- Email: [email protected]
Body wave propagation in rotating elastic media
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 31 (2004) 21–27 www.elsevier.com/locate/mechrescom
Body wave propagation in rotating elastic media J.-L. Auriault
*
Laboratoire ‘‘Sols, Solides, Structures’’, Institut National Polytechnique de Grenoble, Centre National pour la Recherche Scientifique (UMR 5521), Universit e Joseph Fourier, Domaine Universitaire, BP 53X, 38041 Grenoble Cedex 9, France Received 23 July 2003
Abstract We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation X. Wave propagation is shown to be directly related to the Kibel number Ki ¼ x=X, where x is the wave frequency. Two dispersive waves W1 and W2 are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W1 shows a cutoff frequency xc ¼ X below which it does not propagate. The case of small angular rotation X is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order OðKi1 Þ. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Elastic wave; Rotating media; Kibel number; Dispersive wave
1. Introduction The propagation of elastic waves in inertial 3-D unbounded media is a very classical problem. For the sake of simplicity, consider an isotropic medium, of LameÕs coefficients k and l. The wave equation at constant angular frequency x is of the form lDui þ ðk þ lÞ
oe ¼ x2 qui ; oxi
ð1Þ
where u expðixtÞ, i2 ¼ 1, is the displacement vector, e ¼ ouj =oxj is the spherical strain and q is the material density. The investigation of Eq. (1) yields two types of non-dispersive body waves which propagate separately: a dilatational wave and an equivoluminal or shear wave that propagate at speeds cd ¼
k þ 2l q
1=2 ;
cs ¼
1=2 l ; q
respectively. *
Tel.: +33-4-76-82-51-68. E-mail address: [email protected] (J.-L. Auriault).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.07.002
ð2Þ
22
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
In the paper, wave propagation in rotating elastic media is addressed. This problem is of interest, e.g. in geophysics to model seismic waves inside Earth. More precisely, we consider an elastic medium in constant angular rotation X with respect to a fixed axis e3 . The wave propagation of a perturbation displacement in the plane ðe1 ; e2 Þ perpendicular to e3 may be affected by the Coriolis force. The vibrational analysis of elastic rotating structures such as beams, disks or membranes have been thoroughly addressed in the literature, see, e.g., (Advani, 1967; Advani and Bulkeley, 1969; Bulkeley, 1973; Hashemi and Richard, 2001; Huang and Wang, 2001; Lamb and Southwell, 1921; Luo and Mote, 2000; Nowinski, 1981; Southwell, 1922). These studies generally aim at determining the natural frequencies of some particular structure under rotation. The case of small angular rotation, X x, is addressed by geophysicists to investigate the Earth natural frequencies in numerous papers, see e.g. (Aki and Richards, 1980; Backus and Gilbert, 1961; Dahlen, 1968, 1969; Dahlen and Smith, 1975; MacDonald and Ness, 1961), and surface wave propagation, (Backus, 1962a,b). To our knowledge bulk waves have not been yet investigated in rotating media. This is the aim of the paper. The wave equation in rotating media is rapidly presented in the first section. In the following section, wave propagation is investigated as a function of the Kibel number Ki ¼ x=X. Two dispersive waves W1 and W2 are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W1 shows a cutoff frequency xc ¼ X: wave W1 does not propagate when x 6 X. The case of small angular rotation X is then studied. The corrections to be introduced to dilatational and shear waves in this case are shown to be of order Ki1 .
2. Wave equation in rotating media Consider an isotropic elastic medium which rotates at a constant angular velocity X with respect to a galilean axis ðO; e3 Þ. In the frame of the non-inertial medium, the momentum balance at constant frequency is in vectorial form lDu þ ðk þ lÞ grad div u ¼ q½x2 u þ X ðX OMÞ þ 2X ixu ;
ð3Þ
where X ðX OMÞ is the convective acceleration, M is the actual position of the particle under consideration and 2X ixu is the Coriolis acceleration. Taking advantage of the linearity of Eq. (3), the displacement u can be decomposed into a static displacement us , which is time independent, and a vibrational displacement uv , u ¼ us þ uv . The static displacement us is the solution of lDus þ ðk þ lÞ grad div us ¼ qX ðX OMs Þ
ð4Þ
and the vibrational displacement uv verifies the wave equation lDuv þ ðk þ lÞ grad div uv ¼ q½x2 uv þ X ðX uv Þ þ 2X ixuv :
ð5Þ
3. Dispersion relation Let ðe1 ; e2 ; e3 Þ be the rotating orthonormal basis with X ¼ Xe3 . A perturbation uv that is colinear to X is not affected by Coriolis or convective accelerations. Therefore, we limit the analysis to displacements in the plane ðe1 ; e2 Þ which remain constant in the direction e3 . By applying successively the divergence and the curl operators to Eq. (5), and after introducing e ¼ div uv and w ¼ $ uv ¼ w3 e3 , we obtain two coupled wave equations for e and w: ðk þ 2lÞDe ¼ q½ðx2 þ X2 Þe þ 2ixXw3 ;
ð6Þ
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
lDw3 ¼ q½ðx2 þ X2 Þw3 2ixXe :
23
ð7Þ
Therefore, in contrast to wave propagation in an inertial medium, dilatational and shear waves do not propagate separately. When X cancels out, we recover the uncoupled wave equations for propagation in an inertial medium. Consider waves that propagate in the direction e1 of the form e ¼ A1 exp½ikx1 ;
i2 ¼ 1:
w3 ¼ A2 exp½ikx1 ;
ð8Þ
Introducing these expressions into relations (6) and (7) gives two equations for the amplitudes A1 and A2 : ðk þ 2lÞk 2 A1 ¼ qðx2 þ X2 ÞA1 þ 2iqXxA2 ;
ð9Þ
lk 2 A2 ¼ qðx2 þ X2 ÞA2 2iqXxA1 :
ð10Þ
The existence of non-trivial solutions for A1 and A2 yields the dispersion equation 2
lðk þ 2lÞk 4 qðx2 þ X2 Þðk þ 3lÞk 2 þ q2 ðx2 X2 Þ ¼ 0:
ð11Þ
4. Dispersive waves Eq. (11) admits two body waves W1 and W2 of wavenumbers k1 and k2 , respectively. We have 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 2 2 qx X x X2 A 2 2 @ 1þ 2 : k1 ¼ k þ 3l ðk þ 3lÞ 4lðk þ 2lÞ 2lðk þ 2lÞ x x2 þ X2 2
ð12Þ
Waves W1 and W2 are dispersive waves. As X goes to zero, we recover the classical non-dispersive elastic waves, the dilatational wave and the shear wave of speed cd and cs , respectively: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi qx2 k þ 2l 2 2 ; ð13Þ ; cd ¼ lim k1 ¼ kd ¼ X!0 q k þ 2l lim
X!0
k22
¼
ks2
qx2 ; ¼ l
rffiffiffi l : cs ¼ q
ð14Þ
ph The phase velocities cph 1 ¼ x=k1 and c2 ¼ x=k2 of the waves W1 and W2 are easily obtained from Eq. (12). As X 6¼ 0, waves W1 and W2 are coupled dilatational-shear waves. The coupling is measured by the amplitude ratio. For each wave W1 and W2 , respectively, it can be put in the form A1 iKi1 A2 iKi1 or ¼ ¼ ð15Þ 2 2 ; ph A2 1 A1 1 c2 =c 1 1 c2 =cph 2
d
1 2
2
s
1 2
where Ki ¼ x=X is the Kibel number. Due to Coriolis effects, the dilatational and the shear components are in phase quadrature. In dimensionless form, the phase velocities are 1 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2 2ffi 1 ph 2 2 c 2 2Ki 1 1 4 Ki 1 A ; @1 þ þ1 cph ¼ 12 ¼ ð16Þ 1 cd C C C 1 þ Ki2 1 þ Ki2
2 cph 2
1 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 cph 2Ki2 @ Ki 1 A 2 2 2 ¼ 2 ¼ 1 þ C þ ð1 þ CÞ 4C : cs 1 þ Ki2 1 þ Ki2
ð17Þ
24
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
The constant C is a dimensionless parameter defined by C¼
c2s l 1 2g ¼ ; ¼ 2 cd k þ 2l 2ð1 gÞ
ð18Þ
where g is the Poisson ratio. The number C represents the elastic properties of the medium. Velocities cph 1 and cph are shown in Figs. 1 and 2 versus the Kibel number Ki for different values of the Poisson ratio g. 2 We note that the dimensionless phase velocity of the wave W1 slightly depends on the elastic properties of the medium. However, it increases considerably with X at constant x. An important fact is that W1 shows a cutoff frequency xc which is obtained for Ki ¼ 1, xc ¼ X, see Fig. 1. For x < X, the phase velocity of wave W1 becomes complex and the wave does not propagate. This phenomenon is classical for waves in waveguides, see (Miklowitz, 1980), and it is also encountered for waves in elastic composite materials with high property contrast, (Auriault and Bonnet, 1985). In contrast to wave W1 , the dimensionless phase velocity of wave W2 is more strongly dependent on the Poisson ratio, as seen in Fig. 2. As Ki goes to zero, the velocity cph tends to zero. 2
Fig. 1. Dimensionless phase velocity of wave W1 versus Kibel number Ki for different values of the Poisson ratio g.
Fig. 2. Dimensionless phase velocity of wave W2 versus Kibel number Ki for different values of the Poisson ratio g.
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
25
Fig. 3. Dimensionless group velocity of wave W1 versus Kibel number Ki for different values of the Poisson ratio g.
Fig. 4. Dimensionless group velocity of wave W2 versus Kibel number Ki for different values of the Poisson ratio g. gr The group velocities of waves W1 and W2 are defined by cgr 1 ¼ ox=ok1 and c2 ¼ ox=ok2 , respectively. gr gr gr gr Their dimensionless forms c1 ¼ c1 =cd and c2 ¼ c2 =cs are shown in Figs. 3 and 4 versus the Kibel number for different values of the Poisson ratio. In contrast to the phase velocity, the group velocity of wave W1 now strongly depends on the Poisson ratio and it tends to zero as x reaches to the cutoff frequency xc . We also see that the group velocity of wave W2 goes to infinity as Ki goes to zero.
5. Large Kibel number We now consider large Kibel numbers, i.e. small angular rotation X relatively to the wave frequency x. This is the case, e.g., for waves in a slowly rotating Earth. Relations (12), (16) and (17) are then approximated in the form k12 Ki2 ðC þ 5Þ ; 1 ð1 CÞ kd2
cph 1þ 1
Ki2 ðC þ 5Þ ; ð1 CÞ
ð19Þ
26
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
k22 Ki2 ð1 þ 3CÞ ; 1 þ ð1 CÞ ks2
cph 1 1
Ki2 ð1 þ 3CÞ : ð1 CÞ
ð20Þ
These relations show that Coriolis corrections are of order Ki2 . Concerning the amplitudes, we obtain from (15) the two relations A2 iKi1 A1 iKi1 : ð21Þ ; A1 1 1 c2s =c2d A2 2 c2d =c2s 1 The Coriolis amplitude corrections are of order Ki1 . Finally, we have to the first-order approximation iKi1 e uv1 A1 exp½iks x1 e1 þ ; ð22Þ 2 1 c2s =c2d uv2 A2 exp½ikd x1
iKi1 e þ e 1 2 : c2d =c2s 1
ð23Þ
Such an order X correction has been obtained by Backus (1962a,b), for Rayleigh waves on the Earth surface.
6. Conclusions Free wave propagation in non-Galilean rotating media gives rise to two dispersive waves W1 and W2 which are coupled dilatational-shear waves. Due to Coriolis effects, the dilatational and the shear components of each wave W1 or W2 are in phase quadrature. As the angular rotation of the elastic medium goes to zero, waves W1 and W2 tend to the classical dilatational and shear waves, respectively. Wave W1 displays a cutoff frequency xc ¼ X, below which it does not propagate. Dimensionless phase and group velocities depend on the Kibel number Ki ¼ x=X and on the elastic properties of the medium through a dimensionless number C ¼ c2s =c2d , which is related to the Poisson ratio only. The case of large Kibel number, i.e., small angular velocity, shows small corrections to dilatational and shear wave velocities that are of OðKi1 Þ.
References Advani, S.H., 1967. Stationary waves in a thin spinning disk. Int. J. Mech. Sci. 9, 307–313. Advani, S.H., Bulkeley, P.Z., 1969. Nonlinear transverse vibrations and waves in spinning membrane discs. Int. J. Non-linear Mech. 4, 123–127. Aki, K., Richards, P.G., 1980. Quantitative Seismology, Theory and Methods. W.H. Freeman & Co., San Francisco. Auriault, J.-L., Bonnet, G., 1985. Dynamique des composites elastiques periodiques. Arch. Mech. 37 (4–5), 269–284 (in French). Backus, G.E., 1962a. The propagation of short elastic surface waves on a slowly rotating Earth. Bull. Seismologic. Soc. Am. 52 (4), 823–846. Backus, G.E., 1962b. Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths. Bull. Seismologic. Soc. Am. 54 (2), 571–610. Backus, G., Gilbert, F., 1961. The rotational splitting of the free oscillations of the Earth. Proc. Nat. Academy Sci. United States Am. 47, 362–373. Bulkeley, P.Z., 1973. Stability of transverse waves in a spinning membrane disk. ASME J. Appl. Mech. 40, 133–136. Dahlen, F.A., 1968. The normal modes of a rotating, elliptical Earth. Geophys. J. R. Astr. Soc. 16, 329–367. Dahlen, F.A., 1969. The normal modes of a rotating, elliptical Earth-II Near-resonance multiplet coupling. Geophys. J. R. Astr. Soc. 18, 397–436. Dahlen, F.A., Smith, M.L., 1975. The influence of rotation on the free oscillations of the Earth. Trans. Roy. Soc. Lon. A 279, 583–624. Hashemi, S.M., Richard, M.J., 2001. Natural frequencies of rotating uniform beams with Coriolis effects. J. Vibr. Acoust. 123, 444– 455.
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
27
Huang, Y.M., Wang, C.-M., 2001. Combined methodology for analysis of rotary systems. J. Vibr. Acoust. 123, 428–434. Lamb, H., Southwell, R.V., 1921. The vibrations of a spinning disk. Proc. Roy. Soc. Lon. A: Math. Phys. Sci. 99, 272–280. Luo, A.C.J., Mote, C.D., 2000. Nonlinear vibration of rotating thin disks. J. Vibr. Acoust. 122, 376–383. MacDonald, G.J.F., Ness, N.F., 1961. A study of the free oscillations of the Earth. J. Geophys. Res. 66 (6), 1865–1911. Miklowitz, J., 1980. The Theory of Elastic Waves and Waveguides. North-Holland, Amsterdam. Nowinski, J.L., 1981. Stability of thermoelastic waves in membrane-like spinning disks. J. Thermal Sci. 4, 1–11. Southwell, R.V., 1922. On the free transverse vibrations of a uniform circular disc clamped at its centre; and on the effects of rotation. Proc. Roy. Soc. Lon. A: Math. Phys. Sci. 101, 133–153.
Mechanics Research Communications 31 (2004) 21–27 www.elsevier.com/locate/mechrescom
Body wave propagation in rotating elastic media J.-L. Auriault
*
Laboratoire ‘‘Sols, Solides, Structures’’, Institut National Polytechnique de Grenoble, Centre National pour la Recherche Scientifique (UMR 5521), Universit e Joseph Fourier, Domaine Universitaire, BP 53X, 38041 Grenoble Cedex 9, France Received 23 July 2003
Abstract We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation X. Wave propagation is shown to be directly related to the Kibel number Ki ¼ x=X, where x is the wave frequency. Two dispersive waves W1 and W2 are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W1 shows a cutoff frequency xc ¼ X below which it does not propagate. The case of small angular rotation X is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order OðKi1 Þ. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Elastic wave; Rotating media; Kibel number; Dispersive wave
1. Introduction The propagation of elastic waves in inertial 3-D unbounded media is a very classical problem. For the sake of simplicity, consider an isotropic medium, of LameÕs coefficients k and l. The wave equation at constant angular frequency x is of the form lDui þ ðk þ lÞ
oe ¼ x2 qui ; oxi
ð1Þ
where u expðixtÞ, i2 ¼ 1, is the displacement vector, e ¼ ouj =oxj is the spherical strain and q is the material density. The investigation of Eq. (1) yields two types of non-dispersive body waves which propagate separately: a dilatational wave and an equivoluminal or shear wave that propagate at speeds cd ¼
k þ 2l q
1=2 ;
cs ¼
1=2 l ; q
respectively. *
Tel.: +33-4-76-82-51-68. E-mail address: [email protected] (J.-L. Auriault).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2003.07.002
ð2Þ
22
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
In the paper, wave propagation in rotating elastic media is addressed. This problem is of interest, e.g. in geophysics to model seismic waves inside Earth. More precisely, we consider an elastic medium in constant angular rotation X with respect to a fixed axis e3 . The wave propagation of a perturbation displacement in the plane ðe1 ; e2 Þ perpendicular to e3 may be affected by the Coriolis force. The vibrational analysis of elastic rotating structures such as beams, disks or membranes have been thoroughly addressed in the literature, see, e.g., (Advani, 1967; Advani and Bulkeley, 1969; Bulkeley, 1973; Hashemi and Richard, 2001; Huang and Wang, 2001; Lamb and Southwell, 1921; Luo and Mote, 2000; Nowinski, 1981; Southwell, 1922). These studies generally aim at determining the natural frequencies of some particular structure under rotation. The case of small angular rotation, X x, is addressed by geophysicists to investigate the Earth natural frequencies in numerous papers, see e.g. (Aki and Richards, 1980; Backus and Gilbert, 1961; Dahlen, 1968, 1969; Dahlen and Smith, 1975; MacDonald and Ness, 1961), and surface wave propagation, (Backus, 1962a,b). To our knowledge bulk waves have not been yet investigated in rotating media. This is the aim of the paper. The wave equation in rotating media is rapidly presented in the first section. In the following section, wave propagation is investigated as a function of the Kibel number Ki ¼ x=X. Two dispersive waves W1 and W2 are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W1 shows a cutoff frequency xc ¼ X: wave W1 does not propagate when x 6 X. The case of small angular rotation X is then studied. The corrections to be introduced to dilatational and shear waves in this case are shown to be of order Ki1 .
2. Wave equation in rotating media Consider an isotropic elastic medium which rotates at a constant angular velocity X with respect to a galilean axis ðO; e3 Þ. In the frame of the non-inertial medium, the momentum balance at constant frequency is in vectorial form lDu þ ðk þ lÞ grad div u ¼ q½x2 u þ X ðX OMÞ þ 2X ixu ;
ð3Þ
where X ðX OMÞ is the convective acceleration, M is the actual position of the particle under consideration and 2X ixu is the Coriolis acceleration. Taking advantage of the linearity of Eq. (3), the displacement u can be decomposed into a static displacement us , which is time independent, and a vibrational displacement uv , u ¼ us þ uv . The static displacement us is the solution of lDus þ ðk þ lÞ grad div us ¼ qX ðX OMs Þ
ð4Þ
and the vibrational displacement uv verifies the wave equation lDuv þ ðk þ lÞ grad div uv ¼ q½x2 uv þ X ðX uv Þ þ 2X ixuv :
ð5Þ
3. Dispersion relation Let ðe1 ; e2 ; e3 Þ be the rotating orthonormal basis with X ¼ Xe3 . A perturbation uv that is colinear to X is not affected by Coriolis or convective accelerations. Therefore, we limit the analysis to displacements in the plane ðe1 ; e2 Þ which remain constant in the direction e3 . By applying successively the divergence and the curl operators to Eq. (5), and after introducing e ¼ div uv and w ¼ $ uv ¼ w3 e3 , we obtain two coupled wave equations for e and w: ðk þ 2lÞDe ¼ q½ðx2 þ X2 Þe þ 2ixXw3 ;
ð6Þ
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
lDw3 ¼ q½ðx2 þ X2 Þw3 2ixXe :
23
ð7Þ
Therefore, in contrast to wave propagation in an inertial medium, dilatational and shear waves do not propagate separately. When X cancels out, we recover the uncoupled wave equations for propagation in an inertial medium. Consider waves that propagate in the direction e1 of the form e ¼ A1 exp½ikx1 ;
i2 ¼ 1:
w3 ¼ A2 exp½ikx1 ;
ð8Þ
Introducing these expressions into relations (6) and (7) gives two equations for the amplitudes A1 and A2 : ðk þ 2lÞk 2 A1 ¼ qðx2 þ X2 ÞA1 þ 2iqXxA2 ;
ð9Þ
lk 2 A2 ¼ qðx2 þ X2 ÞA2 2iqXxA1 :
ð10Þ
The existence of non-trivial solutions for A1 and A2 yields the dispersion equation 2
lðk þ 2lÞk 4 qðx2 þ X2 Þðk þ 3lÞk 2 þ q2 ðx2 X2 Þ ¼ 0:
ð11Þ
4. Dispersive waves Eq. (11) admits two body waves W1 and W2 of wavenumbers k1 and k2 , respectively. We have 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 2 2 qx X x X2 A 2 2 @ 1þ 2 : k1 ¼ k þ 3l ðk þ 3lÞ 4lðk þ 2lÞ 2lðk þ 2lÞ x x2 þ X2 2
ð12Þ
Waves W1 and W2 are dispersive waves. As X goes to zero, we recover the classical non-dispersive elastic waves, the dilatational wave and the shear wave of speed cd and cs , respectively: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi qx2 k þ 2l 2 2 ; ð13Þ ; cd ¼ lim k1 ¼ kd ¼ X!0 q k þ 2l lim
X!0
k22
¼
ks2
qx2 ; ¼ l
rffiffiffi l : cs ¼ q
ð14Þ
ph The phase velocities cph 1 ¼ x=k1 and c2 ¼ x=k2 of the waves W1 and W2 are easily obtained from Eq. (12). As X 6¼ 0, waves W1 and W2 are coupled dilatational-shear waves. The coupling is measured by the amplitude ratio. For each wave W1 and W2 , respectively, it can be put in the form A1 iKi1 A2 iKi1 or ¼ ¼ ð15Þ 2 2 ; ph A2 1 A1 1 c2 =c 1 1 c2 =cph 2
d
1 2
2
s
1 2
where Ki ¼ x=X is the Kibel number. Due to Coriolis effects, the dilatational and the shear components are in phase quadrature. In dimensionless form, the phase velocities are 1 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2 2ffi 1 ph 2 2 c 2 2Ki 1 1 4 Ki 1 A ; @1 þ þ1 cph ¼ 12 ¼ ð16Þ 1 cd C C C 1 þ Ki2 1 þ Ki2
2 cph 2
1 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 cph 2Ki2 @ Ki 1 A 2 2 2 ¼ 2 ¼ 1 þ C þ ð1 þ CÞ 4C : cs 1 þ Ki2 1 þ Ki2
ð17Þ
24
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
The constant C is a dimensionless parameter defined by C¼
c2s l 1 2g ¼ ; ¼ 2 cd k þ 2l 2ð1 gÞ
ð18Þ
where g is the Poisson ratio. The number C represents the elastic properties of the medium. Velocities cph 1 and cph are shown in Figs. 1 and 2 versus the Kibel number Ki for different values of the Poisson ratio g. 2 We note that the dimensionless phase velocity of the wave W1 slightly depends on the elastic properties of the medium. However, it increases considerably with X at constant x. An important fact is that W1 shows a cutoff frequency xc which is obtained for Ki ¼ 1, xc ¼ X, see Fig. 1. For x < X, the phase velocity of wave W1 becomes complex and the wave does not propagate. This phenomenon is classical for waves in waveguides, see (Miklowitz, 1980), and it is also encountered for waves in elastic composite materials with high property contrast, (Auriault and Bonnet, 1985). In contrast to wave W1 , the dimensionless phase velocity of wave W2 is more strongly dependent on the Poisson ratio, as seen in Fig. 2. As Ki goes to zero, the velocity cph tends to zero. 2
Fig. 1. Dimensionless phase velocity of wave W1 versus Kibel number Ki for different values of the Poisson ratio g.
Fig. 2. Dimensionless phase velocity of wave W2 versus Kibel number Ki for different values of the Poisson ratio g.
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
25
Fig. 3. Dimensionless group velocity of wave W1 versus Kibel number Ki for different values of the Poisson ratio g.
Fig. 4. Dimensionless group velocity of wave W2 versus Kibel number Ki for different values of the Poisson ratio g. gr The group velocities of waves W1 and W2 are defined by cgr 1 ¼ ox=ok1 and c2 ¼ ox=ok2 , respectively. gr gr gr gr Their dimensionless forms c1 ¼ c1 =cd and c2 ¼ c2 =cs are shown in Figs. 3 and 4 versus the Kibel number for different values of the Poisson ratio. In contrast to the phase velocity, the group velocity of wave W1 now strongly depends on the Poisson ratio and it tends to zero as x reaches to the cutoff frequency xc . We also see that the group velocity of wave W2 goes to infinity as Ki goes to zero.
5. Large Kibel number We now consider large Kibel numbers, i.e. small angular rotation X relatively to the wave frequency x. This is the case, e.g., for waves in a slowly rotating Earth. Relations (12), (16) and (17) are then approximated in the form k12 Ki2 ðC þ 5Þ ; 1 ð1 CÞ kd2
cph 1þ 1
Ki2 ðC þ 5Þ ; ð1 CÞ
ð19Þ
26
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
k22 Ki2 ð1 þ 3CÞ ; 1 þ ð1 CÞ ks2
cph 1 1
Ki2 ð1 þ 3CÞ : ð1 CÞ
ð20Þ
These relations show that Coriolis corrections are of order Ki2 . Concerning the amplitudes, we obtain from (15) the two relations A2 iKi1 A1 iKi1 : ð21Þ ; A1 1 1 c2s =c2d A2 2 c2d =c2s 1 The Coriolis amplitude corrections are of order Ki1 . Finally, we have to the first-order approximation iKi1 e uv1 A1 exp½iks x1 e1 þ ; ð22Þ 2 1 c2s =c2d uv2 A2 exp½ikd x1
iKi1 e þ e 1 2 : c2d =c2s 1
ð23Þ
Such an order X correction has been obtained by Backus (1962a,b), for Rayleigh waves on the Earth surface.
6. Conclusions Free wave propagation in non-Galilean rotating media gives rise to two dispersive waves W1 and W2 which are coupled dilatational-shear waves. Due to Coriolis effects, the dilatational and the shear components of each wave W1 or W2 are in phase quadrature. As the angular rotation of the elastic medium goes to zero, waves W1 and W2 tend to the classical dilatational and shear waves, respectively. Wave W1 displays a cutoff frequency xc ¼ X, below which it does not propagate. Dimensionless phase and group velocities depend on the Kibel number Ki ¼ x=X and on the elastic properties of the medium through a dimensionless number C ¼ c2s =c2d , which is related to the Poisson ratio only. The case of large Kibel number, i.e., small angular velocity, shows small corrections to dilatational and shear wave velocities that are of OðKi1 Þ.
References Advani, S.H., 1967. Stationary waves in a thin spinning disk. Int. J. Mech. Sci. 9, 307–313. Advani, S.H., Bulkeley, P.Z., 1969. Nonlinear transverse vibrations and waves in spinning membrane discs. Int. J. Non-linear Mech. 4, 123–127. Aki, K., Richards, P.G., 1980. Quantitative Seismology, Theory and Methods. W.H. Freeman & Co., San Francisco. Auriault, J.-L., Bonnet, G., 1985. Dynamique des composites elastiques periodiques. Arch. Mech. 37 (4–5), 269–284 (in French). Backus, G.E., 1962a. The propagation of short elastic surface waves on a slowly rotating Earth. Bull. Seismologic. Soc. Am. 52 (4), 823–846. Backus, G.E., 1962b. Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths. Bull. Seismologic. Soc. Am. 54 (2), 571–610. Backus, G., Gilbert, F., 1961. The rotational splitting of the free oscillations of the Earth. Proc. Nat. Academy Sci. United States Am. 47, 362–373. Bulkeley, P.Z., 1973. Stability of transverse waves in a spinning membrane disk. ASME J. Appl. Mech. 40, 133–136. Dahlen, F.A., 1968. The normal modes of a rotating, elliptical Earth. Geophys. J. R. Astr. Soc. 16, 329–367. Dahlen, F.A., 1969. The normal modes of a rotating, elliptical Earth-II Near-resonance multiplet coupling. Geophys. J. R. Astr. Soc. 18, 397–436. Dahlen, F.A., Smith, M.L., 1975. The influence of rotation on the free oscillations of the Earth. Trans. Roy. Soc. Lon. A 279, 583–624. Hashemi, S.M., Richard, M.J., 2001. Natural frequencies of rotating uniform beams with Coriolis effects. J. Vibr. Acoust. 123, 444– 455.
J.-L. Auriault / Mechanics Research Communications 31 (2004) 21–27
27
Huang, Y.M., Wang, C.-M., 2001. Combined methodology for analysis of rotary systems. J. Vibr. Acoust. 123, 428–434. Lamb, H., Southwell, R.V., 1921. The vibrations of a spinning disk. Proc. Roy. Soc. Lon. A: Math. Phys. Sci. 99, 272–280. Luo, A.C.J., Mote, C.D., 2000. Nonlinear vibration of rotating thin disks. J. Vibr. Acoust. 122, 376–383. MacDonald, G.J.F., Ness, N.F., 1961. A study of the free oscillations of the Earth. J. Geophys. Res. 66 (6), 1865–1911. Miklowitz, J., 1980. The Theory of Elastic Waves and Waveguides. North-Holland, Amsterdam. Nowinski, J.L., 1981. Stability of thermoelastic waves in membrane-like spinning disks. J. Thermal Sci. 4, 1–11. Southwell, R.V., 1922. On the free transverse vibrations of a uniform circular disc clamped at its centre; and on the effects of rotation. Proc. Roy. Soc. Lon. A: Math. Phys. Sci. 101, 133–153.