Existence of second slip waves in anisotropic elastic media

Wave Motion 18 (1993) 79-99 Elsevier

79

Existence of second slip waves in anisotropic elastic media Litian Wang and Jens Lothe Department of Physics. University of Oslo, P.O. Box 1048, Blindern, 0316 Oslo, Norway Received 24 September 1992, Revised 21 October 1992

Existence of subsonic second slip waves, a variant of interfacial (Stoneley) waves, is investigated using the Stroh sextic formalism. We have examined the second slip wave propagatingalong an interface between two identical half-infiniteanisotropic media which have the same orientation. Special attention is paid to exceptional transonic states where the appearance of a certain bulk wave may prevent the existence of the second slip wave. By examining the impedancetensor at various transonic states, we formulate an existence theorem for second slip waves. Numerical examples are shown for materials with various symmetries, which give convincing evidence for the existence of second slip waves.

1. Introduction The existence theories of subsonic free surface (Rayleigh) waves and interfacial (Stoneley) waves in anisotropic media are well established. The subsonic surface wave [ 1 ] is a wave that propagates along a surface o f a halfinfinite homogeneous linear elastic medium leaving the surface free of traction and having a displacement that attenuates exponentially with depth into the medium. The Stoneley wave [2], a typical interfacial wave, is a wave propagating in an interface separating two half-infinite elastic media, producing continuous displacements across the interface, attenuating with distance from the interface in both media. The existence theorems of subsonic surface waves have been developed by Barnett and Lothe [ 3,4 ] and Chadwick and Smith [5,6], and they have shown that the existence of subsonic surface waves is solely dependent on the properties of a so-called surface impedance tensor at a limiting transonic state whose velocity 0 is the lowest velocity for a bulk wave propagation for a given surface geometry. Using the same methodology in terms of which the subsonic surface wave theory is established, Barnett et ai. [ 7 ] have formulated the existence and uniqueness theorems for Stoneley waves. The slip wave, a variant of the Stoneley waves, is an interfacial wave that supports only normal stress across the interface and allows relative shear movement between the two media along the interface. It has been investigated by Barnett et al. ( B G L ) [8], and they revealed that there is a possibility to find two slip waves, and that the appearance of the second slip wave is solely due to elastic anisotropy. This also raises the fundamental problem whether there is a second zero in the subdeterminant ]]Bp,II for anisotropic media, which is a question of general interest in wave theory and dislocation theory [9]. A realistic situation where our investigation is important is the Scholte wave at the interface between a half-infinite anisotropic medium and a non-viscous compressible liquid. The second zero of [IBpll] implies that two Scholte waves may be found if the sound velocity in the liquid is higher than the velocity of the limiting bulk wave, and the reflection of sound waves can only produce shear displacements in the interface when IBp, II s 0 [8]. From a practical point of view, the (pure) slip wave can be regarded as an ideal case in contrast to realistic slip waves for which various interface conditions have been suggested. Chevalier et al. [ 10] have examined interfacial 0165-2125/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

L. Wang. J. Lothe / l:.'xistence of second slip wave~ in anisotropic elastic media

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waves propagating along a symmetry axis of two orthotropic media under four types of interface conditions with or without friction. For the so-called elastic-slip interface condition, they introduced a parameter to characterize the elastic-slip nature, and they showed that the interface conditions for pure Stoneley waves (perfect bond) and for the (pure) slip wave which we have considered in this study, are two limits of the elastic slip condition. For the case of Coulomb friction, only a pure slip wave solution is guaranteed. Thus, investigation of pure slip waves can help us understand basic implications of slip interface conditions, and the method that we have used here could also be applied to slip waves for various types of realistic interface conditions. For the case of pure slip waves, where two media are identical and have the same orientation, Barnett et ai. [8] have proved that the subsonic surface wave can be regarded as a slip wave itself and may be called afirst slip wat'e, and that another slip wave, a so-called second slip war,e, has higher velocity than the first slip wave. They also derived a series of theorems concerning the existence and non-existence of the second slip wave in this simple case by analyzing an interface impedance tensor which is related to the surface impedance tensor. However, the examples given by them are not very convincing because of the extremely narrow interval between the velocities of the second slip waves and the limiting waves, of the order 1 0 - " ) - 1 0 - t3 m/s. In this paper, we will only deal with the second slip waves within the interface between two identical elastic media with identical orientation. Instead of using the integral formalism for the interface impedance tensor which B G L ' s investigation employed, we will apply the Stroh sextic formalism I 11 ] and explore various exceptional transonic states and give more explicit results for the existence criteria for the second slip wave. We will also give some examples of second slip waves which are consistent with our theoretical analysis.

2. General theory The general theoretical considerations are mainly based upon the Stroh sextic formalism. Consider a slip wave propagating along the direction m within a planar interface with normal n. The displacement field in each medium can be described by the so-called inhomogeneous waves which have the form u(l)=A,~(i)

exp[ik(m.x+p~(1)n.x-l,,t)]

u(2)=A*(2)exp[ik(m.x+p*(2)n.x-rt)]

forn-x>~0, forn.x~<0,

(2.1a) (2.1b)

where the asterisk * means complex conjugate, k is the wavenumber, p,, and p * are scalar complex numbers and A,~( 1 ) and A * ( 2 ) are the polarization vectors associated with p,,( 1 ) and p,*, (2) in the upper medium ( 1 ) and the lower medium (2), respectively, m and n span the reference plane FI, and m and t, where t = x × n, span the interface plane P. Thus an interface geometry is defined by the orthogonal triad (m, n, t) (See Fig. I ). The equations of motion for elastodynamics are ~uk 02u, C°~t Oxj ~xt - p ~~t

(2.2)

where C0k~is the elastic stiffness tensor, and p is the density of the medium. Ignore the indices ( 1 ) and (2) in (2. I a) and (2. I b), and substitute them into (2.2). We will get {(ram) +p,,[ (ran) + ( n m ) ] + p ~ ( n n ) -pt,21}A,, = 0

(2.3)

where the notation (mm), or generally, (ab), is a real 3 by 3 matrix whose matrix elements are defined as (ab)jk = a~Cokt b~, and I is the 3 by 3 unit matrix. Solutions for p in (2.3) can be determined from the characteristic equation

L Wang, J. Lothe/Existence of second slip waves in anisotropic elastic media

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r

Fig. 1. An interface geometry for interracial wave propagation, n is the normal for the planar interface P. m is the propagation vector, and t is a unit vector normal to the paper.

II(ram) +p,,[ (ran) + (nm)] +pZ~(nn) --pc21ll = 0

(2.4)

where II [J means determinant. (2.4) will give six roots of p,~ which appear in three pairs of conjugate complex numbers. Conventionally, we assign Imp,,>0

forot=l,2,3,

P~*---P,~+3 f o r o t = l , 2 , 3 .

(2.5a, b)

For the displacement fields described by (2. I a) and (2. I b), we have to impose a restriction on the p,~'s when finite displacements in both media are required. Therefore, we have I m p ~ , ( l ) > 0 for the upper medium ( l ) and I m p * ( 2 ) < 0 for the lower medium (2) in order to keep the displacements attenuated with distance from the interface into each medium. This is the reason why we have p * in (2. I b). The traction T exerted on the planes parallel to n .x = 0 by the inhomogeneous wave with the displacement A can then be expressed by

aUk

(2.6)

Tj = rlityij = Cijktni Oxi

where o'i~is the stress tensor associated with the displacement field u. From (2.6), the traction in each medium can be described by T,~(I)= -ikL(l)~exp[ik(m.x+p,,(l)n.x-vt)]

forn.x>~0

(2.7a)

T,(2) =ikL*(2),~exp[ik(m.x+p*(2)n.x-vt)]

forn.x~<0.

(2.7b)

The auxiliary vector L is usually defined as the traction vector instead of T. The traction vector L,~ has a simple relation with the polarization vectorA,,

L~(l)=-[(nm)+p~,(I)(nn)]A~(l) L * ( 2 ) = - [ ( n m ) +p*(2)(nn)]A*(2)

for n.x~>0 forn.x~<0

(2.8a) (2.8b)

By defining six-dimensional eigenvectors ~:,, = (A,~, L,~)T,

a = 1. . . . . 6 ,

(2.9)

Stroh [ 11 ] reformulated (2.3), (2.8a) and (2.8b) into a six-dimensional eigenvalue problem Ns¢,, =p,,s¢,,,

or= 1. . . . . 6

(2.10)

L. Wang, J. I~;the /Existence of second slip waves in anisotropic elastic media

82

where

( N=-

(nn)-'(nrn) (nn)-' ) (mn)(nn)_l(nm)_(mm)+peZ 1 (mn)(nn)- l

(2.11)

and consistent with the convention (2.5a) and (2.5b). ~:* = ~c,,, 3

for a = l .

2, 3 .

(2.12)

Generally, when there is no non-semisimple degeneracy among the six eigenvaluesp.'s, the eigenvectors ~:,,= (A,,, L . ) T satisfy the following orthogonality and completeness relations

A,,.L~+A~.L,=6,,~

(2.13)

and 6

b

A.®A.=O,

~

L,,®L,,=O.

L,,®A,,=I. cr = I

(2.14a)

A.®L.=I

(2.14b)

(~t~ I

where ® represents the dyadic product. The displacement field in the interface region can be characterized by a linear combination of the six eigenvectors, each of which represents an inhomogeneous wave. For the upper medium, the eigenvectors ~, = (A~, L,) x for ot = 1,2, 3 are the physical ones and hence the displacement field u within the upper medium ( I ) and the traction T on the planes parallel to n .x = 0 can be expressed by 3

u(l)=

~

T(1)=

~

E,~A,exp[ik(m.x+p,n.x-ct)],

(2.15a)

-ikE,,L, exp[ik(m.x+p,,n.x-ct)],

(2.15h)

3

(r ~ I

respectively. Similarly. for the lower medium (2), we use the eigenvectors (A.. L,,) tor a = 4.5, 6, because of the relation (2.5b) and (2.12). and we have 3

u(2)=

E'.A*exp[ik(m.x+p*n.x-t.t)],

~ r~-

(2.16a)

I

3

T(2)=

~ ' lkE,exp[ ' ' lk(m ' .x + p ~* n .x - ct) ] •

(2.16b)

Concisely, we will denote the resultant displacement amplitude A and traction L in each medium as 3

A(I)=

~

3

E,e4,,,

~r-- I

o~1

respectively.

y" E , L , , ~-

3

A(2) = ~

L(I)=

forn.x>~0,

(2.17a)

I 3

E", , A .*.

L ( 2 ) = y" E ,' , L .*. a--I

forn.x~<0.

(2.17b)

L Wang, Z Lothe/Ex~tence ~second sl~ waves in anisotropic e~stic med~

83

TheinterfaceimpedancetensorZforeach medium canthenbeintroduced by L(I) = -iZ(I)A(I),

(2.18a)

L(2) = i Z ( 2 ) * A ( 2 ) .

(2.18b)

It has been proven by Ingebrigtsen and Tonning [ 12] and Barnett and Lothe [3,4] that the impedance tensor Z is hermitian, that it is positive definite at v = 0 and that its eigenvalues are monotonically decreasing functions with increasing velocity. From the orthogonality relation (2.13), the impedance tensor Z can be explicitly expressed in terms of A,~'s and L~'s as

Z = i L" ® (A p ×AT") + L # ® (AT' ×A,,) + L r ® (A,~ ×A~) [A,,, Ate, A~,] =i

Y [A,~, A~, A~]

(2.19a)

(2.19b)

where [ ] means the triple product and a,/3 and 7 equal 1, 2 and 3, respectively. Such an expression for Z is independent of the normalization scheme of the eigenvectors A ~ and L,, (2.13). So, A,, and L~ can be regarded as either normalized or unnormalized vectors and the representation of Z (2.19a) is valid as long as there is no nonsemisimple degeneracy among the six eigenvalues p,,' s. From now on we regard A,, and L,, as unnormalized vectors unless otherwise explicitly indicated. For two perfectly bonded anisotropic media, a Stoneley wave may propagate along a planar interface where both the displacement A and the traction L are continuous across the interface. Therefore, an interface condition can be described A(1) - A ( 2 ) = 0 ,

(2.20a)

L( 1 ) - L ( 2 ) = 0 ,

(2.20b)

at the interface. As shown by Barnett et al. [7] an existence condition of Stoneley waves can be expressed as II(Z(l) +Z*(2)II = 0 .

(2.21)

For a slip wave propagating along the direction m within an interface with normal n, the displacement A and the traction L for each medium must fulfil interface conditions L(1) ---L(2) = k n ,

(2.22a)

n. (A(1) - A ( 2 ) ) = 0 ,

(2.22b)

at the interface. Then the existence condition of the slip wave becomes n.(Z-I(l)

+Z-l*(2)).n=0

(2.23)

When we consider two identical materials with identical crystallographic orientations, because Z is hermitian the slip wave condition (2.23) becomes

n . Z - ' - n = IlzPl H = 0

IIZII

(2.24)

where IIZptll stands for the determinant of so-called planar impedance Zpl which is minor for the n ®n element of the impedance tensor Z.

L. Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

84

BGL [ 8 ] have shown that the condition for the slip wave can be reformulated to n.Z

- ~.n = n . B

- t.n =0

(2.25)

where B is a real matrix and is defined as 3

B=2i

]~

L,~®L~,.

(2.26)

Noticing that the surface wave at t:R is itself the first slip wave, and ~ZII = 0 at t,k [3,4], the condition (2.24) will not be valid for the first slip wave which occurs at vn. Since the subsonic surface wave is related to a double zero of the eigenvalues of B, the first slip wave can then be determined by the planar part of the B matrix [ 8 ] IIBp, II = 0 .

(2.27)

The condition (2.27) is also valid for the second slip wave due to (2.25). Thus, (2.24) will be a condition only for the second slip wave and (2.27) is a condition for both the first slip wave and the second slip wave. BGL 18] have also shown that Zp~ has similar behaviour as Z, i.e., it is positive definite at v = 0 and it is a monotonically decreasing function of L, within the subsonic region t, = (0, t~). In order to examine the existence of second slip waves, we can concentrate on the behaviour Of Zpt at transonic states t) which are the limits for the Stroh eigenvalue problem to have solutions with all six eigenvalues complex. So, the condition for a second slip wave becomes

IlZp, II < 0

(2.28)

at the transonic state t?. However, when we assume that the two media can have relative shear slide with each other at the interface, there are many types of boundary conditions besides the pure slip condition shown in (2.22). A typical one is the elastic slip condition proposed by Chevalier et al. [ 101 for the case where there is no detachment between the two media and the slip is characterized by a parameter b. Such an interface condition can be written as n-(A(1) -A(2)) =0,

(2.29a)

n . (L( 1 ) - L ( 2 ) ) = 0 ,

(2.29b)

L(1)T =L(2)T =b(A(1) --A(2))T,

(2.29C)

at the interface, where the subscript T denotes shear component and the parameter b is a positive real number that characterizes the nature of elastic-slip. It is clear that the case for b = ~ will yield Stoneley waves and the case for b = 0 will result in (pure) slip waves. Under the elastic slip condition, Chevalier et al. [ 10] have demonstrated, in a simple interface geometry where propagation direction is along the symmetry axis of two orthotropic media, that elastic-slip waves can be found and that the wave is of pure slip type for small b and of Stoneley type for large b in an Aluminum-Tungsten interface. In the subsequent sections, we will concentrate on the pure slip wave and examine the existence of such a slip wave in general direction and interface orientations. We will explain more about transonic states and other special waves related to the slip wave problem. Then, we will derive a series of existence criteria for the second slip waves by examining IlZptJJ at various kinds of transonic states.

L. Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

85

3. Exceptional transonic states The transonic state is a limiting state where a certain pair of complex conjugate eigenvalues of the Stroh eigenvalue problem coalesce into a real number and the inhomogeneous waves associated with the eigenvalues like (2.1) turn into a real bulk wave without damping. If such a bulk wave with the wave vector k = m +pn leaves the interface traction free, we call the transonic state exceptional, otherwise, normal. Chadwick and Smith [5] have classified all possible cases into six types of transonic states, from Type 1 to Type 6. The velocity of a transonic state is usually called the limiting velocity and is conventionally denoted as ~3. Concerning the slip wave problem, there are two kinds of exceptional waves according to the definition by Barnett et al. [8]. For brevity, here we just quote their definitions: • The exceptional wave of the first kind is a wave for which there exists a plane P which is left traction free by the wave; • The exceptional waves of the second kind are waves for which among the planes parallel with the amplitude A there is a plane P on which the wave exerts no tangential tractions. The exceptional wave of the first kind is, in fact, the usual exceptional bulk wave as defined in the surface wave theory. The exceptional wave of the second kind is, however, a special bulk wave for which the polarization A, the group velocity g and the wave propgation direction k are all lying in the same plane P. Consider a bulk wave propagating along a wave vector k. The displacement can be described by

u(x, t) =A exp[ ik( k . x - vt) ]

(3.1)

and consequently, the equation of motion (2.2) will yield the Christoffel equation (kk)A = p v 2 A .

(3.2)

The group velocity is given by [ 8,13 ] (kA)A

(3.3)

g= pv(k) and the traction T exerted on a plane with normal n by the bulk wave is given by

L = pal

(3.4)

where/x is a real 3 by 3 matrix with elements/~0 = Cijr,Arks. Thus, the condition for an exceptional wave of the first kind is

fl~ll--O

(3.5)

and the condition for an exceptional wave of the second kind is

[A,k,g] =0.

(3.6)

There has been a lot of discussion about these two kinds of exceptional waves as related to the bulk wave theory [ 14]. Here we list some basic facts about these two kinds of exceptional waves. Generally, when the reference plane R lies in a symmetry plane, in-plane polarized bulk waves (A. t = 0) will always be exceptional of the second kind, and transversely polarized bulk waves (also perpendicular to FI) will always be exceptional of the first kind. The introduction of an exceptional wave of the second kind can be understood from the slip wave definition (2.22a) and (2.22b). In fact, if a limiting wave is exceptional of either the first kind ( L = 0 ) or the second kind (L = kn, k is a constant), because also the usual exceptional wave satisfies n .A = 0, the limiting bulk wave itself will satisfy the slip wave condition, that is, the subsonic slip wave is degraded into a limiting bulk wave which is

L Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

86

without damping with distance normal to the interface in each medium. Such a case is very similar to that in the surface wave theory. The distinction among these two kinds of exceptional waves plays an important role in our following discussions on the second slip wave since they both prevent the existence of subsonic second slip waves. Concerning other types of transonic states, Types 2-6 in Chadwick and Smith's terminology [ 5 I, they are always normal, but they may contain one exceptional wave or they may possibly be composite exceptional, i.e., a superposition of two or three bulk waves related to a transonic state leaves an interface traction free. Also a certain superposition may leave an interface free of tangential traction and produce only tangential displacement, that is a composite exceptional wave of the second kind. The development in computer graphics has made it possible to visualize the relation among various exceptional waves. We utilize FOTO graphics software, and apply the criteria (3.5) and (3.6) for both kinds of exceptional waves, and produce three dimensional visualizations for the exceptional waves in some cubic materials. ( See Section 5 and Fig. 3).

4. Existence criteria Now we resume our analysis on the planar impedance matrix Zpv With the sextic representation of Z (2.19a), we will calculate I[Zp~l[at various types of transonic states Starting with Y in (2.19b), one can easily show that

II Yp~ II IlZp, II = - [ A , , A a , Ar]2

( 4. I )

where

II r~, II = ~ . , r , , -

r'.,,r.,,

= { (m.L,~) (m . A , XA ~) + (m . L , ) (m .A~ XA~) + (m .L~) (m .A,~ XA ~) }

X { (t.L~) ( t . A , XA ~) + (t.L/j) (t.A y XA~) + (t.Ly) (t.A~ XA/3) } - { (m .L~) (t.A~ XA ~) + (m . L , ) (t.A ~ XA~) + (m .L~) (t.A~ XA/3) }

X { ( t . L , ) (m .A a XAy) + ( t . L , ) ( m . A ~ XA,,) + (t.L~) (m.A,, XA/3) }

(4.2)

Since there are as many as six types of transonic states, we are going to discuss them separately.

4.1. Type 1 transonic state At a Type I transonic state, one of the inhomogeneous waves in (2.15a), say (A ~, L~), becomes a real bulk wave (A,/~), and it obeys the self-orthogonality relation [3-5] .4./]=0.

(4.3)

Notice that ,/, a n d / ] are unnormalized vectors. Then we can derive the four elements of Illp~(~,~)II as follows

Y,,,, = m ' { L ~ ® ( A tj + fil ) } ' m + m ' { L a ® ( , 4 XA,,) } ' m + m ' { £ ® ( A ~ XA a) }'m = {(t'A)Xm, - (n-,4)X,,, } + ( m . / ] ) ( r e . A , , × A a) where

(4.4)

L Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

X=L~®Aa-LB®A~,

87

(4.5)

and similarly, we get Y,,, = - {(m ",'~)X,,n - (n ..4) X,,,,, } + ( m - / ~ ) ( t . A ~ X A a)

(4.6)

Y.. = + { ( t ' A ) X , n - (n . / [ ) X . } + ( t . L ) (m .A~. × A a )

(4.7)

Y. = - { (m .A )X,n - (n .A)X.~ } + ( t . L ) ( t . A ~ × A a)

(4.8)

From (4.2), IIYp,(~)U can be expressed as

IIY~,(~)II = IA~, A a, .4 ] { [Xn, n,/~] + (n .A) [n, L,,, La] },

(4.9)

and IlZvl(13) II becomes ilZv,(~) ii = _

IXn, n, £] + (n.A) [n, L,~, Lp] [ A ~ , A a , A]

(4.10)

In order to obtain some definite results, we consider a simple case. Suppose that the interface is a plane which contains the polarization,4 of the limiting bulk wave,

n.,4=O.

(4.11)

Then (4.10) is simplified to IlZpl(6) II = -

[Xn, n , £ ] [A,,, A a, .41 "

(4.12)

Now, we can examine different cases under this circumstance. First, when the transonic state is exceptional either first kind

£=0

(4.13)

or second kind

/~=kn

(4.14)

we get

IlZo,(~) II = 0 ,

(4.15)

(4.15) means that there would be no second slip wave. This agrees with B G L ' s Theorem 4. This also means that the limiting bulk wave satisfies the slip wave condition (2.24), in other words, the second slip wave degenerates to a bulk wave. When the conditions (4.11 ) and (4.14) are fulfilled, the condition for the exceptional wave of the second kind (3.6) can be obtained from relation [4,5] ( m + lSn ) . L + pv2n..4 = 0

(4.16)

which leaves as one condition (m + l~n ) . L ~ k . n = O

(4.17)

where m +/~n is the wave vector k of the limiting bulk wave. Secondly, when the transonic state is not exceptional of either kind, we can prove that there will be a second slip wave for one interface geometry agreeing with (4.11 ). In fact, at a normal transonic state with the interface geometry (4.11 ) the self-orthogonality relation (4.3) implies that n X/_~= AA

(4.18)

88

L. Wang. J. l.z~the / Existence of second slip waves in anisotropic elastic media

where n, ,4 and/~ are real vectors and A is a real constant. If A = 0 , then (4.18) implies that the transonic state is either exceptional of the first kind or exceptional of the second kind, consistent with the preceding arguments. From (4.18) and (2.13), we find that

[Xn, n, L] = (n X£) .(L,~®A~ - L ~ ® A , . ) .n = A(/~ .A , ) ( A . . n )

- A(/~ -A,,) (A ~ .n )

=A(nXL).(A,,XA,) = A2IA,,, A/j, ,4]

(4.19)

and we finally come to the conclusion that IIZM ~7) II = - ,~ 2.

(4.20)

This guarantees the existence of a second slip wave and agrees with BGL's Theorem 5.

4.2. Type 2 and Type 4 transonic states At a Type 2 or Type 4 transonic state, the interface plane is fixed, and two of inhomogeneous waves, say (A s, L~) and (Aw L , ) , become bulk waves, and the two polarization vectors A s and A~, remain linearly independent [ 3-5 ]. Consider a superposition

A=A/z+A~.

(4.21)

so that with respect to the plane P n -,,~ = 0 .

( 4.22 )

The traction associated with ,,~ is

L=La+L ~

(4.23)

Similar to the case of a Type 1 transonic state, we can find again that

Y,,,,=(t.,4)X,,n+(m.l(,)(m.A, x A a ) , Y,,,=(t.,4)X,~+(t.I,)(m.A, xA/j),

Y,,,,= -(m.,4)X,,,,+(m.£)(t.A,,xA~) Y,,= -(m.,4)X,,,+(t.£)(t.A,xXAa).

(4.24a, b) (4.24c, d)

Equations (4.24a)-(4.24d) have the same forms as (4.4) and ( 4 . 6 ) - ( 4 . 8 ) . Thus,

IIYp,(;)II

= [ A , , A a, At] [Xn, n,/~]

(4.25)

and IIzp, (;) II -

[Xn.n,l,] [A,~, A a. A~]

( 4.26 )

Therefore, for a Type 2 or Type 4 transonic state, when the superposition (4.21) is exceptional of either the first kind,/~ = 0, or the second kind, L = kn, with respect to the plane P, we again obtain IlZp,(c) II = 0.

(4.27)

Then the second slip wave is precluded. This is the same as B G L ' s Theorem 6. When the transonic state is not

L. Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

89

exceptional of either kind with respect to the plane P, there will be second slip wave, since a similar argument to (4.18)-(4.20) still holds, so that we will have nX/_]=~,n~

and

~ Z p , ( f ) ~ = - A z.

(4.28)

Beyond agreement with BGL's Theorem 9, (4.28) also shows that a normal Type 2 or Type 4 transonic state can also guarantee the existence of a second slip wave.

4.3. Type 3, 5 and 6 transonic states For a Type 3, 5 or 6 transonic state, since all three inhomogeneous waves in (2.19a) become real bulk waves, and their polarizations A~, A ~ and A ~ are necessarily linearly independent, we can find such a linear combination

tI=A~ +Aa + A ~

(4.29)

(in fact several) that in the plane P n.P: = 0 .

(4.30)

The traction associated with ,4 is then

I,=L,, +La +L r.

(4.31)

Both the ,,[ and the/~ are real vectors. Again, we can obtain Y,,~ etc, with the same expressions as (4.24a)-(4.24d), and thus we obtain again

IIZp~(6) ~ =

[Xn, n,£1 [A~, A a, At]

(4.32)

So, at a Type 3, 5 or 6 transonic state, if the superposition (4.29) is exceptional of either kind, IlZp~(0) II = 0

(4.33)

which means that the second slip wave is precluded. If the transonic state is normal, we again have nX/~,=AA

and

I Z p , ( ~ ) l ~ = - h 2.

(4.34)

This guarantees the existence of a second slip wave. Notice that a Type 3 transonic state is always composite exceptional, that means A = 0, so that there will never be a second slip wave in this situation [ 8 ]. This agrees with BGL's Theorem 7. Equation (4.34) also indicates that a normal Type 5 or Type 6 transonic state can also ensure the existence of a second slip wave. From the above discussions, we can see that, as long as the interface plane P contains the polarization of limiting waves, simple (4.11 ) or composite (4.22), (4.30), the existence and uniqueness is secured. We will denote such a plane as the proper plane, the corresponding interface geometry as the proper configuration. Therefore, we can summarize the above existence criteria for the second slip wave as the following existence theorem: Theorem: When a transonic state is not simple/composite exceptional of the first/second kind with respect to a proper plane, a second slip wave is guaranteed.

90

L. Wang. J. Lothe / Existence of second slip waves in anisotropic elastic media

4.4. Exceptional wave of the third kind For a Type 1 transonic state, when we tilt the interface (about an axis parallel to the slowness surface normal at the limiting wave, so that the limiting wave remains the same) from the proper plane P that contains polarization ,,i, the second slip wave may possibly disappear. In the next section, we will demonstrate that there is a very narrow window in the vicinity of n . A = 0 where a second slip wave can be observed. At the limit of such a window, we find that IlZp,(t ~) II gradually vanishes even though the transonic state remains normal. This is a case similar to the exceptional transonic state in the surface wave theory. From the definition of slip waves (2.22a) and (2.22b), we can reason as follows about such a limit: when such a normal Type 1 transonic state is approached, the superposition of the two inhomogeneous waves and the limiting bulk wave will satisfy the slip wave condition, giving

n.A'=n.(EoA,,+EaAa+E~:i) L'=E,L,

=0

+E~L/3 + E r L = k n .

(4.35) (4.36)

Here we use the normalized eigenvectors for the inhomogeneous waves. From (4.35) and (4.36), we can show that

IlYp,(:)ll=0

and

IlZp,(,')ll=0

(4.37)

meaning that there is no second slip wave. Therefore, we describe such a transonic state as an exceptional transonic state of the third kind. From the conditions (4.35) and (4.36), it is clear that A '.L' =0.

(4.38)

Using the normalization condition (2.13), we obtain

E.A,~2

. L . +E~A/3.L~+E-~A" ^ . L = 0 ^ .

(4.39)

Notice that ( A . , L,,), (A~, L v) are normalized eigenvectors, and ,4 ./~ = 0 , (4.39) can be reduced E,Z~+ k,'~, = 0 .

(4.40)

Therefore, we can rewrite (4.36) the form

L ' = E,,( L,, _+iLt0 + Ez, lf_,= kn ,

(4.41)

and consequently, a necessary condition for the third kind of exceptional wave can be explicitly established from the co-planar requirement as [n,/~,, ( L : , - t - i L a ) ] = 0 .

(4.42)

The exceptional waves o f the third kind versus slip waves are very different from the ordinary (or the first kind) of exceptional waves versus surface waves. The difference lies in that the ordinary exceptional waves for surface waves do not contain an inhomogeneous wave. The surface wave degenerates into a pure bulk wave at the exceptional transonic state, and near the exceptional wave, there can be a quasi bulk wave [ 17,18]. A possible exception is the point on the exceptional wave branch where a subsonic surface wave crosses to a supersonic surface wave [ 15]. Then, at the crossing point, an exceptional bulk wave and a two component surface wave (inhomogeneous) are present simultaneously. The exceptional wave of the third kind, however, contains a bulk-wave part and an inhomogeneous-wave part, but the two parts are not exceptional separately, in distinction from the surface wave case. Also, the exceptional waves of the third kind form a two-dimensional area on the unit sphere Ilkll = 1, while the other exceptional waves form curves.

L. Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

91

The arguments above indicate that a normal transonic state is not a sufficient requirement for the existence of the subsonic second slip wave in any general interface geometry. But it is always true than an exceptional transonic state of any kind will prevent the existence of a subsonic second slip wave. For the proper configuration where the interface is a proper plane, a normal transonic state becomes a sufficient and necessary condition for the subsonic second slip wave as stated in the Theorem.

5. Numerical calculations and discussions Numerically, we calculate two typical cases where the transonic states are Type 1 and Type 4 because Type 3 transonic states are intrinsically composite exceptional of the first kind and Types 2, 5 and 6 are usually found to be composite exceptional in most materials. The example materials that we choose are cubic, trigonal and triclinic materials. The following general scheme is established in order to examine the existence of second slip waves systematically (see Fig. 2). Consider a transonic state related to a wave vector k, which belongs to a point on the outer sheet of the showness surface, k = sin 0 cos ~pe~ + sin 0 sin ~pe~ + cos Oez ,

(5.1)

where ex, ey and e~ are the orthogonal crystallographic basis. We first find the group velocity g (or equivalently the normal of the slowness surface) associated with the bulk wave k, and construct the interface plane P containing g and the polarizationA of the bulk wave, then construct the reference plane Iq containing k and which is perpendicular to the plane P. In this way, the interface geometry, an orthogonal triad (m, n, t), is defined and it is the proper configuration consistent with the requirement of the Theorem, i.e., n..4 = 0. If the transonic state is normal, a subsonic second slip wave should be found. Next, we tilt the interface plane P around the normal of the slowness surface g at an angle 3' from the proper plane. Under such a new interface geometry (m,, n ' , t ' ) , i.e., a non-proper configuration, we expect the second slip wave still to exist if the rotation is small. All the interface planes defined above for a given k, proper or non-proper, contain g, i.e., all the configurations are related to the same limiting wave. The above scheme leads us to an important concept, the e x p o s e d transonic state. For a given wave vector k and

(a)

(b)

n

g

eZ

t

~

(c),/~~ t

nI

tt Fig. 2. Determination of the interface geometry. (a) the proper configuration related to the bulk wave k; (b) the interface geometry (m, n, t) for ( a ) : (c) the interface geometry (m ', n ', t ' ) for the non-proper configuration in the neighbourhood of (b) after a rotation around the normal to the slowness surface g. The vector r is an auxiliary vector where r = g × n, T = arccos ( r • r ' ) .

92

L Wang. J. Lothe / Existence of second slip waves in anisotropic elastic media

an interface orientation y, if the transonic state is a first transonic state, that is, a bulk wave k related to a point on the slowness surface whose tangential line L parallel to n does not intersect with the slowness surface itself, we describe such a state as the exposed transonic state, otherwise, we call it the unexposed transonic state. Particularly, for any fixed k, when the transonic states for the entire set of y = [ - 9 0 °, 90 °] are exposed transonic states, such a k will be located in the so-called exposed region (EX region). Correspondingly, a partially exposed region (PE region) and an unexposed region ( U E region) can also be introduced. The partially exposed region is the set o f k where the transonic states for 3, = [ - 9 0 °, 90 °] consist of exposed transonic states and unexposed transonic states, while the unexposed region is the set of k where all transonic states tbr y = [ - 9 0 °, 90 ° ] are unexposed transonic states. There is an important distinction between EX and PE regions. For a k in the EX region, the transonic states for the entire 3/domain are Type I. However, in the PE region there is a transition from the domain of exposed transonic states, say, 3,= [ %, % ) to the domain of unexposed transonic states, "y= ( %, y,.], and such a point % is, in fact, a Type 4 transonic state. Such a Type 4 transonic state is usually asymmetric with respect to the interface P. The search for second slip waves with the above scheme over all the exposed transonic states either within the exposed region or within the partially exposed region will cover all the possible interface geometries. We will consider interface geometries around the surface normal for a certain k in either of those two regions and carry out numerical calculations. We calculate the velocities of the transonic states, the second slip waves and the subsonic surface waves, and [[Zpl(C)[[ for a set of orientations associated with the k=k(~o, O) as defined in (5.1) with successive tilting of the interface around g. The tilting angle 3~is in the range of 3, = [ - 90 °, 90°]. Numerically, we take Cu and TlaTaSe4 (cubic), Quartz (trigonal) and an artificial triclinic material [ 16] as examples. W e first calculate the outer slowness surface, then find the first and the second kind exceptional waves and the EX, PE and UE regions. Figures 3 and 4 present three dimensional realizations of these results for Cu and Tl3TaSe4, respectively. These two cubic materials represent main characteristics of the slowness surface for cubic materials [6]. Figure 3 shows the locations o f exceptional waves. It is clearly shown that, the lines of the first kind of exceptional waves consist of only purely transverse bulk waves or SH waves, and the lines of the second kind of exceptional waves are always found lying in the symmetry planes and they are related to the non-transverse bulk waves. Figure 4 illustrates EX, PE and UE regions. The exposed ( E X ) regions are clearly defined within the most bulging parts of the slowness surface and there we can find intersections between the lines of the first and the second kind

Fig. 3. Outer sheet of the slowness surface for cubic materials, (a) Cu and (b) TI3TaSe4.with illustrations of the exceptional waves of the first kind and the exceptional waves of the second kind. The exceptional waves of the first kind are represented by the black lines in (a) and white lines in (b), and the exceptional waves of the second kind are denoted by the white lines in (a) and black lines in (b).

L. Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

93

Fig. 4. Illustration of the exposed region (dark grey), partially exposed region ( bright grey) and the unexposed region (mid-grey) for the cubic materials. (a) Cu, (b) TI~TaSe4.

exceptional waves. The slowness surfaces for trigonal and triclinic materials look more complicated, and we will not include them here.

5.1. Type 1 transonic state Consider the case related to Type I transonic states. Noticing the difference between the EX and PE regions, we first confine our computation within the exposed region. We find that, at the normal transonic states related to the proper plane P, i.e., 7 = 0 °, there is always a unique subsonic second slip wave. As we tilt the interface away from the proper plane P with the angle 3' to non-proper configurations, we observe that there exists a very narrow window A3' = ( 3"0,3'd ) in the vicinity of 3'-- 0 ° where the second slip waves are present. At the limits 3'0± where the second slip waves coincide with transonic states which are neither first nor second exceptional, IlZp,(0)II approaches zero, and so does [n, L, ( L , , + iLt3)]. This indicates that 3'~: are exceptional transonic states of the third kind. Beyond such a window, we observe that IlZpt(0) II becomes positive and increases monotonically. In Cu, we choose a k = k ( q~, 0), where q~= 8 ° and 0 = 4 8 °, in the exposed region. The existence window for the second slip waves is a 3 ' = ( - 0 . 7 1 °, 0.69°). The numerical results are illustrated Fig. 5. Figure 5 (a) demonstrates the differences between the velocities of the limiting wave t3 and that of the second slip wave v2, which are ranging from 10- 5 m / s to zero. Although the differences are very small it demonstrates the existence of an existence window systematically. Figure 5 (b) shows that IIZpt(~3)IIgradually vanishes as 3' approaches the limits 3'0± which indicates that the existence window for the second slip waves is extremely narrow. In TI~TaSe4, because the exposed regions are very narrow, we can not reach the second slip wave in the proper configuration when the computation has a precision limit of 10- ,6 When we come to Quartz and the triclinic material, we can still define the exposed region though the outer slowness surfaces look more complicated. In Quartz, a k=k(~o, O) (~p=3 °, 0 = 3 0 °) is selected. The existence window for second slip waves is A3'= ( --2.74 °, 2.49 °) (See Fig. 6 ( a ) ) . The maximum of Av=3--v2 is about 5 x 10 - 3 m / s and that is very large compared with the example in Cu. In the triclinic material, calculation for a k = k( ~0, 0) ( ~o= - 30 °, 0 = 90 °) gives the window for second slip waves in the range of A3' = ( - 0.096 °, 0.1'65 °) (See Fig. 6(b) ). The maximum of A~, = t3-- v2 is as small as about 5 × 10 -8 m/s. Now, we consider the transonic state in the partially exposed region where a Type 4 transonic state appears at, say, 3'4, as we tilt the interface away from the proper configuration.

94

L Wang, J. I_x~the/ Existence of second slip waves in anisotropic elastic media 0.025

J

I

0.5

I

(a)

'

'

'

-0.5

0.0

0.5

(b)

0.020 E

~" E

"z

0.015

0.0

C

• 0.010 <3

e~

0.005 0.000 -1.0

-0.5

-0.5

0.0

0.5

.0

-I.0

7 (deg)

1.0

7(deg)

Fig. 5. (a) Difference between the velocities of the limiting waves £ and the second slip waves v2. (b) Variation of ]Z~,(f)it as a function of rotation angle 7for the case ¢ = 8°, 0=48 ° within ~/= [ - 1.0°, 1.0°] in Cu.

8

I

6 E

I

I

I

(a)

)

4

~z

I

2 0

~

I

I

(b)

/

2

E

~z

•~

i

0

-2 -4

I

I

I

I

I

I

-6

-4

-2

0

2

4

y (deg)

-I

6

I

-0.6

1

I

-0.2

I

0.2

I

0.6

7 (deg)

Fig. 6. Variation of nZr,(D)ll as a function of the rotation angle "yfor the case of (a) Quartz: ¢ = 3°, 0 = 30° within 7 = [ - 5.0°, 5.0°1, and (b) the artificial triclinic material: ~0= - 30°, 0 = 90° within 3,= [ - 0.5 °, 0.5°].

W e p e r f o r m c a l c u l a t i o n s in the n e i g h b o u r h o o d o f the p r o p e r c o n f i g u r a t i o n 7 = 0. A t a n o r m a l e x p o s e d T y p e 1 t r a n s o n i c state, t h e r e still e x i s t s a s e c o n d slip w a v e a n d the c o r r e s p o n d i n g e x i s t e n c e w i n d o w . H o w e v e r , b e y o n d the w i n d o w , IlZp,(0) II is not a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n o f 7 as it is in the e x p o s e d region. It r e a c h e s zero a g a i n at a n o t h e r third k i n d o f e x c e p t i o n a l t r a n s o n i c state a n d b e c o m e s n e g a t i v e u p to the T y p e 4 t r a n s o n i c state as 7774is a p p r o a c h e d . A f t e r b y p a s s i n g the T y p e 4 t r a n s o n i c state, it b e c o m e p o s i t i v e a g a i n b y c r o s s i n g o n e m o r e t h i r d k i n d e x c e p t i o n a l t r a n s o n i c state a n d t h e r e f o r e a n o t h e r e x i s t e n c e w i n d o w in the v i c i n i t y o f the T y p e 4 t r a n s o n i c state at 74 occurs. W i t h i n s u c h a w i n d o w , all the s e c o n d slip w a v e s are related to n o r m a l T y p e 1 t r a n s o n i c states o f n o n p r o p e r c o n f i g u r a t i o n s e x c e p t the o n e at 7774that is r e l a t e d to a n a s y m m e t r i c n o r m a l T y p e 4 t r a n s o n i c state w h i c h is presumably a proper configuration. In Fig. 7 ( a ) , w e p r e s e n t a n e x a m p l e o f s u c h a T y p e 4 t r a n s o n i c state in C u r e l a t e d to ~ o = 8 °, 0 = 3 5 ° w i t h 7f

= 14.205° a n d 7774 = - 8 0 . 8 6 °. T h r e e e x i s t e n c e w i n d o w s are o b s e r v e d . W i n d o w 0 is in A 7 = ( -- 2.23 °, 4 . 3 6 °+ )

95

L Wang, J. Lothe/Existence of second slip waves in anisotropic elastic media

(' n ex p c,r-~-.5

Exposed

Unexposed

Unexposed

IlZp,II

N\\

(a) Window 2 Fig. 7.

l

Window 0

Expo~l

Unexposed

Ilzp,II

% \\

/////

Window 1

(b)

///

Window 0

/

Window 1

chematic illustration of IZ,,,(C)~ as a function of the rotation angle y for the exposed transonic state in the partially exposed region. (a) Cu: if, = 8 °, 0 = 35 °, ( b ) T13TaSe4: ~o= 30 °. 0 = 30 °, within the range of 3' = [ - 90.0°, 90.0° ].

in the vicinity of the normal Type 1 transonic state at 3, = 0. Window 1, A T = ( 14.025 °, 14.205°), and Window 2, A T = ( - 18.056 °, - 8 0 . 6 1 5 ° ) , are related to the normal Type 4 transonic states at y4+ and 74-, respectively. The maximum of Av = C - v 2 in Window 2 is very large and is of the order 2.0 m / s , the largest in our limited examples. Similar results are also obtained for TI3TaSe4 as shown in Fig. 7 ( b ) . Two Type 4 transonic states, for example, are found at "/4+ = 10.7529 ° and 3'4- = - 4-8° related to tp = 30 °, 0 = 30 °. The width of the existence window around 3'2 is extremely small ( 1 0 - 7 ° ) . The window in the vicinity of 7 = 0 ° is located in A T = ( - 4 . 8 0 ° - , 2.60 °÷ ), that is, it is connected with that o f the Type 4 transonic state at 3'4- = - 4.80 °. The maximum of Av = C-- v2 in Window 0, for example, is about l × 10 -3 m / s . In the partially exposed region, the proper configuration 3'o is frequently not found within the exposed domain. That will result in the disappearance of the window in the vicinity of 3~= 0 or Window 0, but the windows related to the asymmetric Type 4 transonic states will remain. Comparing with the results of BGL were A v = C - v 2 is in the range from l0 - ' ° m / s to l0 -~3 m / s for some selected cases, our results seem more systematic, and they also indicate that B G L ' s computation is just an illustration of an existence window in the neighbourhood of a Type 4 transonic state.

5.2. Type 4 transonic state W e have noticed that the above-mentioned Type 4 transonic states are asymmetric with respect to the interface I°, and that the transonic states in their neighbourhood are Type 1 transonic states. Besides the cases of asymmetric Type 4 transonic states, in cubic materials there are many symmetric Type 4 transonic states. In Cu, for example, instead of using the general scheme presented above, we can construct the interface geometry as m =e:,

n = c o s Te~ + s i n 7ey

(5.2)

which means that the propagation direction m is fixed along the four-fold symmetry axis ez, and as the interface normal n is rotating around ez, the transonic states in the entire range 3, = (0 °, 360 °) are of Type 4. At symmetric orientations, such as 7 = 0 °, 45°, 90°, etc, the transonic states are exceptional of the second kind as illustrated in Fig. 3 ( a ) . The second slip waves are found in all the other cases where Type 4 transonic states are normal, and their

96

L Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

velocities v2 have well pronounced distances from the velocities of the transonic states t~. Figure 8 shows the differences between v2 and 0 and ]]Zpl(t~)l] in the range of ~/= [0 °, 45°]. These are very special cases. An important feature of these configurations is that all the interfaces are proper planes and the corresponding Type 4 transonic states are normal. Thus, we find no non-proper configurations nor the existence window. The advantage of selecting these configurations is that the difference between the velocity of the second slip wave and the limiting velocity of the transonic state is as much as 2.0 m/s, much greater than what it is in most cases related to Type 1 transonic states. As a matter of fact, all these interface geometries can be reached as we apply the general scheme in the partially exposed region as we did in the preceding subsection.

5.3. Exceptional transonic states

When the Type 1 transonic states are exceptional under the proper configuration in the exposed region, ]lZp](:) H will be zero consistent with the theorems. That means no second slip wave is present. In the neighbouring nonproper configuration we observe that llZpt(:)II is positive definite in the entire y domain and thus the second slip waves are precluded. W h e n we are in the partially exposed region, such exceptional transonic states will preclude the existence of second slip waves in the proper configuration and its near neighbourhood. But the existence of second slip waves appearing in the more distant neighbouring non-proper configurations can not automatically be excluded, because the occurrence of normal Type 4 transonic states, no matter where they appear, imply an existence window as we observe in the case of in Fig. 7. The Type 4 exceptional transonic states observed under the interface geometry (5.2) are also covered by our general scheme. In other words, we can reach those configurations when we adopt our scheme in the partially exposed region. Therefore, the numerical results indicate that, when the transonic state is exceptional of any kind, no second slip wave exists under the proper configuration, nor in the near neighbouring non-proper configurations.

2.5 V--" t

i

i

i

i

0.0

(a)

b)

2.0

E >

1.5

~z

E

-0.5

N,~

-l.0

1.0

0.5

-1.5

0.0

0

10

20

30

(deg)

40

50

0

I

I

I

I

10

20

30

40

50

7 (deg)

Fig. 8. (a) Difference between the velocities of the limiting wave f and the second slip wave v2. (b) Variation of IZp~(t~) ii as a function of rotation angle 3' for the case of the symmetric Type 4 exposed transonic states in Cu.

L Wang, J. Lothe/Existence of second slip waves in anisotropic elastic media

97

5.4. Other types of transonic states We have also examined some cases of Type 5 and Type 6 transonic states. For these types of transonic states, the interface is fixed, and they are usually located in certain symmetry planes which result in composite exceptional transonic states of the second kind. For example, in a modified monoclinic material suggested by Chadwick and Wilson [ 16], there is a Type 5 transonic state lying in the symmetry plane. Such a Type 5 transonic state is actually a Type 4 exceptional transonic state of the second kind which coincides with a Type 1 transonic state that is exceptional of the first kind with respect to the other plane. In Cu, we can find a Type 6 transonic state under the interface geometry, m = cos ~pex+ sin ~ e y , n = ¢z where ~p= 22.5 ° [ 15 ]. Such a Type 6 transonic state is composite exceptional with respect to the symmetry plane. To our knowledge, there will be no second slip wave for such a configuration. The remaining configurations are the cases related to the unexposed transonic states. Our investigation scheme ensures that all those configurations are covered when we go through all the exposed transonic states.

5.5. Summary Above all, we demonstrate the existence of second slip waves related to the normal transonic states both in the exposed and in the partially exposed region. When the transonic states under the proper configurations are exceptional of either the first or second kind, no second slip wave is found, consistently with the theoretical analysis. When the transonic states are not exceptional of either the first or second kind under the proper configurations, there will be a unique subsonic second slip wave and an existence window can be found for the neighbouring non-proper configurations. For a non-proper configuration, introduction of the exceptional wave of the third kind, together with the observation of an existence window, enables us to conclude the numerical results for non-proper configurations with the following statement: When the transonic state is normal, i.e., neither first nor second kind exceptional, second slip waves can only be found in a narrow vicinity of the proper configuration, a window which is confined between two exceptional transonic states of the third kind. This statement refers to configurations with same limiting wave when the interface is tilted as the general scheme specifies. Because the first slip waves are actually the subsonic surface waves in our discussions, we can deduce from the surface wave theory [3,4] that, when an exceptional transonic state is of the first kind, there may or may not be a first slip wave and there is no second slip wave; and when the exceptional transonic state is of the second kind, there is always an unique first slip wave but no second slip wave. Therefore, we can summarize our numerical results on the existence of both the first and the second subsonic slip waves associated with the exposed transonic states as in Table 1. The left block is for the proper configurations and the right block is for the non-proper configurations in the neighbourhood of the proper configurations.

6. Conclusions In this paper, we have formulated an existence theorem for the subsonic second slip wave propagating along the interface separating two identical materials with the same orientation. By examining the behaviour of the planar impedance tensor at various types of transonic states, using the Stroh sextic formalism, we are able to obtain more

98

L Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

Table 1 The existence of slip waves in the proper and its near neighbouring non-proper configurations in cases associated with different features of the limiting transonic state. O means 'exist', X means 'does not exist' and * means that the slip wave exists only within the existence windows in the vicinity of the proper configurations. The superscript ** implies that when the exceptional transonic states are Type I there may or may not be first slip waves [4]. --*

proper configurations feature of transonic state

existence of slip waves

Ist excep.

X Ist** X 2nd Q Ist x 2nd Q 1st (S) 2nd

2nd excep. normal

non-proper configurations feature of transonic state

existence of slip waves

~

normal

~

normal

~

normal

C) 1st x 2nd Q Ist x 2nd 6_3 Ist * 2nd

explicit existence criteria for the existence of second slip waves and with natural introduction of the exceptional wave of the second kind. The numerical calculation yields results consistent with the theoretical analysis on the proper configurations, and it is observed that there is a narrow window for the existence of the second slip waves for non-proper configurations in the vicinity of the proper configuration. We have also revealed the existence of a third kind of exceptional wave related to the existence windows. The numerical results are more systematic and demonstrative than previous investigations. Some cases, like non-proper configurations where n.A ~ 0, are still lacking general theoretical analysis. More research is needed on the behaviour of the planar impedance IlZp,(c) II at transonic states in non-proper configurations, and which illuminate the observed fact that IlZo,(C)II is generally not monotonic under these circumstances. Applying computer graphics techniques, we have been able to build up three-dimensional graphics to demonstrate the relation between the exceptional waves of the first and the second kinds for cubic materials. There is a great potential of possible applications of computer graphics to surface wave theory and interfacial wave theory.

Acknowledgement The authors would like to thank STATOIL and the Norwegian Research Council for Science and the Humanities (NAVF) for the research supports.

References [ 1 ] Lord Rayleigh, " O n waves propagated along the plane surface of an elastic solid". Proc. Lond. Math. Soc. 17, 4--11 (1885). [2] R. Stoneley, "Elastic waves at the surface of the separation of two solids", Proc. Roy. Soc. Lond. A 106, 416-428 (1924). [ 3 ] D.M. Barnett and J. Lothe, "Consideration of the existence of surface wave ( Rayleigh wave) solutions in anisotropic elastic crystals", J. Phys. F 4 , 671-678 (1974). [4] D.M. Barnett and J. Lothe, "'Free surface (Rayleigh) waves in anisotropic elastic half spaces: the surface impedance method", Proc. Roy. 5oc. Lond. A 402, 135-152 (1985). [5] P. Chadwick and G.D. Smith, "Foundations of the theory of surface waves in anisotropic elastic materials", in: Advances in Applied Mechanics, 17, 303-376, Academic Press, NY (1977). [6] P. Chadwick and G.D. Smith, "Surface waves in cubic elastic materials", in: Mechanic's of Solids, The Rodney Hill 60th. Annit'ersary Volume, 47-100 (1982).

L Wang, J. Lothe / Existence of second slip waves in anisotropic elastic media

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