Effect of Anisotropic Thermal Conductivity on Deformation of a Thermoelastic Half-Space Subjected to Surface Loads

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1 18 IJSRST Volume 4 Iue Print ISSN: Online ISSN: 395-6X Themed Section: Science and Technology Effect of Aniotropic Thermal Conductivity on Deformation of a Thermoelatic Half-Space Subjected to Surface Load Kavita Rani* 1, Anil K. Vahihth, Kuldip Singh 3 *1 Department of Mathematic, CMG Govt. College for Women, Bhodia Khera, Fatehabad, Haryana, India Department of Mathematic, Kuruhetra Univerity, Kuruhetra, Haryana, India 3 Department of Mathematic, Guru Jambhehwar Univerity of Science and Technology, Hiar, Haryana, India ABSTRACT The tudy i motivated by a deire to develop an analytical technique to tudy quai-tatic plane train deformation of a thermoelatic medium due to urface load by taing into account the aniotropy of thermal conductivity. By applying the Laplace and Fourier tranform to the tate variable involved in the baic governing equation, the olution for the tree, diplacement, temperature difference and heat flux are obtained. Conidering the boundary condition, the problem i olved in the tranformed domain. The actual olution of the problem in the phyical domain are acquired by inverting the Laplace-Fourier tranform. Finally, ome numerical example are given to demontrate the influence of the urface load and the aniotropy of thermal conductivity on the thermo-elatic repone. Keyword: Aniotropy, Thermal conductivity, Surface load, Thermoelatic. I. INTRODUCTION The thermo-elatic problem of the time-dependent behaviour of material with urface load are ignificant and have received extenive attention in the field of geology, environmental engineering, oil cience and civil engineering. The problem for thermal repone and deformation caued by the thermal and mechanical load have been tudied by numerou invetigator [1-1]. It i oberved from practice that the typical depoit proce of natural geomaterial may lead to clear difference in thermal conductivity between different direction, epecially for the horiontal and vertical thermal conductivity. Therefore, ome reearcher focued their attention on the tudy of the thermoelatic repone of the material with aniotropic thermal conductivity. Pan [13] tudied the tranient thermoelatic deformation in a tranverely iotropic and layered half-pace by urface load and internal ource. Wei and Yu-Qiu [14] dicued plane problem of orthotropic quai-tatic thermoelaticity. Sharma and Kumar [15] invetigated the plane train problem in generalied theory of thermo-elaticity in a homogeneou tranverely iotropic medium. Kögl and Gaul [16] preented a boundary element method for dynamic aniotropic coupled thermoelaticity and the reult for quai tatic and tationary thermoelaticity were alo deduced. Youef and El- Bary [17] dicued the thermal hoc problem to the general cae with variable thermal conductivity. Aouadi [18] tudied the problem of variable electrical and thermal conductivity in the theory of generalied thermoelatic diffuion and dicued thermoelatic diffuion interaction in an infinitely long olid cylinder ubjected to a thermal hoc on it urface, which i in contact with a permeating ubtance. IJSRST18419 Received : 5 Feb 18 Accepted : 15 Feb 18 January-February-18 [ (4) : ] 98

2 Youef and Abba [19] ued a finite element method to analye tranient phenomena in an infinitely long annular cylinder in the context of the theory of generalied thermoelaticity with one relaxation time conidering the thermal conductivity to be variable. Ai et al. [] derived an analytical olution for the axiymmetric thermo-elatic problem of multilayered material with aniotropic thermal diffuivity due to a buried heat ource. Ai and Wang [1] dicued time dependent analyi of thermo-mechanical behaviour of a layered thermoelatic half-pace with aniotropic thermal diffuivity. Ai and Wu [] tudied a thermal conolidation problem of a multilayered porou thermo-elatic medium with aniotropic thermal diffuivity and permeability due to a heat. Eat and El-Bary [3] ued the theory of generalied magnetothermoelaticity with variable thermal conductivity and fractional order of heat tranfer and olved the problem of an infinite long hollow cylinder in the preence of an axial uniform magnetic field. It i noted that although many problem involving thermal conolidation and thermomechanical repone due to heat ource or thermal loading have been invetigated, tudie dealing with the quai tatic plane train deformation of a thermoelatic medium due to urface load by taing into account the aniotropy of thermal conductivity are rather limited. The objective of thi paper i to introduce the analytical method to tudy the behaviour of thermoelatic material with aniotropic thermal diffuivity due to urface load. The baic equation are governed by coupled theory thermoelaticity. The olution in the Laplace Fourier tranformed domain are obtained by olving equation atifying the boundary condition. The actual olution in the phyical domain can be acquired by the inverion of the Laplace-Fourier tranform. Finally, numerical example are preented to invetigate the effect of different material parameter on temperature difference, heat flux and variation of diplacement. The importance of the problem conidered lie in the fact that the crut of the earth i aniotropic in different direction and therefore it i ueful to tudy the effect of aniotropy on the tatic field due to urface load. II. FORMULATION OF THE PROBLEM In thi tudy, we conider the temperature ditribution in a deformable body and therefore the correponding model evoe the olution of a coupled thermomechanical problem. A homogeneou, thermoelatic half-pace with aniotropic thermal conductivity i conidered. The origin of a Carteian coordinate ytem ( x, y, ) i placed at the boundary of the half-pace and the -axi i drawn vertically into the medium. Conider a trip L y L of infinite length on the urface of thermoelatic half-pace. Let a normal load of force denity acting in poitive direction of be uniformly ditributed over the trip (Fig1 a). For hear trip loading (Fig 1b), the force denity i taen a. Since the ource i long in one direction in comparion to other, two dimenional approximation i jutified. So, a plane train problem i tudied in -plane and the diplacement vector can be written a u ( u, u ). (1) y In cae the material i homogeneou with aniotropic thermal conductivity, the heat flow obey Fourier law a q, () where q ( qy, q) i heat flux vector; y, are thermal conductivitie in y and direction repectively; y y, and i the temperature deviation from the reference temperature T. (.) repreent partial differentiation with repect to y. y 983

3 By conidering the coupling of elatic and thermal procee, the heat conduction equation with aniotropic thermal conductivity can be written a C Te, (3) where, e Ce i the pecific heat, i the denity, t(3 ), and are Lame contant, t i the coefficient of linear thermal expanion and e u u. y, y, The Duhamel-Neumann relation etablih the relationhip between the tate of tre, train and temperature and can be written a C. I, (4) where 1 yy and yy T u u are tre and train tenor repectively, y, (1 ) C (1 ), ratio., = (1 ) t, 1 I= 1 and i Poion The tatic equilibrium, with body force being neglected, i given by., (5) and the compatibility equation i. (6) yy,, yy, Figure 1. Geometry of the model (a) Normal trip load (b) Shear trip load III. DERIVATION OF ANALYTICAL SOLUTION The Airy tre function U i defined a U, U, U. (7) get yy yy Uing Eq. (4)-(5) and (7) in Eq. (3) and (6), we ( U ), (8) T T Ce ( U), (9) (1 ) where,.. (1 ) Taing Laplace tranform of Eq. (8) and (9), we obtain ( ) c, (1) 4 ( ), c U (11) where, e i T ci,, C C c, c c y y and i variable of Laplace tranform. Solving Eq. (1) and (11), we have in y R d coy, in y U F d coy, (1) e 984

4 m where R ( Ae 1 Ae ), F ( B e ( B B ) e ), m 1 3 and AB, may be function of. i i cy m c c Maing ue of Eq. (1) in Eq. (8)-(9), we get a A1 B1, a (1 ) B3 1 A, (13) c ( ) (1 ) where c ( m ), a Eq. (7) give in y N d coy, coy S d in y, m where N ( B1e ( B B3) e ), S mb e B B B e. m ( 1 (( 3) 3 ) ). 1 (14) The diplacement component can now be written a coy uy V d in y, where in y u W d coy, (15) m V B1e ( B B3( ) A ) e, m W ( mb1 e ( B B3(1 ) A ) e ) IV. BOUNDARY CONDITIONS For precribed urface load, auming the urface to be non adiabatic iothermal, and nown at. So, are ( ), and at. (16) in y Let ( ) N d coy, ( ) coy S d in y. (17) Uing Eq. (16)-(17) in Eq. (14) and olving, we get A ( )( m )( S N) A, 1 ( )( SN) B1 ( m) N,, B B1 ( m )( S N) B3 ( ) a (1 ), (18) where ( )( m) (1 )( m). a Uing thee value of A i and Bi ' in equation (1), (14)-(15), the expreion for the temperature difference, tree, diplacement and heat flux are obtained. Value of S and N for different type of load are given in Table 1. Table 1. Value of S, N for different type of load Normal trip loading on the urface. Normal line loading hear trip loading S in L L N in L L Upper olution/ Lower olution Lower olution Lower olution hear line loading Upper olution Upper V. SOLUTIONS FOR SURFACE LOADS A. Normal Strip Loading olution The expreion of diplacement, tree, temperature difference and heat flux olution for normal trip loading are given by ( ) m uy [ e e { ( ) ( m ) 1 a (1 )} ( m) e { ( )( m ) in L in y a(1 )(1 )( m )} e ] d, L m( ) m u [ e e { ( ) ( m ) (1 )} ( m) e a (19) in L coy a(1 )(1 )( m ) e ] d, L () 985

5 m( ) m [ e e { ( ) ( m ) in L in y a(1 )} ( m) e ] d, L ( ) m [ e me { ( ) ( m ) in L coy a(1 )}(1 )( m) e ] d, L m (1) () ( ) in L coy ( e e )( m) d, (3) L ( ) in in ( )( ), (4) y m L y qy e e m d L ( ) m in L co y q me e m d L B. Normal Line Loading ( )( ). (5) Taing the limit a L with fixed, the diplacement, tree, temperature difference and heat flux for normal line loading can be obtained. The expreion for the olution can be obtained from equation (19)-(5) by removing the factor in L L that. Taing the limit t, uing the reult lim f ( t) lim f ( ), and evaluating the t integral given in Eq. (19)-(5) analytically, we have q and the diplacement and tree obtained for thi limiting cae match with the correponding olution of elatic medium [4]. C. Shear Strip Loading The olution for hear trip loading are obtained a ( ) m uy [ e e { ( ) ( m ) (1 )} ( m) e a in L coy a(1 )(1 )( m ) e ] d, L (6) ( ) m u [ me e { ( ) ( m ) 1 a (1 )} ( m) e { ( ) in L in y a(1 )(1 )( m )} e ] d, L (7) ( ) m [ me e { ( ) ( m ) in L coy a(1 )}( 1)( m) e ] d, L (8) ( ) m [ e e { ( ) ( m ) in L in y a(1 )} ( m) e ] d, L m (9) ( ) in L in y ( e e )( m) d, (3) L ( ) in co ( )( ), (31) y m L y qy e e m d L ( ) m in L in y q me e m d L D. Shear line loading ( )( ). (3) A dicued earlier, the olution for hear line loading can be obtained from equation (6)-(3 by in L removing the factor. For the limitt, L thee reult match with the correponding reult of elatic medium [4]. VI. NUMERICAL RESULTS AND DISCUSSION It i een that all the olution are in Laplace- Fourier tranformed domain. For computation of the invere Laplace tranform, Schapery method [5] i ued and the emi-infinite integral due to Fourier tranform i computed numerically uing Gau quadrature formula. A an example, we have computed the temperature difference, heat flux and component of diplacement due to normal trip loading on the urface of a thermoelatic half-pace with 986

6 aniotropic thermal conductivity. The parameter for thermoelatic medium are taen a (1) g m, (1) g m, (1) K, 33 g m, t 1 T 6 K, C 155 m K. e (33) To mae the quantitie dimenionle, the following are defined y ct ui Y, Z, T, U,, i L L L L (34) ij Lq c y ij, Q, r. c Figure how the effect of aniotropy thermal conductivity on the variation of temperature difference with time at Y= due to normal trip loading. We notice that, near the ource, aniotropy in thermal conductivity ha only a mall effect (Fig. a). A the depth increae, thi effect become more ignificant (Fig.. b-d). At a given time, the temperature difference decreae a the thermal conductivity aniotropy parameter r increae. When the horiontal thermal conductivity i greater than the vertical thermal conductivity i.e. r 1, at a given time; the temperature difference for an aniotropic iothermal half-pace i le than the temperature difference for the correponding iotropic iothermal half-pace. A the depth increae, effect of r occur delayed. In Fig., curve for different value of the aniotropy parameter r coincide for very mall and for very large time. Thi how that the thermal conductivity aniotropy ha no effect in the adiabatic and iothermal condition. Figure. Effect of the value of aniotropic conductivity parameter r on temperature difference with time T for Y= and at depth (a) Z=.1; (b) Z=1; (c) Z=; (d) Z=1. Figure 3 depict the variation of the temperature difference with time at different depth for a mall and large value of r. A the depth increae, temperature difference decreae along time and approache to ero. For maller Z, the variation i fater. Fig. 3 alo indicate that the variation of temperature difference can reach it extremum value on the urface and become maller and to be table with the increae of time. Figure 4 reveal the depth profile of the temperature difference at four time for mall and large value of r. The temperature difference i ero at the urface, attain a maximum value at a depth depending upon r and T and then tend to ero a Z increae. Maximum value of i hifted with time along Z. 987

7 The heat flux pattern near the urface (Z=.1) i ignificantly different from the pattern at depth. Figure 7 repreent time hitory of the heat flux in vertical direction at different depth for four value of r.1,1,,1. Figure 3. Variation of temperature difference with time T at four depth for (a) r=.1, (b) r=1. Figure 4. Depth profile of temperate difference at T=.1,.1, 1, 1 for (a) r=.1, (b) r=1. Figure 5(a) how the depth profile of the temperature difference for T =.1 for four value of r. Initially, a very large temperature difference develop near the urface reulting in a teep gradient of the temperature difference. No ignificant effect for different value of r i noticed. Fig 5(b) (d) depict depth profile of for T =.1, 1, 1. A T increae, the maximum value of the temperature difference decreae and the location of the maximum value travel down and hift right. The effect of increae of r i prominently increae with increae in T. Fig. 6 diplay time hitory of the heat flux in the vertical direction at four depth (Z=.1, 1,, 1) for r =.1, 1. Figure 5. Effect of the value of aniotropic conductivity parameter r on temperature difference with depth for Y= and time (a) T=.1; (b) T=.1; (c) T=1; (d) T=1. 988

8 Figure 6. Variation of heat flux in vertical direction at four depth for (a) r=.1, (b) r=1. Figure 7. Effect of the value of aniotropic conductivity parameter r on temperature difference with time T at Y= and depth (a) Z=.1; (b) Z=1; (c) Z=; (d) Z=1. 989

9 Figure 8. Effect of the value of aniotropic conductivity parameter r on horiontal diplacement with time T for Y= and depth (a) Z=.1; (b) Z=1; (c) Z=; (d) Z=1. Figure 9. Depth profile of horiontal diplacement at r =.1, r =1 for (a) T=.1, (b) T=1. Figure 1. Depth profile of vertical diplacement at r=.1, r=1 for (a) T=.1, (b) T=1. 99

10 Figure 11. Effect of the value of aniotropic conductivity parameter r on vertical diplacement with time T for Y= and depth (a) Z=.1; (b) Z=1; (c) Z=; (d) Z=1. Time hitory of horiontal diplacement at Y=1 for different depth r.1,1,,1 i plotted in Figure 8. A the depth increae, horiontal diplacement decreae along time and large value of r mae it maller. Depth profile of horiontal diplacement at Y=1 i plotted in Figure 9 for two value of r. It increae quicly near the urface and then become table and approache to ero. Variation of vertical diplacement at Y=1 along Z i preented in Figure 1. It decreae with depth for all time. No ignificant effect of r eem on it. Time hitory of vertical diplacement and effect of value of aniotropic thermal conductivity on it i hown in Fig 11 for four depth Z=.1, 1,, 1 at Y=1. The vertical diplacement increae with increae in time and increae in horiontal thermal conductivity. The curve for different value of r merge after a time. 7. VII. CONCLUSION In thi paper, the quai-tatic plane train deformation of a thermoelatic half pace with aniotropic thermal conductivity due to urface load i tudied. The problem of normal trip and line loading and hear trip and line loading are dicued in detail. Explicit analytical expreion in the Fourier-Laplace domain for the temperature function, heat flux, tree and diplacement have been obtained. From numerical computation, it i found that the aniotropy of thermal conductivity ha a ignificant effect on deformation, temperature ditribution a well a on heat tranfer of the medium. However, the aniotropy ha no effect for very mall and large time (adiabatic and iothermal condition). 991

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