Lðn2 þ m22 Þ cos m2 z þ Mðn2 þ m23 Þ cos m3 zÞgA1 einðxctÞ ; ðeÞasy ¼ fðn2 þ m21 Þ sin m1 z þ L0 ðn2 þ m22 Þ sin m2 z þ M 0 ðn2 þ m23 Þ sin m3 zgB1 einðxctÞ ;
d2 f4 m4 s4 ðb2 m24 Þ sin m4 z ðb2 m25 Þ ð/2 Þsy ¼ sin m5 z A4 einðxctÞ ; p f5 m5 s5
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ð/2 Þasy
d2 2 2 2 2 f4 m4 c4 ðb m4 Þ cos m4 z ðb m5 Þ ¼ cos m5 z B4 einðxctÞ ; p f5 m5 c5
ð/ Þsy ¼ fV 1 cos m1 z þ V 2 L cos m2 z þ V 3 M cos m3 zgA1 einðxctÞ ; ð/ Þasy ¼ fV 1 sin m1 z þ V 2 L0 sin m2 z þ V 3 M 0 sin m3 zgB1 einðxctÞ ðT Þsy ¼ fS 1 cos m1 z þ S 2 L cos m2 z þ S 3 M cos m3 zgA1 einðxctÞ ; ðT Þasy ¼ fS 1 sin m1 z þ S 2 L0 sin m2 z þ S 3 M 0 sin m3 zgB1 einðxctÞ : where ðV 3 S 1 V 1 S 3 Þm1 s1 ðV 3 S 1 V 1 S 3 Þm1 c1 ; L0 ¼ ; ðV 2 S 3 V 3 S 2 Þm2 s2 ðV 2 S 3 V 3 S 2 Þm2 c2 ðV 1 S 2 V 2 S 1 Þ m1 s1 ðV 1 S 2 V 2 S 1 Þ m1 c1 M¼ ; M0 ¼ : ðV 2 S 3 V 3 S 2 Þ m3 s3 ðV 2 S 3 V 3 S 2 Þ m3 c3 L¼
10. Numerical results and discussion With the view of illustrating theoretical results obtained in the preceding sections and comparing these in the context of various theories of thermoelasticity, some numerical results will be presented. The material chosen for this purpose is magnesium crystal (microstretch thermoelastic solid), the physical data for which is given below q ¼ 1:74 103 kg=m3 ;
k ¼ 9:4 1010 N=m2 ;
K ¼ 1:0 1010 N=m2 ;
c ¼ 0:779 109 N;
s0 ¼ 6:131 1013 s;
2¼ 0:028;
K ¼ 0:6 106 J=m s ;
l ¼ 4:0 1010 N=m2 ; j ¼ 0:2 1019 m2 ;
21 ¼ 0:069;
m ¼ m1 ¼ 0:084;
j0 ¼ 0:185 1019 m2
T 0 ¼ 300 K; C ¼ 0:23 103 J=kg ;
k0 ¼ 0:5 1010 N=m2 ;
k1 ¼ 0:5 1010 N=m2 ;
a0 ¼ 0:779 109 N; d ¼ 0:01 m: The non-dimensional phase velocity and attenuation coefficient of symmetric and skew symmetric modes of wave propagation in the context of L–S and CT theories of thermoelasticity have been computed for various values of non-dimensional wave number from dispersion Eq. (33) for stress free thermally insulated boundaries and have been represented graphically for different modes (n = 0 to n = 2) in Figs. 2–5. The solid curves correspond to CT theory and broken-line curves correspond to L–S theory of thermoelasticity. The
Fig. 2. Variation of phase velocity C of symmetric mode of wave propagation.
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
Fig. 3. Variation of attenuation coefficient Q of symmetric mode of wave propagation.
Fig. 4. Variation of phase velocity C of skew-symmetric mode of wave propagation.
Fig. 5. Variation of attenuation coefficient Q of skew-symmetric mode of wave propagation.
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amplitudes of dilatation, microrotation, microstretch and temperature distribution for symmetric and skew symmetric modes in the context of L–S and CT theories of thermoelasticity are presented graphically in Figs. 6–13. The phase velocities of lowest symmetric mode of propagation become dispersionless i.e. remain constant with variation in wave number. The phase velocities of higher modes of propagation, symmetric and skew symmetric attain quite large values at vanishing wave number which sharply slashes down to become steady and asymptotic to the reduced Rayleigh wave velocity with increasing wave number. The reason for this asymptotic approach is that for short wavelengths (or high frequencies) the material plate behaves increasingly like a thick slab and hence the coupling between upper and lower boundary surfaces is reduced and as a result the properties of symmetric and skew symmetric waves become more and more similar.
Fig. 6. Amplitude of symmetric dilatation e.
Fig. 7. Amplitude of skew-symmetric dilatation e.
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253
Fig. 8. Amplitude of symmetric microrotation /2.
Fig. 9. Amplitude of skew-symmetric microrotation /2.
It is observed that in the context of various theories of thermoelasticity (L–S and CT), various symmetric modes of propagation have nearly same velocities for n = 0, n = 1, and n = 2. For skew symmetric modes of wave propagation, we observe the following (a) for lowest mode n = 0, phase velocity profiles for L–S and CT theory coincide for wave number nd P 5.0and for wave number nd 6 5.0. Phase velocity for L–S theory is less than in case of CT theory (b) for n = 1, phase velocity profiles for L–S and CT theory coincide for wave number nd P 6.2 and for wave number nd 6 6.2, phase velocity for L–S theory is more than in case of CT theory (c) for n = 2, phase velocity profiles for L–S and CT theory coincide for wave number nd P 5.2 and for wave number nd 6 5.2, phase velocity for L–S theory is more than in case of CT theory.
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Fig. 10. Amplitude of symmetric microstretch /*.
Fig. 11. Amplitude of skew-symmetric microstretch /*.
The attenuation coefficients of symmetric and skew symmetric modes have been plotted in the context of coupled thermoelasticity and that in [4] for thermoelasticity. In general, wave number and phase velocity of the waves are complex quantities, therefore, the waves are attenuated in space. Let C 1 ¼ V 1 þ ix1 Q
ð54Þ
then n = R + iQ, where R = x/V and Q are real numbers. This shows that V is the propagation speed and Q is attenuation coefficient of waves. Upon using (54) in secular Eq. (33), the value of propagation speed V and attenuation coefficient Q for different modes of wave propagation can be obtained.
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255
Fig. 12. Amplitude of symmetric temperature T.
Fig. 13. Amplitude of skew-symmetric temperature T.
The variation of attenuation coefficient with wave number for symmetric and skew symmetric modes is represented graphically in Figs. 2 and 4 respectively. For lowest symmetric mode, the magnitude of attenuation coefficient for L–S theory has maxima up to 108.9 in region 0.2 6 nd 6 4.2 at nd = 1.2 and approaches to zero with increase in wave number. For first symmetric mode, the magnitude of attenuation coefficient for L–S theory has negligible variation with wave number in region 0.2 6 nd 6 4.2, the magnitude of attenuation coefficient for L–S theory have maxima up to 127.7 and 158.4 in region 4.2 6 nd 6 6.2 at nd = 5.2 and 6.2 6 nd 6 8.2 at nd = 7.2 respectively, the magnitude of attenuation coefficient decreases from 15.16 to 5.325 in the region 8.2 6 nd 6 9.2. For lowest symmetric mode, the magnitude of attenuation coefficient for CT theory have maxima upto 34.96 and 34.39 in region 0.2 6 nd 6 2.2 at nd = 1.2 and 2.2 6 nd 6 4.2 at nd = 3.2 respectively and has
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negligible variation with wave number in region 4.2 6 nd 6 9.2. For first symmetric mode, the magnitude of attenuation coefficient has negligible variation with wave number in region 0.2 6 nd 6 4.2, has maxima upto 115.7 in region 4.2 6 nd 6 6.2 at nd = 5.2 and increases to 30.4 at nd = 7.2 and decreases to 9.417 at nd = 9.2 in region 6.2 6 nd 6 9.2 for CT theory. The attenuation coefficient has negligible variation with wave number for second symmetric mode (n = 2) in case of CT and L–S theory of thermoelasticity. For lowest skew symmetric mode, the magnitude of attenuation coefficient for L–S theory decreases from 49.99 to 2.238 in the region 0.2 6 n d 6 1.2, has maxima upto 10.57 in region 1.2 6 nd 6 3.2 at nd = 2.2, has maximum values 68.99 and 82.23 at nd = 4.2 and nd = 5.2 respectively in the region 3.2 6 nd 6 6.2 and has negligible variation with wave number in region 6.2 6 nd 6 9.2. For first skew symmetric mode, attenuation coefficient for L–S theory has negligible variation with wave number in region 0.2 6 n d 6 1.2, has maximum values 23.49 and 20.62 at nd = 2.2 and n d = 3.2 respectively in the region 1.2 6 nd 6 3.2, has maxima up to 218.5 in region 3.2 6 nd 6 7.2 at nd = 5.2 and has negligible variation with wave number in region 7.2 6 nd 6 9.2. For first skew symmetric mode, attenuation coefficient for L–S theory has negligible variation with wave number in regions 0.2 6 nd 6 4.2 and 8.2 6 nd 6 9.2, have maxima up to 5.776 and 20.18 in region 4.2 6 nd 6 6.2 at nd = 5.2 and 6.2 6 n d 6 8.2 at nd = 7.2 respectively. For lowest skew symmetric mode, attenuation coefficient for CT theory attains the maxima up to 89.57 at nd = 2.2 in the region 0.2 6 nd 6 3.2and has negligible variation with wave number in region 3.2 6 nd 6 9.2. For first skew symmetric mode, attenuation coefficient for CT theory increases up to 6.358 at nd = 2.2 in the region 0.2 6 nd 6 3.2and has negligible variation with wave number in region 3.2 6 nd 6 9.2. The attenuation coefficient has negligible variation with wave number for second skew symmetric mode (n = 2) in CT theory of thermoelasticity. The effect of stress free thermally insulated boundaries of the plate is quite pertinent and can be easily noticed from dispersion and attenuation curves plotted in Figs. 2–5. The effect of micropolarity and microstretch can be seen from the comparison of figures. 10.1. Amplitudes Figs. 6 and 7 depict the variations of symmetric and skew symmetric amplitudes of dilatation (e) in the context of L–S and C-T theories of thermoelasticity for stress free thermally insulated boundary. The dilatation (e) of the plate is minimum at the centre and maximum at the surfaces for symmetric mode and zero at the centre and maximum at the surfaces for skew symmetric mode as evident from Figs. 6 and 7 respectively. Figs. 8–13 show the variations of symmetric and skew symmetric amplitudes of microrotation (/2), microstretch (/*) and temperature distribution (T) in the context of L–S and C-T theories of thermoelasticity for stress free thermally insulated boundary. It is evident from Figs. 8–13 that the values of microrotation(/2), microstretch (/*) and temperature distribution (T) of the plate is minimum at the centre and maximum at the surfaces for symmetric mode and zero at the centre and maximum at the surfaces for skew symmetric mode. (e)sym, (e)asym, (/2)sym, (/2)asym, (/*)sym, (/*)asym, (T)sym and(T)asym correspond to the values of (e),(/2), (/*) and (T) for symmetric and skew symmetric modes respectively. It is observed that behavior and trend of variations of (e)sym, (/2)sym,(/*)sym and (T)sym is same; whereas the behavior and trend of variations of (e)asym, (/2)asym, (/*)asym and(T)asym is similar. The values of the dilatation, microstretch and temperature distribution of the plate in case of L–S theory are less in comparison to C-T Theory for symmetric and skew symmetric modes. The values of the microrotation(/2) of the plate are same in case of L–S and C-T theories of thermoelasticity for symmetric and skew symmetric modes. 11. Conclusions The propagation of waves in microstretch thermoelastic plate subjected to stress free thermally insulated and isothermal boundary is investigated in the context of conventional coupled thermoelasticity (CT) and Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skewsymmetric wave mode propagation have been obtained. The secular equations for microstretch coupled thermoelastic, micropolar thermoelastic, and thermoelastic plates have been deduced as particular cases from the
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derived secular equations. At short wavelength limits, the secular equations for symmetric and skew-symmetric waves in stress free thermally insulated and isothermal microstretch thermoelastic plate reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical results are found to be in close agreement. The significant response of different theories of thermoelasticity i.e coupled thermoelasticity and thermoelasticity with one relaxation time for different modes of propagation (symmetric and skewsymmetric) on phase velocity, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch and temperature distribution have been observed. References [1] A.C. Eringen, Linear theory of micropolar elasticity, J. Math. Mech. 15 (1966) 909–923. [2] A.C. Eringen, Theory of micropolar elasticity, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, p. 622 (Chapter 7). [3] D. Iesan, Some applications of micropolar mechanics to earthquake problem, Int. J. Eng. Sci. 19 (1981) 855–864. [4] H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309. [5] A.C. Eringen, Micropolar elastic solids with stretch, Ari Kitabevi Matbassi, Istanbul 24 (1971) 1–18. [6] W. Nowacki, Couple stress theory in the theory of thermoelasticity, Proceedings of IUTAM symposia, Springer-Verlag, Vienna, 1966, pp. 259–278. [7] A.C. Eringen, Foundations of micropolar thermoelasticity, International Centre For Mechanical Studies, Course and Lectures, vol. 23, Springer-Verlag, Wein, 1970. [8] T.R. Touchert, W.D. Claus Jr., T. Ariman, The linear theory of micropolar thermoelasticity, Int. J. Eng. Sci. 6 (1968) 37–47. [9] D.S. Chandrasekharaiah, Heat flux dependent micropolar thermoelasticity, Int. J. Eng. Sci. 24 (1986) 1389–1395. [10] A.C. Eringen, Theory of thermo-microstretch elastic solids, Int. J. Eng. Sci. 28 (1990) 1291–1301. [11] A.C. Eringen, Microcontinuum Field Theories I: Foundations and Solids, Springer-Verlag, New York, 1999. [12] Xiaonong Liu, Gengkai Hu, Inclusion problem of microstretch continuum, Int. J. Eng. Sci. 42 (2004) 849–860. [13] Merab Svanadze, Fundamental solution of the system of equations of steady oscillations in the theory of microstretch elastic solids, Int. J. Eng. Sci. 42 (2004) 1897–1910. [14] S. De Cicco, Stress concentration effects in microstretch elastic solids, Int. J. Eng. Sci. 41 (2003) 187–199. [15] R. Kumar, Geeta Partap, Reflection of plane waves in a heat flux dependent microstretch thermoelastic solid half space, Int. J. Appl. Mech. Eng. 10 (2005) 2, pp. 253–266. [16] S.K. Tomer, Monica Garg, Reflection and transmission of waves from a plane interface between two microstretch solid half spaces, Int. J. Eng. Sci. 43 (2005) 139–169. [17] R. Kumar, Baljeet Singh, Wave propagation in a generalized thermo-microstretch elastic solid, Int. J. Eng. Sci. 36 (1998) 891–912. [18] R. Kumar, Sunita Deswal, Wave propagation through cylindrical bore contained in a microstretch elastic medium, J. Sound Vibr. 250 (2002) 711–722. [19] H. Lamb, On waves in an elastic plate, Phil. Trans.Roy. Soc., London, Ser. A 93 (1917) 114–128. [20] K.F. Graff, Wave Motion in Elastic Solids, Dover publications, Inc., New York, 1991.
Theoretical and Applied Fracture Mechanics 48 (2007) 238–257 www.elsevier.com/locate/tafmec
Free vibration of microstretch thermoelastic plate with one relaxation time R. Kumar
a,*
, G. Partap
b
a
b
Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India Department of Applied Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, Punjab, India Available online 20 September 2007
Abstract The propagation of waves in microstretch thermoelastic homogeneous isotropic plate subjected to stress free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skew-symmetric wave mode propagation have been obtained. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical computations are found to be in close agreement. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Microstretch; Secular equations; Phase velocity; Attenuation
1. Introduction The theory of micropolar elasticity introduced and developed in [1,2] has aroused much interest in recent years because of its possible utility in investigating deformation properties of solids for which the classical (Hookean) theory is inadequate. This theory is believed to be particularly useful in investigating materials consisting of barlike molecules which exhibit microrotational effects and which can support body and surface couples. Furthermore, the work in [3] considered the micropolar elastic model to be more realistic than classical elastic model in studying earth science problems. Considered in [4] is a wave-type heat equation by postulating a new law of heat conduction (the Maxwell– Cattaneo equation) to replace the classical Fourier law. Because the heat equation of this theory is of the wave-type, it automatically ensures finite speeds of propagation for heat and elastic waves. The remaining
*
Corresponding author. E-mail addresses: rajneesh_kuk@rediffmail.com (R. Kumar), [email protected] (G. Partap).
0167-8442/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2007.08.003
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
239
governing equations for this theory, namely, the equations of motions and constitutive relations, remain the same as those for the coupled and the uncoupled theories. Microstretch [5] theory is a generalization of the micropolar theory, for such a material, a homogeneous stretch microdeformation is added to every particle i.e. besides the translation and rigid rotation, each particle can have an independently breathe-like degree of freedom. Such a generalized media can catch more detailed information about the microdeformation inside a material point. The material points of microstretch solids can stretch and contract independently of their translations and rotations. The microstretch continua are used to characterize composite materials and various porous media. The linear theory of micropolar thermoelasticity was developed by extended the theory micropolar continua to include thermal effects in the works of [6–8]. Derived also are the basic equations of linear theory of micropolar coupled thermoelasticity. Formulated in [9] a theory of micropolar elasticity which includes the heat-flux among the constitutive. The work in [10] extended the theory of microstretch elastic solids to include the heat conduction. For the detailed study in microstretch thermoelastic solid, one can refer the book of [11]. Investigated in [12] is the inclusion problem of microstretch. Constructed in [13] is the fundamental solution of the system of equations of steady oscillations in the theory of microstretch elastic solids. It was investigated in [14] that the stress concentration effects in microstretch elastic bodies. It was shown in [15] that the reflection of plane waves in a heat flux depends on the microstretch thermoelastic solid half spaces. Discussed in [16] is the reflection and transmission of waves from a plane interface between two microstretch elastic solid half spaces. Studied in [17] is the wave propagation in a generalized thermo-microstretch elastic solid and in [18] the wave propagation through cylindrical bore contained in a microstretch elastic medium. The present investigation is concerned to study the propagation of waves in an infinite homogeneous isotropic microstretch generaliized thermoelastic plate of finite thickness. More general dispersion equations of microstretch thermoelastic Lamb type waves are derived and discussed. The secular equations for different conditions of solutions have been deduced from the present one. Numerical solutions of the dispersion equations and attenuation coefficients for symmetric and skew symmetric modes are presented graphically. 2. Basic equations The equations of motion and the constitutive relations in a homogeneous isotropic microstretch generalized thermoelastic solid in the absence of body forces, body couples, stretch force and heat sources are given in [4,11]. 2
o~ u ðk þ 2l þ KÞrðr ~ uÞ ðl þ KÞr r ~ u þ Kr ~ / mrT þ k0 r/ ¼ q 2 ; ot 2~ * / o ða þ b þ cÞrðr ~ /Þ cr ðr ~ /Þ þ Kr ~ u 2K / ¼ qj 2 ; ot 2 qj o / a0 r2 / þ m1 T k1 / k0 r ~ u¼ 0 ; 2 ot2 oT o2 T o o2 o o2 þ s0 2 þ mT 0 þ s0 2 ðr ~ þ s0 2 / ; K r2 T ¼ qC uÞ þ m1 T 0 ot ot ot ot ot ot tij ¼ kur;r dij þ lðui;j þ uj;i Þ þ Kðuj;i eijr /r Þ mT dij þ k0 dij / ; mij ¼ a/r;r dij þ b/i;j þ c/j;i þ
b0 emji /;m ;
ki ¼
a0 /;i
þ b0 eijm /j;m :
ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ
where k, l, a, b, c, K, a0, k0, k1 are material constants, q is the density, j is the microinertia, j0 is the microinertia of microelements, tij and mij are the components of force stress and couple stress tensors respectively. ~ u ¼ ðu1 ; u2 ; u3 Þ is the displacement vector, ~ / ¼ ð/1 ; /2 ; /3 Þ is the microrotation vector and /* is the scalar point microstretch function, ki is the component of microstretch tensor, T is the temperature change, T0 is uniform temperature, m ¼ ð3k þ 2l þ KÞat1 , m1 ¼ ð3k þ 2l þ KÞat2 , at1 and at2 are the coefficients of linear thermal expansion and K* is the coefficient of thermal conductivity, C* is specific heat at constant strain, dij is Kronecker delta. The comma notation denotes spatial derivatives.
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3. Formulation of the problem and its solution We consider an infinite homogeneous isotropic thermally conducting microstretch elastic plate of thickness 2d initially undisturbed and at uniform temperature T0. The plate is axisymmetric with the z-axis as the axis of the symmetry. The origin of the co-ordinate system (x, y,z) is taken as the middle surface of the plate and z-axis normal to it along the thickness. The surface z = ±d is subjected to different boundary conditions. The x–yplane is chosen to coincide with the middle surface and z-axis normal to it along the thickness as illustrated in Fig. 1. For two dimensional problem, we take ~ u ¼ ðu1 ; 0; u3 Þ and ~ / ¼ ð0; /2 ; 0Þ:
ð7Þ
Define the non-dimensional quantities x0 ¼
x x x z qx c1 qx c1 qc2 qc2 T ; z0 ¼ ; u01 ¼ u1 ; u03 ¼ u3 ; t0 ¼ x t; /02 ¼ 1 /2 ; /0 ¼ 1 / ; T 0 ¼ ; c1 c1 T0 mT 0 mT 0 mT 0 mT 0
s00 ¼ x s0 ; t0ij ¼ c22 ¼ 2
1 x mij c1 h k þ 2l þ K ; tij ; m0ij ¼ ; h0 ¼ ; c21 ¼ mT 0 x q c1 mT 0
lþK c 2a0 K k1 k0 c2 c2 c2 ; c23 ¼ ; c24 ¼ ; p ¼ 2 ; p1 ¼ 2 ; p0 ¼ 2 ; d2 ¼ 22 ; d21 ¼ 32 ; d22 ¼ 42 ; q qj qc1 qj0 qc1 qc1 c1 c1 c1
d ¼
Kc21 qc41 m2 T 0 qC c21 m1 x ki 0 ; x ; k ; d ¼ ; 2¼ ¼ ; m ¼ ¼ : 1 i cx qC ðk þ 2l þ KÞ K m c1 mT 0 a0 x2 ð8Þ
where x* is the characteristic frequency of the medium. With the help of non-dimensional quantities defined by Eq. (8), Eqs. (1)–(4) can be recast into following dimensionless form after suppressing the primes 2 o u1 o2 u3 o/ o/2 oT o2 u1 2 2 2 ¼ 2 ; p þ r u þ p ð9Þ ð1 d Þ þ d 1 0 ox ox2 oxoz ox oz ot 2 o u1 o2 u3 o/ o/ oT o2 u3 ¼ 2 ; þ 2 þ d2 r 2 u3 þ p 0 þp 2 ð1 d2 Þ ð10Þ oz oxoz oz oz ox ot 2 ou3 1 o2 /2 2 ou1 2 r /2 þ d ; ð11Þ 2d /2 ¼ 2 oz ox d1 ot2 ou3 1 o2 / ou1 2 þ ; ð12Þ r / þ md1 T p1 d1 / p0 d1 ¼ 2 ox oz d2 ot2
Fig. 1. Geometry of the problem.
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
oT o2 T o o2 o o2 ou1 ou3 þ s0 2 ¼ m 2 þ s0 2 / þ 2 þ s0 2 þ rT : ot ot ot ot ot ot ox oz 2
241
ð13Þ
where
r2 ¼
o2 o2 þ ; ox2 oz2
Introducing the potential functions / and w through the relations u1 ¼
o/ ow þ ; ox oz
u3 ¼
o/ ow ; oz ox
ð14Þ
in Eqs. (9)–(13), we obtain o2 r2 2 / þ p0 / T ¼ 0; ot r2 w
ð15Þ
p/2 1 o2 w 2 2 ¼ 0; d ot d2 2
2
r2 /2 þ d r2 w 2d /2
ð16Þ 1 o2 /2 ¼ 0; d21 ot2
r2 / p1 d1 / p0 d1 r2 / þ md1 T
1 o2 / ¼ 0; d22 ot2
€ þ m 2 r T ðT_ þ s0 T€ Þ ¼2 r2 ð/_ þ s0 /Þ 2
ð17Þ
ð18Þ
o/ o2 / þ s0 2 : ot ot
ð19Þ
Assume the solutions of Eqs. (15)–(19) of the form ð/; w; /2 ; T ; / Þ ¼ ½f ðzÞ; gðzÞ; wðzÞ; hðzÞ; gðzÞeinðxctÞ ;
ð20Þ
where x is the circular frequency and n is the wave number. Using Eq. (20) in Eqs. (15)–(19) and solving the resulting differential equations, the expressions for /, w, /2, T and /* are obtained as / ¼ ðA1 cos m1 z þ A2 cos m2 z þ A3 cos m3 z þ B1 sin m1 z þ B2 sin m2 z þ B3 sin m3 zÞeinðxctÞ ; w ¼ ðA4 cos m4 z þ B4 sin m4 z þ A5 cos m5 z þ B5 sin m5 zÞe /2 ¼
inðxctÞ
;
d2 2 ½ðb m24 ÞðA4 cos m4 z þ B4 sin m4 zÞ þ ðb2 m25 ÞðA5 cos m5 z þ B5 sin m5 zÞeinðxctÞ ; p
T ¼ ½S 1 ðA1 cos m1 z þ B1 sin m1 zÞ þ S 2 ðA2 cos m2 z þ B2 sin m2 zÞ þ S 3 ðA3 cos m3 z þ B3 sin m3 zÞeinðxctÞ ;
ð21Þ ð22Þ ð23Þ ð24Þ
/ ¼ ½V 1 ðA1 cos m1 z þ B1 sin m1 zÞ þ V 2 ðA2 cos m2 z þ B2 sin m2 zÞ þ V 3 ðA3 cos m3 z þ B3 sin m3 zÞeinðxctÞ : ð25Þ
242
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
where m2i ¼ n2 ðc2 a2i 1Þ;
i ¼ 1; 2; 3; 4; 5: 2 2 2 2 2 c 2 a ¼ n ðc 1Þ; b ¼ n 1 ; d2 k 0 ¼ s0 þ ix1 ; k 1 ¼ ix1 ; 1 1 d ðp 2d2 Þ a24 þ a25 ¼ 2 þ 2 þ ; d d1 x2 d2 1 1 2d a24 a25 ¼ 2 ð 2 2 Þ; d d1 x X 1 p d p 2 d a21 ¼ 1 þ k 0 þ 2 1 21 þ 0 21 ix 2 k 1 k 0 ; x x d2 X 1 þ k d ix 2 k 0 k 1 0 a21 a22 ¼ k 0 þ 12 ½ð1 þ k 0 Þp1 p20 k 0 þ ix 2 k 0 k 1 ðmp0 p1 Þ ix 2 k 0 k 1mðm p0 Þ ; 2 x d2 d22 ! 1 p d ik 0 k 1 2 d1m2 ; a21 a22 a23 ¼ k 0 2 1 21 x x d2 d1 ðm p0 Þm2i mn2 ðc2 þ pm0 1Þ h i ; i ¼ 1; 2; 3 Vi ¼ d m2i n2 c2 d12 x12 ðmp0 p1 Þ 1 2 h i d ½m2i n2 ðc2 1Þ m2i n2 ðc2 d12 x12 ðmp0 p1 Þ 1 þ d1 p0 ðm p0 Þm2i mn2 ðc2 þ pm0 1Þ h2 i Si ¼ d ixk 1 m2i n2 ðc2 d12 x12 ðmp0 p1 Þ 1Þ 2
ð26Þ With the help of Eqs. (20) and (21), the displacement components u1 and u3 are obtained: u1 ¼ ½inðA1 cos m1 z þ A2 cos m2 z þ A3 cos m3 z þ B1 sin m1 z þ B2 sin m2 z þ B3 sin m3 zÞ þ m4 ðB4 cos m4 z A4 sin m4 zÞ þ m5 ðB5 cos m5 z A5 sin m5 zÞeinðxctÞ ;
ð27Þ
u3 ¼ ½ðm1 A1 sin m1 z þ m1 B1 cos m1 z m2 A2 sin m2 z þ m2 B2 cos m2 z m3 A3 sin m3 z þ m3 B3 cos m3 zÞ inðA4 cos m4 z þ B4 sin m4 z þ A5 cos m5 z þ B5 sin m5 zÞeinðxctÞ :
ð28Þ
3.1. Boundary conditions Consider the following mechanical and thermal boundary conditions at the plate surfaces z = ± d. 3.2. Mechanical conditions The non-dimensional mechanical boundary conditions at z = ± d are given as follows t33 ¼ 0;
t31 ¼ 0;
m32 ¼ 0;
k;3 ¼ 0:
ð29Þ
where ou3 ou1 þk mT þ k0 / ; oz ox ou1 ou3 þl K/2 ; t31 ¼ ðl þ KÞ oz ox o/ o/ ; m32 ¼ c 2 þ b0 oz ox o/ o/ k;3 ¼ a0 b0 2 : oz ox t33 ¼ ðk þ 2l þ KÞ
ð30Þ
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243
3.3. Thermal conditions The thermal boundary conditions at z = ±d are given by oT þ hT ¼ 0: oz
ð31Þ
where h is the surface heat transfer coefficient. Here h ! 0 corresponds to thermal insulated boundaries and h ! 1 refers to isothermal one. 4. Derivation of the secular equations Using the boundary conditions (29) and (31) on the surfaces z = ±d of the plate and with the help of Eqs. (21)–(25), there results a system of ten simultaneous equations P ðA1 c1 þ A2 c2 þ A3 c3 þ B1 s1 þ B2 s2 þ B3 s3 Þ þ QfðA4 m4 s4 þ A5 m5 s5 Þ þ ðB4 m4 c4 þ B5 m5 c5 Þg ¼ 0;
ð32aÞ
P ðA1 c1 þ A2 c2 þ A3 c3 B1 s1 B2 s2 B3 s3 Þ þ QfðA4 m4 s4 þ A5 m5 s5 Þ þ ðB4 m4 c4 þ B5 m5 c5 Þg ¼ 0;
ð32bÞ
QfðA1 m1 s1 þ A2 m2 s2 þ A3 m3 s3 Þ þ ðB1 m1 c1 þ B2 m2 c2 þ B3 m3 c3 Þg þ P ðA4 c4 þ A5 c5 þ B4 s4 þ B5 s5 Þ ¼ 0; ð32cÞ QfðA1 m1 s1 þ A2 m2 s2 þ A3 m3 s3 Þ þ ðB1 m1 c1 þ B2 m2 c2 þ B3 m3 c3 Þg þ P ðA4 c4 þ A5 c5 B4 s4 B5 s5 Þ ¼ 0; ð32dÞ RðV 1 A1 c1 þ V 2 A2 c2 þ V 3 A3 c3 þ V 1 B1 s1 þ V 2 B2 s2 þ V 3 B3 s3 Þ þ SðA4 f4 s4 m4 A5 f5 s5 m5 þ B4 f4 c4 m4 þ B5 f5 c5 m5 Þ ¼ 0;
ð32eÞ
RðV 1 A1 c1 þ V 2 A2 c2 þ V 3 A3 c3 V 1 B1 s1 V 2 B2 s2 V 3 B3 s3 Þ þ SðA4 f4 s4 m4 þ A5 f5 s5 m5 þ B4 f4 c4 m4 þ B5 f5 c5 m5 ¼ 0
ð32fÞ
U ½V 1 m1 s1 A1 V 2 m2 s2 A2 V 3 m3 s3 A3 þ V 1 m1 c1 B1 þ V 2 m2 c2 B2 þ V 3 m3 c3 B3 þ V ½f4 c4 A4 þ f5 c5 A5 þ f4 s4 B4 þ f5 s5 B5 ¼ 0;
ð32gÞ
U ½V 1 m1 s1 A1 þ V 2 m2 s2 A2 þ V 3 m3 s3 A3 þ V 1 m1 c1 B1 þ V 2 m2 c2 B2 þ V 3 m3 c3 B3 þ V ½f4 c4 A4 þ f5 c5 A5 f4 s4 B4 f5 s5 B5 ¼ 0;
ð32hÞ
S 1 ½ðm1 s1 þ hc1 ÞA1 þ ðm1 c1 þ hs1 ÞB1 þ S 2 ½ðm2 s2 þ hc2 ÞA2 þ ðm2 c2 þ hs2 ÞB2 þ S 3 ½ðm3 s3 þ hc3 ÞA3 þ ðm3 c3 þ hs3 ÞB3 ¼ 0;
ð32iÞ
S 1 ½ðm1 s1 þ hc1 ÞA1 þ ðm1 c1 hs1 ÞB1 þ S 2 ½ðm2 s2 þ hc2 ÞA2 þ ðm2 c2 hs2 ÞB2 þ S 3 ½ðm3 s3 þ hc3 ÞA3 þ ðm3 c3 hs3 ÞB3 ¼ 0:
ð32jÞ
where pn2 p P ¼ b n þ 2 ; Q ¼ 2in 1 2 ; f i ¼ b2 m2i ; 2d d 2 cd b0 ind2 R ¼ inb0 ; S ¼ ; U ¼ a0 ; V ¼ ; p p si ¼ sin mi z; ci ¼ cos mi z ; i ¼ 1; 2; 3; 4; 5: 2
2
i ¼ 4; 5;
The system of Eqs. (32a)–(32j) has a non-trivial solution if the determinant of the coefficients of amplitudes [A1, A2, A3, A4, A5, B1, B2, B3, B4, B5]T vanishes. This after lengthy algebraic reductions and manipulations leads to the secular equations for the plate with force stress free and couple stress free thermally insulated and isothermal boundaries, there results
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R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
T1 T4
m1 ðV 1 S 3 V 3 S 1 Þ T 2 m1 ðV 1 S 2 V 2 S 1 Þ T 3 RV ðf5 f4 Þ m1 S 1 ðV 3 V 2 Þ þ þ m2 ðV 2 S 3 V 3 S 2 Þ T 4 m3 ðV 2 S 3 V 3 S 2 Þ T 4 SU ðm5 f5 T 4 m4 f4 T 5 Þ m2 m3 ðV 2 S 3 V 3 S 2 Þ ( ) T 2T 3 m2 S 2 ðV 3 V 1 Þ T 1 T 3 m3 S 3 ðV 2 V 1 Þ T 1 T 2 þ m1 S 1 ðV 3 V 2 Þ T 24 T 5 m1 S 1 ðV 3 V 2 Þ T 24 T 5 T 24 T 5 ( " # m4 m5 TT 54 Q V f5 f4 ðS 3 S 2 Þ T1 m1 ðS 3 S 1 Þ T 2 m1 ðS 2 S 1 Þ T 3 þ þ U ðV S V S Þ T P m2 ðS 3 S 2 Þ T 4 m3 ðS 3 S 2 Þ T 4 2 3 3 2 4 m5 f5 m4 f4 TT 54 R þ S
¼
"
T1 T4
#) m1 V 2 ðV 1 S 3 V 3 S 1 Þ T 2 m1 V 3 ðV 1 S 2 V 2 S 1 Þ T 3 þ m2 V 1 ðV 2 S 3 V 3 S 2 Þ T 4 m3 V 1 ðV 2 S 3 V 3 S 2 Þ T 4
Q2 m1 m4 m5 ððV 1 V 2 ÞðS 2 S 3 Þ ðV 2 V 3 ÞðS 1 S 2 ÞÞðf5 f4 Þ Q2 RV ðm5 f4 T 4 m4 f5 T 5 Þ 2 P SU ðm5 f5 T 4 m4 f4 T 5 Þ 2 P ðV 2 S 3 V 3 S 2 Þ m5 f5 m4 f4 TT 54 V 1 ðS 3 S 2 Þ ðV 2 S 3 V 3 S 2 Þ
( ) T1 m1 V 2 ðS 3 S 1 Þ T 2 m1 V 3 ðS 2 S 1 Þ T 3 þ m2 V 1 ðS 3 S 2 Þ T 4 m3 V 1 ðS 3 S 2 Þ T 4 T4
ð33Þ
for stress free insulated boundaries (h ! 0) of the plate.
m1 S 2 ðV 3 V 1 Þ T 2 m1 S 3 ðV 2 V 1 Þ T 3 RV ðm5 f4 T 4 m4 f5 T 5 Þ m1 ðV 2 S 3 V 3 S 2 Þ þ þ m2 S 1 ðV 3 V 2 Þ T 4 m3 S 1 ðV 3 V 2 Þ T 4 SU m4 m5 ðf5 f4 Þ m2 m3 S 1 ðV 3 V 2 Þ ( ) T 2T 3 m2 ðV 1 S 3 V 3 S 1 Þ T 1 T 3 m3 ðV 1 S 2 V 2 S 1 Þ T 1 T 2 P ðm4 T 5 m5 T 4 Þ þ þ m1 ðV 2 S 3 V 3 S 2 Þ T 4 m1 ðV 2 S 3 V 3 S 2 Þ T 4 Q m4 m5 ðf5 f4 Þ T4 ( " # U m1 f5 f4 ðS 3 S 2 Þ T 2T 3 m2 ðS 3 S 1 Þ T 1 T 3 m3 ðS 2 S 1 Þ T 1 T 2 þ V m2 m3 S 1 ðV 3 V 2 Þ m1 ðS 3 S 2 Þ T 4 m1 ðS 3 S 2 Þ T 4 T4 " #) R m1 V 1 ðV 2 S 3 V 3 S 2 Þ T 2 T 3 m2 V 2 ðV 1 S 3 V 3 S 1 Þ T 1 T 3 m3 V 3 ðV 1 S 2 V 2 S 1 Þ T 1 T 2 þ þ S m2 m3 S 1 ðV 3 V 2 Þ m1 V 1 ðV 2 S 3 V 3 S 2 Þ T 4 m1 V 1 ðV 2 S 3 V 3 S 2 Þ T 4 T4 ) V 2 ðS 3 S 1 Þ 1 mm45 ff45 TT 54 f5 P 2 ( T 1T 3 m1 V 1 ðS 3 S 2 Þ T 2 T 3 m3 V 3 ðS 2 S 1 Þ T 1 T 2 ¼ m2 V 2 ðS 3 S 1 Þ T 24 m2 V 2 ðS 3 S 1 Þ T 24 T 24 Q2 m3 m4 S 1 ðV 3 V 2 Þðf5 f4 Þ
P 2 RV S 2 ðV 3 V 1 Þ T 1T 2T 3 S 1 ðV 3 V 2 Þ S 3 ðV 2 V 1 Þ þ 2 1 ð34Þ S 2 ðV 3 V 1 Þ S 2 ðV 3 V 1 Þ Q SU S 1 ðV 3 V 2 Þm2 m3 m4 m5 T 24 T 5
T1 T4
for stress free isothermal boundaries (h ! 1)of the plate. Where T i ¼ tan mi d;
i ¼ 1; 2; 3; 4; 5:
Here the superscript +1 refers to skew symmetric and 1 refers to symmetric modes. Eqs. (33) and (34) are the most general dispersion relations involving wave number and phase velocity of various modes of propagation in a microstretch generalized thermoelastic plates under the considered situations. These equations can be recognized as modified Rayleigh-Lamb equations which respectively govern the symmetric and anti symmetric modes of wave propagation for force stress and couple stress free, thermally insulated and isothermal
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
245
microstretch thermoelastic plate. Refer to such waves as microstretch thermoelastic plate waves rather than Lamb waves whose properties were derived in[19] for isotropic elastic solids in elastokinetics. 4.1. Particular cases 4.1.1. Microstretch coupled thermoelastic plate In case of coupled theory of thermoelasticity (CT), the thermal relaxation times vanish i.e s0 = 0 so that k0 = k1 = ix1. In this case, the secular equations are given by the Eqs. (33) and (34) with changed values of characteristic roots. 4.1.2. Micropolar thermoelastic plate In the absence of microstretch effect, let R = V = U = V1 = V3 = 0, V2 = 1 and S2 = 0, S i ¼ 2 2 ix1 k 1 1 ðmi a Þ; i ¼ 1; 3 and consequently, the secular Eqs. (33) and (34) reduce to tan m1 d m1 ðm21 a2 Þ tan m3 d ¼ tan m4 d m3 ðm23 a2 Þ tan m4 d
2 4n2 1 2dp2 m1 m4 m5 ðm23 m21 Þðm24 m25 Þ 2 ; 2 2 2 2 pn2 2 2 2 2 b n þ d2 ðm3 a Þ m5 ðb m5 Þ m4 ðb m4 Þ TT 54 2 2
m ðb2 m2 Þ
b2 n2 þ pnd2 ðb2 m25 Þðm23 m21 Þ 1 m4 ðb2 m42 Þ 5 tan m1 d m1 ðm23 a2 Þ tan m3 d 5 ¼ 2 2 2 tan m4 d m3 ðm1 a Þ tan m4 d 2 p 2 2 2 4n 1 2d2 m3 m4 ðm5 m4 Þðm1 a2 Þ
T5 T4
ð35Þ
T 1T 3 T 24
;
ð36Þ 4.1.3. Thermoelastic plate In the absence of micropolarity effect (K = p = 0), there prevails ! 2 c m24 ¼ b2 ; m25 ¼ n2 2 1 ; d1 and consequently, the secular Eqs. (35) and (36) for insulated and isothermal boundaries reduce to tan m1 d m1 ðm21 a2 Þ tan m3 d 4n2 m1 bðm23 m21 Þ ; ¼ 2 2 2 2 tan m4 d m3 ðm3 a Þ tan m4 d ðb n2 Þ ðm23 a2 Þ 2 tan m1 d m1 ðm23 a2 Þ tan m3 d ðb2 n2 Þ ðm23 m21 Þ : ¼ 2 2 tan m4 d m3 ðm1 a2 Þ tan m4 d 4n m3 bðm21 a2 Þ
ð37Þ
ð38Þ
5. Discussion of the secular equation 5.1. Regions of the secular equation In order to explore various regions of the secular equations, here consider the Eq. (33) as an example for the purpose of discussion. Depending upon whether m1, m2, m3, m4, m5, b being real, purely imaginary or complex, the frequency Eq. (33) is correspondingly altered as follows: Region I. When the characteristic roots are of the type, a2 = a 0 2, b2 = b 0 2, m2k ¼ a2k , k = 1, 2, 3, 4, 5 so that a = ia 0 , b = ib 0 , mk = iak, i = 1, 2, 3, 4, 5 are purely imaginary or complex numbers. This ensures that the superposition of partial waves has the property of exponential decay. The secular Eq. (33) becomes
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tanh a1 d a1 ðV 01 S 03 V 03 S 01 Þ tanh a2 d a1 ðV 01 S 02 V 02 S 01 Þ tanh a3 d þ tanh a4 d a2 ðV 02 S 03 V 03 Sv2 Þ tanh a4 d a3 ðV 02 S 03 V 03 S 02 Þ tanh a4 d 0 0 RV ðf5 f4 Þ a1 S 01 ðV 03 V 02 Þ þ SU ða f 0 tanh a4 d a4 f40 tanh a5 dÞ a2 a3 ðV 02 S 03 V 03 S 02 Þ # " # (" 5 5 tanh a2 d tanh a3 d a2 S 02 ðV 03 V 01 Þ tanh a1 d tanh a3 d a1 S 01 ðV 03 V 02 Þ ðtanh a4 dÞ2 tanh a5 d ðtanh a4 dÞ2 tanh a5 d " # ) a5 d ða4 a5 ðtanh Þ Þ a3 S 03 ðV 02 V 01 Þ tanh a1 d tanh a2 d Q tanh a4 d þ þ a1 S 01 ðV 03 V 02 Þ ðtanh a4 dÞ2 tanh a5 d P tanh a5 d a5 f50 a4 f40 tanh a4 d ( " # 0 0 0 0 0 0 V f5 f4 ðS 3 S 2 Þ tanh a1 d a1 ðS 3 S 1 Þ tanh a2 d a1 ðS 02 S 01 Þ tanh a3 d þ U ðV 0 S 0 V 0 S 0 Þ tanh a4 d a2 ðS 03 S 02 Þ tanh a4 d a3 ðS 03 S 02 Þ tanh a4 d " 2 3 3 2 #) R tanh a1 d a1 V 02 ðV 01 S 03 V 03 S 01 Þ tanh a2 d a1 V 03 ðV 01 S 02 V 02 S 01 Þ tanh a3 d þ þ S tanh a4 d a2 V 01 ðV 02 S 03 V 03 S 02 Þ tanh a4 d a3 V 01 ðV 02 S 03 V 03 S 02 Þ tanh a4 d Q2 RV ða5 f40 tanh a4 d a4 f50 tanh a5 dÞ V 01 ðS 03 S 02 Þ P 2 SU ða5 f50 tanh a4 d a4 f40 tanh a5 dÞ ðV 02 S 03 V 03 S 02 Þ ( ) tanh a1 d a1 V 02 ðS 03 S 01 Þ tanh a2 d a1 V 03 ðS 02 S 01 Þ tanh a3 d þ tanh a4 d a2 V 01 ðS 03 S 02 Þ tanh a4 d a3 V 01 ðS 03 S 02 Þ tanh a4 d
¼
Q2 a1 a4 a5 ððV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 ÞÞða24 a25 Þ ; a5 d P 2 ðV 02 S 03 V 03 S 02 Þ a5 f50 a4 f40 tanh tanh a4 d
ð39Þ
where V 0i , S 0i , f 0 can be obtained from the Eq. (26) for the above cases. Region II. In case, the characteristic roots a2 = a 0 2, m2k ¼ a2k , k = 1, 2, 3 and m2k ¼ m2k for k = 4, 5 are real, the frequency equation can be obtained from Eq. (33) as tanh a1 d a1 ðV 01 S 03 V 03 S 01 Þ tanh a2 d a1 ðV 01 S 02 V 02 S 01 Þ tanh a3 d þ tan m4 d a2 ðV 02 S 03 V 03 S 02 Þ tan m4 d a3 ðV 02 S 03 V 03 S 02 Þ tan m4 d RV ðm24 m25 Þ a1 S 01 ðV 03 V 02 Þ þ SU ðm f tan m4 d m4 f4 tan m5 dÞ a2 a3 ðV 02 S 03 V 03 S 02 Þ # " # (" 5 5 tanh a2 d tanh a3 d a2 S 02 ðV 03 V 01 Þ tanh a1 d tanh a3 d a3 S 03 ðV 02 V 01 Þ þ 0 0 0 a1 S 1 ðV 3 V 2 Þ ðtan m4 dÞ2 tan m5 d a1 S 01 ðV 03 V 02 Þ ðtan m4 dÞ2 tan m5 d m5 d ( " " # ) m4 m5 tan d tan m 4 tanh a1 d tanh a2 d Q V f5 f4 ðS 03 S 02 Þ tanh a1 d þ U ðV 0 S 00 V 0 S 0 Þ tan m d 2 P 4 ðtan m4 dÞ tan m5 d m5 d 2 3 3 2 m5 f5 m4 f4 tan tan m4 d " # a1 ðS 03 S 01 Þ tanh a2 d a1 ðS 02 S 01 Þ tanh a3 d R tanh a1 d þ þ a2 ðS 03 S 02 Þ tan m4 d a3 ðS 03 S 02 Þ tan m4 d S tan m4 d #) 0 0 0 0 0 0 0 0 0 0 a1 V 2 ðV 1 S 3 V 3 S 1 Þ tanh a2 d a1 V 3 ðV 1 S 2 V 2 S 1 Þ tanh a3 d þ a2 V 01 ðV 02 S 03 V 03 S 02 Þ tan m4 d a3 V 01 ðV 02 S 03 V 03 S 02 Þ tan m4 d ( Q2 RV ðm5 f4 tan m4 d m4 f5 tan m5 dÞ V 01 ðS 03 S 02 Þ tanh a1 d ¼ 2 tan m4 d P SU ðm5 f5 tan m4 d m4 f4 tan m5 dÞ ðV 02 S 03 V 03 S 02 Þ ) 0 0 0 0 0 0 a1 V 2 ðS 3 S 1 Þ tanh a2 d a1 V 3 ðS 2 S 1 Þ tanh a3 d þ 0 0 0 a2 V 1 ðS 3 S 2 Þ tan m4 d a3 V 01 ðS 03 S 02 Þ tan m4 d
Q2 a1 m4 m5 ððV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 ÞÞðm24 m25 Þ
m5 d P 2 ðV 02 S 03 V 03 S 02 Þðm5 f5 m4 f4 ðtan Þ Þ tan m4 d
:
ð40Þ
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Region III. In this case, the characteristic roots are given by m2k , k = 1, 2, 3, 4, 5 and the secular equation is given by Eq. (33). Similar regions can be characterized for secular Eq. (34). 6. Thin plate results Consider the case when transverse wavelength with respect to thickness of the plate is quite large, so that nd 6 1. Regions I and II yield the results of interest in this case. In Region I, the symmetric case has no roots. In Region II, skew symmetric case has no roots. The skew symmetric case and symmetric case, upon retaining the first two terms in the expansion of hyperbolic tangents, the secular Eqs. (39) and (40) reduce to 2 G3 þ d3 GG4 G d 2 G RV 1 þ SU b02 d 2 1 a2 a2 d 2 ð1 a24 d 2 Þ 1 a25 d 2 3G 5 4 3 3 3 !) d 2 ða24 þa25 Þ ( 2 R G2 d3 G2 3 P 1 V d 2 G1 02 2 02 2 2 2 b ða4 þ a5 b Þ a4 a5 þ þ Q a5 b02 d 2 1 U 3 G S Ga4 V 01 3
Q2 RV ¼ 2 P SU
h i 2 d 2 ða24 þa25 b02 Þ a24 d 2 2 3 d2 1 Q2 a4 1 3 02 2 G2 G2 2 02 2 b d b d 3 P 1 G 1 3
ð41Þ
3
RV G V 01 a21 ða22 S 02 a23 S 03 Þ þ V 02 a22 ða23 S 03 a21 S 01 Þ þ V 03 a23 ða21 S 01 a22 S 02 Þ SU 3
Q 1 V 2 2 R G ½m m b2 ðm24 þ m25 b2 Þ þ 0 G5 P m4 ðm4 þ m5 Þ U 4 4 5 SV 1 ¼
Q2 a21 a22 a23 G Q2 RV G þ 2 P2 P SU 5
ð42Þ
where G ¼ ðV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 Þ; G ¼ a21 ðV 02 S 03 V 03 S 02 Þ a22 ðV 01 S 03 V 03 S 01 Þ þ a23 ðV 01 S 02 V 02 S 01 Þ; G1 ¼ a21 ðS 03 S 02 Þ a22 ðS 03 S 01 Þ þ a23 ðS 03 S 01 Þ; G2 ¼ V 01 ðV 02 S 03 V 03 S 02 Þ V 02 ðV 01 S 03 V 03 S 01 Þ þ V 03 ðV 01 S 02 V 02 S 01 Þ; G2 ¼ a21 V 01 ðV 02 S 03 V 03 S 02 Þ a22 V 02 ðV 01 S 03 V 03 S 01 Þ þ a23 V 03 ðV 01 S 02 V 02 S 01 Þ; G3 ¼ V 01 ðS 03 S 02 Þ V 02 ðS 03 S 01 Þ þ V 03 ðS 02 S 01 Þ; G3 ¼ a21 V 01 ðS 03 S 02 Þ a22 V 02 ðS 03 S 01 Þ þ a23 V 03 ðS 02 S 01 Þ; a22 a23 2 0 0 a2 a2 d ÞS 1 ðV 3 V 02 Þ ða21 þ a23 1 3 d 2 ÞS 02 ðV 03 V 01 Þ 3 3 2 2 aa þ ða21 þ a22 1 2 d 2 ÞS 03 ðV 02 V 01 Þ 3 0 0 2 2 G4 ¼ a2 a3 ðS 3 S 2 Þ a21 a23 ðS 03 S 01 Þ þ a21 a22 ðS 02 S 01 Þ; G4 ¼ ða22 þ a23
G5 ¼ a22 a23 V 01 ðV 02 S 03 V 03 S 02 Þ a21 a23 V 02 ðV 01 S 03 V 03 S 01 Þ þ a21 a22 V 03 ðV 01 S 02 V 02 S 01 Þ; G5 ¼ a22 a23 V 01 ðS 03 S 02 Þ a21 a23 V 02 ðS 03 S 01 Þ þ a21 a22 V 03 ðS 02 S 01 Þ; In the absence of microstretch effect, there prevails R = V = U = V1 = V3 = 0, V2 = 1 and S2 = 0, 2 2 G 2 S i ¼ ix1 k 1 1 ðmi a Þ; i ¼ 1; 3, G ¼ a and consequently, the secular Eqs. (41) and (42) reduce to
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02 2 2 2 pn2 2 d2 2 b d p a24 d 2 2 2 a 1 ¼ 4n Þ 1 1 a 1 ; 4 3 3 3 2d2 d2 2 4n2 1 2dp2 a21 a23 ða23 þ a21 a02 Þ ¼ : 2 2 b02 n2 þ pnd2
ðb02 þ n2
ð43Þ
ð44Þ
If we ignore the micropolarity effect i.e. (K = p = 0), then Eqs. (43) and (44) reduce to ðb02 þ n2 Þ2 4n2 b02 ¼ ðb02 þ n2 Þ2 ða23 þ a21 a02 Þ ¼
4n2 a21 a23 ðb02 þ n2 Þ
2
a2 d 2 4 2 04 2 nb d ; 3 3
:
ð45Þ ð46Þ
In general, here the wave modes depend upon the micropolar parameter. 7. Waves of short wavelength Some information on the asymptotic behavior is obtainable when the transverse wavelength with respect to the thickness of the plate is quite small, so that nd 1. Then the characteristic roots a, b, mi, i = 1, 2, 3, 4 ,5 lie in Region I and secular Eqs. (33) and (34) reduce to ðb02 a24 a25 a4 a5 Þ a2 a3 ðV 02 S 03 V 03 S 02 Þ a1 a3 ðV 01 S 03 V 03 S 01 Þ þ a1 a2 ðV 01 S 02 V 02 S 01 Þ RV ða4 þ a5 Þ a1 S 01 ðV 03 V 02 Þ a2 S 02 ðV 03 V 01 Þ þ a3 S 03 ðV 02 V 01 Þ þ SU
QV 2 2 a4 a5 b02 a24 þ a25 b02 a3 a2 ðS 03 S 02 Þ a1 a3 ðS 03 S 01 Þ þ a1 a2 ðS 02 S 01 Þ þ PU QR a2 a3 V 01 ðV 02 S 03 V 03 S 02 Þ a1 a3 V 02 ðV 01 S 03 V 03 S 01 Þ þ a1 a2 V 03 ðV 01 S 02 V 02 S 01 Þ þ PS Q2 ða4 a5 b02 Þ RV a3 a2 V 01 ðS 03 S 02 Þ a1 a3 V 02 ðS 03 S 01 Þ þ a1 a2 V 03 ðS 02 S 01 Þ ¼ 2 SU P Q2 ð47Þ 2 a1 a2 a3 a4 a5 ða4 þ a5 Þ ðV 01 V 02 ÞðS 02 S 03 Þ ðV 02 V 03 ÞðS 01 S 02 Þ P a2 a3 S 01 ðV 03 V 02 Þ a1 a3 S 02 ðV 03 V 01 Þ þ a1 a2 S 03 ðV 02 V 01 Þ RV ðb2 a4 a5 Þ a1 ðV 02 S 03 V 03 S 02 Þ a2 ðV 01 S 03 V 03 S 01 Þ þ a3 ðV 01 S 02 V 02 S 01 Þ SU a4 a5 ða4 þ a5 Þ
02 02 PU a24 a25 b a24 þ a25 b þ a1 ðS 03 S 02 Þ a2 ðS 03 S 01 Þ þ a3 ðS 02 S 01 Þ QV a4 a5 ða4 þ a5 Þ 2 2
02 02 PR a4 a5 b a24 þ a25 b a1 V 01 ðV 02 S 03 V 03 S 02 Þ a2 V 02 ðV 01 S 03 V 03 S 01 Þ þ a3 V 03 ðV 01 S 02 V 02 S 01 Þ þ QS a4 a5 ða4 þ a5 Þ P 2 ðb02 a24 a25 a4 a5 Þ P 2 RV G3 : ð48Þ þ ¼ a2 V 02 ðS 03 S 01 Þ a3 V 03 ðS 02 S 01 Þ a1 V 01 ðS 03 S 02 Þ Q2 a4 a5 ða4 þ a5 Þ Q2 a4 a5 SU þ
The Eqs. (47) and (48) are respectively the Rayleigh surface wave equations for a force stress and couple stress thermally insulated and isothermal, microstretch generalized thermoelastic half space. In the absence of micro2 2 stretch effect, we have V1 = V3 = 0, V2 = 1 and S2 = 0, S i ¼ ix1 k 1 1 ðmi a Þ, i = 1,3; R = V = U = 0 so that the Eqs. (47) and (48) reduce to
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
2 p 4n2 1 2 a1 a3 a4 a5 ða1 þ a3 Þða4 þ a5 Þ 2d 2 p ¼ b2 n2 1 2 ða21 þ a23 þ a1 a3 a02 Þðb02 a24 a25 a4 a5 Þ; 2d 2 2 p p 2 2 2 02 4n 1 2 a4 a5 ða þ a1 a3 Þða4 þ a5 Þ ¼ b n 1 2 ða1 þ a3 Þðb02 a24 a25 a4 a5 Þ: 2d 2d
249
ð49Þ ð50Þ
The Eqs. (49) and (50) are respectively the Rayleigh surface wave equations for a force stress and couple stress free thermally insulated and isothermal, micropolar generalized thermoelastic solid half space. If the micropolarity effect is neglected i.e. (p = 0), the Eqs. (49) and (50) reduce to 2
4n2 a1 a3 a4 a5 ða1 þ a3 Þða4 þ a5 Þ ¼ ðb02 þ n2 Þ ða21 þ a23 þ a1 a3 a02 Þðb02 a24 a25 a4 a5 Þ; 2
02
02
2 2
02
4n a4 a5 ða þ a1 a3 Þða4 þ a5 Þ ¼ ðb þ n Þ ða1 þ a3 Þðb
a24
a25
a4 a5 Þ:
ð51Þ ð52Þ
for thermally insulated and isothermal stress free thermoelastic plates. The Eqs. (51) and (52) are merely Rayleigh surface wave equations. The Rayleigh results enter here since at such wave lengths the finite thickness plate appears as a semi-infinite medium and hence vibrational energy is transmitted mainly along the surface of the plate. 8. Lame modes A special class of exact solutions, called theLamemodes, but evidently first identified in [19] can be obtained by considering the special case b2 ¼ n2 1 dp2 , the characteristic roots for this case are in Region II and the frequency Eq. (33) reduces to Symmetric modes: tan m4d = 1, ) m4 ¼ np , n = 1, 3, 5, . . . 2d Anti symmetric modes: tan m4d = 0, ) m4 ¼ np , n = 0, 2, 4, . . . 2d Here, the frequency is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4b2 d 2 þ n2 p2 ð1 dp2 Þ qffiffiffiffiffiffiffiffiffiffiffiffi x¼ : 2da4 1 dp2
ð53Þ
It is obvious that these modes depend upon the micropolar parameter (K or p) and thickness of the plate. However in the absence of stretch and micropolarity effect and consequently the frequency is given by ffiffi which agrees with the result in [20]. As these modes don’t depend upon the thermomechanical coux ¼ pnpd 2d pling and thermal relaxation time. However, it is obvious that these modes depend upon the micropolar and stretch parameters in addition to thickness of the plate. 9. Amplitudes of dilatation, microrotation, microstretch and temperature In this section, the amplitudes of dilatation, microrotation, microstretch and temperature distribution for symmetric and skew symmetric modes of plate waves have been computed for stress free microstretch thermoelastic plate. Upon using Eqs. 27,28 and (32a)–(32j), there results ðeÞsy ¼ fðn2 þ m21 Þ cos m1 z þ Lðn2 þ m22 Þ cos m2 z þ Mðn2 þ m23 Þ cos m3 zÞgA1 einðxctÞ ; ðeÞasy ¼ fðn2 þ m21 Þ sin m1 z þ L0 ðn2 þ m22 Þ sin m2 z þ M 0 ðn2 þ m23 Þ sin m3 zgB1 einðxctÞ ;
d2 f4 m4 s4 ðb2 m24 Þ sin m4 z ðb2 m25 Þ ð/2 Þsy ¼ sin m5 z A4 einðxctÞ ; p f5 m5 s5
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ð/2 Þasy
d2 2 2 2 2 f4 m4 c4 ðb m4 Þ cos m4 z ðb m5 Þ ¼ cos m5 z B4 einðxctÞ ; p f5 m5 c5
ð/ Þsy ¼ fV 1 cos m1 z þ V 2 L cos m2 z þ V 3 M cos m3 zgA1 einðxctÞ ; ð/ Þasy ¼ fV 1 sin m1 z þ V 2 L0 sin m2 z þ V 3 M 0 sin m3 zgB1 einðxctÞ ðT Þsy ¼ fS 1 cos m1 z þ S 2 L cos m2 z þ S 3 M cos m3 zgA1 einðxctÞ ; ðT Þasy ¼ fS 1 sin m1 z þ S 2 L0 sin m2 z þ S 3 M 0 sin m3 zgB1 einðxctÞ : where ðV 3 S 1 V 1 S 3 Þm1 s1 ðV 3 S 1 V 1 S 3 Þm1 c1 ; L0 ¼ ; ðV 2 S 3 V 3 S 2 Þm2 s2 ðV 2 S 3 V 3 S 2 Þm2 c2 ðV 1 S 2 V 2 S 1 Þ m1 s1 ðV 1 S 2 V 2 S 1 Þ m1 c1 M¼ ; M0 ¼ : ðV 2 S 3 V 3 S 2 Þ m3 s3 ðV 2 S 3 V 3 S 2 Þ m3 c3 L¼
10. Numerical results and discussion With the view of illustrating theoretical results obtained in the preceding sections and comparing these in the context of various theories of thermoelasticity, some numerical results will be presented. The material chosen for this purpose is magnesium crystal (microstretch thermoelastic solid), the physical data for which is given below q ¼ 1:74 103 kg=m3 ;
k ¼ 9:4 1010 N=m2 ;
K ¼ 1:0 1010 N=m2 ;
c ¼ 0:779 109 N;
s0 ¼ 6:131 1013 s;
2¼ 0:028;
K ¼ 0:6 106 J=m s ;
l ¼ 4:0 1010 N=m2 ; j ¼ 0:2 1019 m2 ;
21 ¼ 0:069;
m ¼ m1 ¼ 0:084;
j0 ¼ 0:185 1019 m2
T 0 ¼ 300 K; C ¼ 0:23 103 J=kg ;
k0 ¼ 0:5 1010 N=m2 ;
k1 ¼ 0:5 1010 N=m2 ;
a0 ¼ 0:779 109 N; d ¼ 0:01 m: The non-dimensional phase velocity and attenuation coefficient of symmetric and skew symmetric modes of wave propagation in the context of L–S and CT theories of thermoelasticity have been computed for various values of non-dimensional wave number from dispersion Eq. (33) for stress free thermally insulated boundaries and have been represented graphically for different modes (n = 0 to n = 2) in Figs. 2–5. The solid curves correspond to CT theory and broken-line curves correspond to L–S theory of thermoelasticity. The
Fig. 2. Variation of phase velocity C of symmetric mode of wave propagation.
R. Kumar, G. Partap / Theoretical and Applied Fracture Mechanics 48 (2007) 238–257
Fig. 3. Variation of attenuation coefficient Q of symmetric mode of wave propagation.
Fig. 4. Variation of phase velocity C of skew-symmetric mode of wave propagation.
Fig. 5. Variation of attenuation coefficient Q of skew-symmetric mode of wave propagation.
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amplitudes of dilatation, microrotation, microstretch and temperature distribution for symmetric and skew symmetric modes in the context of L–S and CT theories of thermoelasticity are presented graphically in Figs. 6–13. The phase velocities of lowest symmetric mode of propagation become dispersionless i.e. remain constant with variation in wave number. The phase velocities of higher modes of propagation, symmetric and skew symmetric attain quite large values at vanishing wave number which sharply slashes down to become steady and asymptotic to the reduced Rayleigh wave velocity with increasing wave number. The reason for this asymptotic approach is that for short wavelengths (or high frequencies) the material plate behaves increasingly like a thick slab and hence the coupling between upper and lower boundary surfaces is reduced and as a result the properties of symmetric and skew symmetric waves become more and more similar.
Fig. 6. Amplitude of symmetric dilatation e.
Fig. 7. Amplitude of skew-symmetric dilatation e.
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Fig. 8. Amplitude of symmetric microrotation /2.
Fig. 9. Amplitude of skew-symmetric microrotation /2.
It is observed that in the context of various theories of thermoelasticity (L–S and CT), various symmetric modes of propagation have nearly same velocities for n = 0, n = 1, and n = 2. For skew symmetric modes of wave propagation, we observe the following (a) for lowest mode n = 0, phase velocity profiles for L–S and CT theory coincide for wave number nd P 5.0and for wave number nd 6 5.0. Phase velocity for L–S theory is less than in case of CT theory (b) for n = 1, phase velocity profiles for L–S and CT theory coincide for wave number nd P 6.2 and for wave number nd 6 6.2, phase velocity for L–S theory is more than in case of CT theory (c) for n = 2, phase velocity profiles for L–S and CT theory coincide for wave number nd P 5.2 and for wave number nd 6 5.2, phase velocity for L–S theory is more than in case of CT theory.
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Fig. 10. Amplitude of symmetric microstretch /*.
Fig. 11. Amplitude of skew-symmetric microstretch /*.
The attenuation coefficients of symmetric and skew symmetric modes have been plotted in the context of coupled thermoelasticity and that in [4] for thermoelasticity. In general, wave number and phase velocity of the waves are complex quantities, therefore, the waves are attenuated in space. Let C 1 ¼ V 1 þ ix1 Q
ð54Þ
then n = R + iQ, where R = x/V and Q are real numbers. This shows that V is the propagation speed and Q is attenuation coefficient of waves. Upon using (54) in secular Eq. (33), the value of propagation speed V and attenuation coefficient Q for different modes of wave propagation can be obtained.
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Fig. 12. Amplitude of symmetric temperature T.
Fig. 13. Amplitude of skew-symmetric temperature T.
The variation of attenuation coefficient with wave number for symmetric and skew symmetric modes is represented graphically in Figs. 2 and 4 respectively. For lowest symmetric mode, the magnitude of attenuation coefficient for L–S theory has maxima up to 108.9 in region 0.2 6 nd 6 4.2 at nd = 1.2 and approaches to zero with increase in wave number. For first symmetric mode, the magnitude of attenuation coefficient for L–S theory has negligible variation with wave number in region 0.2 6 nd 6 4.2, the magnitude of attenuation coefficient for L–S theory have maxima up to 127.7 and 158.4 in region 4.2 6 nd 6 6.2 at nd = 5.2 and 6.2 6 nd 6 8.2 at nd = 7.2 respectively, the magnitude of attenuation coefficient decreases from 15.16 to 5.325 in the region 8.2 6 nd 6 9.2. For lowest symmetric mode, the magnitude of attenuation coefficient for CT theory have maxima upto 34.96 and 34.39 in region 0.2 6 nd 6 2.2 at nd = 1.2 and 2.2 6 nd 6 4.2 at nd = 3.2 respectively and has
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negligible variation with wave number in region 4.2 6 nd 6 9.2. For first symmetric mode, the magnitude of attenuation coefficient has negligible variation with wave number in region 0.2 6 nd 6 4.2, has maxima upto 115.7 in region 4.2 6 nd 6 6.2 at nd = 5.2 and increases to 30.4 at nd = 7.2 and decreases to 9.417 at nd = 9.2 in region 6.2 6 nd 6 9.2 for CT theory. The attenuation coefficient has negligible variation with wave number for second symmetric mode (n = 2) in case of CT and L–S theory of thermoelasticity. For lowest skew symmetric mode, the magnitude of attenuation coefficient for L–S theory decreases from 49.99 to 2.238 in the region 0.2 6 n d 6 1.2, has maxima upto 10.57 in region 1.2 6 nd 6 3.2 at nd = 2.2, has maximum values 68.99 and 82.23 at nd = 4.2 and nd = 5.2 respectively in the region 3.2 6 nd 6 6.2 and has negligible variation with wave number in region 6.2 6 nd 6 9.2. For first skew symmetric mode, attenuation coefficient for L–S theory has negligible variation with wave number in region 0.2 6 n d 6 1.2, has maximum values 23.49 and 20.62 at nd = 2.2 and n d = 3.2 respectively in the region 1.2 6 nd 6 3.2, has maxima up to 218.5 in region 3.2 6 nd 6 7.2 at nd = 5.2 and has negligible variation with wave number in region 7.2 6 nd 6 9.2. For first skew symmetric mode, attenuation coefficient for L–S theory has negligible variation with wave number in regions 0.2 6 nd 6 4.2 and 8.2 6 nd 6 9.2, have maxima up to 5.776 and 20.18 in region 4.2 6 nd 6 6.2 at nd = 5.2 and 6.2 6 n d 6 8.2 at nd = 7.2 respectively. For lowest skew symmetric mode, attenuation coefficient for CT theory attains the maxima up to 89.57 at nd = 2.2 in the region 0.2 6 nd 6 3.2and has negligible variation with wave number in region 3.2 6 nd 6 9.2. For first skew symmetric mode, attenuation coefficient for CT theory increases up to 6.358 at nd = 2.2 in the region 0.2 6 nd 6 3.2and has negligible variation with wave number in region 3.2 6 nd 6 9.2. The attenuation coefficient has negligible variation with wave number for second skew symmetric mode (n = 2) in CT theory of thermoelasticity. The effect of stress free thermally insulated boundaries of the plate is quite pertinent and can be easily noticed from dispersion and attenuation curves plotted in Figs. 2–5. The effect of micropolarity and microstretch can be seen from the comparison of figures. 10.1. Amplitudes Figs. 6 and 7 depict the variations of symmetric and skew symmetric amplitudes of dilatation (e) in the context of L–S and C-T theories of thermoelasticity for stress free thermally insulated boundary. The dilatation (e) of the plate is minimum at the centre and maximum at the surfaces for symmetric mode and zero at the centre and maximum at the surfaces for skew symmetric mode as evident from Figs. 6 and 7 respectively. Figs. 8–13 show the variations of symmetric and skew symmetric amplitudes of microrotation (/2), microstretch (/*) and temperature distribution (T) in the context of L–S and C-T theories of thermoelasticity for stress free thermally insulated boundary. It is evident from Figs. 8–13 that the values of microrotation(/2), microstretch (/*) and temperature distribution (T) of the plate is minimum at the centre and maximum at the surfaces for symmetric mode and zero at the centre and maximum at the surfaces for skew symmetric mode. (e)sym, (e)asym, (/2)sym, (/2)asym, (/*)sym, (/*)asym, (T)sym and(T)asym correspond to the values of (e),(/2), (/*) and (T) for symmetric and skew symmetric modes respectively. It is observed that behavior and trend of variations of (e)sym, (/2)sym,(/*)sym and (T)sym is same; whereas the behavior and trend of variations of (e)asym, (/2)asym, (/*)asym and(T)asym is similar. The values of the dilatation, microstretch and temperature distribution of the plate in case of L–S theory are less in comparison to C-T Theory for symmetric and skew symmetric modes. The values of the microrotation(/2) of the plate are same in case of L–S and C-T theories of thermoelasticity for symmetric and skew symmetric modes. 11. Conclusions The propagation of waves in microstretch thermoelastic plate subjected to stress free thermally insulated and isothermal boundary is investigated in the context of conventional coupled thermoelasticity (CT) and Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skewsymmetric wave mode propagation have been obtained. The secular equations for microstretch coupled thermoelastic, micropolar thermoelastic, and thermoelastic plates have been deduced as particular cases from the
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derived secular equations. At short wavelength limits, the secular equations for symmetric and skew-symmetric waves in stress free thermally insulated and isothermal microstretch thermoelastic plate reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical results are found to be in close agreement. The significant response of different theories of thermoelasticity i.e coupled thermoelasticity and thermoelasticity with one relaxation time for different modes of propagation (symmetric and skewsymmetric) on phase velocity, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch and temperature distribution have been observed. References [1] A.C. Eringen, Linear theory of micropolar elasticity, J. Math. Mech. 15 (1966) 909–923. [2] A.C. Eringen, Theory of micropolar elasticity, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, p. 622 (Chapter 7). [3] D. Iesan, Some applications of micropolar mechanics to earthquake problem, Int. J. Eng. Sci. 19 (1981) 855–864. [4] H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309. [5] A.C. Eringen, Micropolar elastic solids with stretch, Ari Kitabevi Matbassi, Istanbul 24 (1971) 1–18. [6] W. Nowacki, Couple stress theory in the theory of thermoelasticity, Proceedings of IUTAM symposia, Springer-Verlag, Vienna, 1966, pp. 259–278. [7] A.C. Eringen, Foundations of micropolar thermoelasticity, International Centre For Mechanical Studies, Course and Lectures, vol. 23, Springer-Verlag, Wein, 1970. [8] T.R. Touchert, W.D. Claus Jr., T. Ariman, The linear theory of micropolar thermoelasticity, Int. J. Eng. Sci. 6 (1968) 37–47. [9] D.S. Chandrasekharaiah, Heat flux dependent micropolar thermoelasticity, Int. J. Eng. Sci. 24 (1986) 1389–1395. [10] A.C. Eringen, Theory of thermo-microstretch elastic solids, Int. J. Eng. Sci. 28 (1990) 1291–1301. [11] A.C. Eringen, Microcontinuum Field Theories I: Foundations and Solids, Springer-Verlag, New York, 1999. [12] Xiaonong Liu, Gengkai Hu, Inclusion problem of microstretch continuum, Int. J. Eng. Sci. 42 (2004) 849–860. [13] Merab Svanadze, Fundamental solution of the system of equations of steady oscillations in the theory of microstretch elastic solids, Int. J. Eng. Sci. 42 (2004) 1897–1910. [14] S. De Cicco, Stress concentration effects in microstretch elastic solids, Int. J. Eng. Sci. 41 (2003) 187–199. [15] R. Kumar, Geeta Partap, Reflection of plane waves in a heat flux dependent microstretch thermoelastic solid half space, Int. J. Appl. Mech. Eng. 10 (2005) 2, pp. 253–266. [16] S.K. Tomer, Monica Garg, Reflection and transmission of waves from a plane interface between two microstretch solid half spaces, Int. J. Eng. Sci. 43 (2005) 139–169. [17] R. Kumar, Baljeet Singh, Wave propagation in a generalized thermo-microstretch elastic solid, Int. J. Eng. Sci. 36 (1998) 891–912. [18] R. Kumar, Sunita Deswal, Wave propagation through cylindrical bore contained in a microstretch elastic medium, J. Sound Vibr. 250 (2002) 711–722. [19] H. Lamb, On waves in an elastic plate, Phil. Trans.Roy. Soc., London, Ser. A 93 (1917) 114–128. [20] K.F. Graff, Wave Motion in Elastic Solids, Dover publications, Inc., New York, 1991.