Effect of Rotation, Magnetic Field and Initial Stresses on Propagation of Plane Waves in Transversely Isotropic Dissipative Half Space

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1 Applied Mathematis Published Online August ( Effet of otation Magneti Field and Initial Stresses on Propagation of Plane Waves in Transversely Isotropi Dissipative Half Spae Sushant Shekhar Imtiyaz A. Parvez Aademy of Sientifi and Innovative esearh New Delhi India CSI Center for Mathematial Modelling and Computer Simulation NAL Belur Campus Bangalore India sushant.shekhar85@gmail.om parvez@mmas.ernet.in eeived May ; revised June ; aepted June Copyright Sushant Shekhar Imtiyaz A. Parvez. This is an open aess artile distributed under the Creative Commons Attribution Liense whih permits unrestrited use distribution and reprodution in any medium provided the original work is properly ited. ABSTACT The problem regarding the refletion of plane waves in a transversely isotropi dissipative medium is onsidered in whih we are studying about the refletion of inidene waves in initially stressed dissipative half spae. After solving the governing equations we find the two omplex quasi-p (qp) and quasi-sv (qsv) waves. The ourrene of refleted waves is studied to alulate the refletion oeffiient and the energy partition of inidene wave at the plane boundary of the dissipative medium. Numerial example is onsidered for the refletion oeffiient and the partition of inident energy in whih we study about the effet of rotation initial stresses and magneti field. Keywords: Magneti Field; Initial Stress; otation; efletion of Plane Waves; efletion Coeffiients; Energy atios. Introdution In reent years most of the researhers are interested to solve the problems related to eletrially onduting elasti media permeated by uniform magneti fields. The seismi wave propagation has been used for various studies related to magneto-elastiity on the Earth s mantle and ores. By the knowledge related to refletion and refration the plane waves are the soure of information used to image the Earth s interior. Dissipation of the medium depends upon the internal struture. The initial stress in the medium may be developed due to some reasons suh as slow proess of reep gravity external fores differene in temperature et. Problem related to plane waves in transversely isotropi medium is very important for the possible appliation in various branhes of siene and tehnology suh as earthquake siene aousti geophysis and optis et. Biot [] observed that the initial stresses have notable effet on the propagation on elasti waves in a medium. The problem related to refletion and refration of elasti waves from the boundaries of different media has been disussed in the famous book by Ahenbah []. Borejko [3] introdued the refletion and transmission oeffiients for three dimensional plane waves in elasti media. By using the Biot s theory [4] several researhers [5-9] have studied extensively the propagation of elasti waves. The problem related to refletion and refration of qp and qsv waves at the interfae of Fiber-reinfored medium has been disussed by Chattopadhyay and Venkateswarlu []. A huge amount of mathematial work has been performed for the propagation of elasti waves in dissipative medium; see for instane [-4]. The problem related to thermo-magneto-dynami stresses and perturbation of magneti field vetor in a non-homogeneous hollow ylinder has been studied by Kong et al. [5]. In the present study we have used the Biot s inremental deformation theory [4] for deriving the algebrai expressions of the refletion oeffiients and energy ratios when plane waves of qp and qsv type are inident on the initially stressed dissipative half-spae with stress free boundaries. The dispersion equations for refletion oeffiients and energy ratios of inident qp and qsv waves on the free surfae of the initially stressed dissipative medium have been derived and observed that in presene of initial stresses magneti field and rotation of the medium must affet the refletion oeffiients and energy shares of refleted plane waves. Copyright Sies.

2 8 S. SHEKHA I. A. PAVEZ. Formulation of the Problem We onsider the medium is a perfet eletri ondutor. The Maxwell s equations of eletromagneti field for perfetly onduting elasti medium are given by ayhoudhuri and Banerjee [7]. h h J E e t h E huh H H h where is the Hamilton s operator is the magneti permeability J is the urrent density vetor E and h h are the indued eletri field and the indued magneti field both are developed due to appliation of initial magneti field H H. We onsider a transversely isotropi dissipative half spae rotating about y-axis Ω under onstant magneti field (along y-axis) and initial ompressive stress P along x-axis (Figure ). The equations of motion in x-z plane for the present problem may be written in the forms given by Biot [4] s s u w P eh x z z x xz u w t t s3 s33 u w P eh x z x xz z w u t t u w w u where is density is the rotational x z omponent and s i j are inremental stress tensor in x-z plane. () () The stress-strain relation for transversely isotropi medium an be given by Biot [] s P e P e 33 s s e s e e where are the elasti oeffiients. For dissipative medium the elasti oeffiients must be of omplex onstants an be written as I I i i (4) I i i I 44 I I I I where i are all real onstants. The inremental strain omponents e i j are related to the displaement omponents uw whih are given by the relation u w w u e e33 e (5) x z x z Let the harmoni solution of Equation () for the propagation of plane waves the displaement omponents are written as follows: n n n n u w Ad n Ad n 3 Where the index n is assigns an arbitrary diretion of propagation of the plane waves. n n p and p 3 are osine of angle made by diretion of propagation with the normal (z-axis) n n ik n xp zp3 vnt (3) e (6) n n 3 d d are the omponent of unit displaement vetor A n is the arbitrary onstant and v n and k n are the veloity of propagation and orresponding wave number. Substituting Equation (6) in () with the help of Equations (3)-(5) then we get the veloities orrespond to quasi-p(qp) and quasi-sv(qsv) wave respetively. n n L vn And L ivn n 3 A d (7) n n L ivn Ad n L3 vn Ad n 3 (8) For non-trivial solution of Equations (7) and (8) the determinant must be equal to zero. That gives two values of v whih are given by n Figure. Geometry of the problem. v v n p L L L L 4L 3 3 (9) Copyright Sies.

3 S. SHEKHA I. A. PAVEZ 9 where v v n SV L L L L 4L 3 3 P L P H p p n n e I n I n 44 3 I n I n i p p P L H p p i p p P n n L3 44 p 33 eh p3 () n n n n I I 44 e i p p The real and imaginary parts of Equations (9) and () represents the phase veloities and damping veloities of qp and qsv waves respetively. From the Equations (9) and () we an say that both v P and v SV depend on initial stresses rotation magneti field damping and dire- n n tion of propagation p p. Also relation between the unit displaement vetors an be given as L v n L iv 3 d. n d 3 L v v L i () 3. efletion of Plane Waves from Stress Free Surfae We onsider an initially stressed rotating magneto-elasti transversely isotropi dissipative half spae absorbing the region z. In this setion we are disussing about the refletion oeffiients for inident qp and qsv waves. The displaement omponents of solid partiles in transversely isotropi dissipative medium due to presene of an inidene wave and refleted waves are expressed as follows: 3 3 u x z t A d exp ik p x p x v t j j j 3 3 Ajd exp i k p x p x vjt j () w x z t A d exp ik p x p x v t j j j j j 3 exp i 3 3 A d k p x p x v t j We are assuming the supersript for inident qp and qsv waves () for refleted qp waves and () for refleted qsv waves. The boundary onditions for the free surfaes are vanishing of inremental boundary fores. So the two bound- ary onditions required to be satisfied at the plane z are n n n z 33 f s e P f s () x Equation () an be written as s s s e e e P s s s (4) Equation () will satisfy the boundary ondition s (4) if the following Snell s law holds p p p v v v with the relations A A A A A A where kl p3 d p d3 kl p3 d p d3 kl p3 d p d3 3 k 33 p3 d3 p 4 k 33 p3 d3 p d 5 k 33 p3 d3 p d (5) (6) d (7) P L 44 The amplitude ratios for inidene qp and qsv waves an be given by A 5 3 A 4 3 (8) A 5 4 A 4 5 The real part of Equation (8) allow to determine the refletion oeffiients of the refleted qp and qsv waves at a given inident qp and qsv waves. The refletion oeffiients are denoted by Z PP (the inident qp wave is refleted as the qp wave) Z PS (the inident qp wave is refleted as the qsv wave) Z SP (the inident qsv wave is refleted as the qp wave) and Z SS (the inident qsv wave is refleted as the qsv wave). 4. Energy atios We are onsidering the surfae element of unit area at the Copyright Sies.

4 S. SHEKHA I. A. PAVEZ half spae z. Our main purpose is to alulate the distribution of energy (of the inident wave) among the two refleted waves on this half spae. Following Ahenbah [] the rate at whih the inident energy transmitted per unit area is given by the salar produt of surfae tration and partile veloity denoted by P. The time average of P ov er a period denoted by P represents the average energy transmission per unit area per unit time. The average energy flux for a given surfae with normal along z-diretion is represented through the omponents P is given by i j i j P s u s u (9) The distribution of inident energy is given in the form of a matrix at the free surfae of the dissipative medium. E i j () P P The element of matrix P is defined as follows: i 3 i i j i 3 i 44 j i 3 i 3 d i p d 33 i p d 3 j vjkikj P AA d p d p d where bar over entity implies omplex onjugate. The sum of all non-diagonal entries of this matrix alulates the share of interation energy in the medium. The di- the energy shares of refleted qp and qsv waves in the inident energy. The energy due to interation be- agonal entries E and E of the matrix (Equation ()) denote tween inident wave and two refleted waves is given by E E E I i i i. The energy due to the interation 5. Numerial esults and Disussion () among the two refleted waves is given by E E Eii. The onservation of the energy i j at the free-surfae is given by the relation. E E E E () I Material parameter for opper-alloy is hosen for the numerial model of magneto-elasti medium. The elasti and dynami onstants are given by [8].68 N m.56 N m N m N m I I.5 N m N m I I.45 N m N m 7.4 kg m H e.4 N m The given values of various parameters are used to alulate the omplex veloities (qp and qsv waves) refletion oeffiients (Z PP Z PS Z SP and Z SS ) and energy shares (E E E I and E ) in the dissipative medium. The variation of refletion oeffiients and Energy ratios with angle of inidene (θ) are shown in Figures -5 (for inidene qp wave) and in Figures 6-9 (for inidene qsv wave). Figure. efletion oeffiients of qp and qsv waves with variations in inident angle (θ) and initial stress P for inident qp wave. Figure 3. efletion oeffiients of qp and qsv waves with variations in inident angle (θ) and Effet of rotation parameter Ω ω for inident qp wave. Figure 4. Energy shares of refleted (qp qsv) waves inident and refleted waves interation and interation among refleted waves variations with inident angle (θ) and Initial stress (P) for inident qp wave. Copyright Sies.

5 S. SHEKHA I. A. PAVEZ Figure 5. Energy shares of refleted (qp qsv) waves inident and refleted waves interation and interation among refleted waves variations with inident angle (θ) and rotational parameter Ω ω for inident qp wave. Figure 8. Energy shares of refleted (qp qsv) waves inident and refleted waves interation and interation among refleted waves variations with inident angle (θ) and Initial stress ( P) for inident qsv wave. Figure 6. efletion oeffiients of qp and qsv waves with variations in inident angle (θ) and initial stress (P) for inident qsv wave. Figure 7. efletion oeffiients of qp and qsv waves with variations in inident angle (θ) and Effet of rotation parameter Ω ω for inident qsv wave. Figure shows that the variation in the refletion oeffiients (Z PP and Z PS ) with respet to angle of inidene (θ) for different values of initial stress (P). The omparison of the solid and dashes lines shows that the refletion oeffiients Z PP and Z PS hange in presene of initial stresses for eah angle of inidene of qp wave exept grazing inidene. The effet of initial stresses is mini- Figure 9. Energy shares of refleted (qp qsv) waves inident and refleted waves interation and interation among refleted waves variations with inident angle (θ) and rotational parameter Ω ω for inident qsv wave. mum in the range of 5 < θ < 85 for Z PP and < θ < 8 for Z PS. From Figure 3 it is notied that Z PP and Z PS hanges in the presene of rotation of the medium at eah angle of the inidene of qp wave exept some grazing points. The Figure 4 for inidene qp wave shows that the energy variations with respet to angle of inidene (θ) for different values of initial stress P. The refleted qp and qsv waves reflet strongly in the presene of initial stress. We an see that as inrease of initial stress the refleted qp wave reflets slowly in the range of < θ < 3 but the refleted qsv wave reflets strongly in this range. For normal inidene interation between inidene wave and refleted waves is maximum whereas interation between refleted waves is absent. For the inidene of qp wave effet of different values of rotation parameter on energy partition is shown in Figure 5. It is lear that at inident angle θ = no effet of hange in rotation parameter is shown. Also both qp and qsv waves reflet weakly with inrease the rotation parameter in the range (6 < θ < 9 ) The ritial angle for inidene qp wave is around 6. Copyright Sies.

6 S. SHEKHA I. A. PAVEZ From Figures 6 and 7 it an be easily seen that for the inidene of qsv wave the refletion oeffiients Z SP = and Z SS = at the inidene angle θ = and θ = 9 It shows that there is only one refleted qp wave. For the inidene of qsv wave the hange in energy shares of refleted waves with hange in initial stress is shown in Figure 8. The refleted qp and qsv waves are not more affeted by the hange in initial stress; the small hange is shown in the range (5 < θ < 7 ) for qsv wave. Also the hange in initial stress affets the interation between inidene and refleted waves in the range (5 < θ < 8 ) but very less for the interation between refleted waves. The Figure 9 for inidene qsv wave shows that energy variations with respet to inident angle (θ) for different values on rotation parameter. As inreasing the rotation parameter results is stronger the refleted qsv wave but weaken the refleted qp wave. For normal inidene qsv wave reflets strongly but qp wave is almost absent. A signifiant effet of the magneti field on the propagation of plane waves in dissipative medium is observed by the omparison of our results with results of the study see []. From the omparison of Figure with Figure of the study see [] we find that no signifiant effet has been shown in the study see [] but in our results it is lear that how does initial stress affet the refletion oeffiients. If we neglet the effet of magneti field then our results oinide with the study see [] exept the energy ratios. From the omparison of the Figure 3 with Figure 3 of the study see [] it is lear that in presene of magneti field the refletion oeffiient dereases. Comparison of Figure 6 is made with Figure 4 of the study see [] the maximum value of refletion oeffiients for angle of inidene in between ( < θ < 45 ) is around θ = 3 but it is obtained maximum at an angle θ = for the study see []. From the omparison of these results with the study see [] we find that physis of the problem will hange in the presene of the magneti field. 6. Conlusion The refletion fro m the stress free surfae of an initially stressed rotating magneto-elasti transversely isotropi dissipative medium is onsidered. The derived expression of refletion oeffiients and energy ratios are obtained for inidene of qp and qsv waves. For opper alloy material refletion oeffiients and energy ratios are omputed and presented graphially with angle of inidene for different values of initial stress and rotation parameter in presene of magneti field. From the graphial representation it is observed that the presene of initial stresses rotation parameter and magneti field affet signifiantly to the refletion oeffiients and energy ratios. 7. Aknowledgements The authors onvey their sinere thanks to Counil of Sientifi and industrial researh (CSI) New Delhi for providing SF to Mr. Sushant Shekhar and also thanks to Sientist in-harge CMMACS Bangalore for providing all failities related to this researh work. EFEENCES [] M. A. Biot The Influene of Initial Stress on Elasti Waves Journal of Applied Physis Vol. No pp doi:.63/.787 [] J. D. Ahenbah Wave Propagation in Elasti Solids North-Holland Pub. Co. New York 973. [3] P. Borejko efletion and Transmission Coeffiients for Three-Dimensional Plane Waves in Elasti Media Wave Motion Vol. 4 No pp doi:.6/s65-5(96)6- [4] M. A. Biot Mehanis of Inremental Deformation Wiley New York 965. [5] S. Dey and D. Dutta Propagation and Attenuation of Seismi Body Waves in Initially Stressed Dissipative Medium Ata Geophysia Vol. 46 No pp [6]. W. Ogden and D. Sotirropoulos efletion of Plane Waves from the Boundary of a Pre-Stressed Compressible Elasti Half-Spae IMA Journal of Applied Mathematis Vol. 6 No. 998 pp doi:.93/imamat/6..6 [7] S. K. oyhoudhuri and S. Banerjee Magneto-Thermoelasti Interation in an Infinite Visoelasti Cylinder of Temperature-ate Dependent Material Subjeted to a Periodi Loading International Journal of Engineering Siene Vol. 36 No pp doi:.6/s-75(97)96-7 [8] M. D. Sharma Effet of Initial Stress on efletion at the Free Surfaes of Anisotropi Elasti Medium Journal of Earth System Siene Vol. 6 No. 6 7 pp doi:.7/s [9]. D. Tooly T. W. Spener and H. F. Sagoi efletion and Transmission of Plane Compressional Waves Geophysis Vol. 3 No pp doi:.9/.4396 [] A. Chattopadhyay and. L. K. Venkateswarlu efletion and efration of Quasi P and SV Waves at the Interfae of Fiber-einfored Media Advaned Studies in Theoretial Physis Vol. No. 7 pp [] A. N. Norris Propagation of Plane Waves in a Pre- Stressed Elasti Media Journal of the Aoustial Soiety of Ameria Vol. 74 No pp doi:./.39 [] B. Singh and J. Arora efletion of Plane Waves from a Free Surfae of an Initially Stressed Transversely Isotropi Dissipative Medium Applied Mathematis Vol. pp [] B. Singh and J. Arora efletion of Plane Waves from Free Surfae of an Initially Stressed otating Orthotropi Copyright Sies.

7 S. SHEKHA I. A. PAVEZ Dissipative Solid Half Spae Engineering Vol. 4 No. 3 pp doi:.436/eng..43 [4] M. M. Selim efletion of Plane Waves at Free Surfae of an Initially Stressed Dissipative Medium World Aademy of Siene Engineering and Tehnology Vol. 3 8 pp. -7. [5] T. Kong D. X. Li and X. Wang Thermo-Magneto-Dynami Stresses and Perturbation of Magneti Field Vetor in a Non-Homogeneous Hollow Cylinder Applied Mathematial Modelling Vol. 33 No. 9 pp doi:.6/j.apm.8..3 Copyright Sies.