Biographies
5. CONCLUSIONS
For Peer Review Only
(a) (b)
Figure 13. Influence of the degree of anisotropy tr tz on long-term temperature changes of the strata with (a) sr sz 1.0 and (b) sr sz 10.0.
The long-term normalized temperature increment of the strata were calculated from equation (23c) for values of various anisotropic ratio E E , r z sr sz , tr tz, and the results are shown in Figures 11 to 13. Figures 11 and 12 display the anisotropic ratio E E and r z sr sz, and they have no effect on the long-term temperature increment of the strata due to a point heat source. However, Figure 13 illustrates that the ratio of anisotropic thermal conductivity tr tz has the most significant effect on temperature increment of the strata. In all cases, the temperature increment of the strata is larger when the location is closer to the point heat source.
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ACKNOWLEDGEMENTS
The authors are grateful for the funds provided by National Science Council of Taiwan, R.O.C., Grant No. NSC100-2221-E-216-025, and Chung Hua University, Grant No. CHU-99-A-03.
REFERENCES
1. Gonzalez-Romero EM. Impact of partitioning and transmutation on the high level waste management. Nuclear Engineering and Design 2011; 241:3436-3444.
2. Tong F, Jing L, Zimmerman RB. A fully coupled thermo-hydro-mechanical model for simulating multiphase flow, deformation and heat transfer in buffer material and rock masses. International Journal of Rock Mechanics and Mining Sciences 2010; 47:205-217.
3. Booker JR, Savvidou C. Consolidation around a spherical heat source. International Journal of Solids and Structures 1984; 20(11-12):1079-1090.
4. Booker JR, Savvidou C. Consolidation around a point heat source. International Journal for Numerical and Analytical Methods in Geomechanics 1985; 9(2):173-184.
5. Savvidou C, Booker JR. Consolidation around a heat source buried deep in a porous thermoelastic medium with anisotropic flow properties. International Journal for Numerical and Analytical Methods in Geomechanics 1989; 13(1):75-90.
6. Lu J CC, Lin FT. The transient ground surface displacements due to a point sink/heat source in an elastic half-space. Geotechnical Special Publication No. 148 (ASCE) 2006; 210-218.
7. Lin FT, Lu J CC. Golden ratio in the point heat source induced horizontal and vertical displacements of an isotropic elastic half space. Geotechnical Special Publication No. 204 (ASCE) 2010; 87-94.
8. Wang X, Sudak LJ. 3D Green’s functions for a steady point heat source interacting with a homogeneous imperfect interface. Journal of Mechanics of Materials and Structures 2006; 1(7):1269-1280.
9. Hudson JA, Stephansson O, Andersson J. Guidance on numerical modelling of thermo-hydro-mechanical coupled processes for performance assessment of radioactive waste repositories. International Journal of Rock Mechanics and Mining Sciences 2005; 42(5-6):850-870.
10. Chao CK, Chen FM, Shen MH. Green’s functions for a point heat source in circularly cylindrical layered media.
Journal of Thermal Stresses 2006; 29(9):809-847.
11. Amadei B, Swolfs HS, Savage WZ. Gravity-induced stresses in stratified rock masses. Rock Mechanics and Rock Engineering 1988; 21(1):1-20.
12. Tarn JQ, Lu CC. Analysis of subsidence due to a point sink in an anisotropic porous elastic half space.
International Journal for Numerical and Analytical Methods in Geomechanics 1991; 15(8):573-592.
13. Sheorey PR. A theory for in situ stresses in isotropic and transversely isotropic rock. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 1994; 31(1):23-34.
14. Lee SL, Yang JH. Modeling of effective thermal conductivity for a nonhomogeneous anisotropic porous medium. International Journal of Heat and Mass Transfer 1998; 41(6-7):931-937.
15. Wang CD, Tzeng CS. Displacements and stresses due to nonuniform circular loadings in an inhomogeneous cross-anisotropic material. Mechanics Research Communications 2009; 36:921-932.
16. Love AEH. A Treatise on the Mathematical Theory of Elasticity. Dover Publications: New York, 1944; 643.
17. Sneddon IN. Fourier Transforms. McGraw-Hill: New York, 1951; 48-70.
18. Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG. Tables of Integral Transforms, vol. 1 & 2. McGraw-Hill:
New York, 1954.
19. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products (7th edn). Academic Press: New York, 2007;
1171.
20. Booker JR, Carter JP. Analysis of a point sink embedded in a porous elastic half space. International Journal for Numerical and Analytical Methods in Geomechanics 1986; 10(2):137-150.
21. Poulos HG, Davis EH. Elastic Solutions for Soil and Rock Mechanics. John Wiley & Sons: New York, 1974;
183-192.
22. Lee KM, Rowe RK. Deformations caused by surface loading and tunnelling: The role of elastic anisotropy.
Geotechnique 1989; 39(1):125-140.
23. Wang CD, Pan E, Tzeng CS, Han F, Liao JJ. Displacements and stresses due to a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space. International Journal of Geomechanics (ASCE) 2006;
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6(1):1-10.
NOMENCLATURE
1, 2,3
a ii Constants defined in equations (21a) to (21c) (ºC-1) A , C , F , L , N Material constants defined by Love (Pa)
1, 2,3
b ii Constants defined in equations (21d) to (21f) (ºC-1) E , r E z Young’s modulus in horizontal/vertical direction (Pa)
,
f i r zi Body forces of the strata (N/m3)
G Shear modulus of the isotropic strata (Pa)
G rz Modulus of shear deformation in vertical plane (Pa) h Heat flux vector (J/sm2)
ir, iz Unit vector parallel to the radial/vertical direction(Dimensionless)
J x First kind of the Bessel function of order (Dimensionless) q h Internal (or external) heat sources (J/sm3)
Q Strength of the point heat source (J/s)
r, , z
Cylindrical coordinates system (m, radian, m) R Parameter, R r2z2 (m) 1, 2,3
R ii Parameter, Ri r2i2 2z (m) R* Parameter, R* r2z2 (m) z
* 1, 2,3
R ii Parameter, Ri*Rii z (m)
,
u i r zi Displacement components of the strata (m)
U , r U z Hankel transforms of u and r u , equations (13a) and (13b) (mz 3) Ur, Uz Fourier transforms of U and r U , equations (16a) and (16b) (mz 4) Greek letters
s Linear thermal expansion coefficient of the isotropic strata (ºC-1)
sr, sz Linear thermal expansion coefficient of the cross-anisotropic strata in horizontal/vertical
direction (ºC-1)
r, z Thermal expansion factors of the cross-anisotropic strata (Pa/ºC)
*
r, *z Thermal expansion factors of the isotropic strata (Pa/ºC)
x
Dirac delta function (m-1)
Parameter, 1 1 2 (Dimensionless)
Temperature change of the strata (ºC)
Hankel transform of , equation (13c) (ºCm2)
Fourier transform of , equation (16c) (ºCm3)
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Lame constant of the isotropic strata (Pa)
t Thermal conductivity of the isotropic thermoelastic medium (J/smC)
tr, tz Thermal conductivity of the cross-anisotropic thermoelastic medium in the horizontal/
vertical direction (J/smC)
Characteristic root, tr tz (Dimensionless)
1, 2 Characteristic roots of characteristic equation (22) (Dimensionless)
3 Characteristic root, 3 tr tz (Dimensionless)
Poisson’s ratio of the isotropic strata (Dimensionless)
rz Poisson’s ratio for strain in the vertical direction due to a horizontal direct stress (Dimensionless)
r
Poisson’s ratio for strain in the horizontal direction due to a horizontal direct stress (Dimensionless)
zr Poisson’s ratio for strain in the horizontal direction due to a vertical direct stress (Dimensionless)
Hankel transform parameter (m-1)
, , ,
ij i j r z
Thermal stress components of the strata (Pa)
1, 2,3, 4
i i
Functions defined in equations (26a) to (26d) (Dimensionless)
5, ,13
i i
Functions defined in equations (26e) to (26m) (m-1)
Fourier transform parameter (m-1) Page 23 of 23
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研究成果投稿後已發表 於 EI 等級之國際會議 論文(第一篇)
Lu, John C.-C. (Session Chair) and Feng-Tsai Lin, 2012/6/25~27, “Modelling of a Buried Deep Horizontal Line Heat Source in a Cross-Anisotropic Thermoelastic Medium,”
Proceedings of the 20 th IASTED International Conference on Applied Simulation and Modelling, CD ISBN: 978-0-88986-925-7, Napoli, Italy, pp.
150-157. (This work is supported by the National
Science Council through grants NSC100-2221-E-216-025.) (EI)
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1. Accession number: 20124115537529
Title: Modelling of a buried deep horizontal line heat source in a cross-anisotropic thermoelastic medium
Authors:
Author affiliation:
Corresponding author: Lu, J.C.-C. (cclu@chu.edu.tw)
Source title: Proceedings of the IASTED International Conference on Applied Simulation and Modelling, ASM 2012
Abbreviated source title: Proc. IASTED Int. Conf. Appl. Simul. Model., ASM
Monograph title: Proceedings of the IASTED International Conference on Applied Simulation and Modelling, ASM 2012
Issue date: 2012 Publication year: 2012 Pages: 150-157 Language: English
ISBN-13: 9780889869257 Document type: Conference article (CA)
Conference name: 20th IASTED International Conference on Applied Simulation and Modelling, ASM 2012
Conference date: June 25, 2012 - June 27, 2012 Conference location: Napoli, Italy
Conference code: 92890
Publisher: Acta Press, Building B6, Suite 101, 2509 Dieppe Avenue S.W., Calgary, AB, T3E 7J9, Canada
Abstract: In this paper, the deep buried line heat source of constant strength affects the thermally mechanical responses of the stratum are presented. To simulate the stratified earth medium, the soil mass is modeled as cross-anisotropic with different properties in the horizontal and vertical directions. On the basis of fundamental solutions caused by a deep point heat source, the analytic solutions of ground deformation, thermal stresses and temperature changes of the thermoelastic medium due to deep line heat source are presented by using appropriate line integral techniques. The anisotropic soil shows significant effect on long-term thermally elastic responses compared with the results from isotropic soil. Besides, the derived solutions illustrated that shear modulus does not have influence on long-term displacements and temperature increment of the strata for the case of isotropic properties.
Number of references: 19
Main heading: Anisotropic media
Controlled terms: Anisotropy - Modal analysis - Soils - Thermoelasticity
Uncontrolled terms: Analytic solution- Closed form solutions- Elastic response- Fundamental solutions- Ground deformations- Isotropic property- Line heat sources- Line integrals - Mechanical response- Point heat source- Soil mass - Temperature changes- Temperature increment- Thermoelastics - Vertical direction Classification code: 483.1 Soils and Soil Mechanics -921Mathematics -931.2 Physical Properties of
Gases, Liquids and Solids -951 Materials Science DOI: 10.2316/P.2012.776-040
Database: Compendex
Compilation and indexing terms, © 2012 Elsevier Inc.
Lu, John C.-C.1 ; Lin, Feng-Tsai2
1Department of Civil Engineering, Chung Hua University, No. 707, WuFu Rd., Hsinchu 30012, Taiwan
2Department of Naval Architecture, National Kaohsiung Marine University, No. 142, Haijhuan Rd., Kaohsiung 81157, Taiwan
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MODELLING OF A BURIED DEEP HORIZONTAL LINE HEAT SOURCE IN A CROSS-ANISOTROPIC THERMOELASTIC MEDIUM
John C.-C. Lu1and Feng-Tsai Lin2
1Department of Civil Engineering, Chung Hua University No. 707, Sec. 2, WuFu Rd., Hsinchu 30012, Taiwan, R.O.C.
cclu@chu.edu.tw
2Department of Naval Architecture, National Kaohsiung Marine University No. 142, Haijhuan Rd., Kaohsiung 81157, Taiwan, R.O.C.
ftlin@mail.nkmu.edu.tw ABSTRACT
In this paper, the deep buried line heat source of constant strength affects the thermally mechanical responses of the stratum are presented. To simulate the stratified earth medium, the soil mass is modeled as cross-anisotropic with different properties in the horizontal and vertical directions. On the basis of fundamental solutions caused by a deep point heat source, the analytic solutions of ground deformation, thermal stresses and temperature changes of the thermoelastic medium due to deep line heat source are presented by using appropriate line integral techniques. The anisotropic soil shows significant effect on long-term thermally elastic responses compared with the results from isotropic soil. Besides, the derived solutions illustrated that shear modulus does not have influence on long-term displacements and temperature increment of the strata for the case of isotropic properties.
KEY WORDS
Point Heat Source, Line Heat Source, Fundamental Solution, Closed-form Solution.
1. Introduction
The deep buried line heat source of constant strength affects the thermally mechanical responses of the stratum.
The heat source such as a canister of radioactive waste can cause temperature rise in the soil, and thus the solid skeleton and pore fluid can expand. This leads to increase in pore water pressure and reduction in effective stress, because the volume increment of the pore water is greater than that of the voids of solid matrix. Therefore, thermal failure of soil can occur as a result of losing shear resistance due to reduction in effective stress.
Booker and Savvidou [1,2], Savvidou and Booker [3]
presented the solutions of thermo-consolidation around spherical and point heat sources. In their solutions, the thermal properties were considered as isotropic [1,2] or cross-anisotropic [3] whereas the elastic properties of the soil were treated as isotropic [1-3]. Moreover, the stratum was modeled in full space to simulate the deep buried heat sources. Georgiadis et al. [4] analyzed the transient
dynamic coupled thermoelasticity paradigm of a half-space under the action of a buried thermal/mechanical source. Shendeleva [5] theoretically presented a model comprising an instantaneous line heat source situated parallel to the interface between two semi-infinite heat-conductive media in perfect thermal contact. Three-dimensional Green’s functions for a steady point heat source were derived by Wang et al. [6]. Lu and Lin [7]
displayed the transient ground surface displacement produced by a point heat source or fluid sink through analog quantities between thermoelasticity and poroelasticity. Lu et al. [8] presented the closed-form solutions of a homogeneous isotropic elastic half space subjected to circular plane heat source on the basis of the fundamental solutions of half space due to a point heat source. Analytical solutions of the transient and long-term horizontal and vertical displacements due to a point heat source were presented by Lin and Lu [9].
Soils in general are deposited through process of sedimentation over a long period of time. Under the accumulative overburden pressure, soils display significant anisotropic mechanical and thermal properties.
In order to describe the anisotropic nature of soils, it can be modeled as cross-anisotropic medium whose properties are symmetric about the vertical axis. For the heat source buried at a great depth, the effects of half space boundary on thermally response can be neglected.
In general, soils or rocks are deposited through a geologic process of sedimentation over a long period of time. Under the accumulative overburden pressure, strata display significant anisotropic mechanical, seepage and thermal properties. Both stratified soil and rock masses can show the phenomenon of anisotropy. For this reason, theoretical and numerical models should be able to simulate the layered soils and rocks as cross-anisotropic media [10-14].
The investigation is focused on long-term thermally elastic mechanical behaviors of the stratum. On the basis of the derived deep point heat source induced fundamental solutions, the closed-form solutions of long-term ground deformation, thermal stresses, and temperature changes of the soil mass due to a deep line heat source are obtained by using appropriate line integral techniques. Results are simplified to isotropic case to Proceedings of the IASTED International Conference
June 25 - 27, 2012 Napoli, Italy Applied Simulation and Modelling (ASM 2012)
DOI: 10.2316/P.2012.776-040 附 66
provide better understanding of the thermally induced mechanical responses of the stratum. The solutions can be used to test numerical models and evaluate numerical simulations of the thermoelastic responses near the line heat source.