REFLECTION AND TRANSMISSION OF PLANE WAVES AT AN INTERFACE BETWEEN ELASTIC AND MICROPOLAR THERMOELASTIC DIFFUSION MEDIA

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 20, Number 3, Fall 2012 REFLECTION AND TRANSMISSION OF PLANE WAVES AT AN INTERFACE BETWEEN ELASTIC AND MICROPOLAR THERMOELASTIC DIFFUSION MEDIA RAJNEESH KUMAR AND VIJAY CHAWLA ABSTRACT. The present investigation is concerned with the study of reflection and transmission phenomenon due to longitudinal and transverse waves at a plane interface between isotropic elastic solid half-space and micopolar thermoelastic diffusion half space. It is found that reflections and transmission coefficients are functions of angle of incidence, frequency of the incident wave and are influenced by the elastic properties of the media. Closed form expressions of amplitudes ratio and energy ratios are derived. Numerical computations have been performed for a particular model and the obtained results are depicted graphically against the angle of incident. It is verified that during transmission there is no dissipation of energy at the interface. From the present investigation, a special case of interest is also deduced to depict the effect of micropolarity. 1 Introduction The classical theory deals with the basic assumption that the effect of the microstructural of a material is not essential for describing mechanical behavior. In reality, almost all materials posses microstructure and in such materials, microstrctural motions (instrinsic rotations of grain) cannot be ignored at high frequency. Eringen [9] developed the linear theory of micropolar elasticity which takes into account the microstructural motions. The difference between classical theory of elasticity and that of micropolar elasticity is the introduction of an additional kinematic variable corresponding to microrotation. Tomar and Gogna [29] investigated the problem of reflection and refraction of longitudinal wave at an interface between two micropolar elastic solids in welded contact. Tomar and Kumar [30] discussed wave propagation at micropolar elastic solid interface. The Generalized theory of ther- Keywords: Micropolar thermoelastic diffusion, amplitudes, reflection, transmission, energy ratio. Copyright c Applied Mathematics Institute, University of Alberta. 375

2 376 R. KUMAR AND V. CHAWLA moelasticity is one of the modified version of classical uncoupled and coupled theory of thermoelasticity and have been developed in order to remove the paradox of physical impossible phenomena of infinite velocity of thermal signals in the classical coupled thermoelasticity. Lord and Shulman [17] formulated a generalized theory of thermoelasticity with one thermal relaxation time, who obtained a wave-type equation by postulating a new law of heat conduction instead of classical Fouriers law. Green and Lindsay [10] developed a temperature rate-dependent thermoelasticity that includes two thermal relaxation times and does not violate the classical Fouriers law of heat conduction, when the body under consideration has a center of symmetry. One can refer to Hetnarski and Ignaczak [11] for a review and presentation of generalized theories of thermoelasticity. The linear theory of micropolar elasticity has been extanded to include thermal effects by Eringen [7, 8] and Nowacki [18, 19, 20]. Boschi and Ieasan [5] extanded a generalized theory of micropolar thermoelasticity. When thermal effects are considered, Sinha and Elsibai [27] discussed the reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two relaxation times. Sinha and Sinha [28] investigated the problem of reflection of thermoelastic waves at a solid half space with relaxation times. Kumar and Sarthi [12] discussed the reflection and refraction of thermoelastic plane waves at an interface of two thermoelastic media without energy dissipation. Diffusion is defined as the spontaneous movement of the particles from a high concentration region to the low concentration region and it occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas to accomplish isotope separation. Today, thermal diffusion remains a practical process to separate isotopes of noble gases (e.g., xexon) and other light isotopes (e.g., carbon) for research purposes. Nowacki [21 24] developed the theory of thermoelastic diffusion by using coupled thermoelastic model. Sherief et al. [25] proved the uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion, in isotropic media. Sherief and Saleh [26] investigated the problem of a thermoelastic half-space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. Auodi [2] proved the uniqueness and reciprocity theorems for the generalized problem in anisotropic media, under the restriction that the elastic, thermal conductivity and diffusion tensors are positive definite. Recently, Auodi [13] derived the uniqueness and reciprocity theorems for the general-

3 REFLECTION AND TRANSMISSION OF PLANE WAVES 377 ized micropolar thermoelastic diffusion problem in anisotropic media. Kumar and Kansal [3] derived the basic equations for generalized thermoelastic diffusion and discussed the Lamb waves. Kumar and Chawla [14] discussed the plane wave propagation in anisotropic three-phaselag model. Kumar and Chawla [15] derived the Green s functions for two-dimensional problem in orthotropic thermoelastic diffusion media. However, the important reflection and transmission at an interface between an elastic solid medium and micropolar thermoelastic diffusion solid medium has not been discussed so far and also verified the law of conservation of energy at the interface. Keeping in view of the above applications, we have investigated the problem of reflection and transmission at a plane interface between an elastic solid medium and a micropolar thermoelastic diffusion solid medium. The graphical representation is given for these energy ratios for different direction of propagation. The law of conservation of energy at the interface is verified. 2 Basic equations Following Aouadi [3] and Kumar et al. [16], the basic equations for homogeneous isotropic generalized micropolar thermoelastic diffusion, in the absence of body forces, body couple, heat and mass diffusion sources are (1) (2) (3) (4) ( (λ + µ) ( u) + (µ + K) 2 u + K ϕ β τ 1 t ( β τ 1 ) C = ρ 2 u t t 2, ) T (α + β + γ) ( ϕ) γ ϕ + K u 2Kϕ = ρj 2 ϕ t 2, ( ρc E t + τ 2 ) ( 0 t 2 T + β 1 T 0 t + Ωτ 2 ) 0 t 2 u ( ) + at 0 t + ε 2 t 2 C = K 2 T, ( ) ( ) Dβ 2 2 u + Da 1 + τ T + + Ωτ t t t 2 C ( Db 1 + τ 1 t ) 2 C = 0, (5) t ij = λu r,r δ ij + µ(u i,j + u j,i ) + K(u j,i ε ijr φ r )

4 378 R. KUMAR AND V. CHAWLA (6) ( ( T β 1 T + τ 1 )δ ij β 2 C + τ 1 C ) δ ij, t t m ij = αφ r,r + βφ i,j + γφ j,i, where β 1 = (3λ + 2µ + K)α t, β 2 = (3λ + 2µ + K)α c, α t and α c are, respectively, the coefficients of linear thermal expansion and diffusion expansion. λ, µ, α, β, γ and K are material constants, ρ is the density, C E is the specific heat at constant strain, j is the microinertia density, K is the coefficient of thermal conductivity, u = (u 1, u 2, u 3 ) is the displacement vector and ϕ = (φ 1, φ 2, φ 3 ) is the microrotation vector, T = Θ T 0 is the small temperature increment, Θ is the absolue temperature of the medium, T 0 is the reference temperature of the body chosen such that T/T 0 1, C is the concentration of the diffusive material in the elastic body, t ij are the components of stress tensor, K is the coefficient of thermal conductivity, D is the thermoelastic diffusion constant, and 2 are, respectively, the gradient and Laplacian operators, δ ij is Kronecker delta, τ 0 and τ 1 are thermal relaxation times with τ 1 τ 0 0, and τ 0 and τ 1 are diffusion relaxation times with τ 1 τ 0 0. Here, τ 1 = τ 1 = 0, Ω = 1, ε = τ 0 for Lord- Shulman (L-S) model and Ω = 0, ε = τ 0 for Green-Lindsay model. In the above equations, the symbol ( ) followed by a suffix denotes differentiation with respect to spatial coordinate. The equations of motion in homogeneous isotropic elastic solid medium are (7) (λ e + µ e ) ( u e ) + µ 2 u e = ρ e 2 u e t 2, where λ e and µ e are Lame s constants, u e = (u e 1, ue 2, ue 3 ) and ρe are, respectively, the displacement vector and density corresponding to the isotropic elastic solid. The stress-strain relation in the isotropic elastic medium is given by (8) σ e ij = λ e e e k,kδ ij + 2µ e e e ij, where σij e and ee ij = 1 2 (ue i,j + ue j,i ) are, respectively, components of stress and strain tensor, and e e k,k is the dilatation. 3 Formulation of the problem We consider an isotropic elastic solid half-space (medium I) lying over a homogenous isotropic, generalized micropolar thermoelastic diffusion solid half-space (medium II).

5 REFLECTION AND TRANSMISSION OF PLANE WAVES 379 The origin of the cartesian coordinate system (x 1, x 2, x 3 ) is taken at any point on the plane surface and the x 3 -axis points vertically downwards into the micropolar thermoelastic diffusion solid half-space. The elastic solid half-space occupies the region x 3 0 and the region x 3 0 is occupied by the dissipative micropolar thermoelastic diffusion solid half-space as shown in Figure 1. FIGURE 1: Geometry of the problem. We consider plane waves in the x 1 x 3 -plane with wave front parallel to the x 2 -axis. For the two-dimensional problem we assume the displacement vector u e in medium I and the displacement vector u and microrotation vector ϕ in medium II are, respectively, of the form (9) u e = (u e 1, 0, u e 3), u = (u 1, 0, u 3 ), ϕ = (0, φ 2, 0). We define the following nondimensional quantities x i = x i ν 1, u i = ρ ν 1 β 1 T 0 u i, φ 2 = ρν2 1 β 1 T 0 φ 2, T = T T 0,

6 380 R. KUMAR AND V. CHAWLA C = β 2C β 1 T 0, t = t, τ 1 = τ 1, τ 0 = τ 0, τ 0 = τ 0, τ 1 = τ 1, t ij = 1 β 1 T 0 t ij, m ij = ν 1 β 1 T 0 m ij, Pij = ρν 1 β1 2T Pij, P e = ρν 1 0 β1 2T P e, 0 where u e i = ρ ν 1 u e i β 1 T, 0 (10) = ρc Eν1 2 K, ν1 2 = λ + 2µ + K. ρ te ij = 1 t e ij β 1 T, 0 Making use of equation (10) in equations (1) (4), with the aid of (9) and after suppressing the primes, we obtain (11) (12) (13) (14) (15) δ 1 (u 1,11 + u 1,33 ) + δ 2 (u 1,11 + u 3,13 ) δ 1 (u 3,11 + u 3,33 ) + δ 2 (u 1,31 + u 3,33 ) δ 3 φ 2,3 τ 1 t T,1 τ 1 c C,1 = ü 1, δ 3 φ 2,1 τ 1 t T,3 τ 1 c C,3 = ü 3, 2 φ 2 + a 1 (u 1,3 u 3,1 ) 2a 1 φ 2 = a 2 φ2, ζ 1 τ 0 tt( u 1,1 + u 3,3 ) + τ 0 t T + ζ 2 τ 0 c Ċ = 2 T, q 1 (u 1,111 + u 3,311 + u 1,133 + u 3,333 ) + q 2 τ 1 t (T,11 + T,33 ) q 3τ 1 c (C,11 + C,33 ) + τ 0 ccċ = 0, where (δ 1, δ 2, δ 3 ) = 1 ρν 2 1 (a 1, a 2 ) = ν2 1 γ (µ + K, λ + µ, K), ( K 2, ρ j (a 3, a 4 ) = 1 µ + K (ρν2 1, K), ( (τt 1, τ c 1, τ t 0, τ c 0, τ tt 0, τ cc 0 ) = 1 + τ 1 t, 1 + τ 1 t, 1 + τ 0 t, ), 1 + ε t, 1 + Ωτ 0 t, 1 + Ωτ 0 t ),

7 REFLECTION AND TRANSMISSION OF PLANE WAVES 381 ( (q1, q 2, q 3 ) = D β 2 2 ν1 2 ρν1 2, β 2a (ζ 1, ζ 2 ) = β ( 1T 0 β1 K ρ, aν2 1 β 2 ), b, β 1 We introduce the potential functions φ and ψ through the relations ). (16) u 1 = φ x 1 ψ x 3, u 3 = φ x 3 + ψ x 1. Substituting equation (16) in equations (11) (15), we obtain ( (17) 2 2 t 2 ( 2 2 (18) a 3 t 2 ( 2 2 (19) 2a 1 a 2 t 2 ( (20) 2 τt 0 (21) where ) φ τ 1 t T τ 1 c C = 0, ) ψ a 4 φ 2 = 0, ) φ 2 + a 1 2 ψ = 0, ) T ζ 2 τc 0 t Ċ ζ 1τtt 0 2 φ = 0, q 1 4 φ + q 2 τ 1 t 2 T q 3 τ 1 c 2 C + τ 0 ccċ = 0, For harmonic motion, we assume 2 = 2 x x 2. 3 (22) {φ, ψ, φ 2, T, C}(x 1, x 3, t) = {φ, ψ, φ 2, T, C}e it, where is the angular frequency of the vibrations of material particles. Substituting equation (22) in equations (17) (21), we obtain (23) ( )φ τt 11 T τc 11 C = 0, (24) ( 2 + a3 2 )ψ a4φ2 = 0, (25) (26) ( 2 2a 1 + a 2 2 )φ 2 + a 1 2 ψ = 0, ζ 1 τ 0 tt 2 φ + ( 2 τ 10 t )T ζ 2 τ 10 c C = 0,

8 382 R. KUMAR AND V. CHAWLA (27) q 1 4 φ + q 2τ 11 t 2 T [q 3τ 11 c 2 τ 10 cc ]C = 0, where (τ 11 t, τ 11 c, τ 10 t, τ 10 c, τ 10 tt, τ 10 cc ) = (1 iτ 1, 1 iτ 1, 1 iτ 0, i(1 iε), i(1 iτ 0 ), i(1 iωτ 0 )). Equations (26) and (27) of this system are solved into two relations, given by (28) (29) [{q 3 τ 11 c ζ 1τ 10 tt + q 1 τ 10 c ζ 2} 4 ζ 1 τ 10 tt τ 10 cc 2 ]φ = [q 3 τ 11 c 4 {q 3 τ 11 c τ 10 t + τ 10 cc + q 2 τ 11 t τ 10 c ζ 2} 2 + τ 10 t τ 10 cc ]T, [q {[q 2τ 11 t ζ 1 τ 10 tt q 1τ 10 t ] 4 }φ = [q 3 τ 11 c 4 {q 3 τ 11 c τ 10 t + τ 10 cc + q 2 τ 11 t τ 10 c ζ 2} 2 + τ 10 t τ 10 cc ]C. Making use of equations (28) and (29) in equation (23), we have (30) A A A A 4 φ = 0, where A 1 = (q1 q 11 3 )]τc, A 2 = τcc 10 + (q 1 + q 10 2 )τc τ t 11 ζ 2 + (q2 + q 10 3 )τtt τ t 11 τc 11 ζ 1 A 3 = ( 2 τ 10 t ζ 1 τ 10 tt τ 11 t )τ 10 cc + q 2ζ 2 τ 11 t τ 10 c + 2 q 3τ 11 c τ 10 t, A 4 = 2 τ 10 cc τ 10 tt. Substituting the value of φ 2 from equation (25) into equation (24), we obtain (31) 4 + H H 2 ψ = 0, where H 1 = a 1 a 4 2a 1 + (a 2 + a 3 ) 2, H 2 = (a 2 2 2a 1 ) 2. The general solution of equation (30) can be written as (32) φ = 3 φ i, i=1

9 REFLECTION AND TRANSMISSION OF PLANE WAVES 383 where the potentials φ i, i = 1, 2, 3 are solutions of wave equations, given by (33) ] [ Vi 2 φ i = 0, i = 1, 2, 3. Here, Vi 2 (i = 1, 2, 3) are the velocities of three longitudinal waves, that is, longitudinal displacement wave, MD (mass diffusion) and T (thermal) waves and derived from the roots of cubic equation in V 2, given by (34) A 4 V 6 A 3 2 V 4 + A 2 4 V 2 A 4 6 = 0. The general solution of equation (31) can be written as (35) ψ = 2 ψ i, i=1 where the potentials ψ i, i = 4, 5 are solutions of wave equations, given by (36) ] [ Vi 2 ψ i = 0, i = 4, 5. Here, Vi 2 (i = 4, 5) are the velocities of two transverse displacement waves coupled with transverse microrotation (CD I, CD II) and are derived from the roots of the equation in V 2, given by (37) H 2 V 4 H 1 2 V = 0. Making use of equation (32) in the equations (28), (29) with the aid of equations (22) and (33), the general solutions for φ, T and C are obtained as (38) {φ, T, C} = where 3 {1, r 1i, r 2i }φ i, i=1 r 1i = (q 3 τ 11 c ζ 1τ 10 tt + q 1 τ 10 c ζ 2) 4 + ζ 1 τ 10 tt τ 10 cc 2 V 2 i p 1i,

10 384 R. KUMAR AND V. CHAWLA and r 2i = q (q2 τ t 11 ζ 1 τtt 10 q 1 τ t 10 ) 4 Vi 2 Vi 2p, ij p ij = q3 τ c (q3 τ c 11 τ t 10 +τcc 10 +q 2 τ t 11 τc 10 ζ 2) 2 Vi τt τcc 10 V i 4, i = 1, 2, 3. Substituting equation (35) in equations (25) (26) with the aid of equations (22) and (36), the general solutions for ψ and φ 2 are obtained as (39) (40) {ψ, φ 2 } = p 1i = 5 {1, p 1i }ψ i, i=4 a 1 2 (a 2 2a 1 )Vi 2, i = 4, 5. 2 Applying the dimensionless quantities defined by (10) in equation (7) and with the aid of equation (9) after suppressing the primes, we obtain (41) (42) η e2 χ e2 ν 2 1 η e2 χ e2 ν 2 1 [u e 1,11 + u e 3,13] + χe2 ν1 2 [u e 1,11 + u e 1,33] = ue 1 t 2, [u e 1,13 + ue 3,33 ] + χe2 ν1 2 [u e 3,11 + ue 3,33 ] = ue 3 t 2, λ e +2µ e µ e where η e = ρ and χ e = e ρ are, respectively, the velocities of e longitudinal and transverse waves of the medium I. The components u e 1 and ue 3 are related by the potential functions as (43) u 1 = φe x 1 ψe x 3, u 3 = φe x 3 + ψe x 1, where φ e and ψ e are solutions of wave equations (44) 2 φ e = φ η 2, 2 ψ e = ψ e χ 2, where η = ηe ν 1, χ = χe ν 1.

11 REFLECTION AND TRANSMISSION OF PLANE WAVES Reflection and transmission We consider a plane wave (P or SV) propagating through the isotropic elastic solid half-space and is incident at the interface x 3 = 0, as shown in Figure 1. Corresponding to this incident wave, two homogeneous waves (P and SV) are reflected in isotropic elastic solid half space and five inhomogeneous waves (Longitudinal displacement waves, MD, T, CD I and CD II) are transmitted in isotropic micropolar thermoelastic diffusion solid half-space. The potential functions satisfying equation (44) can be written as (45) φ e = A e 0e i{(x1 sin θ0+x3 cos θ0)/η t} + A e 1 ei{(x1 sin θ1+x3 cos θ1)/η t}, (46) ψ e = B e 0 ei{(x1 sin θ0+x3 cos θ0)/χ t} + B e 1e i{(x1 sin θ1+x3 cos θ1)/χ t}. The coefficients A e 0, B e 0, A e 1 and B e 1 represent the amplitude of the incident P (or SV), reflected P and reflected SV waves, respectively. Following Borcherdt [4], in an isotropic micropolar thermoelastic diffusion half-space, the potential functions satisfying equations (33) and (36) can be written as (47) (48) {φ, T, C} = {ψ, φ 2 } = 3 {1, r 1i, r 2i }B i e ( A i, r ) e i( P i, r t), i=1 5 {1, p 1i, }B i e ( A i, r ) e i( P i, r t). i=4 The coefficients B i, i = 1, 2, 3, 4, 5, represent the amplitudes of transmitted waves. The propagation vector P i, i = 1, 2, 3, 4, 5 and attenuation factor A i, i = 1, 2, 3, 4, 5 are given by (49) P i = ξ Rˆx 1 + dv ir ˆx 3, A i = ξ 1ˆx 1 dv ii ˆx 3, i = 1, 2, 3, 4, 5, where (50) dv i = dv ir + idv ii = p.v. 2 Vi 2 ξ 2, i = 1, 2, 3, 4, 5. and ξ = ξ R + iξ I is the complex wave number. The subscripts R and I denote the real and imaginary parts of the corresponding complex

12 386 R. KUMAR AND V. CHAWLA number and p.v. stands for the principal value of the complex quantity obtained from square root. ξ R 0 ensures propagation in positive x 1 -direction. The complex wave number ξ in the isotropic micropolar thermoelastic diffusion medium is given by (51) ξ = P i sin θ i i A i sin(θ i γ i), i = 1, 2, 3, 4, 5, where γ i, i = 1, 2, 3, 4, 5 is the angle between the propagation and attenuation vector and, θ i, i = 1, 2, 3, 4, 5, is the angle of refraction in medium II. 5 Boundary conditions The boundary conditions to be satisfied at the interface x 3 = 0 as follows: (I) continuity of Stress components (52a) (52b) t e 33 = t 33, t e 31 = t 31 ; (II) continuity of displacement components (52c) (52d) u e 3 = u 3, u e 1 = u 1 ; (III) vanishing of the tangential couple stress components (52e) m 32 = 0; (IV) thermally insulated boundary (52f) T x 3 = 0; (V) impermeable boundary (52g) C x 3 = 0.

13 REFLECTION AND TRANSMISSION OF PLANE WAVES 387 Making use of equations (45) (48) in equations (52a) (52g), we find that the boundary conditions are satisfied if and only if (53) ξ R = sin θ 0 V 0 = sin θ 1 η = sin θ 2 χ, and (54) ξ 1 = 0, where (55) V 0 = { η, χ, for incident P -wave for incident SV -wave. It means that waves are attenuating only in x 3 -direction. From equation (51), it implies that if A i 0, then γ i = θ i, i = 1, 2, 3, 4, 5, that is, attenuated vectors for the five transmitted waves are directed along the x 3 -axis. Substituting equations (45) (48) in equations (52a) (52g) and with the aid of the equations (5), (6), (8), (9), (10), (16), (43), (53) and (54), we get a system of seventh nonhomogeneous equations which can be written as (56) 7 f ij Z j = d j, j=1 where Z j = Z j e iψ j, Zj and ψj, j = 1, 2, 3, 4, 5, are respectively, the ratios of amplitudes and phase shift of reflected P-, reflected SVtransmitted longitudinal displacement wave, MD, T, CDI and CDII waves to that of incident wave. f 11 = 2µ e ( ξr ) 2 ρ e ν 2 1, f 12 = 2µ e ξ R dv β, f 16 = (2µ + K) ξ R dv β, f 17 = (2µ + K) ξ R dv β, f 21 = 2µ e ξ [( ) 2 ( ) 2 ] R dv η, f 22 = µ e dvχ ξr, [ ( ) 2 ( ) 2 ξr dv4 f 26 = µ (µ + K) Kp 14],

14 388 R. KUMAR AND V. CHAWLA and f 27 = [ µ ( ξr ) 2 (µ + K) ( dv5 ) 2 Kp 15], f 31 = ξ R, f dvχ 32 =, f 36 = dv 4, f 37 = dv 5, f 41 = dv η, f 42 = ξ R, f 46 = ξ R, f 47 = ξ R, f 51 = 0, f 52 = 0, f 56 = 0, f 57 = 0, f 61 = 0, f 62 = 0, f 66 = 0, f 67 = 0, f 71 = 0, f 72 = 0, f 76 = 0, f 77 = 0, f 2j = 2µ ξ R dv j f 5j = r 1j, dv η = 1 η 2 dv χ = dv j, f 3j = ξ R, f dv j 6j = r 2j, ( ) 2 ξr = 1 χ 2 sin2 θ 0 V0 2, dv j = p.v. 1 V 2 j f 4j = dv j, f 6j = p 1j dv j, 1 η 2 sin2 θ 0 V0 2, sin2 θ 0 V 2 0, j = 1, 2, 3, 4, 5. Here, p.v. is evaluated with restriction dv ji 0 to satisfy decay condition in the micropolar thermoelastic diffusion medium. The coefficients d i, i = 1, 2,..., 7 on the right side of equation (56) are given by (I) for incident P-wave (57) d i = ( 1) i f i1 for i = 1, 2, 3, 4 and d i = 0 for i = 5, 6, 7; (II) for incident SV-wave (58) d i = ( 1) i+1 f i2 for i = 1, 2, 3, 4 and d i = 0 for i = 5, 6, 7. We consider a surface element of unit area at the interface between two media. The reason for this consideration is to calculate the partition of energy of the incident wave among the reflected and refracted waves on the both sides of surface. Following Achenbach [1], the energy flux across

15 REFLECTION AND TRANSMISSION OF PLANE WAVES 389 the surface element, that is, rate at which the energy is communicated per unit area of the surface is represented as (59) P = t qr q r u q, where t qr are the components of stress tensor, q r are the direction cosines of the unit normal ˆq outward to the surface element and u q are the components of the particle velocity. The time average of P over a period, denoted by P, represents the average energy transmission per unit surface area per unit time. Thus, on the surface with normal along the x 3 -direction, the average energy intensities of the waves in the elastic solid are given by (60) P e = Re t e 31Re ( u e 1) + Re t e 33 Re ( u e 3). Following Achenbach [1], for any two complex functions f and g, we have (61) Re (f) Re (g) = 1 Re (f g). 2 The expressions for the energy ratios E i (i = 1, 2) for the reflected P and reflected SV are given by (62) E i = P i e P0 e, i = 1, 2, where and (I) for incident P-wave P1 e = 4 ρ e ν1 2 η Z 1 2 Re (cos θ 1 ), P2 e = 4 ρ e ν1 2 χ Z 2 2 Re (cos θ 2 ), (63) P0 e = 4 ρ e ν1 2 η cos θ 0, (II) for incident SV-wave (64) P 0 e = 4 ρ e ν 2 1 χ cos θ 0,

16 390 R. KUMAR AND V. CHAWLA are the average energy intensities of the reflected P-, reflected SV-, incident P- and incident SV-waves, respectively. In equation (62), negative sign is taken because the direction of reflected waves is opposite to that of incident waves. For micropolar thermoelastic diffusion solid half-space, the average energy intensities of the waves on the surface with normal along x 3 - direction, are given by (65) P ij = Re t (i) 31 Re ( u(j) 1 ) + Re t (i) 33 Re ( u(j) 3 ) + Re m (i) 32 Re ( φ (j) 2 ). The expressions for the energy ratios E ij, i, j = 1, 2, 3, 4, 5 for the transmitted waves are given by (66) E ij = P ij P0 e, i, j = 1, 2,..., 5, where [ Pij = 4 Re (2µ + K) dv i + ρν 2 1 ( dvi [ Pi4 = 4 Re (2µ + K) dv i ( + ρν1 2 dvi [[ ( P4j = ξr 4 Re µ ] ξr ξ R + { λ ( ) 2 ξr ξ R ) 2 + ρν2 1 (r 1iτt 11 + r 2i τc 11) 2 ξ R ξ R + { λ ) 2 + ρν2 1 (r 1iτ 11 t + r 2i τ 11 c ) 2 ) 2 (µ + K) 2 1 ( dv4 dv j ( ) 2 ξr ) 2 + Kp (2µ + K)ξ R dv 4 dv j [{( ( ) 2 ( P5j = ξr dvj 4 Re µ (µ + K) + (2µ + K) ξ R dv 4 [{( ( ) 2 P4j = 4 dv4 Re (µ + K) µ ξ R ], ) 2 ) ξr }Z i+2 Z j+2 ], }Z i+2 Z 6 ], } dv j Z 7 Z j+2, i, j = 1, 2, 3, ( ξr ) 2 Kp ) 14 dv j 2

17 REFLECTION AND TRANSMISSION OF PLANE WAVES 391 }Z 6 Z j+2 ], + (2µ + K) ξ R dv 4 dξ R + γ 2 dv 4 ρν1 42 p 14p 1j [{( ( ) 2 ( ) 2 P5j = dv5 ξr 4 Re (µ + K) µ Kp ) 15 dv j 2 + (2µ + K) ξ R dv 5 dξ R + γ 2 dv 5 ρν1 42 p 15p 1j = 4, 5. }Z 6 Z j+2 ], The diagonal entries of energy matrix E ij in equation (66) represent the energy ratios of the longitudinal displacement wave, MD, T, CD I and CD II waves, whereas sum of the nondiagonal entries of E ij gives the share of interaction energy among all the transmitted waves in the medium and is given by (67) E RR = 5 ( 5 E ij E ii ). i=1 j=1 The energy ratios E i, i = 1, 2, diagonal entries and sum of nondiagonal entries of the energy matrix E ij, that is, E 11, E 22, E 33 E 44, E 55 and E RR yield the conservation of the incident energy across the interface, through the relation E 1 + E 2 + E 11 + E 22 + E 33 + E 44 + E 55 + E RR = 1. Special case: In the absence of micropolarity effect, that is, if we take K = 0 in equations (56) and (62), we obtain corresponding expressions amplitude and energy ratios of reflected P-, reflected SV-, refracted P-, refracted T-, refracted C-, and refracted SV-waves to that of incident wave. In these expressions, the velocities are derived from equation (34) and is the velocity of the transverse wave and coupling constant given by equation (38). 6 Numerical results and discussion With the view of illustrating theoretical results obtained in the preceding sections, we now present some numerical results; the values of the following relevant parameters are given as λ = Kgm 1 s 2, µ = Kgm 1 s 2,

18 392 R. KUMAR AND V. CHAWLA T 0 = K, C E = JKg 1 K 1, α t = K 1, α C = Kg 1 m 3, a = m 2 s 2 K 1, b = m 5 s 2 Kg 1, D = m 3 skg, ρ = Kgm 3, K = Wm 1 K 1, K = 10 dyn/cm 2, j = 0.2 cm 2, γ = 7.79 dyn, and the values of relaxation times are τ 0 = 0.4 s, τ 1 = 0.9 s, τ 0 = 0.5 s, τ 1 = 0.8 s. Following Bullen [6], the numerical data of granite for elastic medium is given by ρ e = Kgm 3, η e = ms 1, χ e = ms 1. The Matlab software 7.04 has been used to determine the values of energy ratios E i, i = 1, 2 and energy matrix E ij, i, j = 1,..., 5 defined in the previous section for different values of incident angle (θ 0 ) ranging from 0 to 90 for fixed frequency = 2 π 100 Hz. Corresponding to incident P, the variation of energy ratios with respect to angle of incident have been plotted in Figures (2) (9). Similarly, corresponding to SV waves, the variation of energy ratios with respect to angle of incident have been plotted in Figures (10) (17). In all figures, micropolar thermoelastic diffusion is represented by the word MTED and TDE correspond to the thermoelastic diffusion. Incident P-wave: Figure 2 exhibits the variation of energy ratio E 1 with the angle of incidence (θ 0 ). It shows that the values of E 1 for both cases TED and MTED decrease with the increase in θ 0 from 0 to 80 and then increase as θ 0 increases further. Figure 3 depicts the variation of energy ratio E 2 with θ 0 and it indicates that the values of E 2 increase for smaller values of θ 0, whereas for higher values of θ 0, it increase for both cases TED and MTED. Figure 4 depicts the variation of energy ratio E 11 with θ 0 and it shows that the values of E 11 for the case of MTED increase slightly for smaller values of θ 0 whereas for the higher values of θ 0 the values of E 11 decreases, but for the case TED the values of E 11 decrease for all values of θ 0. Figure 5 exhibits the variation of energy ratio E 22 with θ 0. The similar type of behavior and

19 REFLECTION AND TRANSMISSION OF PLANE WAVES 393 variation is noticed for E 22 as E 11 with difference in their magnitude values. Figure 6 depicts the variation of energy ratio E 33 with θ 0 and it indicates the values of E 33 for the case of TED show an oscillating behavior in the initial stage, but after that it decreases, whereas for the case of MTED, the values of E 33 slightly increase for smaller values of θ 0 although for higher values of θ 0 the values of E 33 decrease. Figure 7 depicts the variation of energy ratio E 44 with θ 0. It shows that the values of E 44 for both cases TED and MTED increase with the increase θ 0 from 0 to 75 and then decrease as θ 0 increase further. Figure 8 exhibits the variation of energy ratio E 55 with θ 0 and it indicates the values E 55 of slightly increase at the initial stage, but after that it decrease. Figure 9 depicts the variation of interaction energy ratio E RR with θ 0 and it indicates the values of E RR for the case of MTED decrease, although for the case of TED, the values of E RR decreases for smaller values of θ 0, whereas for higher values of θ 0, the values of E RR increase. If we compare TED and MTED in all figures, we find that the values of E 1, E 33 and E 44 are higher in TED theory in comparison to MTED, but the values of E 2, E 11, E 22 and E RR are higher in MTED theory (for higher values of θ 0 ) in comparison to TED theory. FIGURE 2: Variation of energy ratio E 1 with respect to the angle of incident P -wave.

20 394 R. KUMAR AND V. CHAWLA FIGURE 3: Variation of energy ratio E 2 with respect to the angle of incident P -wave. FIGURE 4: Variation of energy ratio E 11 with respect to the angle of incident P -wave.

21 REFLECTION AND TRANSMISSION OF PLANE WAVES 395 FIGURE 5: Variation of energy ratio E 22 with respect to the angle of incident P -wave. FIGURE 6: Variation of energy ratio E 33 with respect to the angle of incident P -wave.

22 396 R. KUMAR AND V. CHAWLA FIGURE 7: Variation of energy ratio E 44 with respect to the angle of incident P -wave. FIGURE 8: Variation of energy ratio E 55 with respect to the angle of incident P -wave.

23 REFLECTION AND TRANSMISSION OF PLANE WAVES 397 FIGURE 9: Variation of energy ratio E RR with respect to the angle of incident P -wave. Incident SV-wave: Figures depict the variation of energy ratios with the angle of incidence (θ 0 ) for SV waves. Figure 10 represents the variation of energy ratio E 1 with θ 0 and it indicates that the values of E 1 for both cases TED and MTED increase for smaller values of θ 0, whereas for higher values of θ 0, the values of E 1 decrease. Figure 11 shows the variation of energy ratio E 2 with θ 0. The values of E 2 for both cases TED and MTED decrease with the increase of θ 0 from 0 to 30 and then increase as θ 0 increases further. Figure 12 shows that the values of E 11 for both cases TED and MTED show an oscillatory behavior for initial values of θ 0, whereas for higher values of θ 0, the values of E 11 decrease. Figure 13 exhibits the variation of energy ratio E 22 with θ 0, and it is noticed that the behavior and variation of E 22 is similar to E 11 with a difference in their magnitude values. Figure 14 exhibits that the values of E 33 for the case of MTED oscillate, but for the case of TED, the values of E 33 increase slightly. Figure 15 shows the variation of energy ratio E 44 with θ 0, and it found that the behavior and variation of E 44 is similar to E 33 with difference in their magnitude values. Figure 16 indicates that the values of E 55 increase at the initial stage but after that, it decrease. Figure 17 represents the variation of E RR with θ 0, and it shows that the values of E RR for both cases TED and MTED initially decrease, but for higher values of θ 0, the values of E RR decrease for the case of MTED, but for the case of TED, the values slightly increase. If we compare TED and MTED in all figures, we find

24 398 R. KUMAR AND V. CHAWLA that the values of E 1, E 2, E 33 and E 44 are higher in TED theory in comparison to MTED whereas the values of E RR are higher in MTED theory (for higher values of θ 0 ) compared to TED theory. FIGURE 10: Variation of energy ratio E 1 with respect to the angle of incident SV -wave. FIGURE 11: Variation of energy ratio E 2 with respect to the angle of incident SV -wave.

25 REFLECTION AND TRANSMISSION OF PLANE WAVES 399 FIGURE 12: Variation of energy ratio E 11 with respect to the angle of incident SV -wave. FIGURE 13: Variation of energy ratio E 22 with respect to the angle of incident SV -wave.

26 400 R. KUMAR AND V. CHAWLA FIGURE 14: Variation of energy ratio E 33 with respect to the angle of incident SV -wave. FIGURE 15: Variation of energy ratio E 44 with respect to the angle of incident SV -wave.

27 REFLECTION AND TRANSMISSION OF PLANE WAVES 401 FIGURE 16: Variation of energy ratio E 55 with respect to the angle of incident SV -wave. FIGURE 17: Variation of energy ratio E RR with respect to the angle of incident SV -wave.

28 402 R. KUMAR AND V. CHAWLA 7 Conclusion The phenomena of reflection and refraction of obliquely incident elastic waves at the interface between an elastic solid half space and a micropolar thermoelastic diffusion solid half space has been investigated. The expressions of amplitudes ratio and energy ratios are presented in compact form. The variations of energy ratios against the angle of incident are computed numerically and depicted graphically for a specific model. The law of conservation is verified at the interface. From the present investigation, a special case of interest is also deduced to depict the effect of micropolarity. Appreciable relaxation time and micropolarity effect are noticed on energy ratios. From the numerical results, we conclude that the values of E 1, E 33 and E 44 are higher in TED theory in comparison to MTED whereas the values of E 2, E 11, E 22 and E RR are higher in MTED theory (for higher values of θ 0 ) in comparison to TED theory in the case of incident P- wave, although for the case of incident SV-wave we find that the values of E 1, E 2, E 33 and E 44 are higher in TED theory in comparison to MTED whereas the values E RR are higher in MTED theory (for higher values of θ 0 ) compared to TED theory. The sum of energy ratios of the reflected waves, transmitted waves and interface between transmitted waves for both incident P- and SV-waves is verified to be unity, which shows the law of conservation of energy at the interface. REFERENCES 1. J. D. Achenbach, Wave Propagation in Elastic Solids, North Holland, Amsterdam, M. Auodi, Generalized theory of thermoelastic diffusion for anisotropic media, J. Thermal Stress. 31 (2008), M. Auodi, Theory of generalized micropolar thermoelastic diffusion under Lord Shulman model, J. Thermal Stresses 32 (2009), R. D. Borcherdt, Reflection refraction of general P and type-1 S waves in elastic and anelastic solids, Geophy. J. Roy. Astron. Soc. 70 (1982), E. Boschi and D. Isean, A generalized theory of linear micropolar thermoelasticity, Mechanica 8 (1973), K. E. Bullen, An Introduction to the Theory of Seismology, Cambridge University Press, England, A. C. Eringen, Foundations of micropolar thermoelasticity, Int. Centre for Mech. Sci., Course and Lectures, no. 23, Springer, Berlin, A. C. Eringen, Microcontinuum field theory I: Foundations and Solids, Springer- Verlag, Berlin. 9. A. C. Eringen, Linear theory of micropolar elasticity, J. Math. Mech. 16 (1966) A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity 2 (1972), 1 7.

29 REFLECTION AND TRANSMISSION OF PLANE WAVES R. B. Hetnarski and J. Ignaczak, Soliton-like waves in a low temperature nonlinear thermoelastic solid, Int. J. Eng. Sci. 34 (1996), R. Kumar and P. Sarthi, Reflection and refraction of thermoelastic plane waves at an interface between two thermoelastic media without energy dissipation, 58 (2006), R. Kumar and T. Kansal, Propagation of Lamb waves in transversely isotropic thermoelastic diffusive plate, Int. J. Solids Struct. 45 (2008), R. Kumar and V. Chawla, A study of plane wave propagation in anisotropic three-phase lag and two-phase-lag model, Inter. Comm. Heat Mass Transfer 38 (2011), R. Kumar and V. Chawla, Green s functions in orthotropic thermoelastic diffusion media, Eng. Anal. Boundary Elements 36 (2012), R. Kumar, S. Kausal and A. Miglani, Disturbance due to concentrated source in micropolar thermodiffusive medium, J. Mech. Eng. Sci. 225 (2011), H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solid 15 (1967), W. Nowacki, Couple stress in the theory of thermoelasticity I, Bull. Pol. Acad. Sci. Ser. Sci. Tech. 14 (1966), W. Nowacki, Couple stress in the theory of thermoelasticity II, Bull. Pol. Acad. Sci. Ser. Sci. Tech. 14 (1966), W. Nowacki, Couple stress in the theory of thermoelasticity III, Bull. Pol. Acad. Sci. Ser. Sci. Tech. 14 (1966), W. Nowacki, Dynamical problems of thermodiffusion in solid I, Bull. Pol. Acad. Sci. Ser. Sci. Tech. 22 (1974), W. Nowacki, Dynamical problems of thermodiffusion in solid II, Bull. Pol. Acad. Sci. Ser. Sci. Tech. 22 (1974), W. Nowacki, Dynamical problems of thermodiffusion in solid III, Bull. Pol. Acad. Sci. Ser. Sci. Tech. 14 (1974), W. Nowacki, Dynamical problems of thermodiffusion in solid, Eng. Fract. Mech. 8 (1976), H. H. Sherief, F. A. Hamza and H. A. Saleh, The theory of generalized Thermoelastic diffusion, Int. J. Eng. Sci. 42 (2004), H. H. Sherief and H. Saleh, A half space problem in the theory of generalized thermo-elastic diffusion, Int. J. Solids Struct. 42 (2005), S. B. Sinha and K. A. Elsibai, Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two relaxation times, J. Thermal Stress. 20 (1997), A. N. Sinha and S. B. Sinha, Reflection of thermoelastic waves at a solid half space with thermal relaxation, Phys. Earth 22 (1974), S. K. Tomar and M. L. Gogna, Reflection and refraction of longitudinal waves at an interface between two micropolar elastic media in welded contact, J. Acoust. Soc. Amer. (1995), S. K. Tomar and R. Kumar, Wave propagation at liquid/micropolar elastic solid interface, J. Sound Vibr. 222 (1999), Department of Mathematics, Kurukshetra University, Kurukshetra , Haryana, India address: rajneesh kuk@rediffmail.com address: vijay1 kuk@rediffmail.com

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