on rayleigh waves in self reinforced layer embedded over an incompressible half space with varying rigidity and density

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 173 (2017) 1021 – 1028 11th International Symposium on Plasticity and Impact Mechanics, Implast 2016 On Rayleigh waves in self-reinforced layer embedded over an incompressible half-space with varying rigidity and density Shishir Gupta a , Mostaid Ahmed a,∗ a Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, India Abstract In this paper a mathematical model to study the Rayleigh wave propagation in a self-reinforced layer over an incompressible inhomogeneous elastic half-space with quadratically varying rigidity and linearly varying density has been represented The displacement components for both the media have been deduced separately Under suitable boundary conditions the secular equation has been established in closed form Numerical results analyzing the equation are discussed and presented graphically It is quite ocular that the reinforcement in the upper medium and variations in density and rigidity related to the lower semi-infinite medium has remarkable effect on the Rayleigh wave propagation Due to the importance of self-reinforced material, the above problem has certain realistic applications in geophysics and civil engineering ©c 2017 Published by Elsevier Ltd This 2016The TheAuthors Authors Published by Elsevier B.V.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 Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: Rayleigh wave; Self-reinforced material; Incompressible, Whittaker function Introduction The study of elastic waves and the nature of earthquakes remained as paramount interest to the geophysicists The primary work on elastic waves received it impetus from the earlier studies by the scientists around the middle of the nineteenth century Owing to slower attenuation of energy compare to the body waves, Rayleigh type surface waves are the most destructive seismic surface waves that can unobstructedly transmit along the stress free surface with the phase velocities lying within the subsonic range and its amplitude decaying exponentially with depth from the surface In case of layered earth media, measuring Rayleigh waves is one of the versatile and convenient tool to predict disasters caused by earthquakes and to find out various material properties of the layers present in earth’s crust The study of these surface waves in heterogeneous medium with different boundaries is of great importance to the seismologists Rayleigh waves are generally non-dispersive in nature but exhibit dispersion in case of stratified semi-infinite medium Lord Rayleigh [1] came out with an pioneering work where he showed the how these kind of waves propagate in isotropic elastic solid Bromwich [2] demonstrated the Rayleigh type surface wave propagation in isotropic layered media which was found as dispersive ∗ Corresponding author Tel.: +91-7870591051 E-mail address: mostaidahmed@yahoo.in 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.2016.12.178 1022 Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 Two brittle materials of different physical and chemical properties are brought together to form a composite material Fiber composites produces a composition of much preferred mechanical properties that can not be achieved by either of the ingredients alone and having fatigue resistance with high strength and stiffness of fibers Due to excessive temperature and pressure these fiber materials are transformed into self-reinforced fibers The ocular property retained by fiber-reinforced concretes is that its components act together as a single body such that there can be no relative displacement between them under any external forces Nylon, aramid, vinylester etc lie on the class of fiber-reinforced composites Massive applications of fiber-reinforced composites can be found in the field of structural and civil engineering, aerospace engineering and automobile industries Belfield et al [3] first introduced the idea of continuous self-reinforcement at each point of elastic solid Later on, Spencer [4] established the constitutive equation for fiber-reinforced linearly anisotropic elastic solid in preferred direction Kaur et al [5] formulated the effect of reinforcement, gravity and liquid loading on Rayleigh wave propagation The layered earth medium comprises of various hard/soft rocks with dissimilar elastic properties When these type of rocks persist in the way of seismic waves they influence the transmission of waves Any disturbance in the earth’s interior may serve as the basic cause of seismic wave propagation The inhomogeneity plays a pivotal role in study of dynamic problems on seismic waves In a pioneering work, Birch [6] noted down the nature of seismic waves with increasing depth below the earth’s crust Singh [7] discussed about Rayleigh waves in an inhomogeneous medium Vishwakarma and Gupta [8] demonstrated a case study on Rayleigh wave propagation in an incompressible crustal layer under rigid boundary over a dry sandy and elastic half-space with void pores Recently Vishwakarma and Xu [9] have shown the effect of quadratically varying rigidity and linearly varying density on the Rayleigh wave propagation Sharma and Sharma [10] modeled the phenomenon of thermoelastic Rayleigh waves in a solid over a fluid layer with varying temperature With these geophysical notions and motivated by certain practical situations we have studied the propagation of Rayleigh waves in self-reinforced layer embedded over an incompressible inhomogeneous half-space by deducing the secular equation The effect of inhomogeneity parameter and thickness of layer on the phase velocity of Rayleigh wave has been identified Within the theoretical framework the outcome of the study bears a great significance in diverse fields like geophysics, geotechnical engineering and civil engineering etc Formulation of the problem In this present investigation, we consider an incompressible inhomogeneous half-space with quadratically varying rigidity and density covered with a self-reinforced layer of thickness H Referring to the rectangular Cartesian coordinate system, O lies at the common interface of the layer and the semi-infinite medium, x-axis is taken along the propagation of Rayleigh wave and z-axis directed vertically downwards Owing to give it an algebraic formation, the lower semi-infinite medium is considered to occupy the region z ≥ 0, whereas the upper layer occupies the region −H < z < The uppermost boundary surface of the fiber-reinforced layer is defined as z = −H Fig Geometry of the Problem Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 1023 Dynamics of self-reinforced medium The elasto-dynamical eqns of motion for a three dimensional elastic solid medium are given by ∂2 u1 ∂τ xx ∂τ xy ∂τ xz + + = ρ1 ∂x ∂y ∂z ∂t ∂τyx ∂τyy ∂τyz ∂2 v1 + + = ρ1 ∂x ∂y ∂z ∂t ∂τzx ∂τzy ∂τzz ∂2 w1 + + = ρ1 ∂x ∂y ∂z ∂t (1) (2) (3) where ρ1 is the density of the material, (u1 , v1 , w1 ) are the displacement components and (x, y, z) be the coordinates at any time t The governing eqn for a self-reinforced linearly elastic medium with preferred direction ‘a’is given by Belfield [3] as τi j = λekk δi j + 2μT ei j + α(ak am ekm δi j + a j ekk ) + 2(μL − μT )(ai ak ek j + a j ak eki ) + β(ak am ekm a j ) (4) where i, j, k, m = 1, 2, τi j are the components of stress, ei j = 12 ui, j + u j,i represent the components of infinitesimal −a , all referred to rectangular Cartesian coordinates x , strain, δi j denotes the Kronecker delta are components of → i → −a = (a , a , a ) is the preferred directions of reinforcement under the restrictive assumption of a + a + a = 1 3 −a may be a function of position The coefficients λ, μ , α, β & μ are the elastic constants with dimension The vector → T L of stress μT refers to the shear modulus in transverse shear across the preferred direction, μL as the shear modulus in longitudinal shear in preferred direction α, β are the specific stress components that are taken into account in regard of the composite material −a is chosen so that its components are (1, 0, 0) i.e., the preferred direction of reinforcement is everywhere along If → x− axis, then (4) leads to τ xx = (λ + 2α + 4μL − 2μT + β) e xx + (λ + α) eyy + (λ + α) ezz τyy = (λ + α) e xx + (λ + 2μT ) eyy + λezz τzz = (λ + α) e xx + λeyy + (λ + 2μT ) ezz τyz = 2μT eyz , τ xz = 2μL e xz τ xy = 2μL e xy The Rayleigh waves are propagating in the x− direction, so we may write u1 = u1 (x, z, t) , v1 = 0, w1 = w1 (x, z, t) and ∂ ≡0 ∂y With the aid of the above relation and eqn (4), the eqns (1)-(3) reduce to ∂2 u1 ∂2 w1 ∂2 u1 ∂2 u1 + P2 + μL = ρ1 2 ∂x∂z ∂x ∂z ∂t ∂2 w1 ∂2 u1 ∂ w1 ∂2 w1 μ L + P2 + P3 = ρ1 ∂x∂z ∂x ∂z ∂t P1 where P1 = (λ + 2α + 4μL − 2μT + β), P2 = (α + λ + μL ) and P3 = (λ + 2μT ) We assume that the solutions of (5) & (6) as ⎫ u1 = Ae−skz+ik(x−ct) ⎪ ⎪ ⎬ ⎪ −skz+ik(x−ct) ⎪ ⎭ w1 = Be (5) (6) (7) Thus using (7) in (5) and (6) we get s2 μL + ρ1 c2 − P1 A − isP2 B = (8) −iS P2 A + s2 P3 + ρ1 c2 − μL B = (9) 1024 Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 For non-trivial solution of eqns (8) & (9) we get P3 s + M1 s + N = μL The roots of the above eqn is sj = where M1 = −M1 ± M1 − 4N1 P3 μL ( j = 1, 2) P3 μL P2 c2 P3 c2 P1 c2 P1 − − + − , N = − 2 2 μ μ μ μ β1 β1 L β1 L L L c2 − , β1 = β1 (10) μL P1 Thus (8) & (9) yield s j + βc − w1 = u1 i s j μPL2 P1 μL = ηj ( j = 1, 2) (11) In light of (10) & (11), the displacement components of the fiber-reinforced layer is given by u1 = A1 e−ks1 z + A2 e−ks2 z + A3 e−ks3 z + A4 e−ks4 z eik(x−ct) (12) w1 = η1 A1 e−ks1 z − A3 e−ks1 z + η2 A2 e−ks2 z − A4 e−ks2 z eik(x−ct) (13) Dynamics of incompressible inhomogeneous half-space The two dimensional eqn of motion of an incompressible inhomogeneous semi-infinite medium are given by ∂ ∂u2 ∂w2 ∂ ∂u2 λΔ + 2μ2 + μ2 + ∂x ∂x ∂z ∂z ∂x ∂u2 ∂w2 ∂w2 ∂ ∂ μ2 λΔ + 2μ2 + + ∂x ∂z ∂x ∂z ∂z ∂2 u2 ∂t2 ∂2 u2 = ρ2 ∂t2 = ρ2 (14) (15) where λ is the Lame’s constant, Δ refers to dilatation, μ2 is shear modulus ρ2 is the density of the medium, u2 , w2 are the displacement components in the x & z directions respectively So, for an incompressible medium, we have Δ= ∂u2 ∂w2 + =0 ∂x ∂z (16) We may write the displacement components u2 andw2 in terms of scalar and vector potentials as u2 = ∂φ ∂ψ ∂φ ∂ψ + and w2 = − ∂x ∂z ∂z ∂x (17) where φ and ψ are the scalar and vector potentials respectively We consider μ2 = μ2 (1 + γz)2 , ρ2 = ρ2 (1 + γz) with μ2 & ρ2 are the rigidity and the density of the medium γ is chosen as an inhomogeneity parameter which has the dimension i.e., inverse of length Now we have the compressibility condition Δ = together with lim → −P1 , P1 be the hydrostatic stress Now eqns (14) and (15) can be λ→∞, Δ→0 written as ∂ ∂ ∂2 φ ∂2 ψ −P1 + 4μ2 γ (1 + γz) w2 − ρ2 + μ2 ∇2 ψ − ρ2 = ∂x ∂z ∂t ∂t ∂ ∂2 φ ∂ ψ ∂ −P1 + 4μ2 γ (1 + γz) w2 − ρ2 − μ2 ∇2 ψ − ρ2 = ∂z ∂x ∂t ∂t (18) (19) Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 1025 which are identically satisfied by P1 = 4μ2 γ (1 + γz) w2 − ρ2 μ2 ∇ ψ − ρ ∂2 ψ =0 ∂t2 ∂2 φ ∂t2 (20) (21) together with ∇2 φ = where ∇2 ≡ ∂2 ∂2 ∂2 + + ∂x2 ∂y2 ∂z2 (22) Here Rayleigh waves are confined to propagate along the x− direction with phase velocity ‘c’ and wave length 2π k u1 , w2 are the functions of z only apart from a factor eik(x−ct) , thus we may consider ⎫ φ = φ1 (z) eik(x−ct) ⎪ ⎪ ⎬ (23) ⎪ ik(x−ct) ⎪ ⎭ ψ = ψ1 (z) e Using the relations (23), eqn (20) and (21) now become equivalent to ρ2 c2 d ψ1 − ψ1 = + k2 μ2 (1 + γz) dz d φ1 − k φ1 = dz2 In order to solve eqn (24), we assume r = to 2k γ (24) (25) (1 + γz) , z > and with the aid of this transformation eqn (24) reduces d2 ψ1 Q + − ψ1 = r dr2 (26) where Q = 12 cc1 γk and c1 = μρ22 Equation (26) is commonly known as Whittaker’s eqn and the solution can be written as ψ1 = B1 WQ, 12 (r) + B2 WQ, 12 (−r) (27) where WQ, 12 (r) is the Whittaker’s function and B1 , B2 are the arbitrary constants We have considered the following asymptotic expansion of Whittaker’s function [11] upto three terms as ⎡ ⎢⎢⎢ m2 − k − ⎢⎢ Wk, m (z) = e zk ⎢⎢⎢⎢1 + z.1! ⎣⎢ − 2z 2 + m2 − k − 2 m2 − k − z2 2! 2 ⎤ ⎥⎥⎥ ⎥⎥ + ⎥⎥⎥⎥ ⎦⎥ where the value z dominates the values of k, leading to each term very small compared to the previous term The solution of (25) can be written as φ1 = C1 ekz + C2 e−kz (28) With the help of (28) & (27), the eqn (23) becomes ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ik(x−ct) ⎪ ⎪ ⎭ (−r) e φ = C1 ekz + C2 e−kz eik(x−ct) ψ = B1 WQ, 12 (r) + B2 WQ, 12 (29) Since the displacement at z → ∞ should be zero, the arbitrary constant B2 & C1 must vanishes Therefore (19) takes of the form ⎫ ⎪ φ = C2 e−kz eik(x−ct) ⎪ ⎬ (30) ⎪ ik(x−ct) ⎪ ⎭ ψ = B1 WQ, (r) e 1026 Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 Thus we will have the displacement components of the form u2 = C2 (ik) e−kz + 2B1 k −kz w2 = C2 (−k) e d W (r) eik(x−ct) dr Q, (31) − B1 (ik) WQ, 12 (r) e ik(x−ct) Boundary conditions and Secular equation i) At z = −H the surface remains stress free, i.e., (τ xz )I = and (τzz )II = ii) At the interface i.e., z = of the two different elastic solid the displacement components remains continuous, u1 = u2 and w1 = w2 iii) At the interface of the two media i.e., z = 0, the stress components also remains continuous, therefore, (τ xz )I = (τ xz )II and (τzz )I = (τzz )II From the first boundary condition and using (12) & (13) we have (−ks1 + ikη1 ) eks1 H A1 + (−ks2 + ikη2 ) eks2 H A2 + (ks1 − ikη1 ) e−ks1 H A3 + (ks2 − ikη2 ) e−ks2 H A4 = (32) Similarly from the second condition and using (12) & (13) we get (λ + α) ik − ks1 η1 (λ + 2μT ) eks1 H A1 + (λ + α) ik − ks2 η2 (λ + 2μT ) eks2 H A2 + (λ + α) ik − ks1 η1 (λ + 2μT ) e−ks1 H A3 + (λ + α) ik − ks2 η2 (λ + 2μT ) e−ks2 H A4 = Using (12) & (31) and substituting r = 2k γ (33) we have A1 + A2 + A3 + A4 − 2k Similarly using (13) & (31) and substituting r = 2k γ d W (r) B1 − ikC2 = dr Q, r= 2k γ we have η1 A1 + η2 A2 − η1 A3 − η2 A4 + ik WQ, 12 (r) With the help of (12), (13) & (31) and putting r = (34) 2k γ r= 2k γ B1 + kC2 = (35) the fifth boundary condition reduces into μL (−ks1 + ikη1 ) A1 + μL (−ks2 + ikη2 ) A2 + μL (ks1 − ikη1 ) A3 ⎡ ⎤ ⎢⎢ d2 ⎥⎥⎥ 2⎢ ⎢ ⎢ + μL (ks2 − ikη2 ) A4 − μ2 k ⎢⎣4 WQ, 12 (r) + WQ, 12 (r) 2k ⎥⎥⎥⎦ B1 + i.2μ2 k2C2 = r= γ dr r= 2k (36) γ Similarly with an aid of (12), (13) & (31) and putting r = 2k γ, the fifth boundary condition reduces into (λ + α) ik − ks1 η1 (λ + 2μT ) A1 + (λ + α) ik − ks2 η2 (λ + 2μT ) A2 + (λ + α) ik − ks1 η1 (λ + 2μT ) A3 + (λ + α) ik − ks2 η2 (λ + 2μT ) A4 + i4μ2 k2 d W (r) 2k B1 − 2μ2 k2C2 = r= γ dr Q, (37) Now eliminating the arbitrary constants A1 , A2 , A3 , A4 , B1 &C2 we have the equation in determinant form θ11 Real θ61 θ16 =0 θ66 (38) where θi j ’s (i, j = 1, 2, , 6) are given in Appendix-I We consider the real one and ignoring the imaginary part for its attenuating nature Hence the eqn (38) refers the desired secular equation of Rayleigh waves in a self-reinforced layer over an inhomogeneous incompressible half-space 1027 Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 Numerical Computation and Discussion In order to illustrate to the effect of reinforcement, inhomogeneity in rigidity and density on phase velocity and wave number, we perform some numerical computations taking the following data: i) For self-reinforced medium [Markham [12]]: μL = 5.66 × 109 N/m2 , μT = 2.46 × 109 N/m2 , α = −1.28 × 109 N/m2 , β = 220.9 × 109 N/m2 λ = 5.65 × 109 N/m2 , ρ1 = 2660Kg/m3 ii) For incompressible, inhomogeneous semi-infinite solid [Gubbins [13]]: μ2 = 78.4 × 109 N/m2 , ρ2 = 3.5350 × 106 Kg/m3 0.10 Γ = 0.18 Γ = 0.20 Γ = 0.22 0.08 Phase Velocity c Β1 Phase Velocity c Β1 14.0 16.0 18.0 1 H H H 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0.30 0.35 0.40 0.45 0.50 0.55 Wave Number kH 0.15 0.20 0.25 0.30 0.35 0.40 Wave Number kH Fig Variation in dimensionless phase velocity βc1 against dimensionless wave number kH for increasing values of inhomogeneity parameter γ Fig Variation in dimensionless phase velocity βc1 against dimensionless wave number kH for increasing values of the thickness of layer H Due to the variation in elastic properties of various materials present in the earth’s media, the phase velocity of Rayleigh type waves is varying with wave number and the whole incident is known as dispersion In Fig and 3, the dispersion curves expound the variation of non-dimensional quantity βc1 against non-dimensional wave number From figure 2, the dispersion curves delineate that the gradual increase in the inhomogeneity parameter γ present in the lower semi-infinite medium increases the phase velocity of the Rayleigh type surface waves In figure 3, it is clearly observed that the phase velocity shows a dual character with the increase of the thickness of the upper layer With an increase of thickness H, first the phase velocity decreases and after a certain while it shows an increasing nature With an overview of this graphical representation, a conclusion can drawn that the inhomogeneity and thickness of layer substantially affects the phase velocities concerned with the proposed earth model Conclusions An analytical and numerical approach has been adopted to investigate the two layered earth model consists of self-reinforced layer and incompressible inhomogeneous half-space Accumulating the real terms, a closed form expression of the dispersion relation in terms of sixth order determinantal form has been established We have ignored the imaginary terms due to its attenuating nature The observations from the helps to arrive at the conclusions (i) The inhomogeneity present in the lower half-space influences the phase velocity of the Rayleigh waves by gradually increasing it (ii) The increasing value of average thickness of layer first affects the phase velocity by decreasing it and then shows an increasing nature after a certain wave number (iii) The phase velocity of Rayleigh waves is greatly affected by the wave number 1028 Shishir Gupta and Mostaid Ahmed / Procedia Engineering 173 (2017) 1021 – 1028 The outcome of the model has certain practical applications in order to predict the damage caused by the surface waves As the reinforced concrete material is considered to be the basic ingredient in heavy constructions, it holds an important aspect for setting up highly efficient and disaster resistant buildings and bridges The study also bears a great significance in diverse fields like geotechnical and civil engineering also Acknowledgements The authors are thankful to University Grants Commission, New Delhi for providing fellowship to Mr Mostaid Ahmed through grant number F 7-79/2007(BSR) References [1] L Raleigh, J.W Strutt, On waves propagating along the plane surfaces of an elastic solid, Proc London Math Soc., 17 (1885) 4–11 [2] T.J Bromwich, On the influence of gravity on elastic waves, and, in particular on the vibrations of an elastic globe, Proceedings of the London Mathematical Society, (1898) 98–120 [3] A.J Belfield, T.G Rogers, A.J.M Spencer, Stress in elastic plates reinforced by fibres lying in concentric circles, J Mech Phys Solids, 31 (1983) 25–54 [4] A.J.M Spencer, Deformations of Fiber-Reinforced Materials, Oxford University Press, London 1972 [5] T Kaur, S.K Sharma, A.K.Singh, Effect of reinforcement, gravity and liquid loading on Rayleigh-type wave propagation, Meccanica, 51 (2016) 2449–2458 [6] F Birch, Elasticity and constitution of the Earth’s interior, Journal of Geophysical Research, 57 (1952) 227–286 [7] P.C Singh, Rayleigh waves in an inhomogeneous medium, Pure and Applied Geophysics, 98 (1972) 87–101 [8] S.K Vishwakarma, S Gupta, Rayleigh wave propagation:A case wise study in a layer over a half space under the effect of rigid boundary, Archives of Civil and Mechanical Engineering, 14 (2014) 181–189 [9] S.K Vishwakarma, R Xu, Impact of quadratically varying rigidity and linearly varying density on the Rayleigh wave propagation: An analytic solution, International Journal of Solids and Structures, 97 (2016) 182–188 [10] J.N Sharma, R Sharma, Modelling of thermoelastic Rayleigh waves in a solid underlying a fluid layer with varying temperature, Applied Mathematical Modelling, 33 (2009) 1683–1695 [11] E.T Whittaker, G.N Watson, A Course in Modern Analysis, Cambridge University Press, Cambridge, 1990 [12] M.F Markham, Measurement of the elastic constants of fibre composites by ultrasonics, Composites, (1969) 145–149 [13] D Gubbins, Seismology and Plate Tectonics, Cambridge University Press, Cambridge, 1990 Appendix-I θ11 θ21 θ23 θ31 = (−s1 + iη1 ) eks1 H , θ12 = (−s2 + iη2 ) eks2 H , θ13 = (s1 − iη1 ) e−ks1 H , θ14 = (s2 − iη2 ) e−ks2 H , θ15 = θ16 = = (λ + α) i − s1 η1 (λ + 2μT ) eks1 H , θ22 = (λ + α) i − s2 η2 (λ + 2μT ) eks2 H = (λ + α) i − s1 η1 (λ + 2μT ) e−ks1 H , θ24 = (λ + α) i − s2 η2 (λ + 2μT ) e−ks2 H , θ25 = θ26 = = 1, θ32 = 1, θ33 = 1, θ34 = 1, θ35 = −2k drd WQ, 12 (r) 2k , θ36 = −ik r= θ41 = η1 , θ42 = η2 , θ43 = η3 , θ44 = η4 , θ45 = −ik WQ, 12 (r) γ r= 2k γ , θ46 = k θ51 = μL (−s1 + iη1 ) , θ52 = μL (−s2 + iη2 ) , θ53 = μL (s1 − iη1 ) , θ54 = μL (s2 − iη2 ) θ55 = −μ2 k drd WQ, 12 (r) r= 2k γ + WQ, 21 (r) r= 2k γ , θ56 = i.2μ2 k θ61 = (λ + α) i − s1 η1 (λ + 2μT ) , θ62 = (λ + α) i − s2 η2 (λ + 2μT ) θ63 = (λ + α) i − s1 η1 (λ + 2μT ) , θ64 = (λ + α) i − s2 η2 (λ + 2μT ) θ65 = i4μ2 k d dr WQ, 12 (r) r= 2k γ , θ66 = −2μ2 k ... engineering and civil engineering etc Formulation of the problem In this present investigation, we consider an incompressible inhomogeneous half- space with quadratically varying rigidity and density. .. propagation of Rayleigh waves in self- reinforced layer embedded over an incompressible inhomogeneous half- space by deducing the secular equation The effect of inhomogeneity parameter and thickness of layer. .. Rayleigh waves in an inhomogeneous medium Vishwakarma and Gupta [8] demonstrated a case study on Rayleigh wave propagation in an incompressible crustal layer under rigid boundary over a dry sandy and