Deformation Due to Inclined Loads in Thermoporoelastic Half Space

Transcription

1 Journal of Solid Mechanic Vol. 8, No. (6) pp Deformation Due to Inclined Load in Thermoporoelatic Half Space R. Kumar, S.Kumar, *, M. G. Gorla Department of Mathematic, Kurukhetra Univerity, Kurukhetra, Haryana, India Department of Mathematic, Govt. Degree College Chowari (Chamba), Himachal Pradeh, India Department of Mathematic, Himachal Pradeh Univerity, Shimla-75, India Received 8 June 6; accepted 5 Augut 6 ABSTRACT The preent invetigation i concerned with the deformation of thermoporoelatic half pace with incompreible fluid a a reult of inclined load of arbitrary orientation. The inclined load i aumed to be linear combination of normal load and tangential load. The Laplace and Fourier tranform technique are ued to olve the problem. The concentrated force, uniformly ditributed force and a moving force in time and frequency domain are taken to illutrate the utility of the approach. The tranformed component of diplacement, tre, pore preure and temperature change are obtained and inverted by uing a numerical inverion technique. The variation of reulting quantitie are depicted graphically. A particular cae ha alo been deduced..all right reerved. Keyword : Inclined load;time and frequency domain; Laplace and fourier tranform. INTRODUCTION P OROELASTISITY i the mechanic of poroelatic olid with pore filled with fluid. Mathematical theory of poroelatiity deal with the mechanical behaviour of fluid aturated porou medium. Pore fluid generally include ga, water and oil. Due to different motion of olid and fluid phae and complicated geometry of pore tructure, it i very difficult to tudy the mechanical behaviour of a fluid aturated porou medium. The dicovery of fundamental mechanical effect in aturated porou olid and the formulation of the firt porou media theorie are mainly Fillunger [],Terzaghi [,,4] and their ucceor. Baed on the work of Von Terzaghi [,], Biot [5] propoed a general theory of three dimenional conolidation. Taking the compreibility of the oil into conideration, the water contained in the pore wa taken to be incompreible. Biot [6,7] developed the theory for the propagation of tre wave in porou elatic olid containing a compreible vicou fluid and demontrated the eitence of two type of compreional wave (a fat and a low wave) along with one hare wave. Biot model wa broadly accepted and ome of hi reult have been taken a tandard reference and the bai for ubequent analyi in acoutic, geophyic and other uch field. For the thermoporoelatiity problem, coupled thermal and poro-mechanical procee play an important role in a number of problem of interet in the geomechanic uch a tability of borehole and permeability enhancement in geothermal reervoir. A thermoporoelatic approach combine the theory of heat conduction with poroelatic contitutive equation and coupling the temperature field with the tree and pore preure. Rice and Cleary [8] preented ome baic tre-diffuion olution for fluid aturated elatic porou media with compreible contituent. There eit a ubtantial literature treating the etenion of the well known iothermal * Correponding author. addre: atinderkumar@gmail.com (S. Kumar).. All right reerved.

2 R. Kumar et al. 66 theory to account for the effect of thermal epanion of both the pore fluid and the elatic matri [eg. Schiffman[9], Bowen[], Noorihad []]. McTigue [] developed a linear theory of fluid aturated porou thermoelatic material and thi theory allow compreibility and thermal epanion of both the fluid and olid contituent. He preented a general olution cheme in which a diffuion equation with temperature dependent ource term govern a combination of the mean total tre and the fluid pore preure. Kurahige[] etend the Rice and Cleary [8] theory to incorporate the heat tranportation by a pore fluid flow in addition to the effect of difference in epanibility between the pore fluid and the keletal olid and preented a thermoelatic theory of fluid-filled porou material. Thi theory how that the diplacement field i completely coupled with the pore preure and temperature field in general, however, for irrotational diplacement; the firt field i decoupled from the lat two, which are till coupled to each other. Thi pore preure-temperature coupling involve nonlinearity. Abouleiman and Ekbote [4] obtained the olution for the inclined borehole in a porothermoelatic tranverely iotropic medium. Bai [5] tudied the fluctuation repone of porou media ubjected to cyclic thermal loading. Bai and Li [6] obtained the olution for cylindrical cavity in a aturated thermoporoelatic medium. Jabbari and Dehbani [7] conidered the claical coupled thermoporoelatic model of hollow and olid cylinder under radial ymmetric loading condition and preented a unique olution. Ganbin et al. [8] obtained the olution in aturated porou thermovicoelatic medium, with cylindrical cavity that i ubjected to time dependent thermal load by uing Laplace tranform technique. Gatmiri et al. [9] preented the two-dimenional fundamental olution for non-iothermal unaturated deformable porou medium ubjected to quai- tatic loading in time and frequency domain. Li et al. [] preented the tudy tate olution for tranverely iotropic thermoporoelatic media in three dimenion. Jabbari and Dehbani [] conidered the quai- tatic porothermoelaticity model of hollow and olid phere and obtained the diplacement, temperature ditribution and preure ditribution due mechanical, thermal and preure ource. Liu et al. [ ] tudied the relaation effect of a aturated porou media uing the two dimenional generalized thermoelatic theory. Beloterkovet and Prevot [] obtained an analytical olution of thermoporoelatic repone of fluid-aturated porou phere. Bai [4] derived an analytical method for the thermal conolidation of layered aturated porou material ubjected to eponential decaying thermal loading. Mied variation principal for dynamic repone of thermoelatic and poroelatic continua wa dicued by Apotolaki and Dargu [5]. Hou, et al. [6] dicued the three dimenional Green function for tranverely iotropic thermoporoelatic biomaterial. Jabbari et al. [7] preented the thermal buckling analyi of functionally graded thin circular plate made of aturated porou material and obtained the cloed form olution for circular plate ubjected to temperature load. Liu and Chain [8] dicued a micromechanical analyi of the fracture propertie of aturated porou media. He et al. [9] tudied the dynamic imulation of landlide baed on thermoporoelatic approach. Nguyen et al. [] dicued the analytical tudy of freezing behaviour of a cavity in thermoporoelatic medium. Wu et al. [] preented a refined theory of aiymmetric thermoporoelatic circular cylinder. Kumar and Ailawalia [, ] tudied the repone of moving inclined load in orthotropic elatic half-pace and micropolar elatic half-pace with void repectively. Kumar and Rani [4] tudied the general plane train problem of thermoelatic half-pace with void a a reult of inclined load different ource. Sharma [5] invetigated the deformation in a homogeneou iotropic thermodiffuive elatic half-pace a a reult of inclined load by auming the inclined load a a linear combination of normal load and tangential load. Stre-train tate of a inclined elliptical defect in a plate under biaial loading wa dicued by Otemin and Utkin [6]. Stre-train tate of an elatic half plane under a ytem of inclined piecewie - linear load wa tudied by Bogomolov and Uhakov [7]. In the preent paper, the invetigation i concerned with the deformation of thermoporoelatic half pace with incompreible fluid a a reult of inclined load of arbitrary orientation. The inclined load i aumed to be linear combination of normal load and tangential load. The component of diplacement, tre, pore preure and temperature change are obtained in time and frequency domain. Numerical inverion technique i applied to obtain the reulting quantitie in a phyical domain. The reulting quantitie are hown graphically to depict the effect of poroity The reult of the problem may be applied in the field of engineering and geophyical problem involving temperature change. The phyical application are encounter in the contet of problem like ground eploion, oil indutrie etc. Thi problem i alo ueful in the field of geomechanic, where the interet i about the variou phenomenon occurring in the earthquake and meauring the component of diplacement, tre, pore preure and

3 67 Deformation Due to Inclined Load in Thermoporoelatic Half Space temperature change the preence of certain ource. Alo the preent invetigation i ueful to tudy the deformation field around mining tremor and drilling into the crut of the earth. It alo contribute to the theoretical conideration of the eimic ource becaue it can account for the deformation field in the entire volume urrounding the ource region. BASIC EQUATIONS Following Jabbari and Dehbani [8], the baic equation are u u u p T. t () k w p p p Y T div u () K T ZTT Y Tp T div u () u ( u u ) p T (4) ij k, k ij i, j j, i ij ij where u C i the diplacement component, p i the pore preure, i the bulk ma denity, i the C ( ) Biot coefficient, C i the coefficient of volumetric compreion of olid grain, with E and E ( ) being the elatic modulu and Poion ratio of olid grain, C i the coefficient of volumetric E compreion of olid keleton, with E and v being the elatic modulu and Poion ratio of olid keleton, T i initial reference temperature, i the thermal epanion factor, i the coefficient of linear thermal C epanion of olid grain, Y ( nw ( n) ) and p n( Cw C ) C are coupling parameter, w and C w are the coefficient of linear thermal epanion and volumetric compreion of pure water, n i the poroity, k i ( n) the hydraulic conductivity, w i the unit of pore water and C n w C Z w i coupling parameter, T w and are denitie of pore water and olid grain and C w and C are heat capacitie of pore water and olid grain and K i the coefficient of heat conductivity. FORMULATION OF THE PROBLEM We conider homogeneou, iotropic, poroelatic thermal conducting half pace of a rectangular Carteian coordinate ytem (,, ) having origin at the urface and ai pointing vertically downward in the medium ha been taken. We take an inclined load F, per unit length, i acting along the interface on ai and it inclination with (e). ai i. The complete geometry of the problem i hown in the Fig. (a), (b),(c),(d) and

4 R. Kumar et al. 68 F P Inclined load of magnitude P Fluid aturated thermoporoelatic half pace (a) (b) Concentrated Source Uniformly ditributed force (c) (d) vt (e) Fig. a) Inclined load on the fluid aturated thermoporoelatic half pace. b) Component of the inclined load. c) Concentrated force on the fluid aturated thermoporoelatic half pace. d) Uniformly ditributed force on the fluid aturated thermoporoelatic half pace.e) Moving force on the fluid aturated thermoporoelatic half pace. For two dimenional problem, we aume the diplacement vector u a: (5) u ( u,, u ) For further conideration it i convenient to introduce the dimenionle quantitie defined a: c c p,, u u, u u, p, c, t * * * * c c T T T * T * ZTOc t, T,,, T K T T (6) * where i a contant having the dimenion of frequency. The diplacement component u and u are related to the potential function (,, t ), (,, t ) a:

5 69 Deformation Due to Inclined Load in Thermoporoelatic Half Space u, u (7) Making ue of (5) on ()-() and uing the dimenionle quantitie given by (6), on the reulting quantitie and with the aid of (7), after uppreing the prime, yield ( a ) a p a T a t (8) a a t (9) p T b p b b t t t [ ] () T p b T b b t t t [ ] () where c * c c k p,,,, w * Y c K Z c Y c u u, 4, 5, 6, T a a a b b b b b b e We define the Laplace and Fourier tranform a follow: t f (,, ) f (,, t ) e dt () f (,, ) f (,, ) e i d () Applying Laplace and Fourier tranform defined by () and () on Eq. (8)-() and eliminating p and T from reulting equation, we obtain 6 4 d d d [ A A A A ] (4) d d d d [ A ] 5 d (5) where A b b ( a ), 4 A b b ( a ) ( a )( b b b b ) a b b a b a b, A b b ( a ) ( a )( b b b b ) ( a )( b b b b ) a b,

6 R. Kumar et al. 6 A b b ( a ) ( a )( b b b b ) ( a )( b b b b ) a b b, a A 5. a T a: Solving (4) and (5) and auming that,, p and T a we obtain the value of,,p and p T ri i i -mi (6) i (,, ) (,, )B e -m 4 B 4e (7) where m, m, m are the root of the Eq. (6) and m4 A5. The coupling contant are given by b ( m ) ( b b )( m ) 4 i 5 i i bb 4 mi bb 5 bb4 mi bb5 bb6 r ( ) ( )( ) ( ) (8) b ( m ) ( b b )( m ) ( i,,) ( ) ( )( ) ( ) i 6 i i bb 4 mi bb 5 bb4 mi bb5 bb 6 (9) The diplacement component u and u are obtained with the aid of ()-() and (6)-(7) a: u B ie B ie B ie B m e () m m m m4 4 4 u B m e B m e B m e B i e () m m m m4 4 4 BOUNDARY CONDITIONS The boundary condition at are T P F (, t ), P F (, t ), p, () where P, P are the magnitude of the force and F(, t ) i a known function of and t. Applying Laplace and Fourier tranform defined by () and () on () and with the aid of (4),(6) along with P P P', P' T T (after uppreing the prime), we obtain where T P F (, ), P F (, ), p,, at () du (4) Ri u R p T d

7 6 Deformation Due to Inclined Load in Thermoporoelatic Half Space du R [ i u ] (5) d. and R, R, R c c c Subtituting the value of u, u, p and T from (),() and (6) in the boundary condition () and with help of (4) and (5), after ome implification, we obtain F [ d e d e d e d e ] m m m m4 44 (6) F [ d e d e d e d e ] m m m m (7) F m i ( p, T ) ( ri, i ) ie (8) i where d d ( m r m r ) d d ( m r m r ) d d ( m r m r ) d d ( m r m r ) d d ( m r m r ) d d ( m r m r ) d R R m r where ( i,,), d 4 im4( R R) d i i i i j i m R where ( j 5,6,7), d R ( m ) j 8 4 and,,, 4 are obtained by replacing [ P, P,,] T in. 5 APPLICATIONS 5. Inclined load For an inclined load F, per unit length, we have P F co, P F in (9) Uing (9) in Eq. (6)-(8), we obtain the correponding epreion for tree, pore preure and temperature in cae of inclined load on the urface of half pace. 5.. Time domain The epreion of tre component, pore preure and temperature change are in tranformed variable in which F(, t ) i an unknown function. A different cla of ource i repreented by etting, F(, t ) F ( ) ( t ) () where ( t ) H ( t ),where H () Heaviide unit tep function, F( ) i a known function and take two type of value repreenting the two different ource a,

8 R. Kumar et al. 6 Concentrated force: F ( ) ( ) () where () i the Dirac-delta function. Uniformly ditributed force: F ( ) H ( a) H ( a) () Applying the Laplace and Fourier tranform defined by () and () on (), with the aid of () and (), yield F (, ) / () For concentrated force, and F (, ) [(in a) / ] (4) For ditributed force. Moving force: F(, t ) ( t ) ( t ) (5) where v i the uniform peed of the impulive force at. Applying the Laplace and Fourier tranform defined by () and () on (5), we obtain F (, ) /( i) (6) The epreion for tre component, pore preure and temperature change can be obtained for concentrated force, ditributed force and moving force by replacing F(, ) from (),(4) and (6) in (6)-(8). 5.. Frequency domain In thi cae, we aume the time harmonic behaviour a, ( u, u, p, T )(,, t ) ( u, u, p, T )(, ) e it In frequency domain, we take i t () t e (7) The epreion for tree, pore preure and thermal ource in frequency domain can be obtained by replacing i t by i in the epreion of time domain along with ( ) to be replaced by e for concentrated and ditributed force. The olution impulive harmonic force, moving with uniform dimenionle peed v at i obtained by replacing F(, t ) with ( t), whoe Fourier tranform ie i t. 5. Special cae In the abence of poroity effect, the boundary condition reduce to

9 6 Deformation Due to Inclined Load in Thermoporoelatic Half Space T P F (, ), P F (, ), (8) and we obtain the correponding epreion for tre component and temperature change in thermoelatic elatic half pace a: F [ d e d e d e ] m5 m6 m (9) F [ d e d e d e ] m5 m6 m (4) F m5 m6 T [ r5 5e r6 6e ] (4) where d d m r d d m r d d m r d d m r d R R m r, d R R m r, d i m R, d i m R, m ( i ) i 4( i ) 5 r ( i 5, 6) b m b and 5, 6, 7 are obtained by replacing [ P, P,] T in. The reult obtained in (9)-(4) are imilar if we olve the problem in thermoelatic half pace inclined load of arbitrary orientation. 5. Inverion of the tranform The tranformed tree, pore preure and temperature are function of the parameter of the Laplace and Fourier tranform and repectively and hence are of the form f (,, ). To obtain the olution of the problem in the phyical domain, we invert the Laplace and Fourier tranform by uing the method decribed by Kumar et al. [9]. 6 NUMERICAL RESULTS AND DISCUSSION With the view of illutrating the theoretical reult and for numerical dicuion we take a model for which the value of the variou phyical parameter i taken from Jabbari and dehbani [8]: E 6 Pa,., T 9 K, K Pa, K 5 Pa, K.5, 5 9 w / C, w / C, c.8j/g C, cw 4.J/g C,.6 g / m, w 6 g / m, n.4,, P P, t.5 The variation of normal tre, tangential tre, pore preure p and temperature change T for incompreible fluid aturated thermoporoelatic medium (FSPM) and empty porou thermoelatic medium (EPM) are hown in Fig. -5 concentrated force (CS), uniformly ditributed force (UDS), and moving force. In all thee figure, olid line ( ), olid line with central ymbol (- o - o -) and olid line with central ymbol (- -) correpond to the variation at,45,9 repectively. Similarly mall dahed line (------), mall dahed line

10 R. Kumar et al. 64 with central ymbol (- o - o -) and mall dahed line with central ymbol (- -) correpond to the variation at,45,9 repectively. The computation i carried out at for the range. All the reult are hown for one value of dimenionle width a and one value of dimenionle velocity V. 6. Time domain Fig. - how the variation concentrated force, uniformly ditributed force and moving force repectively. Fig. how the variation of normal tre component for both FSPM and EPM. The value of for FSPM increae in the range for, 45 and then ocillate with large magnitude, wherea for 9 firt increae and then decreae increae. The value of for EPM, firt increae in the range. and then ocillate with mall magnitude when,45 and for 9 it value converge near the boundary urface. Fig. how the variation of tangential tre component for both FSPM and EPM. The value of for FSPM ocillate for all value of with large magnitude a increae, while for EPM, the value of decreae in the range. and then ocillate with mall magnitude for all value of a increae. Fig.4 and Fig.8 how the variation of pore preure p for FSPM. The value of p for FSPM firt decreae in the range. when,45 and increae in the range. 9 and then decreae a increae wherea for etreme angle, with mall initial increae, the value of p decreae in the range.5 4 and then increae a increae. Behaviour of temperature T for FSPM and EPM i hown in Fig.5. The value of T for FSPM, when,45, increae in the range. and then ocillate wherea, for etreme angle, the value of T ocillate about the origin a increae. For EPM, the value of T decreae a increae when,45 wherea for decreae in the range. and then converge near the boundary urface a increae. Fig.6 depict the variation of normal tre component for both FSPM and EPM. The value of increae in the range for increae and then decreae a increae. In cae of EPM, the value of, when,45 and then ocillate with large magnitude wherea for 9, it for FSPM 9 it,45 firt increae in the range. and then ocillate about the origin and for 9 it value firt increae and then converge near the boundary urface. Fig.7 how the variation of tangential tre component for both FSPM and EPM. The value of for FSPM increae in the range for initial angle and increae in the range.5 for intermediate and etreme angle and then ocillate with large magnitude a increae. In cae of EPM, the value of decreae in the range and then ocillate with mall magnitude for all value of a increae. Fig.9 how the variation of temperature T for both FSPM and EPM. The value of T for FSPM increae in the range.5 for initial and intermediate angle and then ocillate wherea for etreme angle it value ocillate a increae. In cae of EPM, the value of T decreae eponentially for initial and intermediate angle and converge near the boundary urface for etreme angle a increae. Fig. depict the variation of tangential tre component for both FSPM and EPM. The value of for FSPM increae in the range for,45 and then ocillate with large magnitude wherea for 9 firt increae and then decreae a increae. The value of for EPM, when,45 firt increae in the range. and then ocillate about the origin and for 9 it value firt increae and then converge near the boundary urface a increae. The variation of tangential tre component for both FSPM and EPM i hown in Fig.. The value of for FSPM increae in the range for initial angle and increae in the range.5 for intermediate and etreme angle and then ocillate with large magnitude a increae. In cae of EPM, the value of

11 Pore Preure p 65 Deformation Due to Inclined Load in Thermoporoelatic Half Space decreae in the range and then ocillate with mall magnitude for intermediate and etreme angle wherea for initial angle it value converge near the boundary urface a increae. Fig. how the variation of pore preure p for FSPM. The value of p when,45 for FSPM firt decreae in the range 4 and increae in the range 4 9 and then decreae a increae wherea for etreme angle, the value of p decreae in the range 4 and then increae a increae. Behaviour of temperature T for FSPM and EPM i hown in Fig.. The value of T for FSPM increae in the range.5 for,45 and then ocillate wherea for etreme angle it value ocillate about the origin a increae. In cae of EPM, the value of T decreae gradually for,45 and converge near the boundary urface for 9 a increae..4. Normal Stre -. FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of normal tre w.r.t ditance concentrated force. 4 Tangential Stre - FSPM( ) FSPM( ) FSPM( ) -4-6 EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of tangential tre w.r.t ditance concentrated force..5 FSPM( ) FSPM( ) FSPM( ) Ditance Fig.4 Variation of pore preure p w.r.t ditance concentrated force.

12 Pore Preure p Temperature T R. Kumar et al FSPM( ) FSPM( ) - - FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.5 Variation of temperature T w.r.t ditance concentrated force..8.4 Normal Stre -.4 FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.6 Variation of normal tre w w.r.t ditance ditributed force. 4 Tangential tre -4 FSPM( ) FSPM( ) -8 FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.7 Variation of tangential tre w.r.t ditance ditributed force. FSPM( ) FSPM( ) FSPM( ) Ditance Fig.8 Variation of pore preure p w.r.t ditance ditributed force.

13 Pore Preure p Temperature T 67 Deformation Due to Inclined Load in Thermoporoelatic Half Space 4 FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.9 Variation of temperature T w.r.t ditance ditributed force... Normal Stre FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of normal tre w w.r.t ditance moving force. Tangential Stre - FSPM( ) FSPM( ) FSPM( ) - - EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of tangential tre w.r.t ditance moving force...8 FSPM( ) FSPM( ) FSPM( ) Ditance Fig. Variation of pore preure p w.r.t ditance moving force.

14 Temperature T R. Kumar et al. 68 FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of temperature T w.r.t ditance moving force. 6. Frequency domain Fig.4-5 how the variation concentrated force, uniformly ditributed force and moving force repectively. Behaviour of normal tre component for FSPM and EPM i hown in Fig. 4. The value of for FSPM, when,9 ocillate oppoitely in the range 5.5 and then how the ame behaviour wherea for intermediate angle it value ocillate about the origin a increae. In cae of EPM, the value of increae in the range. and then ocillate with large magnitude wherea for initial angle it value ocillate about the origin a increae. Fig.5 depict the variation of tangential tre component for both FSPM and EPM. The value of for intermediate and etreme angle, for FSPM, ocillate oppoitely wherea for initial angle it value ocillate with large magnitude a increae. In cae of EPM, with mall initial decreae, the value of converge near the boundary urface for wherea for 45,9, the value of decreae in the range 4. and then ocillate a increae. Fig.6 and Fig. how the variation of pore preure p for FSPM. The value of p for FSPM, when,9, ocillate oppoitely wherea for 45 with mall initial increae it value decreae in the range.5 5.5, increae in the range and then decreae a increae. Behaviour of temperature T for FSPM and EPM i hown in Fig.7. The value of T for FSPM, when,9, ocillate oppoitely wherea for 45 it value ocillate about the origin a increae. In cae of EPM, for all value of, the value of T increae in the range 4., decreae in the range and then increae a increae. Fig.8 depict the variation of normal tre component for both FSPM and EPM. The value of, for FSPM, when decreae in the range. and then ocillate wherea for 45,9 it value, for both FSPM and EPM, firt increae and then ocillate a increae and when in cae of EPM, it value firt increae and then ocillate a increae. Fig.9 how the variation of tangential tre component for both FSPM and EPM. The value of for intermediate and etreme angle, for FSPM, ocillate oppoitely wherea for initial angle it value ocillate with large magnitude a increae. In cae of EPM, with mall initial decreae, the value of converge near the boundary urface for wherea for 45,9, the value of decreae in the range 4. and then ocillate a increae. Behaviour of temperature T for FSPM and EPM i hown in Fig.. The value of T for FSPM, when,9, ocillate oppoitely wherea for 45 it value ocillate about the origin a increae. In cae of EPM, for

15 69 Deformation Due to Inclined Load in Thermoporoelatic Half Space all value of, the value of T increae in the range 4., decreae in the range and then increae a increae. Behaviour of normal tre component for FSPM and EPM i hown in Fig.. The value of, for FSPM, when decreae in the range. and then ocillate, wherea for 45,9, it value, for both FSPM and EPM, firt increae and then ocillate a increae and when in cae of EPM, it value firt increae and then ocillate a increae. Fig. depict the variation of tangential tre component for both FSPM and EPM. The value of for all value of ocillate with mall magnitude a increae. In cae of EPM, when 45,9, the value of decreae in the range 4. and then ocillate wherea for with mall initial decreae it value ocillate a increae. Fig.4 how the variation of pore preure p for FSPM. The value of p for FSPM ocillate with large magnitude for,9 and when 45, with mall initial increae, it value decreae in the range.6 5.5, increae in the range and then decreae a increae. Behaviour of temperature T for FSPM and EPM i hown in Fig. 5. The value of T for FSPM, when,9, how the oppoite behaviour wherea, for 45, with mall initial increae, it value decreae in the range 5. and then ocillate a increae. In cae of EPM, for all value of, the value of T increae in the range 4. and then ocillate a increae. 4 Normal tre - FSPM( ) FSPM( ) -4-6 FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.4 Variation of normal tre w.r.t ditance concentrated force (frequency domain). 4 FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Tangential tre Ditance Fig.5 Variation of tangential tre w.r.t ditance concentrated force (frequency domain).

16 Temperature T Pore Preure p R. Kumar et al FSPM( ) FSPM( ) FSPM( ) Ditance Fig.6 Variation of pore preure p w.r.t ditance concentrated force (frequency domain) FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.7 Variation of temperature T w.r.t ditance concentrated force(frequency domain) Normal Stre FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.8 Variation of normal tre w.r.t ditance ditributed force(frequency domain). 4 Tangential tre FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Diatance Fig.9 Variation of tangential tre w.r.t ditance ditributed force(frequency domain).

17 Temperature T Pore Preure p 64 Deformation Due to Inclined Load in Thermoporoelatic Half Space.8 FSPM( ) FSPM( ).4 FSPM( ) Ditance Fig. Variation of pore preure p w.r.t ditance ditributed force (frequency domain) FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of temperature T w.r.t ditance ditributed force(frequency domain) Normal tre FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig. Variation of normal tre w w.r.t ditance moving force(frequency domain). FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Tangential tre Ditance Fig. Variation of tangential tre w.r.t ditance moving force(frequency domain).

18 Temperature T Pore Preure p R. Kumar et al FSPM( ) FSPM( ) FSPM( ) Ditance Fig.4 Variation of pore preure p w.r.t ditance moving force (frequency domain) FSPM( ) FSPM( ) FSPM( ) EPM( ) EPM( ) EPM( ) Ditance Fig.5 Variation of temperature T w.r.t ditance moving force (frequency domain). 7 CONCLUSIONS Analyi of diplacement, tre, pore preure and temperature change concentrated force, uniformly ditributed force and a moving force inclined load in time and frequency domain i a ignificant problem of continuum mechanic. Integral tranform technique ha been ued which i applicable to wide range of problem in thermoporoelaticity. The preent invetigation i concerned with the deformation of thermoporoelatic half pace with incompreible fluid a a reult of inclined load of arbitrary orientation. The component of diplacement, tre, pore preure and temperature change are obtained in thermoporoelatic medium the variou ource by uing the Laplace and Fourier tranform. Appreciable poroity effect and effect of change in angle of inclination of inclined load are oberved on the component of normal tre, tangential tre, pore preure and temperature change. It i oberved that for time domain the value of,, pt, for FSPM ocillate with large magnitude a compared to the value for EPM. Alo on the point of application of concentrated force, ditributed force and moving force, the poroity effect increae the value of and p while revere behavior i oberved in the value of and T. In frequency domain, it i oberved that the trend of variation of tree, pore preure and temperature change on the application of concentrated force, ditributed force and moving force are imilar in nature with ignificant difference in their magnitude. All the field quantitie are oberved to be very enitive toward the angle of inclination and poroity parameter. Angle of inclination and poroity parameter have ocillatory effect on the numerical value of the phyical quantitie. The reult obtained a a conequence of thi reearch work hould be beneficial for reearcher working on thermoporoelatic olid. The preent tudy preent a more realitic model for further invetigation.

19 64 Deformation Due to Inclined Load in Thermoporoelatic Half Space ACKNOWLEDGEMENTS The author are thankful to the referee for hi valuable comment to improve the manucript. REFERENCES [] Fillunger P., 9, Der auftrieb in talperren, Oterr Wochenchrift Offentl Baudient 9: [] Terzaghi K.V., 9, Die berechnug der durchlaigkeitziffer de tone au dem verlauf der hydromechanichen pannungercheinungen, Sitzungber Akad Wi Wien, Math Naturwi KI, Abt, IIa :5-8. [] Terzaghi K.V., 95, Erdbaumechanik auf Bodenphyikalicher Grundlage, Leipzig-wien, Franz Deuticke. [4] Terzaghi K.V., 9, Auftrieb und kapillardruck an betonierten talperren, Die Waerwirtchaft 6: [5] Biot M.A., 94, General theory of three dimenional conolidation, Journal of Applied Phyic (): [6] Biot M.A., 956, Theory of propagation of elatic wave in fluid aturated porou olid I-low frequency range, Journal of the Acoutical Society of America 8: [7] Biot M.A., 956, Theory of propagation of elatic wave in fluid aturated porou olid II-higher frequency range, Journal of the Acoutical Society of America 8:79-9. [8] Rice J.R.,Cleary M.P.,976, Some baic tre diffuion olution for fluid aturated elatic porou media with compreible contituent, Review of Geophyic and Space Phyic 4:7-4. [9] Schiffman R. L., 97, A Thermoelatic Theory of Conolidation, in Environmental and Geophyical Heat Tranfer, American Society of Mechanical Engineer, New York. [] Bowen R. M., 98, Compreible porou media model by ue of the theory of miture, International Journal of Engineering Science : [] Noorihad J., Tang C.F., Witherpoon P. A., 984, Coupled thermohydraulic-mechanical phenomena in aturated fractured porou rock: Numerical approach, Journal of Geophyical Reearch 89:65-7. [] McTigue D.F., 986, Thermal repone of fluid-aturated porou rock, Journal of Geophyical Reearch 9(B9): [] Kurahige M., 989, A thermoelatic theory of fluid-filled porou material, International Journal of Solid and Structure 5:9-5. [4] Abouleiman Y., Ekbote S., 5, Solution for the inclined borehole in a porothermoelatic tranverely iotropic medium, ASME Journal Applied Mechanic 7: -4. [5] Bai B., 6, Fluctuation repone of porou media ubjected to cyclic thermal loading, Computer and geotechnic :96-4. [6] Bai B., Li T., 9, Solution for cylindrical cavity in aturated thermoporoelatic medium, Acta Mechanica Solida Sinica ():85-9. [7] Jabbari M., Dehbani H.,, An eact olution for claic coupled thermoelaticity in aiymmetric cylinder, Journal of Solid Mechanic ():9-4. [8] Ganbin L., Kanghe X., Rongyue Z.,, Thermo-elatodynamic repone of a pherical cavity in a aturated poroelatic medium, Applied Mathematical Modeling 4:-. [9] Gatmiri B., Maghoul P., Duhamel D.,, Two-dimenional tranient thermo-hydro-mechanical fundamental olution of multiphae porou media in frequency and time domain, International Journal of Solid and Structure 47: [] Li X., Chen W., Wang H.,, General tudy tate olution for tranverely iotropic thermoporoelatic media in three dimenion and it application, European Journal of Mechanic - A/Solid 9(): 7-6. [] Jabbari M., Dehbani H.,, An eact olution for quai-tatic poro- thermoelaticity in pherical coordinate, Iranian Journal of Mechanical Engineering (): [] Liu G., Ding S.,YE R., Liu X.,, Relaation effect of a aturated porou media uing the two dimenional generalized thermoelatic theory, Tranport in Porou Media 86:8-. [] Beloterkovet b. A., Prevot J. H.,, Thermoporoelatic repone of fluid-aturated porou phere: An analytical olution, International Journal of Engineering Science 49(): [4] Bai B.,, Thermal repone of aturated porou pherical body containing a cavity under everal boundary condition, Journal of Thermal Stree 6(): 7-. [5] Apotolaki G., Dargu G.F.,, Mied variation principal for dynamic repone of thermoelatic and poroelatic continua, International Journal of Solid and Structure 5(5): [6] Hou P.F., Zhao M., Jiann-Wen J.U.,, The three dimenional green function for tranverely iotropic thermoporoelatic biomaterial, Journal of Applied Geophyic 95: [7] Jabbari M., Hahemitaheri M., Mojahedin A., Elami M.R., 4, Thermal buckling analyi of functionally graded thin circular plate made of aturated porou material, Journal of Thermal Stree 7:-. [8] Liu M., Chain C., 5, A micromechanical analyi of the fracture propertie of aturated porou media, International Journal of Solid and Structure 6:-8.

20 R. Kumar et al. 644 [9] He S.M., Liu W., Wang J., 5, Dynamic imulation of landlide baed on thermoporoelatic approach, Computer and Geocience 75: 4-. [] Nguyen H.T., Wong H., Fabbri A., Georgin J.F., Prudhomme E., 5, Analytical tudy of freezing behaviour of a cavity in thermoporoelatic medium, Computer and Geotechnic 67: -45. [] Wu D., Yu L., Wang Y., Zhao B., Gao Y., 5, A refined theory of aiymmetric thermoporoelatic circular cylinder, European Journal of Mechanic - A/Solid 5: [] Kumar R., Ailawalia P., 5, Moving inclined load at boundary urface, Applied Mathematic and Mechanic 6: [] Kumar R., Ailawalia P., 5, Interaction inclined load at micropolar elatic half-pace with void, International Journal of Applied Mechanic and Engineering :9-. [4] Kumar R., Rani L., 5, Deformation due inclined load in thermoelatic half-pace with void, Archive of Mechanic 57:7-4. [5] Sharma K.,, Analyi of deformation inclined load in generalized thermodiffuive elatic medium, International Journal of Engineering, Science and Technology (): 7-9. [6] Otemin A.A., Utkin P.B.,, Stre-train tate of a inclined elliptical defect in a plate under biaial loading, Journal of Applied Mechanic and Technical Phyic 5(): [7] Bogomolov A. N.,, Uhakov A. N., Stre-train tate of an elatic half plane under a ytem of inclined piecewie-linear load, Soil Mechanic and Foundation Engineering 5(): [8] Jabbari M., Dehbani H., 9, An eact olution for claic coupled thermoporoelaticity in cylindrical coordinate, Journal of Solid Mechanic (4): [9] Kumar R., Ailawalia P., 5, Elatodynamic of inclined load in micropolar cubic crytal, Mechanic and Mechanical Engineering 9():