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.

References

References
1.
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
.
2.
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
.
3.
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
.
4.
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
.
5.
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
.
6.
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
.
7.
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
.
8.
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
.
9.
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
.
10.
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
.
11.
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
.
12.
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
.
13.
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
.
14.
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
.
15.
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
.
16.
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
.
17.
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
.
18.
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
.
19.
Yosibash
,
Z.
,
2011
,
Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection With Failure Initiation
,
Springer
,
New York
.
20.
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
.
21.
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
.
22.
Chao
,
C. K.
, and
Chang
,
R. C.
,
1992
, “
Thermal Interface Crack Problems in Dissimilar Anisotropic Media
,”
J. Appl. Phys.
,
72
(
7
), pp.
2598
2604
.
23.
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
.
24.
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
.
25.
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
.
26.
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
.
27.
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
.
28.
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
.
29.
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
.
30.
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
.
31.
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
.
32.
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
.
33.
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
.
34.
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
.
35.
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
.
36.
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
.
37.
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
.
38.
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
.
39.
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
.
40.
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
.
41.
Lim
,
C. W.
, and
Xu
,
X. S.
,
2010
, “
Symplectic Elasticity: Theory and Applications
,”
Appl. Mech. Rev.
,
63
(
5
), p.
050802
.
42.
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
.
43.
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
.
44.
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
.
45.
Hahn
,
D. W.
, and
Özişik
,
M.
,
2012
,
Heat Conduction
,
3rd ed.
,
Wiley
,
Hoboken, NJ
.
You do not currently have access to this content.