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2D Green’s functions for semi-infinite orthotropic thermoelastic plane
Applied Mathematical Modelling 33 (2009) 1674–1682
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
2D Green’s functions for semi-infinite orthotropic thermoelastic plane Peng-Fei Hou *, Li Wang, Tao Yi Department of Engineering Mechanics, Hunan University, Changsha 410082, PR China
a r t i c l e
i n f o
Article history: Received 29 July 2007 Received in revised form 27 February 2008 Accepted 4 March 2008 Available online 21 March 2008
a b s t r a c t Based on the 2D general solutions of orthotropic thermoelastic material, the Green’s function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic plane is constructed by three newly introduced harmonic functions. All components of coupled field in semi-infinite thermoelastic plane are expressed in terms of elementary functions. Numerical results are given graphically by contours. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: 2D Green’s function Point heat source Semi-infinite Orthotropic Thermoelastic
1. Introduction Fundamental solutions or Green’s functions play an important role in both applied and theoretical studies on the physics of solids. They are foundations of a lot of further works. For example, fundamental solutions can be used to construct many analytical solutions of practical problems when boundary conditions are imposed. They are essential in the boundary element method as well as the study of cracks, defects and inclusions. For isotropic materials, there is well-known closed-form Kelvin Green’s function [1]. For transversely isotropic materials, Lifshitz and Rozentsveig [2] and Lejcek [3] presented the Green’s functions using the Fourier transform method. Elliott [4], Kroner [5] and Willis [6] obtained the Green’s functions using the direct method. Sveklo [7] obtained Green’s functions using the complex method. Pan and Chou [8] and Ding et al. [9] presented the Green’s function in form of compact elementary functions. For anisotropic materials, Pan and Yuan [10] and Pan [11] obtained the three-dimensional Green’s functions for bimaterials with perfect and imperfect interfaces, respectively. The thermal effects are not considered in all above works. Sharma [12] gave the fundamental solution of transversely isotropic thermoelastic materials in an integral form. Yu et al. [13] gave the Green’s function for a point heat source in two-phase isotropic thermoelastic materials. Chen et al. [14] derived a compact 3D general solution for transversely isotropic thermoelastic materials. In this general solution, all components of thermoelastic field are expressed by three harmonic functions. In this paper, 2D Green’s functions for a steady point heat source in a semi-infinite orthotropic thermoelastic plane is investigated. For this object, the 2D general solution, which is parallel to 3D general solution of Chen et al. [14], is presented in Section 2. In Section 3, three new suitable harmonic functions are constructed in form of elementary functions with undetermined constants by the method of trial-and-error. The corresponding coupled field can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions on plane
* Corresponding author. Tel./fax: +86 731 8822330. E-mail address: [email protected] (P.-F. Hou). S0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.03.004
P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
1675
z = h, equilibrium conditions of a rectangle within 0 < a1 < z < a2 and b 6 x 6 b and boundary conditions on surface z = 0. Numerical examples are presented in Section 4. Finally, the paper is concluded in Section 5. 2. 2D General solution for orthotropic thermoelastic material The 3D equations of orthotropic thermoelastic materials can be found in Kumara and Singh. [15]. If all components are independent of coordinate y, one can have the so-called plane problem. The constitutive equations in 2D Cartesian coordinates (x, z) can be simplified as ou ow þ c13 k11 h; ox oz ou ow þ c33 k33 h; rz ¼ c13 ox oz ou ow szx ¼ c44 þ ; oz ox rx ¼ c11
ð1Þ
where u and w are components of the mechanical displacement in x and z directions, respectively; rij are the components of stress; h is temperature increment, respectively; cij and kii are elastic constants and thermal moduli, respectively. In the absence of body forces, the mechanical and heat equilibrium equations are orx oszx oszx orz þ ¼ 0; þ ¼ 0; ox oz ox oz ! 2 2 o o b11 2 þ b33 2 h ¼ 0; o x o z
ð2aÞ ð2bÞ
where bii(i = 1, 3) are coefficients of heat conduction. By virtue of the parallel method of Chen et al. [14], 2D general solution for Eqs. (1) and (2) can be obtained as follows: u¼
3 X owj ; ox j¼1
rx ¼
3 X
w¼
sj k1j
j¼1
s2j xj
j¼1
3 X
o2 wj ; oz2j
rz ¼
owj ; ozj 3 X j¼1
xj
h ¼ k23
o2 w3 ; oz23
o2 wj ; oz2j
szx ¼
ð3aÞ
3 X j¼1
sj xj
o2 wj ; oxozj
ð3bÞ
where zj ¼ sj z ðj ¼ 1; 2; 3Þ: ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s3 ¼ b11 =b33 ; sj ðj ¼ 1; 2Þ satisfying Re(sj) > 0 are the two eigenvalues of the fourth degree polynomial Eq. (11) in Chen et al. [14]. Functions wj(j = 1, 2, 3) satisfy, respectively, the following harmonic equations: ! o2 o2 wj ¼ 0 ðj ¼ 1; 2; 3Þ; þ ð5Þ ox2 oz2j and k1j ¼ aj1 =sj ;
k2j ¼ aj2 ;
ð6aÞ
xj ¼ ½c11 c13 k1j s2j þ k11 k2j =s2j ¼ c44 ð1 þ k1j Þ ¼ c13 þ c33 k1j s2j k33 k2j ;
ð6bÞ
where ajm (m = 1,2) are constants defined in Eq. (18) of Chen et al. [14]. It should be noted that the general solutions given in Eq. (3) are only valid for the case when the eigenvalues sj (j = 1, 2, 3) are distinct, which is the most common case. 3. Green’s functions for a point heat source in the interior of a semi-infinite orthotropic thermoelastic plane Consider a semi-infinite orthotropic thermoelastic plane z P 0 (Fig. 1). A point heat source of strength H is applied at the point (0, h) in 2D Cartesian coordinate (x, z). The surface (z = 0) is free and thermally insulated. Based on the general solution Eq. (3), the coupled field in the semi-infinite thermoelastic plane is derived in this section. The boundary conditions on the surface (z = 0) are in the form of rz ¼ szr ¼ 0;
oh=oz ¼ 0:
ð7Þ
For future reference, following denotations are introduced: zj ¼ sj z;
hk ¼ sk h;
zjk ¼ zj þ hk ; zjk ¼ zj hk ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ z2jk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r jk ¼ x2 þ zjk ; r jk ¼
ð8Þ ðj; k ¼ 1; 2; 3Þ:
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b
o
x
a1
H
(0, h)
a2
z 3
Fig. 1. Semi-infinite thermoelastic plane applied by a point heat source of strength H.
By virtue of trial-and-error method, Green’s functions in semi-infinite plane are assumed in following form: X 3 1 3 x 1 3 x þ ðj ¼ 1; 2; 3Þ; Ajk ðz2jk x2 Þ ln r jk xzjj arctan xzjk arctan wj ¼ Aj ðz2jj x2 Þ ln r jj zjj 2 2 2 2 zjk k¼1
ð9Þ
where Aj and Ajk ðj; k ¼ 1; 2; 3Þ are twelve constants to be determined. Substitution of Eq. (9) into general solution Eq.(3) yields the expressions of coupled field as follows: 3 X x Aj xðln r jj 1Þ þ zjj arctan u¼ zjj j¼1 3 3 XX x ; Ajk xðln r jk 1Þ þ zjk arctan zjk j¼1 k¼1 3 X x w¼ sj k1j Aj zjj ðln r jj 1Þ x arctan zjj j¼1 3 X 3 X x ; sj k1j Ajk zjk ðln rjk 1Þ x arctan þ zjk j¼1 k¼1 h ¼ k23 A3 ln r 33 þ k23
3 X
A3k ln r3k ;
k¼1
rx ¼
3 X
s2j x1j Aj ln r jj
j¼1
rz ¼
3 X
s2j x1j Ajk ln r jk ;
j¼1 k¼1
xj Aj ln rjj þ
j¼1
szx ¼
3 X 3 X
3 X 3 X
xj Ajk ln r jk ;
j¼1 k¼1
3 X
sj xj Aj arctan
j¼1
3 X 3 x X x sj xj Ajk arctan : zjj z jk j¼1 k¼1
ð10Þ
Consideration of the continuity on plane z = h for w and szx yields 3 X
sj k1j Aj ¼ 0;
ð11Þ
sj xj Aj ¼ 0:
ð12Þ
j¼1 3 X j¼1
Substitution of xj in Eq. (6b) into Eq. (12) gives 3 X
c44 ð1 þ k1j Þsj Aj ¼ 0:
ð13Þ
j¼1
By virtue of Eq. (11), Eq. (13) can be simplified to 3 X
sj Aj ¼ 0:
ð14Þ
j¼1
When the mechanical and thermal equilibrium for a cylinder of a1 6 z 6 a2 (0 < a1 < h < a2) and 0 6 r 6 b are considered (Fig. 1), two additional equations can be obtained
P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
X
Z¼0:
Z
b
½rz ðx; a2 Þ rz ðx; a1 Þdx þ b
X
Heat flux ¼ H : b33
Z
a2
½szx ðb; zÞ szx ðb; zÞdz ¼ 0;
ð15aÞ
a1
Z a2 b oh oh oh oh ðx; a2 Þ ðx; a1 Þ dx b11 ðb; zÞ ðb; zÞ dz ¼ H; oz ox ox b oz a1
Z
1677
where b11 and b33 are coefficients of heat conduction along with x axis and z axis, respectively. Some useful integrals are listed as follows: Z x ln rjj dx ¼ xðln rjj 1Þ þ zjj arctan ; zjj Z x ln rjk dx ¼ xðln r jk 1Þ þ zjk arctan ; zjk Z x 1 x ; x ln r jj þ zjj arctan arctan dz ¼ zjj zjj sj Z x 1 x arctan dz ¼ ; x ln rjk þ zjk arctan zjk sj zjk ! Z Z 3 z33 X oh z3k A3 2 þ A3k 2 dx dx ¼ s3 k23 r33 oz r 3k k¼1 ! 3 X x x ; þ A3k arctan ¼ s3 k23 A3 arctan z33 z3k k¼1 ! Z Z 3 X oh x x A3k 2 dz dz ¼ k23 A3 2 þ r33 ox r 3k k¼1 ! 3 X k23 x x : A3 arctan þ A3k arctan ¼ z33 s3 z3k k¼1 It is noted that integral (16d) is not continuous at z = h, following expression should be used: Z a2 Z h Z a2 oh oh oh dz ¼ dz þ dz: ox a1 ox a1 hþ ox
ð15bÞ
ð16aÞ
ð16bÞ
ð16cÞ
ð16dÞ
ð17Þ
Substituting Eq. (10) into Eq. (15a) with using integrals (16a), (16b), one can obtain 3 X
xj Aj I1 þ
j¼1
3 X
xj
3 X
j¼1
Ajk I2 ¼ 0;
ð18Þ
k¼1
where " " z¼a2 #x¼b x¼b #z¼a2 x x xðln r jj 1Þ þ zjj arctan x ln r jj þ zjj arctan ¼ 0; I1 ¼ zjj z¼a zjj x¼b 1 x¼b z¼a1 " " z¼a2 #x¼b x¼b #z¼a2 x x xðln r jk 1Þ þ zjk arctan x ln rjk þ zjk arctan ¼ 0: I2 ¼ zjk z¼a1 zjk x¼b x¼b
ð19aÞ
ð19bÞ
z¼a1
i.e. Eqs. (18) and (15a) are satisfied automatically. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting Eq. (10) into Eq. (15b) with using s3 ¼ b11 =b33 and integrals (16c, d, 17), one can obtain A3 I3 þ
3 X
A3k I4 ¼
k¼1
k23
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b11 b33
ð20Þ
where " " z¼a2 #x¼b x¼b #z¼h " x¼b #z¼a2 x x x þ arctan þ arctan ¼ 2p; I3 ¼ arctan z33 z¼a z33 x¼b z33 x¼b 1 x¼b z¼a1 z¼hþ " x¼b #z¼a2 " z¼a2 #x¼b x x arctan arctan ¼ 0: I4 ¼ z3k x¼b z3k z¼a1 z¼a1
ð21aÞ
ð21bÞ
x¼b
Thus A3 can be determined by Eqs. (20) and (21) as follows: A3 ¼
2pk23
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : b11 b33
At last, when the couple field on surface z = 0 is considered, one has
ð22Þ
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P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
hk ¼ sk h; z0j ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 zjk ¼ hk ; r jk ¼ x2 þ hk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zjk ¼ hk ; r jk ¼ x2 þ h2k : zj ¼ 0;
ð23Þ
Substituting Eq. (10) into boundary condition (7) with using s3 ¼ sj xj Aj þ
3 X
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b11 =b33 and Eq. (23), one can obtain
sk xk Akj ¼ 0;
ð24Þ
k¼1
xj Aj þ
3 X
xk Akj ¼ 0;
ð25Þ
k¼1
A3 A33 ¼ 0;
A3k ¼ 0;
ðk ¼ 1; 2Þ;
ð26Þ
where j = 1, 2, 3. Thus, twelve constants Aj and Ajk ðj; k ¼ 1; 2; 3Þ can be determined by twelve equations including Eqs. (11), (14), (22), (24), (25), (26). Concerning the problems of point forces Px, Pz in x, z direction in semi-infinite orthotropic thermoelastic plane, because the foundational Eqs. (1) and (2) are single-direction coupling, i.e. the thermal loading can change elastic field, while on the contrary, the mechanical loading cannot change thermal field (h = 0), so the corresponding solutions under mechanical loading degenerate to those when thermal effects are not considered. The corresponding solutions are listed in Appendix. 4. Numerical results The contours of temperature increment and stress of coupled field in semi-infinite orthotropic thermoelastic plane induced by a point heat source of strength H are evaluated numerically and plotted in Figs. 2–5. The material properties listed in Table 1 are taken from hexagonal zinc [16]. In additions, the following non-dimensional components are used in figures: ri sij ; skl ¼ ; c33 ar T 0 c33 ar T 0 h x z ; n ¼ ; f ¼ ; ði; j ¼ x; z; k; l ¼ n; fÞ; #¼ T0 r1 r1
rk ¼
ð27Þ
where r1 is a non-zero dimension, ar and T0 are thermal expansion coefficient and reference temperature, respectively. And Eq. (22) should be rewritten in following non-dimensional form: A3 ¼
d ; 2ps3 k23
ð28Þ
0
10
-1 0
0
-2 5 -3 0
-2 0
1
ζ
2
5 -3
0 -4 5 -4
3
0 -5 5 -5
4
0 -6
5 5 -6
6
0
1
2
3
4
5
6
ξ Fig. 2. Contour of non-dimensional temperature increment # 102 in a semi-infinite plane under point heat source of strength d = 1 acted at (0, 1).
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P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
0 20
40
0
-2 0
1
-4 0
-70
-50 8 -6 -7 0
ζ
2
-6 0
-7 0
3
-6 8 -7 5
-8 0
4
-8 5
5 -9 0
6
0
1
2
3
4
5
6
ξ Fig. 3. Contour of non-dimensional stress rn 102 in a semi-infinite plane under point heat source of strength d = 1 acted at (0, 1).
0
0
0. 1 0.5
1 -3 0
1
1. 5
0 -2
3
6 -1
3
2.3 2
-1 4
ζ
2
4
-2
-6
-8
-1 0
-1 2
0.1
-1
-4
4
1. 5
1
0 0 .5
5
6
0
1
2
3
4
5
6
ξ Fig. 4. Contour of non-dimensional stress rf 102 in an infinite plane under point heat source of strength d = 1 acted at (0, 1).
where d is a non-dimensional strength of point heat source as follows: d¼
H ; r 1 T 0 b33
ð29Þ
and here let d = 1 acted at (0, 1). On the basis of the contours plotted above, following conclusions can be drawn: (1) Fig. 2 shows that the contours of temperature increment #(h) for Hexagonal zinc are similar to normal circle. This is because the heat conduction coefficients b11 and b33 are equal for Hexagonal zinc (Table 1).
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P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
0
1
2
4 16
1
6
12
8
10
2
8.5
-4
ζ
9
8.5
-2
3
4
8
6
-1 4
5 2
0
1
2
1
6
0
-0 .5
3
4
5
6
ξ Fig. 5. Contour of non-dimensional stress sfn 102 in a semi-infinite plane under point heat source of strength d = 1 acted at (0, 1).
Table 1 Material properties of hexagonal zinc [16] Elastic constants (109N m2) c13 36.2 Heat conduction coefficients (W K1 m1)
c11 162.8 Thermal moduli (105N K1 m2)
c33 62.7 Thermal expansion coefficient (106 K1)
c44 38.5
k11
k33
b11
b33
ar
az
17.9839
13.8367
124
124
5.818
15.35
(2) Figs. 3–5 show that there is a zero common tangent for stresses rn(rx), rf(rz) and szx(sfn), respectively. They are positive above these zero common tangents and are negative on the other side. All stresses are singular at point (0, h/r1 = 1) at which the point heat source is located. This is an important contribution for Green’s functions. In some cases, Green’s functions are also called singular solutions. In additions, one can find that the grads of all stresses are very large near the point heat source, which should attract our attentions in practice. (3) Figs. 4 and 5 show that stresses rf(rz) and szx(sfn) tend to zero near the surface f = 0 and satisfy the boundary condition (7). We here give a interesting imagine that when h/r1 tend to larger and larger, the effect of boundary condition (7) on stresses in the vicinity of the load is less and the obtained solution near the load for semi-infinite plane will tend to the solution of infinite plane. So boundary condition (7) is one of the most important attribution factors for the distinctive features of the profiles in semi-infinite and infinite plane. 5. Conclusions By virtue of the compact 2D general solution of orthotropic thermoelastic materials, which is parallel to those of Chen et al. [14], three harmonic functions wj (j = 1, 2, 3) in Eq. (9) are constructed and the corresponding coupled field for a point heat source acted in the interior of a semi-infinite orthotropic thermoelastic plane is obtained. For the sake of concise expressions and convenient derivations, point heat source is loaded at z axis in this paper. The solution in case of point heat source loaded at arbitrary point can be easily generalized from the obtained solution by coordination parallel transform. Because all the components are expressed in terms of elementary functions, it is convenient to use them. Typical numerical examples are presented. All these show that the general solution of Chen et al. [14] is an important foundation for the studies of steady problems thermoelastic materials. It should be noted that the obtained Green’s functions are valid for distinct eigenvalues sj (j = 1, 2, 3), which is the most common case for 2D problems of orthotropic materials. The general solutions and Green’s functions for cases of multiple eigenvalues, such as s1 – s2 = s3 and s1 = s2 = s3 are different from those in this paper. So the cor-
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P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
responding solutions for isotropic material with s1 = s2 = s3 = 1 cannot be obtained by direct degeneration from the obtained solutions. Acknowledgement The authors thankfully acknowledge the financial support from the National Natural Science Foundation of China, Foundation in Hunan Province, SRG grant of City University of Hong Kong, National 985 Special Foundation of China and the reviewers for their constructive comments, which lead to improvements of the earlier version of this article. Appendix A. Solutions for other point loadings in semi-infinite orthotropic thermoelastic material A.1. Solution to the problem of point force Pz Introduce the following harmonic functions: X 3 x x þ wj ¼ Bj zjj ðln rjj 1Þ x arctan Bjk zjk ðln r jk 1Þ x arctan zjj zjk k¼1 where zjj ; r jj ; zjk and rjk are functions defined in Eq. (8), Bj and Bjk
ðj ¼ 1; 2Þ;
w3 ¼ 0;
ðj; k ¼ 1; 2Þ are constants to be determined by
Bj ¼ aj Pz ; ðj ¼ 1; 2Þ; " # " #1 " # a1 0 1 1 1 ¼ ; 2p x1 x2 a2 1
xj Bj
2 X
xk Bkj ¼ 0;
ðA:1Þ
ðA:2aÞ ðA:2bÞ
ðj ¼ 1; 2Þ;
ðA:3Þ
k¼1
sj xj Bj þ
2 X
sk xk Bkj ¼ 0;
ðj ¼ 1; 2Þ:
ðA:4Þ
k¼1
A.2. Solution to the problem of point force Px Introduce the following harmonic functions: X 2 x x þ wj ¼ C j xðln rjj 1Þ þ zjj arctan C jk xðln r jk 1Þ þ zjk arctan zjj zjk k¼1
ðj ¼ 1; 2Þ;
w3 ¼ 0;
ðA:5Þ
where zjj ; r jj ; zjk and rjk are functions defined in Eq. (8), Cj and Cjk (j,k = 1, 2) are constants to be determined by C j ¼ kj P x ; ðj ¼ 1; 2Þ; " # " #1 " # k1 0 1 s1 k11 s2 k12 ¼ ; 2p s1 k21 s2 k22 k2 1
xj C j þ
2 X
xk C kj ¼ 0;
ðj ¼ 1; 2Þ;
ðA:6aÞ ðA:6bÞ
ðA:7Þ
k¼1
sj xj C j
2 X
sk xk C kj ¼ 0;
ðj ¼ 1; 2Þ:
ðA:8Þ
k¼1
References [1] P.K. Banerjee, R. Butterfield, Boundary Element Methods in Engineering Science, McGraw-Hill, London, 1981. [2] I.M. Lifshitz, L.N. Rozentsveig, Construction of the Green tensor for the fundamental equation of elasticity theory in the case of an unbounded elastically anisotropic medium, Zhurnal Eksperimental, noi I Teioretical Fiziki 17 (1947) 783–791. [3] J. Lejcek, The Green function of the theory of elasticity in an anisotropic hexagonal medium, Czech. J. Phys. B19 (1969) 799–803. [4] H.A. Elliott, Three-dimensional stress distributions in hexagonal aeolotropic crystals, Proc. Cambridge Philos. Soc. 44 (1948) 522–533. [5] E. Kroner, Das fundamentalintegral der anisotropen elastischen diferentialgleichungen, Z. Phys. 136 (1953) S.402–S.410. [6] J.R. Willis, The elastic interaction energy of dislocation loops in anisotropic media, Quarter. J. Mech. Appl. Math. 18 (1965) 419–433. [7] V.A. Sveklo, Concentrated force in a transversely isotropic half-space and in a composite space, PMM 33 (1969) 532–537. [8] Y.C. Pan, T.W. Chou, Point force solution for an infinite transversely isotropic solid, ASME J. Appl. Mech. 98 (E) (1976) 608–612. [9] H.J. Ding, W.Q. Chen, L. Zhang, Elasticity of Transversely Isotropic Materials, Springer, Netherlands, 2006.
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E. Pan, F.G. Yuan, Three-dimensional Green’s functions in anisotropic bimaterials, Int. J. Solids Struct. 37 (2000) 5329–5351. E. Pan, Three-dimensional Green’s functions in anisotropic elastic bimaterials with imperfect interfaces, J. Appl. Mech. ASME 70 (2003) 180–190. B. Sharma, Thermal stresses in transversely isotropic semi-Infinite elastic solids, ASME J. Appl. Mech. 25 (1958) 86–88. H.Y. Yu, S.C. Sanday, B.B. Rath, Thermoelastic stresses in bimaterials, Philos. Mag. A 65 (1992) 1049–1064. W.Q. Chen, H.J. Ding, D.S. Ling, Thermoelastic field of transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution, Int. J. Solids Struct. 41 (2004) 69–83. [15] R. Kumara, M. Singh, Reflection/transmission of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic halfspaces, Mater. Sci. Eng. A 472 (2008) 83–96. [16] J.N. Sharma, P.K. Sharma, Free vibration of homogeneous transversely isotropic cylindrical panel, J. Therm. Stresses 25 (2002) 169–182.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
2D Green’s functions for semi-infinite orthotropic thermoelastic plane Peng-Fei Hou *, Li Wang, Tao Yi Department of Engineering Mechanics, Hunan University, Changsha 410082, PR China
a r t i c l e
i n f o
Article history: Received 29 July 2007 Received in revised form 27 February 2008 Accepted 4 March 2008 Available online 21 March 2008
a b s t r a c t Based on the 2D general solutions of orthotropic thermoelastic material, the Green’s function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic plane is constructed by three newly introduced harmonic functions. All components of coupled field in semi-infinite thermoelastic plane are expressed in terms of elementary functions. Numerical results are given graphically by contours. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: 2D Green’s function Point heat source Semi-infinite Orthotropic Thermoelastic
1. Introduction Fundamental solutions or Green’s functions play an important role in both applied and theoretical studies on the physics of solids. They are foundations of a lot of further works. For example, fundamental solutions can be used to construct many analytical solutions of practical problems when boundary conditions are imposed. They are essential in the boundary element method as well as the study of cracks, defects and inclusions. For isotropic materials, there is well-known closed-form Kelvin Green’s function [1]. For transversely isotropic materials, Lifshitz and Rozentsveig [2] and Lejcek [3] presented the Green’s functions using the Fourier transform method. Elliott [4], Kroner [5] and Willis [6] obtained the Green’s functions using the direct method. Sveklo [7] obtained Green’s functions using the complex method. Pan and Chou [8] and Ding et al. [9] presented the Green’s function in form of compact elementary functions. For anisotropic materials, Pan and Yuan [10] and Pan [11] obtained the three-dimensional Green’s functions for bimaterials with perfect and imperfect interfaces, respectively. The thermal effects are not considered in all above works. Sharma [12] gave the fundamental solution of transversely isotropic thermoelastic materials in an integral form. Yu et al. [13] gave the Green’s function for a point heat source in two-phase isotropic thermoelastic materials. Chen et al. [14] derived a compact 3D general solution for transversely isotropic thermoelastic materials. In this general solution, all components of thermoelastic field are expressed by three harmonic functions. In this paper, 2D Green’s functions for a steady point heat source in a semi-infinite orthotropic thermoelastic plane is investigated. For this object, the 2D general solution, which is parallel to 3D general solution of Chen et al. [14], is presented in Section 2. In Section 3, three new suitable harmonic functions are constructed in form of elementary functions with undetermined constants by the method of trial-and-error. The corresponding coupled field can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions on plane
* Corresponding author. Tel./fax: +86 731 8822330. E-mail address: [email protected] (P.-F. Hou). S0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.03.004
P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
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z = h, equilibrium conditions of a rectangle within 0 < a1 < z < a2 and b 6 x 6 b and boundary conditions on surface z = 0. Numerical examples are presented in Section 4. Finally, the paper is concluded in Section 5. 2. 2D General solution for orthotropic thermoelastic material The 3D equations of orthotropic thermoelastic materials can be found in Kumara and Singh. [15]. If all components are independent of coordinate y, one can have the so-called plane problem. The constitutive equations in 2D Cartesian coordinates (x, z) can be simplified as ou ow þ c13 k11 h; ox oz ou ow þ c33 k33 h; rz ¼ c13 ox oz ou ow szx ¼ c44 þ ; oz ox rx ¼ c11
ð1Þ
where u and w are components of the mechanical displacement in x and z directions, respectively; rij are the components of stress; h is temperature increment, respectively; cij and kii are elastic constants and thermal moduli, respectively. In the absence of body forces, the mechanical and heat equilibrium equations are orx oszx oszx orz þ ¼ 0; þ ¼ 0; ox oz ox oz ! 2 2 o o b11 2 þ b33 2 h ¼ 0; o x o z
ð2aÞ ð2bÞ
where bii(i = 1, 3) are coefficients of heat conduction. By virtue of the parallel method of Chen et al. [14], 2D general solution for Eqs. (1) and (2) can be obtained as follows: u¼
3 X owj ; ox j¼1
rx ¼
3 X
w¼
sj k1j
j¼1
s2j xj
j¼1
3 X
o2 wj ; oz2j
rz ¼
owj ; ozj 3 X j¼1
xj
h ¼ k23
o2 w3 ; oz23
o2 wj ; oz2j
szx ¼
ð3aÞ
3 X j¼1
sj xj
o2 wj ; oxozj
ð3bÞ
where zj ¼ sj z ðj ¼ 1; 2; 3Þ: ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s3 ¼ b11 =b33 ; sj ðj ¼ 1; 2Þ satisfying Re(sj) > 0 are the two eigenvalues of the fourth degree polynomial Eq. (11) in Chen et al. [14]. Functions wj(j = 1, 2, 3) satisfy, respectively, the following harmonic equations: ! o2 o2 wj ¼ 0 ðj ¼ 1; 2; 3Þ; þ ð5Þ ox2 oz2j and k1j ¼ aj1 =sj ;
k2j ¼ aj2 ;
ð6aÞ
xj ¼ ½c11 c13 k1j s2j þ k11 k2j =s2j ¼ c44 ð1 þ k1j Þ ¼ c13 þ c33 k1j s2j k33 k2j ;
ð6bÞ
where ajm (m = 1,2) are constants defined in Eq. (18) of Chen et al. [14]. It should be noted that the general solutions given in Eq. (3) are only valid for the case when the eigenvalues sj (j = 1, 2, 3) are distinct, which is the most common case. 3. Green’s functions for a point heat source in the interior of a semi-infinite orthotropic thermoelastic plane Consider a semi-infinite orthotropic thermoelastic plane z P 0 (Fig. 1). A point heat source of strength H is applied at the point (0, h) in 2D Cartesian coordinate (x, z). The surface (z = 0) is free and thermally insulated. Based on the general solution Eq. (3), the coupled field in the semi-infinite thermoelastic plane is derived in this section. The boundary conditions on the surface (z = 0) are in the form of rz ¼ szr ¼ 0;
oh=oz ¼ 0:
ð7Þ
For future reference, following denotations are introduced: zj ¼ sj z;
hk ¼ sk h;
zjk ¼ zj þ hk ; zjk ¼ zj hk ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ z2jk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r jk ¼ x2 þ zjk ; r jk ¼
ð8Þ ðj; k ¼ 1; 2; 3Þ:
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b
o
x
a1
H
(0, h)
a2
z 3
Fig. 1. Semi-infinite thermoelastic plane applied by a point heat source of strength H.
By virtue of trial-and-error method, Green’s functions in semi-infinite plane are assumed in following form: X 3 1 3 x 1 3 x þ ðj ¼ 1; 2; 3Þ; Ajk ðz2jk x2 Þ ln r jk xzjj arctan xzjk arctan wj ¼ Aj ðz2jj x2 Þ ln r jj zjj 2 2 2 2 zjk k¼1
ð9Þ
where Aj and Ajk ðj; k ¼ 1; 2; 3Þ are twelve constants to be determined. Substitution of Eq. (9) into general solution Eq.(3) yields the expressions of coupled field as follows: 3 X x Aj xðln r jj 1Þ þ zjj arctan u¼ zjj j¼1 3 3 XX x ; Ajk xðln r jk 1Þ þ zjk arctan zjk j¼1 k¼1 3 X x w¼ sj k1j Aj zjj ðln r jj 1Þ x arctan zjj j¼1 3 X 3 X x ; sj k1j Ajk zjk ðln rjk 1Þ x arctan þ zjk j¼1 k¼1 h ¼ k23 A3 ln r 33 þ k23
3 X
A3k ln r3k ;
k¼1
rx ¼
3 X
s2j x1j Aj ln r jj
j¼1
rz ¼
3 X
s2j x1j Ajk ln r jk ;
j¼1 k¼1
xj Aj ln rjj þ
j¼1
szx ¼
3 X 3 X
3 X 3 X
xj Ajk ln r jk ;
j¼1 k¼1
3 X
sj xj Aj arctan
j¼1
3 X 3 x X x sj xj Ajk arctan : zjj z jk j¼1 k¼1
ð10Þ
Consideration of the continuity on plane z = h for w and szx yields 3 X
sj k1j Aj ¼ 0;
ð11Þ
sj xj Aj ¼ 0:
ð12Þ
j¼1 3 X j¼1
Substitution of xj in Eq. (6b) into Eq. (12) gives 3 X
c44 ð1 þ k1j Þsj Aj ¼ 0:
ð13Þ
j¼1
By virtue of Eq. (11), Eq. (13) can be simplified to 3 X
sj Aj ¼ 0:
ð14Þ
j¼1
When the mechanical and thermal equilibrium for a cylinder of a1 6 z 6 a2 (0 < a1 < h < a2) and 0 6 r 6 b are considered (Fig. 1), two additional equations can be obtained
P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
X
Z¼0:
Z
b
½rz ðx; a2 Þ rz ðx; a1 Þdx þ b
X
Heat flux ¼ H : b33
Z
a2
½szx ðb; zÞ szx ðb; zÞdz ¼ 0;
ð15aÞ
a1
Z a2 b oh oh oh oh ðx; a2 Þ ðx; a1 Þ dx b11 ðb; zÞ ðb; zÞ dz ¼ H; oz ox ox b oz a1
Z
1677
where b11 and b33 are coefficients of heat conduction along with x axis and z axis, respectively. Some useful integrals are listed as follows: Z x ln rjj dx ¼ xðln rjj 1Þ þ zjj arctan ; zjj Z x ln rjk dx ¼ xðln r jk 1Þ þ zjk arctan ; zjk Z x 1 x ; x ln r jj þ zjj arctan arctan dz ¼ zjj zjj sj Z x 1 x arctan dz ¼ ; x ln rjk þ zjk arctan zjk sj zjk ! Z Z 3 z33 X oh z3k A3 2 þ A3k 2 dx dx ¼ s3 k23 r33 oz r 3k k¼1 ! 3 X x x ; þ A3k arctan ¼ s3 k23 A3 arctan z33 z3k k¼1 ! Z Z 3 X oh x x A3k 2 dz dz ¼ k23 A3 2 þ r33 ox r 3k k¼1 ! 3 X k23 x x : A3 arctan þ A3k arctan ¼ z33 s3 z3k k¼1 It is noted that integral (16d) is not continuous at z = h, following expression should be used: Z a2 Z h Z a2 oh oh oh dz ¼ dz þ dz: ox a1 ox a1 hþ ox
ð15bÞ
ð16aÞ
ð16bÞ
ð16cÞ
ð16dÞ
ð17Þ
Substituting Eq. (10) into Eq. (15a) with using integrals (16a), (16b), one can obtain 3 X
xj Aj I1 þ
j¼1
3 X
xj
3 X
j¼1
Ajk I2 ¼ 0;
ð18Þ
k¼1
where " " z¼a2 #x¼b x¼b #z¼a2 x x xðln r jj 1Þ þ zjj arctan x ln r jj þ zjj arctan ¼ 0; I1 ¼ zjj z¼a zjj x¼b 1 x¼b z¼a1 " " z¼a2 #x¼b x¼b #z¼a2 x x xðln r jk 1Þ þ zjk arctan x ln rjk þ zjk arctan ¼ 0: I2 ¼ zjk z¼a1 zjk x¼b x¼b
ð19aÞ
ð19bÞ
z¼a1
i.e. Eqs. (18) and (15a) are satisfied automatically. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting Eq. (10) into Eq. (15b) with using s3 ¼ b11 =b33 and integrals (16c, d, 17), one can obtain A3 I3 þ
3 X
A3k I4 ¼
k¼1
k23
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b11 b33
ð20Þ
where " " z¼a2 #x¼b x¼b #z¼h " x¼b #z¼a2 x x x þ arctan þ arctan ¼ 2p; I3 ¼ arctan z33 z¼a z33 x¼b z33 x¼b 1 x¼b z¼a1 z¼hþ " x¼b #z¼a2 " z¼a2 #x¼b x x arctan arctan ¼ 0: I4 ¼ z3k x¼b z3k z¼a1 z¼a1
ð21aÞ
ð21bÞ
x¼b
Thus A3 can be determined by Eqs. (20) and (21) as follows: A3 ¼
2pk23
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : b11 b33
At last, when the couple field on surface z = 0 is considered, one has
ð22Þ
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hk ¼ sk h; z0j ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 zjk ¼ hk ; r jk ¼ x2 þ hk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zjk ¼ hk ; r jk ¼ x2 þ h2k : zj ¼ 0;
ð23Þ
Substituting Eq. (10) into boundary condition (7) with using s3 ¼ sj xj Aj þ
3 X
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b11 =b33 and Eq. (23), one can obtain
sk xk Akj ¼ 0;
ð24Þ
k¼1
xj Aj þ
3 X
xk Akj ¼ 0;
ð25Þ
k¼1
A3 A33 ¼ 0;
A3k ¼ 0;
ðk ¼ 1; 2Þ;
ð26Þ
where j = 1, 2, 3. Thus, twelve constants Aj and Ajk ðj; k ¼ 1; 2; 3Þ can be determined by twelve equations including Eqs. (11), (14), (22), (24), (25), (26). Concerning the problems of point forces Px, Pz in x, z direction in semi-infinite orthotropic thermoelastic plane, because the foundational Eqs. (1) and (2) are single-direction coupling, i.e. the thermal loading can change elastic field, while on the contrary, the mechanical loading cannot change thermal field (h = 0), so the corresponding solutions under mechanical loading degenerate to those when thermal effects are not considered. The corresponding solutions are listed in Appendix. 4. Numerical results The contours of temperature increment and stress of coupled field in semi-infinite orthotropic thermoelastic plane induced by a point heat source of strength H are evaluated numerically and plotted in Figs. 2–5. The material properties listed in Table 1 are taken from hexagonal zinc [16]. In additions, the following non-dimensional components are used in figures: ri sij ; skl ¼ ; c33 ar T 0 c33 ar T 0 h x z ; n ¼ ; f ¼ ; ði; j ¼ x; z; k; l ¼ n; fÞ; #¼ T0 r1 r1
rk ¼
ð27Þ
where r1 is a non-zero dimension, ar and T0 are thermal expansion coefficient and reference temperature, respectively. And Eq. (22) should be rewritten in following non-dimensional form: A3 ¼
d ; 2ps3 k23
ð28Þ
0
10
-1 0
0
-2 5 -3 0
-2 0
1
ζ
2
5 -3
0 -4 5 -4
3
0 -5 5 -5
4
0 -6
5 5 -6
6
0
1
2
3
4
5
6
ξ Fig. 2. Contour of non-dimensional temperature increment # 102 in a semi-infinite plane under point heat source of strength d = 1 acted at (0, 1).
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P.-F. Hou et al. / Applied Mathematical Modelling 33 (2009) 1674–1682
0 20
40
0
-2 0
1
-4 0
-70
-50 8 -6 -7 0
ζ
2
-6 0
-7 0
3
-6 8 -7 5
-8 0
4
-8 5
5 -9 0
6
0
1
2
3
4
5
6
ξ Fig. 3. Contour of non-dimensional stress rn 102 in a semi-infinite plane under point heat source of strength d = 1 acted at (0, 1).
0
0
0. 1 0.5
1 -3 0
1
1. 5
0 -2
3
6 -1
3
2.3 2
-1 4
ζ
2
4
-2
-6
-8
-1 0
-1 2
0.1
-1
-4
4
1. 5
1
0 0 .5
5
6
0
1
2
3
4
5
6
ξ Fig. 4. Contour of non-dimensional stress rf 102 in an infinite plane under point heat source of strength d = 1 acted at (0, 1).
where d is a non-dimensional strength of point heat source as follows: d¼
H ; r 1 T 0 b33
ð29Þ
and here let d = 1 acted at (0, 1). On the basis of the contours plotted above, following conclusions can be drawn: (1) Fig. 2 shows that the contours of temperature increment #(h) for Hexagonal zinc are similar to normal circle. This is because the heat conduction coefficients b11 and b33 are equal for Hexagonal zinc (Table 1).
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0
1
2
4 16
1
6
12
8
10
2
8.5
-4
ζ
9
8.5
-2
3
4
8
6
-1 4
5 2
0
1
2
1
6
0
-0 .5
3
4
5
6
ξ Fig. 5. Contour of non-dimensional stress sfn 102 in a semi-infinite plane under point heat source of strength d = 1 acted at (0, 1).
Table 1 Material properties of hexagonal zinc [16] Elastic constants (109N m2) c13 36.2 Heat conduction coefficients (W K1 m1)
c11 162.8 Thermal moduli (105N K1 m2)
c33 62.7 Thermal expansion coefficient (106 K1)
c44 38.5
k11
k33
b11
b33
ar
az
17.9839
13.8367
124
124
5.818
15.35
(2) Figs. 3–5 show that there is a zero common tangent for stresses rn(rx), rf(rz) and szx(sfn), respectively. They are positive above these zero common tangents and are negative on the other side. All stresses are singular at point (0, h/r1 = 1) at which the point heat source is located. This is an important contribution for Green’s functions. In some cases, Green’s functions are also called singular solutions. In additions, one can find that the grads of all stresses are very large near the point heat source, which should attract our attentions in practice. (3) Figs. 4 and 5 show that stresses rf(rz) and szx(sfn) tend to zero near the surface f = 0 and satisfy the boundary condition (7). We here give a interesting imagine that when h/r1 tend to larger and larger, the effect of boundary condition (7) on stresses in the vicinity of the load is less and the obtained solution near the load for semi-infinite plane will tend to the solution of infinite plane. So boundary condition (7) is one of the most important attribution factors for the distinctive features of the profiles in semi-infinite and infinite plane. 5. Conclusions By virtue of the compact 2D general solution of orthotropic thermoelastic materials, which is parallel to those of Chen et al. [14], three harmonic functions wj (j = 1, 2, 3) in Eq. (9) are constructed and the corresponding coupled field for a point heat source acted in the interior of a semi-infinite orthotropic thermoelastic plane is obtained. For the sake of concise expressions and convenient derivations, point heat source is loaded at z axis in this paper. The solution in case of point heat source loaded at arbitrary point can be easily generalized from the obtained solution by coordination parallel transform. Because all the components are expressed in terms of elementary functions, it is convenient to use them. Typical numerical examples are presented. All these show that the general solution of Chen et al. [14] is an important foundation for the studies of steady problems thermoelastic materials. It should be noted that the obtained Green’s functions are valid for distinct eigenvalues sj (j = 1, 2, 3), which is the most common case for 2D problems of orthotropic materials. The general solutions and Green’s functions for cases of multiple eigenvalues, such as s1 – s2 = s3 and s1 = s2 = s3 are different from those in this paper. So the cor-
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responding solutions for isotropic material with s1 = s2 = s3 = 1 cannot be obtained by direct degeneration from the obtained solutions. Acknowledgement The authors thankfully acknowledge the financial support from the National Natural Science Foundation of China, Foundation in Hunan Province, SRG grant of City University of Hong Kong, National 985 Special Foundation of China and the reviewers for their constructive comments, which lead to improvements of the earlier version of this article. Appendix A. Solutions for other point loadings in semi-infinite orthotropic thermoelastic material A.1. Solution to the problem of point force Pz Introduce the following harmonic functions: X 3 x x þ wj ¼ Bj zjj ðln rjj 1Þ x arctan Bjk zjk ðln r jk 1Þ x arctan zjj zjk k¼1 where zjj ; r jj ; zjk and rjk are functions defined in Eq. (8), Bj and Bjk
ðj ¼ 1; 2Þ;
w3 ¼ 0;
ðj; k ¼ 1; 2Þ are constants to be determined by
Bj ¼ aj Pz ; ðj ¼ 1; 2Þ; " # " #1 " # a1 0 1 1 1 ¼ ; 2p x1 x2 a2 1
xj Bj
2 X
xk Bkj ¼ 0;
ðA:1Þ
ðA:2aÞ ðA:2bÞ
ðj ¼ 1; 2Þ;
ðA:3Þ
k¼1
sj xj Bj þ
2 X
sk xk Bkj ¼ 0;
ðj ¼ 1; 2Þ:
ðA:4Þ
k¼1
A.2. Solution to the problem of point force Px Introduce the following harmonic functions: X 2 x x þ wj ¼ C j xðln rjj 1Þ þ zjj arctan C jk xðln r jk 1Þ þ zjk arctan zjj zjk k¼1
ðj ¼ 1; 2Þ;
w3 ¼ 0;
ðA:5Þ
where zjj ; r jj ; zjk and rjk are functions defined in Eq. (8), Cj and Cjk (j,k = 1, 2) are constants to be determined by C j ¼ kj P x ; ðj ¼ 1; 2Þ; " # " #1 " # k1 0 1 s1 k11 s2 k12 ¼ ; 2p s1 k21 s2 k22 k2 1
xj C j þ
2 X
xk C kj ¼ 0;
ðj ¼ 1; 2Þ;
ðA:6aÞ ðA:6bÞ
ðA:7Þ
k¼1
sj xj C j
2 X
sk xk C kj ¼ 0;
ðj ¼ 1; 2Þ:
ðA:8Þ
k¼1
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