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Wave propagation in a generalized thermoelastic material with voids
Applied Mathematics and Computation 189 (2007) 698–709 www.elsevier.com/locate/amc
Wave propagation in a generalized thermoelastic material with voids Baljeet Singh Department of Mathematics, Government College, Sector-11, Chandigarh 160 011, India
Abstract The governing equations for two-dimensional homogeneous, isotropic generalized thermoelastic half-space with voids are solved in context of Lord–Shulman theory. Three compressional waves and a shear vertical (SV) wave are shown to exist. The reflection phenomena of compressional or shear wave from the free surface of a thermoelastic solid with voids is considered. The boundary conditions at stress-free thermally insulated surface are satisfied to obtain a system of four equations in the reflection coefficients of various reflected waves. These reflection coefficients depend on the angle of incidence of striking wave, void, thermo-void and other elastic parameters. Numerical values of the complex modulus of the reflection coefficients are also visualized graphically to observe the effects of void parameters. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Thermoelasticity; Voids; Compressional; Reflection; Reflection coefficients; Thermal relaxations
1. Introduction Theory of linear elastic materials with voids is an important generalization of the classical theory of elasticity. This theory is used for investigating various types of geological and biological materials for which classical theory of elasticity is not adequate. The theory of linear elastic materials with voids deals the materials with a distribution of small pores or voids, where the volume of void is included among the kinematics variables. This theory reduces to the classical theory in the limiting case of volume of void tending to zero. Nonlinear theory of elastic materials with voids was developed by Nunziato and Cowin [20]. Cowin and Nunziato [8] developed a theory of linear elastic materials with voids to study mathematically the mechanical behavior of porous solids. Puri and Cowin [22] studied the behaviour of plane waves in a linear elastic materials with voids. Iesan [15] developed linear theory of thermoelastic material with voids. Ciarletta and Scalia [5] developed the nonlinear theory of non-simple thermoelastic materials with voids. Ciarletta and Scarpetta [6] studied some results on thermoelasticity for dielectric materials with voids. Dhaliwal and Wang [11] developed a heat flux dependent theory of thermoelasticity with voids. Marin [18,19] studied uniqueness and domain of E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.123
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influence results in thermoelastic bodies with voids. Chirita and Scalia [4] studied the spatial and temporal behavior in linear thermoelasticity of materials with voids. Pompei and Scalia [21] studied the asymptotic spatial behavior in linear thermoelasticity of materials with voids. In the classical theory of thermoelasticity, when an elastic solid is subjected to a thermal disturbance, the effect is felt at a location far from the source, instantaneously. This implies that the thermal wave propagates with infinite speed, a physically impossible result. In contrast to conventional thermoelasticity, non-classical theories came into existence during the last part of 20th century. These theories, referred to as generalized thermoelasticity, were introduced in the literature in an attempt to eliminate the shortcomings of the classical dynamical thermoelasticity. For example, Lord and Shulman [17], by incorporating a flux-rate term into Fourier’s law of heat conduction, formulated a generalized theory which involves a hyperbolic heat transport equation admitting finite speed for thermal signals. Green and Lindsay [13], by including temperature rate among the constitutive variables, developed a temperature-rate-dependent thermoelasticity that does not violate the classical Fourier law of heat conduction, when body under consideration has center of symmetry and this theory also predicts a finite speed for heat propagation. Chandrasekharaiah [3] referred to this wavelike thermal disturbance as ‘‘second sound’’. The Lord and Shulman theory of generalized thermoelasticity was further extended by Dhaliwal and Sherief [10] to include the anisotropic case. A survey article of representative theories in the range of generalized thermoelasticity is due to Hetnarski and Ignaczak [14]. A theory of thermoelastic materials with voids and without energy dissipation is discussed by Cicco and Diaco [7]. Sinha and Sinha [25] and Sinha and Elsibai [26,27] studied the reflection of thermoelastic waves from the free surface of a solid half-space and at the interface between two semi-infinite media in welded contact, in the context of generalized thermoelasticity. Abd-Alla and Al-Dawy [1] studied the reflection phenomena of SV waves in a generalized thermoelastic medium. Sharma et al. [23] investigated the problem of thermoelastic wave reflection from the insulated and isothermal stress-free as well as rigidly fixed boundaries of a solid half-space in the context of different theories of generalized thermoelasticity. Singh [24] discussed the reflection of plane waves from a stress-free surface of an elastic solid with thermodiffusion. The present investigation is motivated by the well-established theories given by Lord and Shulman [17] and Iesan [15]. The paper is organized as follows: in Section 2, the governing equations for homogeneous isotropic generalized thermoelastic solid with voids are obtained. From these equations, a serious error in Eq. (3) of Kumar and Rani [16] is detected for L–S case. The governing equations are then solved for x–z plane to show the existence of a shear and three compressional waves. In Section 3, the boundary conditions at stress-free thermally insulated surface are satisfied to obtain a system of four non-homogeneous equations in reflection coefficients for the incidence of compressional or shear wave at a thermally insulated free surface. A numerical example is given in last section to discuss and visualize the dependence of reflection coefficients upon angle of incidence of striking wave and effects of void parameters on these coefficients and other material parameters. 2. Governing equations and solution Following, Lord and Shulman [17] and Iesan [15], the constitutive equations for isotropic generalized thermoelastic material with voids become: rij ¼ 2leij þ dij ½kekk bH þ bU; qi þ s0 q_i ¼ KH;i ;
ð1Þ ð2Þ
hi ¼ aU;i ;
ð3Þ
qg ¼ bekk þ aH þ mU; g ¼ bekk nU þ mH;
ð4Þ ð5Þ
where k; l are Lame’s constants, H ¼ T T 0 ; T 0 is the temperature of the medium in its natural state assumed to be such that jH=T 0 j 1, T is the absolute temperature, rij are the components of the stress tensor, eij ¼ ðui;j þ uj;i Þ=2, ui are the components of the displacement vector, g is the entropy per unit mass, K is the coefficient of thermal conductivity, s0 is the thermal relaxation time. a; b; n are void material parameters, m is thermo-void coefficient, b ¼ ð3k þ 2lÞat , at is the coefficient of linear thermal expansion, dij is Kronecker
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delta, qi are components of heat flux vector, hi are components of equilibrated stress vector, U is change in volume fraction field, g is the intrinsic equilibrated body force and a is thermal constant. In the absence of heat sources, the energy equation for linear theory of thermoelastic material with voids is ð6Þ
qT 0 g_ ¼ qi;i :
Following, Lord and Shulman [17] and Iesan [15], the field equations in terms of the displacement, volume fraction and temperature, for homogeneous isotropic generalized thermoelastic material with voids in the absence of body forces, heat sources and extrinsic equilibrated body forces are lui;jj þ ðk þ lÞuj;ij bH;i þ bU;i ¼ qu€i ; _ þ s0 HÞ € þ bT 0 ðu_ k;k þ s0 € € ¼ KH;ii ; qcE ðH uk;k Þ þ mT 0 ðU_ þ s0 UÞ
ð7Þ
€ aU;ii buk;k nU þ mH ¼ qvU;
ð9Þ
ð8Þ
The superposed dots denote derivative with respect to time. cE ð¼ aTq 0 Þ is specific heat at constant strain. v is the equilibrated inertia. From Eq. (8), it may be pointed here that Eq. (3) of Kumar and Rani [16] is not a correct L–S generalization. Therefore, the further analysis of Kumar and Rani [16] should be changed accordingly. For x–z plane, Eqs. (7)–(9) become ðk þ 2lÞu1;11 þ ðl þ kÞu3;13 þ lu1;33 bH;1 þ bU;1 ¼ qu€1 ; lu3;11 þ ðl þ kÞu1;13 þ ðk þ 2lÞu3;33 bH;3 þ bU;3 ¼ qu€3 ; _ þ bT 0 su_ 1;1 þ bT 0 su_ 3;3 þ mT 0 sU; _ Kr2 H ¼ qcE sH
ð12Þ
€ ar U nU bðu1;1 þ u3;3 Þ þ mH ¼ qvU;
ð13Þ
2
ð10Þ ð11Þ
where s ¼ 1 þ s0
o ; ot
r2 ¼
o2 o2 þ : ox2 oz2
ð14Þ
The displacement components u1 and u3 may be written in terms of potential functions / and w as u1 ¼
o/ ow ; ox oz
u3 ¼
o/ ow þ : oz ox
ð15Þ
Using (15) into Eqs. (10)–(13) we obtain lr2 w ¼ q
o2 w ; ot2
ð16Þ
o2 / ðk þ 2lÞr2 / bH þ bU ¼ q 2 ; ot oH o oU þ r2 / þ 1 x r2 H ¼ s ; ot ot ot € ar2 U nU þ mH br2 / ¼ qvU;
ð17Þ ð18Þ ð19Þ
where x ¼
K ; qcE
¼
bT 0 ; qcE
1 ¼
mT 0 ; qcE
ð20Þ
Eq. (16) is uncoupled, whereas Eqs. (17)–(19) are coupled in /, H and U. From Eqs. (17)–(19), we see that while the P-wave is affected due to the presence of thermal and void fields, whereas the pSV ffiffiffiffiffiffiffiffiremains unaffected. The solution of Eq. (16) corresponds to the propagation of SV wave with velocity l=q. Solutions of Eqs. (17)–(19) are now sought in the form of the harmonic travelling wave f/; H; Ug ¼ f/0 ; H0 ; U0 geikðx sin hþz cos hvtÞ ;
ð21Þ
where v is the phase speed, k is the wave number and ðsin h; cos hÞ denotes the projection of the wave normal onto x–z plane.
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The homogeneous system of equations in /0 ; H0 ; and U0, obtained by inserting (21) into Eqs. (17)–(19), admits non-trivial solutions and enables to conclude that C satisfies the cubic equation C3 þ LC2 þ MC þ N ¼ 0;
ð22Þ
where C ¼ v2 ; x þ 1 m L ¼ þ c21 þ c3 2 þ b ; s k2 m m x m 2 þ 1 c2 m b b 2 þ bc þ þ b ; M ¼ c21 þ c3 2 þ c21 c3 2 1 3 1 2 s k k k2 k2 x m ; N ¼ c21 c3 2 þ b s k2 and n k þ 2l a b ; c3 2 ¼ þ 2 ; m ¼ ; q qv k qv b b m i ¼ ; m ¼ ; s ¼ s0 þ : b¼ ; b q q qv x
c21 ¼
n n¼ ; qv
Eq. (22) is a cubic equation with complex coefficients. To solve Eq. (22), we calculate Q¼
L2 3M ; 9
R¼
2L3 9LM þ 27N : 54
ð23Þ
If Q and R are real (always true when L, M, N are real) and R2 < Q3 , then cubic Eq. (22) has three real roots. These three roots are u L pffiffiffiffi ð24Þ C1 ¼ 2 Q cos ; 3 3 pffiffiffiffi u þ 2p L ð25Þ C2 ¼ 2 Q cos ; 3 3 pffiffiffiffi u 2p L ð26Þ C3 ¼ 2 Q cos ; 3 3 where R u ¼ arccos pffiffiffiffiffi3ffi Q
! ð27Þ
If Q and R are non-real, then the three roots are L C1 ¼ ðA þ BÞ ; 3 pffiffiffi 1 L 3 ðA BÞ; C2 ¼ ðA þ BÞ þ i 2 3 2 pffiffiffi 1 L 3 ðA BÞ; C3 ¼ ðA þ BÞ i 2 3 2
ð28Þ ð29Þ ð30Þ
where "
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#1=3 A ¼ R þ R2 Q 3 ;
ð31Þ
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B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
where sign of the square root is chosen to make qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re R R2 Q3 P 0;
ð32Þ
asterisk denotes complex conjugation and B¼
Q ðA 6¼ 0Þ; A
B ¼ 0 ðA ¼ 0Þ:
ð33Þ
The squared velocities C1, C2 and C3 are dominated by displacement, volume fraction and temperature fields, respectively, and show the existence of three compressional waves. Let us name these waves as P1, P2 and P3, respectively, for displacement, temperature displacement and volume fraction fields. When we solve Eq. (22) by using a computer program, it is also observed that P1 wave is fastest wave whereas P3 is slowest wave. If we neglect void effects, i.e. for m ¼ 0; b ¼ 0; a ¼ 0; 1 ¼ 0; n ¼ 0; c23 ¼ 0; c3 2 ¼ 0, the cubic equation (22) reduces to a quadratic equation whose roots are as 2v2 ¼ d 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 21 4c21 x ;
ð34Þ
where d1 ¼
x þ c21 þ b; s
ð35Þ
and the positive and negative signs correspond to P1 wave and P2 wave, respectively. Similarly, if we neglect the thermal effects, i.e. for b ¼ 0; x ¼ 0; K ¼ 0; ¼ 0; m ¼ 0; Eq. (22) reduces to a quadratic equation whose roots are as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 2 2 2 ð36Þ 2v ¼ d 2 d 2 4 c1 c3 þ b 2 ; k where d 2 ¼ c21 þ c3 2
ð37Þ
and the positive and negative signs correspond to P1 wave and P3 waves, respectively. In absence of thermal and void effects, Eqs. (34) and (36) reduce to sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ 2l ; ð38Þ v¼ q which is the velocity of classical longitudinal wave. 3. Reflection coefficients In the previous section, it has been discussed that there exists a shear and three compressional waves in an isotropic generalized thermoelastic material with voids. Any incident wave at the interface between two elastic solid bodies, in general, produce compressional and distortional waves in both media (see for example, Ewing et al. [12]; Ben-Menahem and Singh [2]). Let us consider an incident P1 or SV wave (Fig. 1). The boundary conditions at the free surface z ¼ 0 are satisfied, if the incident plane wave (P1 or SV) gives rise to a reflected SV and three reflected compressional waves (i.e. P1, P2 and P3 waves). The surface z ¼ 0 is assumed to be traction free and thermally insulated so that there is no variation of temperature and volume fraction on it. Therefore, the boundary conditions on z ¼ 0 are written as rzz ¼ 0;
rzx ¼ 0;
oH ¼ 0; oz
oU ¼ 0; oz
on z ¼ 0
ð39Þ
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z z > 0 ( Vaccum)
x Thermoelastic Half-space with voids
θ0 θ 4 θ1 θ2 θ3
A3 A2
A0 or B0
A1 B1
Fig. 1. Schematic diagram for the problem (for incident P1 wave h0 = h1, for incident SV wave h0 = h4).
The appropriate displacement potentials / and w, temperature H and volume fraction U are taken in the form w ¼ B0 exp½ik 4 ðx sin h0 þ z cos h0 Þ ixt þ B1 exp½ik 4 ðx sin h4 z cos h4 Þ ixt;
ð40Þ
/ ¼ A0 exp½ik 1 ðx sin h0 þ z cos h0 Þ ixt þ A1 exp½ik 1 ðx sin h1 z cos h1 Þ ixt þ A2 exp½ik 2 ðx sin h2 z cos h2 Þ ixt þ A3 exp½ik 3 ðx sin h3 z cos h3 Þ ixt; H ¼ f1 A0 exp½ik 1 ðx sin h0 þ z cos h0 Þ ixt þ f1 A1 exp½ik 1 ðx sin h1 z cos h1 Þ ixt
ð41Þ
þ f2 A2 exp½ik 2 ðx sin h2 z cos h2 Þ ixt þ f3 A3 exp½ik 3 ðx sin h3 z cos h3 Þ ixt; U ¼ g1 A0 exp½ik 1 ðx sin h0 þ z cos h0 Þ ixt þ g1 A1 exp½ik 1 ðx sin h1 z cos h1 Þ ixt
ð42Þ
þ g2 A2 exp½ik 2 ðx sin h2 z cos h2 Þ ixt þ g3 A3 exp½ik 3 ðx sin h3 z cos h3 Þ ixt;
ð43Þ
where the wave normal of the incident P1 or SV wave makes angle h0 with the positive direction of the z-axis, and those of reflected P1, P2, P3 and SV waves make h1 ; h2 ; h3 and h4 with the same direction, and k 2 ðv2 c2 Þ ði ¼ 1; 2; 3Þ; fi ¼ i i 1 b bGi k 2 ðv2 c2 Þ ði ¼ 1; 2; 3Þ; gi ¼ Gi i i 1 b bGi
ð44Þ ð45Þ
where
Gi ¼
b ðv2i c21 Þð1 sx v2 Þ i
b 1 ðv2i c21 Þ þ
:
ð46Þ
The ratios of the amplitudes of the reflected waves to the amplitude of the incident P1 wave, namely B1 =A0 ; A1 =A0 ; A2 =A0 and A3 =A0 give the reflection coefficients for reflected SV, reflected P1, reflected P2 and reflected P3 waves, respectively. Similarly for incident SV wave, B1 =B0 ; A1 =B0 ; A2 =B0 and A3 =B0 are the reflection coefficients for reflected SV, reflected P1, reflected P2 and reflected P3 waves, respectively. The wave number k 1 ; k 2 ; k 3 ; k 4 and the angles h1 ; h2 ; h3 ; h4 are connected by the relation k 1 sin h1 ¼ k 2 sin h2 ¼ k 3 sin h3 ¼ k 4 sin h4 ;
ð47Þ
at surface z ¼ 0. Relation (47) may also be written in order to satisfy the boundary conditions (39) as sin h1 sin h2 sin h3 sin h4 ¼ ¼ ¼ ; v1 v2 v3 v4
ð48Þ
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B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
pffiffiffiffiffiffiffiffi where v4 ¼ l=q is the velocity of SV wave and vi ; ði ¼ 1; 2; 3Þ are the velocities of three kinds of reflected compressional waves. Using the potentials given by (40)–(43) in boundary conditions Eq. (39), we obtain a system of four non-homogeneous equations Raij Z j ¼ bi ; where
ði; j ¼ 1; 2; . . . 4Þ;
ð49Þ
"
a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33
# 2 f1 g1 k 1 ¼ k þ 2l cos h1 þ b 2 b 2 ; l k1 k1 " # 2 f2 g2 k 2 2 ¼ k þ 2l cos h2 þ b 2 b 2 ; l k2 k2 " # 2 f3 g3 k 3 2 ¼ k þ 2l cos h3 þ b 2 b 2 ; l k3 k3 2 k4 ¼ l sin 2h4 ; l 2 k1 ¼ sin 2h1 ; l 2 k2 ¼ sin 2h2 ; l 2 k3 ¼ sin 2h3 ; l 2 k4 ¼ cos 2h4 ; l 3 f k1 ¼ cos h1 12 ; k1 l 3 f2 k 2 ¼ cos h2 2 ; k2 l 3 f k3 ¼ cos h3 32 ; k3 l 2
a34 ¼ 0;
3 g1 k 1 ; k 21 l 3 g2 k 2 a42 ¼ cos h2 2 ; k2 l 3 g k3 a43 ¼ cos h3 23 ; k3 l a41 ¼ cos h1
a44 ¼ 0: For incident P1 wave b1 ¼ a11 ;
b2 ¼ a21 ;
b3 ¼ a31 ;
b4 ¼ a41 ;
l ¼ k1:
b3 ¼ a34 ;
b4 ¼ a44 ;
l ¼ k4
For incident SV wave b1 ¼ a14 ;
b2 ¼ a24 ;
and Z 1 ; Z 2 ; Z 3 ; Z 4 are reflection coefficients of reflected P1, P2, P3 and SV waves, respectively.
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
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In the absence of thermal and void parameters, these reflection coefficients reduce to For incident P wave 2
A1 sin 2h1 sin 2h4 ðv1 =v4 Þ cos2 2h4 ¼ ; A0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4 B1 2ðv1 =v4 Þ sin 2h1 cos 2h4 ¼ : A0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4
ð50Þ ð51Þ
For incident SV wave A1 ðv1 =v4 Þ sin 4h4 ¼ ; B0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4
ð52Þ
2
B1 sin 2h1 sin 2h4 ðv1 =v4 Þ cos2 2h4 ¼ ; B0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4
ð53Þ
which are same as those given by Ben-Menahem and Singh [2], if h1, h4, v1 and v4 are replaced by e, f, a and b, respectively. 4. Numerical example To study the above theoretical expressions numerically, we consider an example where Magnesium Crystal like material is modelled as an isotropic generalized thermoelastic material with voids. The following elastic and thermal constants at T0 = 298 K are considered [9]: k ¼ 2:17 1010 N m2 ;
l ¼ 3:278 1010 N m2 ;
cE ¼ 1:04 103 J kg1 degree1 ;
q ¼ 1:74 103 kg m3 ;
K ¼ 1:7 102 W m1 degree1 ;
b ¼ 2:68 106 N m2 degree1 : The following void parameters are also considered for numerical computations v ¼ 1:753 1015 m2 ;
a ¼ 3:688 105 N;
b ¼ 1:13849 1010 N m2 ;
n1 ¼ 1:475 1010 N m2 ;
m ¼ 2 106 N m2 degree2 :
A Fortran program is developed to solve the system of Eq. (49) in terms of reflection coefficients for incidence of P1 as well as SV waves. The variations of these reflection coefficients with the angle of incidence of striking wave are shown graphically in Figs. 2–5 for incident P1 wave and in Figs. 6–9 for incident SV wave.
0.10
Reflection coefficients
Reflected SV wave 0.08
0.06
0.04
0.02
0.00 0
18
36
54
72
90
Angle of incidence Fig. 2. Effects of the presence of voids on reflected SV wave for incidence of P1 wave.
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
1.06
Reflection coefficients
Reflected P1 wave 1.04
1.02
1.00
0.98
0.96 0
18
36
54
72
90
Angle of incidence Fig. 3. Effects of the presence of voids on reflected P1 wave for incidence of P1 wave.
0.008
Reflection coefficients
Reflected P2 wave
0.004
0.000 0
18
36
54
72
90
Angle of incidence Fig. 4. Effects of the presence of voids on reflected P2 wave for incidence of P1 wave.
0.0015 Reflected P3 wave Reflection coefficients
706
0.0012
0.0009
0.0006
0.0003
0.0000 0
18
36
54
72
90
Angle of incidence Fig. 5. The dependence of reflected P3 wave on the angle of incidence of P1 wave.
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
1.2
Reflection coefficients
Reflected SV wave
1.0
0.8 0
9
18
27
36
45
Angle of incidence Fig. 6. Effects of the presence of voids on reflected SV wave for incidence of SV wave.
2.5
Reflection coefficients
Reflected P1 wave 2.0
1.5
1.0
0.5
0.0 0
9
18
27
36
45
Angle of incidence Fig. 7. Effects of the presence of voids on reflected P1 wave for incidence of SV wave.
0.015
Reflection coefficients
Reflected P2 wave 0.012
0.009
0.006
0.003
0.000 0
9
18
27
36
45
Angle of incidence Fig. 8. Effects of the presence of voids on reflected P2 wave for incidence of SV wave.
707
708
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
0.003
Reflection coefficients
Reflected P3 wave
0.002
0.001
0.000 0
9
18
27
36
45
Angle of incidence Fig. 9. The dependence of reflected P3 wave on the angle of incidence of SV wave.
The solid curve in these figures represent the variations of reflection coefficients with voids, whereas the dotted curves represent the variations without voids. All of these reflection coefficients are dependent on the angle of incidence. Figs. 2–5 are plotted for the range 0 < h0 6 90, where as Figs. 6–9 are plotted for the range 0 < h0 6 45. For incident P1 wave, the effect of voids on reflected SV is observed smaller as compared to the effects of voids on reflected P1 and P2 waves. The reflected P3 wave shown in Fig. 5 is purely due to change in volume fraction field, which disappears in absence of voids. For incident SV wave, the critical angle is observed near h0 ¼ 14 . The reflected SV, P1 and P2 waves are affected due to voids. The reflected P3 wave shown in Fig. 9 is again due to change in volume fraction field and will not appear in the absence of voids. 5. Conclusions The correct governing equations of thermoelasticity with voids are derived in context of Lord–Shulman theory. These equations are solved to show the existence of three compressible and one shear wave for a two-dimensional model. One of the compressible wave; namely P3 is due to change in volume fraction field and is slowest wave. A problem of reflection from free surface is considered to obtain the reflection coefficients of various reflected waves. The numerical computations of these reflection coefficients show the effect of voids on various reflected waves. Acknowledgement This work was supported by Indian National Science Academy, New Delhi, under INSA Young Scientist Project (No. BS/YSP/2005 1489). References [1] A.N. Abd-alla, A.A.S. Al-dawy, The reflection phenomena of SV waves in a generalized thermoelastic medium, Int. J. Math. Math. Sci. 23 (2000) 529–546. [2] A. Ben-menahem, S.J. Singh, Seismic Waves and Sources, Springer-Verlag, New York, 1981, p. 89. [3] D.S. Chandresekharaiah, Thermoelasticity with second sound: a review, Appl. Mech. Rev. 39 (1986) 355–376. [4] S. Chirita, A. Scalia, On the spatial and temporal behavior in linear thermoelasticity of materials with voids, J. Thermal Stresses 24 (2001) 433–455. [5] M. Ciarletta, A. Scalia, On the nonlinear theory of nonsimple thermoelastic materials with voids, Z. Angew. Math. Mech. 73 (1993) 7–75.
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[6] M. Ciarletta, E. Scarpetta, Some results on thermoelasticity for dielectric materials with voids, Z. Angew. Math. Mech. 75 (1995) 707– 714. [7] S.D. Cicco, M. Diaco, A theory of thermoelastic materials with voids without energy dissipation, J. Thermal Stresses 25 (2002) 493– 503. [8] S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, J. Elasticity 13 (1983) 125–147. [9] R.S. Dhaliwal, A. Singh, Dynamic Coupled Thermoelasticity, Hindustan Publ. Corp, New Delhi, India, 1980, pp. 726. [10] R.S. Dhaliwal, H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math. 33 (1980) 1–8. [11] R.S. Dhaliwal, J. Wang, A heat-flux dependent theory of thermoelasticity with voids, Acta Mech. 110 (1995) 33–39. [12] W.B. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York, 1957, pp. 76. [13] A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elasticity 2 (1972) 1–7. [14] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses 22 (1999) 451–476. [15] D. Iesan, A theory of thermoelastic materials with voids, Acta Mech. 60 (1986) 67–89. [16] R. Kumar, L. Rani, Response of generalized thermoelastic half-space with voids due to mechanical and thermal sources, Meccanica 39 (2004) 563–584. [17] H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309. [18] M. Marin, An uniqueness result for body with voids in linear thermoelasticity, Rend. Mat. Appl. 17 (7) (1997) 103–113. [19] M. Marin, On the domain of influence in thermoelasticity of bodies with voids, Arch. Math. (Brno) 33 (1997) 301–308. [20] J.W. Nunziato, S.C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal. 72 (1979) 175–201. [21] A. Pompei, A. Scalia, On the asymptotic spatial behavior in linear thermoelasticity of materials with voids, J. Thermal Stresses 25 (2002) 183–193. [22] P. Puri, S.C. Cowin, Plane waves in linear elastic materials with voids, J. Elasticity 15 (1985) 167–183. [23] J.N. Sharma, V. Kumar, D. Chand, Reflection of generalized thermoelastic waves from the boundary of a half-space, J. Thermal Stresses 26 (2003) 925–942. [24] B. Singh, Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion, J. Earth Syst. Sci. 114 (2005) 159–168. [25] A.N. Sinha, S.B. Sinha, Reflection of thermoelastic waves at a solid half-space with thermal relaxation, J. Phys. Earth 22 (1974) 237– 244. [26] S.B. Sinha, K.A. Elsibai, Reflection of thermoelastic waves at a solid half-space with two thermal relaxation times, J. Thermal Stresses 19 (1996) 763–777. [27] S.B. Sinha, K.A. Elsibai, Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two thermal relaxation times, J. Thermal Stresses 20 (1997) 129–146.
Wave propagation in a generalized thermoelastic material with voids Baljeet Singh Department of Mathematics, Government College, Sector-11, Chandigarh 160 011, India
Abstract The governing equations for two-dimensional homogeneous, isotropic generalized thermoelastic half-space with voids are solved in context of Lord–Shulman theory. Three compressional waves and a shear vertical (SV) wave are shown to exist. The reflection phenomena of compressional or shear wave from the free surface of a thermoelastic solid with voids is considered. The boundary conditions at stress-free thermally insulated surface are satisfied to obtain a system of four equations in the reflection coefficients of various reflected waves. These reflection coefficients depend on the angle of incidence of striking wave, void, thermo-void and other elastic parameters. Numerical values of the complex modulus of the reflection coefficients are also visualized graphically to observe the effects of void parameters. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Thermoelasticity; Voids; Compressional; Reflection; Reflection coefficients; Thermal relaxations
1. Introduction Theory of linear elastic materials with voids is an important generalization of the classical theory of elasticity. This theory is used for investigating various types of geological and biological materials for which classical theory of elasticity is not adequate. The theory of linear elastic materials with voids deals the materials with a distribution of small pores or voids, where the volume of void is included among the kinematics variables. This theory reduces to the classical theory in the limiting case of volume of void tending to zero. Nonlinear theory of elastic materials with voids was developed by Nunziato and Cowin [20]. Cowin and Nunziato [8] developed a theory of linear elastic materials with voids to study mathematically the mechanical behavior of porous solids. Puri and Cowin [22] studied the behaviour of plane waves in a linear elastic materials with voids. Iesan [15] developed linear theory of thermoelastic material with voids. Ciarletta and Scalia [5] developed the nonlinear theory of non-simple thermoelastic materials with voids. Ciarletta and Scarpetta [6] studied some results on thermoelasticity for dielectric materials with voids. Dhaliwal and Wang [11] developed a heat flux dependent theory of thermoelasticity with voids. Marin [18,19] studied uniqueness and domain of E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.123
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influence results in thermoelastic bodies with voids. Chirita and Scalia [4] studied the spatial and temporal behavior in linear thermoelasticity of materials with voids. Pompei and Scalia [21] studied the asymptotic spatial behavior in linear thermoelasticity of materials with voids. In the classical theory of thermoelasticity, when an elastic solid is subjected to a thermal disturbance, the effect is felt at a location far from the source, instantaneously. This implies that the thermal wave propagates with infinite speed, a physically impossible result. In contrast to conventional thermoelasticity, non-classical theories came into existence during the last part of 20th century. These theories, referred to as generalized thermoelasticity, were introduced in the literature in an attempt to eliminate the shortcomings of the classical dynamical thermoelasticity. For example, Lord and Shulman [17], by incorporating a flux-rate term into Fourier’s law of heat conduction, formulated a generalized theory which involves a hyperbolic heat transport equation admitting finite speed for thermal signals. Green and Lindsay [13], by including temperature rate among the constitutive variables, developed a temperature-rate-dependent thermoelasticity that does not violate the classical Fourier law of heat conduction, when body under consideration has center of symmetry and this theory also predicts a finite speed for heat propagation. Chandrasekharaiah [3] referred to this wavelike thermal disturbance as ‘‘second sound’’. The Lord and Shulman theory of generalized thermoelasticity was further extended by Dhaliwal and Sherief [10] to include the anisotropic case. A survey article of representative theories in the range of generalized thermoelasticity is due to Hetnarski and Ignaczak [14]. A theory of thermoelastic materials with voids and without energy dissipation is discussed by Cicco and Diaco [7]. Sinha and Sinha [25] and Sinha and Elsibai [26,27] studied the reflection of thermoelastic waves from the free surface of a solid half-space and at the interface between two semi-infinite media in welded contact, in the context of generalized thermoelasticity. Abd-Alla and Al-Dawy [1] studied the reflection phenomena of SV waves in a generalized thermoelastic medium. Sharma et al. [23] investigated the problem of thermoelastic wave reflection from the insulated and isothermal stress-free as well as rigidly fixed boundaries of a solid half-space in the context of different theories of generalized thermoelasticity. Singh [24] discussed the reflection of plane waves from a stress-free surface of an elastic solid with thermodiffusion. The present investigation is motivated by the well-established theories given by Lord and Shulman [17] and Iesan [15]. The paper is organized as follows: in Section 2, the governing equations for homogeneous isotropic generalized thermoelastic solid with voids are obtained. From these equations, a serious error in Eq. (3) of Kumar and Rani [16] is detected for L–S case. The governing equations are then solved for x–z plane to show the existence of a shear and three compressional waves. In Section 3, the boundary conditions at stress-free thermally insulated surface are satisfied to obtain a system of four non-homogeneous equations in reflection coefficients for the incidence of compressional or shear wave at a thermally insulated free surface. A numerical example is given in last section to discuss and visualize the dependence of reflection coefficients upon angle of incidence of striking wave and effects of void parameters on these coefficients and other material parameters. 2. Governing equations and solution Following, Lord and Shulman [17] and Iesan [15], the constitutive equations for isotropic generalized thermoelastic material with voids become: rij ¼ 2leij þ dij ½kekk bH þ bU; qi þ s0 q_i ¼ KH;i ;
ð1Þ ð2Þ
hi ¼ aU;i ;
ð3Þ
qg ¼ bekk þ aH þ mU; g ¼ bekk nU þ mH;
ð4Þ ð5Þ
where k; l are Lame’s constants, H ¼ T T 0 ; T 0 is the temperature of the medium in its natural state assumed to be such that jH=T 0 j 1, T is the absolute temperature, rij are the components of the stress tensor, eij ¼ ðui;j þ uj;i Þ=2, ui are the components of the displacement vector, g is the entropy per unit mass, K is the coefficient of thermal conductivity, s0 is the thermal relaxation time. a; b; n are void material parameters, m is thermo-void coefficient, b ¼ ð3k þ 2lÞat , at is the coefficient of linear thermal expansion, dij is Kronecker
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delta, qi are components of heat flux vector, hi are components of equilibrated stress vector, U is change in volume fraction field, g is the intrinsic equilibrated body force and a is thermal constant. In the absence of heat sources, the energy equation for linear theory of thermoelastic material with voids is ð6Þ
qT 0 g_ ¼ qi;i :
Following, Lord and Shulman [17] and Iesan [15], the field equations in terms of the displacement, volume fraction and temperature, for homogeneous isotropic generalized thermoelastic material with voids in the absence of body forces, heat sources and extrinsic equilibrated body forces are lui;jj þ ðk þ lÞuj;ij bH;i þ bU;i ¼ qu€i ; _ þ s0 HÞ € þ bT 0 ðu_ k;k þ s0 € € ¼ KH;ii ; qcE ðH uk;k Þ þ mT 0 ðU_ þ s0 UÞ
ð7Þ
€ aU;ii buk;k nU þ mH ¼ qvU;
ð9Þ
ð8Þ
The superposed dots denote derivative with respect to time. cE ð¼ aTq 0 Þ is specific heat at constant strain. v is the equilibrated inertia. From Eq. (8), it may be pointed here that Eq. (3) of Kumar and Rani [16] is not a correct L–S generalization. Therefore, the further analysis of Kumar and Rani [16] should be changed accordingly. For x–z plane, Eqs. (7)–(9) become ðk þ 2lÞu1;11 þ ðl þ kÞu3;13 þ lu1;33 bH;1 þ bU;1 ¼ qu€1 ; lu3;11 þ ðl þ kÞu1;13 þ ðk þ 2lÞu3;33 bH;3 þ bU;3 ¼ qu€3 ; _ þ bT 0 su_ 1;1 þ bT 0 su_ 3;3 þ mT 0 sU; _ Kr2 H ¼ qcE sH
ð12Þ
€ ar U nU bðu1;1 þ u3;3 Þ þ mH ¼ qvU;
ð13Þ
2
ð10Þ ð11Þ
where s ¼ 1 þ s0
o ; ot
r2 ¼
o2 o2 þ : ox2 oz2
ð14Þ
The displacement components u1 and u3 may be written in terms of potential functions / and w as u1 ¼
o/ ow ; ox oz
u3 ¼
o/ ow þ : oz ox
ð15Þ
Using (15) into Eqs. (10)–(13) we obtain lr2 w ¼ q
o2 w ; ot2
ð16Þ
o2 / ðk þ 2lÞr2 / bH þ bU ¼ q 2 ; ot oH o oU þ r2 / þ 1 x r2 H ¼ s ; ot ot ot € ar2 U nU þ mH br2 / ¼ qvU;
ð17Þ ð18Þ ð19Þ
where x ¼
K ; qcE
¼
bT 0 ; qcE
1 ¼
mT 0 ; qcE
ð20Þ
Eq. (16) is uncoupled, whereas Eqs. (17)–(19) are coupled in /, H and U. From Eqs. (17)–(19), we see that while the P-wave is affected due to the presence of thermal and void fields, whereas the pSV ffiffiffiffiffiffiffiffiremains unaffected. The solution of Eq. (16) corresponds to the propagation of SV wave with velocity l=q. Solutions of Eqs. (17)–(19) are now sought in the form of the harmonic travelling wave f/; H; Ug ¼ f/0 ; H0 ; U0 geikðx sin hþz cos hvtÞ ;
ð21Þ
where v is the phase speed, k is the wave number and ðsin h; cos hÞ denotes the projection of the wave normal onto x–z plane.
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The homogeneous system of equations in /0 ; H0 ; and U0, obtained by inserting (21) into Eqs. (17)–(19), admits non-trivial solutions and enables to conclude that C satisfies the cubic equation C3 þ LC2 þ MC þ N ¼ 0;
ð22Þ
where C ¼ v2 ; x þ 1 m L ¼ þ c21 þ c3 2 þ b ; s k2 m m x m 2 þ 1 c2 m b b 2 þ bc þ þ b ; M ¼ c21 þ c3 2 þ c21 c3 2 1 3 1 2 s k k k2 k2 x m ; N ¼ c21 c3 2 þ b s k2 and n k þ 2l a b ; c3 2 ¼ þ 2 ; m ¼ ; q qv k qv b b m i ¼ ; m ¼ ; s ¼ s0 þ : b¼ ; b q q qv x
c21 ¼
n n¼ ; qv
Eq. (22) is a cubic equation with complex coefficients. To solve Eq. (22), we calculate Q¼
L2 3M ; 9
R¼
2L3 9LM þ 27N : 54
ð23Þ
If Q and R are real (always true when L, M, N are real) and R2 < Q3 , then cubic Eq. (22) has three real roots. These three roots are u L pffiffiffiffi ð24Þ C1 ¼ 2 Q cos ; 3 3 pffiffiffiffi u þ 2p L ð25Þ C2 ¼ 2 Q cos ; 3 3 pffiffiffiffi u 2p L ð26Þ C3 ¼ 2 Q cos ; 3 3 where R u ¼ arccos pffiffiffiffiffi3ffi Q
! ð27Þ
If Q and R are non-real, then the three roots are L C1 ¼ ðA þ BÞ ; 3 pffiffiffi 1 L 3 ðA BÞ; C2 ¼ ðA þ BÞ þ i 2 3 2 pffiffiffi 1 L 3 ðA BÞ; C3 ¼ ðA þ BÞ i 2 3 2
ð28Þ ð29Þ ð30Þ
where "
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#1=3 A ¼ R þ R2 Q 3 ;
ð31Þ
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where sign of the square root is chosen to make qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re R R2 Q3 P 0;
ð32Þ
asterisk denotes complex conjugation and B¼
Q ðA 6¼ 0Þ; A
B ¼ 0 ðA ¼ 0Þ:
ð33Þ
The squared velocities C1, C2 and C3 are dominated by displacement, volume fraction and temperature fields, respectively, and show the existence of three compressional waves. Let us name these waves as P1, P2 and P3, respectively, for displacement, temperature displacement and volume fraction fields. When we solve Eq. (22) by using a computer program, it is also observed that P1 wave is fastest wave whereas P3 is slowest wave. If we neglect void effects, i.e. for m ¼ 0; b ¼ 0; a ¼ 0; 1 ¼ 0; n ¼ 0; c23 ¼ 0; c3 2 ¼ 0, the cubic equation (22) reduces to a quadratic equation whose roots are as 2v2 ¼ d 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 21 4c21 x ;
ð34Þ
where d1 ¼
x þ c21 þ b; s
ð35Þ
and the positive and negative signs correspond to P1 wave and P2 wave, respectively. Similarly, if we neglect the thermal effects, i.e. for b ¼ 0; x ¼ 0; K ¼ 0; ¼ 0; m ¼ 0; Eq. (22) reduces to a quadratic equation whose roots are as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 2 2 2 ð36Þ 2v ¼ d 2 d 2 4 c1 c3 þ b 2 ; k where d 2 ¼ c21 þ c3 2
ð37Þ
and the positive and negative signs correspond to P1 wave and P3 waves, respectively. In absence of thermal and void effects, Eqs. (34) and (36) reduce to sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ 2l ; ð38Þ v¼ q which is the velocity of classical longitudinal wave. 3. Reflection coefficients In the previous section, it has been discussed that there exists a shear and three compressional waves in an isotropic generalized thermoelastic material with voids. Any incident wave at the interface between two elastic solid bodies, in general, produce compressional and distortional waves in both media (see for example, Ewing et al. [12]; Ben-Menahem and Singh [2]). Let us consider an incident P1 or SV wave (Fig. 1). The boundary conditions at the free surface z ¼ 0 are satisfied, if the incident plane wave (P1 or SV) gives rise to a reflected SV and three reflected compressional waves (i.e. P1, P2 and P3 waves). The surface z ¼ 0 is assumed to be traction free and thermally insulated so that there is no variation of temperature and volume fraction on it. Therefore, the boundary conditions on z ¼ 0 are written as rzz ¼ 0;
rzx ¼ 0;
oH ¼ 0; oz
oU ¼ 0; oz
on z ¼ 0
ð39Þ
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
703
z z > 0 ( Vaccum)
x Thermoelastic Half-space with voids
θ0 θ 4 θ1 θ2 θ3
A3 A2
A0 or B0
A1 B1
Fig. 1. Schematic diagram for the problem (for incident P1 wave h0 = h1, for incident SV wave h0 = h4).
The appropriate displacement potentials / and w, temperature H and volume fraction U are taken in the form w ¼ B0 exp½ik 4 ðx sin h0 þ z cos h0 Þ ixt þ B1 exp½ik 4 ðx sin h4 z cos h4 Þ ixt;
ð40Þ
/ ¼ A0 exp½ik 1 ðx sin h0 þ z cos h0 Þ ixt þ A1 exp½ik 1 ðx sin h1 z cos h1 Þ ixt þ A2 exp½ik 2 ðx sin h2 z cos h2 Þ ixt þ A3 exp½ik 3 ðx sin h3 z cos h3 Þ ixt; H ¼ f1 A0 exp½ik 1 ðx sin h0 þ z cos h0 Þ ixt þ f1 A1 exp½ik 1 ðx sin h1 z cos h1 Þ ixt
ð41Þ
þ f2 A2 exp½ik 2 ðx sin h2 z cos h2 Þ ixt þ f3 A3 exp½ik 3 ðx sin h3 z cos h3 Þ ixt; U ¼ g1 A0 exp½ik 1 ðx sin h0 þ z cos h0 Þ ixt þ g1 A1 exp½ik 1 ðx sin h1 z cos h1 Þ ixt
ð42Þ
þ g2 A2 exp½ik 2 ðx sin h2 z cos h2 Þ ixt þ g3 A3 exp½ik 3 ðx sin h3 z cos h3 Þ ixt;
ð43Þ
where the wave normal of the incident P1 or SV wave makes angle h0 with the positive direction of the z-axis, and those of reflected P1, P2, P3 and SV waves make h1 ; h2 ; h3 and h4 with the same direction, and k 2 ðv2 c2 Þ ði ¼ 1; 2; 3Þ; fi ¼ i i 1 b bGi k 2 ðv2 c2 Þ ði ¼ 1; 2; 3Þ; gi ¼ Gi i i 1 b bGi
ð44Þ ð45Þ
where
Gi ¼
b ðv2i c21 Þð1 sx v2 Þ i
b 1 ðv2i c21 Þ þ
:
ð46Þ
The ratios of the amplitudes of the reflected waves to the amplitude of the incident P1 wave, namely B1 =A0 ; A1 =A0 ; A2 =A0 and A3 =A0 give the reflection coefficients for reflected SV, reflected P1, reflected P2 and reflected P3 waves, respectively. Similarly for incident SV wave, B1 =B0 ; A1 =B0 ; A2 =B0 and A3 =B0 are the reflection coefficients for reflected SV, reflected P1, reflected P2 and reflected P3 waves, respectively. The wave number k 1 ; k 2 ; k 3 ; k 4 and the angles h1 ; h2 ; h3 ; h4 are connected by the relation k 1 sin h1 ¼ k 2 sin h2 ¼ k 3 sin h3 ¼ k 4 sin h4 ;
ð47Þ
at surface z ¼ 0. Relation (47) may also be written in order to satisfy the boundary conditions (39) as sin h1 sin h2 sin h3 sin h4 ¼ ¼ ¼ ; v1 v2 v3 v4
ð48Þ
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B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
pffiffiffiffiffiffiffiffi where v4 ¼ l=q is the velocity of SV wave and vi ; ði ¼ 1; 2; 3Þ are the velocities of three kinds of reflected compressional waves. Using the potentials given by (40)–(43) in boundary conditions Eq. (39), we obtain a system of four non-homogeneous equations Raij Z j ¼ bi ; where
ði; j ¼ 1; 2; . . . 4Þ;
ð49Þ
"
a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33
# 2 f1 g1 k 1 ¼ k þ 2l cos h1 þ b 2 b 2 ; l k1 k1 " # 2 f2 g2 k 2 2 ¼ k þ 2l cos h2 þ b 2 b 2 ; l k2 k2 " # 2 f3 g3 k 3 2 ¼ k þ 2l cos h3 þ b 2 b 2 ; l k3 k3 2 k4 ¼ l sin 2h4 ; l 2 k1 ¼ sin 2h1 ; l 2 k2 ¼ sin 2h2 ; l 2 k3 ¼ sin 2h3 ; l 2 k4 ¼ cos 2h4 ; l 3 f k1 ¼ cos h1 12 ; k1 l 3 f2 k 2 ¼ cos h2 2 ; k2 l 3 f k3 ¼ cos h3 32 ; k3 l 2
a34 ¼ 0;
3 g1 k 1 ; k 21 l 3 g2 k 2 a42 ¼ cos h2 2 ; k2 l 3 g k3 a43 ¼ cos h3 23 ; k3 l a41 ¼ cos h1
a44 ¼ 0: For incident P1 wave b1 ¼ a11 ;
b2 ¼ a21 ;
b3 ¼ a31 ;
b4 ¼ a41 ;
l ¼ k1:
b3 ¼ a34 ;
b4 ¼ a44 ;
l ¼ k4
For incident SV wave b1 ¼ a14 ;
b2 ¼ a24 ;
and Z 1 ; Z 2 ; Z 3 ; Z 4 are reflection coefficients of reflected P1, P2, P3 and SV waves, respectively.
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
705
In the absence of thermal and void parameters, these reflection coefficients reduce to For incident P wave 2
A1 sin 2h1 sin 2h4 ðv1 =v4 Þ cos2 2h4 ¼ ; A0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4 B1 2ðv1 =v4 Þ sin 2h1 cos 2h4 ¼ : A0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4
ð50Þ ð51Þ
For incident SV wave A1 ðv1 =v4 Þ sin 4h4 ¼ ; B0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4
ð52Þ
2
B1 sin 2h1 sin 2h4 ðv1 =v4 Þ cos2 2h4 ¼ ; B0 sin 2h1 sin 2h4 þ ðv1 =v4 Þ2 cos2 2h4
ð53Þ
which are same as those given by Ben-Menahem and Singh [2], if h1, h4, v1 and v4 are replaced by e, f, a and b, respectively. 4. Numerical example To study the above theoretical expressions numerically, we consider an example where Magnesium Crystal like material is modelled as an isotropic generalized thermoelastic material with voids. The following elastic and thermal constants at T0 = 298 K are considered [9]: k ¼ 2:17 1010 N m2 ;
l ¼ 3:278 1010 N m2 ;
cE ¼ 1:04 103 J kg1 degree1 ;
q ¼ 1:74 103 kg m3 ;
K ¼ 1:7 102 W m1 degree1 ;
b ¼ 2:68 106 N m2 degree1 : The following void parameters are also considered for numerical computations v ¼ 1:753 1015 m2 ;
a ¼ 3:688 105 N;
b ¼ 1:13849 1010 N m2 ;
n1 ¼ 1:475 1010 N m2 ;
m ¼ 2 106 N m2 degree2 :
A Fortran program is developed to solve the system of Eq. (49) in terms of reflection coefficients for incidence of P1 as well as SV waves. The variations of these reflection coefficients with the angle of incidence of striking wave are shown graphically in Figs. 2–5 for incident P1 wave and in Figs. 6–9 for incident SV wave.
0.10
Reflection coefficients
Reflected SV wave 0.08
0.06
0.04
0.02
0.00 0
18
36
54
72
90
Angle of incidence Fig. 2. Effects of the presence of voids on reflected SV wave for incidence of P1 wave.
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
1.06
Reflection coefficients
Reflected P1 wave 1.04
1.02
1.00
0.98
0.96 0
18
36
54
72
90
Angle of incidence Fig. 3. Effects of the presence of voids on reflected P1 wave for incidence of P1 wave.
0.008
Reflection coefficients
Reflected P2 wave
0.004
0.000 0
18
36
54
72
90
Angle of incidence Fig. 4. Effects of the presence of voids on reflected P2 wave for incidence of P1 wave.
0.0015 Reflected P3 wave Reflection coefficients
706
0.0012
0.0009
0.0006
0.0003
0.0000 0
18
36
54
72
90
Angle of incidence Fig. 5. The dependence of reflected P3 wave on the angle of incidence of P1 wave.
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
1.2
Reflection coefficients
Reflected SV wave
1.0
0.8 0
9
18
27
36
45
Angle of incidence Fig. 6. Effects of the presence of voids on reflected SV wave for incidence of SV wave.
2.5
Reflection coefficients
Reflected P1 wave 2.0
1.5
1.0
0.5
0.0 0
9
18
27
36
45
Angle of incidence Fig. 7. Effects of the presence of voids on reflected P1 wave for incidence of SV wave.
0.015
Reflection coefficients
Reflected P2 wave 0.012
0.009
0.006
0.003
0.000 0
9
18
27
36
45
Angle of incidence Fig. 8. Effects of the presence of voids on reflected P2 wave for incidence of SV wave.
707
708
B. Singh / Applied Mathematics and Computation 189 (2007) 698–709
0.003
Reflection coefficients
Reflected P3 wave
0.002
0.001
0.000 0
9
18
27
36
45
Angle of incidence Fig. 9. The dependence of reflected P3 wave on the angle of incidence of SV wave.
The solid curve in these figures represent the variations of reflection coefficients with voids, whereas the dotted curves represent the variations without voids. All of these reflection coefficients are dependent on the angle of incidence. Figs. 2–5 are plotted for the range 0 < h0 6 90, where as Figs. 6–9 are plotted for the range 0 < h0 6 45. For incident P1 wave, the effect of voids on reflected SV is observed smaller as compared to the effects of voids on reflected P1 and P2 waves. The reflected P3 wave shown in Fig. 5 is purely due to change in volume fraction field, which disappears in absence of voids. For incident SV wave, the critical angle is observed near h0 ¼ 14 . The reflected SV, P1 and P2 waves are affected due to voids. The reflected P3 wave shown in Fig. 9 is again due to change in volume fraction field and will not appear in the absence of voids. 5. Conclusions The correct governing equations of thermoelasticity with voids are derived in context of Lord–Shulman theory. These equations are solved to show the existence of three compressible and one shear wave for a two-dimensional model. One of the compressible wave; namely P3 is due to change in volume fraction field and is slowest wave. A problem of reflection from free surface is considered to obtain the reflection coefficients of various reflected waves. The numerical computations of these reflection coefficients show the effect of voids on various reflected waves. Acknowledgement This work was supported by Indian National Science Academy, New Delhi, under INSA Young Scientist Project (No. BS/YSP/2005 1489). References [1] A.N. Abd-alla, A.A.S. Al-dawy, The reflection phenomena of SV waves in a generalized thermoelastic medium, Int. J. Math. Math. Sci. 23 (2000) 529–546. [2] A. Ben-menahem, S.J. Singh, Seismic Waves and Sources, Springer-Verlag, New York, 1981, p. 89. [3] D.S. Chandresekharaiah, Thermoelasticity with second sound: a review, Appl. Mech. Rev. 39 (1986) 355–376. [4] S. Chirita, A. Scalia, On the spatial and temporal behavior in linear thermoelasticity of materials with voids, J. Thermal Stresses 24 (2001) 433–455. [5] M. Ciarletta, A. Scalia, On the nonlinear theory of nonsimple thermoelastic materials with voids, Z. Angew. Math. Mech. 73 (1993) 7–75.
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[6] M. Ciarletta, E. Scarpetta, Some results on thermoelasticity for dielectric materials with voids, Z. Angew. Math. Mech. 75 (1995) 707– 714. [7] S.D. Cicco, M. Diaco, A theory of thermoelastic materials with voids without energy dissipation, J. Thermal Stresses 25 (2002) 493– 503. [8] S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, J. Elasticity 13 (1983) 125–147. [9] R.S. Dhaliwal, A. Singh, Dynamic Coupled Thermoelasticity, Hindustan Publ. Corp, New Delhi, India, 1980, pp. 726. [10] R.S. Dhaliwal, H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math. 33 (1980) 1–8. [11] R.S. Dhaliwal, J. Wang, A heat-flux dependent theory of thermoelasticity with voids, Acta Mech. 110 (1995) 33–39. [12] W.B. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York, 1957, pp. 76. [13] A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elasticity 2 (1972) 1–7. [14] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses 22 (1999) 451–476. [15] D. Iesan, A theory of thermoelastic materials with voids, Acta Mech. 60 (1986) 67–89. [16] R. Kumar, L. Rani, Response of generalized thermoelastic half-space with voids due to mechanical and thermal sources, Meccanica 39 (2004) 563–584. [17] H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967) 299–309. [18] M. Marin, An uniqueness result for body with voids in linear thermoelasticity, Rend. Mat. Appl. 17 (7) (1997) 103–113. [19] M. Marin, On the domain of influence in thermoelasticity of bodies with voids, Arch. Math. (Brno) 33 (1997) 301–308. [20] J.W. Nunziato, S.C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal. 72 (1979) 175–201. [21] A. Pompei, A. Scalia, On the asymptotic spatial behavior in linear thermoelasticity of materials with voids, J. Thermal Stresses 25 (2002) 183–193. [22] P. Puri, S.C. Cowin, Plane waves in linear elastic materials with voids, J. Elasticity 15 (1985) 167–183. [23] J.N. Sharma, V. Kumar, D. Chand, Reflection of generalized thermoelastic waves from the boundary of a half-space, J. Thermal Stresses 26 (2003) 925–942. [24] B. Singh, Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion, J. Earth Syst. Sci. 114 (2005) 159–168. [25] A.N. Sinha, S.B. Sinha, Reflection of thermoelastic waves at a solid half-space with thermal relaxation, J. Phys. Earth 22 (1974) 237– 244. [26] S.B. Sinha, K.A. Elsibai, Reflection of thermoelastic waves at a solid half-space with two thermal relaxation times, J. Thermal Stresses 19 (1996) 763–777. [27] S.B. Sinha, K.A. Elsibai, Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two thermal relaxation times, J. Thermal Stresses 20 (1997) 129–146.