Chapter-3 Propagation of Surface Waves in Thermoelastic and Liquid-Saturated Porous Media

Transcription

1 Chapter-3 Propagation of Surfae Waves in Thermoelasti and Liquid-Saturated Porous Media

2 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3. Propagation of waves in a generalized thermo-elasti orthotropi medium * 3.. Introdution Thermo-elasti deformations have a signifiant effet on the ontat between elasti bodies, partiularly in ase where the thermal boundary onditions at the interfae are influened by the ontat pressure. Many important stress analysis problems involve strutures that are subjeted to both mehanial and thermal loadings. Thermal effet within an elasti solid produed heat transfer by ondution, and this flow of thermal energy establishes a temperature field within the material. If a body is made up a large number of small ubial elements and temperature of all these elements are uniform raised and boundary of the body is unonstrained, then all the ubial elements will expand uniformly and all will fit together to form a ontinuous body. Most solids exhibit a volumetri hange with temperature variation, and thus the presene of a temperature distribution generally indues stresses reated from boundary or internal onstraints. The generalized theory of thermo-elastiity developed by Lord and Shulman (967) has been extended to homogeneous anisotropi media by Dhaliwal and Sherief (98). Hawwa and Nayfeh (995) have disussed the problem of thermoelasti waves in anisotropi periodially laminated omposites. Sharma (986) has disussed the propagation of thermo-elasti waves in transversely isotropi materials. Sharma and Singh (99) have disussed the propagation of generalized thermo-elasti waves in ubi rystals respetively. Sharma et al. () have disussed the plane harmoni generalized thermoelasti weaves in orthorhombi materials. Sharma et al. () have disussed the propagation of plane wave in an infinite homogeneous, isotropi thermoelasti plate of thikness d in the ontext of various theories of thermo elastiity. * Published in International Journal of Applied Mathematis and Mehanis, (), 7-34,. ~ 5 ~

3 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Sharma (6-7) has extended suh a orrespondene for three-dimensional wave propagation and propagation of inhomogeneous, in generalized thermoelasti anisotropi media. Sherief (993) has investigated one dimensional problems whih inlude heat soures in the ontext of generalized thermo-elastiity with one relaxation time for an infinite spae and a half spae by using state spae approah. Verma () has onsidered the thermo-elasti vibrations of transversely isotropi plate with thermal relaxations. Verma and Hasebe (-4) have disussed the wave propagation in plates of general anisotropi media in generalized thermo-elastiity and extensional thermo-elasti waves in orthotropi plates with two thermal relaxation times. Kumar and Singh (7) have onsidered the refletion and transmission of thermoelasti plane wave in thermally onduting ubi rystal media with two relaxation times. This problem deals with the refletion of quasi-p, quasi-s and thermal wave from stress-free with thermal relaxations. The distint features of variation of veloity and the ratio of refletion oeffiients of P & S waves in orthotropi media with angle of propagation and angular frequeny are shown graphially for the onsidered waves. 3.. Formulation of the problem Consider a orthotropi thermo-elasti body at a uniform temperature T. The governing equation in orthotropi generalized thermoelasti media are given by [vide Verma and Hasebe (4)]: xx 66 yy 55 zz 66 xy 3 55 xz x 66 xy 66 xx yy 44 zz 3 44 yz y xz yz xx yy zz KT xx KTyy K33Tzz e T T T u x v y 3w z u u u v w T T u u v v v w T T v u v w w w T T w z (3..) ~ 5 ~

4 Propagation of surfae waves in thermoelasti and liquid-saturated porous media where, and ij are the orthotropi elastiities, ρ and e are density and speifi heat at onstant strain,, are thermal relaxation time, K ij and i are thermal ondutivities and the thermal expansion tensor, T is the temperature and j are thermal modulii respetively. For two dimensional motion in xz-plane, the governing equation for generalized thermoelastiity in the absene of body fore and heat soure are xx 55 zz 3 55 xz u u w T T u (3..) 3 55 xz 55 xx 33 zz 3 u w w T T w (3..3) K T K T T T T u w xx 33 zz e x 3 z z x (3..4) where,,, (3..5) Method of solution Consider a irular frequeny, wave number k, and phase veloity, for wave inident at z= at an angle with the z-axis, we may assume where, A, B and C are the amplitude fators and u Aexp( ip), w Bexp( ip ), T C exp( ip ) (3..6) P t k xsin z os (3..7) is the phase fator with inident wave. ~ 53 ~

5 Propagation of surfae waves in thermoelasti and liquid-saturated porous media For wave refleted at z=, we assume u Aexp( ip ), w B exp( ip ), T C exp( ip ) (3..8) where, P t k xsin z os (3..9) is the phase fator assoiated with refleted waves. Using the equations (3..5) or (3..7), in equation (3..) to (3..4), we obtain i ' D A 3 55 sin os B sin C k (3..) ' 3 55 sin os A D B i os C k (3..) * sin 3 os 3 e T A T B D C (3..) where, D sin 55 os D 55 sin 33 os D3 K sin K33 os ' ik i * k (3..3) Eliminating A, B and C, from the equations (3..) to (3..), we an have the oeffiients of real and imaginary parts as A A A A (3..4) 3 3 and A4 A5 (3..5) where, ~ 54 ~

6 Propagation of surfae waves in thermoelasti and liquid-saturated porous media * A * * A D D D4 * * A D D D D4 D D sin os A3 D DD4 D sin os ' A4 T 3 os sin ' A5 T D 3 os D sin sin os D3 and, D4 e (3..6) Equation (3..5), gives the ratio A5 A, refletion oeffiient of P & S waves in 4 orthotropi media. We use the Cardan s method to solve the equation (3..4), and obtain where, 3 3H G (3..7) * * A 3 A A, H 3 9 * * 3 7 A3 9 A A A G 7 (3..8) The three roots of equation (3..7) an be written as where, h h, h g h g, h g h g (3..9) 3 h G G 4H h G G 4H ~ 55 ~

7 Propagation of surfae waves in thermoelasti and liquid-saturated porous media and i 3 g, a ube root of unity. Therefore, the three roots of Eq. (3..4) are A A A , *, * 3 (3..) * Equation (3..) gives the veloities of propagation of quasi-p, thermal and quasi-s waves respetively. Partiular ase- For isotropi elasti solid, we take,,, λ and μ are Lame s onstant K K K, 33 then D4, A3 Equation (3..4) an be redued to quadrati equation in as where, B (3..) B D D B ' ' B ' ' D D sin os D sin os D ' os sin ' The solution of equation (3..) gives the expression of propagation of P and S waves in two-dimensional model for isotropi media. ~ 56 ~

8 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3..4 Numerial results and disussion We have onsidered the following data for orthotropi material (f. Srinath (9)): =8.8 Gpa, 3 =.89 Gpa, 55 =.34 Gpa, 33 =7.7Gpa. Thermal ondutivities, speifi heat at onstant strain and thermal relaxation time are taken as K =7.8-4 w/m/deg, K 33 =5. -4 w/m/deg, e =3.9 J/kg/deg, =.5. Moreover, the following data are also used in thermal modulii = Nm - deg - and 3 = Nm - deg -. The variation of the veloities of quasi-p, thermal, quasi-s and isotropi (P & S) waves with the angle of propagation are shown in fig. 3.., when =, =8.4, and =3.6. The veloities of propagation of these plane waves are ompared for isotropi elasti ase. The numerial values of the veloities of propagation of quasi-p, thermal and quasi-s waves are alulated for frequeny range when diretion of propagation makes 45 with vertial axis. The variations of these veloities with frequeny are shown in fig quasi-p thermal quasi-s P (isotropi) S (isotropi) Veloity (m/s) Angle of Propagation Fig. 3..: Variation of veloity as a funtion of angle of propagation. ~ 57 ~

9 Propagation of surfae waves in thermoelasti and liquid-saturated porous media A 5 /A Angle of propagation Fig. 3..: Variation of ratio of refletion oeffiients (P & S waves) as a funtion of angle of propagation. 9 8 quasi-p thermal quasi-s 7 Veloity (m/s) Angular frequeny Fig. 3..3: Variation of veloity as a funtion of angular frequeny. ~ 58 ~

10 Propagation of surfae waves in thermoelasti and liquid-saturated porous media From Fig.3.., it is inferred that the variation of veloity and angle of propagation in quasi- S and thermal waves behaves almost same nature. Veloity of quasi-p wave in orthotropi media oinides with the veloity of S-wave (isotropi) up to ertain degree of angle of propagation where as veloity of P-wave and S-wave for isotropi ases are parallel to eah other. The variation of this with angle of propagation is depited in Fig.3... The value of ratio of refletion oeffiients inreases as angle of propagation inreases. The value attains maximum at 45 (degree) and then it dereases with angle of propagation. From Fig. 3..3, it is inferred that the variation of veloity with angular frequeny behaves same for thermal and quasi S-waves. Thus orthotropi thermoelasti solid has pronouned effets on the propagation of quasi-p, quasi-s and thermal waves. ~ 59 ~

11 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3. Wave propagation in an inhomogeneous anisotropi generalized thermoelasti solid * 3.. Introdution When an isotropi and homogeneous elasti solid is subjeted to a thermal disturbane, the effet is instantaneous at a loation distant from the soure in the lassial linear elasti theory. This means that the thermal wave propagates at infinite veloity whih is a physially unreasonable result. Two generalized thermo-elasti theories are proposed to eliminate that paradox and orret the lassial theory on the assumption that a thermal wave propagates at finite veloity. The first generalization, due to Lord and Shulman (967) and Fox (969), modifies the well known Fourier s law of heat ondution but was until now restrited to isotropi and homogeneous media. The seond generalization, due to Green and Lindsay (97), does not violate Fourier s law of heat ondution when the body under onsideration has a entre of symmetry, and was derived for both isotropi and anisotropi media. Pal and Aharya (998) have disussed effets of inhomogeneity on surfae waves in anisotropi media taking exponential inhomogeneity. Hawwa and Nayfeh (995) have disussed the problem of thermoelasti waves in anisotropi periodially laminated omposites. Sharma and Singh (99) have disussed the propagation of generalized thermo-elasti waves in ubi rystals respetively. Sharma et al. () have disussed the plane harmoni generalized thermoelasti weaves in orthorhombi materials. * Aepted for publiation in Journal of Thermal Stresses (Taylor & Franis) ~ 6 ~

12 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Sharma (6-7) have extended suh a orrespondene for three-dimensional wave propagation and propagation of inhomogeneous waves, in generalized thermoelasti anisotropi media. Sherief (993) has investigated one dimensional problems whih inlude heat soures in the ontext of generalized thermo-elastiity with one relaxation time for an infinite spae and a half spae by using state spae approah. Verma and Hasebe (, 4) have disussed the wave propagation in plates of general anisotropi media in generalized thermo-elastiity and extensional thermo-elasti waves in orthotropi plates with two thermal relaxation times. Singh (33) has disussed the wave propagation in an anisotropi generalized thermoelasti solid with two relaxation times. Our intention in this paper is to study the propagation of plane waves in inhomogeneous anisotropi generalized thermoelasti solid with two thermal relaxations. The analytial expressions for veloities of these plane waves are derived by an approah used by Singh (3). For numerial results a partiular material is taken to obtain the veloities of plane waves and results are shown by plotting graphs between different veloities of plane wave with angular frequeny and the diretion of propagation. It is found that, due to thermal relaxation times, the amplitude of quasi-p, quasi SV waves and thermal waves are higher than that of onventional theories. 3.. Formulation of the problem Consider an inhomogeneous anisotropi thermally onduting elasti medium at uniform temperature T.The origin is taken on the thermally insulated and stress free plane surfae ~ 6 ~

13 Propagation of surfae waves in thermoelasti and liquid-saturated porous media and z axis normally into the half-spae whih is represented by z. The fundamental equations of generalized thermo-elastiity are given by: Equation of motion: x ij j u t i (3..) Heat ondution equation: T T T ekk e kk Kij Ce T ij i j x x t t t t (3..) Stress-displaement-temperature relations: T ij Cijklekl ij T t ` (3..3) and C, i, j, k, l,,3, (3..4) ij ijkl kl where is the density, t is the time, ui is the displaement in the x i diretion, thermal ondutivities, K ij are the C e and are, respetively, the speifi heat at onstant strain and thermal relaxation time, ij and eijare the stress and strain tensor, respetively, ij are thermal modulii, fourth-order tensor of the elastiity ij is thermal expansion tensor, T is the absolute temperature, and the C ijkl satisfies the (Green) symmetry onditions: C C C C,,, K K. (3..5) ijkl klij ijlk jikl ij ji ij ji ij ji The parameters and are the thermal-mehanial relaxation time and the thermal relaxation time of the G-L theory, and they satisfy the inequality. The use of ~ 6 ~

14 Propagation of surfae waves in thermoelasti and liquid-saturated porous media symbol, in eq. (3..) makes these fundamental equations possible for the two different theories of the generalized thermo-elastiity. For the L-S (Lord-Shulman) theory, and for G-L theory and. If C,, K and are assumed as i.e. C ij ij ij ij az ije, ij e az, az ij Kij K ije and e az then equations of motion in x-z plane for generalized thermo-elastiity are u w u w u w u w a5 a55 x x z z xz xz x x a a a T z z x xt z zt t t u w T T T T T u (3..6) u w u w u w u w a3 a35 x x z z xz xz x x u w T T T T T w a35 a a 33 T z z x xt z zt t t (3..7) T T T T T T T K K 3 K 33 ak 3 ak 33 Ce x xz z x z t t 3 3 u u w w T 33 xt xt zt zt (3..8) where, ~ 63 ~

15 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3..3 Method of solution For wave inident at z at an angle with the z-axis, we may assume the solution as u x, z, t A w x, z, t Bexp i t k x sin z os T x, z, t C (3..9) where, is the irular frequeny and k is the wave number and A, B and C are the amplitude fators and t k x sin z os is the phase fator. For wave refleted at z, we assume the same solution as (3..9) exept the phase fator is given by t k x sin z os (3..) If the phase fator be positive then inident, refleted P, SV and thermal waves travel in positive z diretions with time. Using the equation (3..9) and (3..) in equations (3..6) to (3..8), we obtain (3..) D A D B D3C D A D B D C (3..) * e D A D B D C C (3..3) where the upper sign orresponds to the inident waves [equation (3..9)] and the lower sign orresponds to the refleted waves [equation (3..)] and * i a a D sin 55 os 5 sin os i 5 sin i 55 os k k ~ 64 ~

16 Propagation of surfae waves in thermoelasti and liquid-saturated porous media a a D 5 sin 35 os 3 55 sin os i 55 sin i 35 os k k D i i a sin sin os os i k k k k k a a D4 5 sin 35 os 3 55 sin os i 3 sin i 35 os k k a a D5 55 sin 33 os 35 sin os i 35 sin i 33 os k k D i i a sin sin os os i k k k k k D7 T sin i sin k k D8 T 33 os i os k k a a D9 K sin K 3 sin os K 33 os i K 3 sin i K33 os k k Equations (3..) to (3..3) in A, B and C an have a nontrivial solution only if the determinant of their oeffiients vanishes, i.e. A A A A (3..4) 3 3 where, A * C e A C D D D * e 5 9 A D D D D D D C D D C D D D D * * e 5 e ~ 65 ~

17 Propagation of surfae waves in thermoelasti and liquid-saturated porous media A D D D D D D D D D D D D D D D D D D Using Cardan s method to solve the equation (3..4), we obtain 3 3H G (3..5) where, * * A 3 A A, H 3 9 * * 3 7 A3 9 A A A G 7 (3..6) The three roots of equation (3..5) an be written as h h, h g h g, h g h g (3..7) 3 where, h G G 4H h G G 4H and i 3 g, a ube root of unity. Therefore, the three roots of equation (3..4) are A A A , *, * 3 (3..8) * Equation (3..8) gives the veloities of propagation of quasi-p, thermal and quasi-s waves respetively. It may be noted that whether we take the upper sign or lower sign in equations (3..) to (3..3), we get the same three values of given by equation (3..8). Thus, in ~ 66 ~

18 Propagation of surfae waves in thermoelasti and liquid-saturated porous media general, in two dimensional inhomogeneous anisotropi generalized thermoelasti media with two relaxation times, there are three types of plane waves, whose phase veloities vary with diretion of propagation and frequeny Redution to homogeneous and isotropi ase: For simply homogeneous transversely isotropi solid, we have inhomogeneity parameter a and elasti oeffiients, 3 3, 33 33, and Also, 33 3, 3 and K K, K 33 K3, K 3.Then equation (3..4) redues to A A A A (3..9) 3 3 where, * A C e A C D D D * e 5 9 A DD DD DD C DD C DD DD * * e 5 e A DDD DDD DDD DDD DDD DDD ' D sin os D sin os i D 3 sin sin k k D sin os D sin os ~ 67 ~

19 Propagation of surfae waves in thermoelasti and liquid-saturated porous media i D 6 3 os os k k D 7 T sin i sin k k D 8 T 3 os i os k k D K sin K os 9 3 Equation (3..9) gives the veloities of propagation of quasi-p, thermal and quasi-s waves respetively in homogeneous transversely isotropi thermally onduting elasti medium at uniform temperature T as onsidered by Singh (3). For isotropi elasti solid, we take,,, λ and µ are Lame s onstant , K K3 K, Equation (3..9) redued to quadrati equation in as B (3..) B where, B D D B DD sin os D sin os D os sin The solution of equation (3..) gives the expression of propagation of P and SV waves in two-dimensional model for isotropi elasti media. ~ 68 ~

20 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3..5 Numerial results and disussion In order to show the effet of inhomogeneity and dependeny of quasi-p, quasi-s and thermal waves on frequeny and angle of propagation, we have taken data from Rasolofosaon and Zinszner () and Singh (3). 6.8Gpa, 99. Gpa, Gpa, 7.Gpa Gpa,.3Gpa,.8Gpa,.Gpa Gpa,.49Gpa,.3Gpa,.58Gpa Gpa,.69Gpa,.75Gpa, 5.97 Gpa Gpa,.43Gpa, 5.5Gpa, 37.8Gpa Gpa,.77 kg/m Nm deg, 5.45 Nm deg, 5.7 Nm deg C 3.9 J kg deg, K K K.4 Wm deg e T o 96 K,.5,., k.5, 5, 45. o The real parts of expression for variation of veloities() are onsidered. The variations of veloity for quasi-p, quasi-s and thermal waves with angle of propagation are plotted in figures 3.. to 3... Figs 3.. to 3..6 are drawn on the basis of L-S theory whereas figs 3..7 to 3.. are drawn on the basis of G-L theory. ~ 69 ~

21 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig.3.. Fig.3.. Fig.3..3 From fig. 3.., it is observed that as we inrease the inhomogeneity parameter the veloity of quasi-p inreases with the inrease of frequeny. Fig.3.. shows that quasi-s wave veloity inreases with the inreasing value of inhomogeneity parameter. Fig shows that thermal wave inreases with the inreasing value of inhomogeneity parameter but nature is different from the variation of quasi-p wave and quasi-s wave veloity. ~ 7 ~

22 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig.3..4 Fig.3..5 Fig.3..6 Figs to 3..6 show the variation of different wave veloities against the diretion of propagation. Fig shows that the veloity of quasi-p wave inreases then dereases rapidly and with jump in veloity at ertain angle and then dereases rapidly with inreasing value angle of propagation and inhomogeneity parameter. The angle at whih jump is observed for wave is shifting towards the normal angle with minor hange in the inhomogeneity parameter. Fig shows that quasi-s wave dereases rapidly with angle of propagation. As we inrease the inhomogeneity parameter the rate of dereasing of the veloity of the quasi-s wave inreases rapidly. So, all the urves interset at a point and urves show disontinuity at ertain angle and with jump in veloity. In fig.3..6, we ~ 7 ~

23 Propagation of surfae waves in thermoelasti and liquid-saturated porous media observe that veloity of thermal wave inreases with angle of propagation and inhomogeneity parameter and beome oinident and show disontinuity as well. Fig.3..7 Fig.3..8 Fig.3..9 Figs to 3..9, the graphs for veloities of different waves against frequeny are plotted. From these graphs we an observe that same type of nature is exhibited as L-S theory for both relaxation times. ~ 7 ~

24 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig.3.. Fig.3.. Fig.3.. In figs. 3.. to 3.. the graphs for veloities of different wave against diretion of propagation are plotted. From these graphs we an observe that same type of nature is exhibited as L-S theory for data we have onsidered for both relaxation times. ~ 73 ~

25 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig.3..3 Fig.3..4 Fig.3..5 Fig.3..6 Figs to 3..5 are plotted for different veloities in a homogeneous transversely isotropi thermally onduting elasti medium at uniform temperature T against the angle of propagation. From all the three graphs we observe that the nature exhibited by both theories L-S and G-L are oiniding for quasi-p, quasi-s and thermal waves. In fig. 3..3, the veloity of quasi-p wave dereases with angle of propagation after ertain angle of propagation veloity inreases rapidly. Fig shows that the veloity of quasi-s wave inreases rapidly with angle of propagation. Fig shows that the veloity of thermal wave dereases rapidly and after ertain value of angle of propagation inreases rapidly. In Fig. 3..6, the graph has been plotted for isotropi ase against the angle of propagation. The veloity of P and SV wave are onstant with the hange of angle of propagation. ~ 74 ~

26 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3..6 Conlusions In this analysis the propagation of plane waves in inhomogeneous anisotropi generalized thermoelasti solid is onsidered on the basis of L-S theory and G-L theory. The problem is generalized with two thermal relaxations. The analytial expression and relation are obtained for veloities of quasi-p, quasi-s and thermal waves and the numerial results are disussed through figures onsidering a partiular model of rystalline rok. It is observed that when thermal dissipative properties of a half-spae are taken into onsideration, the undamped harateristi features of plane waves do not remain valid. The variations of quasi-p, quasi-s and thermal waves with frequeny and angle of propagation for different values of inhomogeneity parameter are onsidered. These variations are shown in figures 3.. to It is onluded from figures that the inhomogeneity, frequeny and angle of propagation have immense effet in the veloity of quasi-p, quasi-s & thermal waves. Figures aption: Fig. 3.. to 3..3: Variation of phase veloity against angular frequeny. Fig to 3..6: Variation of phase veloity against angle of propagation. Fig to 3..9: Variation of phase veloity against angular frequeny. Fig. 3.. to 3..: Variation of phase veloity against angle of propagation. Fig to 3..6: Variation of phase veloity against angle of propagation. ~ 75 ~

27 Propagation of surfae waves in thermoelasti and liquid-saturated porous media 3.3 Rayleigh wave propagation in a liquid saturated porous layer overlying an orthotropi solid half-spae and lying under a uniform liquid layer * 3.3. Introdution Porous media theories play an important role in many branhes of engineering inluding material siene, petroleum industries, hemial engineering, biomehanis, soil mehanis and other suh field of engineering. Liquid-saturated porous material are often present on and below the surfae of the earth in the form of sandstone, limestone and other sediments permeated by groundwater or oil are present in the earth s rust. The porosity of rok also affet veloity, whih an be useful. Some extra properties like porosity, lithology and permeability are required for knowing wave propagation through solid/porous roks (Domenio, (984)). The porosity is affeted in sedimentary roks by their maximum depth of burial. Veloity variations an affet seismi data in a variety of ways, produing, for example false antilines whih are not there (Christensen and Szymansk, (99)). The relation between veloity and density may be found for partiular rok types. Porosity is, inturn, also involved through the effet on elastiity and density. The propagation of Love waves in a fluid-saturated porous layer have been disussed by Konzak (989). It was Deresiewiz (96) who first, using the theory of Biot (956) for the wave propagation in a statistially isotropi fluid-saturated porous medium, studied the propagation of Love waves in a porous layer resting on an elasti, homogeneous and isotropi semi-infinite spae. Deresiewiz and Rie (96) investigated the various aspets of the presene of boundaries on the propagation of plane harmoni seismi waves in liquid-saturated porous solids. Gogna (979) onsidered the surfae wave propagation in a homogeneous anisotropi layer lying over a homogeneous isotropi elasti half-spae and under a uniform layer of liquid. Sharma et al. (99) have disussed the surfae wave propagation in a liquid-saturated porous solid layer, overlying an impervious, transversely isotropi, elasti, solid half-spae and under a uniform layer of liquid. * Communiated ~ 76 ~

28 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Kumar and Hundal (3; 5; 7) have studied the problem of wave propagation in fluid-saturated inompressible porous media. Liu and Liu (4) have studied the influene of anisotropy of the solid skeleton on the propagation of harateristi of Rayleigh waves in orthotropi fluid-saturated porous media. Pal (6) has studied on shear wave propagation in a multilayered medium inluding a fluid saturated porous solid stratum. Rangelov et. al. () have onsidered a speial ase of wave propagation problem in a restrited lass of orthotropi inhomogeneous half-spae. The method of solution is a hybrid approah and investigated the effet of the inident wave angle and frequeny, type of material inhomogeneity and orthotropi on the free-field wave motion. Present problem deals with the Rayleigh wave propagation in a liquid-saturated porous elasti solid lying over an orthotropi solid half-spae and under a uniform layer of homogeneous liquid. This appears to be of pratial interest as the sediments deposited under water may be assumed to be orthotropi. It is also a more realisti model for the oean bottom. It is relevant to study the Rayleigh wave at the upper surfae of the oean. Some speial ases are disussed Formulation of the problem We onsider a medium onsisting of a liquid-saturated porous solid layer, of thikness h, resting on an orthotropi solid half-spae and under a uniform liquid layer of thikness h. We onsider a retangular oordinate system, suh that the z-axis is hosen in the diretion of inreasing depth and z= is taken as the interfae between the two layers, orthotropi solid (medium III) oupies the region z>h, liquid-saturated porous solid (medium II) oupies the region <z<h and the region h <z<is oupied by the liquid layer (medium I) as shown in fig.3... We are disussing the two dimensional problem (xz-plane) where the wave front is taken parallel to yz-plane, so the omponents of displaements parallel to x and z-axis will be independent of y and the omponents in y-diretion are taken to be zero. ~ 77 ~

29 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig.3.3.: Geometry of the problem. For liquid layer (medium I), the equation of motion in terms of displaement potential is given by (f. Ewing et al. (957)) x z t (3.3.a) where, / is the veloity of the dilatational wave in liquid, is the density and is the elasti onstant of the liquid. The displaement omponents u, w and pressure p are given by and u, w, x z p (3.3.b) z z e and substituting in (3.3.) and solving, we get ikxt Assuming kz / kz / ( z) Ae Be where, A and B are arbitrary onstants. ~ 78 ~

30 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Thus we an obtain kz / kz / ik ( xt) Ae Be e (3.3.) For the liquid-saturated porous solid (medium II), the field equation for suh medium in the absene of dissipation are given by Biot (956) as and [( ) ] ( ), t grad Qdivu RdivU u U, t N u grad D N divu QdivU u U (3.3.) where, u and U are the displaements in the solid and liquid parts; D,N,Q and R, all nonnegative, are elasti Modulii ;, and are dynamial oeffiients. The Modulii enter into the relations between the stress in the solid, ij and that in the liquid, σ, on the one hand, and the strain in the solid, and the dilatations, e u u ij i, j j, i div u and divu, on the other:, ij De Q ij Nij Qe R (3.3.3) where, ij is kroneker delta. If and represent the mass of the solid and liquid per unit volume of the aggregate s f respetively, s and aggregate, then we have represent their respetive mass densities and β, the porosity of f and s - s f f Following Biot (956), we find that the following relations are also satisfied (3.3.4) o,,,, PR Q and R P Q, (3.3.5) ~ 79 ~

31 Propagation of surfae waves in thermoelasti and liquid-saturated porous media where, P=D+N. We now onsider a Helmholtz resolution of eah of two displaement vetors, of the form u grad urlh U grad urlg (3.3.6) Intersetion of (3.3.6) in (3.3.), yields a pair of equations whih are satisfied identially by setting P Q t t Q R (3.3.7) and Assuming t H G t N H H G ( z) e ( z) e ik ( x- t ) ik ( x- t) (3.3.8) (3.3.9) where, k is the wave number and is the phase veloity. Substituting these values of (3.3.9) in (3.3.7), yields '' '' '' '' P k Q k k Q k R k k where, prime denote the differentiation with respet to z. (3.3.) eliminating '' from (3.3.), we get ~ 8 ~

32 Propagation of surfae waves in thermoelasti and liquid-saturated porous media A '' A R Q Q R k (3.3.) Substituting this value of in (3.3.) will give us where, and IV '' 4 A B A k k A B C (3.3.) A PR - Q, C - B R - Q P For A, B, C being nonnegative, the solution of (3.3.) may be written as (3.3.3) With and kz / kz / A e Ae kz / kz / A3e A4e (3.3.4) where, A,A,A 3,A 4 are arbitrary onstants and Thus we obtain where, 4 4 / / B B AC B B AC, (3.3.5) A A ik ( xt ) e (say) (3.3.6) and k k (3.3.7) ~ 8 ~

33 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Hene an unbounded solution of (3.3.6) and (3.3.7) orresponds to the two dilatational waves. The wave orresponding to being the faster one, alled fast P (or P f ) wave propagation with the phase veloity and that orresponding to being the slower one, alled slow P (or P s ) wave propagating with the phase veloity. From equations (3.3.) and (3.3.6), we get where, similarly, the assumption and (3.3.8) A R Q Q R A R Q Q R H H z e G G z e ik x-t ik x-t (3.3.9) when inserted in equation (3.3.8), we get N where, 3 C and G H H H t 3 i.e., in an unbounded medium 3 is the phase veloity of only shear wave. Now for two dimensional motion in xz-plane the displaements u u,, w U U,, W in liquid are given as follows: (3.3.) in the solid and ~ 8 ~

34 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Where, -H y u, x x z w, z z x U -, x x z W, z z x (3.3.) Stress omponent will be given by and z P Q P Q N, x x xz zx N, xz xz z x Q R Q R (3.3.a) As the displaement omponents are proportional to e expressions for the potentials, and as ik ( xt), we therefore take the kz / kz / ik ( xt ) A e Ae e kz / kz / ik ( xt ) A3e A4e e kz / 3 kz / 3 ik ( xt ) A5 e A6e e where, A,A,A 3,A 4 A 5 and A 6 are all arbitrary onstants. (3.3.b) For orthotropi elasti solid (medium III), the stress-strain relations ~ 83 ~

35 Propagation of surfae waves in thermoelasti and liquid-saturated porous media C e C e C e x xx yy 3 zz C e C e C e y xx yy 3 zz C e C e C e z 3 xx 3 yy 33 zz yz zx xy C e 44 C e 55 C e 66 yz zx xy (3.3.3) The equation of motion where there are no body fore, are and * * * * u u w * u C C 55 C3 C55 x z x z t * * * * w w u * w C55 C 33 C3 C55 x z xz t (3.3.4) Where, * is the density of the orthotropi solid. We seek harmoni solution of (3.3.4) in the form of u U ( z) e * * ik ( x- t) w W ( z) e * * ik ( x- t) and find the equation (3.3.4) in the form of ' ' C U ik C C W C k U *'' * * * C W ik C C U C k W *'' * * * (3.3.5) (3.3.6) Solving simultaneous linear equations with onstant oeffiients, as displaement omponents tend to zero as z tends to infinity, we have U W P e * * Q e * * ksz ksz (3.3.7) Substituting these values in (3.3.6), we obtain s C R P isj Q * * * * 55 s C S Q isj P * * * * 33 (3.3.8) ~ 84 ~

36 Propagation of surfae waves in thermoelasti and liquid-saturated porous media * * * * * where, J C3 C55, R C3, S C55 In order to get a nontrivial solution of (3.3.8), we must have 4 * * * * * C C s R C S C J s R S (3.3.9) This eq. being quadrati in s has the solution as * * * where, R C S C J * * 4R S C C C C (3.3.3) The ratio of displaement omponent U, W from (3.3.8) orresponding to s=s j is * * j j * * * Wj Qj C55s j R m ( ) 4,5 * * * j say j (3.3.3) U P is J j j j Thus the solution of equation (3.3.8) an be written as ks4z ks5 z ks4z ks5 z ik xt 4 4 ks5z ks5z u P e P e P e P e e * * * * * ( ) 3 4 w m P e P e m P e P e e * * ks z * ks z * * ik ( xt) 3 4 * * * where, p, p, p and * 3 p are arbitrary onstants and 4 (3.3.3) s 4R S C C 4R S C C * * * * , s5 C33C55 C33C 55 Sine the displaements tends to zero when z tends to infinity, we therefore take the expressions for * u and w * as ~ 85 ~

37 Propagation of surfae waves in thermoelasti and liquid-saturated porous media u P e P e e * * ks4z * ks5 z ik ( xt) w m P e m P e e * * ks4z * ks5 z ik ( xt ), (3.3.33) where s 4 and s 5 are assumed to be real and positive Boundary Conditions The free surfae z h of the liquid layer, the normal stress omponent vanishes i.e. ( P) (3.3.34) z At z= i.e. at the interfae of homogeneous liquid layer and fluid saturated porous solid, aording to Deresiewiz and Skalak (963) boundary onditions for open pore are as follows: z I II z z II zx II... ( ) u U u II I (3.3.35) At z=h i.e. at the interfae of fluid saturated porous solid and orthotropi elasti solid halfspae, stresses and displaements are ontinuous where as normal veloity of the liquid relative to the solid vanishes i.e. ~ 86 ~

38 Propagation of surfae waves in thermoelasti and liquid-saturated porous media * z II z * zx II zx III * w II w III * u u III II III.. w-w II (3.3.36) Making use of (3.3.), (3.3.), (3.3.a), (3.3.3), (3.3.33) in above boundary onditions * * ( ), we obtain the ten homogeneous equations in A, A, A, A, A, A, P, P, A and B. For nontrivial solution of the system of ten homogeneous equations we have where a ij, the entries of the tenth-order square matrix are as follows a, (3.3.37) kh kh T T, a N e a N e kh kh T T 3, 4 a N e a N e kh kh 3 3 5, a in e a in e ij, a ic m s C a ic m s C a9 a kh kh, a in e a i N e kh kh 3, 4 a in e a in e kh kh 3 3 5, a N e a N e ~ 87 ~

39 Propagation of surfae waves in thermoelasti and liquid-saturated porous media, a C im s a C im s a9 a kh kh 3, 3 a e a e kh kh 33, 34 a e a e kh kh , 36 a ie a ie a m, a m, a a kh kh 4, 4 a ie a ie kh kh 43, 44 a e a e kh kh , a e a e a a, a a kh kh 5, 5 a e a e kh kh 53, 54 a e a e kh kh , 56 a i e a i e a5 j, j 7,..., Q R Q R a a, a a a6 j, j 5,...,8 a a T a7 a 7 N a a N T ~ 88 ~

40 Propagation of surfae waves in thermoelasti and liquid-saturated porous media a a in , 79 7 a a a a a a in 8 8 a a in a85 a86 N 3 a a a a a a 9 9 a a a a i a a, a a a j, j,...,8 kh kh 9, a e a e Speial ases Case I. If kh the overlying liquid layer will be treated as liquid half-spae and the frequeny equation (3.3.37) redues to b (3.3.38) Where, b are the entries of square matrix of order 9 and are given by ij ij b a ( i, j,,...,9) ij ij This brings about the dispersion equation for surfae wave propagation in liquid-saturated porous solid bounded by liquid half-spae and orthotropi solid half-spae. ~ 89 ~

41 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Case II. Reduing the depth of the liquid layer to zero, i.e. h = we get the frequeny equation for surfae wave propagation in a liquid-saturated porous solid lying over an orthotropi solid half-spae as where d ij are entries of the square matrix of order 8. d (3.3.39) ij Case III. Substituting h h, we get the equation (3.3.37) reduing to i3 ms 433 i3 ms533 im s im s 4 5 (3..4) Giving the propagation of Rayleigh waves over the free surfae of an orthotropi solid halfspae Numerial results and disussion Sine a large number of parameters enter into the final expressions, then in order to disuss the possibility of propagation of surfae waves disussed above along the x-diretion, a partiular model is onsidered. For the water layer, following Gubaidullin et al. (4), the seismi body wave veloity, density and elasti parameters are taken as follows: 55m/se, 65 kg/m,.6gpa 3 Keeping in view the experimental results given by Doyle (995), we have the data for kerosene saturated sandstone (medium II) are as follows: P 6.8GPa, Q 5.36GPa, R 36.5GPa, N.GPa 6kg/m, 998kg/m, 556kg/m, The veloities of the P f, P s and SV waves for the above onstants are 943.m/se, 576.9m/se, 484.7m/se 3 ~ 9 ~

42 Propagation of surfae waves in thermoelasti and liquid-saturated porous media For orthotropi solid half-spae (ubi rystal) the elasti modulii are as follows: 85.73GPa, C GPa C GPa, 77kg/m For Epoxy- rystals the elasti modulii are taken as C C C GPa, C * 3 5.6GPa.GPa, kg/m * 3 Eq. (3.3.37) is a omplex equation and from where the real part is extrated and graphs are plotted between phase veloity and non dimensional wave number. Atually, we onsider two models. Medium I is a liquid layer. The third layer is an orthotropi half-spae. In the first model, medium II is taken to be water-saturated sandstone layer. In the seond model, medium II is taken to be layer of kerosene saturated sandstone. We take two different sets of value ofc3, C33 andc 55. It is reported that (Liu and Bassett (986)) mantle is made up of perovskite-type ferromagnesian siliates. It is also shown that ommon rok type (peridotite) oupies a plae below upper most mantles down to depth of 5 km below oeani bed. Olivine minerals (rystal) are believed to be omposed of at least 65% of the upper mantle. For this reason, it is our aim to see the behavior of Rayleigh wave dispersion urves for two different sets of data of rystals. Dispersion urves for Rayleigh waves are drawn in figs to Figs 3.3. and show the variation of phase veloity with non-dimensional wave number kh for three different ratios of depth of layers i.e. h. Figs 3.3. and show that phase veloity dereases as the wave number h inreases. Same pattern follows for both the rystals. But marginal hanges in the magnitude of phase veloity are observed for both types of rystals. The variations of phase veloity with non-dimensional wave number kh for three different ratios of depth ~ 9 ~

43 Propagation of surfae waves in thermoelasti and liquid-saturated porous media h of layers i.e. are shown in figs and The behaviors of variation are quite h different. The urves derease upto ertain value of kh and again starts inreasing. The natures are quite different for both the rystals. The variations observed from figures 3.3. to are quite lear. The effet of depth of layering in vertial diretions auses the hange in phase veloities in both types of rystals. The variations are signifiant and variations of veloity in horizontal diretions are observed almost 5% of veloity in vertial diretion in both the rok types. It is also onluded that effet of layering in oremantle region plays an important role to observe the hange of phase veloity and wave numbers in these regions. Fig Variation of phase veloity against non dimensional wave number k h ubi rystal. for ~ 9 ~

44 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig Variation of phase veloity against non dimensional wave number k h Epoxy-rystal. for Fig Variation of phase veloity against non dimensional wave number k h for ubi rystal. ~ 93 ~

45 Propagation of surfae waves in thermoelasti and liquid-saturated porous media Fig Variation of phase veloity against non dimensional wave number k h for Epoxy-rystal Conlusions An exat solution for a transient analysis of two dimensional problems onsisting of a liquid-saturated porous layer and orthotropi half-spae is onsidered. The fluid saturated porous material is modeled as two phase system omposed of an inompressible solid phase and in ompressible fluid phase, thus meeting the assumption of many problems in engineering pratie e.g., in soil mehanis. Therefore, it is hoped that this work may be useful in further studies, both theoretial and observational, of wave propagation in liquid saturated porous solid in orthotropi solid half-spae. ~ 94 ~