## Abstract

Steady waves propagating in a plate that consists of one or more layers of general anisotropic elastic material are studied. The surface of the plate can be a traction-free (F), rigid (R) or slippery surface (S). The interface between any two layers in the plate can be perfectly bonded (b) or in sliding contact (s). The thickness of the layers need not be the same. The purpose of this paper is to present dispersion equations for all possible combinations of the boundary and interface conditions. If the thickness *h* of one of the layers is very small, the dispersion equation allows us to expand the solution in an infinite series in the power of *h* from which an approximate solution can be obtained by keeping the terms up to *O*(*h*^{n}) for any *n*. The special case of a sandwich plate that consists of a centre layer and two identical outside layers is studied. In the literature, the dispersion equations for a sandwich plate were studied for special elastic materials. The results presented here are for elastic materials of general anisotropy.

## 1. Introduction

Steady waves propagating in a sandwich plate in which the centre layer is isotropic while the two identical outer layers are monoclinic materials with the symmetry plane parallel to the layers were studied by Niklasson *et al*. (2000*a*). They assumed that the thickness of the outer layers is very small and obtained an approximate dispersion equation. They also considered the case in which the centre layer is monoclinic and very thin while the two identical outer layers are isotropic (Niklasson *et al*. 2000*b*). Again, an approximate dispersion equation was obtained, which is valid when the thickness of the centre layer is very thin.

We study a more general case in which all layers are of general anisotropic materials. The thickness of the layers need not be small and identical. In fact, we consider steady waves in a layered plate that can have any number of layers. The surface of the layered plate can be a traction-free, rigid or slippery surface. The interface between any two layers can be perfectly bonded or in sliding contact. If the thickness of any layer is very thin, the dispersion equation can be expanded in power series in the thickness of the thin layer so that an approximate dispersion equation can be obtained that is valid to any order of the thickness of the thin layer.

The basic equations for steady waves in an anisotropic elastic media based on the Stroh formalism (Stroh 1962; Ingebrigtsen & Tonning 1969; Barnett & Lothe 1973, 1985; Chadwick & Smith 1977; Ting 1996) are outlined in §2. For steady waves in a plate, the Stroh eigenvalues *p* need not be all complex. There are several approaches in deriving the dispersion equations for a plate (Gilbert & Backus 1966; Ting & Chadwick 1988; Nayfeh 1995; Alshits & Kirchner 1995; Shuvalov 2000, 2004; Alshits *et al*. 2001, 2003). The propagator matrix ** E** that transfers the solution at one

*x*

_{2}to another

*x*

_{2}is presented in §3. The matrix

**can be expressed in an infinite power series in**

*E**h*. In §4, we derive the dispersion equations in a homogeneous elastic plate. Depending on whether the top or bottom surface of the plate is a traction-free (F), rigid (R) or slippery surface (S), there are six cases. Dispersion equations for all six cases are presented. The dispersion equations for a plate that consists of two layers are studied in §5. There are nine cases each depending on whether the interface between the two layers is perfectly bonded (b) or in sliding contact (s). The dispersion equations are given for all cases. In §6, we discuss the case when the plate consists of three or more layers. The special case when the plate is a sandwich plate is investigated in §7. A sandwich plate is a three-layer plate that has one centre layer and two identical outside layers. The materials in the layers considered in this paper are all elastic materials of general anisotropy.

## 2. Basic equations

In a fixed rectangular coordinate system *x*_{i} (*i*=1, 2, 3), the equation of motion is(2.1)where *σ*_{ij} is the stress; *u*_{i} is the displacement; *ρ* is the mass density; the dot denotes differentiation with time *t*; and a comma denotes differentiation with *x*_{i}. The stress–strain relation is(2.2)(2.3)where *C*_{ijks} is the elastic stiffness. *C*_{ijks} is positive definite and possesses the full symmetry shown in (2.3). The third equality in (2.3) is redundant because the first two imply the third (Ting 1996, p. 32).

For a two-dimensional steady state motion in the *x*_{1}-direction with a constant wave speed *υ*>0, a general solution for the displacement ** u** in (2.1) and (2.2) is(2.4)where

*k*>0 is the real wavenumber;

*t*is the time; and

*p*and

**satisfy the equation(2.5)In (2.5), the superscript T denotes the transpose,**

*a***is the identity matrix and(2.6)Introducing the vector(2.7)in which the second equality follows from (2.5), the stress computed from (2.2) and (2.4) can be written as(2.8)In (2.8), the vector(2.9)is the stress function. The two equations in (2.7) can be written as(2.10)where(2.11)and(2.12)The matrix**

*I*

*N*_{2}is symmetric and positive definite while −

*N*_{3}is symmetric and positive semi-definite (Ting 1996).

There are six eigenvalues *p* and six associated eigenvectors ** ξ**=(

**,**

*a***). Let**

*b**p*

_{m}(

*m*=1, 2, …, 6) be the eigenvalues and

*ξ*_{m}=(

*a*_{m},

*b*_{m}) (

*m*=1, 2, …, 6) be the associated eigenvectors. The general solution obtained by superposing the solutions (2.4) and (2.9) associated with

*p*=

*p*

_{1},

*p*

_{2}, …,

*p*

_{6}is(2.13)In the above equations,

**and**

*q*

*q*^{*}are arbitrary constant vectors to be determined and(2.14)It can be shown (Ting 1996) that, after a proper normalization of the eigenvector (

**,**

*a***),(2.15)The product of the two 6×6 matrices commutes. Hence,(2.16)Useful relations obtained from (2.16)**

*b*_{3,4}are(2.17)This suggests that (−i

^{−1}

^{*}) and (i

^{−1}

^{*}) are orthogonal tensors (Ting 1993; see also Ting 1996, p. 148). They are Hermitian if all the eigenvalues

*p*are complex.

Equation (2.17) remains valid without the normalization of the eigenvector (** a**,

**).**

*b*## 3. The propagator matrix *E*

*E*

Equation (2.13) can be written as(3.1)in which we write ** w**(

*x*

_{2}) instead of

**(**

*w**x*

_{1},

*x*

_{2}) for simplicity. Hence,(3.2)or, using (2.15),(3.3)(3.4)Equation (3.3) transfers the solution

**(**

*w**x*

_{2}) to

**(**

*w**x*

_{2}−

*h*) through

**(**

*E**h*). Hence,

**(**

*E**h*) is the

*propagator matrix*(Gilbert & Backus 1966; Ting & Chadwick 1988; Ting in press). If we write

**(**

*E**h*) as(3.5)we have(3.6)Equation (3.3) can be written more explicitly as(3.7)When

*h*is small, we can expand in an infinite series as(3.8)By keeping the terms up to

*h*

^{n}, one obtains an approximation of

*E*_{k}(

*h*), which is valid for

*O*(

*h*

^{n}). It should be noted that

**(0)=**

*E***so that(3.9)**

*I*An alternative expression of ** E**(

*h*) is (Ting & Chadwick 1988)(3.10)which can be proved from (3.4) using the diagonalization of the matrix

**. Equation (3.10) has one advantage over (3.8) and (3.4) for an approximation of**

*N***(**

*E**h*). It does not require the computation of the eigenvalues and eigenvectors of

**. Equation (3.10) shows that**

*N***(**

*E**h*) is a real function of (i

*kh*). Shuvalov (2000) has shown that the determinant |

*E*_{3}(

*h*)| is a real function of (i

*kh*).

From (3.4) and (2.15), one can show that(3.11)In particular, when we have(3.12)This means that(3.13)Elimination of *E*_{1}(−*h*) leads to(3.14)In the sequel, the l.h.s. appears in some of the dispersion equations. It can be replaced by the r.h.s.

## 4. Dispersion equations for a homogeneous plate

In this section, the dispersion equations for steady waves in a homogeneous anisotropic elastic plate are considered. Let the plate of thickness *h* occupy the region −*h*/2≤*x*_{2}≤*h*/2. Setting *x*_{2}=*h*/2 in (3.7), we have(4.1)The top and bottom surfaces of the plate can be a traction-free (F), rigid (R) or slippery surface (S). If the top surface of the plate is traction free and the bottom surface is slippery, we call it the F/S plate. The R/S plate is the one for which the top surface of the plate is rigid and the bottom surface is slippery. There are a total of nine cases but three of them are the mirror images of the other three.

*The F/F plate*. If the top and bottom surfaces of the plate are traction free, we have . Equation (4.1)_{2} has a non-trivial solution for ** u**(

*h*/2) if(4.2)This means, from (3.6),(4.3a)It is not difficult to show with the use of (2.16) that this is equivalent to(4.3b)(4.3c)(4.3d)Equations (4.3

*c*) and (4.3

*d*) recover the ones obtained by Shuvalov (2000). He also gave a review (Shuvalov 2006) on the impact of anisotropy on the dispersion spectra of acoustic waves in plates. One advantage of (4.3

*b*)–(4.3

*d*) over (4.3

*a*) is that the eigenvectors

**=(**

*ξ***,**

*a***) need not be normalized.**

*b**The F/R plate*. If the top surface of the plate is traction free and the bottom surface is rigid, . Equation (4.1)_{1} has a nontrivial solution for ** u**(

*h*/2) if(4.4)This means, from (3.6)(4.5a)It can be shown with the use of (2.16) that this is equivalent to(4.5b)(4.5c)(4.5d)Again, the eigenvectors

**=(**

*ξ***,**

*a***) need not be normalized for (4.5**

*b**b*)–(4.5

*d*).

*The F/S plate*. When the top surface of the plate is traction free and the bottom surface is slippery, we have ** ϕ**(

*h*/2)=

**0**and(4.6)where

*γ*is an unknown constant and(4.7)Equation (4.1)

_{2}gives(4.8)Substituting (4.8) into (4.1)

_{1}and (4.6)

_{2}leads to(4.9a)This means that(4.9b)Hence, the dispersion equation is(4.10)

*The R/F plate*. It can be shown that the dispersion equation for the R/F plate is identical to that for the F/R plate shown in (4.4) and (4.5*a*)–(4.5*d*).

*The R/R plate*. If the top and bottom surfaces of the plate are rigid, . Equation (4.1)_{1} has a non-trivial solution for ** ϕ**(

*h*/2) if(4.11)This means, from (3.6),(4.12a)It is not difficult to show with the use of (2.16) that this is equivalent to(4.12b)(4.12c)(4.12d)As before, the eigenvectors

**=(**

*ξ***,**

*a***) need not be normalized for (4.12**

*b**b*)–(4.12

*d*).

*The R/S plate*. When the top surface of the plate is rigid and the bottom surface is slippery, we have ** u**(

*h*/2)=

**0**and (4.6) applies. Equation (4.1)

_{2}gives(4.13)Substituting (4.13) into (4.1)

_{1}and (4.6)

_{2}leads to(4.14a)This means that(4.14b)Hence, the dispersion equation is(4.15)

*The S/F plate*. The dispersion equation for the S/F plate is identical to that for the F/S plate shown in (4.10).

*The S/R plate*. The dispersion equation for the S/R plate is identical to that for the R/S plate shown in (4.15).

*The S/S plate*. When both the top and bottom surfaces of the plate are slippery, we have in addition to (4.6)(4.16)where *γ*^{+} is an unknown constant. Equations (4.1)_{2} and (4.1)_{1}, in that order, give(4.17a)(4.17b)where we have employed the identity (3.14). Substitution of (4.17*a*) and (4.17*b*) into (4.16)_{2} and (4.6)_{2} yields(4.18a)(4.18b)A non-trivial solution for *γ* and *γ*^{+} exists if(4.19)This is the dispersion equation when both the top and bottom surfaces of the plate are slippery.

In view of the mirror image of each other, the dispersion equations for the R/F, S/F or S/R plates are identical, respectively, to those for the F/R, F/S or S/R plates, as expected.

## 5. Two layers in a plate

In this section, we consider the case when the plate consists of two layers of thickness *h* and . Let the top layer occupy the region 0≤*x*_{2}≤*h* and the bottom layer occupy the region . Making use of (3.3), we have(5.1)where the hat refers to the bottom layer. As discussed in §4, the top and bottom surfaces of the plate can be a traction-free (F), rigid (R) or slippery surface (S). The interface between the two layers at *x*_{2}=0 can be perfectly bonded (b) or a slippery surface (s). A plate that is traction free on the top surface has a slippery interface and a rigid surface on the bottom is called the F/s/R plate. We now study each case separately.

First, consider the case when the interface at *x*_{2}=0 is perfectly bonded. This means that and (5.1) gives(5.2)(5.3)Equation (5.2) can be written in full as(5.4)This is similar to (4.1) if *E*_{1}(*h*), *E*_{2}(*h*), *E*_{3}(*h*) and are replaced by *K*_{1}, *K*_{2}, *K*_{3} and *K*_{4}, respectively. Thus, the dispersion equations for the F/b/F, F/b/R, F/b/S, etc. plates can be obtained by modifying the dispersion equations for the F/F, F/R, F/S, etc. plates presented in §4 as shown below.(5.5)(5.6)(5.7)(5.8)(5.9)(5.10)(5.11)(5.12)(5.13)If the materials in the two layers are identical, (5.3) simplifies to , and (5.5)–(5.13) recover the dispersion equations for a homogeneous plate presented in §4. It should be noted that no simplification is obtained when

We next consider the case when the interface at *x*_{2}=0 is in slippery contact. This means that(5.14)Equation (5.1), after written in full, has the expression(5.15)In (5.15), we have written *E*_{m} for *E*_{m}(*h*) and for for simplicity. There are again nine cases.

*The F/s/F plate*. This means that . Solving ** u**(

*h*) from (5.15)

_{2}and substituting the result into (5.15)

_{1}yields(5.16)Equation (5.15)

_{4}gives(5.17)Substitution of (5.16) and (5.17) into (5.14)

_{2}leads to the dispersion equation(5.18)

*The F/s/R plate*. In this case, . Equation (5.16) remains valid here, while (5.15)_{3} gives(5.19)Substitution of (5.16) and (5.19) into (5.14)_{2} leads to the dispersion equation(5.20)

*The F/s/S plate*. We have(5.21)where is an unknown constant. Equation (5.16) remains valid here, while (5.15)_{4} gives(5.22)With (5.22), (5.15)_{3} becomes(5.23)where we have employed the identity (3.14). Substitution of (5.23) into (5.21)_{2} provides(5.24)Substitution of (5.16) and (5.22) into (5.14)_{2} leads to(5.25)The dispersion equation obtained from (5.24) and (5.25) is(5.26)

*The R/s/F plate*. In this case, we have . Elimination of ** ϕ**(

*h*) between (5.15)

_{1}and (5.15)

_{2}gives us(5.27)Equation (5.17) applies here. Substitution of (5.17) and (5.27) into (5.14)

_{2}leads to(5.28)This is identical to the dispersion equation for the F/s/R plate in (5.20) if we interchange the notations with and without a hat.

*The R/s/R plate*. This means that . Equations (5.27) and (5.19) apply here. Substitution of (5.27) and (5.19) into (5.14)_{2} gives(5.29)*The R/s/S plate*. In this case, ** u**(

*h*)=0 and (5.21) applies here. Hence, (5.22), (5.24) and (5.27) remain valid here. Substitution of (5.27) and (5.22) into (5.14)

_{2}leads to(5.30)The dispersion equation obtained from (5.24) and (5.30) is(5.31)

*The S/s/F plate*. The surface *x*_{2}=*h* is a slippery surface so that(5.32)where *γ*^{+} is an unknown constant. Equation (5.15)_{2} gives(5.33)Substituting (5.33) into (5.15)_{1} and (5.32)_{2} leads to(5.34)(5.35)The surface is traction free so that (5.17) applies here. Application of (5.17) and (5.34) into (5.14)_{2} yields(5.36)The dispersion equation obtained from (5.35) and (5.36) is(5.37)This is identical to the dispersion equation for the F/s/S plate in (5.26) if we interchange the notations with and without a hat.

*The S/s/R plate*. It can be shown that the dispersion equation is identical to that for the R/s/S plate shown in (5.31) if we interchange the notations with and without a hat.

*The S/s/S plate*. In this case, (5.22), (5.24), (5.34) and (5.35) apply here. Substitution of (5.22) and (5.34) into (5.14)_{2} leads to(5.38)Together with (5.24) and (5.35), we have three linear homogeneous equations for *γ*, *γ*^{+} and . A non-trivial solution exists if(5.39)

## 6. Three or more layers in a plate

It is clear that the derivation can be extended to a plate that consists of three or more layers, but the algebra becomes more complicated when the interfaces between the layers have a slippery surface. The derivation is much simpler for multilayers if all the interfaces between the layers are perfectly bonded. We study this case here.

Let the plate of thickness *h* occupy the region . It consists of *n* layers whose thicknesses are, from top to bottom, *h*_{1}, *h*_{2}, …, *h*_{n} with(6.1)Equation (5.2) can be written for each layer as(6.2)The superscript (*m*), refers to the *m*th layer . If all the interfaces are perfectly bonded,(6.3)We then have(6.4)(6.5)This is similar to the two-layer plates given in (5.2). Hence, the dispersion equations presented in (5.5)–(5.13) apply here.

We could consider the case when one or more of the interfaces is a slippery surface. The derivation is straightforward, but the algebra can be rather involved.

## 7. A sandwich plate

Of particular interest in multilayered plates is a sandwich plate that consists of one centre layer and two identical outer layers.

Let be the thickness of the centre layer that occupies the region . Two identical outer layers have the thickness *h*. Employing (5.2) we have(7.1)where the hat refers to the centre layer. The centre and the outer layers are perfectly bonded so that(7.2)We then obtain from (7.1)(7.3)(7.4)Since (** JE**) is symmetric, it is readily shown that (

**) is also symmetric. Hence, we may write(7.5)Carrying out the matrix products in (7.4) leads to(7.6)Equation (7.3) can be written in full as(7.7)**

*JK*Equation (7.7) is similar to (4.1) for a homogeneous plate. Thus, the derivations presented in (4.2)–(4.19) apply here if we replace *E*_{1}, *E*_{2} and *E*_{3} by *K*_{1}, *K*_{2} and *K*_{3}, respectively. For instance, if the top and bottom surfaces of the sandwich plate are traction free, we obtain from (4.2) |*K*_{3}|=0 so that, by (7.6)_{3},(7.8)This is the dispersion equation for the sandwich plate when both surfaces of the plate are traction free. If both surfaces are rigid, we obtain from (4.11) |*K*_{2}|=0 so that, by (7.6)_{2},(7.9)

It is clear that if the materials in the centre layer and the two outer layers are identical, (7.4) using (3.11) reduces to(7.10)The sandwich plate is a homogeneous plate. Equations (7.8) and (7.9) recover, respectively, and .

When the thickness *h* of the outer layers is very small compared with some reference length, we may use (3.10) to get an approximate expression of the propagator matrix ** E**. If we keep the first order term in

*h*, we have(7.11a)or(7.11b)Ignoring the terms of order higher than

*h*, (7.8) reduces to(7.12)This is the dispersion equation for the traction-free sandwich plate in which the two outer layers are very thin. This problem has been studied by Niklasson

*et al*. (2000

*a*), who assumed that the centre layer is isotropic, while the thin outer layers are monoclinic with the symmetry plane parallel to the plate.

In the limiting case when *h*=0, (7.12) reduces to . This recovers the dispersion equation (4.2) for a plate of homogeneous material with traction-free surfaces.

If the thickness of the centre layer is very small compared with some reference length, (7.8) reduces to, after ignoring the terms of order higher than ,(7.13)This is the dispersion equation for the traction-free sandwich plate in which the centre layer is very thin. This problem has also been studied by Niklasson *et al*. (2000*b*), who assumed that the outer layers are isotropic, while the thin centre layer is monoclinic with the symmetry plane parallel to the plate.

In the limiting case when =0, (7.13) simplifies to(7.14)This should be the dispersion equation for a homogeneous plate of thickness (2*h*) with traction-free surfaces shown in (4.2). Indeed, if we set =*h* in (3.11), we have(7.15)from which we obtain(7.16)Hence, (7.14) is(7.17)which is (4.2) if the thickness of the plate is *h*.

## 8. Remarks

We have presented the dispersion equations for a plate that consists of one or more layers of different general anisotropic elastic materials. The surfaces of the plate can be a traction-free, rigid or slippery surface. The interface between any two layers can be perfectly bonded or in slippery contact. If none of the surfaces and the interfaces of the layered plate is slippery, the dispersion equation is the vanishing of the determinant of a 3×3 matrix. If there is one slippery surface, the dispersion equation is the vanishing of the centre element, the (22)-element, of a matrix. If there are *n* slippery surfaces, the dispersion equation is the vanishing of an *n*×*n* determinant whose elements are the centre elements of some 3×3 matrices (see (4.19) and (5.39) for example).

The characteristics of the dispersion equation for a traction-free homogeneous plate have been extensively studied in the literature. However, the same cannot be said of the dispersion equations for the other plates presented here. Of particular interests are the dispersion equations for a sandwich plate when the central layer is very thin or when the two outer layers are very thin. These have been studied in the literature but only for special anisotropic elastic materials for the thin layers. The dispersion equations presented here for the sandwich plates are for general anisotropic elastic materials. Moreover, higher-order approximations can be obtained by keeping the terms up to the *n*th order of the thickness of the layer in the infinite series of the propagator matrix.

Finally, the dispersion equations presented here do not necessarily mean that all steady waves are dispersive. It is shown in Ting (submitted) that one-component waves can propagate in a homogeneous or layered plate. These waves are non-dispersive.

## Footnotes

↵† Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.

- Received October 1, 2007.
- Accepted November 22, 2007.

- © 2007 The Royal Society