Numerical analysis of the propagation characteristics of Stoneley waves at an interface between microstretch thermoelastic diffusion solid half spaces

This paper is concerned with the study of propagation of Stoneley waves at the interface of two dissimilar isotropic microstretch thermoelastic diffusion medium in the context of generalized theories of thermoelasticity. The dispersion equation of Stoneley waves is derived in the form of a determinant by using the boundary conditions. The dispersion curves giving the phase velocity and attenuation coefficients with wave number are computed numerically. Numerically computed results are shown graphically to depict the diffusion effect alongwith the relaxation times in microstretch thermoelastic diffusion solid half spaces for thermally insulated and impermeable boundaries, respectively. The components of displacement, stress, couple stress, microstress, and temperature change are presented graphically for two dissimilar microstretch thermoelastic diffusion half-spaces. Several cases of interest under different conditions are also deduced and discussed.

Latin American Journal of Solids and Structures 11 (2014) 2408-2425 SAW devices and more recently nondestructive evaluation.The study of surface wave propagation is of much practical importance in various fields such as earthquake engineering, soil dynamics, aeronautics, nuclear reactors, high energy particle accelerator etc. Rayleigh (1885) discussed the surface wave propagation along the free boundary of an elastic half-space, non-attenuated in their direction of propagation and damped normal to the boundary.Stoneley (1924) studied the existence of waves, which are similar to surface waves and propagating along the plane interface between two distinct elastic solid half spaces in perfect contact.Stoneley waves can also propagate on interfaces either two elastic media or a solid medium and a liquid medium.Stoneley (1924) derived the dispersion equation for the propagation of Stoneley waves.Tajuddin (1995) investigated the existence of Stoneley waves at an interface between two micropolar elastic half spaces.Eringen (1966bEringen ( , 1968) developed the theory of micromorphic bodies by considering a material point as endowed with three deformable directions.Subsequently, he developed the theory of microstretch elastic solid (1971) which is a generalization of micropolar elasticity (1966a).The material points in microstretch elastic body can stretch and contract independently of the translational and rotational processes.The difference between these solids and micropolar elastic solids stems from the presence of scalar microstretch and a vector first moment.These solids can undergo intrinsic volume change independent of the macro volume change and is accompanied by a non deviatoric stress moment vector.Eringen(1990) also developed the theory of thermo microstretch elastic solids.The microstretch continuum is a model for Bravias lattice with a basis on the atomic level and a two phase dipolar solid with a core on the macroscopic level.For example, composite materials reinforced with chopped elastic fibres, porous media whose pores are filled with gas or inviscid liquid, asphalt or other elastic inclusions and 'solid-liquid' crystals, etc., should be characterizable by microstretch solids.A comprehensive review on the micropolar continuum theory has been given in his book by Eringen(1999).Iesan and Pompei (1995) discussed the equilibrium theory of microstretch elastic solids.The problem of wave propagation through a cylindrical bore contained in a microstretch elastic medium has been studied by Kumar and Deswal(2002).Tomar and Singh (2006) discussed the propagation of Stoneley waves at an interface between two microstretch elastic half-spaces.Kumar and Pratap (2009) studied the propagation of free vibrations in microstretch thermoelastic homogeneous isotropic, thermally conducting plate bordered with layers of inviscid liquid on both sides subjected to stress free thermally insulated and isothermal conditions.Markov (2009) discussed the propagation of Stoneley elastic wave at the boundary of two fluid-saturated porous media and determined the velocity and attenuation of the Stoneley surface waves.Ahmed and Abo-Dahab (2012) studied the propagation of Rayleigh and Stoneley waves in a thermoelastic orthotropic granular half-space supporting a different layer under the influence of initial stress and gravity field.
Diffusion can be defined as the random walk to accumulate the particles from region of high concentration to that of low concentration.At the present time, there is a great deal of interest in the study of this phenomenon due to its application in geophysics and electronic industry.Study of phenomenon of diffusion is utilized to enhance the conditions of oil extraction (searching ways of more efficiently recovering oil from its deposits).
Latin American Journal of Solids and Structures 11 (2014) 2408-2425 Nowacki (1974aNowacki ( , 1974bNowacki ( , 1974cNowacki ( , 1976) ) in a series of papers presented the theory of thermoelastic diffusion by using coupled thermoelastic model.Uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem, in isotropic media, was proved by Sherief et al.(2004)on the basis of the variational principle equations, under restrictive assumptions on the elastic coefficients.Kumar and Kansal (2008) derived the basic equation of anisotropic thermoelastic diffusion based upon Green-Lindsay model and discussed the Lamb waves.Kumar and Chawla (2009) discussed the wave propagation at the imperfect boundary between transversely isotropic themoelastic diffusive half space and an isotropic elastic layer.Kumar and Kansal(2011) construct the fundamental solution of system of differential equations in the theory of thermomicrostretch elastic diffusive solids in case of steady oscillations in terms of elementary functions.Sharma (2007Sharma ( , 2008) ) discussed the plane harmonic generalized thermoelastic diffusive waves and elasto thermodiffusive surface waves in heat-conducting solids.Recently Kumar et al. (2013)studied the reflection and transmission of plane waves at the interface between a microstretch thermoelastic diffusion solid half-space and elastic solid half space.
Keeping in view of these applications, dispersion equations for Stoneley waves at the interface of two dissimilar isotropic microstretch thermoelastic diffusion medium in the context of generalized theories of thermoelasticity have been derived.It is found that Stoneley waves in a microstretch thermoelastic diffusion solid medium are dispersive.Numerical computations are performed for a particular model to study the variations of phase velocity and attenuation coefficient with respect to wave number.The present work is novel and have not been discussed earlier in the literature.The results presented in this paper should prove useful for researchers in material science, designers of new materials as well as for those working on the development of theory of elasticity.

BASIC EQUATIONS
Following Eringen (1999), Sherief et al. (2004) and Kumar & Kansal (2008), 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 Latin American Journal of Solids and Structures 11 (2014) 2408-2425 and constitutive relations are where 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.

FORMULATION OF THE PROBLEM
We consider two homogeneous isotropic microstretch generalized thermoelastic diffusion halfspaces 1 M and 2 M connecting at the interface 3 0 x  .The origin of the coordinate system x x x is taken at any point on the plane horizontal surface with 3 x  axis, pointing vertically downward to the half-space, which is thus represented by 3 0 x  .We choose the 1 x axis in the direction of wave propagation in such a way that all the particles on a line parallel to the 2 x axis Latin American Journal of Solids and Structures 11 (2014) 2408-2425 are equally displaced.Therefore, all field quantities are independent of the 2 x coordinate.Medium 2 M occupies the region 3 0 x    and the region 3 0 x  is occupied by the half-space (medium 1 M ).The plane 3 0 x  represents the interface between two media 1 M and 2 M .For the two dimensional problem, we take ( , , ) ( ,0, ), (0, ,0), ( , , ), ( , , ), ( , , ), We define the following dimensionless quantities where  is the characteristic frequency of the medium, Upon introducing the quantities (10) in equations ( 1)-( 5), with the aid of ( 9) and after suppressing the primes, we obtain , where We introduce the potential functions and   through the relations in the equations ( 11)-( 16), we obtain

SOLUTION OF THE PROBLEM
We assume the solutions of the form where  is the wave number, c    is the angular frequency, and c is phase velocity of the wave.Using (24) in equations ( 18), ( 21)-( 23) and satisfying the radiation condition where are the roots of the equation and are the roots of the equation where Similarly, we assume the solutions of the field equations as Using (29) in equations ( 19) and ( 20), and satisfying the radiation condition The roots of equation ( 27) in the descending order corresponds to the velocities of propagation of four possible waves, namely longitudinal displacement wave (LD), thermal wave (T), mass diffusion wave (MD) and longitudinal microstretch wave (LM), respectively.Similarly, two roots of the equation (32) in the descending order corresponds to the velocities of propagation of two coupled transverse displacement and transverse microrotational waves (CD I, CD II), respectively.

(i)
In the absence of diffusion effect, equation ( 27) leads to sixth order differential equation The roots of the equation ( 34) correspond to the LD, T and LM waves, respectively.
Clearly, we can notice here that, on neglecting the diffusion effect, the wave corresponding to this parameter namely mass diffusion wave (MD) become deceased.Therefore, it is observed from the equation ( 27) and ( 34), that there exist a new type of wave namely MD wave.

(ii)
On neglecting the diffusion, micropolarity and microstretch effects, equation ( 27) and (32) simultaneously leads to the forth and second order differential equations as where and are given in appendix A A B .
The roots of the equation ( 35) correspond to the Longitudinal wave (P-wave), and T waves, and (36) relate to the SV-wave, respectively.Therefore, it is again observed that there exist new type of wave in (34) namely Longitudinal microstretch wave (LM) and transverse microrotational waves (CD II) in ( 32) which become decoupled in this case .

e m Ae m Ae e u m A e m A e m A e m A e i A e
A e e Latin American Journal of Solids and Structures 11 (2014) 2408-2425 For medium

e m Ae m Ae e u m A e m A e m A e m A e i A e
A e e

BOUNDARY CONDITIONS
The following boundary conditions must be satisfied on the boundary between two microstretch thermoelastic diffusion media.Mathematically, these can be written (at the surface 3 0 x  ) as

DERIVATIONS OF THE SECULAR EQUATIONS
Making use of equations ( 25)-( 26), ( 30) and ( 31) in the equations ( 37)-( 48), we obtain a system of twelve simultaneous linear equations: where the values of Equation ( 50) is the dispersion equation for the propagation of Stoneley waves at an interface between microstretch thermoelastic diffusion solid half spaces.This equation has complete information about the phase velocity, wave number, and attenuation coefficient of the surface waves propagating in such a medium.

PARTICULAR CASES (i)
In the absence of diffusion effect, the dispersion equation for the propagation of Stoneley waves at an interface between microstretch thermoelastic solid half spaces is obtained as In the absence of thermal and diffusion effects, the dispersion equation ( 50) reduced to the propagation of Stoneley waves at an interface between microstretch elastic solid half spaces.The resulting dispersion equation in reduced form is similar as obtained by Tomer and Singh (2006).(iii) Take 0 0, 50), yield the expression of secular equation for the propagation of Stoneley waves at an interface between microstretch thermoelastic diffusion solid half spaces with two relaxation times.(iv) Using 50), gives the corresponding results for the propagation of Stoneley waves at an interface between microstretch thermoelastic diffusion solid half spaces with with one relaxation time.
(v) On taking 50), provide the corresponding expression of secular equation for the propagation of Stoneley waves at an interface between microstretch thermoelastic diffusion solid half spaces with Coupled Thermoelastic (CT) theory.

NUMERICAL RESULTS AND DISCUSSION
The analysis is conducted for a magnesium crystal-like material.Following Eringen (1984), the values of micropolar parameters for medium M 1 are given by 10 -2 10 -2 10 2 3 3 1 9 2 9 9.4 10 Nm , 4.0 10 Nm , 1.0 10 , 1.74 10 , 0.2 10 , 0.779 10 Thermal and diffusion parameters are given by and, the microstretch parameters are taken as Thermal and diffusion parameters are given by and, the microstretch parameters are taken as ures, for the thermally insulated boundary and impermeable boundary, the words LSWD and GLWD symbolize the graphs for L-S and G-L theories in microstretch thermoelastic diffusion medium, respectively and the words LSWTD and GLWTD symbolize the graphs for L-S and G-L theories in microstretch thermoelastic medium, respectively.

Phase Velocity
Figure 1 depicts the variation of phase velocity with frequency.In the presence of diffusion effect, the values of phase velocity for LSWD decrease monotonically and a significant difference in the values is noticed when compared in absence of diffusion LSWTD, for smaller values of frequency.The trend of variation and behavior of GLWD and GLWTD is similar to LSWD and LSWTD, respectively and for higher values of frequency, the values of phase velocity become dispersionless.

Attenuation Coefficient
Figure 2 shows that the values of attenuation coefficient increase linearly for whole range of frequency for LSWD, GLWD, LSWTD and GLWTD respectively; nevertheless, a significant difference in the values of attenuation coefficient is noticed for LSWTD, GLWTD when compared to LSWD, GLWD respectively, for whole range of frequency.Figure3 depicts the variation of component of vertical displacement with wave number.It is noticed that, the value of 3 u decrease monotonically and oscillates afterward for smaller values of wave number in case of LSWD and GLWD, respectively, which finally become stationary for higher values of  .An incredible effect in LSWTD is noticed when compared to LSWD for the initial values of wave number and similar trend of variation is also noticed for GLWTD when compared to GLWD, respectively.

CONCLUSION
The propagation of Stoneley waves at the interface of two dissimilar isotropic microstretch thermoelastic diffusion medium has been investigated in the context of generalized theories of thermoelasticity.Dispersion equations of Stoneley waves for surface wave propagation are derived for the considered mathematical model.Numerical computations are performed for a particular model to study the variation of phase velocity and attenuation coefficient with respect to wave number.The components of displacement, stress, couple stress, microstress, and temperature change are computed numerically and shown graphically to depict the effect of diffusion for Lord-Shulman (1967) and Green-Lindsay (1972) theories of thermoelasticity.It is concluded that the effect of diffusion for all the resulting quantities in medium M 1 is significant for smaller values of wave number, on the other hand, the effect of diffusion in medium M 2 is more for the higher values of wave number.Furthermore, the resulting quantities for LSWD, GLWD, LSWTD and GLWTD try to converge towards zero in medium M 1, but the same quantities attempt to diverge in medium M 2. Notable impact due to relaxation times is also revealed due to the presence of diffusion effect for every resulting quantity in both the mediums M 1 and M 2 , respectively.


is the scalar microstretch function, T and 0 T are the small temperature increment and the reference temperature of the body chosen such that 0 1, T T  C is the concentration of the diffusion material in the elastic body.* K is the coefficient of the thermal conductivity, * C the specific heat at constant strain, D is the thermoelastic diffusion constant.a, b are, respectively, coefficients describing the measure of thermodiffusion and of mass diffusion effects,

,B
coupling constant given in appendix B.We attach bar for the variables in the medium 2 M and write the appropriate values of 2 are given in appendix A, , and are given in appendix A 04 has been used for numerical computation of the resulting quantities.The values of phase velocity and attenuation coefficient with wave number at the stress free boundary with thermally insulated and impermeable boundaries along with the relaxation times are shown in fig.1 and fig.2.Components of displacements, normal stress, tangential couple stress, microstress, and temperature change with wave number has been determined at the surface 3 1 x  , and is shown in figs.3-9 for medium M 1 and in figs.10-16 for medium M 2 , respectively.In all fig-

Figure 3 :
Figure 3: Variation of 3 u w.r.t wave number.Figures 4-5 shows the trend of variation and behavior of the component of normal stress and couple stress, respectively.It is clear from the figures that due to the absence of diffusion effect, T 33 and m 32 remains oscillatory and become linear due to the presence of diffusion effect, for smaller values of wave number.The oscillation of T 33 is opposite to that of m 32 , in case of T 33 the values first increases sharply whereas the values of m 32 decrease monotonically for 0 5    .In Figures 4-5, LSWTD and GLWTD show a notable diffusion effect when compared to LSWD and GLWD for16 30   

Figure 10
Figure 10 depicts the variation of the component of vertical displacement with wave number.It is noticed that the trend of variation and behavior for LSWD, GLWD, LSWTD and GLWTD is similar for 0 18    with difference in their magnitude values, but for 18 30    the values of LSWD and GLWD decrease sharply due to presence of the diffusion effect.The similar behavior of curves is also noticed for the component of normal stress as revealed in Figure11.A significance difference in the corresponding values for LSWD and GLWD in comparison to LSWTD and