## Abstract

When a very thin elastic layer is bonded to an elastic body, it is desirable to have *effective boundary conditions* for the interface between the layer and the body that take into account the existence of the layer. In the literature, this has been done for special anisotropic elastic layers. We consider here the layer that is a general anisotropic elastic material. The mechanics of a thin layer is studied for elastostatics as well as steady state waves. It is shown that one-component surface waves cannot propagate in a semi-infinite thin layer. We then present Love waves in an anisotropic elastic half-space bonded to a thin anisotropic elastic layer. The dispersion equation so obtained is valid for long wavelength. Finally, effective boundary conditions are presented for two thin layers bonded to two surfaces of a plate and a thin layer bonded between two anisotropic elastic half-spaces.

## 1. Introduction

When an elastic layer is bonded to another elastic body, one has to find the solutions in the layer as well as in the body and impose the continuity conditions across the interface. If the layer is very thin compared with some reference length, it is desirable to replace the existence of the layer by *effective boundary conditions* to avoid finding the solution in the layer. This was first considered by Bovik (1994) who assumed that the layer is an isotropic elastic material. Niklasson *et al*. (2000) studied the case when the layer is a monoclinic material with the symmetry plane parallel to the plane of the layer. In this paper, we consider the case in which the layer is a general anisotropic elastic material.

The basic governing equations for an elastic layer are presented in §2. There are several matrices in the governing equations that depend on the elastic constants of the layer. Explicit expression of the matrices is given in §3 in terms of the elastic stiffness *C*_{αβ}. Some of the matrices require computation of 4×4 determinants. In §4, we present these matrices in terms of the elastic compliance *s*_{αβ}. The problem of computing 4×4 determinants for some matrices in §3 is avoided, but this is traded off with one matrix that now requires computation of 4×4 determinants. The mechanics of a thin layer are studied in §5. All quantities are expanded in terms of the thickness *h* of the layer, and the terms of order higher than *h* are ignored. A general solution for elastostatics of thin layer is presented. The surface waves propagating in the semi-infinite thin layer are also presented. It is shown that one-component surface waves do not exist for the semi-infinite thin layer. In §6, we consider the case when the thin layer is bonded to a half-space. The effective boundary conditions for the interface are presented. The results are applied to a generalization of Love waves. A dispersion equation is presented that is a good approximation for wavelength much larger than the thickness of the layer. The case when the top and bottom surfaces of a plate are bonded to two thin layers of different anisotropic elastic materials is considered briefly in §7. Finally in §8, we present effective boundary conditions for the case when a thin layer is bonded between two anisotropic elastic half-spaces.

## 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; a comma denotes differentiation with *x*_{i}; and the dot denotes differentiation with time *t*. The stress–strain relation is(2.2)(2.3)where *C*_{ijks} is the elastic stiffness. The *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 equalities imply the third (Ting 1996, p. 32).

Consider a layer of thickness *h* that is parallel to the plane *x*_{2}=0. If the layer is bonded to an elastic body, the stress components *σ*_{i2} (*i*=1, 2, 3) are continuous across the interface between the layer and the body but not *σ*_{11}, *σ*_{13}, *σ*_{33}. Hence, let(2.4)With (2.4), the equation of motion (2.1) can be rewritten as(2.5)where(2.6)Likewise, the stress–strain law (2.2) can be rewritten as(2.7)where, using the contracted notation *C*_{αβ} for *C*_{ijks} (Voigt 1910),(2.8)The three equations in (2.5) and (2.7) are the governing equations for ** u**,

*t*and . When the layer is bonded to an elastic body,

**,**

*u***,**

*t***,**

*u*_{1},

**,**

*u*_{3},

**,**

*t*_{1}and

**,**

*t*_{3}are continuous across the interface but not ,

**,**

*u*_{2}and

**,**

*t*_{2}. The idea is to eliminate ,

**,**

*u*_{2}and

**,**

*t*_{2}. But we have only three equations. Thus, we can only eliminate two out of the three.

The matrix *C*_{2} is positive definite so that (2.7)_{1} can be solved for ** u**,

_{2}as(2.9)where the superscript T denotes the transpose and(2.10)Substitution of (2.9) into (2.7)

_{2}yields(2.11)in which(2.12)Substitution of from (2.11) into (2.5) gives(2.13)In the above,(2.14)The

*D*_{1}and

*D*_{3}defined in (2.10) and (2.14) can be shown to be equivalent using the identities(2.15)Equations (2.9), (2.11) and (2.13) are the key equations for an anisotropic elastic layer. It should be pointed out that the above derivation did not assume that the layer is thin. Hence, the results are valid regardless of whether the layer is thin or not.

The six matrices *C*_{k} and (*k*=1, 2, 3) given in (2.8) are not independent of each other. Let(2.16)which is symmetric and positive definite. It can be shown that(2.17)Hence, only *C*_{0}, *C*_{2} and are independent. *C*_{1}, *C*_{3}, and can be computed from (2.15) and (2.17).

We will show that the second columns of *E*_{1} and *E*_{3} vanish. Also, the matrices *G*_{k} (*k*=1, 2, 3) are symmetric, and the second row and the second column of *G*_{k} (*k*=1, 2, 3) vanish. To this end, let(2.18)The matrix ** w** is symmetric. We will show in appendix A that it is positive definite. With the use of (2.15) and (2.17), the

*E*_{1}and

*E*_{3}in (2.12) can be written in terms of

**as(2.19)The (**

*w**k*=1, 2, 3) in (2.14) can also be written in terms of

**as(2.20)**

*w*In conclusion, *D*_{1} and *D*_{3} shown in (2.14) depend only on *E*_{0} while *E*_{k} (*k*=1, 3) and *G*_{k} (*k*=1, 2, 3) given in (2.19) and (2.20) depend only on ** w**. Hence, it suffices to compute

*D*_{0},

*E*_{0}and

**. Since**

*w*

*D*_{0}and

**are symmetric and**

*w*

*E*_{0}is not, there are a total of twenty-one independent parameters in

*D*_{0},

*E*_{0}and

**, same as**

*w**C*

_{αβ}. Before we study various problems associated with a thin anisotropic elastic layer, we compute

*D*_{0},

*E*_{0}and

**explicitly in terms of**

*w**C*

_{αβ}in §3.

## 3. Explicit expression of *D*_{0}, *E*_{0} and *w* in terms of *C*_{αβ}

*D*

*E*

*w*

We introduce the following notation for a minor of the 6×6 matrix *C*_{αβ}. Let *C* ( be the *k*×*k* minor of *C*_{αβ}, the elements of which belong to the rows of *C*_{αβ} numbered and the columns numbered , 1 (Barnett & Chadwick 1991). The row and column numbers can be interchanged and a minor with two equal row or column numbers is zero. A principal minor is of the form , which will be denoted by for simplicity.

It is readily shown that the inverse of the symmetric matrix *C*_{2} is(3.1)From (2.12) and (2.18), we obtain(3.2)In the above,(3.3)Except the second column and the second row of ** w**,

*w*

_{km}=

*β*

_{km}. For the second row (which is identical to the second column) of

**, we have =, = and =. Using the**

*w*

*E*_{0}in (3.2),

*D*_{1}and

*D*_{3}in (2.14) have the expression(3.4)With

**given in (3.2), (2.19) and (2.20) provide (**

*w**k*=1, 3) and (

*k*=1, 2, 3).

As an example, consider the special case of monoclinic materials with the symmetry plane at =0. The 6×6 matrix has the structure(3.5)Only the upper triangle of the matrix is shown because is symmetric. One can then show that(3.6a)(3.6b)(3.6c)(3.6d)

Equation (3.3) tells us that the needed in *w* require computation of 4×4 determinants. Although they simplify substantially for monoclinic materials with the symmetry plane at =0 as shown in (3.6*c*), this is not the case for other monoclinic materials with the symmetry plane at =0 or =0. By considering the elastic compliance instead of the elastic stiffness , no 4×4 determinants need to be computed for . This is presented next.

## 4. Explicit expression of *D*_{0}, *E*_{0} and *w* in terms of *s*_{αβ}

*D*

*E*

*w*

The stress–strain relation can be expressed in terms of the elastic compliance as(4.1)(4.2)where is the strain and is the elastic compliance tensor whose contracted notation is . The 6×6 matrices of and are the inverse of each other, i.e.(4.3)in which ** I** is the identity matrix. The strain is related to the displacement by(4.4)

Equation (4.1), after inserting from (4.4), can be rewritten as(4.5)where, using the contracted notation for ,(4.6)The matrices and *s*_{3} are both symmetric and positive definite. The first equation in (4.5) gives (2.11) if we set(4.7)Substitution of (2.11) into the second equation of (4.5) yields (2.9) if we let(4.8)Using (4.7), it can be shown that the *D*_{1} and *D*_{3} in (2.14) and (4.8) are equivalent. Equation (2.13) and (2.14) remain valid here.

We now proceed to compute *D*_{0}, *E*_{0} and ** w**. Comparison of

*E*_{1}and

*E*_{3}in (4.7) and (2.19) tells us that(4.9)We show in appendix A that

**and are indeed the inverse of each other. Since is symmetric and positive definite, so is**

*w***. The**

*w*

*E*_{0}computed from (4.7) is(4.10)where(4.11)The

*D*_{0}in (4.8) is(4.12)(4.13)where

*k*and

*m*take the value of 2, 4 or 6. With the given in (4.10),

*D*_{1}and

*D*_{3}in (2.14) have the expression(4.14)Thus, while

**does not require computation of 4×4 determinants,**

*w*

*D*_{0}does.

The above results could have been obtained by converting the minors of the elastic stiffness to those of elastic compliance using a relation similar to Jacobi's theorem (Mirsky 1990; see also Ting 1997).

## 5. Mechanics of a thin layer

For the rest of the paper, we assume that the thickness *h* of the layer is very thin. We will expand all quantities in term of *h* and retain only the first-order term in *h*, ignoring all terms that are of order higher than *h*.

Consider a thin anisotropic elastic layer located at . The surfaces of the layer at are traction free. This means that ** t**=

**0**at . In fact

**=**

*t***0**for all if we ignore terms of order higher than

*h*. Equations (2.13) and (2.11) reduce to(5.1)(5.2)Equation (5.1) consists of three scalar equations. Since the second rows and the second columns of

*G*_{1},

*G*_{2},

*G*_{3}contain only zero elements, the second equation in (5.1) gives(5.3)The remaining two equations are the equations of motion for the thin layer for the displacement(5.4)After solving for , from (5.1) and from (5.2), can be computed from (2.9) with

**=**

*t***0**. and give the shear strains, while provides a measure of the change in the thickness of the thin layer.

With the *G*_{1}, *G*_{2}, *G*_{3} given in (2.20), (5.1) without the second equation can be written as(5.5)where(5.6)Equation (5.2) can be written as(5.7)Thus, (5.5) and (5.7) are the governing equations for the thin layer. They are similar to the Stroh (1958) formalism for two-dimensional deformation of anisotropic elastic bodies (see also Chadwick & Smith 1977; Ting 1996).

In the case of elastostatics, let(5.8)where *f* is an arbitrary function of *z*; and *p* and ** a** are constants to be determined. Equation (5.5) with is satisfied if(5.9)A non-trivial solution for

**exists when the determinant of the 2×2 matrix in (5.9) vanishes, i.e.(5.10)This is a quartic equation in**

*a**p*. We will show that

*p*cannot be real.

Carrying out the computation of the determinant in (5.10) gives(5.11)It can be rewritten as(5.12)in which(5.13)However, according to (4.9). Hence, (5.12) can be replaced by(5.14)and (5.11) is replaced by(5.15)The matrix is symmetric and positive definite. If *p* were real, *g* is real and cannot vanish. Hence, *p* cannot be real. The four roots of *p* consist of two pairs of complex conjugates.

Following the Stroh formalism, let (Stroh 1958; Ting 1996)(5.16)where the second equality comes from (5.9). The stress obtained from (5.7) and (5.8) is(5.17)where(5.18)is the stress function. Let , be the roots of (5.15) with positive imaginary part. The ** a**,

**associated with and are denoted by**

*b*

*a*_{1},

*b*_{1}and

*a*_{2},

*b*_{2}, respectively. The general solution for the displacement

**and the stress function are obtained from (5.8) and (5.18) by superposing the solutions associated with , and taking the real part. We have(5.19)where**

*u***is an arbitrary constant vector and(5.20)**

*q*The above derivation can be modified for a steady-state wave with wave speed *υ* propagating in the direction . Equations (5.8)–(5.10) are replaced by(5.21)(5.22)(5.23)Equations (5.16)–(5.18) have the expression(5.24)(5.25)(5.26)The general solution (5.19) remains valid if all *p* computed from (5.23) are complex.

As an application, consider surface waves in the half-plane 0. The edge =0 is traction free. This means that =0 or, from (5.17), is a constant at =0. Without loss in generality, we take =0 at =0. The function *f*(*z*) in (5.21) and (5.26) takes the form(5.27)where *k* is the real wavenumber. From (5.19), =0 at =0 leads to(5.28)A non-trivial solution for ** q** exists if(5.29)This is the secular equation for the wave speed

*υ*.

The surface wave solution given by (5.19) consists of two components. It is known that one-component surface wave exists for an anisotropic elastic half-space (Barnett & Chadwick 1991; Barnett *et al*. 1991). Does one-component surface wave exist in the thin anisotropic elastic half-plane? To answer this question, we rewrite the two equations in (5.24) as(5.30)where(5.31)It can be shown that(5.32)Since ** b**=

**0**for one-component surface waves, (5.30) leads to(5.33)Equation (5.33) is satisfied when(5.34)and is arbitrary. However, since

*p*is real, a one-component surface wave does not exist for the thin half-plane.

## 6. A thin layer bonded to a half-space

Let a thin anisotropic elastic layer be bonded to a half-space of different anisotropic elastic material. The layer occupies the region . The surface at of the layer is traction free. The surface traction at the interface can be approximated by(6.1)where ** t** and are evaluated at of the layer. Equation (6.1) is correct if we ignore terms of order higher than

*h*. With (6.1), (2.13) has the expression(6.2)All quantities in (6.2) are evaluated at of the layer. The

**and**

*t***are continuous across the interface , so are , , , and . Hence, (6.2) is the ‘effective’ boundary condition for the half-space at .**

*u*Niklasson *et al*. (2000) derived the effective boundary condition for which the thin layer is a monoclinic material with the symmetry plane parallel to the layer, which is at in their formulation. Equation (6.2) with the use of (3.6*c*), (3.6*d*) and (2.20) recovers their effective boundary conditions if we interchange the coordinate and . Bovik (1996) studied Love waves for which the thin layer and the half-space are both isotropic. He obtained the dispersion equation using the effective boundary condition (6.2) for isotropic materials. Excellent agreements were found with the exact theory for long wavelength (or small wavenumber).

More general Love waves have been studied by Shuvalov & Every (2002). For a very thin layer bonded to the half-space, (6.2) can be employed to replace the thin layer and used as the effective boundary condition at the interface between the thin coating and the half-space. The procedure is illustrated below.

In order to be self-contained, we briefly derive the Stroh formalism for surface waves in a half-space. Let the surface wave be propagating in the -direction in the half-space with wave speed *υ* and let(6.3)The stress obtained from (2.2) is(6.4)where(6.5)The equation of motion (2.1) is satisfied if(6.6)Introducing the new vector(6.7)where the second equality comes from (6.6), (6.4) can be replaced by(6.8)in which(6.9)is the stress function. For a non-trivial solution of ** a** in (6.6), the determinant of the 3×3 matrix in (6.6) must vanish. This leads to a sextic equation for

*p*. Let , , be the roots of the sextic equation with a positive imaginary part and

*a*_{1},

*a*_{2},

*a*_{3}be the associated solution from (6.6). The

**computed from (6.7) for**

*b*

*a*_{1},

*a*_{2},

*a*_{3}will be denoted by

*b*_{1},

*b*_{2},

*b*_{3}. The general solution for the displacement

**and the stress function**

*u**ϕ*obtained from (6.3) and (6.9) by superposition of the three solutions associated with , , can be written as(6.10)where

**is an arbitrary constant vector and(6.11)Since the imaginary parts of , , are positive,**

*q***vanishes at .**

*u*We now impose the boundary condition (6.2) at . Noticing that == and ** u** and do not depend on , substitutions of (6.10) into (6.2) for leads to(6.12a)or(6.12b)in which(6.13)where is the impedance tensor (Barnett & Lothe 1985). For a non-trivial solution of

**, we must have(6.14)This is the dispersion equation for Love waves when the coating layer is very thin compared with the wavelength, i.e. (**

*q**hk*) is very small. It should be noted that the elastic constants implicit in , refer to the material in the coating layer.

Let *m*_{1}, *m*_{2}, *m*_{3} denote the columns of and *y*_{1}, *y*_{2}, *y*_{3} denote the columns of ** Y**. For a very small (

*hk*), (6.14) can be approximated by(6.15)Let be the adjoint of so that(6.16)Equation (6.14) can then be approximated by(6.17)

## 7. Two thin layers bonded to the top and bottom of a plate

Let an anisotropic elastic plate of thickness 2*a* occupy the region . An elastic layer of thickness with different anisotropy occupies the region and is bonded to the top of the plate. A different anisotropic elastic layer of thickness occupies the region and is bonded to the bottom of the plate. If the thickness is sufficiently small, the effective boundary condition (6.2) can be applied at . We have(7.1)The ± sign on (*k*=1, 3) and (*k*=1, 2, 3) refer to the materials in the layer on top and bottom of the plate, respectively.

The special case when the top and bottom layers are the same monoclinic material with the symmetry plane parallel to the layers and of the same thickness while the plate is isotropic has been studied by Niklasson *et al*. (2000). They obtained guided waves in the plate using the effective boundary condition (7.1) and found good agreement with the exact theory for long wavelength. One could employ the effective boundary condition (7.1) for the more general problem in which the plate, the top layer and the bottom layer are all different anisotropic elastic materials. The thickness of the top and the bottom layers need not be the same.

## 8. A thin layer bonded between two half-spaces

Let a thin anisotropic elastic layer of thickness *h* occupy the region . It is bonded to the half-space on the top and to the half-space on the bottom . The two anisotropic materials in the two half-spaces can be different from each other and can be different from the material in the thin layer.

Ignoring the terms of order higher than *h*, the difference in the traction *t* and the displacement *u* at of the layer can be written as(8.1)The ± sign on the variable denotes the value evaluated at of the layer. We will write (8.1) as(8.2)where d and a stand for the difference and average, respectively, of the values at . Substitution of and from (2.13) and (2.9) into (8.1) yields(8.3)In (8.3), ** u**,

**, and their derivatives with respect to , and time**

*t**t*are continuous across the interfaces between the layer and the half-spaces. Thus, their values at are identical to those of upper half-space at and their values at are identical to those of lower half-space at . Hence, (8.3) is the effective interfacial condition between the two half-spaces. The (

*k*=1, 3), (

*k*=1, 2, 3) and (

*k*=1, 2, 3) refer to the material in the thin layer.

The special case when the upper and lower half-space are the same isotropic elastic material and the thin layer is a different isotropic elastic material was considered by Bovik (1994).

## 9. Concluding remarks

We have presented governing equations for an anisotropic elastic layer that can be employed to derive effective boundary conditions at an interface between a thin anisotropic elastic layer and an anisotropic elastic body. The effective boundary conditions essentially avoid the necessity of finding the solution inside the thin layer. The results are valid when the thickness of the layer is sufficiently small.

With the exception of the mechanics of thin layer that is studied in reasonable detail in §5 and the Love waves in an anisotropic elastic half-space bonded to a thin anisotropic elastic layer discussed in §6, §§7 and 8 present little analysis beyond the effective boundary conditions. This is so because each topic in §§ 7 and 8 can be a full-length paper by itself (Ting submitted, in preparation). It is hoped that the effective boundary conditions presented here provide means for analysing practical problems that involve a thin layer coating on an elastic body.

Note added in proof. The author is grateful to a reviewer for pointing out a relevant work by Benveniste (2006) who derived effective boundary conditions for a curved thin anisotropic elastic layer. While the effective boundary conditions are more general than the ones presented here because the layer is a curved surface, the elastic constants involved in the effective boundary conditions are less explicit than the ones given in §§ 2–4 here. It should be pointed out that Benveniste (2006) also studied unsteady heat conduction.

## Footnotes

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

- Received February 17, 2007.
- Accepted May 21, 2007.

- © 2007 The Royal Society