ENERGY OF RAYLEIGH WAVES

Transcription

1 ENERGY OF RAYLEIGH WAVES Sergey V. Kuznetsov Institute for Problems in Mechanics Prosp. Vernadsogo, Moscow 756 Russia Abstract A general theory of Rayleigh waves propagating in media with arbitrary elastic anisotropy is developed on the basis of combination of six- and three-dimensional complex formalism. Case of multiple roots for the Christoffel equation is studied, numerical results for the inetic and strain energy functions are analysed. Problem of existence of the forbidden directions for Rayleigh waves is discussed.. Introduction. Conventionally, Rayleigh wave means the surface wave propagating in the half-space and exponentially attenuating with depth. Following Rayleigh wor (Strutt; 885 where propagation of the surface waves in isotropic medium was studied, such waves are composed of three partial waves u( x C m e i r( xnxct ( where u is the displacement field; boundary Fig.. n of the regarded half-space; C are complex coefficients determined up to a multiplier by the traction-free boundary conditions; m are complex eigenvectors (amplitudes of the Christoffel equation, these eigenvectors correspond to complex roots of the characteristic polynomial; r is the wave number; is an outward normal to the n is the unit vector determining direction of propagation, refer to Fig. ; and c is the phase speed. Terms i r( xnxct m e (

2 in representation ( are called partial waves. REMARK.. Attenuation with depth at the lower half-space ( x can be achieved, if complex roots in the representation ( have negative imaginary part, that will be assumed. It should be noted that while the third-order polynomial equation for determination of the surface wave speed was obtained by Rayleigh (Rayleigh equation, analytical expression for the speed, giving it as a function of elastic parameters, is absent until now. Approximate linear-fractional formula for the speed of Rayleigh wave propagating in an isotropic medium was proposed by Bergmann (97, some times this formula is attributed to Victorov (967. The representation similar to ( was used by Stoneley (955 in an analytical study of Rayleigh waves propagating in planes of elastic symmetry of cubic crystals in directions coinciding with the axes of elastic symmetry. For such a case, the thirdorder polynomial equation analogous to the Rayleigh equation, was derived. Further, Dieulesaint and Royer (98, and Royer and Dieulesaint (985 extrapolated Stoneley s analytical approach on Rayleigh waves propagating in planes of elastic symmetry of orthorhombic and tetragonal crystals. One, interesting result flowing out from these studies is concerned with the constructive proof of non-existence of forbidden directions which coincide with the axes of elastic symmetry of cubic, orthorhombic and tetragonal crystals. Here the forbidden direction means a direction along which Rayleigh wave cannot propagate. Numerical study of Rayleigh waves propagating in anisotropic crystals was undertaen by Farnell (97, see also Lim and Farnell (968, 969, on the bases of three-dimensional complex formalism which utilizes representation (. One of these papers (Lim and Farnell, 968 was devoted for search of forbidden directions. Numerical analysis revealed that for all the tested materials and directions Rayleigh waves existed (with an exception when Rayleigh wave degenerates into bul shear wave, so the non-degenerate forbidden directions were not found. Sextic formalism which establishes similarity between moving line dislocation and propagation of Rayleigh wave was proposed by Stroh (958, 96. In the course of theoretical study based on this formalism theorems of existence and

3 uniqueness for Rayleigh waves were proved (Barnett and Kirchner, 997; Barnett and Lothe, 97, 974; Lothe and Barnett, 976; Chadwic et al., 977, 979, 987, 99; Gunderson, Barnett and Lothe, 987; Ting and Barnett, 997. The developed theory also accounts the exceptional case of the degeneracy studied by Lim and Farnell (968. Numerical algorithms based on the sextic formalism were wored out by Barnett, Nishioa and Lothe (97, Mase (987, Mase and Johnson (987. Corresponding calculations confirmed absence of the forbidden directions. Still, there remains a chance for existence of forbidden directions in anisotropic materials, when multiple roots arise in the Christoffel equation with simultaneous appearing the Jordan blocs in a matrix associated with the Christoffel equation. Presumably, all the regarded theories beginning from Stoneley s wor (955 neglected appearance of multiple roots with the corresponding Jordan blocs. Following analysis based on combination of three- and six-dimensional complex formalism, covers cases of both primitive and multiple roots. In the course of theoretical and numerical study it is shown that the forbidden directions exist, and they can even coincide with the axes of elastic symmetry of some cubic crystals. Moreover, analysis of the stored energy function indicates stability of the forbidden direction with respect to small variations of the direction of propagation.. Basic notations. Equations of motion for anisotropic medium can be written in the form A( x, t u div xcxu u ( where is the density of the medium; and C is the fourth-order elasticity tensor assumed to be positive definite: A Asym( R R, A ijmn ( A C A Aij C Amn i, j, m, n (4 Along with condition (4 supplementary positive definite condition (SPDcondition will be imposed on the elasticity tensor:

4 A Asym( R R, A ijmn ( AC A Ain C Ajm i, n, j, m (5 It will be shown later that generally, neither (4, nor (5 implies from each other. In addition to (4, (5 the material is assumed to be hyperelastic, this ensures symmetry of the elasticity tensor regarded as linear operator acting in the sixdimensional space sym( R R : A, B A, B sym( R R BCA ACB (6 Direct analysis shows, that under condition (5 following lemma holds a LEMMA.. For any orthogonal vectors Ca, b Cb and a Cb bca a, b R symmetric matrices do not commute with each other.. The Christoffel equation. Substituting partial wave ( in Eq. ( produces the Christoffel equation ( n C( n I G( m c m (7 where I is the unit diagonal matrix. Equation (7 can be represented in the equivalent form det G ( (8 Left-hand side of Eq. (8 is a polynomial of degree 6 with respect to.

5 REMARK.. Representation ( and, hence, the Christoffel equation in the form (7 are valid for the case of primitive roots. Situation when multiple roots arise will be considered in the following section. For the subsequent analysis following definition for the limiting speed lim c,,, is needed (Stroh; 96: lim с inf cos ( ( w C w (9 ; where w sin( cos( n ( and ( w C w,,, are eigenvalues of the matrix ( w C w arranged in the descending mode. Reason for introduction of the limiting speed is motivated by the following a If b If PROPOSITION.. Let c be the phase speed: lim c c, then all roots of the Christoffel equation (8 are real; lim lim c c c, then the Christoffel equation has four real and one pair of complex-conjugate roots; c If lim lim c c c complex-conjugate roots; lim, then the Christoffel equation has two real and two pairs of d If c c, then the Christoffel equation has three pairs of complexconjugate roots. PROOF. Polynomial coefficients in Eq. (8 are all real, this ensures that if is a complex root, then is a root also. Assumption that is real allows to introduce a

6 new parameter ( / ; /, such that tan. In terms of the parameter Eq. (8 can be represented in the form det cos ( C w I w c ( where w is the real-valued vector defined by (. Reduction of the matrix ( w C w to the diagonal form gives c cos ( w C w, ( Actually, relation ( completes the proof. PROPOSITION.. For any complex matrix G in (7 is not normal. Re( G PROOF. Analysis of the matrix G produces: ncn c I (Re ( Im ( ( C Re( ( nc Cn Im( G Im( Re( ( C ( Cn nc But in view of Lemma., the Hermitian components Re(G, Im(G do not commute, so matrix G is not normal. ( COROLLARY. For any complex orthonormal basis in C. eigenvectors of the matrix G do not form REMARK.. The following analysis is confined to the interval lim c с, where three pairs of the complex-conjugate roots arise.

7 4. Six-dimensional complex formalism. To analyze the case when multiple roots of the Christoffel equation arise, some modification of representation ( is needed. Let partial waves be of the form i r( nxct v ( x e (4 where x ( x and v (x is unnown vector function. The exponential multiplier in (4 corresponds to propagation of the plane wave along the direction n with the phase speed c. It is clear, that representation (4 generalises (. Substituting (4 in Eq. ( produces ( C ir( C n ( n C r n C n c I v( x x To simplify the analysis of eigensolutions of Eq. (5, reduction to the first order (matrix differential equation is desirable. Let reduction to the first-order equation in 6 C : x w (5 x v, then Eq. (5 admits v v x I6 R6 (6 w w where I 6 denotes the unit (diagonal matrix in 6 R, and R 6 r ( C ( ncn c I ir ( C I ( Cn nc (7 In (7 I stands for the unit (diagonal matrix in R. lim c PROPOSITION 4.. Let c (; : a If is an eigenvalue of the matrix R 6, then Re ;

8 b If i is an eigenvalue and m ( m, m is corresponding eigenvector of the matrix R 6, where m, m C, then i is also an eigenvalue with corresponding eigenvector m ( m, m. PROOF. If is an eigenvalue with corresponding eigenvector m ( m, m, then due to (7, action of the matrix R 6 on this eigenvector gives: m, m er ( C ir ( Cn n C r ( n Cn The latter expression is equivalent c I (8 m, m er ( C ( Cn n C ( n Cn c I (9 where it is denoted ir. Now, it is clear that, matrix in right-hand side of (9 coincides with one in the Christoffel equation (7. To complete the proof it remains to note that the polynomial in (8 has real coefficients. Following flows out directly from the proof of the preceding Proposition COROLLARY. a Eigenvalues of the matrix R 6 and roots of the Christoffel equation are connected by the relation ir ; b Six-dimensional eigenvectors m ( m, m of the matrix R 6 correspond to the three-dimensional ernel vectors m, m C of the Christoffel equation. According to the general theory of ordinary differential equations, see Hartman (964, structure of eigensolutions of the system (6 depends on the Jordan form of lim c the matrix R 6. Directly from Proposition 4. it follows, that for c (; there may be three inds of the Jordan normal form of the matrix R 6 :

9 ,, (II 6 (I 6 ir ir J J (III 6 ir J ( In ( 6 J denotes the Jordan normal form of the matrix 6 R. REMARK. 4.. In view of expressions ( and the theory of ordinary differential equations (Harman; 964, following three types of representations for the surface waves can occur: (i for the Jordan normal form (I 6 J corresponding representation is well nown and is given by (; (ii for the Jordan normal form (II 6 J corresponding representation is as follows: ( ( ( ( ct r i ct r i e C e r C C x n x x n x m m x x u ( where C m corresponds to the eigenvalue ir of (II 6 J, and C m corresponds to the eigenvalue ir (multiplier r in the term x r C is included to mae coefficient C dimensionless; (iii for the Jordan normal form (III 6 J the representation is as follows:

10 u( x ( C C i r( xnxct r x C ( r x m e ( where C m corresponds to the eigenvalue ir of (III J 6. Thus, expressions (, ( determine the unnown vector-function v in representation (4. Following proposition restricts possible types of the Jordan normal form. PROPOSITION 4.. The Jordan normal form (III J 6 cannot arise. PROOF. Substituting the term ( into Eq. ( and taing ( x ( i r xnxct m e ( r x from representation, one gets: Cm inconsistent with the positive definiteness of the tensor C.. But the latter is REMARK 4.. It will be assumed that both three-dimensional vectors m and complex coefficients C are normalised, i.e. m, where denotes l - norm in C, and C. Expressions (, ( along with Remar 4. ensure PROPOSITION 4.. The displacement field given by ( or (, does not exceed unity at any x belonging to the regarded half-space. 5. Boundary conditions. In this section the three-dimensional complex formalism is adopted. The traction-free boundary conditions on the surface be written in the form: can

11 t C u ( x According to Proposition 4. following two cases can occur when substituting corresponding representations for the surface wave into boundary conditions (: (i for the Jordan normal form (I J 6 and representation ( C C C n m (4 (ii for the Jordan normal form (II J 6 and representation ( C C C m C m C n ( C m C m (( (5 where C ic. Expressions (4, (5 can be regarded as linear equations with respect to unnown coefficients C. Existence of non-trivial solutions of (4, (5 is equivalent to vanishing of the corresponding determinants, and this also provides necessary and sufficient condition for existence of Rayleigh wave. Matrices in (4, (5 are not Hermitian, while from theoretical and computational point of view it is more convenient to analyse eigenvalues of Hermitian matrices. These matrices can be obtained through multiplication of matrices in (4, (5 by corresponding conjugates. For system (4 it gives: m j n C C C C n j j,, m C (6 Analogous procedure applied to system (5 gives following components of the Hermitian matrix:

12 H j m j m j m j n C C C C n C C j m, n C C C j m, m,,; j, ; j ; j, (7 REMARK 5.. Normalised (complex three-dimensional vectors C C ; C ; belonging to the ernel spaces of matrices in (4 and (5, ( C coincide with the vectors belonging to the ernel spaces of the respective matrices (6 and (7, see Marcus and Min (964; Propositions.7. and Energy functions. Let x x x U( x u( x C u( (8 be the strain energy function; and, respectively u( x r u( K( x c x (9 be the inetic energy function. Integration of these functions over specific volume and averaging over time domain produces strain and inetic energy respectively (see, Gurtin; 97. Corresponding fluxes of strain and inetic energy can be defined by following formulas: F U ( x U( x; F ( x K( x ( x K x Vectors F U, Rayleigh waves. F K play the most important role in description of transmitting energy of

13 REMARK 6.. a Traditionally, (see Fedorov; 968 energy functions are defined by expressions analogous to (8, (9 with substituting Re ( x or Im ( x instead of complex gradient fields. It can be easily shown that for time-periodic processes, energy functions in the traditional approach being averaged over time domain produce (up to a multiplier xu / c expressions coinciding with (8, (9; b Similarly, formulas ( define the time-averaged fluxes of energy, if compared with more traditional approaches exploiting real or imaginary part of the displacement gradient field. But for these, two different time-averaging procedures can tae place: xu and F U T ( x dt x Re xu( x C Re xu( x ( F U T u( x C Re u( ( x x Re x x x dt ( where T / c; and dependence of the displacement fields on t -variable is assumed implicitly. Of course, the same time average procedures can be applied to the flux of the inetic energy. In view of the preceding remar, the first formula in ( corresponds (up to a multiplier / c to the averaging (. At the same time, averaging ( generally, does not correspond to (, (. Formula ( is used for definition of the group speed for the bul waves analysis; c Due to positive definiteness of the elasticity tensor, it flows out from (8 that U ( x, provided the (infinitesimal strain tensor ( x sym( x u ( x does not vanish at x. Similarly, K ( x when u ( x, or due to (9, when u ( x ; d Analysis of displacement fields (, ( shows that the well-nown relation U( x K( x (

14 valid for bul waves (see, Fedorov; 968, generally does not hold for Rayleigh waves; e Direct analysis of (8, (9 shows that for Rayleigh waves energy functions depend upon a spatial variable x x, while corresponding time-averaged functions for bul waves are independent of spatial variables. Substitution of representation ( in (8 gives U I ( x r C, l C ( l ir( l x n m C m ( n e (4 l l In (4 and further upper Roman indices correspond to the Jordan normal forms of the matrix R 6. Similarly, substituting ( in (8 produces U II ( x r ( C C r x ( n C me ir x C ( n m e ir x C ir x ir x ( C C r x C e C e ( n m ( n m (5 I I I I x x ( ;] x ( ;] Let U sup U ( x ; K sup K (. Expressions (4, (5 and Remar 6. ensure lim c PROPOSITION 6.. Let c (;, then a Strain energy functions are independent of time-variable and variables x xn in the -plane; b At any x x strain energy functions represent positive definite quadratic functionals with respect to complex coefficients C ; c Strain energy functions exponentially attenuate with depth (at x. Similarly,

15 lim c PROPOSITION 6.. Let c (;, then a Kinetic energy functions are independent of time-variable and variables x xn in the -plane; b At any x inetic energy functions represent positive definite quadratic functionals with respect to complex coefficients C ; c Kinetic energy functions exponentially attenuate with depth (at x. variables COROLLARY. Under conditions of Propositions 6., 6.: a Strain and inetic energy fluxes are independent of time-variable and x xn in the -plane; b At any x strain and inetic energy fluxes are oriented along vector ; c Energy fluxes exponentially attenuate with depth. REMARK 6.. For both representations ( and ( energy functions U, K and corresponding energy fluxes can attain extremums at points other than x. 7. Anisotropic material having forbidden directions. For arbitrary cubic crystal the elasticity tensor can be represented in the form C e e e e e e em em m 4 sym ( e em sym( e em m (6 where the orthonormal vectors e,,, are oriented along directions of elastic symmetry, and, and are elastic constants. REMARK 7.. a Supposition leads to isotropic materials with Lamé s constants and ; b Positive definite condition (4 applied to (6 gives

16 ,, (7 while the SPD-condition (5 produces,, (8 Both these conditions ensure:, and it is clear that neither of them is implied from the other. Let material constants of the cubic crystal be as follows ;.4868;.4 ; (9 where is material density.direct verification shows, that for such a crystal both conditions (7 and (8 are satisfied. It is clear, that material constants (9 only slightly differ from Lamè s constants. and. 4 (with corresponding to an isotropic material. Suppose now, that vectors and n are oriented along crystallographical axes of the cubic crystal. Analysis of structure of matrix R 6 and the boundary conditions (, shows that at the phase speed c following conditions hold: (i Matrix R 6 admits the Jordan normal form representation ( for the displacement field; (II J 6 with corresponding (ii Boundary conditions (5 cannot be satisfied by non-trivial coefficients C,,, corresponding to representation (; (iii At other values of the phase speed: c ( ; c, c c, matrix R 6 admits the Jordan normal form lim (I J 6 ; (iv At c ( ; c, c c boundary conditions (4 cannot be satisfied by non-trivial coefficients C,,, corresponding to representation (. lim

17 Thus, conditions (i - (iv ensure PROPOSITION 7.. a Cubic crystal (9 possesses forbidden directions in respect of Rayleigh wave propagation; b Forbidden directions coincide with the crystallographical axes. 8 Now, for the regarded cubic crystal variation of the strain and inetic energy functions at x due to variation of direction of propagation of Rayleigh wave in the analyzed. -plane is Corresponding numerical results are presented in Fig., where ( stands for strain, and ( for inetic energy function, both these functions plotted at the wave number r. In Fig. orientation of the vector n in the -plane is determined by the angle, which is the angle between vector n and one of the crystallographical axes belonging to the the 5 -plane. It is also assumed that normal to -plane is oriented along one of the crystallographical axes. tend to zero as Plots in Fig. shows that both strain and inetic energy functions at x n, nz. More detailed analysis reveals, that the energy functions tend to zero uniformly on x (;], so forbidden directions can be characterized as directions along which both strain and inetic energy cannot be transmitted. The last remar concerns stability of forbidden directions in respect of small variations of direction of propagation, this obviously flows out from the plots in Fig Fig. 6

18 Acnowledgements. Author thans Dr. Kaptsov for discussions.

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