Stoneley Waves at the Boundary Surface of Modified Couple Stress Generalized Thermoelastic with Mass Diffusion

Size: px

Start display at page:

Download "Stoneley Waves at the Boundary Surface of Modified Couple Stress Generalized Thermoelastic with Mass Diffusion"

Marlene Flowers
5 years ago
Views:

1 Journal of Applied Science and Engineering, Vol. 21, No. 1, pp. 18 (2018) DOI: /jase _21(1).0001 Stoneley Waves at the Boundary Surface of Modified Couple Stress Generalized Thermoelastic with Mass Diffusion Rajneesh Kumar 1, Shaloo Devi 2 * and S. M. Abo-Dahab 3,4 1 Department of Mathematics, Kurukshetra University, Kurukshetra, India 2 Department of Mathematics & Statistics, Himachal Pradesh University, Shimla, India 3 Mathematics Department, Faculty of Science, Taif University, Saudi Arabia 4 Mathematics Department, Faculty of Science, SVU, Qena, Egypt Abstract The present problem deals with the study of propagation of Stoneley waves at the interface of two dissimilar isotropic modified couple stress thermoelastic with mass diffusion medium in the context of Lord-Shulman (L-S), Green-Lindsay (G-L) theories of thermoelasticity. The dispersion equation of Stoneley waves is derived in the form of determinant by using appropriate boundary conditions. The dispersion curves giving the values of determinant of secular equation, Stoneley waves velocity and attenuation coefficient with respect to angular velocity for different values of wave number in the absence and presence of mass diffusion are computed numerically and shown graphically. Key Words: Stoneley Waves, Modified Couple Stress Theory, Thermoelastic Diffusion, Stoneley Waves Velocity, Attenuation Coefficient 1. Introduction *Corresponding author. shaloosharma2673@gmail.com Stoneley waves play an important role in the earthquake science, optics, geo-physics, and plasma physics. Stoneley [1] studied the existence of waves, which are 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. Various authors studied Stoneley waves at the interface between different elastic media [210]. The concept of couple stress linear theory of elasticity was originally introduced by Voigt [11] and was extended by Cosserat and Cosserat [12]. Couple-stress theory is an extended continuum theory that includes the effects of a couple per unit area on a material volume, in addition to the classical direct and shear forces per unit area. This immediately admits the possibility of asymmetric stress tensor, since shear stress no longer have to be conjugate in order to ensure rotational equilibrium. Yang et al. [13] modified the classical couple stress theory and proposed a modified couple-stress model, in which the couple stress tensor is symmetrical and only one material length parameter is needed to capture the size effect which is caused by micro-structure. Different problems in modified couple stress theory are investigated by [1416]. Thermo diffusion is used to describe the processes of thermo mechanical treatment of metals (carboning, nitriding steel, etc.) and these processes are thermally activated, and their diffusing substances being, e.g. nitrogen, carbon etc. Nowacki [17] first developed the theory of thermoelastic with mass diffusion. This implies infinite speeds of propagation of thermoelastic waves. Sherief et al. [18] developed the theory of generalized thermoelastic diffusion that predicts finite speeds of propagation for thermoelastic and diffusive waves. Sherief and Saleh [19], Kumar and Kansal [20] were studied different problems in thermoelastic diffusion media. Kumar et al. [21]

2 2 Rajneesh Kumar et al. discussed the propagation characteristics of Stoneley waves at an interface between microstretch thermoelastic diffusion solid half spaces. The present problem deals with the study of propagation of Stoneley waves in modified couple stress generalized thermoelastic mass diffusion medium in the context of Lord-Shulman (L-S), Green-Lindsay (G-L) theories of thermoelasticity. The variations of determinant of secular equation, velocity and attenuation coefficient with respect to angular velocity for different values of wave number in the absence and presence of thermoelastic diffusion are computed numerically and shown graphically. The results of this paper may be useful for those working in the field of couple stress theory of elasticity. (iv) Equation of mass diffusion and (5) (6) (7) 2. Basic Equations Following Yang et al. [13], Kumar and Kansal [20], the constitutive relations and the equations of motion in a modified couple-stress generalized thermoelastic with mass diffusion in the absence of body forces, body couples, heat and mass diffusion sources are given by (i) Constitutive relations (ii) Equations of motion (iii) Equation of heat conduction (1) (2) (3) Here 1 = 1 =0, 0 =1, = 0, for Lord-Shulman (L-S) model and 0 =0, = 0, for Green Lindsay (G-L) model. 3. Formulation of the Problem Consider two homogeneous isotropic modified couple stress generalized thermoelastic half spaces M 1 and M 2 connecting at the interface x 3 = 0. We consider a rectangular Cartesian coordinate system, (x 1, x 2, x 3 ) at any point on the plane horizontal surface and x 1 -axis in the direction of the wave propagation and x 3 -axis taking vertically downward into the half space so that all particles on line parallel to x 2 -axis are equally displaced. Therefore, all the field quantities will be independent of x 2 -coordinates. Medium M 2 occupies the region - < x 3 <0 and 0 < x 3 < is occupied by the half-space (medium M 1 ). The plane x 3 = 0 represents the interface between the two media M 1 and M 2. We denotes all the quantities without bar for medium M 1 and attach bar for medium M 2. For two dimensional problem, the displacement components, temperature change and mass concentration for medium M 1 are given by (8) (4) To facilitated the solution, the following non dimensional quantities are introduced:

3 Stoneley Waves at the Boundary Surface of Modified Couple Stress Generalized Thermoelastic with Mass Diffusion 3 where A j (j = 1, 2, 3) are arbitrary constants in equation (12), the coupling constants b j, d j and A j (j =4,5)arearbitrary constants in equation (12). Figure 1. Geometry of the problem. (9) and The displacement components u 1 and u 3 are related to the scalar potentials and in dimensionless form for medium M 1 as We attach bars for medium M 2 and write the appropriate values of, T, C, for medium M 2 (x 3 < 0) satisfying the radiation condition as 4. Solution of the Problem (10) Making use of (8)(10) in equations (3)(5), after surpassing the primes, yield the equations in non-dimensional form. To solve these equations, we assume (11) Make use of equation (11) in non-dimensional equations and satisfying the radiation condition, T, C, 0asx 3, we obtain the values of, T, C, for medium M 1 (12) (13) where A j (j = 1, 2, 3) are arbitrary constants in equation (13), the coupling constants bj, dj. and A j (j = 4, 5) are arbitrary constants in equation (13). The displacement components for medium M 1 are 5. Boundary Conditions (14) (15) The appropriate boundary conditions at the interface

4 4 Rajneesh Kumar et al. x 3 = 0, can be written as (i) Mechanical conditions Continuity of normal stress, tangential stress, tangential couple stress, tangential displacement, normal displacement and rotation vector components (16) (ii) Thermal and mass concentration conditions The thermal and mass concentration boundary conditions appropriate to the problem are and 6. Derivations of the Secular Equation (17) Using equations (16)(19) in the boundary conditions (20) and (21), we obtain where (18) The system of equations (18) has a nontrivial solution if the determinant of unknowns A j, A j, j = (1, 2, 3, 4, 5) vanishes i.e. 7. Particular Cases (19) (i) If = 0, the dispersion equation (19) reduced to the propagation of Stoneley waves at an interface between generalized thermoelastic diffusion solid half spaces. (iv) In the absence of diffusion ( 2 = a = D = 0) the dispersion equation (19) is reduced for Stoneley waves at an interface between modified couple stress thermoelastic solid half spaces as Q ij = 0 for i = (1, 2, 8 8 3, 4, 5, 6, 7, 8) (v) If 1 = 1 =0, 0 =1, = 0, in equations (19), we obtain the corresponding secular equation for the modified couple stress thermoelastic diffusion solid half spaces for [Lord Shulman (L-S) model]. (vi) If 0 =0, = 0, in equations (19), yield the secular equation for modified couple stress thermoelastic diffusion solid half spaces for [Green Lindsay (G-L) model=. 8. Numerical Results and Discussion For numerical computations, following Sherief and Saleh [19], we take the copper material for medium M 1 as:

5 Stoneley Waves at the Boundary Surface of Modified Couple Stress Generalized Thermoelastic with Mass Diffusion 5 Quantity Material Unit Kg m -1 sec Kg m -1 sec -2 a c e m 2 sec -2 K -1 JK -1 K -1 t K -1 c b m 3 Kg -1 Kg -1 m 5 sec -2 D Kg sec m -3 K Kg m -3 Wm -1 K Kg m sec Kg m -1 sec -2 K -1 m 2 sec sec sec sec sec For medium M 2, following Daliwal and Singh [22], we take the magnesium materialas: Quantity Material Unit Kg m -1 sec Kg m Kg m -1 sec -2 a m -2 sec -2 K -1 t K -1 ce JKg -1 K -1 c m 3 Kg -1 b Kg -1 m 5 sec -2 D Kg sec m Kg m sec -2 K Wm -1 K Kg m -1 sec -2 K m 2 sec sec sec sec sec T K 10 sec -1 t 1 sec Software mathcad has been used for numerical commutation. The values of determinant of secular equations, Stoneley wave velocity and attenuation coefficients with respect to frequency have been computed numerically and shown graphically in Figures 27. In Figures 27, solid line ( ) correspond to Lord-Shulman (LS) model and small dash line (...) correspond to Green-Lindsay (GL) model for different values of wave number ( = 0.1 m -1, 0.12 m -1, 0.13 m -1 ) with respect to wave number ( Hz). 9. Special Case: D = 0 Figure 2 depicts the variation of determinant of Stoneley waves secular equation with respect to for different values of wave number ( = 0.1, 0.12, 0.13). The behavior and variation of determinant of secular equation for all values of wave number are similar for the range and has significant difference in their magnitude values in the remaining range for both L-S and G-L theories. The values of G-L theory are more in comparison to L-S theory for all values of wave number. It is observed that the value of secular equation increases Figure 2. Comparison between LS and GL models on determinant of Stoneley waves secular equation with varies values of = 0.1, 0.12, 0.13 with respect to Hz. Figure 3. Comparison between LS and GL models on Stoneley waves velocity with varies values of = 0.1, 0.12, 0.13 with respect to Hz.

6 6 Rajneesh Kumar et al. with increasing value of frequency as wave number increases further. Figure 3 shows the variation of Stoneley wave velocity with respect to. The values of Stoneley wave velocity decreases with increase in the values of wave number with respect to angular frequency for both theories of thermoelastic diffusion. As increases, the values of Stoneley wave velocity decreases with increasing value of wave number for both theories of thermoelasticity. Figure 4 exhibits the variation of attenuation coefficient with respect to angular frequency. The values of attenuation coefficient for = 0.1, 0.12 and = 0.13 show similar patterns. Also, the values of attenuation coefficient becomes stable for smaller values of angular velocity but for higher values of angular frequency there is significant difference between their values for both the theories of thermoelasticity. It is noticed that the value of attenuation coefficient observed higher value for G-L theory than that of L-S theory. Figure 5 that variation of determinant of Stoneley waves secular equation with respect to. The value of determinant of Stoneley waves are close to each other for different values of for the range and there is sharp difference in their magnitude values in the remaining range for both theories of thermoelasticity. Also, the values of determinant of secular equation for G-L theory are higher in comparison to L-S theory for different values of wave number. The value of secular equation increases for all value of wave number. Figure 6 shows the variation of Stoneley waves velocity with respect to. Stoneley wave velocity increases smoothly with different values of wave number. As wave number increases, the value of Stoneley wave velocity also increases with respect to frequency. The value of Stoneley Figure 4. Comparison between LS and GL models on Stoneley waves attenuation coefficient with variesvaluesof = 0.1, 0.12, 0.13 with respect to Hz. Figure 6. Comparison between LS and GL models on Stoneley waves velocity with varies values of = 0.1, 0.12, 0.13 with respect to Hz. Figure 5. Comparison between LS and GL models on determinant of Stoneley waves secular equation with varies values of = 0.1, 0.12, 0.13 with respect to Hz. Figure 7. Comparison between LS and GL models on Stoneley waves attenuation coefficient with variesvaluesof = 0.1, 0.12, 0.13 with respect to Hz.

7 Stoneley Waves at the Boundary Surface of Modified Couple Stress Generalized Thermoelastic with Mass Diffusion 7 wave velocity is higher for G-L theory in comparison with L-S theory. Figure 7 depicts the variation of Stoneley waves attenuation coefficient with respect to angular velocity. It is noticed that the variation of attenuation coefficient for wave number ( = 0.1, 0.12, 0.13) show similar pattern in the range and opposite pattern is observed for both theories of thermoelasticity in the remaining range. The value of attenuation coefficient increases for L-S theory and decreases for G-L theory for all values of wave number. 10. Conclusions A mathematical model was used to study the propagation of Stoneley waves at the interface of two dissimilar isotropic modified couple stress thermoelastic diffusion medium subjected to mechanical, thermal and mass concentration boundary conditions. Secular equation is derived in the context of L-S and G-L theories of thermoelastic diffusion. Dispersion equations of Stoneley waves propagation are derived mathematically for a particular model. The determinant of secular equation, Stoneley waves velocity and attenuation coefficient are computed numerically and shown graphically for different values of wave number in the absence and presence of mass diffusion. It is observed from the figures that the value of determinant of secular equation increases with increase in the value of wave number for both L-S and G-L theories in the absence and presence of mass diffusion. Similarly, the values of Stoneley wave velocity and attenuation coefficient decreases as wave number increases in the presence of mass diffusion and opposite behavior is observed in the absence of mass diffusion. The results obtained in the study should be beneficial for researchers in material science, designers of new materials as well as for those working on modified couple stress theory of thermoelasticity. Also, the study of thermal and diffusion effects on Stoneley waves plays an important role in earthquakes, seismological processes and geodynamics. ij Kronecker s delta e ij Components of strain tensor e ijk Alternate tensor m ij Components of couple-stress t Coefficients of linear thermal expansion 1 (3 +2) t c Coefficients of linear diffusion expansion 2 (3 +2) c T Temperature change C Mass concentration Couple stress parameter ij Symmetric curvature i Rotational vector b Coefficient describing the measure of mass diffusion effects a Coefficient describing the measure of thermoelastic diffusion u Components of displacement vector Density Laplacian operator Del operator K Coefficient of the thermal conductivity c e Specific heat at constant strain T 0 Reference temperature assumed to be such that T/T 0 <1. D Thermoelastic diffusion constant 0, 1 Thermal relaxation times with , 1 Diffusion relaxation times with *2 ( t 2 ) ( 2) 2 c 1 * c 1 c Characteristic frequency of the medium Longitudinal wave velocity Wave number c Angular frequency Phase velocity of the wave References t ij, List of Symbols Components of stress tensor Material constants [1] Stoneley, R., Elastic Waves at the Surface of Separation of Two Solids, Proceeding of Royal Society of London, Vol. 106, pp (1924). doi: / ch3o

8 8 Rajneesh Kumar et al. [2] Tajuddin, M., Existence of Stoneley Waves at an Unbounded Interface between Two Micropolar Elastic Half Spaces, J. Applied Mechanics, Vol. 62, pp (1995). doi: / [3] Sinha, S. B. and Elsibai, K. A., Effect of the Sound Relaxation Time on the Propagation of Stoneley Waves, J. Thermal Stresses, Vol. 21, No. 1, pp (1998). [4] Ting, T. C. T., Secular Equations for Rayleigh and Stoneley Waves in Exponentially Graded Elastic Materials of General Anisotropy under the Influence of Gravity, J. Elast., Vol. 105, pp (2011). doi: /s [5] Abo-Dahab, S. M., Surface Waves in Coupled and Generalized Thermoelasticity, Advances in Materials and Corrosion, Vol. 2, pp (2013). doi: / _371 [6] Abo-Dahab, S. M., Surface Waves in Coupled and Generalized Thermoelasticity, Encyclopedia of Thermal Stresses, R. Hetnarski (ed.), (2014). doi: / [7] Abo-Dahab, S. M., Propagation of Stoneley Waves in Magneto-thermoelastic Materials with Voids and Two Relaxation Times, J. of Vibration and Control, Vol. 21, No. 6, pp (2015). doi: / [8] Kumar, R., Kumar, K. and Nautiyal, R. C., Propagation of Stoneley Waves in Couple Stress Generalized Thermoelastic Medium, Global J. of Science Frontier Research Mathematics and Decision Sciences, Vol. 13, No. 5, pp. 113 (2013). [9] Hideo, C. and Stanislav, I. R., Interface Wave Propagation and Edge Conversion at a Low Stiffness Interphase Layer between Two Elastic Solids: a Numerical Study, Ultrasonics, Vol. 62, pp (2015). doi: /j.ultras [10] Singh, A., Lakhsman, A. and Chattopadhyay, A., Effect of Internal Friction and the Lamé Ratio on Stoneley Wave Propagation in Viscoelastic Media of Order 1, International Journal of Geomechanics, Vol. 16, No. 4 (2016). doi: /(ASCE)GM [11] Voigt, W., Theoretische Studienuberdie Elasticitatsverhaltnisse der Krystalle, Abh. Ges. Wiss. Gottingen, 34 (1887). [12] Cosserat, E. and Cosserat, F., Theory of Deformable Bodies, Hermann et Fils, Paris (1909). [13] Yang, F., Chong, A. C. M., Lam, D. C. C. and Tong, P., Couple Stress Based Strain Gradient Theory for Elasticity, Int. J. Solids Struct., Vol. 39, pp (2002). doi: /S (02)00152-X [14] Simsek, M. and Reddy, J. N., Bending and Vibration of Functionally Graded Microbeams Using a New Higher Order Beam Theory and the Modified Couple Stress Theory, Int. J. of Engg. Sci., Vol. 64, pp (2013). doi: /j.ijengsci [15] Shaat, M., Mahmoud, F. F., Gao, X. L. and Faheem, A. F., Size-dependent Bending Analysis of Kirchhoff Nano-plates Based on a Modified Couple-stress Theory Including Surface Effects, Int. J. of Mech. Sci., (2014). doi: /j.ijmecsci [16] Darijani, H. and Shahdadi, A. H., A New Shear Deformation Model with Modified Couple Stress Theory for Microplates, Acta Mech., Vol. 226, pp (2015). doi: /s y [17] Nowacki, W., Dynamical Problems of Thermo Diffusion in Solids I, Bull Acad. Pol. Sci. Ser. Sci. Tech., Vol. 22, pp (1974). [18] Sherief, H. H., Saleh, H. and Hamza, F., The Theory of Generalized Thermoelastic Diffusion, Int. J. Engg. Sci., Vol. 42, pp (2004). doi: /j. ijengsci [19] Sherief, H. H. and Saleh, H., A Half-space Problem in the Theory of Generalized Thermoelastic Diffusion, Int. J. of Solid and Structures, Vol. 42, pp (2005). doi: /j.ijsolstr [20] Kumar, R. and Kansal, T., Propagation of Lamb Waves in Transversely Isotropic Thermoelastic Diffusion Plate, Int. J. of Solid and Structures, Vol. 45, pp (2008). doi: /j.ijsolstr [21] Kumar, R., Ahuja, S. and Garg, S. K., Numerical Analysis of the Propagation of Stoneley Waves at an Interface between Microstretch Thermoelastic Diffusion Solid Half Spaces, Latin American Journal of Solids and Structures, Vol. 11, pp (2014). doi: /S [22] Daliwal, R. S. and Singh, A., Dynamical Coupled Thermoelasticity, Hindustan Publishers, Delhi (1980). Manuscript Received: May 10, 2017 Accepted: Nov. 16, 2017

Mathematical modelling of Stoneley wave in a transversely isotropic thermoelastic media

Mathematical modelling of Stoneley wave in a transversely isotropic thermoelastic media Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 1 (June 2017), pp. 319-336 Applications and Applied Mathematics: An International Journal (AAM) Mathematical modelling