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Waves and statics for functionally graded materials and laminates
International Journal of Engineering Science 47 (2009) 1315–1321
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Waves and statics for functionally graded materials and laminates q D.F. Parker * School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, UK
a r t i c l e
i n f o
Article history: Available online 12 May 2009 Communicated by K.R. Rajagopal Keywords: Transverse isotropy Surface wave Functionally graded plates
a b s t r a c t It is shown here that, for any laminated stacking of elastic materials which are transversely isotropic with respect to an axis Oz, having elastic moduli which may depend either continuously or discontinuously on the coordinate z, surface-guided disturbances at any frequency are governed by the reduced membrane equation. An analogous treatment of the statics of plates having the same structure, yields the static theory of functionally graded plates due originally to Spencer et al. Thus, for functionally graded transversely isotropic plates with traction-free surfaces, displacements have a structure closely analogous to dynamic disturbances and are represented in terms of solutions to the biharmonic equation. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The static theory of linearly elastic functionally graded and laminated plates was initiated by Kaprielian et al. [1] and developed by Tony Spencer and co-workers (see [2,3] and references therein). In this theory, the material is, at each point, taken to be isotropic, but with Lamé constants and density depending upon the through-thickness coordinate. More recently, it has been shown [4] that time-harmonic surface waves and plate waves in a transversely isotropic medium may be related to a solution of the reduced membrane equation (the Helmholtz equation in two dimensions). Following the recent [5] recognition that a rotational invariance of material properties is the key to this result, it is now clear that a similar reduction to a scalar equation is possible for waves in all plates and half-spaces which locally are transversely isotropic. The density and elastic moduli may vary continuously or discontinuously as functions of the cartesian coordinate z, so describing functionally graded or laminated media. Besides recording this generalization of Achenbach’s result and of those of Kiselev [6,7] and coworkers, this paper links the dynamic theory to the static theory which owes so much to Tony Spencer. In particular, for traction-free plates formed of any stacking of transversely isotropic, functionally graded materials, displacements are described through a solution to the biharmonic equation and are closely analogous to other solutions found by Spencer [8] and based upon plane and anti-plane strain. The paper also reveals a hierarchy of solutions, as shown by England [9], in which transverse loads are solutions to Laplace’s equation, the biharmonic equation, etc. 2. Uni-directional waves in functionally graded materials Using linear elasticity theory, the components tij of Cauchy stress are related to the displacement components ui ðx; tÞ through tij ¼ cijlm ul;m , where x1 ; x2 and x3 z are Cartesian coordinates, cijlm are the elastic moduli and ;j denotes partial
I In memory of Tony Spencer – for more than a quarter of a century an exemplary Head of Department and an inspirational leader in research. * Corresponding author. Tel.: +44 131 447 6364; fax: +44 131 650 6553. E-mail address: [email protected]
0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.04.001
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
differentiation with respect to xj . The moduli cijlm and density q are allowed to be continuous or discontinuous functions of the coordinate z, either in the half-space z x3 > 0 or within a plate 0 < z < h. In a uniform half-space z P 0, the standard € i in z > 0, the traction-free boundary condition surface wave (a Rayleigh wave) is a solution of the Euler equation tij;j ¼ qu ti3 ¼ 0 at z ¼ 0 and the decay condition u ! 0 as z ! 0, with displacements depending on only the depth z and a travelling wave coordinate x1 ct. Here, a dot denotes partial differentiation with respect to time t and the constant c is the propagation speed. The displacement field and speed c are found by seeking uj ¼ Re U j ðz; kÞeikðx1 ctÞ , so yielding a system of constant coefficient differential equations. c is then determined (uniquely) so as to ensure compatibility between the traction-free and decay conditions. It has been shown [4,5] how these solutions may be generalized to give solutions in which ^ 1 ; x2 ÞU 3 ðz; kÞeikct , where wðx ^ 1 ; x2 Þ is any solution to the reduced membrane equation. Here, following steps used u3 ¼ Re wðx in [5] yields a similar result for transversely isotropic media with Oz as symmetry axis and with density and elastic coefficients being any piecewise continuous functions of z. Let z ¼ zp ; ðp ¼ 1; 2; . . . PÞ be the only locations at which either the density and/or the elastic coefficients are discontinuous. The governing system is then
€i tij;j ¼ qu
in z > 0; z – zp
ðp ¼ 1; 2; . . . ; PÞ;
ð2:1Þ
with traction-free condition
ti3 ci3lm ul;m ¼ 0
at z ¼ 0;
ð2:2Þ
continuity conditions
½½t i3 ¼ 0 ;
½½ui ¼ 0
at z ¼ zp
ð2:3Þ
and the decay condition u ! 0 as z ! 1. Here, ½½ denotes the jump in a quantity at z ¼ zp . For transversely isotropic materials, the elastic moduli are such that the stress components have the form
t11 ¼ C 11 u1;1 þ C 12 u2;2 þ C 13 u3;3 ;
t 23 ¼ t 32 ¼ C 44 ðu2;3 þ u3;2 Þ;
t22 ¼ C 12 u1;1 þ C 11 u2;2 þ C 13 u3;3 ;
t 13 ¼ t 31 ¼ C 44 ðu3;1 þ u1;3 Þ;
t33 ¼ C 13 ðu1;1 þ u2;2 Þ þ C 33 u3;3 ; 1 with C 66 ¼ ðC 11 C 12 Þ: 2
t 12 ¼ t 21 ¼ C 66 ðu1;2 þ u2;1 Þ; ð2:4Þ
Then, seeking travelling waves of the form u ¼ Re Uðz; kÞeih , t ¼ Re Tðz; kÞeih , where h kx1 xt yields (with U ¼ Uðz; kÞe1 þ Vðz; kÞe2 þ Wðz; kÞe3 ) from (2.1) the ordinary differential equations in z > 0 (z – zp ) 2
½C 44 ðU 0 þ ikWÞ0 þ ikC 13 W 0 þ ðx2 q k C 11 ÞU ¼ 0; 0 0
2
2
½C 44 V þ ðx q k C 66 ÞV ¼ 0
ð2:5Þ ð2:6Þ
and 2
½C 33 W 0 þ ikC 13 U0 þ ikC 44 U 0 þ ðx2 q k C 44 ÞW ¼ 0:
ð2:7Þ
Here, primes denote ordinary differentiation with respect to z. At each location z ¼ zp , the continuity conditions (2.3) yield
½½C 44 ðU 0 þ ikWÞ ¼ 0 ¼ ½½C 44 V 0 ¼ ½½C 33 W 0 þ ikC 13 U; ½½U ¼ ½½V ¼ ½½W ¼ 0;
ð2:8Þ
while, at the traction-free surface z ¼ 0, the conditions are
C 44 ðU 0 þ ikWÞ ¼ 0 ¼ C 44 V 0 ¼ C 33 W 0 þ ikC 13 U:
ð2:9Þ
Also, decay is ensured by choosing U; V; W ! 0 as z ! 1. For waves in a plate 0 < z < h, the decay condition is replaced by Eq. (2.9) at z ¼ h. Either for a half-space z > 0 or for a plate, transverse (shear-horizontal) displacements U ¼ Vðz; kÞe2 are seen to uncouple from the in-plane (sagittally-polarized) displacements U ¼ Uðz; kÞe1 þ Wðz; kÞe3 . In a uniform half-space, the only solutions are the Rayleigh wave, with V 0, with U and W specific linear combinations of two decaying exponentials and with c x=k equal to the Rayleigh wave speed cR . In a uniform half-space z > h with a dissimilar adjoining layer 0 < z < h, Love waves have transverse displacement V decaying in z > h, but oscillatory in the layer. The explicit solutions
(
VðzÞ ¼
^ cos bz ^ ebðzhÞ cos bh
0 6 z 6 h; h 6 z;
U ¼ W ¼ 0;
ð2:10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 66 k2 Þ=C ^ 44 and b ¼ ðC 66 k2 qx2 Þ=C 44 (with hats denoting density and elastic moduli in the layer) ^ x2 C ðq ^ 44 b ^ 66 =q ^ 44 =q ^ tan bh ^ ¼ C 44 b. Since, C 44 b2 ¼ ðC 66 qC ^2 , ^ Þk2 þ ðqC ^ Þb show that x is related to k through the dispersion relation C ^ this shows that the waves are dispersive (b=k depends upon k so that x=k is not constant). Also, these waves exist in many
^¼ where b
modes, with Vðz; kÞ having 0; 1; 2; . . . zeros within the layer. Moreover, in these same materials, sagitally-polarized waves generalizing Rayleigh waves also exist – with many modes corresponding to a chosen value of k, each being dispersive.
D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
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More generally, for functionally-graded materials or for laminates with more than one interface zp , waves of both the sagittally-polarized and transverse types may exist. They will be dispersive, with distinct modes corresponding to the various branches of the dispersion relation. 3. Generalized periodic vibrations Just as Achenbach [4], Deutsch et al. [10] and Kiselev et al. [7] have found disturbances related to the reduced membrane equation, we show that all layered and functionally-graded materials summarized in Section 2 permit wavelike solutions with depth-dependence related to solutions of the system (2.5)–(2.9). Because all material properties are invariant under rotations about the Oz direction, sagittally-polarized waves exist of the form
n o ^ þ Wðz; kÞ e3 eiðkxxtÞ u ¼ Re ½Uðz; kÞ k
ð3:1Þ
^ k=k and where k ¼ k1 e1 þ k2 e2 is the surface wave vector having magnitude k jkj, with wave normal the unit vector k with sagittally-polarized displacements Uðz; kÞ and Wðz; kÞ, where x ¼ xðkÞ is the dispersion relation arising from compati^ with displacement having the bility of the system (2.5), (2.7)–(2.9). Similarly, SH waves may propagate in any direction k, form
n o ^ eiðkxxtÞ u ¼ Re Vðz; kÞ e3 k
ð3:2Þ
and with dispersion relation arising from compatibility of (2.6), (2.8) and (2.9). 3.1. General sagittally-polarized waves More generally, disturbances having the same depth dependence as the waves (3.1) are found by seeking displacements of the form
^ yÞWðz; kÞeixt ; x x1 ; y x2 u ¼ Re ½fe1 /ðx; yÞ þ e2 wðx; yÞgUðz; kÞ þ e3 wðx;
ð3:3Þ
with x ¼ xðkÞ satisfying the dispersion relation for sagittally-polarized waves. The corresponding traction components on ^ ;x WÞ, t23 ¼ C 44 ðwU 0 þ w ^ ;y WÞ and t33 ¼ C 13 ð/;x þ w;y ÞU þ C 33 wW. ^ any plane z ¼ constant are t 13 ¼ C 44 ð/U 0 þ w ^ ;y and ^ ;x ; ikw ¼ w Inserting conditions (2.9) into the traction-free boundary conditions (2.2) yields ik/ ¼ w ^ so that wðx; ^ yÞ must satisfy the reduced membrane equation /;x þ w;y ¼ ikw;
^ ;yy ¼ k2 w: ^ w ^ ;xx þ w ^ r22 w
ð3:4Þ
^ yÞ satisfies the two-dimensional Helmholtz Eq. (3.4), the corresponding disThen, it is readily checked that, whenever wðx; placement fields
^ yÞgeixt ^ þ Wðz; kÞe3 wðx; u ¼ RefðikÞ1 Uðz; kÞgrad w
ð3:5Þ
satisfy also the continuity conditions (2.3) at each interface z ¼ zp as well as the partial differential Eq. (2.1). This shows that, ^ yÞ to the reduced membrane Eq. (3.4) and for any piecewise continuous z- dependence of corresponding to any solution wðx; the density qðzÞ and elastic moduli C mn ðzÞ, there exist time-harmonic elastic displacement fields which decay with depth and satisfy the traction-free boundary condition (2.2) at z ¼ 0. 3.2. General SH waves Analogously to the treatment above, it is readily shown that displacements
yÞ eixt u ¼ Re Vðz; kÞe3 grad wðx;
ð3:6Þ
satisfy the field Eq. (2.1), the traction-free boundary condition (2.2) and the continuity conditions (2.3) at each material yÞ satisfies the reduced membrane equation interface z ¼ zp , provided that wðx;
;yy ¼ k2 w; w ;xx þ w r22 w
ð3:7Þ
but with x here related to k through the dispersion relation arising from the solution V to the boundary value problem (2.6), (2.8) and (2.9) for shear-horizontal waves (U ¼ W ¼ 0 in the system (2.5), (2.7)–(2.9) in the functionally graded material. 4. Connections to the static theory of plates The ability to relate both sagittally polarized and shear horizontal surface-guided waves in any functionally graded material to solutions of the reduced membrane equation suggests close links to the static theory of laminated and functionallygraded plates initiated in [1]. For this analogy, first consider displacements independent of x2 , for which the stress components (2.4) become
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
t 11 ¼ C 11 u1;1 þ C 13 u3;3 ; t12 ¼ t 21 ¼ C 66 u2;1 ; t13 ¼ t31 ¼ C 44 ðu1;3 þ u3;1 Þ;
ð4:1Þ
t 22 ¼ C 12 u1;1 þ C 13 u3;3 ; t23 ¼ t 32 ¼ C 44 u2;3 ; t33 ¼ C 13 u1;1 þ C 33 u3;3 :
For displacements without body force and with traction vanishing at both z ¼ 0 and z ¼ h, these stresses must satisfy
ti1;1 þ ti3;3 ¼ 0 in 0 < z < h ;
z – zp for i ¼ 1; . . . ; 3
ð4:2Þ
with
½½t i3 ¼ 0 at z ¼ zp ;
ti3 ¼ 0 at z ¼ 0; h for i ¼ 1; . . . ; 3;
ð4:3Þ
from which it is immediately seen that plane strain deformations u1 ; u3 uncouple from the anti-plane strain deformations u2 . The most basic solutions for plane strain have t13 0 and t33 0 so that t11;1 ¼ 0. Thus, u3;1 ¼ u1;3 with C 13 t11 ¼ ðC 213 C 11 C 33 Þu3;3 , so giving u3 in the form
u3 ¼ mðxÞ þ nðzÞ
u3;3 ¼ n0 ðzÞ:
with
0
Since u1;3 ¼ u3;1 ¼ m0 ðxÞ, then u1 has the form u1 ¼ lðxÞ zm0 ðxÞ, so giving 0 ¼ t33 ¼ C 13 ðzÞ½l ðxÞ zm00 ðxÞ þ C 33 ðzÞn0 ðzÞ. 00 2 Consistency then requires that l ðxÞ ¼ 0 ¼ m000 ðxÞ, so giving l ¼ c1 þ bx and m ¼ c3 þ ax þ dx , thus yielding u1 ¼ c1 þ bx az 2dxz, where vanishing of t33 requires that
C 33 ðzÞn0 ðzÞ ¼ C 13 ðzÞ ð2dz bÞ
for z – zp :
Solving this allows nðzÞ to be written as nðzÞ ¼ bT 1 ðzÞ þ 2dT 2 ðzÞ, where
T 1 ðzÞ
Z
z
D2 ðnÞ dn; T 2 ðzÞ
Z
0
z
nD2 ðnÞ dn; D2 ðzÞ C 13 ðzÞ=C 33 ðzÞ:
ð4:4Þ
0
Here, T 1 ðzÞ and T 2 ðzÞ are direct generalizations to transverse anisotropy of the material parameters defined in Eq. (2.24) of [9]. Then, using u3 ¼ c3 þ ax þ bT 1 ðzÞ þ d½x2 þ 2T 2 ðzÞ, gives the displacements in the form
u1
¼
u3
c1 c3
z 2xz x þa þd 2 ; þb x x þ 2T 2 ðzÞ T 1 ðzÞ
ð4:5Þ
in which the four contributions on the right-hand side describe, respectively, a translation, a rotation, a stretching and a uniform flexure of the strip. Proceeding analogously to the dynamic case, involves first generalizing (4.5) by seeking a sequence of functions fU k ðzÞ; V k ðzÞ; W k ðzÞg such that displacements
u1 ¼
K X k¼0
u3 ¼
Kþ1 X k¼0
1 U k ðzÞxKk ; ðK kÞ!
u2 ¼
K X k¼0
1 V k ðzÞxKk ; ðK kÞ!
1 W k ðzÞxKkþ1 ðK k þ 1Þ!
ð4:6Þ
satisfy the system (4.1), (4.2) and (4.3). First deriving from (2.4) the expressions for t i3 , then imposing the traction-free conditions give (at z ¼ 0; h)
U 0k þ W k ¼ 0 for k ¼ 0; . . . ; K C 33 W 0k
V 0k ¼ 0 for k ¼ 0; . . . ; K ;
þ C 13 U k2 ¼ 0 for k ¼ 2; . . . ; K þ 1 ; W 00 ¼ 0 ¼ W 01 ;
ð4:7Þ
and the continuity equations (at z ¼ zp ) give
½½C 44 ðU 0k þ W k Þ ¼ 0 for k ¼ 0; . . . ; K ; ½½C 44 V 0k ¼ 0 for k ¼ 0; . . . ; K; ½½C 33 W 00 ¼ ½½C 33 W 01 ¼ 0; ½½C 33 W 0k þ C 13 U k2 ¼ 0 for k ¼ 2; . . . ; K þ 1;
ð4:8Þ
while the equilibrium equations give (for z – zp )
C 13 W 0k þ C 11 U k2 þ ½C 44 U 0k þ C 44 W k 0 ¼ 0;
k ¼ 0; . . . ; K;
ðC 44 V 0k Þ0 þ C 66 V k2 ¼ 0;
k ¼ 0; . . . ; K;
C 44 U 0k þ C 44 W k þ ½C 33 W 0kþ2 þ C 13 U k 0 ¼ 0;
k ¼ 0; . . . ; K 1;
ð4:9Þ
together with C 33 W 00 ¼ const: ¼ 0 and C 33 W 01 ¼ const: ¼ 0, showing that W 0 a0 and W 1 ¼ a1 each are constants. Since, likewise, V 0 ¼ c0 and V 1 ¼ c1 are constants, the equation ðC 44 V 02 Þ0 ¼ C 66 c0 together with its boundary and continuity conditions is inconsistent. Thus K ¼ 1, so that the anti-plane displacements simplify. For the functions U k ðzÞ and W k ðzÞ, solving the equation ½C 44 ðU 00 ðzÞ þ a0 Þ0 ¼ const: ¼ 0 with boundary and continuity conditions gives U 0 ¼ b0 a0 z. Then, rearranging C 13 U 0 þ C 33 W 02 ðzÞ ¼ 0 as a formula for W 02 allows solution as
W 2 ðzÞ ¼ a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞ :
ð4:10Þ
D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
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Similarly solving for U 1 ðzÞ and W 3 ðzÞ gives
U 1 ðzÞ ¼ b1 a1 z ;
W 3 ðzÞ ¼ a3 þ a1 T 2 ðzÞ þ b1 T 1 ðzÞ:
Continuing this process gives next ½C 44 ðU 02 þ W 2 Þ0 ¼ C 13 W 02 C 11 U 0 ¼ D1 U 0 for z – zp , where D1 ðzÞ ¼ ðC 213 C 11 C 33 Þ=C 33 . Together with continuity of C 44 ðU 02 þ W 2 Þ at z ¼ zp and vanishing at z ¼ 0; h this gives
C 44 ðzÞ½U 02 ðzÞ þ W 2 ðzÞ ¼ b0 S1 ðzÞ a0 S2 ðzÞ; Rz Rz where S1 ðzÞ ¼ 0 D1 ðnÞ dn and S2 ðzÞ ¼ 0 nD1 ðnÞ dn, which together lead to a compatibility condition which fixes the ratio b0 =a0 c S2 ðhÞ=S1 ðhÞ. 4.1. Iterative determination of U k ðzÞ and W k ðzÞ The continuous functions U k ðzÞ and W k ðzÞ are best determined for all K by rearrangement of (4.9) and its associated continuity and boundary conditions using the continuous functions
Rk ðzÞ C 44 ½U 0k ðzÞ þ W k ðzÞ; in terms of which t 13 ¼
Q 0k ¼ Rk ;
PK
k¼0 x
k
Q k ðzÞ C 13 U k ðzÞ þ C 33 W 0kþ2 ðzÞ;
RKk ðzÞ=k! , t 33 ¼
PKþ1
k¼0 x
k
ð4:11Þ
Q Kkþ1 ðzÞ=k! . This gives the system
R0k ¼ C 13 W 0k C 11 U k2 ¼ D1 U k2 D2 Q k2 ;
which may be integrated (using the traction-free condition t31 ð0Þ ¼ 0) as
Z z Z z W k ðzÞ ¼ ak D2 ðnÞU k2 ðnÞ dn þ Q k2 ðnÞ=C 33 ðnÞ dn; 0 0 Z z Z z U k ðzÞ ¼ bk W k ðnÞ dn þ Rk ðnÞ=C44 ðnÞ dn; 0 Z z 0 Z z Rk ðzÞ ¼ D1 ðnÞUk2 ðnÞ dn D2 ðnÞQ k2 ðnÞ dn; 0 0 Z z Rk ðnÞ dn; with d0 ¼ d1 ¼ 0; Q k ðzÞ ¼ dk
ð4:12Þ
0
(so that R0 ¼ R1 ¼ Q 0 ¼ Q 1 0). Since W k ðzÞ is required for k 6 K þ 1, the additional condition Rk ðhÞ ¼ 0 is accommodated by suitable choice of b0 ; . . . ; bK2 , so that both conditions t13 ð0Þ ¼ 0 ¼ t 13 ðhÞ may be satisfied. However, U k ðzÞ is required for k 6 K, but for k P 2 the conditions Q k ð0Þ ¼ 0 ¼ Q k ðhÞ cannot both be satisfied. Thus, deformations (4.6) cannot arise for K P 3 except for laterally loaded plates, such that t33 ðhÞ – t33 ð0Þ. If there is normal loading, then the conditions Q k ð0Þ ¼ 0 and Q k ðhÞ ¼ 0 may be relaxed. 5. Construction of plate bending theories The system of Eq. (4.9) is closely analogous to Eqs. (2.5), (2.6) and (2.7) (set x ¼ 0 and replace each factor ik by a unit reduction in the power of x in (4.6) – each operation corresponds to differentiation with respect to x). This motivates a search for static solutions to (4.2) and (4.3) in the form u1 ¼ v ;x ðx; y; zÞ v;y ðx; yÞ, u2 ¼ v ;y ðx; y; zÞ þ v;x ðx; yÞ with
v ðx; y; zÞ ¼
K X
Ak ðx; yÞ U k ðzÞ ;
u3 ¼
k¼0
K þ1 X
Ak ðx; yÞ W k ðzÞ
ð5:1Þ
k¼0
analogous to that used in [5] to obtain general vibrations (3.5) and (3.6) from standard travelling waves (setting v c0 B0 þ c2 B1 , so v;x ¼ B0;x V 0 þ B1;x V 1 ). This leads to expressions for stress components which include
t 13 ¼ t 31 ¼
K X
Ak;x C 44 ½U 0k ðzÞ þ W k ðzÞ þ AKþ1;x C 44 W Kþ1 ðzÞ;
k¼0
t 23 ¼ t 32 ¼
K X
Ak;y C 44 ½U 0k ðzÞ þ W k ðzÞ þ AKþ1;y C 44 W Kþ1 ðzÞ;
ð5:2Þ
k¼0
t 33 ¼
K X
C 13 ðAk;xx þ Ak;yy ÞU k ðzÞ þ
k¼0
Kþ1 X
C 33 Ak W 0k ðzÞ;
k¼2
so that use of relations (4.8) shows that the continuity conditions ½½ti3 ¼ 0 at z ¼ zp are satisfied whenever AKþ1 ¼ constant, r22 AK ¼ 0 and Akþ2 ¼ r22 Ak , k ¼ 0; . . . ; K 1. These conditions ensure that the equilibrium Eq. (4.2) also are satisfied, so long as the function vðx; yÞ satisfies r22 v ¼ constant. There thus exist two sequences of functions related to the constant AKþ1 and the harmonic function AK , respectively. Within each sequence
r2s 2 AKþ12s ¼ AKþ1 ¼ const:;
r22sþ2 AK2s ¼ r22 AK ¼ 0:
ð5:3Þ
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
For K even, rKþ2 A0 ¼ 0 and rK2 A1 ¼ AKþ1 ¼ constant; alternatively, for K odd, rKþ1 A0 ¼ AKþ1 ¼ constant and rKþ1 A1 ¼ 0. In 2 2 2 2s A with A ¼ r A . Thus, for example, for K even, the transverse displacement is each case, A2s ¼ r2s 0 2sþ1 1 2 2
u3 ¼ A0 ðx; yÞW 0 ðzÞ þ r22 A0 W 2 ðzÞ þ þ rK2 A0 W K ðzÞ þ A1 ðx; yÞW 1 ðzÞ þ r22 A1 W 3 ðzÞ þ þ rK2 A1 W Kþ1 ðzÞ;
ð5:4Þ
while the in-plane displacements may be written as u1 ¼ @ v =@x @ v=@y and u2 ¼ @ v =@y þ @ v=@x, where
v ðx; y; zÞ ¼ A0 ðx; yÞU 0 ðzÞ þ r22 A0 U2 ðzÞ þ þ rK2 A0 UK ðzÞ þ A1 ðx; yÞU1 ðzÞ þ r22 A1 U 3 ðzÞ þ þ r2K2 A1 U K1 ðzÞ Kþ2 A0 2
K 2 A1
ð5:5Þ
Kþ2 A1 2
with r ¼ 0; r ¼ AKþ1 ¼ constant (so that r ¼ 0). Here, the functions U k ðzÞ and W k ðzÞ are identical to those arising in the two-dimensional displacements of Section 4. P P P However, the traction components at z ¼ 0; h reduce to t 13 ¼ Kk¼0 Ak;x Rk ðzÞ, t 23 ¼ Kk¼0 Ak;y Rk ðzÞ and t33 ¼ Kk¼1 Akþ1 Q k1 ðzÞ, so that, as discussed after Eq. (4.12), it is possible to set t 13 ¼ 0 ¼ t23 at both z ¼ 0 and z ¼ h but displacements (5.1) exist only for
t33 ðx; y; hÞ t 33 ðx; y; 0Þ ¼
Kþ1 X
Ak ðx; yÞ½Q k ðhÞ Q k ð0Þ ¼
k¼4
Kþ1 X
Ak ðx; yÞ
Z
k¼4
h
Rk ðzÞ dz:
ð5:6Þ
0
The form of Eqs. (5.4) and (5.5) is the same for all functionally graded plates, including homogeneous isotropic plates; the layered structure enters only through the definitions of fU k ðzÞ; W k ðzÞg. Specific choices of K give theories related to those developed in [8,9]. 5.1. K ¼ 1 The functions W 0 ¼ a0 and W 1 ¼ a1 are constants, while (since U 2 ðzÞ is not required) b0 is arbitrary in U 0 ¼ b0 a0 z, U 1 ¼ b1 a1 z and W 2 ¼ a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞ. Also, since r22 A0 ¼ A2 ¼ const. and r22 A1 ¼ 0, this gives (after writing v;y ¼ p;x y; v;x ¼ p;y þ x, so that r22 v ¼ 2 ¼ u2;x u1;y )
u1 ¼ ðb0 a0 zÞA0;x þ ðb1 a1 zÞA1;x þ p;x y ; u2 ¼ ðb0 a0 zÞA0;y þ ðb1 a1 zÞA1;y þ p;y þ x; u3 ¼ a0 A0 ðx; yÞ þ a1 A1 ðx; yÞ þ ½a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞr22 A0 : ^ b0 =a0 and defining two new functions After writing c
Wðx; yÞ a0 A0 þ a1 A1 þ a2 r22 A0 ; 2b 2A
2 2
such that r W ¼ constant and r
b yÞ ðb c ^a1 ÞA1 þ p Aðx; 1
¼ 0, the displacements may then be expressed as
b y; u2 ¼ ½ðc b þ x; ^ zÞW þ A ^ zÞW þ A u1 ¼ ½ðc ;x ;y ^T 1 ðzÞr22 W; u3 ¼ Wðx; yÞ þ ½T 2 ðzÞ þ c
ð5:7Þ
2b 2A
2 2
^ and correspond to an in-plane stretching and a uniform where r W ¼ constant and r ¼ 0. Here, the arbitrary constants c b yÞ describes a divergence-free displacement – all with the surfaces z ¼ 0; h traction-free. rotation, respectively, while Aðx; 5.2. K ¼ 2 Plates without lateral loading Using R0 ¼ R1 ¼ Q 0 ¼ Q 1 ¼ 0; W 0 ¼ a0 , W 1 ¼ a1 , U 0 ¼ b0 a0 z, U 1 ¼ b1 a1 z, W 2 ¼ a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞ and W 3 ¼ a3 þ a1 T 2 ðzÞ þ b1 T 1 ðzÞ gives, from (4.12), R2 ðzÞ ¼ b0 S1 ðzÞ a0 S2 ðzÞ. Then, as before, setting b0 ¼ ca0 to make R2 ðhÞ ¼ 0 gives U 2 ðzÞ ¼ b2 a2 z a0 T 3 ðzÞ, where
T 3 ðzÞ
Z
z
fT 2 ðnÞ þ cT 1 ðnÞ þ ½S2 ðnÞ cS1 ðnÞ=C 44 ðnÞgdn:
ð5:8Þ
0
Then, taking
v and u3
from (5.4) and (5.5), writing
a0 A0 þ a1 A1 þ a2 r22 A0 þ a3 r22 A1 Wðx; yÞ; b ðb1 ca1 ÞA1 þ ðb2 ca2 Þr22 A0 þ p A where
ð5:9Þ
v;y y p;x and v;x x þ p;y , reduces displacements to the form b ;x y; u1 ¼ ðc zÞW;x T 3 ðzÞr22 W;x þ A b ;y þ x; u2 ¼ ðc zÞW;y T 3 ðzÞr2 W;y þ A 2
u3 ¼ W þ ½T 2 ðzÞ þ cT 1 ðzÞr22 W þ T 1 ðzÞB;
ð5:10Þ
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
b ¼ B ¼ constant, r4 W ¼ 0. The choice (5.9) identifies the biharmonic function Wðx; yÞ as the where r22 v ¼ 2 ¼ constant, r22 A 2 lateral displacement of the surface z ¼ 0. Here, the functions T 1 ðzÞ, T 2 ðzÞ and T 3 ðzÞ arise from the z–dependence of the elastic moduli. The solutions (5.10) are closely similar to those which Spencer [8] based upon plane and antiplane strain. The special case r22 W ¼ constant gives the in-plane stretching solutions (5.7) (i.e. K ¼ 1). 5.3. K ¼ 4 (including K ¼ 3 as a special case) With W 0 ; W 1 ; W 2 ; W 3 ; U 0 ; U 1 and U 2 as in the case K ¼ 2, setting b1 ¼ ca1 , b2 ¼ ca2 and d2 T 1 ðhÞ ¼ a0 S3 ðhÞ, defining further Rz Rf Rz D4 ðzÞ 0 dn=C33 ðnÞ, functions in terms of elastic moduli through D3 ðzÞ 0 0 ðn cÞD1 ðnÞ dndf, Rz Rz T 4 ðzÞ 0 ½T 3 ðnÞD2 ðnÞ þ D3 ðnÞ=C 33 ðnÞ dn and T 5 ðzÞ 0 fT 4 ðnÞ þ S3 ðnÞ=C 44 ðnÞ þ ðd2 =a0 Þ½D4 ðnÞ T 1 ðnÞ=C 44 ðnÞg dn yields
U 3 ðzÞ ¼ b3 a3 z a1 T 3 ðzÞ;
U 4 ðzÞ ¼ b4 a4 z a2 T 3 ðzÞ a0 T 5 ðzÞ;
W 4 ðzÞ ¼ a4 þ a2 T 2 ðzÞ þ b2 T 1 ðzÞ þ a0 T 4 ðzÞ þ d2 D4 ðzÞ; W 5 ðzÞ ¼ a5 þ a3 T 2 ðzÞ þ b3 T 1 ðzÞ þ a1 T 4 ðzÞ þ d3 D4 ðzÞ; Q 2 ðzÞ ¼ d2 þ a0 D3 ðzÞ;
Q 3 ðzÞ ¼ d3 þ a1 D3 ðzÞ:
ð5:11Þ
Further, R4 ðzÞ ¼ a2 ½cS1 ðzÞ S2 ðzÞ þ d2 T 1 ðzÞ a0 S3 ðzÞ and R5 ðzÞ ¼ a3 ½cS1 ðzÞ S2 ðzÞ þ d3 T 1 ðzÞ a1 S3 ðzÞ satisfy R4 ¼ R5 ¼ 0 at Rz z ¼ 0; h when b3 ¼ ca3 and d3 T 1 ðhÞ ¼ a1 S3 ðhÞ, where S3 ðzÞ ¼ 0 ½D1 ðnÞT 3 ðnÞ þ D2 ðnÞD3 ðnÞ dn. The solution is then conveniently expressed in terms of
Wðx; yÞ a0 A0 þ a1 A1 þ a2 r22 A0 þ a3 r22 A1 þ a4 r42 A0 þ a5 r42 A1
ð5:12Þ
b yÞ p þ ðb ca4 Þr4 A0 Aðx; 4 2 b is harmonic) as (so that r62 W ¼ 0 and A
b ;x y; u1 ¼ ðc zÞW;x T 3 ðzÞr22 W;x T 5 ðzÞr42 W;x þ A 2 4 b ;y þ x; u2 ¼ ðc zÞW;y T 3 ðzÞr W;y T 5 ðzÞr W;y þ A 2
2
u3 ¼ Wðx; yÞ þ ½T 2 ðzÞ þ cT 1 ðzÞr22 W þ ½T 4 ðzÞ þ ðd2 =a0 ÞD4 ðzÞr42 W; where, again, Wðx; yÞ ¼ u3 ðx; y; 0Þ. At z ¼ 0; h, the shear tractions vanish (t31 ¼ t 32 ¼ 0), while the normal tractions are
t33 ðzÞ ¼ Q 4 ðzÞr42 A0 þ Q 5 ðzÞr42 A1
ð5:13Þ 4 2 A0
4 2 A1
so involving the constants Q 4 ð0Þ ¼ d4 and Q 5 ð0Þ ¼ d5 as well as the harmonic function r and the constant r (which combine within r42 W). As in [9], it is possible to set t33 ð0Þ ¼ 0 with t33 ðhÞ a solution of Laplace’s equation, by choosing d4 ¼ 0, A1 0 and then choosing Q 4 ðhÞr42 W ¼ t33 ðx; y; hÞ. Solutions for K ¼ 4 clearly may also be matched to other transverse loadings which are harmonic functions of x and y, while higher values of K will include loadings by biharmonic functions, etc., as discussed in detail in [9]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
P.V. Kaprielian, T.G. Rogers, A.J.M. Spencer, Phil. Trans. R Soc. London A 324 (1988) 565. M.A. Mian, A.J.M. Spencer, J. Mech. Phys. Solids 46 (1998) 2283. A.H. England, A.J.M. Spencer, Math. Mech. Solids 10 (2005) 503. J.D. Achenbach, Wave Motion 28 (1998) 98. D.F. Parker, A.P. Kiselev, Q. J. Mech. Appl. Math. 62 (2009) 19. A.P. Kiselev, Proc. Roy. Soc. London A 460 (2004) 3059. A.P. Kiselev, E. Ducasse, M. Deschamps, A. Darinskii, C.R. Mecanique, 35 (2007) 419. A.J.M. Spencer, IMA J. Appl. Math. 72 (2007) 617. A.H. England, J. Elasticity 82 (2006) 129. W.A.K. Deutsch, A. Cheng, J.D. Achenbach, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 40 (1999) 333.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Waves and statics for functionally graded materials and laminates q D.F. Parker * School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, UK
a r t i c l e
i n f o
Article history: Available online 12 May 2009 Communicated by K.R. Rajagopal Keywords: Transverse isotropy Surface wave Functionally graded plates
a b s t r a c t It is shown here that, for any laminated stacking of elastic materials which are transversely isotropic with respect to an axis Oz, having elastic moduli which may depend either continuously or discontinuously on the coordinate z, surface-guided disturbances at any frequency are governed by the reduced membrane equation. An analogous treatment of the statics of plates having the same structure, yields the static theory of functionally graded plates due originally to Spencer et al. Thus, for functionally graded transversely isotropic plates with traction-free surfaces, displacements have a structure closely analogous to dynamic disturbances and are represented in terms of solutions to the biharmonic equation. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The static theory of linearly elastic functionally graded and laminated plates was initiated by Kaprielian et al. [1] and developed by Tony Spencer and co-workers (see [2,3] and references therein). In this theory, the material is, at each point, taken to be isotropic, but with Lamé constants and density depending upon the through-thickness coordinate. More recently, it has been shown [4] that time-harmonic surface waves and plate waves in a transversely isotropic medium may be related to a solution of the reduced membrane equation (the Helmholtz equation in two dimensions). Following the recent [5] recognition that a rotational invariance of material properties is the key to this result, it is now clear that a similar reduction to a scalar equation is possible for waves in all plates and half-spaces which locally are transversely isotropic. The density and elastic moduli may vary continuously or discontinuously as functions of the cartesian coordinate z, so describing functionally graded or laminated media. Besides recording this generalization of Achenbach’s result and of those of Kiselev [6,7] and coworkers, this paper links the dynamic theory to the static theory which owes so much to Tony Spencer. In particular, for traction-free plates formed of any stacking of transversely isotropic, functionally graded materials, displacements are described through a solution to the biharmonic equation and are closely analogous to other solutions found by Spencer [8] and based upon plane and anti-plane strain. The paper also reveals a hierarchy of solutions, as shown by England [9], in which transverse loads are solutions to Laplace’s equation, the biharmonic equation, etc. 2. Uni-directional waves in functionally graded materials Using linear elasticity theory, the components tij of Cauchy stress are related to the displacement components ui ðx; tÞ through tij ¼ cijlm ul;m , where x1 ; x2 and x3 z are Cartesian coordinates, cijlm are the elastic moduli and ;j denotes partial
I In memory of Tony Spencer – for more than a quarter of a century an exemplary Head of Department and an inspirational leader in research. * Corresponding author. Tel.: +44 131 447 6364; fax: +44 131 650 6553. E-mail address: [email protected]
0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.04.001
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
differentiation with respect to xj . The moduli cijlm and density q are allowed to be continuous or discontinuous functions of the coordinate z, either in the half-space z x3 > 0 or within a plate 0 < z < h. In a uniform half-space z P 0, the standard € i in z > 0, the traction-free boundary condition surface wave (a Rayleigh wave) is a solution of the Euler equation tij;j ¼ qu ti3 ¼ 0 at z ¼ 0 and the decay condition u ! 0 as z ! 0, with displacements depending on only the depth z and a travelling wave coordinate x1 ct. Here, a dot denotes partial differentiation with respect to time t and the constant c is the propagation speed. The displacement field and speed c are found by seeking uj ¼ Re U j ðz; kÞeikðx1 ctÞ , so yielding a system of constant coefficient differential equations. c is then determined (uniquely) so as to ensure compatibility between the traction-free and decay conditions. It has been shown [4,5] how these solutions may be generalized to give solutions in which ^ 1 ; x2 ÞU 3 ðz; kÞeikct , where wðx ^ 1 ; x2 Þ is any solution to the reduced membrane equation. Here, following steps used u3 ¼ Re wðx in [5] yields a similar result for transversely isotropic media with Oz as symmetry axis and with density and elastic coefficients being any piecewise continuous functions of z. Let z ¼ zp ; ðp ¼ 1; 2; . . . PÞ be the only locations at which either the density and/or the elastic coefficients are discontinuous. The governing system is then
€i tij;j ¼ qu
in z > 0; z – zp
ðp ¼ 1; 2; . . . ; PÞ;
ð2:1Þ
with traction-free condition
ti3 ci3lm ul;m ¼ 0
at z ¼ 0;
ð2:2Þ
continuity conditions
½½t i3 ¼ 0 ;
½½ui ¼ 0
at z ¼ zp
ð2:3Þ
and the decay condition u ! 0 as z ! 1. Here, ½½ denotes the jump in a quantity at z ¼ zp . For transversely isotropic materials, the elastic moduli are such that the stress components have the form
t11 ¼ C 11 u1;1 þ C 12 u2;2 þ C 13 u3;3 ;
t 23 ¼ t 32 ¼ C 44 ðu2;3 þ u3;2 Þ;
t22 ¼ C 12 u1;1 þ C 11 u2;2 þ C 13 u3;3 ;
t 13 ¼ t 31 ¼ C 44 ðu3;1 þ u1;3 Þ;
t33 ¼ C 13 ðu1;1 þ u2;2 Þ þ C 33 u3;3 ; 1 with C 66 ¼ ðC 11 C 12 Þ: 2
t 12 ¼ t 21 ¼ C 66 ðu1;2 þ u2;1 Þ; ð2:4Þ
Then, seeking travelling waves of the form u ¼ Re Uðz; kÞeih , t ¼ Re Tðz; kÞeih , where h kx1 xt yields (with U ¼ Uðz; kÞe1 þ Vðz; kÞe2 þ Wðz; kÞe3 ) from (2.1) the ordinary differential equations in z > 0 (z – zp ) 2
½C 44 ðU 0 þ ikWÞ0 þ ikC 13 W 0 þ ðx2 q k C 11 ÞU ¼ 0; 0 0
2
2
½C 44 V þ ðx q k C 66 ÞV ¼ 0
ð2:5Þ ð2:6Þ
and 2
½C 33 W 0 þ ikC 13 U0 þ ikC 44 U 0 þ ðx2 q k C 44 ÞW ¼ 0:
ð2:7Þ
Here, primes denote ordinary differentiation with respect to z. At each location z ¼ zp , the continuity conditions (2.3) yield
½½C 44 ðU 0 þ ikWÞ ¼ 0 ¼ ½½C 44 V 0 ¼ ½½C 33 W 0 þ ikC 13 U; ½½U ¼ ½½V ¼ ½½W ¼ 0;
ð2:8Þ
while, at the traction-free surface z ¼ 0, the conditions are
C 44 ðU 0 þ ikWÞ ¼ 0 ¼ C 44 V 0 ¼ C 33 W 0 þ ikC 13 U:
ð2:9Þ
Also, decay is ensured by choosing U; V; W ! 0 as z ! 1. For waves in a plate 0 < z < h, the decay condition is replaced by Eq. (2.9) at z ¼ h. Either for a half-space z > 0 or for a plate, transverse (shear-horizontal) displacements U ¼ Vðz; kÞe2 are seen to uncouple from the in-plane (sagittally-polarized) displacements U ¼ Uðz; kÞe1 þ Wðz; kÞe3 . In a uniform half-space, the only solutions are the Rayleigh wave, with V 0, with U and W specific linear combinations of two decaying exponentials and with c x=k equal to the Rayleigh wave speed cR . In a uniform half-space z > h with a dissimilar adjoining layer 0 < z < h, Love waves have transverse displacement V decaying in z > h, but oscillatory in the layer. The explicit solutions
(
VðzÞ ¼
^ cos bz ^ ebðzhÞ cos bh
0 6 z 6 h; h 6 z;
U ¼ W ¼ 0;
ð2:10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 66 k2 Þ=C ^ 44 and b ¼ ðC 66 k2 qx2 Þ=C 44 (with hats denoting density and elastic moduli in the layer) ^ x2 C ðq ^ 44 b ^ 66 =q ^ 44 =q ^ tan bh ^ ¼ C 44 b. Since, C 44 b2 ¼ ðC 66 qC ^2 , ^ Þk2 þ ðqC ^ Þb show that x is related to k through the dispersion relation C ^ this shows that the waves are dispersive (b=k depends upon k so that x=k is not constant). Also, these waves exist in many
^¼ where b
modes, with Vðz; kÞ having 0; 1; 2; . . . zeros within the layer. Moreover, in these same materials, sagitally-polarized waves generalizing Rayleigh waves also exist – with many modes corresponding to a chosen value of k, each being dispersive.
D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
1317
More generally, for functionally-graded materials or for laminates with more than one interface zp , waves of both the sagittally-polarized and transverse types may exist. They will be dispersive, with distinct modes corresponding to the various branches of the dispersion relation. 3. Generalized periodic vibrations Just as Achenbach [4], Deutsch et al. [10] and Kiselev et al. [7] have found disturbances related to the reduced membrane equation, we show that all layered and functionally-graded materials summarized in Section 2 permit wavelike solutions with depth-dependence related to solutions of the system (2.5)–(2.9). Because all material properties are invariant under rotations about the Oz direction, sagittally-polarized waves exist of the form
n o ^ þ Wðz; kÞ e3 eiðkxxtÞ u ¼ Re ½Uðz; kÞ k
ð3:1Þ
^ k=k and where k ¼ k1 e1 þ k2 e2 is the surface wave vector having magnitude k jkj, with wave normal the unit vector k with sagittally-polarized displacements Uðz; kÞ and Wðz; kÞ, where x ¼ xðkÞ is the dispersion relation arising from compati^ with displacement having the bility of the system (2.5), (2.7)–(2.9). Similarly, SH waves may propagate in any direction k, form
n o ^ eiðkxxtÞ u ¼ Re Vðz; kÞ e3 k
ð3:2Þ
and with dispersion relation arising from compatibility of (2.6), (2.8) and (2.9). 3.1. General sagittally-polarized waves More generally, disturbances having the same depth dependence as the waves (3.1) are found by seeking displacements of the form
^ yÞWðz; kÞeixt ; x x1 ; y x2 u ¼ Re ½fe1 /ðx; yÞ þ e2 wðx; yÞgUðz; kÞ þ e3 wðx;
ð3:3Þ
with x ¼ xðkÞ satisfying the dispersion relation for sagittally-polarized waves. The corresponding traction components on ^ ;x WÞ, t23 ¼ C 44 ðwU 0 þ w ^ ;y WÞ and t33 ¼ C 13 ð/;x þ w;y ÞU þ C 33 wW. ^ any plane z ¼ constant are t 13 ¼ C 44 ð/U 0 þ w ^ ;y and ^ ;x ; ikw ¼ w Inserting conditions (2.9) into the traction-free boundary conditions (2.2) yields ik/ ¼ w ^ so that wðx; ^ yÞ must satisfy the reduced membrane equation /;x þ w;y ¼ ikw;
^ ;yy ¼ k2 w: ^ w ^ ;xx þ w ^ r22 w
ð3:4Þ
^ yÞ satisfies the two-dimensional Helmholtz Eq. (3.4), the corresponding disThen, it is readily checked that, whenever wðx; placement fields
^ yÞgeixt ^ þ Wðz; kÞe3 wðx; u ¼ RefðikÞ1 Uðz; kÞgrad w
ð3:5Þ
satisfy also the continuity conditions (2.3) at each interface z ¼ zp as well as the partial differential Eq. (2.1). This shows that, ^ yÞ to the reduced membrane Eq. (3.4) and for any piecewise continuous z- dependence of corresponding to any solution wðx; the density qðzÞ and elastic moduli C mn ðzÞ, there exist time-harmonic elastic displacement fields which decay with depth and satisfy the traction-free boundary condition (2.2) at z ¼ 0. 3.2. General SH waves Analogously to the treatment above, it is readily shown that displacements
yÞ eixt u ¼ Re Vðz; kÞe3 grad wðx;
ð3:6Þ
satisfy the field Eq. (2.1), the traction-free boundary condition (2.2) and the continuity conditions (2.3) at each material yÞ satisfies the reduced membrane equation interface z ¼ zp , provided that wðx;
;yy ¼ k2 w; w ;xx þ w r22 w
ð3:7Þ
but with x here related to k through the dispersion relation arising from the solution V to the boundary value problem (2.6), (2.8) and (2.9) for shear-horizontal waves (U ¼ W ¼ 0 in the system (2.5), (2.7)–(2.9) in the functionally graded material. 4. Connections to the static theory of plates The ability to relate both sagittally polarized and shear horizontal surface-guided waves in any functionally graded material to solutions of the reduced membrane equation suggests close links to the static theory of laminated and functionallygraded plates initiated in [1]. For this analogy, first consider displacements independent of x2 , for which the stress components (2.4) become
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
t 11 ¼ C 11 u1;1 þ C 13 u3;3 ; t12 ¼ t 21 ¼ C 66 u2;1 ; t13 ¼ t31 ¼ C 44 ðu1;3 þ u3;1 Þ;
ð4:1Þ
t 22 ¼ C 12 u1;1 þ C 13 u3;3 ; t23 ¼ t 32 ¼ C 44 u2;3 ; t33 ¼ C 13 u1;1 þ C 33 u3;3 :
For displacements without body force and with traction vanishing at both z ¼ 0 and z ¼ h, these stresses must satisfy
ti1;1 þ ti3;3 ¼ 0 in 0 < z < h ;
z – zp for i ¼ 1; . . . ; 3
ð4:2Þ
with
½½t i3 ¼ 0 at z ¼ zp ;
ti3 ¼ 0 at z ¼ 0; h for i ¼ 1; . . . ; 3;
ð4:3Þ
from which it is immediately seen that plane strain deformations u1 ; u3 uncouple from the anti-plane strain deformations u2 . The most basic solutions for plane strain have t13 0 and t33 0 so that t11;1 ¼ 0. Thus, u3;1 ¼ u1;3 with C 13 t11 ¼ ðC 213 C 11 C 33 Þu3;3 , so giving u3 in the form
u3 ¼ mðxÞ þ nðzÞ
u3;3 ¼ n0 ðzÞ:
with
0
Since u1;3 ¼ u3;1 ¼ m0 ðxÞ, then u1 has the form u1 ¼ lðxÞ zm0 ðxÞ, so giving 0 ¼ t33 ¼ C 13 ðzÞ½l ðxÞ zm00 ðxÞ þ C 33 ðzÞn0 ðzÞ. 00 2 Consistency then requires that l ðxÞ ¼ 0 ¼ m000 ðxÞ, so giving l ¼ c1 þ bx and m ¼ c3 þ ax þ dx , thus yielding u1 ¼ c1 þ bx az 2dxz, where vanishing of t33 requires that
C 33 ðzÞn0 ðzÞ ¼ C 13 ðzÞ ð2dz bÞ
for z – zp :
Solving this allows nðzÞ to be written as nðzÞ ¼ bT 1 ðzÞ þ 2dT 2 ðzÞ, where
T 1 ðzÞ
Z
z
D2 ðnÞ dn; T 2 ðzÞ
Z
0
z
nD2 ðnÞ dn; D2 ðzÞ C 13 ðzÞ=C 33 ðzÞ:
ð4:4Þ
0
Here, T 1 ðzÞ and T 2 ðzÞ are direct generalizations to transverse anisotropy of the material parameters defined in Eq. (2.24) of [9]. Then, using u3 ¼ c3 þ ax þ bT 1 ðzÞ þ d½x2 þ 2T 2 ðzÞ, gives the displacements in the form
u1
¼
u3
c1 c3
z 2xz x þa þd 2 ; þb x x þ 2T 2 ðzÞ T 1 ðzÞ
ð4:5Þ
in which the four contributions on the right-hand side describe, respectively, a translation, a rotation, a stretching and a uniform flexure of the strip. Proceeding analogously to the dynamic case, involves first generalizing (4.5) by seeking a sequence of functions fU k ðzÞ; V k ðzÞ; W k ðzÞg such that displacements
u1 ¼
K X k¼0
u3 ¼
Kþ1 X k¼0
1 U k ðzÞxKk ; ðK kÞ!
u2 ¼
K X k¼0
1 V k ðzÞxKk ; ðK kÞ!
1 W k ðzÞxKkþ1 ðK k þ 1Þ!
ð4:6Þ
satisfy the system (4.1), (4.2) and (4.3). First deriving from (2.4) the expressions for t i3 , then imposing the traction-free conditions give (at z ¼ 0; h)
U 0k þ W k ¼ 0 for k ¼ 0; . . . ; K C 33 W 0k
V 0k ¼ 0 for k ¼ 0; . . . ; K ;
þ C 13 U k2 ¼ 0 for k ¼ 2; . . . ; K þ 1 ; W 00 ¼ 0 ¼ W 01 ;
ð4:7Þ
and the continuity equations (at z ¼ zp ) give
½½C 44 ðU 0k þ W k Þ ¼ 0 for k ¼ 0; . . . ; K ; ½½C 44 V 0k ¼ 0 for k ¼ 0; . . . ; K; ½½C 33 W 00 ¼ ½½C 33 W 01 ¼ 0; ½½C 33 W 0k þ C 13 U k2 ¼ 0 for k ¼ 2; . . . ; K þ 1;
ð4:8Þ
while the equilibrium equations give (for z – zp )
C 13 W 0k þ C 11 U k2 þ ½C 44 U 0k þ C 44 W k 0 ¼ 0;
k ¼ 0; . . . ; K;
ðC 44 V 0k Þ0 þ C 66 V k2 ¼ 0;
k ¼ 0; . . . ; K;
C 44 U 0k þ C 44 W k þ ½C 33 W 0kþ2 þ C 13 U k 0 ¼ 0;
k ¼ 0; . . . ; K 1;
ð4:9Þ
together with C 33 W 00 ¼ const: ¼ 0 and C 33 W 01 ¼ const: ¼ 0, showing that W 0 a0 and W 1 ¼ a1 each are constants. Since, likewise, V 0 ¼ c0 and V 1 ¼ c1 are constants, the equation ðC 44 V 02 Þ0 ¼ C 66 c0 together with its boundary and continuity conditions is inconsistent. Thus K ¼ 1, so that the anti-plane displacements simplify. For the functions U k ðzÞ and W k ðzÞ, solving the equation ½C 44 ðU 00 ðzÞ þ a0 Þ0 ¼ const: ¼ 0 with boundary and continuity conditions gives U 0 ¼ b0 a0 z. Then, rearranging C 13 U 0 þ C 33 W 02 ðzÞ ¼ 0 as a formula for W 02 allows solution as
W 2 ðzÞ ¼ a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞ :
ð4:10Þ
D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
1319
Similarly solving for U 1 ðzÞ and W 3 ðzÞ gives
U 1 ðzÞ ¼ b1 a1 z ;
W 3 ðzÞ ¼ a3 þ a1 T 2 ðzÞ þ b1 T 1 ðzÞ:
Continuing this process gives next ½C 44 ðU 02 þ W 2 Þ0 ¼ C 13 W 02 C 11 U 0 ¼ D1 U 0 for z – zp , where D1 ðzÞ ¼ ðC 213 C 11 C 33 Þ=C 33 . Together with continuity of C 44 ðU 02 þ W 2 Þ at z ¼ zp and vanishing at z ¼ 0; h this gives
C 44 ðzÞ½U 02 ðzÞ þ W 2 ðzÞ ¼ b0 S1 ðzÞ a0 S2 ðzÞ; Rz Rz where S1 ðzÞ ¼ 0 D1 ðnÞ dn and S2 ðzÞ ¼ 0 nD1 ðnÞ dn, which together lead to a compatibility condition which fixes the ratio b0 =a0 c S2 ðhÞ=S1 ðhÞ. 4.1. Iterative determination of U k ðzÞ and W k ðzÞ The continuous functions U k ðzÞ and W k ðzÞ are best determined for all K by rearrangement of (4.9) and its associated continuity and boundary conditions using the continuous functions
Rk ðzÞ C 44 ½U 0k ðzÞ þ W k ðzÞ; in terms of which t 13 ¼
Q 0k ¼ Rk ;
PK
k¼0 x
k
Q k ðzÞ C 13 U k ðzÞ þ C 33 W 0kþ2 ðzÞ;
RKk ðzÞ=k! , t 33 ¼
PKþ1
k¼0 x
k
ð4:11Þ
Q Kkþ1 ðzÞ=k! . This gives the system
R0k ¼ C 13 W 0k C 11 U k2 ¼ D1 U k2 D2 Q k2 ;
which may be integrated (using the traction-free condition t31 ð0Þ ¼ 0) as
Z z Z z W k ðzÞ ¼ ak D2 ðnÞU k2 ðnÞ dn þ Q k2 ðnÞ=C 33 ðnÞ dn; 0 0 Z z Z z U k ðzÞ ¼ bk W k ðnÞ dn þ Rk ðnÞ=C44 ðnÞ dn; 0 Z z 0 Z z Rk ðzÞ ¼ D1 ðnÞUk2 ðnÞ dn D2 ðnÞQ k2 ðnÞ dn; 0 0 Z z Rk ðnÞ dn; with d0 ¼ d1 ¼ 0; Q k ðzÞ ¼ dk
ð4:12Þ
0
(so that R0 ¼ R1 ¼ Q 0 ¼ Q 1 0). Since W k ðzÞ is required for k 6 K þ 1, the additional condition Rk ðhÞ ¼ 0 is accommodated by suitable choice of b0 ; . . . ; bK2 , so that both conditions t13 ð0Þ ¼ 0 ¼ t 13 ðhÞ may be satisfied. However, U k ðzÞ is required for k 6 K, but for k P 2 the conditions Q k ð0Þ ¼ 0 ¼ Q k ðhÞ cannot both be satisfied. Thus, deformations (4.6) cannot arise for K P 3 except for laterally loaded plates, such that t33 ðhÞ – t33 ð0Þ. If there is normal loading, then the conditions Q k ð0Þ ¼ 0 and Q k ðhÞ ¼ 0 may be relaxed. 5. Construction of plate bending theories The system of Eq. (4.9) is closely analogous to Eqs. (2.5), (2.6) and (2.7) (set x ¼ 0 and replace each factor ik by a unit reduction in the power of x in (4.6) – each operation corresponds to differentiation with respect to x). This motivates a search for static solutions to (4.2) and (4.3) in the form u1 ¼ v ;x ðx; y; zÞ v;y ðx; yÞ, u2 ¼ v ;y ðx; y; zÞ þ v;x ðx; yÞ with
v ðx; y; zÞ ¼
K X
Ak ðx; yÞ U k ðzÞ ;
u3 ¼
k¼0
K þ1 X
Ak ðx; yÞ W k ðzÞ
ð5:1Þ
k¼0
analogous to that used in [5] to obtain general vibrations (3.5) and (3.6) from standard travelling waves (setting v c0 B0 þ c2 B1 , so v;x ¼ B0;x V 0 þ B1;x V 1 ). This leads to expressions for stress components which include
t 13 ¼ t 31 ¼
K X
Ak;x C 44 ½U 0k ðzÞ þ W k ðzÞ þ AKþ1;x C 44 W Kþ1 ðzÞ;
k¼0
t 23 ¼ t 32 ¼
K X
Ak;y C 44 ½U 0k ðzÞ þ W k ðzÞ þ AKþ1;y C 44 W Kþ1 ðzÞ;
ð5:2Þ
k¼0
t 33 ¼
K X
C 13 ðAk;xx þ Ak;yy ÞU k ðzÞ þ
k¼0
Kþ1 X
C 33 Ak W 0k ðzÞ;
k¼2
so that use of relations (4.8) shows that the continuity conditions ½½ti3 ¼ 0 at z ¼ zp are satisfied whenever AKþ1 ¼ constant, r22 AK ¼ 0 and Akþ2 ¼ r22 Ak , k ¼ 0; . . . ; K 1. These conditions ensure that the equilibrium Eq. (4.2) also are satisfied, so long as the function vðx; yÞ satisfies r22 v ¼ constant. There thus exist two sequences of functions related to the constant AKþ1 and the harmonic function AK , respectively. Within each sequence
r2s 2 AKþ12s ¼ AKþ1 ¼ const:;
r22sþ2 AK2s ¼ r22 AK ¼ 0:
ð5:3Þ
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
For K even, rKþ2 A0 ¼ 0 and rK2 A1 ¼ AKþ1 ¼ constant; alternatively, for K odd, rKþ1 A0 ¼ AKþ1 ¼ constant and rKþ1 A1 ¼ 0. In 2 2 2 2s A with A ¼ r A . Thus, for example, for K even, the transverse displacement is each case, A2s ¼ r2s 0 2sþ1 1 2 2
u3 ¼ A0 ðx; yÞW 0 ðzÞ þ r22 A0 W 2 ðzÞ þ þ rK2 A0 W K ðzÞ þ A1 ðx; yÞW 1 ðzÞ þ r22 A1 W 3 ðzÞ þ þ rK2 A1 W Kþ1 ðzÞ;
ð5:4Þ
while the in-plane displacements may be written as u1 ¼ @ v =@x @ v=@y and u2 ¼ @ v =@y þ @ v=@x, where
v ðx; y; zÞ ¼ A0 ðx; yÞU 0 ðzÞ þ r22 A0 U2 ðzÞ þ þ rK2 A0 UK ðzÞ þ A1 ðx; yÞU1 ðzÞ þ r22 A1 U 3 ðzÞ þ þ r2K2 A1 U K1 ðzÞ Kþ2 A0 2
K 2 A1
ð5:5Þ
Kþ2 A1 2
with r ¼ 0; r ¼ AKþ1 ¼ constant (so that r ¼ 0). Here, the functions U k ðzÞ and W k ðzÞ are identical to those arising in the two-dimensional displacements of Section 4. P P P However, the traction components at z ¼ 0; h reduce to t 13 ¼ Kk¼0 Ak;x Rk ðzÞ, t 23 ¼ Kk¼0 Ak;y Rk ðzÞ and t33 ¼ Kk¼1 Akþ1 Q k1 ðzÞ, so that, as discussed after Eq. (4.12), it is possible to set t 13 ¼ 0 ¼ t23 at both z ¼ 0 and z ¼ h but displacements (5.1) exist only for
t33 ðx; y; hÞ t 33 ðx; y; 0Þ ¼
Kþ1 X
Ak ðx; yÞ½Q k ðhÞ Q k ð0Þ ¼
k¼4
Kþ1 X
Ak ðx; yÞ
Z
k¼4
h
Rk ðzÞ dz:
ð5:6Þ
0
The form of Eqs. (5.4) and (5.5) is the same for all functionally graded plates, including homogeneous isotropic plates; the layered structure enters only through the definitions of fU k ðzÞ; W k ðzÞg. Specific choices of K give theories related to those developed in [8,9]. 5.1. K ¼ 1 The functions W 0 ¼ a0 and W 1 ¼ a1 are constants, while (since U 2 ðzÞ is not required) b0 is arbitrary in U 0 ¼ b0 a0 z, U 1 ¼ b1 a1 z and W 2 ¼ a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞ. Also, since r22 A0 ¼ A2 ¼ const. and r22 A1 ¼ 0, this gives (after writing v;y ¼ p;x y; v;x ¼ p;y þ x, so that r22 v ¼ 2 ¼ u2;x u1;y )
u1 ¼ ðb0 a0 zÞA0;x þ ðb1 a1 zÞA1;x þ p;x y ; u2 ¼ ðb0 a0 zÞA0;y þ ðb1 a1 zÞA1;y þ p;y þ x; u3 ¼ a0 A0 ðx; yÞ þ a1 A1 ðx; yÞ þ ½a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞr22 A0 : ^ b0 =a0 and defining two new functions After writing c
Wðx; yÞ a0 A0 þ a1 A1 þ a2 r22 A0 ; 2b 2A
2 2
such that r W ¼ constant and r
b yÞ ðb c ^a1 ÞA1 þ p Aðx; 1
¼ 0, the displacements may then be expressed as
b y; u2 ¼ ½ðc b þ x; ^ zÞW þ A ^ zÞW þ A u1 ¼ ½ðc ;x ;y ^T 1 ðzÞr22 W; u3 ¼ Wðx; yÞ þ ½T 2 ðzÞ þ c
ð5:7Þ
2b 2A
2 2
^ and correspond to an in-plane stretching and a uniform where r W ¼ constant and r ¼ 0. Here, the arbitrary constants c b yÞ describes a divergence-free displacement – all with the surfaces z ¼ 0; h traction-free. rotation, respectively, while Aðx; 5.2. K ¼ 2 Plates without lateral loading Using R0 ¼ R1 ¼ Q 0 ¼ Q 1 ¼ 0; W 0 ¼ a0 , W 1 ¼ a1 , U 0 ¼ b0 a0 z, U 1 ¼ b1 a1 z, W 2 ¼ a2 þ a0 T 2 ðzÞ þ b0 T 1 ðzÞ and W 3 ¼ a3 þ a1 T 2 ðzÞ þ b1 T 1 ðzÞ gives, from (4.12), R2 ðzÞ ¼ b0 S1 ðzÞ a0 S2 ðzÞ. Then, as before, setting b0 ¼ ca0 to make R2 ðhÞ ¼ 0 gives U 2 ðzÞ ¼ b2 a2 z a0 T 3 ðzÞ, where
T 3 ðzÞ
Z
z
fT 2 ðnÞ þ cT 1 ðnÞ þ ½S2 ðnÞ cS1 ðnÞ=C 44 ðnÞgdn:
ð5:8Þ
0
Then, taking
v and u3
from (5.4) and (5.5), writing
a0 A0 þ a1 A1 þ a2 r22 A0 þ a3 r22 A1 Wðx; yÞ; b ðb1 ca1 ÞA1 þ ðb2 ca2 Þr22 A0 þ p A where
ð5:9Þ
v;y y p;x and v;x x þ p;y , reduces displacements to the form b ;x y; u1 ¼ ðc zÞW;x T 3 ðzÞr22 W;x þ A b ;y þ x; u2 ¼ ðc zÞW;y T 3 ðzÞr2 W;y þ A 2
u3 ¼ W þ ½T 2 ðzÞ þ cT 1 ðzÞr22 W þ T 1 ðzÞB;
ð5:10Þ
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D.F. Parker / International Journal of Engineering Science 47 (2009) 1315–1321
b ¼ B ¼ constant, r4 W ¼ 0. The choice (5.9) identifies the biharmonic function Wðx; yÞ as the where r22 v ¼ 2 ¼ constant, r22 A 2 lateral displacement of the surface z ¼ 0. Here, the functions T 1 ðzÞ, T 2 ðzÞ and T 3 ðzÞ arise from the z–dependence of the elastic moduli. The solutions (5.10) are closely similar to those which Spencer [8] based upon plane and antiplane strain. The special case r22 W ¼ constant gives the in-plane stretching solutions (5.7) (i.e. K ¼ 1). 5.3. K ¼ 4 (including K ¼ 3 as a special case) With W 0 ; W 1 ; W 2 ; W 3 ; U 0 ; U 1 and U 2 as in the case K ¼ 2, setting b1 ¼ ca1 , b2 ¼ ca2 and d2 T 1 ðhÞ ¼ a0 S3 ðhÞ, defining further Rz Rf Rz D4 ðzÞ 0 dn=C33 ðnÞ, functions in terms of elastic moduli through D3 ðzÞ 0 0 ðn cÞD1 ðnÞ dndf, Rz Rz T 4 ðzÞ 0 ½T 3 ðnÞD2 ðnÞ þ D3 ðnÞ=C 33 ðnÞ dn and T 5 ðzÞ 0 fT 4 ðnÞ þ S3 ðnÞ=C 44 ðnÞ þ ðd2 =a0 Þ½D4 ðnÞ T 1 ðnÞ=C 44 ðnÞg dn yields
U 3 ðzÞ ¼ b3 a3 z a1 T 3 ðzÞ;
U 4 ðzÞ ¼ b4 a4 z a2 T 3 ðzÞ a0 T 5 ðzÞ;
W 4 ðzÞ ¼ a4 þ a2 T 2 ðzÞ þ b2 T 1 ðzÞ þ a0 T 4 ðzÞ þ d2 D4 ðzÞ; W 5 ðzÞ ¼ a5 þ a3 T 2 ðzÞ þ b3 T 1 ðzÞ þ a1 T 4 ðzÞ þ d3 D4 ðzÞ; Q 2 ðzÞ ¼ d2 þ a0 D3 ðzÞ;
Q 3 ðzÞ ¼ d3 þ a1 D3 ðzÞ:
ð5:11Þ
Further, R4 ðzÞ ¼ a2 ½cS1 ðzÞ S2 ðzÞ þ d2 T 1 ðzÞ a0 S3 ðzÞ and R5 ðzÞ ¼ a3 ½cS1 ðzÞ S2 ðzÞ þ d3 T 1 ðzÞ a1 S3 ðzÞ satisfy R4 ¼ R5 ¼ 0 at Rz z ¼ 0; h when b3 ¼ ca3 and d3 T 1 ðhÞ ¼ a1 S3 ðhÞ, where S3 ðzÞ ¼ 0 ½D1 ðnÞT 3 ðnÞ þ D2 ðnÞD3 ðnÞ dn. The solution is then conveniently expressed in terms of
Wðx; yÞ a0 A0 þ a1 A1 þ a2 r22 A0 þ a3 r22 A1 þ a4 r42 A0 þ a5 r42 A1
ð5:12Þ
b yÞ p þ ðb ca4 Þr4 A0 Aðx; 4 2 b is harmonic) as (so that r62 W ¼ 0 and A
b ;x y; u1 ¼ ðc zÞW;x T 3 ðzÞr22 W;x T 5 ðzÞr42 W;x þ A 2 4 b ;y þ x; u2 ¼ ðc zÞW;y T 3 ðzÞr W;y T 5 ðzÞr W;y þ A 2
2
u3 ¼ Wðx; yÞ þ ½T 2 ðzÞ þ cT 1 ðzÞr22 W þ ½T 4 ðzÞ þ ðd2 =a0 ÞD4 ðzÞr42 W; where, again, Wðx; yÞ ¼ u3 ðx; y; 0Þ. At z ¼ 0; h, the shear tractions vanish (t31 ¼ t 32 ¼ 0), while the normal tractions are
t33 ðzÞ ¼ Q 4 ðzÞr42 A0 þ Q 5 ðzÞr42 A1
ð5:13Þ 4 2 A0
4 2 A1
so involving the constants Q 4 ð0Þ ¼ d4 and Q 5 ð0Þ ¼ d5 as well as the harmonic function r and the constant r (which combine within r42 W). As in [9], it is possible to set t33 ð0Þ ¼ 0 with t33 ðhÞ a solution of Laplace’s equation, by choosing d4 ¼ 0, A1 0 and then choosing Q 4 ðhÞr42 W ¼ t33 ðx; y; hÞ. Solutions for K ¼ 4 clearly may also be matched to other transverse loadings which are harmonic functions of x and y, while higher values of K will include loadings by biharmonic functions, etc., as discussed in detail in [9]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
P.V. Kaprielian, T.G. Rogers, A.J.M. Spencer, Phil. Trans. R Soc. London A 324 (1988) 565. M.A. Mian, A.J.M. Spencer, J. Mech. Phys. Solids 46 (1998) 2283. A.H. England, A.J.M. Spencer, Math. Mech. Solids 10 (2005) 503. J.D. Achenbach, Wave Motion 28 (1998) 98. D.F. Parker, A.P. Kiselev, Q. J. Mech. Appl. Math. 62 (2009) 19. A.P. Kiselev, Proc. Roy. Soc. London A 460 (2004) 3059. A.P. Kiselev, E. Ducasse, M. Deschamps, A. Darinskii, C.R. Mecanique, 35 (2007) 419. A.J.M. Spencer, IMA J. Appl. Math. 72 (2007) 617. A.H. England, J. Elasticity 82 (2006) 129. W.A.K. Deutsch, A. Cheng, J.D. Achenbach, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 40 (1999) 333.