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Research Papers: Conduction

A Symplectic Analytical Singular Element for Steady-State Thermal Conduction With Singularities in Anisotropic Material

[+] Author and Article Information
X. F. Hu

State Key Laboratory of Structural Analysis for
Industrial Equipment,
Dalian University of Technology,
Dalian 116024, China;
International Center for Computational
Mechanics,
Dalian University of Technology,
Dalian 116024, China
e-mail: hxf@dlut.edu.cn

W. A. Yao

State Key Laboratory of Structural Analysis for
Industrial Equipment,
Dalian University of Technology,
Dalian 116024, China;
International Center for Computational
Mechanics,
Dalian University of Technology,
Dalian 116024, China

S. T. Yang

Department of Civil and Environmental
Engineering,
University of Strathclyde,
Glasgow G1 1XJ, UK

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 24, 2017; final manuscript received April 20, 2018; published online May 22, 2018. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 140(9), 091301 (May 22, 2018) (13 pages) Paper No: HT-17-1631; doi: 10.1115/1.4040085 History: Received October 24, 2017; Revised April 20, 2018

Modeling of steady-state thermal conduction for crack and v-notch in anisotropic material remains challenging. Conventional numerical methods could bring significant error and the analytical solution should be used to improve the accuracy. In this study, crack and v-notch in anisotropic material are studied. The analytical symplectic eigen solutions are obtained for the first time and used to construct a new symplectic analytical singular element (SASE). The shape functions of the SASE are defined by the obtained eigen solutions (including higher order terms), hence the temperature as well as heat flux fields around the crack/notch tip can be described accurately. The formulation of the stiffness matrix of the SASE is then derived based on a variational principle with two kinds of variables. The nodal variable is transformed into temperature such that the proposed SASE can be connected with conventional finite elements (FE) directly without transition element. Structures of complex geometries and complicated boundary conditions can be analyzed numerically. The generalized flux intensity factors (GFIFs) can be calculated directly without any postprocessing. A few numerical examples are worked out and it is proven that the proposed method is effective for the discussed problem, and the structure can be analyzed accurately and efficiently.

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References

Ma, C. C. , and Chang, S. W. , 2004, “ Analytical Exact Solutions of Heat Conduction Problems for Anisotropic Multi-Layered Media,” Int. J. Heat Mass Transfer, 47(8–9), pp. 1643–1655. [CrossRef]
Tsai, T. W. , Lee, Y. M. , and Shiah, Y. C. , 2013, “ Heat Conduction Analysis in an Anisotropic Thin Film Irradiated By an Ultrafast Pulse Laser Heating,” Numer. Heat Transfer, Part A: Appl., 64(2), pp. 132–152. [CrossRef]
Chen, T. , and Kuo, H. Y. , 2005, “ On Linking n-Dimensional Anisotropic and Isotropic Green's Functions for Infinite Space, Half-Space, Bimaterial, and Multilayer for Conduction,” Int. J. Solids Struct., 42(14), pp. 4099–4114. [CrossRef]
Yen, D. H. Y. , and Beck, J. V. , 2004, “ Green's Functions and Three-Dimensional Steady-State Heat-Conduction Problems in a Two-Layered Composite,” J. Eng. Math., 49(3), pp. 305–319. [CrossRef]
Marczak, R. J. , and Denda, M. , 2011, “ New Derivations of the Fundamental Solution for Heat Conduction Problems in Three-Dimensional General Anisotropic Media,” Int. J. Heat Mass Transfer, 54(15–16), pp. 3605–3612. [CrossRef]
Shiah, Y. C. , Hwang, P. W. , and Yang, R. B. , 2006, “ Heat Conduction in Multiply Adjoined Anisotropic Media With Embedded Point Heat Sources,” ASME J. Heat Transfer, 128(2), pp. 207–214. [CrossRef]
Rafiezadeh, K. , and Ataie-Ashtiani, B. , 2013, “ Seepage Analysis in Multi-Domain General Anisotropic Media By Three-Dimensional Boundary Elements,” Eng. Anal. Boundary Elem., 37(3), pp. 527–541. [CrossRef]
Marin, L. , and Munteanu, L. , 2010, “ Boundary Reconstruction in Two-Dimensional Steady State Anisotropic Heat Conduction Using a Regularized Meshless Method,” Int. J. Heat Mass Transfer, 53(25–26), pp. 5815–5826. [CrossRef]
Zhang, Y. M. , Liu, Z. Y. , Chen, J. T. , and Gu, Y. , 2011, “ A Novel Boundary Element Approach for Solving the Anisotropic Potential Problems,” Eng. Anal. Boundary Elem., 35(11), pp. 1181–1189. [CrossRef]
Gu, Y. , Chen, W. , and He, X. Q. , 2012, “ Singular Boundary Method for Steady-State Heat Conduction in Three Dimensional General Anisotropic Media,” Int. J. Heat Mass Transfer, 55(17–18), pp. 4837–4848. [CrossRef]
Yeh, C. S. , Shu, Y. C. , and Wu, K. C. , 1993, “ Conservation Laws in Anisotropic Elasticity II. Extension and Application to Thermoelasticity,” Proc. R. Soc. A, 443(1917), pp. 39–151. [CrossRef]
Kattis, M. A. , Papanikos, P. , and Providas, E. , 2004, “ Thermal Green's Functions in Plane Anisotropic Bimaterials,” Acta Mech., 173(1–4), pp. 65–76. [CrossRef]
Sladek, J. , Sladek, V. , Hellmich, C. , and Eberhardsteiner, J. , 2007, “ Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies By Meshless Local Petrov–Galerkin Method,” Comput. Mech., 39(3), pp. 323–333. [CrossRef]
Buroni, F. C. , Marczak, R. J. , Denda, M. , and Saez, A. , 2014, “ A Formalism for Anisotropic Heat Transfer Phenomena: Foundations and Green's Functions,” Int. J. Heat Mass Transfer, 75, pp. 399–409. [CrossRef]
Berger, J. R. , Martin, P. A. , Mantič, V. , and Gray, L. J. , 2005, “ Fundamental Solutions for Steady-State Heat Transfer in an Exponentially Graded Anisotropic Material,” Z. Für Angew. Mathematik Und Phys. (ZAMP), 56(2), pp. 293–303. [CrossRef]
Shen, M. H. , Chen, F. M. , and Hung, S. Y. , 2010, “ Analytical Solutions for Anisotropic Heat Conduction Problems in a Trimaterial With Heat Sources,” ASME J. Heat Transfer, 132(9), p. 091302. [CrossRef]
Hu, X. F. , Gao, H. Y. , Yao, W. A. , and Yang, S. T. , 2017, “ Study on Steady-State Thermal Conduction With Singularities in Multi-Material Composites,” Int. J. Heat Mass Transfer, 104, pp. 861–870. [CrossRef]
Hu, X. F. , Gao, H. Y. , Yao, W. A. , and Yang, S. T. , 2016, “ A Symplectic Analytical Singular Element for Steady-State Thermal Conduction With Singularities in Composite Structures,” Numer. Heat Transfer, Part B: Fundamentals, 70(5), pp. 406–419. [CrossRef]
Yosibash, Z. , 2011, Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection With Failure Initiation, Springer, New York.
Mantič, V. , Parıs, F. , and Berger, J. , 2003, “ Singularities in 2D Anisotropic Potential Problems in Multi-Material Corners: Real Variable Approach,” Int. J. Solids Struct., 40(20), pp. 5197–5218. [CrossRef]
Hwu, C. , and Lee, W. J. , 2004, “ Thermal Effect on the Singular Behavior of Multibonded Anisotropic Wedges,” J. Therm. Stresses, 27(2), pp. 111–136. [CrossRef]
Chao, C. K. , and Chang, R. C. , 1992, “ Thermal Interface Crack Problems in Dissimilar Anisotropic Media,” J. Appl. Phys., 72(7), pp. 2598–2604. [CrossRef]
Yu, T. T. , Bui, T. Q. , Yin, S. H. , Doan, D. H. , Wu, C. T. , Do, T. V. , and Tanaka, S. , 2016, “ On the Thermal Buckling Analysis of Functionally Graded Plates With Internal Defects Using Extended Isogeometric Analysis,” Compos. Struct., 136, pp. 684–695. [CrossRef]
Liu, P. , Yu, T. , Bui, T. Q. , Zhang, C. Z. , Xu, Y. P. , and Lim, C. W. , 2014, “ Transient Thermal Shock Fracture Analysis of Functionally Graded Piezoelectric Materials By the Extended Finite Element Method,” Int. J. Solids Struct., 51(11–12), pp. 2167–2182. [CrossRef]
Tanaka, S. , Suzuki, H. , Sadamoto, S. , Imachi, M. , and Bui, T. Q. , 2015, “ Analysis of Cracked Shear Deformable Plates By an Effective Meshfree Plate Formulation,” Eng. Fract. Mech., 144, pp. 142–157. [CrossRef]
Lei, J. , Zhang, C. Z. , and Bui, T. Q. , 2015, “ Transient Dynamic Interface Crack Analysis in Magnetoelectroelastic Bi-Materials By a Time-Domain BEM,” Eur. J. Mech.-A/Solids, 49, pp. 146–157. [CrossRef]
Hosseini, S. S. , Bayesteh, H. , and Mohammadi, S. , 2013, “ Thermo-Mechanical XFEM Crack Propagation Analysis of Functionally Graded Materials,” Mater. Sci. Eng.: A, 561, pp. 285–302. [CrossRef]
Yvonnet, J. , He, Q. C. , Zhu, Q. Z. , and Shao, J. F. , 2011, “ A General and Efficient Computational Procedure for Modelling the Kapitza Thermal Resistance Based on XFEM,” Comput. Mater. Sci., 50(4), pp. 1220–1224. [CrossRef]
Marin, L. , 2010, “ A Meshless Method for the Stable Solution of Singular Inverse Problems for Two-Dimensional Helmholtz-Type Equations,” Eng. Anal. Boundary Elem., 34(3), pp. 274–288. [CrossRef]
Marin, L. , 2010, “ Treatment of Singularities in the Method of Fundamental Solutions for Two-Dimensional Helmholtz-Type Equations,” Appl. Math. Modell., 34(6), pp. 1615–1633. [CrossRef]
Marin, L. , Lesnic, D. , and Mantič, V. , 2004, “ Treatment of Singularities in Helmholtz-Type Equations Using the Boundary Element Method,” J. Sound Vib., 278(1–2), pp. 39–62. [CrossRef]
Mera, N. S. , Elliott, L. , Ingham, D. B. , and Lesnic, D. , 2002, “ Singularities in Anisotropic Steady-State Heat Conduction Using a Boundary Element Method,” Int. J. Numer. Methods Eng., 53(10), pp. 2413–2427. [CrossRef]
Mera, N. S. , Elliott, L. , Ingham, D. B. , and Lesnic, D. , 2002, “ An Iterative Algorithm for Singular Cauchy Problems for the Steady State Anisotropic Heat Conduction Equation,” Eng. Anal. Boundary Elem., 26(2), pp. 157–168. [CrossRef]
Gu, Y. , Chen, W. , and Fu, Z. J. , 2014, “ Singular Boundary Method for Inverse Heat Conduction Problems in General Anisotropic Media,” Inverse Probl. Sci. Eng., 22(6), pp. 889–909. [CrossRef]
Wei, X. , Chen, W. , Chen, B. , and Sun, L. , 2015, “ Singular Boundary Method for Heat Conduction Problems With Certain Spatially Varying Conductivity,” Comput. Math. Appl., 69(3), pp. 206–222. [CrossRef]
Mierzwiczak, M. , Chen, W. , and Fu, Z. J. , 2015, “ The Singular Boundary Method for Steady-State Nonlinear Heat Conduction Problem With Temperature-Dependent Thermal Conductivity,” Int. J. Heat Mass Transfer, 91, pp. 205–217. [CrossRef]
Shannon, S. , Yosibash, Z. , Dauge, M. , and Costabel, M. , 2013, “ Extracting Generalized Edge Flux Intensity Functions With the Quasidual Function Method Along Circular 3-D Edges,” Int. J. Fract., 181(1), pp. 25–50. [CrossRef]
Xiao, Q. Z. , and Karihaloo, B. L. , 2006, “ Improving the Accuracy of XFEM Crack Tip Fields Using Higher Order Quadrature and Statically Admissible Stress Recovery,” Int. J. Numer. Methods Eng., 66(9), pp. 1378–1410. [CrossRef]
Leung, A. Y. T. , Xu, X. S. , and Zhou, Z. H. , 2010, “ Hamiltonian Approach to Analytical Thermal Stress Intensity Factors-Part 1: Thermal Intensity Factor,” J. Therm. Stresses, 33(3), pp. 262–278. [CrossRef]
Zhou, Z. H. , Xu, C. H. , Xu, X. S. , and Leung, A. Y. T. , 2015, “ Finite-Element Discretized Symplectic Method for Steady-State Heat Conduction With Singularities in Composite Structures,” Numer. Heat Transfer, Part B: Fundamentals, 67(4), pp. 302–319. [CrossRef]
Lim, C. W. , and Xu, X. S. , 2010, “ Symplectic Elasticity: Theory and Applications,” Appl. Mech. Rev., 63(5), p. 050802. [CrossRef]
Li, J. , 2002, “ Singularity Analysis of Near-Tip Fields for Notches Formed From Several Anisotropic Plates Under Bending,” Int. J. Solids Struct., 39(23), pp. 5767–5785. [CrossRef]
Li, J. , Zhang, X. B. , and Recho, N. , 2001, “ Stress Singularities Near the Tip of a Two-Dimensional Notch Formed From Several Elastic Anisotropic Materials,” Int. J. Fract., 107(4), pp. 379–395. [CrossRef]
Hu, X. F. , Yao, W. A. , and Fang, Z. X. , 2011, “ Stress Singularity Analysis of Anisotropic Multi-Material Wedges Under Antiplane Shear Deformation Using the Symplectic Approach,” Theor. Appl. Mech. Lett., 1(6), p. 061003. [CrossRef]
Hahn, D. W. , and Özişik, M. , 2012, Heat Conduction, 3rd ed., Wiley, Hoboken, NJ. [CrossRef] [PubMed] [PubMed]

Figures

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Fig. 1

A wedge in anisotropic material and the subfields

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Fig. 2

Schematic illustration of a subfield

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Fig. 3

The symplectic analytical singular element

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Fig. 4

The flow chart of the proposed method for wedge problem in anisotropic material

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Fig. 5

Error curves of the eigenvalue for a crack

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Fig. 6

Schematic of a reentrant corner of 90 deg

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Fig. 7

Curves of error of the present solution of eigenvalue for a reentrant

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Fig. 8

A cracked disk of anisotropic material

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Fig. 9

The FE meshes of the cracked disk with a SASE and a few four node isoparametric elements. There are 60, 180, and 300 four node elements in the coarse mesh, middle mesh, and refined mesh, respectively.

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Fig. 10

The angular distribution of ψr from the first five eigenvectors (nonzero)

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Fig. 11

The angular distribution of ψs from calculated the first five eigenvectors (nonzero)

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Fig. 12

Contours of qx and qy in the SASE

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Fig. 13

An edge crack in the plate of anisotropic material and the FE mesh

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Fig. 14

Contours of temperature in the cracked plate

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Fig. 15

Contours of heat fluxes in the cracked plate

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Fig. 16

Configuration of a reentrant corner of 90 deg and the FE mesh

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Fig. 17

Contours of temperature in the plate with a reentrant

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Fig. 18

Contours of heat fluxes in the plate with a reentrant

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