Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 173 (2017) 988 – 995 11th International Symposium on Plasticity and Impact Mechanics, Implast 2016 Influence of heterogeneity and initial stress on the propagation of Rayleigh-type wave in a transversely isotropic layer A K Vermaa*, A Chattopadhyayb, A K Singhc a, b, c Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India Abstract In this paper we study the propagation of Rayleigh-type wave in a heterogeneous transversely isotropic elastic layer with initial stress resting on a rigid foundation Frequency equation is obtained in closed form The frequency equation being a function of phase velocity, wave number, initial stress and heterogeneous parameter associated with the rigidity and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced by above stated parameters In Numerical and graphical computation the significant effects of distortional velocity have been carried out Moreover, the obtained dispersion relation is found in well–agreement to the classical case in isotropic and transversely isotropic layer resting on a rigid foundation ©©2017 Authors Published by Elsevier Ltd This 2016The The Authors Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility ofthe organizing committee of Implast 2016 Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: Rayleigh-type wave, transversely isotropic, initial stress, heterogeneity, rigid boundary Introduction The first theoretical investigations of Rayleigh wave carried out by Lord Rayleigh [1] in isotropic elastic media showed that these waves are particularly important in seismology since their propagation is confined to the surface, and therefore, they not scatter in depth as seismic body waves Later, Biot [2] studied the Rayleigh wave under the influence of initial stresses Rayleigh-type waves are of importance in several fields, from earthquake seismology and geophysical exploration to material science (Parker & Maugin [3]) Explicit secular equations of Rayleigh waves * Corresponding author Tel.: +91-8651344428 E-mail address: amitkverma.ismdhanbad@gmail.com 1877-7058 © 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 doi:10.1016/j.proeng.2016.12.168 989 A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 are important in practical applications Rayleigh wave propagation in layered heterogeneous media has been studied in details by Stoneley [4] and Newlands [5] Dutta [6] illustrated Rayleigh wave propagation in a two layer anisotropic media whereas Chattopadhyay et al [7] investigated Rayleigh wave in a medium under initial stresses Recently Chatterjee et al [8] have studied Rayleigh-type wave propagation in pre-stressed heterogeneous medium It is well known in the literature that the earth medium is not at all isotropic throughout, but it is anisotropic Transverse isotropy is generally regarded as the commonest form of anisotropic symmetry Therefore, such materials are also known as polar anisotropic materials Chattopadhyay [9] studied the strong SH motion in a transverselyisotropic layer lying over an isotropic elastic material due to a momentary point source For heterogeneous and anisotropic media the mathematical formulation of Rayleigh waves becomes very complex and there can be cases of anisotropic media where they not exist at all However in the case of transversely isotropic medium with the free surface parallel to the isotropy plane (common situation for soil systems) Rayleigh waves exist and the analogous of the Lamb solution can be found As far as heterogeneity is concerned, the mechanical properties of the medium are often dependent on the space variable (mostly on depth y ) Motivated by such facts we aimed to study the problem in initially stressed heterogeneous transversely isotropic medium on the propagation of Rayleigh wave Wave propagation in an isotropic surface stratum of the earth resting on an extremely rigid foundation has been investigated by Sezawa and Kanai [10] Assuming that earth crusts resting on rigid foundation, Das Gupta [11] and Dutta [12] studied on the propagation of Rayleigh waves in an isotropic layer and transversely- isotropic layer respectively resting on rigid Medium Till date no attempt has been made to study the Rayleigh-type propagation in an initially stressed heterogeneous transversely isotropic layer with rigid base In this paper we aim to study the propagation of Rayleigh-type wave in an initially stressed heterogeneous transversely isotropic layer in effect of rigid foundation The study of the effect of rigid foundation in layer on propagation of elastic wave has always been of great concern to civil engineers and seismologists Analytical treatment has been done to find the dispersion relation from where the real part of the expressions will give the dispersion relation of phase velocity It is found that the heterogeneity and initial stress has a significant favouring effect on phase velocity of Rayleigh-type wave when the layer has rigid base It is also to be noted that as the phase velocity decreases with increase in heterogeneity parameter Deduced result is found in well-agreement to the established standard results existing in the literature Formulation of the problem We consider a heterogeneous initially stressed transversely isotropic layer of finite thickness H with rigid surface at y H Let us choose a co-ordinate system in such a way that, y axis is directed vertically downwards and x - axis is assumed in the direction of the propagation of Rayleigh-wave The geometry of problem is depicted in Fig.1 O S11 S11c eD y x Heterogeneous transversely isotropic layer H Rigid Surface y Fig 1: Geometry of the problem 2.1 Solution of the Problem 990 A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 According to Biot (1965), in the absence of body forces, the equilibrium equations in the Cartesian coordinate system x, y, z for the unbounded medium with initial stress Sij are wW ij wx j w ª SkjZik Sij e Sik ekj ẳ wx j where Zik U w 2ui , wt ui, k uk ,i and ui, j (1) wui , i, j , k wx j 1, 2,3 For propagation of Rayleigh-type wave in x -direction, it is assumed that w u u x, y, t , v = v x, y, t , w and { wz The heterogeneity in the layer is taken as follows C11 C11c eD y , C12 C12c eD y , S11 S11c eD y In view of equations (2) and (3), equation (1) leads to wW x x wW x y w2 u ½ U ,° wx wy wt ° ¾ wW x y wW y y w2 v ° U ° wx wy wt ¿ where ª wu wv º C12c W x x eD y «C11c ằ, wy ẳ wx Wy y Wx y ê wu wv º C11c eD y «C12c », w yẳ wx ê1 Đ w u w v à eD y ô C11c C12c ă áằ â w y w x ạẳ ơ2 Using (5) in (4) we get ªwv wu º w2 u w2 u w2 v C11c L M DL « » 2 w xw y wx wy ¬w x w y ¼ U w2 u , w t2 w2 u U wt C11c C12c S11c C11c C12c S11c where L and M 2 For the solution of equation (6) and (7); we consider P y eiK ( x ct ) ẵ ắ v x, y , t Q y e °¿ Now using equation (8) in equation (6) and (7); we get ª LD2 D LD U c2 C11c K P y êơiK MD D L ẳ Q y ẳ 2 ª c º c c ¬C11 D D C11 D U c L K ¼ Q y êơiK MD D C12 ẳ P y Now substituting iK ( x ct ) (3) (4) (5) ª w2 v w2 v w 2u wu wv º M D «C12c C11c L C11c » w xw y w w x yẳ wx wy u x, y , t (2) (6) (7) (8) (9) (10) 991 A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 P y K1eim1ky K eim2 ky K 3e im1ky K e im2 ky Q y K1eim1ky K eim2 ky K 3e im1ky K e im2 ky (11) Using (11) in equation (9) and (10); we get eim1ky ª K1 m12 Lk U c C11c K im1D Lk K1 ^iD LK m1MK k ` ẳ eim1ky e ^ ` ª K m Lk U c C c K im D Lk ` K ^m M K k iD LK` ^ ẳ ê K ^m Lk U c C c K im D Lk` K ^iD LK m MK k ` ẳ ê K ^m Lk U c C c K im D Lk ` K ^m M K k iD LK`º ¬ ¼ eim2 ky and 11 im2 ky 2 2 2 11 (12) 2 2 2 11 ^ ^ ^ ª K m MK k` K ^m ¬ ^ ` ` eim1ky ª K1 ^m1 MK k` K1 m12C11c k U c L K im1D C11c k º ¬ ¼ im1ky ª 2 2 e K m MK k` K3 m1 C11c k U c L K im1D C11c k º 3^ ẳ im2 ky ê 2 2 e K m MK k` K m2 C11c k U c L K im2D C11c k 2^ ẳ eim2 ky 4 2 C11c k U c ` L K im D C c k `º ¼ 2 11 (13) Thus from equation (12) and (13), we get K1 m1MK k iD LK K , K1 m12 Lk U c C11c K im1D Lk K m2 Lk U c C11c K im2D Lk m1 MK k iD LK K , m12 Lk U c C11c K im1D Lk K m2 Lk U c C11c K im2D Lk K3 K3 m2 MK k iD LK , m2 MK k iD LK Thus K1 G1 K1 , K G K , K3 b1 K3 , K b K where m j Lk U c C11c K im jD Lk Gj bj m j M K k iD LK m j Lk U c C11c K im jD Lk m j M K k iD LK j And hence, we may take K e v x, y , t K e u x, y , t e im1K y K eim2K y K 3e im1K y K e im2K y eiK ( x ct ) im1K y K eim2K y K 3e im1K y K e im2K y 1 1, where m j j 1, are the roots of iK ( x ct ) ½ ° ¾ °¿ (14) m j LC11c K m j 2iLC11c K 3D m j êơC11c U c C11c K L U c L K LC11c D 2K ^ ` M 2K º¼ im j L U c L C11c U c C11c ML K 3D U c C11c U c L K with positive real part and Kj G jKj, Kj J jKj where (15) 992 A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 Gj m j LK U c C11c K im jD L m j MK iD L and J j m j LK U c C11c K im jD L m j MK iD L 2.2 Boundary Conditions for the proposed model are as: i) W xy at y 0, ½ ° ii ) W yy at y 0, ° ¾ iii ) u at y H, ° iv) v at y H °¿ with help of equations (8), boundary conditions (11) result in K1 m1 G1 K m2 G K3 m1 J K m2 J 0, K1 C12c G1C11c m1 K C12c G 2C11c m2 K C12c J 1C11c m1 K C12c J 2C11c m2 K G e K G e K J e K J e K1 eim1K H K eim2K H K3 e im1K H K e im2K H im1K H im1kH im2 kH 2 (16) (17) 0, (18) 0, im2 kH Now in order to eliminate the arbitrary constants K1 , K2 , K3 , K4 , we will have following determinant Real a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 where a11 m1 G1 , (19) a21 m2 G , a13 m1 J , a14 m2 J , C12c G1C11c m1 , a22 C12c G 2C11c m2 , a23 C12c J 1C11c m1 , a24 C12c J 2C11c m2 , a31 eim1K H , a41 G1eim K H , a42 a12 a32 eim2K H , G eim K H , a43 a33 e im1K H , a34 J 1e im K H , a44 e im2K H , J e im K H Eq (19) is the required dispersion equation of Rayleigh-type wave in an initially stressed heterogeneous transversely isotropic layer resting on rigid base The dispersion relation being a function of phase velocity, wave number, initial stress and heterogeneity parameter, reveals that the fact that Rayleigh-type wave propagation is greatly influenced by above stated parameters However, under the condition when D and S11 0, Eq (19) gives the dispersion equation of Rayleigh-type wave in homogeneous transversely isotropic layer resting on rigid base and in similar fashion for C11c O 2P , C12c O with D 0, S11 0, Eq (19) gives the dispersion equation of Rayleigh-type wave in homogeneous isotropic layer resting on rigid base Numerical discussion: In this section, we carry out numerical computations and graphical demonstrations for the deduced closed form dispersion equation when Rayleigh-type wave is propagating in an initially stressed heterogeneous transversely §c C11c · isotropic layer in effect of rigid foundation Graphical interpretation of phase velocity ăă , where E U áạ âE reflecting the effect of various affecting parameters, viz non-dimensional heterogeneity parameter, without heterogeneity, non-dimensional initial stress parameter ,without initial stress parameter are presented in figures 2-5 The following material constants are taken into considerations: For transversely isotropic Beryl material: [13] A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 c C11 c 26.94 u1011 dyne/cm2 , C12 9.61u1011 dyne/cm , U =2.7 gm/cm3 Fig 2: The variation of non-dimensional phase velocity heterogeneity Đcà ă against non-dimensional wave-number K H for different values of âE D H in layer without initial i.e S11 Fig 3: The variation of non-dimensional phase velocity heterogeneity §c· ¨ ¸ against non-dimensional wave-number K H for different values of âE D H in layer with fixed value of initial stress 993 994 A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 Fig 4: The variation of non-dimensional phase velocity S11 Fig 5: The variation of non-dimensional phase velocity Đcà ă against non-dimensional wave-number K H for different values of initial stress âE without heterogeneity in layer i.e D Đcà ă against non-dimensional wave-number K H for different values of initial stress âE S11 with heterogeneous layer Conclusions As the outcome of the present study, it is found that the propagation of Rayleigh-type wave is greatly influenced by the effect of various dimensionless elastic parameters, initial stress and heterogeneity factor In particular, the following conclusions can be made as Wave-length has a significant favouring effect on phase velocity of Rayleigh-type wave when the layer has rigid A.K Verma et al / Procedia Engineering 173 (2017) 988 – 995 995 base Phase velocity decreases with increase in heterogeneity without initial stress with respect to wave number on the situation that layer is comprised of transversely isotropic material The heterogeneity parameter affects considerably phase velocity It is found that for fixed initial stress phase velocity decreases with increase in heterogeneity parameter with respect to wave number For initially stressed homogeneous transversely isotropic layer, the phase velocity decreases with increase in initial stress with respect to wave number The phase velocities of Rayleigh-type wave decreases in the case when the heterogeneity parameter is fixed with increase in the value of initial stress parameter irrespective of the fact that anisotropy is present in the layer in effect of rigid foundation The present study may have its possible applications in the sphere of civil engineering, earthquake engineering, engineering geology and seismology Acknowledgements The authors convey their sincere thanks to the Indian Institute of Technology (Indian School of Mines), Dhanbad, India, for granting access to its best research facility and providing Junior Research Fellowship to Mr Amit Kumar Verma References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] L Rayleigh, on waves propagating along the plane surface of an elastic solid, Proc R Soc Lond vol 17, pp 4–11, 1885 M A Biot, The influence of initial stress on elastic waves, Journal of Applied Physics, vol 11, no 8, pp 522-530, 1940 D Parker, and G.A Maugin, eds Recent Developments in Surface Acoustic Waves: Proceedings of European Mechanics Colloquium 226, University of Nottingham, UK, September 2–5, 1987, vol Springer Science & Business Media, 2012 R Stoneley, The transmission of Rayleigh waves in a heterogeneous medium, Geophysical Supplements to the Monthly Notices of the Royal Astronomical Society, vol 3, no 6, pp 222-232, 1934 M Newlands, and R Stoneley, Rayleigh waves in a two‐layer heterogeneous medium, Geophysical Journal International, vol 6, no s2, pp 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