Wave propagation in a microstretch thermoelastic diffusion solid

Wave propagation in a microstretch thermoelastic diffusion solid

Rajneesh Kumar

Abstract

The present article deals with the two parts: (i)The propagation of plane waves in a microstretch thermoelastic diffusion solid of infinite extent. (ii)The reflection and transmission of plane waves at a plane interface between inviscid fluid half-space and micropolar thermoelas- tic diffusion solid half-space.It is found that for two-dimensional model, there exist four coupled longitudinal waves, that is, longitudinal dis- placement wave (LD), thermal wave (T), mass diffusion wave (MD) and longitudinal microstretch wave (LM) and two coupled transverse waves namely (CD-I and CD-II waves). The phase velocity, attenuation co- efficient, specific loss and penetration depth are computed numerically and depicted graphically. In the second part, it is noticed that the am- plitude ratios of various reflected and transmitted waves are functions of angle of incidence, frequency of incident wave and are influenced by the microstretch thermoelastic diffusion properties of the media. The expressions of amplitude ratios and energy ratios are obtained in closed form. The energy ratios have been computed numerically for a partic- ular model. The variations of energy ratios with angle of incidence for thermoelastic diffusion media in the context of Lord-Shulman (L-S) [1]

and Green-Lindsay (G-L) [2] theories are depicted graphically. Some particular cases are also deduced from the present investigation.

Key Words: Microstretch , Phase velocity, Attenuation coefficient, Specific loss, Pene- tration depth, Reflection, Transmission, Energy ratios.

Received: 28 April, 2014.

Revised: 20 June, 2014.

Accepted: 28 June, 2014.

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1 Introduction

Eringen [3,4] developed the theory of micromorphic bodies.The theory of mi- crostretch elastic solids developed by Eringen [5] is a generalization of the mi- cropolar theory. Eringen [6] also developed the theory of thermomicrostretch elastic solids. The particles of microstretch materials have seven degree of freedom: three for displacements, three for microrotations and one for mi- crostretch. A microstretch continuum can be model as composite materials reinforced with chopped elastic fibres and various porous solids.The mate- rial points of these bodies can stretch and contract independently of their translations and rotations. A book by Eringen [7] gave an exposition of the development in the microcontinuum field theories for solids (micromorphic, microstretch, and microplar) including electromagnetic and thermal interac- tions.

Various investigators have studied different types of problems in microstretch thermoelastic medium notable among them are Ciarletta and Scalia [8], Iesan and Quintanilla [9], Othman et al [10], Passarella and Tibullo [11], Marin [12, 13], Kumar et al [14], Othman and Lofty [15, 16], Kumar and Rupender [17, 18].

Singh [19] studied the reflection and refraction of plane waves at a liquid/

thermo-microstretch elastic solid interface. Kumar and Pratap [20] discussed the reflection of plane waves in a heat flux dependent microstretch thermoe- lastic solid half space. Propagation of Rayleigh surface waves in microstretch thermoelastic continua under inviscid fluid loading have been investigated by Sharma et al. [21]. The propagation of free vibrations in microstretch ther- moelastic homogeneous, isotropic, thermally conducting plate bordered with layers of inviscid liquid on both sides subjected to stress free thermally insu- lated and isothermal conditions have been investigated by Kumar and Pratap [22].

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 concentra- tion 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. In most of the applications, the concentration is calculated using what is known as Fick’s law. This is a simple law which does not take into consideration the mutual interaction between the introduced substance and the medium into which it is introduced or the effect of temperature on this interaction. However, there is a certain degree of coupling with temperature and temperature gradients as

temperature speeds up the diffusion process. The thermodiffusion in elastic solids is due to coupling of fields of temperature, mass diffusion and that of strain in addition to heat and mass exchange with the environment.

Nowacki [23-26] developed the theory of thermoelastic diffusion by using coupled thermoelastic model. Dudziak and Kowalski [27] and Olesiak and Pyryev [28], respectively, discussed the theory of thermodiffusion and cou- pled quasi-stationary problems of thermal diffusion for an elastic layer. They studied the influence of cross effects arising from the coupling of the fields of temperature, mass diffusion and strain due to which the thermal excitation results in additional mass concentration and that generates additional fields of temperature.

Gawinecki and Szymaniec [29] proved a theorem about global existence of the solution for a non-linear parabolic thermoelastic diffusion. problem. Unique- ness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem, in isotropic media, was proved by Sherief et al. [30] on the basis of the variational principle equations, under restrictive assumptions on the elastic coefficients. Due to the inherit complexity of the derivation of the variational principle equations, Aouadi [31] proved this theorem in the Laplace transform domain, under the assumption that the functions of the problem are continuous and the inverse Laplace transform of each is also unique. Sherief and Saleh [32] investigated the problem of a thermoelastic half-space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. Miglani and Kaushal [33] studied the propagation of transverse and microrotational waves in micropolar generalized thermodiffu- sion elastic half space. Kumar and Kansal [34] developed the basic equation of anisotropic thermoelastic diffusion based upon Green-Lindsay model. Kumar and Kansal [35] investigated the fundamental solution in thermomicrostretch elastic diffusive solids.

The Propagation of plane waves at the interface of an elastic solid half- space and a microstretch thermoelastic diffusion solid half-space was studied by Kumar, Garg and Ahuja [36]. They also discussed the Rayleigh wave propagation in isotropic microstrech thermoelastic diffusion solid half- space [37].

For the boundary value problem considered in the context of dipolar bod- ies with stretch, in the paper [41], the authors use some results from the theory of semigroups of the linear operators in order to prove the existence and uniqueness of a weak solution, for the initial boundary value problem of a porous thermoelastic body, the authors analyze the temporal behaviour of the solutions [43]. In the paper [42] the problem of reflection and transmission of plane waves at an imperfect boundary between two thermally conducting micropolar elastic solid half spaces with two temperature is investigated. In

the paper [ 44] the authors establish some existence results of the problem at resonance under some appropriate conditions.

In the first part of the present paper, the propagation of plane waves in an microstretch generalized thermoelastic diffusion solid have been investigated.

The phase velocity, attenuation coefficient, specific loss and penetration depth have been computed numerically and depicted graphically. In the second part, the reflection and refraction phenomenon at a plane interface between an in- viscid fluid medium and a microstretch thermoelastic diffusion solid medium has been analyzed. In microstretch thermoelastic diffusion solid medium, po- tential functions are introduced to the equations. The amplitude ratios of various reflected and transmitted waves to that of incident wave are derived.

These amplitude ratios are further used to find the expressions of energy ra- tios of various reflected and refracted waves to that of incident wave. 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 Eringen [7], Sherief et al. [30] and Kumar and Kansal [34], the equations of motion and the constitutive relations in a homogeneous isotropic microstretch thermoelastic diffusion solid in the absence of body forces, body couples, stretch force, and heat sources are given by

(λ+ 2µ+K)∇(∇ · −→u)−(µ+K)∇ × ∇ × −→u +K∇ × −→ϕ+λ0∇ϕ^∗

−β1(1 +τ1

∂

∂t)∇T−β2(1 +τ¹∂

∂t)∇C=ρ∂²−→u

∂t² , (1)

(α+β+γ)∇(∇ · −→ϕ)−γ∇ × ∇ × −→ϕ +K∇ × −→u −2K−→ϕ =ρj∂²−→ϕ

∂t² , (2) α0∇²ϕ^∗+ν1(1 +τ1

∂

∂t)T+ν2(1 +τ¹ ∂

∂t)C−λ1ϕ^∗−λ0∇ · −→u = ρj0

2

∂²ϕ^∗

∂t² , (3) K^∗∇²T =β₁T₀(1 +ετ₀∂

∂t)∇ ·−→u˙ +ν₁T₀(1 +ετ₀ ∂

∂t) ˙ϕ^∗ +ρC^∗(1 +τ₀∂

∂t) ˙T+aT₀(1 +γ₁∂

∂t)∂C

∂t, (4)

Dβ2ekk,ii+Dν2ϕ^∗_,ii+Da(1+τ1

∂

∂t)T,ii+(1+ετ⁰ ∂

∂t)∂C

∂t −Db(1+τ¹ ∂

∂t)C,ii= 0, (5)

and constitutive equations are

tij =λur,rδij+µ(ui,j+uj,i) +K(uj,i−eijrϕr) +λ0δijϕ^∗

−β1(1 +τ1

∂

∂t)T δij−β2(1 +τ¹∂

∂t)Cδij, (6)

mij=αϕr,rδij+βϕi,j+γϕj,i+b0emjiϕ^∗_,m, (7) λ^∗_i =α0ϕ^∗_,i+b0eijmϕj,m, (8) where λ, µ, α, β, γ, K, λ0, λ1, α0, b0, are material constants,ρis the mass den- sity, −→u = (u₁, u₂, u₃) is the displacement vector and −→ϕ = (ϕ₁, ϕ₂, ϕ₃) is the microrotation vector,ϕ^∗ is the microstretch scalar function,T andT₀ are the small temperature increment and the reference temperature of the body chosen such that kT /T₀| << 1,C is the concentration of the diffusion ma- terial in the elastic body, K^∗ is the coefficient of the thermal conductivity, C^∗ is the specific heat at constant strain;D is the thermoelatic diffusion con- stant, a, b are, respectively, coefficients describing the measure of thermoe- lastic diffusion effects and of mass diffusion effects,β1 = (3λ+ 2µ+K)αt1, β2 = (3λ+ 2µ+K)αc1,ν1 = (3λ+ 2µ+K)αt2, ν2 = (3λ+ 2µ+K)αc2; αt1, αt2are coefficients of linear thermal expansion andαc1, αc2are coefficients of linear diffusion expansion,j is the microinertia,j0is the microinertia of the microelements,σij andmij are components of stress and couple stress tensors respectively,λ^∗_i is the microstress tensor,eij = (ui,j+uj,i)/2 are components of infinitesimal strain,ekkis the dilatation,δij is the Kronecker delta,τ⁰, τ¹are diffusion relaxation times withτ¹≥τ⁰≥0 and τ₀, τ₁ are thermal relaxation times withτ₁≥τ₀≥0. Hereτ₀=τ⁰=τ₁=τ¹= 0 for Coupled Thermoelas- ticity(CT) model,τ₁ =τ¹ = 0, ε= 1, γ₁ =τ₀ for Lord-Shulman(L-S) model andε= 0, γ₁=τ⁰ for Green-Lindsay(G-L) model.

In the above equations, a comma followed by a suffix denotes spatial derivative and a superposed dot denotes the derivative with respect to time respectively.

For two-dimensional problem, we have

−

→u = (u1, u2, u3),−→ϕ = (0, ϕ2,0). (9) We define the following dimensionless quantities

(x⁰1, x⁰3) = w^∗ c1

(x1, x3),(u⁰1, u⁰3) =ρc1w^∗ β1T0

(u1, u3), t⁰ij= t⁰_ij β1T0

, t⁰=w^∗t, τ0⁰ =w^∗τ0,

τ₁⁰=w^∗τ1, τ⁰

0

=w^∗τ⁰, τ¹

0

=w^∗τ¹, T⁰= T T0

, C⁰ =β2C

ρc²₁ ,(u^f₁⁰, u^f₃⁰) =ρc1w^∗ β1T0

(u^f₁, u^f₃),

ϕ^∗0= ρc²₁ β1T0

ϕ^∗, λ^∗_i⁰ = w^∗ c1β1T0

λ^∗_i, ϕ2⁰ = ρc²₁ β1T0

ϕ2, ϕ^f0= w^∗

c²₁ϕ^f, p^f0 = 1 β1T0

p^f, m⁰ij= w^∗

c1β1T0

mij, Pij^∗0 = ρc1

β₁²T₀²Pij^∗, P^∗f0= ρc1

β²₁T₀²P^∗f. (10) where w^∗ = ^ρC

∗c²₁

K^∗ , c²₁ = ^λ+2µ+K_ρ , w^∗ is the characteristic frequency of the medium.

Upon introducing the quantities (1.10) in equations (1.1)-(1.5), with the aid of (1.9) and after suppressing the primes, we obtain

δ² ∂e

∂x₁ + (1−δ²)∇²u1−ζ₁^∗∂ϕ2

∂x₃ +ζ₃^∗∂ϕ^∗

∂x₁ −τ_t¹∂T

∂x₁ −ζ₂^∗τ_c¹∂C

∂x₁ = ∂²u1

∂t² , (11) δ² ∂e

∂x3

+ (1−δ²)∇²u₃+ζ₁^∗∂ϕ₂

∂x1

+ζ₃^∗∂ϕ^∗

∂x3

−τ_t¹∂T

∂x3

−ζ₂^∗τ_c¹∂C

∂x3

= ∂²u₃

∂t² , (12) ζ1∇²ϕ2+ζ2(∂u₁

∂x3

−∂u₃

∂x1

)−ζ3ϕ2= ∂²ϕ₂

∂t² , (13)

(δ₁²∇²−χ^∗₁)ϕ^∗−χ^∗₂e+χ^∗₃τ_t¹T+χ^∗₄τ_c¹C=∂²ϕ^∗

∂t² , (14)

∇²T =l^∗₁τ_e⁰∂e

∂t +l^∗₂τ_e⁰∂ϕ^∗

∂t +τ_t⁰∂T

∂t +l₃^∗τ_c⁰∂C

∂t, (15)

q₁^∗∇²e+q^∗₄∇²ϕ^∗+q₂^∗τ_t¹∇²T+τ_f⁰∂C

∂t −q^∗₃τ_c¹∇²C= 0, (16) where,

ζ1= γ

jρc²₁, ζ2= K

jρw^∗2, ζ3= 2K

jρw^∗2, ζ₁^∗= K

ρc²₁, ζ₂^∗= ρc²₁ β1T0

, ζ₃^∗= λ₀ ρc²₁,

δ²=λ+µ

ρc²₁ , l₁^∗= T0β₁²

ρK^∗w^∗, l^∗₂= T0β1ν1

ρK^∗w^∗, l^∗₃= ρc⁴₁a

β₂K^∗w^∗, q^∗₁=Dw^∗β²₁

ρc⁴₁ , q₂^∗= Dw^∗β2a β₁c²₁ ,

q^∗₃=Dbw^∗

c²₁ , q^∗₄=Dν₂β₂w^∗

ρc⁴₁ , χ^∗₁= 2λ

ρj0w^∗2, χ^∗₂= 2λ₀

ρj0w^∗2, χ^∗₃= 2ν₁c²₁

j0β1w^∗2, χ^∗₄= 2ν₂ρc⁴₁ j0β1β2T0w^∗2,

δ₁²=c²₂

c²₁, c²₂= 2α0

ρj₀, τ_t¹= 1 +τ₁∂

∂t, τ_c¹= 1 +τ¹∂

∂t, τ_f⁰= 1 +ετ⁰∂

∂t, τ_t⁰= 1 +τ₀∂

∂t,

τ_e⁰= 1 +ετ₀∂

∂t, τ_c⁰= 1 +γ₁∂

∂t, e=∂u₁

∂x1

+∂u₃

∂x3

,∇²= ∂²

∂x²₁+ ∂²

∂x²₃.

Introducing the potential functionsφandψthrough the relations u1= ∂φ

∂x1

− ∂ψ

∂x3

, u3= ∂φ

∂x3

+ ∂ψ

∂x1

, (17)

in the equations (1.11)−(1.16), we obtain

∇²φ+ζ₃^∗ϕ^∗−τ_t¹T−ζ₂^∗τ_c¹C=∂²φ

∂t², (18)

(1−δ²)∇²ψ+ζ₁^∗ϕ2=∂²ψ

∂t², (19)

(ζ1∇²−ζ3)ϕ2−ζ2∇²ψ=∂²ϕ₂

∂t² , (20)

(δ²₁∇²−χ^∗₁)ϕ^∗−χ^∗₂∇²φ+χ^∗₃τ_t¹T+χ^∗₄τ_c¹C= ∂²ϕ^∗

∂t² , (21)

∇²T =τ_e⁰∂

∂t(l^∗₁∇²φ+l^∗₂ϕ^∗) +τ_t⁰∂T

∂t +l^∗₃τ_c⁰∂C

∂t, (22)

q^∗₁∇⁴φ+q^∗₄∇²ϕ^∗+q₂^∗τ_t¹∇²T+τ_f⁰∂C

∂t −q^∗₃τ_c¹∇²C= 0. (23)

3 Plane wave propagation:

For plane harmonic waves, we assume the solution of the form

(φ, ψ, T, C, ϕ^∗, ϕ2) = (φ, ψ, T , C, ϕ^∗, ϕ2) exp[ι((x1l1+x3l3)−ωt)], (24) where ω is the angular frequency.φ, ψ, T , C, ϕ^∗, ϕ2 are undetermined ampli- tude vectors that are independent of timetand coordinatesxm(m= 1,3). l1

and l3 are the direction cosines of the wave normal to thex1x3- plane with the propertyl²₁+l²₃= 1.

Substituting the values of φ, ψ, T, C, ϕ^∗, ϕ₂ from equation (2.1) in the equa- tions (1.18)−(1.23) , we obtain

(w²−ξ²)φ+ζ₃^∗ϕ^∗−τ_t¹¹T−ζ₂^∗τ_c¹¹C= 0, (25)

(w²−(1−δ²)ξ²)ψ+ζ₁^∗ϕ₂= 0, (26)

−ξ²ζ2ψ+ (−w²+ζ1ξ²+ζ3)ϕ₂= 0, (27) χ^∗₂ξ²φ+χ^∗₃τ_t¹¹T+ (w²−δ₁²ξ²−χ^∗₁)ϕ^∗+χ^∗₄(1−ιw)C= 0, (28)

−l^∗₁τ_e¹⁰ξ²φ+ (τ_t¹⁰+ξ²)T+l₂^∗τ_e¹⁰ϕ^∗+l^∗₃τ_c¹⁰C= 0, (29)

q₁^∗ξ⁴φ−q₄^∗ξ²ϕ^∗−q₂^∗τ_t¹¹ξ²T+ (τ_f¹⁰+q₃^∗τ_c¹¹ξ²)C= 0, (30) where

τ_t¹¹= 1−ιωτ₁, τ_c¹¹= 1−ιωτ¹, τ_t¹⁰=−ιw(1−ιωτ₀), τ_c¹⁰=−ιw(1−ιωγ₁), τ_e¹⁰=−ιw(1−ιωετ₀), τ_f¹⁰=−ιw(1−ιωετ⁰).

The system of equations (2.2),(2.5)−(2.7) has a non-trivial solution if the determinant of the coefficients [φ, T , ϕ^∗, C]^T vanishes, which yields to the fol- lowing polynomial characteristic equation inξas:

G1ξ⁸+G2ξ⁶+G3ξ⁴+G4ξ²+G5= 0, (31) where,

G1=M1−M11F13, G2=w²M1+M2−F4M5+F9M8−F13M12, G3=w²M2+M3−F4M6+F9M9−F13M13,

G₄=w²M₃+M₄−F₄M₇+F₉M₁₀, G₅=w²M₄,

and

M1=F6F17, M2=−F6F10F17+F7F17+F6F16−F8F14+F6F12F15, M3=F5(F11F17−F12F14)−F10(F7F17+F6F16−F8F14)+F7F16+F15(F7F12−F8F11),

M4=F16(F5F11−F7F10), M5= (F2F17−F3F14),

M6=F2(F11F17−F12F14)−F10(F1F17−F3F14) +F1F16+F15(F1F12−F3F11), M7=F16(F2F11−F1F10), M8=F6(F2F17−F3F15),

M9= (F7F17+F6F16−F8F14)−F5(F1F17−F3F14) +F15(F1F8−F3F7), M10=F16(F2F7−F1F5), M11=−F3F6, M12= (F2F6F12+F3F6F10−F1F8+F3F7),

M13=F2(F7F12−F8F11)−F5(F1F12−F3F11) +F10(F1F8−F3F7), F1=ζ3^∗, F2=−τt¹¹, F3=−ζ2^∗τc¹¹, F4=−χ^∗2, F5=χ^∗3τt¹¹, F6=δ1², F7=w²−χ^∗1, F8 =χ^∗₄(1−ιw), F9=l^∗₁τ_e¹⁰, F10=τ_t¹⁰, F11=l^∗₂τ_e¹⁰, F12=l^∗₃τ_c¹⁰, F13=q^∗₁, F14=q₂^∗τ_t¹¹,

F15=q^∗4, F16=τf¹⁰, F17=−q3^∗τc¹¹

The system of equations (2.3) and (2.4) has a non-trivial solution if the determinant of the coefficients [ψ, ϕ2]^T vanishes, which yields to the following polynomial characteristic equation

F1ξ⁴+F2ξ²+F3= 0, (32) where,

F₁=−ζ₁(1−δ²), F₂=−ζ₁^∗ζ₂, F₃= (ζ₃−w²)(1−δ²)−w²ζ₁+w²(ζ₃−w²).

Solving (2.8) we obtain eight roots ofξin which four rootsξ₁, ξ₂, ξ₃, ξ₄ corre- sponds to positivex3−direction and represents the four waves in descending order of their velocities, namely LD-wave, T-wave, MD-wave, LM-wave. Like- wise, solving (2.9) we obtain four roots of ξ, in which two roots ξ5 and ξ6

correspond to positive x3− direction and other two roots −ξ5 and −ξ6 cor- respond to negativex3−direction. Now and after, we will restrict our work to positivex3− direction . Corresponding to rootsξ5 and ξ6 there exist two waves in descending order of their velocities, namely CD-I and CD-II waves.

We now derive the expressions for phase velocity, attenuation coefficient, spe- cific loss and penetration depth of these waves.

(i) Phase velocity

The phase velocity is given by V_i= ω

|Re(ξi)|, i= 1,2,3,4,5,6 (33) where V1, V2, V3, V4, V5, V6 are the phase velocities of LD, T, MD, LM, CD-I and CD-II waves respectively.

(ii) Attenuation coefficient

The attenuation coefficient is defined as

Qi =Im(ξi), i= 1,2,3,4,5,6 (34) where Q1, Q2, Q3, Q4, Q5, Q6 are the attenuation coefficients of LD, T, MD, LM, CD-I and CD-II, waves respectively.

(iii) Specific loss

The specific loss is the ratio of energy(∆W) dissipated in taking a specimen through a stress cycle, to the elastic energy(W) stored in the specimen when the strain is a maximum. The specific loss is the most direct method of defining internal friction for a material. For a sinusoidal plane wave of small amplitude,

Kolsky[38], shows that the specific loss ∆W/W equals 4πtimes the absolute value of the imaginary part ofξto the real part ofξ, i.e.

Ri= (∆W

W )i= 4π

Im(ξi) Re(ξi)

, i= 1,2,3,4,5,6 (35) whereR1, R2, R3, R4, R5, R6 are the specific losses of LD, T, MD, LM, CD-I and CD-II, waves respectively.

(iv) Penetration depth

The penetration depth is defined by Si= 1

|Im(ξi)|, i= 1,2,3,4,5,6 (36) whereS₁, S₂, S₃, S₄, S₅, S₆are the attenuation coefficients of LD, T, MD, LM, CD-I and CD-II, waves respectively.

Particular cases

1. In absence of microstretch effect, the equation (2.8) becomes ,

H1ξ⁶+H2ξ⁴+H3ξ²+H4= 0, (37) where,

H1=q^∗₃τ_c¹¹−q₁^∗ζ₂^∗τ_c¹¹,

H2=−q^∗₃τ_t¹⁰τ_c¹¹−τ_f¹⁰−l^∗₃q^∗₂τ_c¹⁰τ_t¹¹−l^∗₁q^∗₃τ_e¹⁰τ_t¹¹τ_c¹¹+w²q₃^∗τ_c¹¹

−l^∗₁τ_e¹⁰ζ₂^∗τ_c¹¹τ_t¹¹q^∗₂−l^∗₃q^∗₁τ_c¹⁰τ_t¹¹+q^∗₁ζ₂^∗τ_c¹¹−l₁^∗τ_e¹⁰τ_t¹¹τ_c¹¹q^∗₃, H3=τ_t¹⁰τ_f¹⁰−w²q^∗₃τ_t¹⁰τ_c¹¹−w²τ_f¹⁰−w²l₃^∗q₂^∗τ_c¹⁰τ_t¹¹+τ_t¹¹τ_f¹⁰l^∗₁τ_e¹⁰,

H4=w²τ_f¹⁰τ_t¹⁰.

Solving (2.14) we obtain six roots of ξ, in which three rootsξ1, ξ2 andξ3

corresponds to positivex3−direction and represents the three waves in de- scending order of their velocities, namely LD-wave, T-wave, MD-wave.

2. In absence of micropolarity effect, the velocity equation (2.9) becomes

(δ²−1)ξ²+w²= 0. (38)

Solving (2.15) we obtain two roots ofξ, in which one root which corresponds to positivex3−direction, represents the SV-wave.

4 Reflection and transmission at the boundary surface

Formulation of the problem

We consider an inviscid fluid half-space (M₁) lying over a homogeneous isotropic, microstretch generalized thermoelastic diffusion solid half-space (M₂).

The origin of the cartesian coordinate system (x₁, x₂, x₃) is taken at any point on the plane surface (interface) andx₃−axis point vertically downwards into the microstretch thermoelastic diffusion solid half-space. The inviscid fluid half-space (M1) occupies the regionx3≤0 and the region x3≥0 is occupied by the microstretch themoelastic diffusion solid half-space (M2) as shown in Figure 1. We consider plane waves in thex1−x3plane with wave front parallel to thex2−axis.

A₆ P₆

A₅ P₅ contact surface x₃= 0

Microstretch thermoelastic diffusion solid half-space (M₂) x₃> 0

Inviscid fluid half-space (M₁) x₃< 0

Incident (P)

Reflected (P)

A₄ A₃

A₂ A₁

P₄ P₃

P₂ P₁

x₃-axis

x₁-axis

Figure 1: Geometry of the problem.

For the propagation of harmonic waves inx₁x₃−plane, we assume

[φ, ψ, T, C, ϕ^∗, ϕ2](x1, x3, t) = [φ, ψ,T ,¯ C, ϕ¯ ^∗, ϕ₂]e^−ιωt, (39) whereω is the angular frequency of vibrations of material particles.

Substituting the expressions of φ, ψ, T, C, ϕ^∗, ϕ₂ given by equation (2.1) in the equations (1.18)-(1.23), we obtain

[∇²+ω²] ¯φ−τ_t¹¹T¯+ζ₃^∗ϕ^∗−ζ₂^∗τ_c¹¹C¯= 0, (40) ((1−δ²)∇²+ω²) ¯ψ+ζ₁^∗ϕ₂= 0, (41) ζ2∇²ψ¯+ (−ω²−ζ1∇²+ζ3)ϕ₂= 0, (42)

−χ^∗₂∇²φ¯+χ^∗₃τ_t¹¹T¯+r1ϕ^∗+r2C= 0, (43) l^∗₁τ_e¹⁰∇²φ¯+ (τ_t¹⁰− ∇²)T+l^∗₂τ_e¹⁰ϕ¯^∗+l₃^∗τ_c¹⁰C¯= 0, (44) q^∗₁∇⁴φ¯+q₂^∗τ_t¹¹∇²T¯+q₄^∗∇²ϕ¯^∗+ (τ_f¹⁰−q^∗₃τ_c¹¹∇²) ¯C= 0, (45) where,r1=δ₁²∇²−χ^∗₁+w², r2=χ^∗₄(1−ιw).

Eliminating [ϕ, T , ϕ^∗, C]^T from the system of equations (3.2), (3.5)-(3.7), we obtain

[∇⁸+B₁∇⁶+B₂∇⁴+B₃∇²+B₄]φ= 0, (46) where,

Bi=Ai

A,(i= 1,2,3,4),

A=g^∗1−a14g14^∗, A1=g^∗2+g1^∗w²−a12g6^∗+a13g^∗9−a14g12^∗,

A2=g3^∗+g2^∗w²−a12g^∗7+a13g10^∗ −a14g13^∗, A3=g^∗4+g3^∗w²−a12g^∗8+a13g^∗11, A4=g4^∗w² g₁^∗=−δ²₁a46,

g₂^∗=a23a46−a24a43+δ²₁(a32a46+a45+a34a42), g^∗₄=a45(a22a33+a23a32), g3^∗=−a33(a22a46+a24a42)−a23(a32a46+a45+a34a42)

+a43(a24a32−a22a34)−δ²1a32a45, g6^∗=δ²1(a31a46+a41a34),

g₇^∗=−a33(a21a46+a24a41)−a23(a31a46+a34a41) +a43(a24a31−a21a34)−δ²₁a31a45, g^∗₈ =a45(a23a31+a21a33), g^∗₉ =a24a41+a21a46,

g^∗10=−a21(a32a46+a45+a34a42) +a22(a31a46+a34a41) +a24(a31a42−a32a41), g^∗11=a45(a21a32−a22a31), g12^∗ =−(a23a41+a21a43) +δ²1(a31a42−a41a32), g₁₃^∗ =a33(a22a41−a21a42) +a23(a32a41−a31a42) +a43(a21a32−a22a31), g^∗₁₄=δ²₁a41,

a11=∇²+w², a21=−χ^∗2, a31=l^∗1τe¹⁰, a41=q^∗1, a12=−τt¹¹, a22=χ^∗3τt¹¹, a32=τt¹⁰, a42=q^∗2τt¹¹, a13=ζ3^∗, a23=χ^∗1−w², a33=l^∗2τe¹⁰,

a14=−ζ2^∗τc¹¹, a24=r2, a34=l3^∗τc¹⁰, a43=q4^∗∇², a45=τf¹⁰, a46=q^∗3τc¹¹, a44= (a45−a46∇²).

The general solution of equation (3.8) can be written as

φ¯= ¯φ1+ ¯φ2+ ¯φ3+ ¯φ4, (47) where the potentials ¯φi, i= 1,2,3,4 are solutions of wave equations, given by

[∇²+ ω²

V_i²] ¯φ_i= 0, i= 1,2,3,4. (48) Here V1, V2, V3 and V4 are the velocities of four longitudinal waves, that is, longitudinal displacement wave (LD), thermal wave (T), mass diffusion wave (MD) and longitudinal microstretch wave (LM) and derived from the roots of the biquadratic equation inV², given by

B₄V⁸−B₃ω²V⁶+B₂ω⁴V⁴−B₁ω⁶V²+w⁸= 0. (49) Making use of equation (3.9) in the equations (3.2), (3.5)-(3.7) with the aid of equations (3.1) and (3.10), the general solutions for φ, T, ϕ^∗, and C are obtained as

[φ, T, ϕ^∗, C] =

4

X

i=1

[1, k_1i, k_2i, k_3i]φ_i, (50) where,

k1i = (g₆^∗w⁶−g^∗₇w⁴V_i²+g₈^∗w²V_i⁴)/k^d, k2i=−(g₉^∗w⁶+g₁₀^∗ w⁴V_i²+g^∗₁₁w²V_i⁴)/k^d, k_3i= (−g₁₄^∗ w⁸+g₁₂^∗ w⁶V_i²−g^∗₁₃w⁴V_i⁴)/(V_i²k^d),

k^d= (g^∗₁w⁶+g₂^∗w⁴V_i²+g^∗₃w²V_i⁴+g₄^∗V_i⁶), i= 1,2,3,4.

Eliminating [ψ, ϕ₂]^T from the system of equations (3.3)-(3.4), we obtain [∇⁴+A^∗∇²+B^∗]ψ= 0, (51) where,

A^∗= (w²ζ₁+ζ₁^∗ζ₂−(1−δ²)(ζ₃+w²))/(1−δ²)ζ₁, B^∗=w²(w²−ζ3)/(1−δ²)ζ1,

The general solution of equation (3.13) can be written as

ψ=ψ₅+ψ₆ (52)

where the potentialsψ_i, i= 5,6 are solutions of wave equations, given by [∇²+ ω²

V_i²]ψ_i= 0, i= 5,6. (53) Here (Vi, i= 5,6) are the velocities of two coupled transverse displacement and microrotational (CD-I, CD-II) waves and derived from the root of quadratic equation inV², given by

B^∗V⁴−A^∗w²V²+w⁴= 0, (54) Making use of equation (3.14) in the equations (3.3)-(3.4) with the aid of equations (3.1) and (3.15), the general solutions forψandϕ2are obtained as

[ψ, ϕ₂] =

6

X

i=5

[1, n_1i]ψ_i, (55)

where,

n1i= ζ2w²

(ζ₃−w²)V_i²+ζ₁w², f or i= 5,6

Following Achenbach [39], the field equations in terms of velocity potential for inviscid fluid are

p^f =−ρ^fφ˙^f, (56)

[∇²− 1 α^f2p

∂²

∂t²]φ^f = 0, (57)

−

→u^f =∇φ^f, (58) whereα^f2_p =λ^f/ρ^f andλ^f is the bulk modulus,ρ^f is the density of the liquid,

−

→u^f is the velocity vector andp^f is the acoustic pressure of the inviscid fluid.

For two dimensional problem, −→u^f = (u^f₁,0, u^f₃) can be written in terms of velocity potential as

u^f₁ =∂φ^f

∂x₁, u^f₃ =∂φ^f

∂x₃. (59)

Applying the dimensionless quantities defined by (1.10) in equations (3.18) and (3.19) and after suppressing the primes, we obtain

p^f =−ζφ˙^f, (60)

[∇²− 1 vp^f2

∂²

∂t²]φ^f= 0, (61)

where,

ζ=ρ^fc²₁/β1T0, v_p^f=α^f_p/c1. We assume the solution of (3.23) as

φ^f(x₁, x₃, t) =φ^fe^−ιwt. (62) Using (3.24) in (3.23), we have

[∇²+ w² vp^f2

]φ^f = 0. (63)

Reflection and transmission

We consider a plane harmonic longitudinal wave (P) propagating through the inviscid fluid half-space and is incident at the interfacex3 = 0 as shown in Figure 1. Corresponding to incident wave, one homogeneous longitudinal wave (P) is reflected in inviscid fluid half-space and six inhomogeneous waves (LD, T, MD, LM, CD-I and CD-II) are transmitted in isotropic microstretch ther- moelastic diffusion solid half-space.

In inviscid fluid half-space, the potential functions satisfying equation (3.25) can be written as

φ^f =A^f₀e^{[ιω((x1sinθ0+x3cosθ0)/vfp)−t]}+A^f₁e^{[ιω((x1sinθ1−x3cosθ1)/vpf)−t]}, (64) The coefficientsA^f₀ andA^e₁represent the amplitudes of the incident P and reflected P waves respectively.

Following Borcherdt [40], in a homogeneous isotropic microstretch thermoe- lastic diffusion half-space, potential functions satisfying equations (3.10) and (3.15) can be written as

[φ, T, ϕ^∗, C] =

4

X

i=1

[1, k1i, k2i, k3i]Bie^(A~i.~r)e^{{ι(P~i.~r−ωt)}}, (65)

[ψ, φ₂] =

6

X

i=5

[1, n_ip]B_ie^(A~i.~r)e^{ι(P~i.~r−ωt)}. (66) The coefficientsBi, i= 1,2,3,4,5,6 represent the amplitudes of transmitted waves. The propagation vector P~i, i = 1,2,3,4,5,6 and attenuation factor A~i, i= 1,2,3,4,5,6 are given by

P~i=ξRxˆ1+dViRxˆ3, ~Ai=−ξIxˆ1−dViIxˆ3, i= 1,2,3,4,5,6 (67)

where,

dV_i =dV_iR+ιdV_iI=p.v.(ω²

V_i² −ξ²)^1/2, i= 1,2,3,4,5,6. (68) andξ=ξR+ιξI is a complex wave number. The subscripts R and I denote the real and imaginary parts of the corresponding complex quantity and p.v.

stands for the principal value of the complex quantity obtained after square root. ξ_R ≥ 0 ensures propagation in the positivex₁-direction. The complex wave number ξ in the microstretch thermoelastic diffusion solid medium is given by

ξ=

P~i

sinθ_i⁰−ι

A~i

sin(θ_i⁰−γi), i= 1,2,3,4,5,6, (69) where γ_i, i = 1,2,3,4,5,6 is the angle between the propagation and atten- uation vector and θ⁰_i, i = 1,2,3,4,5,6 is the angle of refraction in medium II.

Boundary conditions

The boundary conditions are the continuity of stress and displacement com- ponents, vanishing of the gradient of temperature, mass concentration, the tangential couple stress and microstress components. Mathematically these can be written as

(i) Continuity of normal stress component

t33=−p^f, (70)

(ii) Continuity of tangential stress component

t₃₃= 0, (71)

(iii) Continuity of normal displacement component

u^f₃ =u3, (72)

(iv)Thermally insulated boundary

∂T

∂x3

= 0, (73)

(v)Impermeable boundary

∂C

∂x3

= 0. (74)