SURFACE WAVES OF NON-RAYLEIGH TYPE

Transcription

1 SURFACE WAVES OF NON-RAYLEIGH TYPE by SERGEY V. KUZNETSOV Institute for Problems in Mehanis Prosp. Vernadskogo, 0, Mosow, 75 Russia Abstrat. Existene of surfae waves of non-rayleigh type propagating on some anisotropi elasti half-spaes is proved. Conditions for originating the non-rayleigh type waves are analyzed. An example of a transversely isotropi material admitting a surfae wave of the non- Rayleigh type, is onstruted.. Introdution. In our previous paper [] it was shown that some anisotropi elasti materials exhibit property of non existene of the genuine Rayleigh waves: ux ( ) = C k kmke k = ir( γ ν x n x t) (.) where C k are omplex oeffiients determined up to a multiplier by the tration-free boundary onditions; m k are omplex eigenvetors of the Christoffel equation, whih will be introdued further; these eigenvetors orrespond to omplex roots γ k of the harateristi polynomial; r is the (real) wave number; ν is an outward normal to the boundary Π ν of the half-spae along whih the surfae wave propagates; n Πν is the unit vetor determining diretion of propagation of the surfae wave, and is the phase speed. The terms u ( ) k k x mke ir( γ ν x n x t) (.) are alled partial waves. As was shown in [], the existene of the forbidden diretions or forbidden planes along whih the genuine Rayleigh wave annot propagate is due to appearing the Jordan bloks in a speially onstruted -matrix assoiated with the Christoffel equation. The following analysis reveals that the situation regarded in [] appears to be more ompliated. The Jordan bloks in the regarded matrix lead to a qualitative hange of the struture of the partial waves (.) and, while the genuine Rayleigh wave at the situation onsidered in [] does not exist, there remains an exponentially attenuating with depth surfae wave of the non-rayleigh type.

2 . Basi notations. Equations of motion for anisotropi elasti medium an be written in the form A(, ) u div C u ρ u = 0 (.) x t x x where u is the displaement field; ρ is the density of a medium; and C is the fourth-order elastiity tensor assumed to be positive definite: A ( A C A ) AC A > 0 (.) A sym( R R ), A 0 i, j, m, n ij ijmn mn The sign in (.), (.) and heneforth means the salar multipliation in the orresponding unitary or Eulidian vetor spae. Substituting partial waves (.) in Eq. (.) produes the Christoffel equation: k n C n k I mk ( γ ν ) ( γ ν ) ρ = 0 (.) where I is the unit diagonal matrix. Equation (.) an be written in the equivalent form: k k det ( γ ν n) C ( nγ ν ) ρ I = 0 (.4) The left-hand side of Eq. (.4) represents a polynomial of degree with respet to γ k. REMARK.. It an be shown, see [], that if the phase speed does not exeed the so alled lower iting speed ( ): <, (.5) then all the roots of Eq. (.) are omplex with Im( γk ) 0. The inequality (.5) ensures that three partial waves (.) with Im( γ k ) < 0 attenuate with depth in a lower half-spae at ( ν x) < 0. Only attenuating with depth partial waves, as being physially reasonable, will be onsidered further.. Six-dimensional formalism. Following [], a more general representation for the partial wave than (.), will be onsidered: v ( x ) e (.) where x = irν x is the dimensionless omplex oordinate, v ( x ) is an unknown vetor funtion, and the exponential multiplier in (.) orresponds to propagation of the plane

3 wave front along the diretion n with the phase speed. Substituting representation (.) into Eq. (.) yields the following system of ordinary differential equations: ( ) ( ν C ν) x ( ν C n n C ν ) x ( n C n ρ I ) v( x ) = 0 (.) Diret analysis of system (.) is rather diffiult, and redution to the first-order system an simplify it. Introdution of a new vetor-funtion w = v allows us to redue the seondorder system (.) in C to the first-order one in x C : v v x = R w w (.) In (.) the omplex six-dimensional matrix R has the form R 0 I = M N (.4) where three-dimensional matries M and N have the form M = ( ν C ν) ( n C n ρ I ) N = ( ν C ν) ( ν C n n C ν) (.5) In (.4) I stands for the unit (diagonal) matrix in the three-dimensional spae. A surjetive homomorphism I: C C, suh that I ( vw, ) = v (.) will be needed for the subsequent analysis. The following Proposition takes plae []: PROPOSITION.. Let (0; ): a) Spetrum of the matrix R oinides with the set of all roots of polynomial (.4); b) If γ is a omplex eigenvalue and m= ( m, m ), m, m C is the orresponding six-dimensional eigenvetor of the matrix R, then γ is also an eigenvalue with the orresponding eigenvetor m= ( m, m ); ) The matrix R admits the following Jordan normal forms

4 J γ γ 0 γ γ γ γ =, =, γ γ γ γ (I) (II) J γ 0 γ γ 0 0 γ 0 0 γ (III) J = 0 (.7) γ 0 γ 0 0 γ d) Aording to the Jordan normal forms the following three types of representations for surfae waves our: (I) (i) for the Jordan normal form J, the orresponding representation is given by (.); (II) (ii) for the Jordan normal form J, the representation is as follows: ir( γ ν x n x t) ν xm ir( γ ν x n x t) Cm e ir( γ ν x n x t) Cm e ux ( ) ( C irc ) e = (.8) where m =I( m ) C, and m is the eigenvetor of R orresponding to the eigenvalue γ ; m =I( m ) C, and with m, and the eigenvetor C m C is the generalized eigenvetor assoiated m orresponds to the eigenvalue γ ; (III) (iii) for the Jordan normal form J, the representation is as follows: ν ν ir( γ ν x n x t) xm ir( γ ν x n x t) m e ux ( ) ( C irc x C ( ir x) ) me = ir( γ ν x n x t) ( C irc ν ) e (.9) C m =I( m ) C, and, C m is the eigenvetor orresponding to the eigenvalue γ ; and m m are the generalized eigenvetors assoiated with m. 4

5 COROLLARY. For any of the Jordan normal forms of the matrix R the threedimensional omponents m k, m k of the (proper) eigenvetor m k, satisfy the equations m =γ m k k k ( ) γ I γ N M m = 0 k k k (.0) Proof. When the matrix R has no Jordan bloks, the solution of Eq. (.) in view of (.4) leads to Eqs. (.0). Thus, the omponent m k belongs to the kernel spae of the matrix ( γ k γ k ) I N M. (II) 4. Constrution of the generalized eigenvetor for J. In view of [] the solution of Eq. (.) orresponding to a Jordan blok of the seond rank an be represented in the form (, γ ) ( (, ) (, )) x C m m C x m m m m e (4.) where as before x = irν x. PROPOSITION 4.. a) The three-dimensional omponents m, m of the genuine eigenvetor satisfy Eqs. (.0); b) Components m, m of the generalized eigenvetor satisfy the following equations: ( ) ( ) γ Iγ N M m = γ I N m m = m γ m (4.) ) At d) At (0; ) the matrix ( γ I N ) is not degenerate; (0; ) vetors ( γ I N) m and m ( ν C ν ) are orthogonal. Proof. Conditions a) and b) flow out by diret substituting the solution (4.) into Eq. (.). To prove ) it is suffiient to demonstrate that the matrix ( ) ( ν C ν) γ I N = γ ( ν C ν) ( ν C n n C ν ) (4.) 5

6 is not degenerate. Considering multipliation of the right-hand side of (4.) by any nonzero onjugate omplex vetors a, a C and aounting Remark., whih ensures Im( γ) 0, we arrive to ( a ( ν C ν ν C n n C ν ) a) Im γ ( ) ( ) = Im( γ )( a ν C ν a) 0 ( a C n n C a) In obtaining (4.4) we took into onsideration that ( ) the matrix ( ν Cn nc ) (4.4) Im ν ν = 0, sine ν is (real) symmetri. The last inequality in (4.4) ompletes the proof of ondition ). To prove d) Eq. (4.) an be transformed into equivalent one by multiplying both sides by the nondegenerate matrix ( ν C ν ), this gives ( ( ν ν) ( ν ν) ( )) γ C γ C n n C n C n ρ I m = ( ( ν ν) ( ν ν) ) γ C C n n C m (4.5) Now, the vetor m belongs to the kernel spae of the matrix in the left-hand side of Eq. (4.5), whih flows out from Proposition 4..a. Moreover, the regarded matrix is omplex symmetri, hene its left and right eigenvetors oinide. The latter allows us to write for the left-hand side of Eq. (4.5) ( ) m γ ( ν C ν) γ ( ν C n n C ν ) ( n C n ρ I) m = 0 (4.) Similarly, for the right-hand side of Eq. (4.5) ( ) m γ ( ν C ν) ( ν C n n C ν ) m = 0 (4.7) In view of (.5), Eq. (4.7) ompletes the proof. COROLLARY. In the fator-spae C /Ker( γ γ ) admits the following representation ( ) ( ) I N M, the vetor m m = γ Iγ N M γ I N m (4.8) REMARK 4.. a) At the regarded speed interval (0; ) the eigenvetors of the omplex symmetri matrix appearing in Eq. (4.) may not form a set of mutually

7 orthogonal vetors in C, in ontrast to the mutually orthogonal eigenvetors of any real symmetri matrix. b) For supersoni Lamb waves propagating with the phase speed exeeding the greatest iting speed, all eigenvalues of the matrix R beome real. Presumably, in suh a ase the ondition ) of Proposition 4. and the subsequent Corollary an be violated. (II) 5. Dispersion equation for J. The tration-free boundary onditions on the surfae Π ν an be written in the form: tν ν C u = 0 (5.) x Π ν Substituting the displaement field into Eq. (5.) yields Ckt k = 0 (5.) k = where t k are the partial surfae tration. The following two ases for the partial surfae tration fields will be onsidered: (I) (i) For the Jordan normal form J and the representation (.), the partial surfae trations are of the form ir( t) ( ) nx t ν C ν ν C n m (5.) k = γk k e (ii) For the Jordan normal form surfae trations are of the form (II) J and the representation (.8), the partial ( ν ν ν ) e ( ( ν ν) ( ν ν ν ) ) t = γ C C n m t = γ C m γ C C n m ( ν ν ν ) t = γ C C n m e e (5.4) Equations (5.) an be regarded as linear system with respet to the unknown oeffiients C k. The existene of the nontrivial solution of Eqs. (5.) is equivalent to vanishing of the following determinant: t t t = 0 (5.5) Equation (5.5) provides a neessary and suffiient ondition for the existene of the surfae wave. 7

8 Equation (5.5) is known as the dispersion equation despite the fat, that the phase speed determined by this equation does not depend upon the wave number, or the wave frequeny.. Surfae waves of non-rayleigh type in transversely isotropi media. Let the unit vetors e k, k =,, form an orthogonal basis in R, and vetor e is normal to the Πν -basal plane of a transversely isotropi medium. This ensures that vetor e and ν oinide. The orresponding elastiity tensor has the following omponents: (.) where 44 = ( ) and the elastiity tensor is assumed to be positive-definite. The following Proposition is needed for further analysis: PROPOSITION.. If the omponents of the elastiity tensor (.) satisfy the relation ( ) = 0 ( 9 ) ( 7 ) ( ) (5 4 5 ) (.) Then a) At the parameter x =ρ determined by the polynomial equation ( 55) ( ) 55 x 55 x x ( )( ) ( ) = 0 (.) (II) the Jordan normal form J appears in the struture of the matrix R ; b) At any other value of the phase speed (0; ), there is no genuine Rayleigh wave admitting the representation (.) and propagating on a tration-free boundary of the transversely isotropi half-spae. 8

9 Proof of Proposition. an be found in []. REMARK.. Equation (.) for a transversely isotropi half-spae with the elastiity tensor, whih does not satisfy Eq. (.), was obtained in [] by appliation of the three dimensional omplex formalism. It an be shown, that Eq. (.) has the unique positive root in the interval 0; ( ) ρ. Combining Eqs. (5.) with (.) and substituting the orresponding values of the elastiity onstants and the phase speed into Eqs. (.0), (4.8), we arrive to PROPOSITION.. At the onditions (.), (.) of Proposition.: a) The eigenvalues γ k in the representation (.8) take the form: ( 55) ρ 55 γ = i 55 ρ γ = i 55 / / (.4) b) The orresponding amplitudes m k ( m and m are of the unit length) are of the form: ( ν i n), ( i ν n) m = p β α m = sp α β m = ν n, (.5) where /4 ρ, ρ α= β= 55 /4 (.) p is the normalization fator: ( ) / p = α β (.7) and the parameter s is obtained by Eq. (4.8): 9

10 s = β 55α ( αβ) ( ) ( ) (.8) ) The partial surfae trations (5.4) are of the form t t t (( i ) ν ( ( i )) n) e (( ) ) ( ) = γ β α β γ α 55 ( ( ( )) n) = s β iγ sα ν γ sβ i s α e (.9) =γ we Substituting the surfae trations (.9) into the dispersion equation (5.5) yields COROLLARY. At the onditions of Propositions.,. a) The dispersion equation (5.5) takes the form t t 0, C = 0 (.0) b) The nontrivial oeffiients C, C defined up to arbitrary salar multiplier by Eq. (5.), are of the form C (( s) β iγsα) ( γ β i α) =, C = (.) Thus, Propositions.. ompletely haraterize the surfae wave propagating on a basal plane of the transversely isotropi half spae and orresponding to the representation (.8). The question, whether there exists a surfae wave admitting the representation (.9), remains open. 0

11 REFRENCES. S.V. KUZNETSOV, Forbidden planes for Rayleigh waves, Quart. Appl. Math. 0, (00). P. HARTMAN, Ordinary Differential Equations, Wiley, N. Y., 94. D. ROYER and E. DIEULESAINT, Rayleigh wave veloity and displaement in orthorhombi, tetragonal, and ubi rystals, J. Aoust. So. Am. 7, (985)