PLANE WAVE AND FUNDAMENTAL SOLUTION IN THERMOPOROELASTIC MEDIUM

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1 Materials Physics and Mechanics 35 (08 0- Received: June 7, 05 PLANE WAVE AND FUNDAMENTAL SOLUTION IN THERMOPOROELASTIC MEDIUM R. Kumar, S. Kumar *, M.G. Gourla 3 Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India Department of Mathematics, Govt. Degree College Chowari (Chamba, Himachal Pradesh, India 3 Department of Mathematics, Himachal Pradesh University, Shimla-7005, India * satinderkumars@gmail.com Abstract. The present article deals with the study of propagation of plane wave and fundamental solution in the thermoporoelastic medium. It is found that for two dimensional model, their exist three longitudinal waves, namely PP -wave, PP -wave and T-wave in addition to transverse wave. Characteristics of waves like phase velocity, attenuation coefficient, specific loss and penetration depth are computed numerically and depicted graphically. The representation of the fundamental solution of the system of equations in the thermoporoelastic medium in case of steady oscillations is considered in term of elementary functions. Some basic properties of the fundamental solution are established. Some special cases are also deduced. Keywords: plane wave; fundamental solution; thermoporoelastic medium; steady oscillations.. Introduction Poroelastisity is the mechanics of poroelastic solids with pores filled with fluid. Mathematical theory of poroelastisity deals with the mechanical behaviour of fluid saturated porous medium. Pore fluid generally includes gas, water and oil. Due to different motions of solid and fluid phases and complicated geometry of pore structures, it is very difficult to study the mechanical behaviour of a fluid saturated porous medium. The discovery of fundamental mechanical effects in saturated porous solids and the formulation of the first porous media theories are mainly due to Fillunger [],Terzaghi [,3,] and their successors. Based on the work of Von Terzaghi [,3], Biot [5] proposed a general theory of three dimensional consolidation. Taking the compressibility of the soil into consideration, the water contained in the pores was taken to be incompressible. Biot [6,7] developed the theory for the propagation of stress waves in porous elastic solids containing a compressible viscous fluid and demonstrated the existence of two types of compressional waves (a fast and a slow wave along with one share wave. Biot s model was broadly accepted and some of his results have been taken as standard references and the basis for subsequent analysis in acoustic, geophysics and other such fields. For the thermoporoelastisity problems, coupled thermal and poro-mechanical processes play an important role in a number of problems of interest in the geomechanics such as stability of boreholes and permeability enhancement in geothermal reservoirs. A thermoporoelastic approach combines the theory of heat conduction with poroelastic constitutive equations and coupling the temperature fields with the stresses and pore pressure. Rice and Cleary [8] presented some basic stress-diffusion solutions for fluid saturated elastic porous media with compressible constituents. There exists a substantial literature treating the extension of the well known isothermal theory to account for the effects of thermal 08, Peter the Great St. Petersburg Polytechnic University 08, Institute of Problems of Mechanical Engineering RAS

2 0 R. Kumar, S. Kumar, M.G. Gourla expansion of both the pore fluid and the elastic matrix [eg. Schiffman [9], Bowen[0], Noorishad[]]. McTigue [] developed a linear theory of fluid saturated porous thermoelastic material and this theory allows compressibility and thermal expansion of both the fluid and solid constituents. He presented a general solution scheme in which a diffusion equation with temperature dependent source term governs a combination of the mean total stress and the fluid pore pressure. Kurashige [3] extends the Rice and Cleary [8] theory to incorporate the heat transportation by a pore fluid flow in addition to the effect of difference in expansibility between the pore fluid and the skeletal solid and presented a thermoelastic theory of fluid-filled porous materials. This theory shows that the displacement field is completely coupled with the pore pressure and temperature field in general, however, for irrotational displacement, the first field is decoupled from the last two, which are still coupled to each other. This pore pressuretemperature coupling involves nonlinearity. Abousleiman and Ekbote [] obtained the solutions for the inclined borehole in a porothermoelastic transversely isotropic medium. Bai [5] studied the fluctuation responses of porous media subjected to cyclic thermal loading. Bai and Li [6] obtained the solution for cylindrical cavity in a saturated thermoporoelastic medium. Jabbari and Dehbani [7] considered the classical coupled thermoporoelastic model of hollow and solid cylinders under radial symmetric loading conditions and presented a unique solution. Ganbin et al. [8] obtained the solution in saturated porous thermoviscoelastic medium, with cylindrical cavity that is subjected to time dependent thermal load by using Laplace transform technique. Gatmiri et al. [9] presented the two-dimensional fundamental solutions for non-isothermal unsaturated deformable porous medium subjected to quasi- static loading in time and frequency domain. Li et al. [0] presented the study state solutions for transversely isotropic thermoporoelastic media in three dimensions. Jabbari and Dehbani [] considered the quasi- static porothermoelasticity model of hollow and solid sphere and obtained the displacement, temperature distribution and pressure distribution due mechanical, thermal and pressure source. Liu et al. [] studied the relaxation effect of a saturated porous media using the two dimensional generalized thermoelastic theory. Belotserkovets and Prevost [3] obtained an analytical solution of thermoporoelastic response of fluid-saturated porous sphere. Bai [] derived an analytical method for the thermal consolidation of layered saturated porous material subjected to exponential decaying thermal loading. Mixed variation principal for dynamic response of thermoelastic and poroelastic continua was discussed by Apostolakis and Dargus [5]. Hou et al. [6] discussed the three dimensional Green s function for transversely isotropic thermoporoelastic biomaterial. Jabbari et.al. [7] presented the thermal buckling analysis of functionally graded thin circular plate made of saturated porous material and obtained the closed form solutions for circular plates subjected to temperature load. Liu and Chain [8] discussed a micromechanical analysis of the fracture properties of saturated porous media. He et al. [9] studied the dynamic simulation of landslide based on thermoporoelastic approach. Nguyen et al. [30] discussed the analytical study of freezing behaviour of a cavity in thermoporoelastic medium. Wu et al. [3] presented a refined theory of axisymmetric thermoporoelastic circular cylinder. Svanadze [3, 33, 3] constructed the fundamental solutions in thermoelasticity with microtemperature and micromorphic elastic solid with microtemperature. Svanadze and his coworkers [35, 36, 37, 38] also constructed the fundamental solution and basic properties in thermomicrostretch, micropolar thermoelsticity without energy dissipation and full coupled theory of elasticity for solids with double porosity.

3 Plane wave and fundamental solution in thermoporoelastic medium 03 Kumar and Kansal [39, 0, ] obtained fundamental solution in thermomicrostrech, micropolar thermoelastic diffusion, micropolar thermoelastic diffusion with voids. Kumar and Kansal [] also investigated the plane wave and fundamental solution in thermoelastic diffusion. Kumar et.al [3] studied the plane wave and fundamental solution in the theory of an electro- microstretch generalized thermoelastic solid. Kumar and Gupta [] studied the plane wave propagation in an anisotropic thermoelastic medium with fractional order derivative and void. Sharma and Kumar [5] studied the propagation of Plane waves and fundamental solution in thermoviscoelastic medium with voids. Plane wave propagation in microstretch thermoelastic medium with microtemperature was studied by Kumar and Kaur [6]. Fundamental and plane wave solution in swelling porous medium was studied by Kumar et.al [7]. Scarpetta et.al [8] constructed the fundamental solution in the theory of thermoelasticity for solids with double porosity. Kumar and Gupta [9] studied the Plane wave propagation and domain of influence in fractional order thermoelastic material with three phase lag heat transfer. The present study deals with the study of propagation of plane waves and fundamental solution in the thermoporoelastic medium. Characteristics of waves like phase velocity and attenuation coefficient, specific loss and penetration depth are computed numerically and depicted graphically. The representation of the fundamental solution of the system of equations in the thermoporoelastic medium in the case of steady oscillations is considered in term of elementary functions.. Basic Equations Following Jabbari and Dehbani [50], the field equations are given by (λ + µ. uu + μμ uu αα pp ββ TT = ρρ uu, ( kk tt pp αα γγ pp pp YYTT ααααααααuu = 0, ( ww KK TT ZZTT 0 TT + YYTT 0 pp ββtt 0 dddddduu = 0, (3 where uu is the displacement component, pp is the pore pressure, ρρ is the bulk mass density, αα = CC ss is the Biot s coefficient, CC CC ss = 3( vv ss /EE ss is the coefficient of volumetric compression of solid grain, with EE ss and vv ss being the elastic modulus and Poisson s ratio of solid grain, C = 3( vv/ee is the coefficient of volumetric compression of solid skeleton, with EE and vv being the elastic modulus and Poisson s ratio of solid skeleton, TT 0 is initial reference temperature, ββ = 3αα ss is the thermal expansion factor, αα CC ss is the coefficient of linear thermal expansion of solid grain, YY = 3(nnαα ww + (αα nnαα ss and αα pp = nn(cc ww CC ss + ααcc ss are coupling parameters, αα ww and CC ww are the coefficients of linear thermal expansion and volumetric compression of pore water, n is the porosity, kk is the hydraulic conductivity, γγ ww is the unit of pore water and ZZ = ( nnρρ sscc ss +nnρρ ww cc ww is coupling parameter, ρρ TT ww and ρρ ss are densities of pore water 0 and solid grain and cc ww and cc ss are heat capacities of pore water and solid grain and KK is the coefficient of heat conductivity. 3. Formulation of the problem We consider a homogeneous thermoporoelastic medium. For two dimensional problems, we take displacement vector uu as: uu = (uu, 0, uu 3 ( We define the dimensionless quantities as: xx = ωω xx, uu = ωω ρρcc uu, pp = pp, cc cc ββtt 0 ββtt = λ+μμ, 0 ρρ tt = ωω tt, TT = TT, ωω = ZZTT 0cc, (5 TT 0 KK

4 0 R. Kumar, S. Kumar, M.G. Gourla ii =,,3. The displacement components uu and uu 3 are related to the potential functions ΦΦ(xx, xx 3, tt, ΨΨ(xx, xx 3, tt as: uu = ΦΦ ΨΨ, uu xx xx 3 = ΦΦ + ΨΨ (6 3 xx 3 xx Using equation ( on (-(3 and applying the dimensionless quantities defined by (5, with the aid of (6, after suppressing the prime, yield ΦΦ αααα TT ΦΦ tt = 0, (7 δδ ΨΨ ΨΨ tt bb pp bb bb 3 bb TT bb 5 where δδ = µ λλ+µ, = 0, (8 [ ΦΦ] = 0, (9 + bb 6 [ ΦΦ] = 0, (0 bb = kkωω ρρ, bb γγ ww αα = αα ppρρcc, bb αα 3 = YYρρcc, bb αα = KKωω ρρ, bb ββ TT 5 = ZZρρcc, bb 0 ββ 6 = YYYYcc ββ and ee = uu xx + uu 3 xx 3, = +.. Solution of Plane waves For Plane harmonic waves, we assume the solution of the form: (ΦΦ, ΨΨ, pp, TT = ΦΦ, ΨΨ, pp, TT ee ii[ξ(xx ll +xx 3 ll 3 ωt], ( where ωω(= ξξ cc is the frequency and ξξ is the wave number and cc is the phase velocity. ΦΦ, ΨΨ, pp, TT are undetermined constants that are independent of time t and coordinates xx, xx 3. ll and ll 3 are the direction cosines of the wave normal to the xx xx 3 plane with the property ll + ll 3 =. Using ( in (7, (9 and (0, we obtain a system of three homogeneous equations in three unknowns and these equations has non-trivial solution if the determinant of the coefficients of the system vanish, which yields the following characteristic equation in cc as: HH c 6 + HH c + HH 3 c + HH = 0, ( where HH = FF,HH ωω 6 = FF 3,HH ωω 3 = FF,HH ωω = FF, FF = bb bb, FF = bb bb ωω + iiii(bb bb 5 + bb bb + ααbb + bb, FF 3 = iiωω 3 (bb bb 5 + bb bb + ωω (bb bb 5 + bb 3 bb 6 ααbb 3 + ααbb 5 + bb 6 + bb, FF = ωω (bb bb 5 + bb 3 bb 6. The complex coefficient implies that three roots of this equation may be complex. The complex phase velocities of the longitudinal waves, given by cc ii, ii =,, 3, will be varying with the direction of phase propagation. The complex velocity of a longitudinal wave, i.e. cc ii = cc RR + iicc II, defines the phase propagation velocity VV ii = (cc RR + cc II cc RR, attenuation quality factor QQ ii = cc II cc RR, Specific loss RR ii = ππ cc II, Penetration depth SS cc ii = (cc RR + cc II ωωωω II RR for the corresponding wave. Therefore, three waves propagating in such a medium are attenuating. The same directions of wave propagation and attenuation vector of these waves make them homogeneous wave. The waves with phase velocity, attenuation quality factor, specific loss and penetration depth i.e. VV ii, QQ ii, RR ii and SS ii (ii =,, 3, may be named as PP -wave, PP -wave and T-wave that are propagating with the descending order of their velocities, respectively. Substituting the value of ΨΨ from ( in (8, we obtain: (ωω δδ ξ ΨΨ = 0, (3 which yield

5 Plane wave and fundamental solution in thermoporoelastic medium 05 c = ±δδ ( The equation (3 corresponds to transverse wave which travel with phase velocity δδ. 5. Special Case In the absence of porosity effect, characteristic equation reduces to: HH 5 c + HH 6 c + bb = 0, (5 where HH 5 = FF 6,HH ωω 6 = FF 5, FF ωω 5 = bb ωω iiii(bb 50 +, FF 6 = iiωω 3 bb 50, where bb 50 = ρρ ss cc ss cc ββ TT 0 and the equation (3 remains the same because it is not effected by porous effect. 6. Steady Oscillations For steady oscillations, we assume the displacement vector, pressure and temperature change, of the form: (uu(xx, tt, pp(xx, tt, TT(xx, tt = (uu, pp, TT ee iiiiii (6 Making use of dimensionless quantities defined by (5, in (-(3 and with the aid of (6 yields: ( δδ. uu + (δδ + ωω uu αα pp TT = 0, (7 aa pp + iiiiaa pp + ααωωtt + iiiiaa 3 dddddduu = 0, (8 aa TT + iiiiaa 5 TT + iiiiaa 6 pp + iiiiaa 7 dddddduu = 0, (9 where aa = kkωω ββ γγ ww YYcc, aa = αα pp ββ, aa αα ββ YY 3 = YYYYcc, aa = kkωω cc, aa 5 = ZZTT 0, aa 6 = YYββTT 0, aa 7 = ββ TT 0 ρρcc. We introduce the matrix differential operator: FF(DD xx = FF ggh (DD xx, (0 5 5 where FF mmmm (DD xx = (δδ + ωω δδ mmmm + ( δδ, FF xx m xx mm (DD xx = αα, n xx m FF mm5 (DD xx =, FF xx nn (DD xx = iiiiaa 3, FF m xx 5nn (DD xx = iiiiaa 7, n xx n FF (DD xx = iiiiaa + aa, FF 5 (DD xx = iiii, FF 5 (DD xx = iiiiaa 6, FF 55 (DD xx = iiiiaa 5 + aa, mm, nn =,,3, ( and δδ mmmm is the Kronecker delta. The system of equations (7-(9 can be written as: FF(DD xx UU(xx = 0, ( where UU = (uu, pp, TT is a five component vector function on EE 3. We assume that δδ aa aa 0 (3 If the condition (3 is satisfied, then FF is an elliptic differential operator, Hoo rmander [55]. Definition: The fundamental solution of the system of equations (7-(9 (the fundamental matrix of operator FF is the matrix GG(xx = GG ggh (xx, satisfying condition, 5 5 Hoo rmander [5]: FF(DD xx UU(xx = δδ(xxii(xx, ( where δδ( is the Dirac delta function, II = δδ ggh is the unit matrix and xx = EE Now we construct GG(xx in terms of elementary functions. 6. Fundamental solution of system of equations of steady oscillations. We consider the system of equations:

6 06 R. Kumar, S. Kumar, M.G. Gourla δδ uu + ( δδ. uu + iiiiaa 3 pp + iiiiaa 7 TT + ωω uu = HH; (5 ααααααααuu + (iiiiaa + aa pp + iiiiaa 6 TT = LL; (6 ααααααuu + ααωωpp + (iiiiaa 5 + aa TT = MM, (7 where HH is three components of vector function on EE 3 and LL, MM are scalar functions of EE 3.The system of equations (5-(7 may be written in the form: FF tttt (DD xx UU(xx = QQ(xx, (8 where FF tttt is the transpose of FF, QQ = (HH, LL, MM and xx = EE 3. Applying the operator αααααα to (5, we obtain: ( + ωω ααααααuu + iiiiaa 3 pp + iiiiaa 7 TT + ωω dddddduu = ααααααhh (9 The equations (6, (7 and (9 can be written in the form: NN( SS = QQ, (30 where SS = (ααααααuu, pp, TT, QQ = (dd, dd, dd 3 = (ααααααhh, LL, MM, and + ωω iiiiaa 3 NN( = NN mmmm ( 3 3 = αα iiiiaa + aa iiii iiiiaa 7 iiiiaa 6 iiiiaa 5 + aa (3 The equation (30 can also be written as: ΓΓ ( SS = ΨΨ, (3 where ΨΨ = (ΨΨ, ΨΨ, ΨΨ 3, ΨΨ nn = ee 3 mm= NN mmmm dd mm, ΓΓ ( = ee det NN(, ee =, nn =,,3, (33 aa aa and NN mmmm is the cofactor of the elements NN mmmm of the matrix NN. From (3 and (33, we see that ΓΓ ( = 3 mm= ( + λλ mm, (3 where λλ mm, mm =,,3 are the root of the equation ΓΓ ( κκ = 0 (with respect to κκ. Applying operator ΓΓ ( on (5, we have: ΓΓ ( [δδ uu + ( δδ. uu + iiiiaa 3 pp + iiiiaa 7 TT + ωω uu] = ΓΓ ( HH ΓΓ ( [δδ uu + ωω uu] = ΓΓ ( HH ggggad[( δδ ΨΨ + iiiiaa 3 ΨΨ + iiiiaa 7 ΨΨ 3 ] ΓΓ ( + ωω δδ uu = δδ [ΓΓ ( HH ggggad{( δδ ΨΨ + iiiiaa 3 ΨΨ + iiiiaa 7 ΨΨ 3 }] ΓΓ ( [ + λλ ]uu = δδ [ΓΓ ( HH ggggad{( δδ ΨΨ + iiiiaa 3 ΨΨ + iiiiaa 7 ΨΨ 3 }] ΓΓ ( [ + λλ ]uu = ΨΨ, (35 where λλ = ωω δδ, ΨΨ = δδ [ΓΓ ( HH ggggad{( δδ ΨΨ + iiiiaa 3 ΨΨ + iiiiaa 7 ΨΨ 3 }] (36 From equations (3 and (35, we obtain: ΘΘ( UU(xx = ΨΨ (xx, (37 where ΨΨ (xx = (ΨΨ, ΨΨ, ΨΨ 3, and ΘΘ( = Θ ggh ( 5 5 ; Θ mmmm ( = ΓΓ ( [ + λλ ], mm =,,3; Θ ( = Θ 55 ( = ΓΓ (, Θ ggh ( = 0, gg, h =,,3,,5, gg h. The equations (33 and (36 can be written in the form: ΨΨ = [ΓΓ δδ ( HH ggggad( δδ ee 3 NN mm= mm dd mm + iiiiaa 3 ggggad ee 3 mm= NN mm dd mm + iiiiaa 7 ggggad ee 3 mm= NN mm3 dd mm ], ΨΨ = ΓΓ δδ ( HH + δδ ee ggggad dddddd HH{ ( δδ NN + iiiiaa 3 NN + iiiiaa 7 NN 3 } + δδ ee ggggad LL{ ( δδ NN + iiiiaa 3 NN + iiiiaa 7 NN 3 } + δδ ee ggggad MM{ ( δδ NN 3 iiiiaa 3 NN 3 + iiiiaa 7 NN 33 }, ΨΨ = ΓΓ δδ ( JJ + qq ( ggggad dddddd HH + qq ( ggggad LL + qq 3 ( ggggad MM, ΨΨ = qq ( ddddddhh + qq ( LL + qq 3 ( MM, (38 +

7 Plane wave and fundamental solution in thermoporoelastic medium 07 ΨΨ 3 = qq 3 ( ddddddhh + qq 3 ( LL + qq 33 ( MM, where JJ = δδ ggh 3 3 is the unit matrix and qq ( = δδ ee { ( δδ NN + iiiiaa 3 NN + iiiiaa 7 NN 3 }, qq ( = δδ ee { ( δδ NN + iiiiaa 3 NN + iiiiaa 7 NN 3 }, qq 3 ( = δδ ee { ( δδ NN 3 + iiiiaa 3 NN 3 + iiiiaa 7 NN 33 }, qq mm ( = ee NN mm, qq mm3 ( = ee NN mm3, m=,,3. Now from equations (38, we have: ΨΨ (xx = RR tttt (DD xx QQ(xx, (39 where RR tttt is the transpose of RR, and RR = RR mmmm 5 5, RR mmmm (DD xx = δδ ΓΓ ( δδ mmmm + qq, RR mm mm (DD xx = qq, RR nn mm5 (DD xx = qq 3 mm mm, RR nn (DD xx = qq, RR 5nn (DD xx = qq 3, mm, nn =,,3, nn nn RR 5 (DD xx = qq, RR 6 (DD xx = qq 3, RR 5 (DD xx = qq 3, RR 55 (DD xx = qq 33. (0 Now from equations (8, (37 and (39, we have: ΘΘUU = RR tttt FF tttt UU. ( It implies that RR tttt FF tttt = ΘΘ, and hence FF(DD xx RR(DD xx = ΘΘ(. ( We assume that: λλ mm λλ nn 0, mm, nn =,,3,, mm nn. Let YY(xx = YY rrrr (xx 5 5, YY mmmm (xx = nn= rr nn ξ nn (xx, mm =,,3, YY (xx = YY 55 (xx = 3 nn= rr nn ξ nn (xx YY uuuu (xx = 0, uu, vv =,,3,,5, uu vv, where ξ nn (xx = exp(iiλλ ππ xx nn xx, nn =,,3,, rr ll = mm=,mm ll (λλ mm λλ ll, ll =,,3,, rr vv = mm=,mm vv (λλ mm λλ vv, vv =,,3,, (3 We will prove the following Lemma: The matrix YY defined above is the fundamental matrix of operator ΘΘ(, that is ΘΘ( YY(xx = δδ(xxii(xx. ( Proof: To prove the lemma, it is sufficient to prove that: ΓΓ ( ( + λλ YY (xx = δδ(xx, ΓΓ ( YY (xx = δδ(xx. (5 We find that: rr + rr + rr 3 + rr = tt +tt +tt 3 +tt, (6 tt 5 tt = (λλ λλ 3 (λλ λλ (λλ 3 λλ, tt = (λλ λλ 3 (λλ λλ (λλ 3 λλ, tt 3 = (λλ λλ (λλ λλ (λλ λλ, tt = (λλ λλ (λλ λλ 3 (λλ λλ 3, tt 5 = (λλ λλ (λλ λλ 3 (λλ λλ (λλ λλ 3 (λλ λλ (λλ 3 λλ. (7 Using (7 in (6, yield rr + rr + rr 3 + rr = 0, (8 rr (λλ λλ + rr 3 (λλ λλ 3 + rr (λλ λλ = 0, (9 rr 3 (λλ λλ 3 (λλ λλ 3 + rr (λλ λλ (λλ λλ = 0, (50 rr (λλ λλ (λλ λλ (λλ 3 λλ =, (5 ( + λλ ξ nn (xx = δδ(xx + (λλ mm λλ nn ξ nn (xx, mm, nn =,,3,. (5 Now consider: ΓΓ ( ( + λλ YY (xx = ( + λλ ( + λλ ( + λλ 3 ( + λλ nn= rr nn ξ nn (xx = = ( + λλ ( + λλ 3 ( + λλ nn= rr nn [δδ(xx + (λλ λλ nn ξ nn (xx] = = ( + λλ ( + λλ 3 ( + λλ [δδ(xx rr nn + rr nn (λλ λλ nn ξ nn (xx] nn= nn= =

8 08 R. Kumar, S. Kumar, M.G. Gourla = ( + λλ ( + λλ 3 ( + λλ nn= rr nn (λλ λλ nn ξ nn (xx = = ( + λλ 3 ( + λλ rr nn (λλ λλ nn [δδ(xx + (λλ λλ nn ξ nn (xx] = ( + λλ 3 ( + λλ nn= rr nn (λλ λλ nn (λλ λλ nn ξ nn (xx = = ( + λλ rr nn (λλ λλ nn (λλ λλ nn [δδ(xx + (λλ 3 λλ nn ξ nn (xx] nn= = nn=3 = nn=3 = = ( + λλ rr nn (λλ λλ nn (λλ λλ nn (λλ 3 λλ nn ξ nn (xx = ( + λλ ξ nn (xx = δδ(xx. (53 Similarly, ΓΓ ( YY (xx = δδ(xx, can be proved. We introduce the matrix: GG(xx = RR(DD xx YY(xx. (5 From equations (, ( and (5, we have: FF(DD xx GG(xx = FF(DD xx RR(DD xx YY(xx = δδ(xxii(xx, (55 hence GG(xx is the solution to the equations (. Therefore we have proved the following theorem. Theorem: The matrix GG(xx defined by (5 is the fundamental solution of the system of equations (7-(9. 6. Basic properties of the matrix GG(xx. Property. Each column of the matrix GG(xx is the solution of system of equations (7-(9 to every point xx EE 3 except the origin. Property. The matrix GG(xx can be written in the form: GG = GG ggh, 5 5 GG mmmm (xx = RR mmmm (DD xx YY (xx, GG pp (xx = RR pp (DD xx YY (xx, GG pp5 (xx = RR pp5 (DD xx YY (xx, GG pp (xx = RR nn (DD xx YY (xx, GG 5pp (xx = RR 5nn (DD xx YY (xx, mm, nn =,,3, pp =,,3,,5. 7. Numerical results and discussion With the view of illustrating the theoretical results and for numerical discussion we take a model for which the values of the various physical parameters are taken from Jabbari and Dehbani [5]: EE = PPPP, νν = 0.3, TT 0 = 93 KK, KK ss = 0 0 PPPP, KK ww = PPPP, KK = 0.5WW/mm, αα ss = /, αα WW = 0 /, cc ss = 0.8 JJ/gg, cc ww =. JJ/gg, ρρ ss = gg/mm 3, ρρ ww = 0 6 gg/mm 3, nn = 0., αα = The software Matlab 7.0. has been used to determine the values of phase velocity, attenuation coefficient, specific loss and penetration depth of plane waves, i.e. PP -wave, PP -wave and T-wave. The variations of phase velocity, attenuation coefficients, specific loss and penetration depth with respect to frequency are shown in Figs. - respectively. In all the Figures, the solid line corresponds to thermoporoelastic medium ( and dotted line corresponds to thermoelastic medium (. 7.. Phase velocity. Fig. depicts the variations of phase velocity of VV with frequency ωω for and. In case of, the value of VV increases exponentially as ωω increases, whereas in case of, the value of VV oscillates in the range 0 ωω.5 and then increase as ωω increases. Fig. shows the variations of VV with ωω for and. The value of VV for and increases gradually but due to the effect of porosity the value of VV is less for as compared to as ωω increases. Fig.3 shows that the value of VV 3 for when αα = 0.5 increases gradually as ωω increases whereas there is sharp increase in the value of VV 3 when αα = 0.75 as ωω increases. Due to the effect of porosity the value of VV 3, when αα = 0.5 is more as compared to αα = 0.75 in the range 0 ωω 0. and then show the opposite behaviour for higher values of ωω.

9 Plane wave and fundamental solution in thermoporoelastic medium Phase velocity (V Frequency (ω Fig.. Variation of phase velocity w.r.t. frequency. Phase velocity (V Frequency (ω Fig.. Variation of phase velocity w.r.t. frequency α=0.5 α=0.75 Phase velocity (V Frequency (ω Fig. 3. Variation of phase velocity w.r.t. frequency. 7.. Attenuation quality factor. Fig. shows the variation of QQ with ωω for and. The value of QQ for and increases gradually as ωω increases but due to the effect of porosity the value of QQ for is more as compared to as ωω increases. The variation of QQ with ωω for and is shown in Fig.5. The value of QQ is consistent for whereas for, there is a small increase in the value of QQ as ωω increases. The value of QQ for is more as compared to due to the effect of porosity as ωω increases.

10 0 R. Kumar, S. Kumar, M.G. Gourla Fig. 6 depicts the variations of QQ 3 with frequency ωω for. When αα = 0.5, there is a small increase in the value of QQ 3 as ωω increases, whereas for αα = 0.75, the value of QQ 3 decreases exponentially for higher values of ωω. 8 Attenuation quality factor (Q Frequency (ω Fig.. Variation of attenuation quality factor w.r.t frequency..0 Attenuation quality factor (Q α=0.5 α= Frequency (ω Fig. 5. Variation of attenuation quality factor w.r.t frequency..0.8 Attenuation quality factor (Q Frequency (ω Fig. 6. Variation of attenuation quality factor w.r.t. frequency Specific loss. The variation of specific loss RR with frequency ωω is shown in Fig. 7. There is a small increase in the value of RR for as ωω increases whereas for, the value of RR increases exponentially as ωω increases. Due to the effect of porosity the value of RR for is more as compared to as ωω increases. Fig. 8 shows the variations of RR with ωω for and. The value of RR for increases in the range 0 ωω 3.5 then starts decreasing as ωω increases whereas for, the

11 Plane wave and fundamental solution in thermoporoelastic medium value of RR decreases exponentially for higher values of ωω. The value of RR for is less as compared to due to the effect of porosity as ωω increases. Fig. 9 shows the variation of RR 3 with frequency ωω for. There is small increase in the value of RR 3 for when αα = 0.5 as ωω increases whereas for = 0.75, the value of RR 3 decreases sharply for higher values of ωω. 0 Specific loss (R Frequency (ω Fig. 7. Variation of specific loss w.r.t frequency. Specific loss (R Frequency (ω Fig. 8. Variation of specific loss w.r.t frequency. α=0.5 α= Specific loss (R Frequency (ω Fig. 9. Variation of specific loss w.r.t frequency. 7.. Penetration depth. The variation of SS with ωω for and is shown in Fig.0. The value of SS, for, decreases exponentially whereas for, with small initial decrease, the value of SS becomes consistent for higher values of ωω. Due to the effect of porosity the value of SS for is more as compared to as ωω increases.

12 R. Kumar, S. Kumar, M.G. Gourla Fig. depicts the variation of SS with frequency ωω.the value of SS decreases gradually for all values of ωω for and but the value of is less as compared to in the range 0 ωω 3 and then show the opposite behaviour for higher values of ωω due to the effect of porosity. Fig. shows the variations of SS 3 with ωω for. The value of SS 3 for decreases gradually for both values of αα, but due to the effect of porosity, the value of SS 3 is less for αα = 0.5 as compared to the value of αα = Penetration depth (S Penetration depth (S Frequency (ω Frequency (ω Fig. 0. Variation of penetration depth w.r.t frequency. 6 Penetration depth (S Frequency (ω Fig.. Variation of penetration depth w.r.t frequency α=0.5 α=0.75 Penetration depth (S Penetration depth (S α= Frequency (ω Frequency (ω Fig.. Variation of penetration depth w.r.t frequency.

13 Plane wave and fundamental solution in thermoporoelastic medium 3 8. Conclusion The present study deals with the propagation of plane wave and fundamental solution in the thermoporoelastic medium. It is found that for two dimensional model, their exist three longitudinal waves, namely PP -wave, PP -wave and T-wave in addition to transverse wave. The phase velocity, attenuation coefficient, specific loss and penetration depth are computed numerically and depicted graphically. The fundamental solution of the system of equations in the thermoporoelastic medium in the case of steady oscillations is considered in term of elementary functions. Due to the presence of porous effect, the phase velocities of PP -wave for remains more in comparison to, whereas the phase velocity of T-wave is less for in comparison to for all values of frequency. Attenuation quality factor, penetration depth and specific loss of PP -wave for is more in comparison to and reverse behaviour is shown in case of T-wave. The value of attenuation quality factor, penetration depth and specific loss of PP -wave for αα = 0.5 is less as compared to αα = 0.75 and opposite behaviour is shown for penetration depth whereas the value of phase velocity of PP -wave is more for αα = 0.5 as compared to αα = 0.75 for initial values of ωω and shows the opposite behaviour for higher values of ωω. References [] P. Fillunger // Osterr, Wochenschrift fur den offentl. Baudienst 9 ( [] K.V. Terzaghi // Sitzungsber Akad. Wiss. (Wien, Math.-Naturwiss.KI., Abt.IIa 3 (93 5. [3] K.V. Terzaghi, Erdbaumechanik auf Bodenphysikalischer Grundlage (Franz Deuticke, Leipzig-wien, 95. [] K.V. Terzaghi // Die Wasserwirtschaft 6 ( [5] M.A. Biot // Journal of Applied Physics ( (9 55. [6] M.A. Biot // Journal of the Acoustical Society of America 8 (956a 68. [7] M.A. Biot // Journal of the Acoustical Society of America 8 (956b 79. [8] J.R. Rice, M.P. Cleary // Rev. Geophys. Space Phys. ( [9] R.L. Schiffman, A thermoelastic theory of consolidation, In: Environmental and Geophysical Heat Transfer (American Society of Mechanical Engineers, New York, [0] R.M. Bowen // International Journal of Engineering Science 0 ( [] J. Noorishad, C.F. Tsang, P.A. Witherspoon // J. Geophys. Res. 89 ( [] D.F. McTigue // Journal of Geophysical Research 9(B9 ( [3] M. Kurashige // International Journal of Solids and Structures 5 ( [] Y. Abousleiman, S. Ekbote // ASME Journal Applied Mechanics 7 ( [5] B. Bai // Computers and geotechnics 33 ( [6] B. Bai, T. Li // Acta Mechanica Solida Sinica ( ( [7] M. Jabbari, H. Dehbani // Journal of Solid Mechanics ( (00 9. [8] L. Ganbin, X. Kanghe, Z. Rongyue // Applied Mathematical Modelling 3 (00 3. [9] B. Gatmiri, P. Maghoul, D. Duhamel // International Journal of Solids and Structures 7 ( [0] X. Li, W. Chen, H. Wang // European J. Mech. A/Solid 9(3 ( [] M. Jabbari, H. Dehbani // Iranian Journal of Mechanical Engineering ( (0 86. [] G. Liu, S. Ding, R. YE, X. Liu // Transp. Prous Med 86 (0 83. [3] A.B. Belotserkovets, J.H. Prevost // Int. J. Eng. Sci. 9( (0 5. [] B. Bai // Journal of Thermal Stresses 36( (03 7. [5] G. Apostolakis, G.F. Dargus // Int. J. of Solid and Structure 50(5 (03 6. [6] Peng-Fei Hou, Meng Zhao, Jiann-Wen Ju // Journal of Applied Geophysics 95 (03 36.

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