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Three-dimensional Green’s functions for transversely isotropic thermoelastic bimaterials
International Journal of Solids and Structures 45 (2008) 6100–6113
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Three-dimensional Green’s functions for transversely isotropic thermoelastic bimaterials Peng-Fei Hou a,*, Andrew Y.T. Leung b, Yong-Jun He c a b c
Department of Engineering Mechanics, Hunan University, Changsha 410082, PR China Department of Building and Construction, City University of Hong Kong, Hong Kong, PR China College of Civil Engineering, Hunan University, Changsha 410082, PR China
a r t i c l e
i n f o
Article history: Received 15 November 2007 Received in revised form 13 July 2008 Available online 7 August 2008
Keywords: Green’s function Transversely isotropic Thermoelastic Bimaterial
a b s t r a c t Green’s functions for transversely isotropic thermoelastic biomaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoelastic material in terms of harmonic functions and introduce six new harmonic functions. The three-dimensional Green’s function having a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of temperature and stress distributions at the interface. The temperature contours are normal to the interface for the isotropic material but not for the orthotropic one. The normal stress contours are parallel to the interface at the boundary in the isotropic region only and shear failure is most likely at the heat source due to the highly degenerated direction of shear stress contours. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Green’s functions play an important role in both applied and theoretical studies on the physics of solids. They are basic building blocks of a lot of further works. Green’s functions can be used to construct many analytical solutions of practical engineering problems by superposition and are very important in the boundary element method as well as the study of cracks, defects and inclusions. For isotropic materials, the Kelvin Green’s function is well-known (Banerjee and Butterfield, 1981). For transversely isotropic materials, Lifshitz and Rozentsveig (1947) and Lejcek (1969) derived Green’s functions using the Fourier transform method. Elliott (1948), Kroner (1953) and Willis (1965) obtained them using the direct method and Sveklo (1969) found them using the complex method. Pan and Chou (1976) solved Green’s function in the form of compact elementary functions. For anisotropic materials, Pan and Yuan (2000) and Pan (2003) obtained the three-dimensional Green’s functions for biomaterials with perfect and imperfect interfaces, respectively. The thermal effects are not considered in the above works. Sharma (1958) studied Green’s functions of transversely isotropic thermoelastic materials in integral form. Yu et al. (1992) found the solution for a point heat source in isotropic thermoelastic bimaterials. Berger and Tewary (2001) and Kattis et al. (2004) obtained the two-dimensional Green’s functions for anisotropic thermoelastic materials. Chen et al. (2004) derived a compact three-dimensional general solution for transversely isotropic thermoelastic materials. Based on this general solution, Hou et al. (2008) constructed Green’s function for infinite and semi-infinite transversely isotropic thermoelastic materials. As a further extension, the three-dimensional Green’s function for a concentrated heat source in a transversely isotropic thermoelastic bimaterial is investigated in this paper. Only steady state is considered. For completeness, the general solution
* Corresponding author. Tel./fax: +86 731 8822330. E-mail address: [email protected] (P.-F. Hou). 0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2008.07.022
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
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of Chen et al. (2004) is summarized in Section 2. In Section 3, six newly found harmonic functions are constructed in terms of elementary functions with undetermined constants. The unique thermoelastic field can be obtained by substituting these functions into the general solutions after determining the constants by the compatibility and equilibrium conditions at interface. Numerical examples are presented in Section 4. The contours of temperature increment and stress components are shown graphically. Finally, the paper is concluded in Section 5. 2. General solutions We summarize the general solutions of Chen et al. (2004) for use in Section 3. Hou et al. (2008) considered a semi-infinite transversely isotropic thermoelastic material, which is isotropic in the xy-plane. The result will be extended to two such semi-infinite materials joining at the interface z = 0 in Section 3. In the Cartesian coordinate (x, y, z), the constitutive relations are
ov ow þ ; oz oy ou ow ; szx ¼ c44 þ oz ox ou ov ; sxy ¼ c66 þ oy ox
ou ov ow þ c12 þ c13 k11 h; ox oy oz ou ov ow ry ¼ c12 þ c11 þ c13 k11 h; ox oy oz ou ov ow þ c33 rz ¼ c13 þ k33 h; ox oy oz
rx ¼ c11
syz ¼ c44
ð1Þ
where u, v and w are the components of the displacement vector in the x, y and z directions, respectively; rij are the stress components and h is the temperature increment; cij and kii are the elastic and thermal moduli, respectively. c66 = (c11 c12)/2 is held for transversely isotropic thermoelastic materials. In the absence of body forces, the mechanical equilibrium equations are
orx osxy oszx þ þ ¼ 0; ox oy oz osxy ory osyz þ þ ¼ 0; ox oy oz oszx osyz orz þ þ ¼ 0: ox oy oz
ð2aÞ
The heat equilibrium equation is
b11
! o2 h o2 h o2 h þ b ¼ 0; þ 33 ox2 oy2 o2 z
ð2bÞ
where bii (i = 1, 3) are coefficients of heat conduction and b22 = b11 when the body is isotropy in the xy-plane. Chen et al. (2004) gave the general solutions to Eq. (1), (2) as follows:
U ¼ K iw0 þ
3 X
! wj ;
w¼
j¼1
r1 ¼ 2
sj k1j
j¼1
owj ; ozj
h ¼ k23
o2 w3 ; oz23
ð3aÞ
3 3 X X o2 wj ðc66 xj s2j Þ 2 ¼ 2 ðc66 xj s2j ÞDwj ; oz j j¼1 j¼1
r2 ¼ 2c66 K2 ðiw0 þ
3 X
wj Þ;
j¼1
sz
3 X
rz ¼
! 3 owj ow0 X ; ¼ K s0 c44 i þ sj xj oz0 ozj j¼1
3 X j¼1
xj
3 X o2 wj ¼ xj Dwj ; 2 ozj j¼1
ð3bÞ
where the quantities U, r1, r2, sz can be defined in the Cartesian coordinate (x, y, z) and the cylindrical coordinate (r, /, z) in the complex forms as follows:
U ¼ u þ iv ¼ ei/ ður þ iu/ Þ;
r1 ¼ rx þ ry ¼ rr þ r/ ; r2 ¼ rx ry þ 2isxy ¼ e2i/ ðrr r/ þ 2isr/ Þ; sz ¼ sxz þ isyz ¼ ei/ ðszr þ is/z Þ;
ð4Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where i ¼ 1. In additions, for simplicity, we let zj = sjz (j = 0, 1, 2, 3), where s0 ¼ c66 =c44 , s3 ¼ b11 =b33 and s1 and s2 are 4 2 the two eigenvalues of the fourth degree equation a0s b0 s + c0 = 0 that is Eq. (11) of Chen et al. (2004) in which a0 ¼ c33 c44 ; b0 ¼ c11 c33 þ c244 ðc13 þ c44 Þ2 ; c0 ¼ c11 c44 . Let wj (j = 0, 1, 2, 3) be the solutions of the following harmonic equations:
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! o2 wj ¼ 0 ðj ¼ 0; 1; 2; 3Þ; oz2j
Dþ
ð5Þ
where
D¼
o2 o2 þ 2 2 ox oy
in Cartesian coordinates ðx; y; zÞ;
2
D¼
ð6aÞ
2
o o o þ þ or 2 ror r2 o/2
in cylindrical coordinates ðr; /; zÞ:
ð6bÞ
In additions
xj ¼ c44 ð1 þ k1j Þ ¼ c13 þ c33 k1j s2j k33 k2j ¼ ðc11 c13 k1j s2j þ k11 k2j Þ=s2j k1j ¼ aj1 =sj ;
k2j ¼ aj2 ;
K¼
ð7Þ
o o þi ; ox oy
where ajm (m = 1, 2) are the constants defined in Eq. (18) of Chen et al. (2004), that is, ai1 = ki2si /ki1, a12 = a22 = 0, a32 = k33 /k31, and kij ¼ aj þ bj s2i ; k33 ¼ a0 s43 b0 s23 þ c0 . It should be noted that the general solutions given in Eq. (3) are correct when eigenvalues sj (j = 1, 2, 3) are distinct. For non-torsional axisymmetric problem, w0 = 0 and wj (j = 1, 2, 3) are independent of /, so that u/ = 0 and s/z = sr/ = 0. The general solution in the cylindrical coordinate (r, /, z) can be simplified to
ur ¼
3 X owj ; or j¼1
rr ¼ 2c66 r/ ¼ 2c66 rz ¼
3 X j¼1
w¼
3 X
sj k1j
j¼1
owj ; ozj
h ¼ k23
o2 w3 : oz23
ð8aÞ
3 3 X o2 wj 1 owj X s2j xj 2 ; r or ozj j¼1 j¼1
3 3 X o2 wj 1 owj X ðs2j xj 2c66 Þ 2 ; r or ozj j¼1 j¼1
xj
o2 wj ; oz2j
szr ¼
3 X
s j xj
j¼1
o2 wj : orozj
ð8bÞ
For the torsional axisymmetric problem, wj = 0 (j = 1, 2, 3) and w0 is independent of /, so that ur = uz = 0, h = 0 and rr = r/ = rz = srz = 0. The general solution can be simplified to the following form:
u/ ¼
ow0 ; or
sr/ ¼ 2c66
! 1 o2 o2 w ; þ 2 oz20 or2 0
s/z ¼ s0 c44
o2 w0 : oroz0
ð9aÞ
The general solution (8) will be used in Section 3 and the general solutions (3) and (8) will be used in Appendix A. 3. Green’s functions When a concentrated heat source of strength H is applied in the interior of one of the two transversely isotropic thermoelastic semi-infinite materials joining at the interface z = 0 parallel to the plane of isotropy (Fig. 1), either the cylindrical coordinates (r, /, z) or the Cartesian coordinates (x, y, z) can be chosen such that the r/-plane or xy-plane lies in the interface. The concentrated heat source is applied at the point (0, 0, h) in the Cartesian coordinates. The coupled thermoelastic field is derived in this section. This is a non-torsional axisymmetric problem such that u/ = 0 and s/z = sr/ = 0. The thermoelastic material in the two half-spaces z P 0 and z 6 0 are assumed to be perfectly bonded. The compatibility conditions at the interface (z = 0) are in form of
ur ¼ u0r ; 0 zr ;
szr ¼ s h ¼ h0 ;
w ¼ w0 ;
rz ¼ r0z ; b33 oh=oz ¼ b033 oh0 =oz;
ð10aÞ ð10bÞ ð10cÞ
where the primed quantities refer to the variables in the lower half-space z 6 0 and the un-primed quantities refer to those in the upper half-space z P 0. To simplify notations, the following quantities are introduced:
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
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Fig. 1. Thermoelastic bimaterial having a point heat source of strength H in steady state.
zj ¼ sj z;
z0j ¼ s0j z; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rjk ¼ r2 þ z2jk ¼ x2 þ y2 þ z2jk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rjk ¼ r2 þ z2jk ¼ x2 þ y2 þ z2jk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0jk ¼ r2 þ z02 x2 þ y2 þ z02 jk ¼ jk ;
hk ¼ sk h;
zjk ¼ zj hk ; zjk ¼ zj þ hk ; z0jk ¼ z0j hk ; 2
ð11Þ
2
b ; d1j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b þ hj þ hj
b d2j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 b þ hj b þ hj þ hj
ðj; k ¼ 1; 2; 3Þ; where s0j are the eigenvalues of material in the lower half-space z 6 0. We shall show later that Green’s functions in the half-space z P 0 can be assumed in following form:
w0 ¼ 0; wj ¼ Aj ½signðz hÞzjj ln Rjj Rjj þ
3 X
Ajk ½zjk ln Rjk
ð12Þ
Rjk ðj ¼ 1; 2; 3Þ;
k¼1
where Aj and Ajk ðj; k ¼ 1; 2; 3Þ are the twelve constants to be determined, sign() is the signum function, and
Rjj ¼ Rjj þ signðz hÞzjj ;
Rjk ¼ Rjk þ zjk
ðj; k ¼ 1; 2; 3Þ:
The coupled field is found by the substitution of Eq. (12) into general solution (8) as follows:
ur ¼
3 X j¼1
w¼
3 X j¼1
h ¼ k23
! r Aj þ Ajk ; Rjk Rjj k¼1 3 X
r
sj k1j Aj sign ðz hÞ ln Rjj þ
rr ¼ 2c66
j¼1
r/ ¼ 2c66
3 X j¼1
3 1 X 1 Aj þ Ajk R Rjj k¼1 jk
Aj
1 Rjj
þ
3 X k¼1
Ajk
! Ajk ln Rjk ;
k¼1
! 3 X 1 1 ; A3 þ A3k R3k R33 k¼1 3 X
3 X
!
1 Rjk
!
3 X j¼1
s2j
xj
! 3 1 X 1 ; Aj þ Ajk Rjk Rjj k¼1
! 3 3 X 1 X 1 ; s2j xj 2c66 Aj þ Ajk Rjk Rjj k¼1 j¼1
ð13Þ
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rz ¼
3 X
xj Aj
j¼1
szr ¼
3 X
1 Rjj
3 X
þ
Ajk
k¼1
! 1 ; Rjk
! r : Aj sign ðz hÞ þ Ajk Rjk Rjk Rjj Rjj k¼1
s j xj
j¼1
ð14Þ
3 X
r
In the lower half-space z 6 0, assume
w00 ¼ 0;
w0j ¼
3 X
A0jk ðz0jk ln R0jk þ R0jk Þ ðj ¼ 1; 2; 3Þ;
ð15Þ
k¼1
where A0jk ðj; k ¼ 1; 2; 3Þ are the nine constants to be determined and
R0jk ¼ R0jk z0jk
ðj; k ¼ 1; 2; 3Þ:
ð16Þ
Substitution of Eq. (15) into general solution (8) yields 3 X 3 X
u0r ¼
A0jk
j¼1 k¼1
w0 ¼
3 X
0
s0j k1j
3 X
j¼1
h0 ¼
r ; R0jk A0jk ln R0jk ;
k¼1
0 k23
3 X
A03k
k¼1
r0r ¼ 2c066
1 ; R03k
3 X 3 X
A0jk
j¼1 k¼1
r0/ ¼ 2c066
A0jk
3 3 X 1 X 1 þ s02 x0 2c066 A0jk 0 ; R0jk j¼1 j j Rjk k¼1
A0jk
1 ; R0jk
A0jk
r : R0jk R0jk
3 X 3 X j¼1 k¼1
r0z ¼
3 X
x0j
j¼1
s0zr ¼
3 X
s0j x0j
j¼1
3 X k¼1 3 X
3 X 1 1 3 02 0 A0jk 0 ; 0 þ umj¼1 sj xj Rjk R jk k¼1
k¼1
ð17Þ
The compatibility conditions on the plane z = h for w and szr given in Eq. (14) yield 3 X
sj k1j Aj ¼ 0:
ð18Þ
sj xj Aj ¼ 0:
ð19Þ
j¼1 3 X j¼1
Substitution of xj in Eq. (7) into Eq. (19) gives 3 X
sj ð1 þ k1j ÞAj ¼ 0:
ð20Þ
j¼1
From Eqs. (18), (20) can be simplified to 3 X
sj Aj ¼ 0:
ð21Þ
j¼1
The mechanical and the thermal equilibriums for a cylinder of finite length a1 6 z 6 a2 (a1 < h < a2) and 0 6 r 6 b shown in Fig. 2 give two additional equations separately for Fig. 2a and b below. For a1 > 0 shown in Fig. 2a
X X
force in z direction ¼ 0 : 2p Heat flux ¼ H : 2pb33
Z 0
b
Z 0
b
½rz ðr; a2 Þ rz ðr; a1 Þrdr þ 2pb
Z
a2
szr ðb; zÞdz ¼ 0;
ð22aÞ
a1
Z a2 oh oh oh ðr; a2 Þ ðr; a1 Þ rdr 2pbb11 ðb; zÞdz ¼ H; oz oz a1 or
where b11 and b33 are the coefficients of heat conduction in half-space z P 0 along the r and z axes, respectively.
ð22bÞ
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
6105
Fig. 2. A cylinder containing the point heat source of strength H.
For a1 < 0 shown in Fig. 2b
X
Z b ½rz ðr; a2 Þ r0z ðr; a1 Þrdr force in z direction ¼ 0 : 2p 0 Z 0 Z a2 s0zr ðb; zÞdz þ szr ðb; zÞdz ¼ 0; þ 2pb b
a1
ð23aÞ
0
X oh oh0 Heat flux ¼ H : 2p b33 ðr; a2 Þ b033 ðr; a1 Þ rdr oz oz 0 Z 0 0 Z a2 oh oh 0 2pb b11 ðb; zÞdz ¼ H; ðb; zÞdz þ b11 or 0 a1 or Z
ð23bÞ
where b011 and b033 are the coefficients of heat conduction in the lower half-space z 6 0 along the r and z axes, respectively. The following integration formulae are useful:
Z
1 Rjj
Z
Z
rdr ¼ Rjj ;
sign ðz hÞ
Z
1 rdr ¼ Rjk ; Rjk
r Rjj Rjj
dz ¼
Z
1 rdr ¼ R0jk ; R0jk
ð24aÞ
1 r ; sj Rjj
Z r 1 r r 1 r dz ¼ 0 0 ; dz ¼ ; Rjk Rjk sj Rjk sj Rjk R0jk R0jk ! Z Z 3 z33 X oh z3k A3k 3 rdr rdr ¼ s3 k23 A3 3 þ oz R33 k¼1 R3k " # 3 z33 X z3k ¼ s3 k23 A3 ; þ A3k R3k R33 k¼1 ! Z Z 3 X oh r r A3 3 þ A3k 3 dz dz ¼ k23 or R33 k¼1 R3k ! 3 X k23 r r ¼ ; A3 sign ðz hÞ þ A 3k R3k R3k s3 R33 R33 k¼1 Z Z 0 3 3 X X z3k z0 oh0 0 0 A03k rdr ¼ s03 k23 A03k 3k ; rdr ¼ s03 k23 03 oz R03k R3k k¼1 k¼1 Z Z 0 3 3 X oh0 r k X r 0 A03k dz ¼ 23 A03k 0 0 : dz ¼ k23 0 03 or s R R 3 3k R3k k¼1 k¼1 3k
ð24bÞ
ð24cÞ
ð24dÞ ð24eÞ ð24fÞ
It is noted that integral (24d) is not continuous at z = h for the signum function sign(z h), so the integration should be done in two parts:
Z
a2
a1
oh dz ¼ or
Z
h
a1
oh dz þ or
Z
a2
hþ
oh dz: or
ð25Þ
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For a1 > 0, substituting Eq. (14) into Eq. (22a) and using integrals (24a, b), one can obtain 3 X
xj Aj I1 þ
j¼1
3 X
xj
j¼1
3 X
Ajk I2 ¼ 0;
ð26Þ
k¼1
where
r¼b I1 ¼ Rjj ðr; a2 Þ Rjj ðr; a1 Þ r¼0
"
b
#z¼a2
2
¼ 0; Rjj ðb; zÞ z¼a 1 " #z¼a2 2 b ¼ 0; I2 ¼ ½Rjk ðr; a2 Þ Rjk ðr; a1 Þr¼b r¼0 Rjk ðb; zÞ
ð27aÞ
ð27bÞ
z¼a1
i.e. Eqs. (26, 22a) are satisfied automatically. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting Eq. (14) into Eq. (22b) and using s3 ¼ b11 =b33 as well as integrals (24c, d), (25), one can obtain
A3 I3
3 X
A3k I4 ¼
2pk23
k¼1
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b11 b33
ð28Þ
where
I3 ¼
" s3 a2 h3
s3 a1 h3
#r¼b
"
sign ðz hÞb
2
#z¼h
"
2
sign ðz hÞb
#z¼a2
þ þ R33 ðr; a1 Þ r¼0 R33 ðb; zÞR33 ðb; zÞ z¼a R33 ðb; zÞR33 ðb; zÞ 1 #z¼a2 r¼b " 2 s3 a2 þ hk s3 a1 þ hk b þ ¼ 0: I4 ¼ R3k ðr; a2 Þ R3k ðr; a1 Þ r¼0 R3k ðb; zÞR3k ðb; zÞ R33 ðr; a2 Þ
¼ 2
ð29aÞ
z¼hþ
ð29bÞ
z¼a1
Thus A3 can be determined by Eqs. (28, 29) as follows:
A3 ¼
4pk23
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : b11 b33
ð30Þ
For a1 < 0, substituting Eqs. (14), (17) into Eq. (23a) and using integrals (24a, b), one can obtain 3 X
xj Aj I5 þ
j¼1
3 X
xj
j¼1
3 X
Ajk I6 þ
3 X
x0j
j¼1
k¼1
3 X
A0jk I7 ¼ 0;
ð31Þ
k¼1
where
"
#z¼a2 2 b I5 ¼ ¼ d1j ; Rjj ðb; zÞ z¼0 " #z¼a2 2 b I6 ¼ ½Rjk ðr; a2 Þr¼b ¼ d1k ; r¼0 Rjk ðb; zÞ z¼0 " #z¼0 2 b r¼b 0 I7 ¼ ½Rjk ðr; a1 Þr¼0 þ 0 ¼ d1k : Rjk ðb; zÞ ½Rjj ðr; a2 Þr¼b r¼0
ð32aÞ
ð32bÞ
ð32cÞ
z¼a1
From, (32) (31) can be rewritten to 3 X
xj Aj d1j þ
j¼1
3 X
xj
3 X
j¼1
Ajk d1k þ
3 X
x0j
j¼1
k¼1
3 X
A0jk d1k ¼ 0;
ð33aÞ
k¼1
which can be simplified to 3 X
xj Aj þ
j¼1
3 X
xk Akj þ
k¼1
3 X
!
x0k A0kj d1j ¼ 0:
Substituting Eqs. (14), (17) into Eq. (23b) and using s3 ¼ can obtain
A3 I8
3 X k¼1
A3k I9
ð33bÞ
k¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b11 =b33 , s3 ¼ b011 =b033 as well as integrals (24c, d, e, f), (25), one
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 k23 b011 b033 X H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A03k I10 ¼ k23 b11 b33 k¼1 2pk23 b11 b33
ð34Þ
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
6107
where
" #r¼b " #z¼h " #z¼a2 2 2 s3 a2 h3 sign ðz hÞb sign ðz hÞb I8 ¼ þ þ ¼ 2 þ d23 ; R33 ðr; a2 Þ r¼0 R33 ðb; zÞR33 ðb; zÞ z¼0 R33 ðb; zÞR33 ðb; zÞ z¼hþ #z¼a2 r¼b " 2 s3 a2 þ hk b I9 ¼ þ ¼ d2k ; R3k ðr; a2 Þ r¼0 R3k ðb; zÞR3k ðb; zÞ z¼0 #z¼0 0 r¼b " 2 s3 a1 hk b I10 ¼ 0 þ 0 ¼ d2k : R3k ðr; a1 Þ r¼0 R3k ðb; zÞR03k ðb; zÞ
ð35aÞ
ð35bÞ
ð35cÞ
z¼a1
Using Eqs. (35), (34) can be rewritten to
2A3 A3 d23 þ
3 X
A3k d2k
k¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 k23 b011 b033 X H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A03k d2k ¼ k23 b11 b33 k¼1 2pk23 b11 b33
ð36Þ
which can be simplified to
! ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 2 X k23 b011 b033 0 k23 b011 b033 0 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A33 d23 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A3k d2k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2A3 A3 A33 þ A3k 2pk23 b11 b33 k23 b11 b33 k23 b11 b33 k¼1
ð37Þ
Finally, when the couple field in interface z = 0 is considered, one has
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r 2 þ hj ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Rjk ¼ r 2 þ hk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R0jk ¼ r 2 þ hk ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r 2 þ hj þ hj ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Rjk ¼ r 2 þ hk þ hk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R0jk ¼ r 2 þ hk þ hk :
Rjj ¼
Rjj ¼
ð38Þ
Substituting Eqs. (14), (17) into the continuous condition (10) and using s3 ¼ obtains
Aj þ
3 X
Akj ¼
k¼1
3 X
A0kj ;
ð39Þ
k¼1
sj k1j Aj þ
3 X
sk k1k Akj ¼
k¼1
sj xj Aj þ
3 X
3 X
3 X
0
s0k k1k A0kj ;
ð40Þ
s0k x0k A0kj ;
ð41Þ
k¼1
sk xk Akj ¼
k¼1
xj Aj þ
3 X k¼1
xk Akj ¼
k¼1
3 X
x0k A0kj ;
ð42Þ
k¼1 0
0
k23 0 k A ; A3k ¼ 23 A03k ðk ¼ 1; 2Þ; k23 33 k23 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0 0 k23 b011 b033 0 k b011 b033 0 ffi A3k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A33 ; A3k ¼ 23 pffiffiffiffiffiffiffiffiffiffiffiffiffi k23 b11 b33 k23 b11 b33
A3 þ A33 ¼ A3 A33
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b11 =b33 , s3 ¼ b011 =b033 as well as Eq. (38), one
ð43Þ ðk ¼ 1; 2Þ;
ð44Þ
where j = 1, 2, 3. Substituting Eqs. (42), (44) into Eqs. (33b), (37), respectively, one can found that Eqs. (33b), (23a) are satisfied automatically and Eq. (37) degenerates to Eq. (30). Thus, A3 can be determined by Eq. (30), Aj ðj ¼ 1; 2Þ can be determined by Eqs. (18), (21), A3k and A03k ðk ¼ 1; 2; 3Þ can be determined by Eqs. (43), (44) as follows:
A3k ¼ A03k ¼ 0 ðk ¼ 1; 2Þ; ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi b11 b33 b011 b033 ffi A3 ; A33 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi b11 b33 þ b011 b033
A033
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k23 b b33 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A3 : b11 b33 þ b011 b033 k23
ð45aÞ ð45bÞ
Finally, the other twelve constants Ajk and A0jk ðj ¼ 1; 2; k ¼ 1; 2; 3Þ can be determined by the twelve equations listed in Eqs. (39)–(42).
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If the thermoelastic materials in two half-spaces are in smooth contact, the tangential stresses vanish and the radial displacement is not continuous at the interface. Thus the compatibility conditions at the interface are
w ¼ w0 ;
ð46aÞ
szr ¼ s0zr ¼ 0; rz ¼ r0z ;
ð46bÞ ð46cÞ
h ¼ h0 ;
b33 oh=oz ¼ b033 oh0 =oz:
Following the same approach as above, the constants Aj ; Ajk and A0jk can be determined by cancelling Eq. (39) and replacing Eq. (41) with
sj xj Aj þ
3 X
sk xk Akj ¼ 0;
ð47aÞ
k¼1 3 X
s0k x0k A0kj ¼ 0:
ð47bÞ
k¼1
Concerning the problems of point forces Px, Py, Pz in x, y, z direction, respectively, the corresponding Green’s functions are similar to those when thermal effects are not considered. This is because the foundational Eq. (1), (2) are coupling in single-direction, i.e. the thermal loading can change elastic field but the mechanical loadings cannot change thermal field (h = 0). The corresponding solutions are listed in Appendix A. 4. Numerical results Consider the two half-spaces z P 0 and z 6 0 as examples. The material properties for z P 0 are taken as hexagonal zinc (Sharma and Sharma, 2002) and those for z 6 0 are graphite (Noda et al., 1992) shown in Appendix B. The contours of temperature increment and stress components induced by a point heat source of strength H are evaluated numerically and plotted in Figs. 3–7. The following non-dimensional components are used in the figures:
h ri sij ; rk ¼ ; skl ¼ ; T0 c33 ar T 0 c33 ar T 0 r z n¼ ; f¼ ði; j ¼ r; /; z; k; l ¼ n; 1; fÞ; r1 r1 #¼
ð48Þ
where r1 is a non-zero dimension, ar and T0 are the thermal expansion coefficient and reference temperature, respectively. Eq. (30) can be rewritten in non-dimensional form:
Fig. 3. Non-dimensional Green’s temperature increment contours in units of # 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/ r1 = 1).
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
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Fig. 4. Non-dimensional Green’s stress contours in units of rn 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
Fig. 5. Non-dimensional Green’s stress contours in units of r1 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
A3 ¼
d ; 4ps3 k23
ð49Þ
where d is a non-dimensional strength of a concentrated heat source given by
d¼
H : r 1 T 0 b33
ð50Þ
Here let d = 1 positioned at point (0,0,h/r1 = 1). It is observed that all components tend to zero in the far field and are singular at the point of application of the heat source (0,0,h/r1 = 1). Fig. 3 shows that the contours of temperature increment #(h) for Hexagonal zinc in the lower half-space
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Fig. 6. Non-dimensional Green’s stress contour in units of rf 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
Fig. 7. Non-dimensional Green’s stress contour in units of sfn 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
z 6 0 are approximately circular, while those for graphite in the upper half-space z P 0 are approximately elliptical. This is because the heat conduction coefficients b11 and b33 for Hexagonal zinc are equal and those for graphite are different (Appendix B). Since the domains of interest are semi-infinite, the boundary layer effects are not obvious except at the interface. The discontinuous gradients of the contour lines at the interface are caused by the boundary effects. The temperature contour seems to be orthogonal to the interface for the isotropic material but it is inclining to the interface when the material is not isotropic. It is also observed that the contours change rapidly near the point source but are gradual at distance away from it.
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Fig. 4 shows the contours of the normal stress rn(rr) in the Cartesian coordinates when a non-dimensional heat source of strength d = 1 is applied at point (0, 0, h/r1 = 1). For stress near the source, the contours are essentially close-loops in the hexagonal zinc region. However, when near the interface, the contour of the normal stress rn(rr) = 10, say, breaks into two and change directions due to the boundary effects. The contours in the hexagonal zinc region tend to be tangential to the interfacial line but those in the graphite region are not obviously tangential. It is observed that the contours are rather dense near the point (1, 0) that will be difficult to simulate using numerical methods such as finite element where very fine meshes are required. The normal stresses are not continuous at the interface. The strengths of normal stresses in the upper half-space z P 0 are larger than those in lower half-space z 6 0 in general. The stress contours for the normal stress r1 (r/) are plotted in Fig. 5 in the Cartesian coordinates when a non-dimensional heat source of strength d = 1 is applied at point (0, 0, h/r1 = 1). It is interesting to note that the contour lines are much smoother when comparing to those of the normal stress rn(rr). Near the interface in both regions, the normal stress r1(r/) changes from negative to positive. There is no inflection point for all contours in the hexagonal zinc region but there exists inflection points always for all contours in the graphite region. The strengths of stresses in the upper half-space z P 0 are larger than those in lower half-space z 6 0. Fig. 6 depicts the normal stress rf(rz) in the Cartesian coordinates when a non-dimensional heat source of strength d = 1 is applied at point (0, 0, h/r1 = 1). The contours of the normal stress rf(rz) in both regions are normal to the interface that is very different from the previous cases. The curvatures of the contours in the hexagonal zinc region change directions while those in the graphite region do not. No zero common tangents are found for the normal stress rf(rz) which is negative everywhere. The normal stress is continuous across the interface. There is no inflection point for all contours in the graphite region but there exists inflection points always for all contours in the hexagonal zinc region. It is exactly opposite to Fig. 5. Fig. 7 shows the contours of the shear stress szr(sfn) have zero common tangents and are continuous across the common boundary with broken gradients. The shear stress szr (sfn) is positive in most area except at the lower left corner. The strengths of shear stress in the upper half-space z P 0 are larger than those in lower half-space. It is interesting to note that there is a higher order singularity at the heat source where the tangents are degenerated into a single tangent. Shear failure of the material is possible. It will be difficult to simulate the highly degenerated scenario using numerical methods such as finite elements. Figs. 3–7 show that the stresses rn(rr) and r1(r/) are not continuous at the interface z = 0, while the temperature increment #(h) and the stresses rf(rz), sfn(szr) satisfy the interface continuous conditions Eq. (10). 5. Conclusions Based on the compact general solution of Chen et al. (2004), six mono-harmonic functions wj and w0j ðj ¼ 1; 2; 3Þ are constructed in Eqs. (12), (15). The corresponding coupled field induced by a concentrated heat source positioned in the interior of two transversely isotropic thermoelastic half-spaces is derived. Because the solution is expressed explicitly in terms of elementary functions, it is very convenient to use. Typical numerical examples are presented. The corresponding solutions under point forces can be derived similarly and are listed in Appendix A. All these show that Chen’s general solution is an important foundation for the studies of steady state problems for transversely isotropic thermoelastic materials. It should be noted that the obtained Green’s functions are valid for distinct eigenvalues sj and s0j ðj ¼ 1; 2; 3Þ, which is the most common case for transversely isotropic materials. The general solutions and Green’s functions for multiple eigenvalues, s1 – s2 = s3, s1 = s2 = s3 and s01 –s02 ¼ s03 , s01 ¼ s02 ¼ s03 , will be quite different. The corresponding solutions for isotropic material with s1 = s2 = s3 = 1 and s01 ¼ s02 ¼ s03 ¼ 1 cannot be obtained by direct degeneration from the solutions obtained here. Acknowledgements The authors thankfully acknowledge the financial support from the National Natural Science Foundation of China, Foundation in Hunan Province, Research Grant Council of Hong Kong grant #115807, National 985 Special Foundation of China and the reviewers for their constructive comments, which lead to much improvement from the earlier version of this article. Appendix A. Green’s functions for concentrated forces in transversely isotropic thermoelastic bimaterials A.1. Solution to the problem of concentrated force Pz This is a non-torsional axisymmetric problem: u/ = 0 and s/z = sr/ = 0. The harmonic functions are given below. In the half-space z P 0:
w0 ¼ w3 ¼ 0;
wj ¼ Bj sign ðz hÞ ln Rjj þ
2 X k¼1
In the half-space of z 6 0:
Bjk ln Rjk
ðj ¼ 1; 2Þ:
ðA1Þ
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w00 ¼ w03 ¼ 0;
w0j ¼
2 X
B0jk ln R0jk
ðj ¼ 1; 2Þ:
ðA2Þ
k¼1
A.2. Solution to the problem of concentrated forces Px and Py This is not an axisymmetric problem and general solution (3) should be used. The harmonic functions are given below. In the half-space z P 0:
w0 ¼ C 0 wj ¼ C j
Px y Py x R00
Px x þ Py y Rjj
þ C 00
þ
2 X
Px y Py x ; R00
C jk
k¼1
Px x þ Py y ; Rjk
ðA3Þ ðj ¼ 1; 2Þ; w3 ¼ 0:
In the half-space of z 6 0:
w00 ¼ C 000
Px y Py x ; R000
w0j ¼
2 X k¼1
C 0jk
Px x þ Py y R0jk
ðj ¼ 1; 2Þ;
w03 ¼ 0:
ðA4Þ
0 0 where Rjj , Rjk and R0 jk are functions defined in Eqs. (13), (16). Bj, Bjk, Bjk , Cj, Cjk and C jk ðj; k ¼ 0; 1; 2Þ are constants to be determined by the compatibility conditions at interface z = 0 and the mechanical equilibrium conditions for a cylinder of a1 6 z 6 a2 (a1 < h < a2) and 0 6 r 6 b.
Appendix B. Material property Material property
Hexagonal zinc
Graphite
c11 c12 c13 c33 c44
162.8 50.8 36.2 62.7 38.5
10.6708 1.3015 1.3170 12.0897 2.07
Heat conduction coefficients (W K1 m1)
b11 b33
124 124
1.172 1.340
Thermal expansion coefficient (106 K1)
ar az
5.818 15.35
3.9 3.5
Thermal moduli (105 N K1 m2)
k11 = (c11 + c12)ar + c13az k33 = 2c13ar + c33az
17.9839 13.8367
0.5130 0.5259
9
Elastic constants (10 Nm
2
)
References Banerjee, P.K., Butterfield, R., 1981. Boundary Element Methods in Engineering Science. McGraw-Hill, London. Berger, J.R., Tewary, V.K., 2001. Greens functions for boundary element analysis of anisotropic bimaterials. Engineering Analysis with Boundary Elements 25, 279–288. Chen, W.Q., Ding, H.J., Ling, D.S., 2004. Thermoelastic field of transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution. International Journal of Solids and Structures 41, 69–83. Elliott, H.A., 1948. Three-dimensional stress distributions in hexagonal aeolotropic crystals. Proceedings Cambridge Philosophical Society 44, 522–533. Hou, P.F., Leung, A.T.Y., Chen, C.P., 2008. Fundamental solution for transversely isotropic thermoelastic materials. International Journal of Solids and Structures 45, 392–408. Kattis, M.A., Papanikos, P., Providas, E., 2004. Thermal Green’s functions in plane anisotropic bimaterials. Acta Mechanica 173, 65–76. Kroner, E., 1953. Das fundamentalintegral der anisotropen elastischen diferentialgleichungen. Zeitschrift fur Physik 136, S.402–S.410. Lejcek, J., 1969. The Green function of the theory of elasticity in an anisotropic hexagonal medium. Czechoslovak Journal of Physics B19, 799–803. Lifshitz, I.M., Rozentsveig, L.N., 1947. 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, 783–791. Noda, N., Ashida, F., Okumura, I.A., 1992. General solution technique for transient thermoelasticity of transversely isotropic solids in Cartesian coordinates. Archive of Applied Mechanics 62, 291–305. Pan, E., 2003. Three-dimensional Green’s functions in anisotropic elastic bimaterials with imperfect interfaces. Journal of Applied Mechanics of The ASME 70, 180–190. Pan, Y.C., Chou, T.W., 1976. Point force solution for an infinite transversely isotropic solid. ASME Journal of Applied Mechanics E 98, 608–612. Pan, E., Yuan, F.G., 2000. Three-dimensional Green’s functions in anisotropic bimaterials. International Journal of Solids and Structures 37, 5329–5351. Sharma, B., 1958. Thermal stresses in transversely isotropic semi-Infinite elastic solids. ASME Journal of Applied Mechanics 25, 86–88. Sharma, J.N., Sharma, P.K., 2002. Free vibration of homogeneous transversely isotropic cylindrical panel. Journal of Thermal Stresses 25, 169–182. Sveklo, V.A., 1969. Concentrated force in a transversely isotropic half-space and in a composite space. PMM 33, 532–537.
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Willis, J.R., 1965. The elastic interaction energy of dislocation loops in anisotropic media. Quarterly Journal of Mechanics and Applied Mathematics 18, 419– 433. Yu, H.Y., Sanday, S.C., Rath, B.B., 1992. Thermoelastic Stresses in Bimaterials. Philosophical Magazine A 65, 1049–1064.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Three-dimensional Green’s functions for transversely isotropic thermoelastic bimaterials Peng-Fei Hou a,*, Andrew Y.T. Leung b, Yong-Jun He c a b c
Department of Engineering Mechanics, Hunan University, Changsha 410082, PR China Department of Building and Construction, City University of Hong Kong, Hong Kong, PR China College of Civil Engineering, Hunan University, Changsha 410082, PR China
a r t i c l e
i n f o
Article history: Received 15 November 2007 Received in revised form 13 July 2008 Available online 7 August 2008
Keywords: Green’s function Transversely isotropic Thermoelastic Bimaterial
a b s t r a c t Green’s functions for transversely isotropic thermoelastic biomaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoelastic material in terms of harmonic functions and introduce six new harmonic functions. The three-dimensional Green’s function having a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of temperature and stress distributions at the interface. The temperature contours are normal to the interface for the isotropic material but not for the orthotropic one. The normal stress contours are parallel to the interface at the boundary in the isotropic region only and shear failure is most likely at the heat source due to the highly degenerated direction of shear stress contours. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Green’s functions play an important role in both applied and theoretical studies on the physics of solids. They are basic building blocks of a lot of further works. Green’s functions can be used to construct many analytical solutions of practical engineering problems by superposition and are very important in the boundary element method as well as the study of cracks, defects and inclusions. For isotropic materials, the Kelvin Green’s function is well-known (Banerjee and Butterfield, 1981). For transversely isotropic materials, Lifshitz and Rozentsveig (1947) and Lejcek (1969) derived Green’s functions using the Fourier transform method. Elliott (1948), Kroner (1953) and Willis (1965) obtained them using the direct method and Sveklo (1969) found them using the complex method. Pan and Chou (1976) solved Green’s function in the form of compact elementary functions. For anisotropic materials, Pan and Yuan (2000) and Pan (2003) obtained the three-dimensional Green’s functions for biomaterials with perfect and imperfect interfaces, respectively. The thermal effects are not considered in the above works. Sharma (1958) studied Green’s functions of transversely isotropic thermoelastic materials in integral form. Yu et al. (1992) found the solution for a point heat source in isotropic thermoelastic bimaterials. Berger and Tewary (2001) and Kattis et al. (2004) obtained the two-dimensional Green’s functions for anisotropic thermoelastic materials. Chen et al. (2004) derived a compact three-dimensional general solution for transversely isotropic thermoelastic materials. Based on this general solution, Hou et al. (2008) constructed Green’s function for infinite and semi-infinite transversely isotropic thermoelastic materials. As a further extension, the three-dimensional Green’s function for a concentrated heat source in a transversely isotropic thermoelastic bimaterial is investigated in this paper. Only steady state is considered. For completeness, the general solution
* Corresponding author. Tel./fax: +86 731 8822330. E-mail address: [email protected] (P.-F. Hou). 0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2008.07.022
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of Chen et al. (2004) is summarized in Section 2. In Section 3, six newly found harmonic functions are constructed in terms of elementary functions with undetermined constants. The unique thermoelastic field can be obtained by substituting these functions into the general solutions after determining the constants by the compatibility and equilibrium conditions at interface. Numerical examples are presented in Section 4. The contours of temperature increment and stress components are shown graphically. Finally, the paper is concluded in Section 5. 2. General solutions We summarize the general solutions of Chen et al. (2004) for use in Section 3. Hou et al. (2008) considered a semi-infinite transversely isotropic thermoelastic material, which is isotropic in the xy-plane. The result will be extended to two such semi-infinite materials joining at the interface z = 0 in Section 3. In the Cartesian coordinate (x, y, z), the constitutive relations are
ov ow þ ; oz oy ou ow ; szx ¼ c44 þ oz ox ou ov ; sxy ¼ c66 þ oy ox
ou ov ow þ c12 þ c13 k11 h; ox oy oz ou ov ow ry ¼ c12 þ c11 þ c13 k11 h; ox oy oz ou ov ow þ c33 rz ¼ c13 þ k33 h; ox oy oz
rx ¼ c11
syz ¼ c44
ð1Þ
where u, v and w are the components of the displacement vector in the x, y and z directions, respectively; rij are the stress components and h is the temperature increment; cij and kii are the elastic and thermal moduli, respectively. c66 = (c11 c12)/2 is held for transversely isotropic thermoelastic materials. In the absence of body forces, the mechanical equilibrium equations are
orx osxy oszx þ þ ¼ 0; ox oy oz osxy ory osyz þ þ ¼ 0; ox oy oz oszx osyz orz þ þ ¼ 0: ox oy oz
ð2aÞ
The heat equilibrium equation is
b11
! o2 h o2 h o2 h þ b ¼ 0; þ 33 ox2 oy2 o2 z
ð2bÞ
where bii (i = 1, 3) are coefficients of heat conduction and b22 = b11 when the body is isotropy in the xy-plane. Chen et al. (2004) gave the general solutions to Eq. (1), (2) as follows:
U ¼ K iw0 þ
3 X
! wj ;
w¼
j¼1
r1 ¼ 2
sj k1j
j¼1
owj ; ozj
h ¼ k23
o2 w3 ; oz23
ð3aÞ
3 3 X X o2 wj ðc66 xj s2j Þ 2 ¼ 2 ðc66 xj s2j ÞDwj ; oz j j¼1 j¼1
r2 ¼ 2c66 K2 ðiw0 þ
3 X
wj Þ;
j¼1
sz
3 X
rz ¼
! 3 owj ow0 X ; ¼ K s0 c44 i þ sj xj oz0 ozj j¼1
3 X j¼1
xj
3 X o2 wj ¼ xj Dwj ; 2 ozj j¼1
ð3bÞ
where the quantities U, r1, r2, sz can be defined in the Cartesian coordinate (x, y, z) and the cylindrical coordinate (r, /, z) in the complex forms as follows:
U ¼ u þ iv ¼ ei/ ður þ iu/ Þ;
r1 ¼ rx þ ry ¼ rr þ r/ ; r2 ¼ rx ry þ 2isxy ¼ e2i/ ðrr r/ þ 2isr/ Þ; sz ¼ sxz þ isyz ¼ ei/ ðszr þ is/z Þ;
ð4Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where i ¼ 1. In additions, for simplicity, we let zj = sjz (j = 0, 1, 2, 3), where s0 ¼ c66 =c44 , s3 ¼ b11 =b33 and s1 and s2 are 4 2 the two eigenvalues of the fourth degree equation a0s b0 s + c0 = 0 that is Eq. (11) of Chen et al. (2004) in which a0 ¼ c33 c44 ; b0 ¼ c11 c33 þ c244 ðc13 þ c44 Þ2 ; c0 ¼ c11 c44 . Let wj (j = 0, 1, 2, 3) be the solutions of the following harmonic equations:
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! o2 wj ¼ 0 ðj ¼ 0; 1; 2; 3Þ; oz2j
Dþ
ð5Þ
where
D¼
o2 o2 þ 2 2 ox oy
in Cartesian coordinates ðx; y; zÞ;
2
D¼
ð6aÞ
2
o o o þ þ or 2 ror r2 o/2
in cylindrical coordinates ðr; /; zÞ:
ð6bÞ
In additions
xj ¼ c44 ð1 þ k1j Þ ¼ c13 þ c33 k1j s2j k33 k2j ¼ ðc11 c13 k1j s2j þ k11 k2j Þ=s2j k1j ¼ aj1 =sj ;
k2j ¼ aj2 ;
K¼
ð7Þ
o o þi ; ox oy
where ajm (m = 1, 2) are the constants defined in Eq. (18) of Chen et al. (2004), that is, ai1 = ki2si /ki1, a12 = a22 = 0, a32 = k33 /k31, and kij ¼ aj þ bj s2i ; k33 ¼ a0 s43 b0 s23 þ c0 . It should be noted that the general solutions given in Eq. (3) are correct when eigenvalues sj (j = 1, 2, 3) are distinct. For non-torsional axisymmetric problem, w0 = 0 and wj (j = 1, 2, 3) are independent of /, so that u/ = 0 and s/z = sr/ = 0. The general solution in the cylindrical coordinate (r, /, z) can be simplified to
ur ¼
3 X owj ; or j¼1
rr ¼ 2c66 r/ ¼ 2c66 rz ¼
3 X j¼1
w¼
3 X
sj k1j
j¼1
owj ; ozj
h ¼ k23
o2 w3 : oz23
ð8aÞ
3 3 X o2 wj 1 owj X s2j xj 2 ; r or ozj j¼1 j¼1
3 3 X o2 wj 1 owj X ðs2j xj 2c66 Þ 2 ; r or ozj j¼1 j¼1
xj
o2 wj ; oz2j
szr ¼
3 X
s j xj
j¼1
o2 wj : orozj
ð8bÞ
For the torsional axisymmetric problem, wj = 0 (j = 1, 2, 3) and w0 is independent of /, so that ur = uz = 0, h = 0 and rr = r/ = rz = srz = 0. The general solution can be simplified to the following form:
u/ ¼
ow0 ; or
sr/ ¼ 2c66
! 1 o2 o2 w ; þ 2 oz20 or2 0
s/z ¼ s0 c44
o2 w0 : oroz0
ð9aÞ
The general solution (8) will be used in Section 3 and the general solutions (3) and (8) will be used in Appendix A. 3. Green’s functions When a concentrated heat source of strength H is applied in the interior of one of the two transversely isotropic thermoelastic semi-infinite materials joining at the interface z = 0 parallel to the plane of isotropy (Fig. 1), either the cylindrical coordinates (r, /, z) or the Cartesian coordinates (x, y, z) can be chosen such that the r/-plane or xy-plane lies in the interface. The concentrated heat source is applied at the point (0, 0, h) in the Cartesian coordinates. The coupled thermoelastic field is derived in this section. This is a non-torsional axisymmetric problem such that u/ = 0 and s/z = sr/ = 0. The thermoelastic material in the two half-spaces z P 0 and z 6 0 are assumed to be perfectly bonded. The compatibility conditions at the interface (z = 0) are in form of
ur ¼ u0r ; 0 zr ;
szr ¼ s h ¼ h0 ;
w ¼ w0 ;
rz ¼ r0z ; b33 oh=oz ¼ b033 oh0 =oz;
ð10aÞ ð10bÞ ð10cÞ
where the primed quantities refer to the variables in the lower half-space z 6 0 and the un-primed quantities refer to those in the upper half-space z P 0. To simplify notations, the following quantities are introduced:
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
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Fig. 1. Thermoelastic bimaterial having a point heat source of strength H in steady state.
zj ¼ sj z;
z0j ¼ s0j z; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rjk ¼ r2 þ z2jk ¼ x2 þ y2 þ z2jk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rjk ¼ r2 þ z2jk ¼ x2 þ y2 þ z2jk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0jk ¼ r2 þ z02 x2 þ y2 þ z02 jk ¼ jk ;
hk ¼ sk h;
zjk ¼ zj hk ; zjk ¼ zj þ hk ; z0jk ¼ z0j hk ; 2
ð11Þ
2
b ; d1j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b þ hj þ hj
b d2j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 b þ hj b þ hj þ hj
ðj; k ¼ 1; 2; 3Þ; where s0j are the eigenvalues of material in the lower half-space z 6 0. We shall show later that Green’s functions in the half-space z P 0 can be assumed in following form:
w0 ¼ 0; wj ¼ Aj ½signðz hÞzjj ln Rjj Rjj þ
3 X
Ajk ½zjk ln Rjk
ð12Þ
Rjk ðj ¼ 1; 2; 3Þ;
k¼1
where Aj and Ajk ðj; k ¼ 1; 2; 3Þ are the twelve constants to be determined, sign() is the signum function, and
Rjj ¼ Rjj þ signðz hÞzjj ;
Rjk ¼ Rjk þ zjk
ðj; k ¼ 1; 2; 3Þ:
The coupled field is found by the substitution of Eq. (12) into general solution (8) as follows:
ur ¼
3 X j¼1
w¼
3 X j¼1
h ¼ k23
! r Aj þ Ajk ; Rjk Rjj k¼1 3 X
r
sj k1j Aj sign ðz hÞ ln Rjj þ
rr ¼ 2c66
j¼1
r/ ¼ 2c66
3 X j¼1
3 1 X 1 Aj þ Ajk R Rjj k¼1 jk
Aj
1 Rjj
þ
3 X k¼1
Ajk
! Ajk ln Rjk ;
k¼1
! 3 X 1 1 ; A3 þ A3k R3k R33 k¼1 3 X
3 X
!
1 Rjk
!
3 X j¼1
s2j
xj
! 3 1 X 1 ; Aj þ Ajk Rjk Rjj k¼1
! 3 3 X 1 X 1 ; s2j xj 2c66 Aj þ Ajk Rjk Rjj k¼1 j¼1
ð13Þ
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rz ¼
3 X
xj Aj
j¼1
szr ¼
3 X
1 Rjj
3 X
þ
Ajk
k¼1
! 1 ; Rjk
! r : Aj sign ðz hÞ þ Ajk Rjk Rjk Rjj Rjj k¼1
s j xj
j¼1
ð14Þ
3 X
r
In the lower half-space z 6 0, assume
w00 ¼ 0;
w0j ¼
3 X
A0jk ðz0jk ln R0jk þ R0jk Þ ðj ¼ 1; 2; 3Þ;
ð15Þ
k¼1
where A0jk ðj; k ¼ 1; 2; 3Þ are the nine constants to be determined and
R0jk ¼ R0jk z0jk
ðj; k ¼ 1; 2; 3Þ:
ð16Þ
Substitution of Eq. (15) into general solution (8) yields 3 X 3 X
u0r ¼
A0jk
j¼1 k¼1
w0 ¼
3 X
0
s0j k1j
3 X
j¼1
h0 ¼
r ; R0jk A0jk ln R0jk ;
k¼1
0 k23
3 X
A03k
k¼1
r0r ¼ 2c066
1 ; R03k
3 X 3 X
A0jk
j¼1 k¼1
r0/ ¼ 2c066
A0jk
3 3 X 1 X 1 þ s02 x0 2c066 A0jk 0 ; R0jk j¼1 j j Rjk k¼1
A0jk
1 ; R0jk
A0jk
r : R0jk R0jk
3 X 3 X j¼1 k¼1
r0z ¼
3 X
x0j
j¼1
s0zr ¼
3 X
s0j x0j
j¼1
3 X k¼1 3 X
3 X 1 1 3 02 0 A0jk 0 ; 0 þ umj¼1 sj xj Rjk R jk k¼1
k¼1
ð17Þ
The compatibility conditions on the plane z = h for w and szr given in Eq. (14) yield 3 X
sj k1j Aj ¼ 0:
ð18Þ
sj xj Aj ¼ 0:
ð19Þ
j¼1 3 X j¼1
Substitution of xj in Eq. (7) into Eq. (19) gives 3 X
sj ð1 þ k1j ÞAj ¼ 0:
ð20Þ
j¼1
From Eqs. (18), (20) can be simplified to 3 X
sj Aj ¼ 0:
ð21Þ
j¼1
The mechanical and the thermal equilibriums for a cylinder of finite length a1 6 z 6 a2 (a1 < h < a2) and 0 6 r 6 b shown in Fig. 2 give two additional equations separately for Fig. 2a and b below. For a1 > 0 shown in Fig. 2a
X X
force in z direction ¼ 0 : 2p Heat flux ¼ H : 2pb33
Z 0
b
Z 0
b
½rz ðr; a2 Þ rz ðr; a1 Þrdr þ 2pb
Z
a2
szr ðb; zÞdz ¼ 0;
ð22aÞ
a1
Z a2 oh oh oh ðr; a2 Þ ðr; a1 Þ rdr 2pbb11 ðb; zÞdz ¼ H; oz oz a1 or
where b11 and b33 are the coefficients of heat conduction in half-space z P 0 along the r and z axes, respectively.
ð22bÞ
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
6105
Fig. 2. A cylinder containing the point heat source of strength H.
For a1 < 0 shown in Fig. 2b
X
Z b ½rz ðr; a2 Þ r0z ðr; a1 Þrdr force in z direction ¼ 0 : 2p 0 Z 0 Z a2 s0zr ðb; zÞdz þ szr ðb; zÞdz ¼ 0; þ 2pb b
a1
ð23aÞ
0
X oh oh0 Heat flux ¼ H : 2p b33 ðr; a2 Þ b033 ðr; a1 Þ rdr oz oz 0 Z 0 0 Z a2 oh oh 0 2pb b11 ðb; zÞdz ¼ H; ðb; zÞdz þ b11 or 0 a1 or Z
ð23bÞ
where b011 and b033 are the coefficients of heat conduction in the lower half-space z 6 0 along the r and z axes, respectively. The following integration formulae are useful:
Z
1 Rjj
Z
Z
rdr ¼ Rjj ;
sign ðz hÞ
Z
1 rdr ¼ Rjk ; Rjk
r Rjj Rjj
dz ¼
Z
1 rdr ¼ R0jk ; R0jk
ð24aÞ
1 r ; sj Rjj
Z r 1 r r 1 r dz ¼ 0 0 ; dz ¼ ; Rjk Rjk sj Rjk sj Rjk R0jk R0jk ! Z Z 3 z33 X oh z3k A3k 3 rdr rdr ¼ s3 k23 A3 3 þ oz R33 k¼1 R3k " # 3 z33 X z3k ¼ s3 k23 A3 ; þ A3k R3k R33 k¼1 ! Z Z 3 X oh r r A3 3 þ A3k 3 dz dz ¼ k23 or R33 k¼1 R3k ! 3 X k23 r r ¼ ; A3 sign ðz hÞ þ A 3k R3k R3k s3 R33 R33 k¼1 Z Z 0 3 3 X X z3k z0 oh0 0 0 A03k rdr ¼ s03 k23 A03k 3k ; rdr ¼ s03 k23 03 oz R03k R3k k¼1 k¼1 Z Z 0 3 3 X oh0 r k X r 0 A03k dz ¼ 23 A03k 0 0 : dz ¼ k23 0 03 or s R R 3 3k R3k k¼1 k¼1 3k
ð24bÞ
ð24cÞ
ð24dÞ ð24eÞ ð24fÞ
It is noted that integral (24d) is not continuous at z = h for the signum function sign(z h), so the integration should be done in two parts:
Z
a2
a1
oh dz ¼ or
Z
h
a1
oh dz þ or
Z
a2
hþ
oh dz: or
ð25Þ
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For a1 > 0, substituting Eq. (14) into Eq. (22a) and using integrals (24a, b), one can obtain 3 X
xj Aj I1 þ
j¼1
3 X
xj
j¼1
3 X
Ajk I2 ¼ 0;
ð26Þ
k¼1
where
r¼b I1 ¼ Rjj ðr; a2 Þ Rjj ðr; a1 Þ r¼0
"
b
#z¼a2
2
¼ 0; Rjj ðb; zÞ z¼a 1 " #z¼a2 2 b ¼ 0; I2 ¼ ½Rjk ðr; a2 Þ Rjk ðr; a1 Þr¼b r¼0 Rjk ðb; zÞ
ð27aÞ
ð27bÞ
z¼a1
i.e. Eqs. (26, 22a) are satisfied automatically. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting Eq. (14) into Eq. (22b) and using s3 ¼ b11 =b33 as well as integrals (24c, d), (25), one can obtain
A3 I3
3 X
A3k I4 ¼
2pk23
k¼1
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b11 b33
ð28Þ
where
I3 ¼
" s3 a2 h3
s3 a1 h3
#r¼b
"
sign ðz hÞb
2
#z¼h
"
2
sign ðz hÞb
#z¼a2
þ þ R33 ðr; a1 Þ r¼0 R33 ðb; zÞR33 ðb; zÞ z¼a R33 ðb; zÞR33 ðb; zÞ 1 #z¼a2 r¼b " 2 s3 a2 þ hk s3 a1 þ hk b þ ¼ 0: I4 ¼ R3k ðr; a2 Þ R3k ðr; a1 Þ r¼0 R3k ðb; zÞR3k ðb; zÞ R33 ðr; a2 Þ
¼ 2
ð29aÞ
z¼hþ
ð29bÞ
z¼a1
Thus A3 can be determined by Eqs. (28, 29) as follows:
A3 ¼
4pk23
H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : b11 b33
ð30Þ
For a1 < 0, substituting Eqs. (14), (17) into Eq. (23a) and using integrals (24a, b), one can obtain 3 X
xj Aj I5 þ
j¼1
3 X
xj
j¼1
3 X
Ajk I6 þ
3 X
x0j
j¼1
k¼1
3 X
A0jk I7 ¼ 0;
ð31Þ
k¼1
where
"
#z¼a2 2 b I5 ¼ ¼ d1j ; Rjj ðb; zÞ z¼0 " #z¼a2 2 b I6 ¼ ½Rjk ðr; a2 Þr¼b ¼ d1k ; r¼0 Rjk ðb; zÞ z¼0 " #z¼0 2 b r¼b 0 I7 ¼ ½Rjk ðr; a1 Þr¼0 þ 0 ¼ d1k : Rjk ðb; zÞ ½Rjj ðr; a2 Þr¼b r¼0
ð32aÞ
ð32bÞ
ð32cÞ
z¼a1
From, (32) (31) can be rewritten to 3 X
xj Aj d1j þ
j¼1
3 X
xj
3 X
j¼1
Ajk d1k þ
3 X
x0j
j¼1
k¼1
3 X
A0jk d1k ¼ 0;
ð33aÞ
k¼1
which can be simplified to 3 X
xj Aj þ
j¼1
3 X
xk Akj þ
k¼1
3 X
!
x0k A0kj d1j ¼ 0:
Substituting Eqs. (14), (17) into Eq. (23b) and using s3 ¼ can obtain
A3 I8
3 X k¼1
A3k I9
ð33bÞ
k¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b11 =b33 , s3 ¼ b011 =b033 as well as integrals (24c, d, e, f), (25), one
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 k23 b011 b033 X H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A03k I10 ¼ k23 b11 b33 k¼1 2pk23 b11 b33
ð34Þ
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
6107
where
" #r¼b " #z¼h " #z¼a2 2 2 s3 a2 h3 sign ðz hÞb sign ðz hÞb I8 ¼ þ þ ¼ 2 þ d23 ; R33 ðr; a2 Þ r¼0 R33 ðb; zÞR33 ðb; zÞ z¼0 R33 ðb; zÞR33 ðb; zÞ z¼hþ #z¼a2 r¼b " 2 s3 a2 þ hk b I9 ¼ þ ¼ d2k ; R3k ðr; a2 Þ r¼0 R3k ðb; zÞR3k ðb; zÞ z¼0 #z¼0 0 r¼b " 2 s3 a1 hk b I10 ¼ 0 þ 0 ¼ d2k : R3k ðr; a1 Þ r¼0 R3k ðb; zÞR03k ðb; zÞ
ð35aÞ
ð35bÞ
ð35cÞ
z¼a1
Using Eqs. (35), (34) can be rewritten to
2A3 A3 d23 þ
3 X
A3k d2k
k¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 k23 b011 b033 X H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A03k d2k ¼ k23 b11 b33 k¼1 2pk23 b11 b33
ð36Þ
which can be simplified to
! ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 2 X k23 b011 b033 0 k23 b011 b033 0 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A33 d23 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A3k d2k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2A3 A3 A33 þ A3k 2pk23 b11 b33 k23 b11 b33 k23 b11 b33 k¼1
ð37Þ
Finally, when the couple field in interface z = 0 is considered, one has
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r 2 þ hj ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Rjk ¼ r 2 þ hk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R0jk ¼ r 2 þ hk ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r 2 þ hj þ hj ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Rjk ¼ r 2 þ hk þ hk ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R0jk ¼ r 2 þ hk þ hk :
Rjj ¼
Rjj ¼
ð38Þ
Substituting Eqs. (14), (17) into the continuous condition (10) and using s3 ¼ obtains
Aj þ
3 X
Akj ¼
k¼1
3 X
A0kj ;
ð39Þ
k¼1
sj k1j Aj þ
3 X
sk k1k Akj ¼
k¼1
sj xj Aj þ
3 X
3 X
3 X
0
s0k k1k A0kj ;
ð40Þ
s0k x0k A0kj ;
ð41Þ
k¼1
sk xk Akj ¼
k¼1
xj Aj þ
3 X k¼1
xk Akj ¼
k¼1
3 X
x0k A0kj ;
ð42Þ
k¼1 0
0
k23 0 k A ; A3k ¼ 23 A03k ðk ¼ 1; 2Þ; k23 33 k23 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0 0 k23 b011 b033 0 k b011 b033 0 ffi A3k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A33 ; A3k ¼ 23 pffiffiffiffiffiffiffiffiffiffiffiffiffi k23 b11 b33 k23 b11 b33
A3 þ A33 ¼ A3 A33
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b11 =b33 , s3 ¼ b011 =b033 as well as Eq. (38), one
ð43Þ ðk ¼ 1; 2Þ;
ð44Þ
where j = 1, 2, 3. Substituting Eqs. (42), (44) into Eqs. (33b), (37), respectively, one can found that Eqs. (33b), (23a) are satisfied automatically and Eq. (37) degenerates to Eq. (30). Thus, A3 can be determined by Eq. (30), Aj ðj ¼ 1; 2Þ can be determined by Eqs. (18), (21), A3k and A03k ðk ¼ 1; 2; 3Þ can be determined by Eqs. (43), (44) as follows:
A3k ¼ A03k ¼ 0 ðk ¼ 1; 2Þ; ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi b11 b33 b011 b033 ffi A3 ; A33 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi b11 b33 þ b011 b033
A033
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k23 b b33 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A3 : b11 b33 þ b011 b033 k23
ð45aÞ ð45bÞ
Finally, the other twelve constants Ajk and A0jk ðj ¼ 1; 2; k ¼ 1; 2; 3Þ can be determined by the twelve equations listed in Eqs. (39)–(42).
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If the thermoelastic materials in two half-spaces are in smooth contact, the tangential stresses vanish and the radial displacement is not continuous at the interface. Thus the compatibility conditions at the interface are
w ¼ w0 ;
ð46aÞ
szr ¼ s0zr ¼ 0; rz ¼ r0z ;
ð46bÞ ð46cÞ
h ¼ h0 ;
b33 oh=oz ¼ b033 oh0 =oz:
Following the same approach as above, the constants Aj ; Ajk and A0jk can be determined by cancelling Eq. (39) and replacing Eq. (41) with
sj xj Aj þ
3 X
sk xk Akj ¼ 0;
ð47aÞ
k¼1 3 X
s0k x0k A0kj ¼ 0:
ð47bÞ
k¼1
Concerning the problems of point forces Px, Py, Pz in x, y, z direction, respectively, the corresponding Green’s functions are similar to those when thermal effects are not considered. This is because the foundational Eq. (1), (2) are coupling in single-direction, i.e. the thermal loading can change elastic field but the mechanical loadings cannot change thermal field (h = 0). The corresponding solutions are listed in Appendix A. 4. Numerical results Consider the two half-spaces z P 0 and z 6 0 as examples. The material properties for z P 0 are taken as hexagonal zinc (Sharma and Sharma, 2002) and those for z 6 0 are graphite (Noda et al., 1992) shown in Appendix B. The contours of temperature increment and stress components induced by a point heat source of strength H are evaluated numerically and plotted in Figs. 3–7. The following non-dimensional components are used in the figures:
h ri sij ; rk ¼ ; skl ¼ ; T0 c33 ar T 0 c33 ar T 0 r z n¼ ; f¼ ði; j ¼ r; /; z; k; l ¼ n; 1; fÞ; r1 r1 #¼
ð48Þ
where r1 is a non-zero dimension, ar and T0 are the thermal expansion coefficient and reference temperature, respectively. Eq. (30) can be rewritten in non-dimensional form:
Fig. 3. Non-dimensional Green’s temperature increment contours in units of # 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/ r1 = 1).
P.-F. Hou et al. / International Journal of Solids and Structures 45 (2008) 6100–6113
6109
Fig. 4. Non-dimensional Green’s stress contours in units of rn 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
Fig. 5. Non-dimensional Green’s stress contours in units of r1 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
A3 ¼
d ; 4ps3 k23
ð49Þ
where d is a non-dimensional strength of a concentrated heat source given by
d¼
H : r 1 T 0 b33
ð50Þ
Here let d = 1 positioned at point (0,0,h/r1 = 1). It is observed that all components tend to zero in the far field and are singular at the point of application of the heat source (0,0,h/r1 = 1). Fig. 3 shows that the contours of temperature increment #(h) for Hexagonal zinc in the lower half-space
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Fig. 6. Non-dimensional Green’s stress contour in units of rf 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
Fig. 7. Non-dimensional Green’s stress contour in units of sfn 102 under a non-dimensional heat source of strength d = 1 at point (0, 0, h/r1 = 1).
z 6 0 are approximately circular, while those for graphite in the upper half-space z P 0 are approximately elliptical. This is because the heat conduction coefficients b11 and b33 for Hexagonal zinc are equal and those for graphite are different (Appendix B). Since the domains of interest are semi-infinite, the boundary layer effects are not obvious except at the interface. The discontinuous gradients of the contour lines at the interface are caused by the boundary effects. The temperature contour seems to be orthogonal to the interface for the isotropic material but it is inclining to the interface when the material is not isotropic. It is also observed that the contours change rapidly near the point source but are gradual at distance away from it.
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6111
Fig. 4 shows the contours of the normal stress rn(rr) in the Cartesian coordinates when a non-dimensional heat source of strength d = 1 is applied at point (0, 0, h/r1 = 1). For stress near the source, the contours are essentially close-loops in the hexagonal zinc region. However, when near the interface, the contour of the normal stress rn(rr) = 10, say, breaks into two and change directions due to the boundary effects. The contours in the hexagonal zinc region tend to be tangential to the interfacial line but those in the graphite region are not obviously tangential. It is observed that the contours are rather dense near the point (1, 0) that will be difficult to simulate using numerical methods such as finite element where very fine meshes are required. The normal stresses are not continuous at the interface. The strengths of normal stresses in the upper half-space z P 0 are larger than those in lower half-space z 6 0 in general. The stress contours for the normal stress r1 (r/) are plotted in Fig. 5 in the Cartesian coordinates when a non-dimensional heat source of strength d = 1 is applied at point (0, 0, h/r1 = 1). It is interesting to note that the contour lines are much smoother when comparing to those of the normal stress rn(rr). Near the interface in both regions, the normal stress r1(r/) changes from negative to positive. There is no inflection point for all contours in the hexagonal zinc region but there exists inflection points always for all contours in the graphite region. The strengths of stresses in the upper half-space z P 0 are larger than those in lower half-space z 6 0. Fig. 6 depicts the normal stress rf(rz) in the Cartesian coordinates when a non-dimensional heat source of strength d = 1 is applied at point (0, 0, h/r1 = 1). The contours of the normal stress rf(rz) in both regions are normal to the interface that is very different from the previous cases. The curvatures of the contours in the hexagonal zinc region change directions while those in the graphite region do not. No zero common tangents are found for the normal stress rf(rz) which is negative everywhere. The normal stress is continuous across the interface. There is no inflection point for all contours in the graphite region but there exists inflection points always for all contours in the hexagonal zinc region. It is exactly opposite to Fig. 5. Fig. 7 shows the contours of the shear stress szr(sfn) have zero common tangents and are continuous across the common boundary with broken gradients. The shear stress szr (sfn) is positive in most area except at the lower left corner. The strengths of shear stress in the upper half-space z P 0 are larger than those in lower half-space. It is interesting to note that there is a higher order singularity at the heat source where the tangents are degenerated into a single tangent. Shear failure of the material is possible. It will be difficult to simulate the highly degenerated scenario using numerical methods such as finite elements. Figs. 3–7 show that the stresses rn(rr) and r1(r/) are not continuous at the interface z = 0, while the temperature increment #(h) and the stresses rf(rz), sfn(szr) satisfy the interface continuous conditions Eq. (10). 5. Conclusions Based on the compact general solution of Chen et al. (2004), six mono-harmonic functions wj and w0j ðj ¼ 1; 2; 3Þ are constructed in Eqs. (12), (15). The corresponding coupled field induced by a concentrated heat source positioned in the interior of two transversely isotropic thermoelastic half-spaces is derived. Because the solution is expressed explicitly in terms of elementary functions, it is very convenient to use. Typical numerical examples are presented. The corresponding solutions under point forces can be derived similarly and are listed in Appendix A. All these show that Chen’s general solution is an important foundation for the studies of steady state problems for transversely isotropic thermoelastic materials. It should be noted that the obtained Green’s functions are valid for distinct eigenvalues sj and s0j ðj ¼ 1; 2; 3Þ, which is the most common case for transversely isotropic materials. The general solutions and Green’s functions for multiple eigenvalues, s1 – s2 = s3, s1 = s2 = s3 and s01 –s02 ¼ s03 , s01 ¼ s02 ¼ s03 , will be quite different. The corresponding solutions for isotropic material with s1 = s2 = s3 = 1 and s01 ¼ s02 ¼ s03 ¼ 1 cannot be obtained by direct degeneration from the solutions obtained here. Acknowledgements The authors thankfully acknowledge the financial support from the National Natural Science Foundation of China, Foundation in Hunan Province, Research Grant Council of Hong Kong grant #115807, National 985 Special Foundation of China and the reviewers for their constructive comments, which lead to much improvement from the earlier version of this article. Appendix A. Green’s functions for concentrated forces in transversely isotropic thermoelastic bimaterials A.1. Solution to the problem of concentrated force Pz This is a non-torsional axisymmetric problem: u/ = 0 and s/z = sr/ = 0. The harmonic functions are given below. In the half-space z P 0:
w0 ¼ w3 ¼ 0;
wj ¼ Bj sign ðz hÞ ln Rjj þ
2 X k¼1
In the half-space of z 6 0:
Bjk ln Rjk
ðj ¼ 1; 2Þ:
ðA1Þ
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w00 ¼ w03 ¼ 0;
w0j ¼
2 X
B0jk ln R0jk
ðj ¼ 1; 2Þ:
ðA2Þ
k¼1
A.2. Solution to the problem of concentrated forces Px and Py This is not an axisymmetric problem and general solution (3) should be used. The harmonic functions are given below. In the half-space z P 0:
w0 ¼ C 0 wj ¼ C j
Px y Py x R00
Px x þ Py y Rjj
þ C 00
þ
2 X
Px y Py x ; R00
C jk
k¼1
Px x þ Py y ; Rjk
ðA3Þ ðj ¼ 1; 2Þ; w3 ¼ 0:
In the half-space of z 6 0:
w00 ¼ C 000
Px y Py x ; R000
w0j ¼
2 X k¼1
C 0jk
Px x þ Py y R0jk
ðj ¼ 1; 2Þ;
w03 ¼ 0:
ðA4Þ
0 0 where Rjj , Rjk and R0 jk are functions defined in Eqs. (13), (16). Bj, Bjk, Bjk , Cj, Cjk and C jk ðj; k ¼ 0; 1; 2Þ are constants to be determined by the compatibility conditions at interface z = 0 and the mechanical equilibrium conditions for a cylinder of a1 6 z 6 a2 (a1 < h < a2) and 0 6 r 6 b.
Appendix B. Material property Material property
Hexagonal zinc
Graphite
c11 c12 c13 c33 c44
162.8 50.8 36.2 62.7 38.5
10.6708 1.3015 1.3170 12.0897 2.07
Heat conduction coefficients (W K1 m1)
b11 b33
124 124
1.172 1.340
Thermal expansion coefficient (106 K1)
ar az
5.818 15.35
3.9 3.5
Thermal moduli (105 N K1 m2)
k11 = (c11 + c12)ar + c13az k33 = 2c13ar + c33az
17.9839 13.8367
0.5130 0.5259
9
Elastic constants (10 Nm
2
)
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