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Green’s function for torsional waves in a cylindrically monoclinic material
International Journal of Engineering Science 43 (2005) 1283–1291 www.elsevier.com/locate/ijengsci
Greens function for torsional waves in a cylindrically monoclinic material Kazumi Watanabe a b
a,*
, Robert G. Payton
b
Department of Mechanical Engineering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan Department of Mathematics and Computer Science, Adelphi University, Garden City, NY 11530, USA Received 16 December 2004; received in revised form 6 May 2005; accepted 6 May 2005
Abstract Two exact Greens functions for impulsive and time-harmonic torsional waves in a monoclinic material are presented. The impulsive Greens function is expressed in the closed form of simple algebraic functions and its wave front shape is a torus with inclined elliptic cross section. The time-harmonic Greens function is also obtained exactly, but in the form of definite integral. Time development of the wave front for the impulsive wave and amplitude contours for the timeharmonic wave are illustrated. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Waves; Monoclinic material; Wave front; Exact solution; Greens function
1. Introduction Wave propagation in cylindrically anisotropic media has been attracting some attention, however, less information is drawn. As for the in-plane wave, such as quasi-P and SV-waves in polar coordinate systems, the time development of wave front shape is obtained by Payton and co-workers [1–4] by applying the method of characteristics, but no Greens function for these in-plane waves is obtained so far, since the exact solution for the coupled governing equations is not known. This situation is also same as that in the axisymmetric waves in the cylindrically orthotropic materials. Martin [5,6] and Shuvalov [7] have considered its approximate solution. Existing solution for the in-plane wave is only in the case of radially symmetric one-dimensional wave [8]. It is only recent years that the Greens function for SH-wave in the cylindrically orthotropic media has been obtained by Watanabe and co-workers [9–11]. They also obtained the Greens function for the monoclinic material [12]. As a simple extension of academic interests, this paper considers the Greens function for axially symmetric torsional waves in a cylindrically monoclinic material. Two Greens functions for the impulsive and time-harmonic waves are obtained and wave propagation phenomena are also discussed in some details. *
Corresponding author. Tel.: +81 238 26 3210; fax: +81 238 26 3205. E-mail address: [email protected] (K. Watanabe).
0020-7225/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.05.005
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K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
2. Governing equations Let us consider an elastic solid under the axially symmetric pure torsion and assume that the medium is monoclinic and no warping takes place. Hookes law for the monoclinic material for the pure torsion is given by rzh c44 c45 ezh ¼ ð1Þ rrh c45 c55 erh and equation of motion is 1 oðr2 rrh Þ orzh o2 uh dðr aÞ þ dðzÞdðtÞ; ¼ q qH0 r2 or r oz ot2
ð2Þ
where d(Æ) is Diracs delta function and the second term in the right hand side represents an impulsive ring source with magnitude H0 and radius a. The strain components are defined by 1 ouh 1 ouh uh ezh ¼ ; erh ¼ . ð3Þ 2 oz 2 or r Substituting Eq. (1) with Eq. (3) into the equation of motion (2), we have the simple equation for the torsional displacement, 2 o2 uh 1 ouh uh o2 uh b ouh 1 o2 uh H0 dðr aÞ 2 o uh dðzÞdðtÞ; þ þ a þ þ 2b ¼ 2 2 2 2 2 2 r or c ot r or r or oz r oz oz c
ð4Þ
where c ¼ ðc55 =2qÞ
1=2
;
a ¼ ðc44 =c55 Þ
1=2
;
b ¼ c45 =c55
ð5Þ
and a > b [13, p. 70]. In order to obtain the Greens function, which is a particular solution corresponding to the nonhomogeneous source term in Eq. (4), a mathematical procedure is developed in the next section. 3. Solution procedure Let us apply the Laplace transform, Z 1 f ðsÞ ¼ f ðtÞ expðstÞ dt
ð6Þ
0
and Fourier transform, Z 1 ~ f ðnÞ ¼ f ðzÞ expðinzÞ dz
ð7Þ
0
to Eq. (4). It yields to the ordinary differential equation, d2 ~ 1 d~ uh 1 ibn H0 dðr aÞ uh 2 2 2ibn þ fðanÞ þ ðs=cÞ g ~uh ¼ 2 . 2þ þ r r r r dr2 dr c This is one of Bessel equations and its particular solution is given by I 1 ðprÞK 1 ðpaÞ; r < a H0 ~ uh ¼ 2 expfibnðr aÞg ; c I 1 ðpaÞK 1 ðprÞ; r > a
ð8Þ
ð9Þ
where p¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða2 b2 Þn2 þ ðs=cÞ .
ð10Þ
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
In order to have a more convenient form for inversion, we apply the formula [14] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 p a2 þ b2 2ab cos u cos u du; I 1 ðaÞK 1 ðbÞ ¼ K0 p 0 to the product of the Bessel functions in Eq. (9). Then, Z p H0 ~ K 0 ðpRÞ cos u du; uh ¼ 2 expfibnðr aÞg pc 0 where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ r2 2ar cos u.
R¼
1285
ð11Þ
ð12Þ
ð13Þ
The formal inversion integral of Fourier transform is applied to Eq. (12) and the order of integration is exchanged. Z p Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H0 2 uh ¼ cos u du K 0 cR n2 þ ðs=ccÞ cos½fz bðr aÞgn dn; ð14Þ 2 ðpcÞ 0 0 where c¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b2 .
ð15Þ
The inner integral in Eq. (14) can be evaluated exactly by the formula [15] Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p K 0 a x2 þ b2 cosðxyÞ dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp b a2 þ y 2 ; 2 a2 þ y 2 0 we have uh ¼
H0 2pcc2
Z
p
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðs=cÞ R2 þ ðZ=cÞ2 du; 2 R2 þ ðZ=cÞ
ð16Þ
ð17Þ
where Z ¼ z bðr aÞ.
ð18Þ
The Laplace transformed displacement, Eq. (17), has the transform parameter s in the argument of the exponential function and thus we can apply the simple inversion formula, L1 fexpðasÞg ¼ dðt aÞ. That is uh ¼
H0 2pcc
Z 0
p
ð19Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ct R2 þ ðZ=cÞ2 du. 2 R2 þ ðZ=cÞ
ð20Þ
Fortunately, we can evaluate this integral exactly, since the Diracs delta function is included. Introducing the variable transform, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þ u ¼ R2 þ ðZ=cÞ2 ¼ a2 þ r2 2ar cos u þ ðZ=cÞ2 ; Eq. (20) is rewritten as Z R2 2 H0 1 ðR2 u2 Þ ðu2 R21 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðct uÞ du; uh ¼ 4pcc ar R1 ðR22 u2 Þðu2 R21 Þ
ð22Þ
where R1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðr aÞ þ ðZ=cÞ ;
R2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðr þ aÞ þ ðZ=cÞ
ð23Þ
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K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
and a very simple integration formula for the delta function, Z b f ðcÞ; a < c < b; f ðxÞdðx cÞ dx ¼ 0; c < a or b < c a is applied to Eq. (22). We have the closed form for the Greens function, (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 2 2 4pc 1 R22 ðctÞ ðctÞ R21 H ðR2 ctÞH ðct R1 Þ ; uh ðr; z; tÞ ¼ 2 H0 car ðctÞ R21 R22 ðctÞ2 where H(Æ) is Heavisides unit step function and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 R2 ¼ ðr þ aÞ þ fz bðr aÞg =c2 . R1 ¼ ðr aÞ þ fz bðr aÞg =c2 ;
ð24Þ
ð25Þ
ð26Þ
4. Wave front and ray The Greens function of Eq. (25) has two wave fronts. They are derived from the argument in the step function. The first one is W1 : ct = R1. This has the explicit expression as ðctÞ2 ¼ ðr aÞ2 þ fz bðr aÞg2 =c2 .
ð27Þ
The above equation shows an ellipse centered at (r = a, z = 0), but with inclined axis. When we introduce the inclined axis (P, Q) as r a ¼ P cos / Q sin /;
z ¼ P sin / þ Q cos /;
where the angle of the axis inclination is given by 1 2b 1 / ¼ tan ; 2 1 a2
ð28Þ
ð29Þ
Eq. (27) is converted to the simple ellipse in (P, Q) plane, (
P2 Q2 þ ) ( )2 ¼ 1. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2ða2 b2 ÞðctÞ 2ða2 b2 ÞðctÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þa2
ð1a2 Þ2 þ4b2
1þa2 þ
ð30Þ
ð1a2 Þ2 þ4b2
Thus, the wave front W1 is emanating from the source and forms a torus with the inclined elliptic cross section. If we decompose Eq. (27) with introducing a new angle parameter /, the parametric expression for the wave front and ray is given by r a ¼ ðctÞ cos u;
z ¼ ðctÞðc sin u þ b cos uÞ.
ð31Þ
We can draw wave front and ray curves simultaneously by using this equation. Diminishing the time, we learn that the ray is a straight line defined by z = (c tan u + b)(r a), and its radiation angle (ray angle) w is defined by w = tan1 (c tan u + b). In other words, the parameter u is the ray angle defined by u = tan1 [(tan w b)/ c]. The second wave front is also derived. That is W2 : ct = R2 and its explicit form is 2
2
2
ðctÞ ¼ ðr þ aÞ þ fz bðr aÞg =c2 .
ð32Þ
Eq. (32) gives the same ellipse as that for the first wave, but its center is displaced at (r = a, z = 2ab). 5. Special case Case A: isotropic (a = 1, b = 0). When the material is isotropic, the Greens function is reduced to the simple form,
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pc 1 2 2 2 ða þ rÞ þ z ct H ct ða rÞ þ z2 uh ðr; z; tÞ ¼ H H0 ar (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 2 2 2 2 ða þ rÞ þ z2 ðctÞ ðctÞ ða rÞ z2
. 2 2 2 2 ðctÞ ða rÞ z2 ða þ rÞ þ z2 ðctÞ
1287
ð33Þ
Case B: c = 0 (a = b). When elastic constants have the relation, c44 c55 ¼ c245 (a = b), we have to return to Eq. (14), uh ¼
H0 ðpcÞ2
Z
p
K 0 ðsR=cÞ cos u du 0
Z
1
cos½fz bðr aÞgn dn
ð34Þ
0
and apply the integration formula, 1 p
Z
1
cosðxyÞ dx ¼ dðyÞ
ð35Þ
0
to Eq. (34). uh
H0 ¼ 2 dðz bðr aÞÞ pc
Z
p
K 0 ðsR=cÞ cos u du.
ð36Þ
0
Then, as the Laplace transform parameter s is included in the argument of the modified Bessel function, the inversion formula [15], H ðt aÞ L1 fK 0 ðasÞg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 a2 is applied. The Greens function is given by Z p pc H ðct RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u du uh ðr; z; tÞ ¼ dðz bðr aÞÞ H0 2 0 ðctÞ R2 8 > Z u < H ðct RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u du ¼ dðz bðr aÞÞ H ðr þ a ctÞH ðct jr ajÞ > 2 0 : ðctÞ R2 9 > Z p = H ðct RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u du ; þH ðct ðr þ aÞÞ > 0 ; ðctÞ2 R2
ð37Þ
ð38Þ
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðctÞ2 ðr aÞ2 . u ¼ 2sin1 4ar
ð39Þ
The definite integrals in Eq. (38) can be reduced to the standard form of elliptic integrals. The final form of the Greens function is given by 2c dðz bðr aÞÞ pffiffiffiffiffi uh ðr; z; tÞ ¼ H ðct ja rjÞH ða þ r ctÞf2EðkÞ KðkÞg H0 ar 1 2k 2 Kð1=kÞ þ 2kEð1=kÞ ; ð40Þ þH ðct ða þ rÞÞ k where the complete elliptic integral of the first and second kinds and their argument are defined by
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K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
Z ffi 2 p=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k 2 sin2 h dh; p 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðctÞ2 ða rÞ2 . k¼ 4ar
EðkÞ ¼
KðkÞ ¼
2 p
Z
p=2 0
dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 k 2 sin2 h
ð41Þ
ð42Þ
The Greens function of Eq. (40) is multiplied by the delta function. This means that the disturbed region is limited on a plane with the line cross section, z = b(r a). The similar line (plane) disturbance in the limiting case of the monoclinic material has also been found in the case of SH-wave [12]. 6. Time-harmonic Green’s function The time-harmonic Greens function can be derived by performing the convolution integral, Z t0 ðGÞ uh ðr; z; tÞ ¼ expðixtÞ lim uh ðr; z; t0 Þ expðixt0 Þ dt0 ; 0 t !1
ð43Þ
0
where uðGÞ is the impulsive Greens function given by Eq. (25). That is z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Z 4pc2 expðixtÞ R2 R22 s2 s2 R21 expðixs=cÞ ds. uh ðr; z; tÞ ¼ 2 2 car H0 s R1 R22 s2 R1
ð44Þ
This integral can be reduced to the more convenient form for the numerical computation by introducing the variable transform, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 þ R21 R22 R21 s¼ sin /. ð45Þ 2 2 The final form of the time-harmonic Greens function for the torsional wave is given by 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Z p=2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pc2 expðixtÞ sin / ix R þ R 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp @ 1 e sin /A d/; uh ðr; z; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c H0 2 2 2 1 e sin / p=2 c 2ðR þ R Þ 2
ð46Þ
1
where R1 and R2 are given by Eq. (26) and the parameter e is defined by 0
R22 R21 < 1. R22 þ R21
ð47Þ
The integral in Eq. (46) is singular when e = 1. Extracting its singular part, we have the logarithmic singularity on the ring source, 2 pc expðixtÞ 1 log lim uh ðr; z; tÞ . ð48Þ e!1 4ca 1 e2 H0 ðr!a;z!0Þ
7. Numerical examples Shutilov [16] mentioned that an example of the monoclinic materials is Gypsum, but its elastic moduli are not shown in his book. The moduli for some other crystals are listed in it. So, we show the numerical examples Table 1 Material parameters for typical monoclinic crystals [16] Crystal
a = (c44/c55)1/2
b = c45/c55
c = (a2b2)1/2
Dibenzyl Sodium-thiosulfate
1.09 0.97
+0.31 0.45
1.04 0.86
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
ct/a=2
2
2
ct/a=2 1.25
z/a
1
1
1
1.25 1
0.5 0
0
0.5
-1
-1
-2
-2
r/a 0
0.5
1
1.5
2
r/a
2.5
0
3
0.5
(a) Dibenzyl
1
1.5
2
(b) Sodium-thiosulfate
Fig. 1. Time development of wave front.
30
4ρac uθ Θ0
20 (0.5, 0 .5)
10 0 (1.5, 0.5)
-10 (r / a, z / a )
-20
= (0.5, / 0.5)
ct/a
-30 0.5
1
1.5
2
2.5
(a) Dibenzyl 30 20
4ρac uθ Θ0
(r / a, z / a ) = (0.5, / 0.5)
10 0 -10 (0.5, 0.5)
(1.5, 0.5)
-20 -30 0.5
ct/a
1
2.5
1.5
2
2.5
3
(b) Sodium-thiosulfate Fig. 2. Time response of Greens function for torsional wave.
3
1289
1290
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291 3
0.05
0.25
0.1
2
0.5 1
3
1
2
0 z/a -1 -2
r/a 0
0.5
1
1.5
2
2.5
-3 3
(a) Dibenzyl 3
0.05
0.1
0.25
2
0.5
1
1 3
0 z/a
2
-1 -2
r/a 0
0.5
1
1.5
2
2.5
-3 3
(b) Sodium-thiosulfate Fig. 3. Contours of displacement amplitude for time-harmonic Greens function,
pc2 H0
juh ðr; z; tÞj.
for two typical monoclinic crystals, positive and negative values of c45. They are dibenzyl and sodium-thiosulfate and their material parameters are listed in Table 1. Fig. 1 shows the time-development of wave fronts. At early time, its form is the inclined ellipse that is given by Eq. (27). After a unit time ct/a = 1, the second wave, W2, appears as if the reflection at the z-axis. But, it is penetrated from the opposite side due to the axisymmetric nature, not reflected wave. The inclination angle of the elliptic wave form is given by Eq. (29). The time response of the torsional displacement in Fig. 2 shows that there is no substantial difference in the response between positive and negative values of c45, since the response curves are similar to each other. Fig. 3 shows amplitude contours for the time-harmonic Greens function given by Eq. (46). Due to the torsional nature, the displacement vanishes on the center axis (r = 0). The contour at the higher amplitude region is a quasi-elliptic form and its form is gradually deformed at the lower amplitude region. 8. Conclusion Greens function for the impulsive and time-harmonic torsional waves is derived exactly for the monoclinic material. The Greens function for the transient wave is expressed in the closed form of simple algebraic functions. Due to the axially symmetric nature, the wave front shape is a torus, but its cross section is an inclined ellipse due to the monoclinic nature. At the wave front, the torsional displacement has the singularity of the inverse square root. This is the same as that in isotropic media. Time-harmonic Greens function is also derived and its amplitude contour is also quasi-elliptic form due to the monoclinic nature. Acknowledgement The authors would like to express their thanks to the referee for helpful comments in the revision of the paper.
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
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References [1] R.G. Payton, K. Watanabe, Two-dimensional wave front shape for cylindrically anisotropic elastic media, Wave motion 33 (3) (2001) 211–224. [2] Y. Yokoyama, R.G. Payton, K. Watanabe, Wave front shapes in cylindrically anisotropic solids, J. Jpn. Soc. Non-Destruct. Inspect. 51 (5) (2002) 296–303 (in Japanese). [3] R.G. Payton, Wave fronts in wood, Q.J. Mech. Appl. Math. 56 (4) (2003) 527–546. [4] R.G. Payton, Wave front shape for a constrained cylindrically anisotropic solid, ZAMP 55 (2004) 658–677. [5] P.A. Martin, J. Berger, Waves in wood: free vibration of a wooden pole, J. Mech. Phys. Solids 49 (2001) 1155–1178. [6] P.A. Martin, Waves in wood: axisymmetric waves in slender solids of revolution, Wave Motion 40 (2004) 387–398. [7] A.L. Shuvalov, The Frobenius power series solution for cylindrically anisotropic radially inhomogeneous elastic materials, Q. Mech. Appl. Math. 56 (2003) 327–345. [8] K. Watanabe, Some exact closed form solutions for transient response of a cylindrically anisotropic elastic solid subjected to three types of a ring source, ZAMM 81 (11) (2001) 788–792. [9] K. Watanabe, K. Nishinari, SH wave in a cylindrically anisotropic solid, ZAMP 47 (6) (1996) 906–914. [10] K. Watanabe, R.G. Payton, SH wave in a cylindrically anisotropic elastic solid (A general solution for a point source), Wave Motion 25 (2) (1997) 197–212. [11] K. Watanabe, R.G. Payton, Time-harmonic SH-wave in cylindrically anisotropic elastic solid, Geophys. J. Int. 145 (2001) 709–713. [12] K. Watanabe, R.G. Payton, Greens function for SH-waves in a cylindrically monoclinic material, J. Mech. Phys. Solids 50 (2002) 2425–2439. [13] T.C.T. Ting, Anisotropic Elasticity, Oxford, 1996. [14] G.N. Watson, Theory of Bessel Functions, Cambridge, 1966. [15] A. Erdelyi (Ed.), Tables of Integral Transforms, vol. 1, McGraw-Hill, 1954. [16] V.A. Shutilov, Fundamental Physics of Ultrasound, Gordon & Breach Sci. Publ., 1988.
Greens function for torsional waves in a cylindrically monoclinic material Kazumi Watanabe a b
a,*
, Robert G. Payton
b
Department of Mechanical Engineering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan Department of Mathematics and Computer Science, Adelphi University, Garden City, NY 11530, USA Received 16 December 2004; received in revised form 6 May 2005; accepted 6 May 2005
Abstract Two exact Greens functions for impulsive and time-harmonic torsional waves in a monoclinic material are presented. The impulsive Greens function is expressed in the closed form of simple algebraic functions and its wave front shape is a torus with inclined elliptic cross section. The time-harmonic Greens function is also obtained exactly, but in the form of definite integral. Time development of the wave front for the impulsive wave and amplitude contours for the timeharmonic wave are illustrated. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Waves; Monoclinic material; Wave front; Exact solution; Greens function
1. Introduction Wave propagation in cylindrically anisotropic media has been attracting some attention, however, less information is drawn. As for the in-plane wave, such as quasi-P and SV-waves in polar coordinate systems, the time development of wave front shape is obtained by Payton and co-workers [1–4] by applying the method of characteristics, but no Greens function for these in-plane waves is obtained so far, since the exact solution for the coupled governing equations is not known. This situation is also same as that in the axisymmetric waves in the cylindrically orthotropic materials. Martin [5,6] and Shuvalov [7] have considered its approximate solution. Existing solution for the in-plane wave is only in the case of radially symmetric one-dimensional wave [8]. It is only recent years that the Greens function for SH-wave in the cylindrically orthotropic media has been obtained by Watanabe and co-workers [9–11]. They also obtained the Greens function for the monoclinic material [12]. As a simple extension of academic interests, this paper considers the Greens function for axially symmetric torsional waves in a cylindrically monoclinic material. Two Greens functions for the impulsive and time-harmonic waves are obtained and wave propagation phenomena are also discussed in some details. *
Corresponding author. Tel.: +81 238 26 3210; fax: +81 238 26 3205. E-mail address: [email protected] (K. Watanabe).
0020-7225/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.05.005
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2. Governing equations Let us consider an elastic solid under the axially symmetric pure torsion and assume that the medium is monoclinic and no warping takes place. Hookes law for the monoclinic material for the pure torsion is given by rzh c44 c45 ezh ¼ ð1Þ rrh c45 c55 erh and equation of motion is 1 oðr2 rrh Þ orzh o2 uh dðr aÞ þ dðzÞdðtÞ; ¼ q qH0 r2 or r oz ot2
ð2Þ
where d(Æ) is Diracs delta function and the second term in the right hand side represents an impulsive ring source with magnitude H0 and radius a. The strain components are defined by 1 ouh 1 ouh uh ezh ¼ ; erh ¼ . ð3Þ 2 oz 2 or r Substituting Eq. (1) with Eq. (3) into the equation of motion (2), we have the simple equation for the torsional displacement, 2 o2 uh 1 ouh uh o2 uh b ouh 1 o2 uh H0 dðr aÞ 2 o uh dðzÞdðtÞ; þ þ a þ þ 2b ¼ 2 2 2 2 2 2 r or c ot r or r or oz r oz oz c
ð4Þ
where c ¼ ðc55 =2qÞ
1=2
;
a ¼ ðc44 =c55 Þ
1=2
;
b ¼ c45 =c55
ð5Þ
and a > b [13, p. 70]. In order to obtain the Greens function, which is a particular solution corresponding to the nonhomogeneous source term in Eq. (4), a mathematical procedure is developed in the next section. 3. Solution procedure Let us apply the Laplace transform, Z 1 f ðsÞ ¼ f ðtÞ expðstÞ dt
ð6Þ
0
and Fourier transform, Z 1 ~ f ðnÞ ¼ f ðzÞ expðinzÞ dz
ð7Þ
0
to Eq. (4). It yields to the ordinary differential equation, d2 ~ 1 d~ uh 1 ibn H0 dðr aÞ uh 2 2 2ibn þ fðanÞ þ ðs=cÞ g ~uh ¼ 2 . 2þ þ r r r r dr2 dr c This is one of Bessel equations and its particular solution is given by I 1 ðprÞK 1 ðpaÞ; r < a H0 ~ uh ¼ 2 expfibnðr aÞg ; c I 1 ðpaÞK 1 ðprÞ; r > a
ð8Þ
ð9Þ
where p¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða2 b2 Þn2 þ ðs=cÞ .
ð10Þ
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
In order to have a more convenient form for inversion, we apply the formula [14] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 p a2 þ b2 2ab cos u cos u du; I 1 ðaÞK 1 ðbÞ ¼ K0 p 0 to the product of the Bessel functions in Eq. (9). Then, Z p H0 ~ K 0 ðpRÞ cos u du; uh ¼ 2 expfibnðr aÞg pc 0 where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ r2 2ar cos u.
R¼
1285
ð11Þ
ð12Þ
ð13Þ
The formal inversion integral of Fourier transform is applied to Eq. (12) and the order of integration is exchanged. Z p Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H0 2 uh ¼ cos u du K 0 cR n2 þ ðs=ccÞ cos½fz bðr aÞgn dn; ð14Þ 2 ðpcÞ 0 0 where c¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b2 .
ð15Þ
The inner integral in Eq. (14) can be evaluated exactly by the formula [15] Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p K 0 a x2 þ b2 cosðxyÞ dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp b a2 þ y 2 ; 2 a2 þ y 2 0 we have uh ¼
H0 2pcc2
Z
p
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðs=cÞ R2 þ ðZ=cÞ2 du; 2 R2 þ ðZ=cÞ
ð16Þ
ð17Þ
where Z ¼ z bðr aÞ.
ð18Þ
The Laplace transformed displacement, Eq. (17), has the transform parameter s in the argument of the exponential function and thus we can apply the simple inversion formula, L1 fexpðasÞg ¼ dðt aÞ. That is uh ¼
H0 2pcc
Z 0
p
ð19Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ct R2 þ ðZ=cÞ2 du. 2 R2 þ ðZ=cÞ
ð20Þ
Fortunately, we can evaluate this integral exactly, since the Diracs delta function is included. Introducing the variable transform, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð21Þ u ¼ R2 þ ðZ=cÞ2 ¼ a2 þ r2 2ar cos u þ ðZ=cÞ2 ; Eq. (20) is rewritten as Z R2 2 H0 1 ðR2 u2 Þ ðu2 R21 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðct uÞ du; uh ¼ 4pcc ar R1 ðR22 u2 Þðu2 R21 Þ
ð22Þ
where R1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðr aÞ þ ðZ=cÞ ;
R2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðr þ aÞ þ ðZ=cÞ
ð23Þ
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K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
and a very simple integration formula for the delta function, Z b f ðcÞ; a < c < b; f ðxÞdðx cÞ dx ¼ 0; c < a or b < c a is applied to Eq. (22). We have the closed form for the Greens function, (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 2 2 4pc 1 R22 ðctÞ ðctÞ R21 H ðR2 ctÞH ðct R1 Þ ; uh ðr; z; tÞ ¼ 2 H0 car ðctÞ R21 R22 ðctÞ2 where H(Æ) is Heavisides unit step function and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 R2 ¼ ðr þ aÞ þ fz bðr aÞg =c2 . R1 ¼ ðr aÞ þ fz bðr aÞg =c2 ;
ð24Þ
ð25Þ
ð26Þ
4. Wave front and ray The Greens function of Eq. (25) has two wave fronts. They are derived from the argument in the step function. The first one is W1 : ct = R1. This has the explicit expression as ðctÞ2 ¼ ðr aÞ2 þ fz bðr aÞg2 =c2 .
ð27Þ
The above equation shows an ellipse centered at (r = a, z = 0), but with inclined axis. When we introduce the inclined axis (P, Q) as r a ¼ P cos / Q sin /;
z ¼ P sin / þ Q cos /;
where the angle of the axis inclination is given by 1 2b 1 / ¼ tan ; 2 1 a2
ð28Þ
ð29Þ
Eq. (27) is converted to the simple ellipse in (P, Q) plane, (
P2 Q2 þ ) ( )2 ¼ 1. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2ða2 b2 ÞðctÞ 2ða2 b2 ÞðctÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þa2
ð1a2 Þ2 þ4b2
1þa2 þ
ð30Þ
ð1a2 Þ2 þ4b2
Thus, the wave front W1 is emanating from the source and forms a torus with the inclined elliptic cross section. If we decompose Eq. (27) with introducing a new angle parameter /, the parametric expression for the wave front and ray is given by r a ¼ ðctÞ cos u;
z ¼ ðctÞðc sin u þ b cos uÞ.
ð31Þ
We can draw wave front and ray curves simultaneously by using this equation. Diminishing the time, we learn that the ray is a straight line defined by z = (c tan u + b)(r a), and its radiation angle (ray angle) w is defined by w = tan1 (c tan u + b). In other words, the parameter u is the ray angle defined by u = tan1 [(tan w b)/ c]. The second wave front is also derived. That is W2 : ct = R2 and its explicit form is 2
2
2
ðctÞ ¼ ðr þ aÞ þ fz bðr aÞg =c2 .
ð32Þ
Eq. (32) gives the same ellipse as that for the first wave, but its center is displaced at (r = a, z = 2ab). 5. Special case Case A: isotropic (a = 1, b = 0). When the material is isotropic, the Greens function is reduced to the simple form,
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pc 1 2 2 2 ða þ rÞ þ z ct H ct ða rÞ þ z2 uh ðr; z; tÞ ¼ H H0 ar (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 2 2 2 2 ða þ rÞ þ z2 ðctÞ ðctÞ ða rÞ z2
. 2 2 2 2 ðctÞ ða rÞ z2 ða þ rÞ þ z2 ðctÞ
1287
ð33Þ
Case B: c = 0 (a = b). When elastic constants have the relation, c44 c55 ¼ c245 (a = b), we have to return to Eq. (14), uh ¼
H0 ðpcÞ2
Z
p
K 0 ðsR=cÞ cos u du 0
Z
1
cos½fz bðr aÞgn dn
ð34Þ
0
and apply the integration formula, 1 p
Z
1
cosðxyÞ dx ¼ dðyÞ
ð35Þ
0
to Eq. (34). uh
H0 ¼ 2 dðz bðr aÞÞ pc
Z
p
K 0 ðsR=cÞ cos u du.
ð36Þ
0
Then, as the Laplace transform parameter s is included in the argument of the modified Bessel function, the inversion formula [15], H ðt aÞ L1 fK 0 ðasÞg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 a2 is applied. The Greens function is given by Z p pc H ðct RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u du uh ðr; z; tÞ ¼ dðz bðr aÞÞ H0 2 0 ðctÞ R2 8 > Z u < H ðct RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u du ¼ dðz bðr aÞÞ H ðr þ a ctÞH ðct jr ajÞ > 2 0 : ðctÞ R2 9 > Z p = H ðct RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u du ; þH ðct ðr þ aÞÞ > 0 ; ðctÞ2 R2
ð37Þ
ð38Þ
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðctÞ2 ðr aÞ2 . u ¼ 2sin1 4ar
ð39Þ
The definite integrals in Eq. (38) can be reduced to the standard form of elliptic integrals. The final form of the Greens function is given by 2c dðz bðr aÞÞ pffiffiffiffiffi uh ðr; z; tÞ ¼ H ðct ja rjÞH ða þ r ctÞf2EðkÞ KðkÞg H0 ar 1 2k 2 Kð1=kÞ þ 2kEð1=kÞ ; ð40Þ þH ðct ða þ rÞÞ k where the complete elliptic integral of the first and second kinds and their argument are defined by
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Z ffi 2 p=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k 2 sin2 h dh; p 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðctÞ2 ða rÞ2 . k¼ 4ar
EðkÞ ¼
KðkÞ ¼
2 p
Z
p=2 0
dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 k 2 sin2 h
ð41Þ
ð42Þ
The Greens function of Eq. (40) is multiplied by the delta function. This means that the disturbed region is limited on a plane with the line cross section, z = b(r a). The similar line (plane) disturbance in the limiting case of the monoclinic material has also been found in the case of SH-wave [12]. 6. Time-harmonic Green’s function The time-harmonic Greens function can be derived by performing the convolution integral, Z t0 ðGÞ uh ðr; z; tÞ ¼ expðixtÞ lim uh ðr; z; t0 Þ expðixt0 Þ dt0 ; 0 t !1
ð43Þ
0
where uðGÞ is the impulsive Greens function given by Eq. (25). That is z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Z 4pc2 expðixtÞ R2 R22 s2 s2 R21 expðixs=cÞ ds. uh ðr; z; tÞ ¼ 2 2 car H0 s R1 R22 s2 R1
ð44Þ
This integral can be reduced to the more convenient form for the numerical computation by introducing the variable transform, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R22 þ R21 R22 R21 s¼ sin /. ð45Þ 2 2 The final form of the time-harmonic Greens function for the torsional wave is given by 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Z p=2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pc2 expðixtÞ sin / ix R þ R 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp @ 1 e sin /A d/; uh ðr; z; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c H0 2 2 2 1 e sin / p=2 c 2ðR þ R Þ 2
ð46Þ
1
where R1 and R2 are given by Eq. (26) and the parameter e is defined by 0
R22 R21 < 1. R22 þ R21
ð47Þ
The integral in Eq. (46) is singular when e = 1. Extracting its singular part, we have the logarithmic singularity on the ring source, 2 pc expðixtÞ 1 log lim uh ðr; z; tÞ . ð48Þ e!1 4ca 1 e2 H0 ðr!a;z!0Þ
7. Numerical examples Shutilov [16] mentioned that an example of the monoclinic materials is Gypsum, but its elastic moduli are not shown in his book. The moduli for some other crystals are listed in it. So, we show the numerical examples Table 1 Material parameters for typical monoclinic crystals [16] Crystal
a = (c44/c55)1/2
b = c45/c55
c = (a2b2)1/2
Dibenzyl Sodium-thiosulfate
1.09 0.97
+0.31 0.45
1.04 0.86
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
ct/a=2
2
2
ct/a=2 1.25
z/a
1
1
1
1.25 1
0.5 0
0
0.5
-1
-1
-2
-2
r/a 0
0.5
1
1.5
2
r/a
2.5
0
3
0.5
(a) Dibenzyl
1
1.5
2
(b) Sodium-thiosulfate
Fig. 1. Time development of wave front.
30
4ρac uθ Θ0
20 (0.5, 0 .5)
10 0 (1.5, 0.5)
-10 (r / a, z / a )
-20
= (0.5, / 0.5)
ct/a
-30 0.5
1
1.5
2
2.5
(a) Dibenzyl 30 20
4ρac uθ Θ0
(r / a, z / a ) = (0.5, / 0.5)
10 0 -10 (0.5, 0.5)
(1.5, 0.5)
-20 -30 0.5
ct/a
1
2.5
1.5
2
2.5
3
(b) Sodium-thiosulfate Fig. 2. Time response of Greens function for torsional wave.
3
1289
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K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291 3
0.05
0.25
0.1
2
0.5 1
3
1
2
0 z/a -1 -2
r/a 0
0.5
1
1.5
2
2.5
-3 3
(a) Dibenzyl 3
0.05
0.1
0.25
2
0.5
1
1 3
0 z/a
2
-1 -2
r/a 0
0.5
1
1.5
2
2.5
-3 3
(b) Sodium-thiosulfate Fig. 3. Contours of displacement amplitude for time-harmonic Greens function,
pc2 H0
juh ðr; z; tÞj.
for two typical monoclinic crystals, positive and negative values of c45. They are dibenzyl and sodium-thiosulfate and their material parameters are listed in Table 1. Fig. 1 shows the time-development of wave fronts. At early time, its form is the inclined ellipse that is given by Eq. (27). After a unit time ct/a = 1, the second wave, W2, appears as if the reflection at the z-axis. But, it is penetrated from the opposite side due to the axisymmetric nature, not reflected wave. The inclination angle of the elliptic wave form is given by Eq. (29). The time response of the torsional displacement in Fig. 2 shows that there is no substantial difference in the response between positive and negative values of c45, since the response curves are similar to each other. Fig. 3 shows amplitude contours for the time-harmonic Greens function given by Eq. (46). Due to the torsional nature, the displacement vanishes on the center axis (r = 0). The contour at the higher amplitude region is a quasi-elliptic form and its form is gradually deformed at the lower amplitude region. 8. Conclusion Greens function for the impulsive and time-harmonic torsional waves is derived exactly for the monoclinic material. The Greens function for the transient wave is expressed in the closed form of simple algebraic functions. Due to the axially symmetric nature, the wave front shape is a torus, but its cross section is an inclined ellipse due to the monoclinic nature. At the wave front, the torsional displacement has the singularity of the inverse square root. This is the same as that in isotropic media. Time-harmonic Greens function is also derived and its amplitude contour is also quasi-elliptic form due to the monoclinic nature. Acknowledgement The authors would like to express their thanks to the referee for helpful comments in the revision of the paper.
K. Watanabe, R.G. Payton / International Journal of Engineering Science 43 (2005) 1283–1291
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References [1] R.G. Payton, K. Watanabe, Two-dimensional wave front shape for cylindrically anisotropic elastic media, Wave motion 33 (3) (2001) 211–224. [2] Y. Yokoyama, R.G. Payton, K. Watanabe, Wave front shapes in cylindrically anisotropic solids, J. Jpn. Soc. Non-Destruct. Inspect. 51 (5) (2002) 296–303 (in Japanese). [3] R.G. Payton, Wave fronts in wood, Q.J. Mech. Appl. Math. 56 (4) (2003) 527–546. [4] R.G. Payton, Wave front shape for a constrained cylindrically anisotropic solid, ZAMP 55 (2004) 658–677. [5] P.A. Martin, J. Berger, Waves in wood: free vibration of a wooden pole, J. Mech. Phys. Solids 49 (2001) 1155–1178. [6] P.A. Martin, Waves in wood: axisymmetric waves in slender solids of revolution, Wave Motion 40 (2004) 387–398. [7] A.L. Shuvalov, The Frobenius power series solution for cylindrically anisotropic radially inhomogeneous elastic materials, Q. Mech. Appl. Math. 56 (2003) 327–345. [8] K. Watanabe, Some exact closed form solutions for transient response of a cylindrically anisotropic elastic solid subjected to three types of a ring source, ZAMM 81 (11) (2001) 788–792. [9] K. Watanabe, K. Nishinari, SH wave in a cylindrically anisotropic solid, ZAMP 47 (6) (1996) 906–914. [10] K. Watanabe, R.G. Payton, SH wave in a cylindrically anisotropic elastic solid (A general solution for a point source), Wave Motion 25 (2) (1997) 197–212. [11] K. Watanabe, R.G. Payton, Time-harmonic SH-wave in cylindrically anisotropic elastic solid, Geophys. J. Int. 145 (2001) 709–713. [12] K. Watanabe, R.G. Payton, Greens function for SH-waves in a cylindrically monoclinic material, J. Mech. Phys. Solids 50 (2002) 2425–2439. [13] T.C.T. Ting, Anisotropic Elasticity, Oxford, 1996. [14] G.N. Watson, Theory of Bessel Functions, Cambridge, 1966. [15] A. Erdelyi (Ed.), Tables of Integral Transforms, vol. 1, McGraw-Hill, 1954. [16] V.A. Shutilov, Fundamental Physics of Ultrasound, Gordon & Breach Sci. Publ., 1988.