## Abstract

Stroh's sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium/motion is written as a system of first-order differential equations for the displacement and traction in terms of one of the spatial variables. The so-called fundamental elasticity matrix *N* appearing in this formulation has the property that, when partitioned as a 2×2 block matrix, its 12- and 21-blocks are symmetric matrices and its 11-block is the transpose of its 22-block. This property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of the Stroh formalism in solving a large variety of problems in general anisotropic elasticity. First, we show that the matrix *N* is guaranteed to have the above property by the fact that the Stroh formulation is in fact a Hamiltonian formulation with one of the spatial variables acting as the time-like variable. This interpretation provides a much desired guide in dealing with other problems for which the governing equations are different, such as incompressible elasticity and problems associated with anisotropic elastic plates as described by the Kirchhoff plate theory. We show that for the last two problems the Hamiltonian interpretation simplifies the derivations significantly, leading to a Stroh formulation in each case which is equivalent to, but much simpler than, what is available in the existing literature.

## 1. Introduction

The equilibrium equations for anisotropic elasticity without any kinematic constraints (such as incompressibility) are given by(1.1)in terms of the displacement components *u*_{k}, the stress tensor *σ*_{ij}, the Cartesian coordinates *x*_{j} and the elastic moduli *c*_{ijkl}. The above equations also apply to steady-state motions in the *x*_{1}-direction such as free-surface waves if *c*_{ijkl} are replaced by the equivalent elastic moduli , where *ρ* is the material density and *v* is the wave speed (Ting 1996, p. 441). The traction vector on planes *x*_{2}= constant with normal *n*_{j}=*δ*_{2j} is given by(1.2)where here and hereafter the notation (*a*_{i}) denotes the vector with components *a*_{i} and (*a*_{ij}) denotes the matrix whose *ij*-component is *a*_{ij}.

It is straightforward to show that in terms of the matrices *T*, *R*, *Q* defined by(1.3)respectively, the traction vector and the governing equation (1.1) can be rewritten as(1.4)(1.5)where we have assumed that all the dependent variables are independent of *x*_{3}. With strong ellipticity ensuring that *T* is invertible, we obtain from (1.4) the first-order differential equation . On substituting this expression into (1.5)_{2}, we obtain another differential equation for ** t**. Hence, we have(1.6)or equivalently,(1.7)where(1.8)The vector function

**defined by (1.8)**

*ϕ*_{2}is introduced in order that all the derivative terms in (1.6) have the same order. It is in fact a ‘stream’ function for the equilibrium equation , and is usually referred to as the stress function.

Equation (1.7) together with (1.8) is known as the Stroh formulation for static or steady problems in anisotropic elasticity (Stroh 1958, 1962). We observe that with defined bywhere *I* is the 3×3 identity matrix, the matrix is symmetric. This property, originally observed as a matter of fact, leads to a large number of orthogonality and closure relations and is fundamental to the success story of the Stroh formalism (Chadwick & Smith 1977; Ting 1996). Whereas this property is seen at the end of a short derivation in the above context, it is not always clear whether the same property will emerge when the derivation gets more involved, as is the case for incompressible materials and anisotropic elastic plates. In fact, for anisotropic elastic plates, it was not even clear what dependent variables should be used and a trial and error approach was used in the literature in order to obtain *N* with this property; see Lu (1994, 2004), Lu & Mahrenholtz (1994), Cheng & Reddy (2002, 2005) and Hwu (2003*a*–,*c*). Although the Hamiltonian nature of the Stroh formulation has previously been observed in passing by a number of researchers (see Barnett 2000, p. 49), the full potential of the Hamiltonian interpretation does not seem to have been exploited. We note, however, that it has previously been observed by Fu (2003) and Fu & Brookes (2006) that the fundamental property of *N* is guaranteed if the two dependent variables (such as the ** u** and

**in the above context) are conjugate variables in the Hamiltonian interpretation. This observation will be the point of departure in this paper. In §2, we shall first elaborate on this observation and show precisely how it works. In §§3 and 4, we shall consider incompressible elastic materials and anisotropic elastic plates, respectively, and show how the Hamiltonian interpretation takes the guesswork out of the derivations and gives in each case a Stroh formulation that is simpler and more transparent than what is already known in the literature. In particular, in §4 it is shown that the shear force**

*t**q*

_{2}in the effective transverse shear force

*V*

_{2}=

*q*

_{2}+

*M*

_{12,1}may be interpreted as the derivative of a Lagrange multiplier in the variational formulation. It is then found that there is no need to expand it and it may be left as a passive variable in the derivation of the Stroh formulation. This observation leads to significant simplifications. The paper is then concluded with a few additional remarks.

## 2. Hamiltonian formulation for elastic materials without constraints

Consider an elastic body that occupies the cylindrical domain *Ω*=*I*×*Σ* with *x*_{2}∈*I* and (*x*_{1}, *x*_{3})∈*Σ* relative to a Cartesian coordinate system, where *I*⊂ is a possibly infinite or semi-infinite interval and the cross-section *Σ*⊂^{2} is a bounded domain. We assume that the boundary of *Ω* consists of two parts, *S*_{u} and *S*_{t}, on which zero displacement ** u**=

**0**and a non-zero traction are prescribed, respectively. The total energy in the sense of Eshelby (strain energy plus loss of potential energy by the loading mechanism) is then given by(2.1)where in obtaining the second expression above we have used the divergence theorem (assuming the displacement field is everywhere continuous within the volume) to convert the surface integral into a volume integral. As is well known, the variational principle yields the equilibrium equations (1.1) and the boundary condition on

*S*

_{t}.

We now treat *x*_{2} as a time-like variable and rewrite (2.1) as(2.2)where(2.3)and a superimposed dot signifies partial differentiation with respect to *x*_{2}. The Euler–Lagrange equation (i.e. the equation of equilibrium/motion) then takes the form(2.4)We note that(2.5)which shows that the displacement ** u** and the traction

**defined by (1.4) are work-conjugate. Thus, viewing the governing equations as a Hamiltonian dynamical system, we may define the Legendre transform of**

*t**L*(2.6)and write(2.7)where () is given by (1.6)

_{1}. It is well known that the Hamiltonian for the (spatial) dynamical system must necessarily be a constant,(2.8)We note thatwhere we have made use of the relations (2.4) and (2.5). It then follows from the definition of variational derivatives and that(2.9)or equivalently,(2.10)On the other hand, with the aid of (2.6) and (2.7), we find that(2.11)It is seen that (2.10) together with (2.11) is equivalent to (1.6).

We now show that *N* has the fundamental property not only for the specific *H* given by (2.6) but also for all *H* that is quadratic in *u*_{,1} and ** t**. To this end, we consider a general

*H*given bywhere

*A*,

*B*,

*C*are 3×3 matrices which are known for each given problem. The matrices

*A*and

*C*may be assumed as symmetric since otherwise they can be symmetrized (e.g. ). It then follows thatThuswhich has the same structure as (2.11). It is then clear that the fundamental property of

*N*is a consequence of the intrinsic structure of the Hamiltonian equations (2.10). Thus, as pointed out by Fu (2003) and Fu & Brookes (2006), as long as the dependent variables are chosen to be conjugate pairs, the resulting matrix

*N*is guaranteed to have the desired property.

We observe that, in terms of the vector ** ξ** defined in (1.8), equation (2.8) may be written as(2.12)For a travelling wave solution for which

**takes the form(2.13)where a superimposed bar signifies complex conjugation and**

*ξ**v*is the wave speed, the above integral reduces to(2.14)This is one of the integrals used by Taziev (1989) and Destrade (2003) to derive explicit secular equations for the wave speed.

We conclude this section by noting that it has been a popular approach to formulate a static or steady-state problem as a (spatial) Hamiltonian dynamical system. For instance, Mielke (1988*a*,*b*) has previously used such an approach in studying Saint-Venant's problem associated with a semi-infinite elastic strip, and Guz & Shulga (1992) have taken advantage of this approach in studying waves in laminated composites.

## 3. Hamiltonian formulation for incompressible materials

The deformation of an incompressible elastic material is subjected to the constraint , which, when *u*_{i} are independent of *x*_{3}, may be rewritten as(3.1)where and . For later use, we observe that (3.1) implies(3.2)The governing equations can again be derived from the variational principle , but now the *L* in (2.2) is replaced by(3.3)where *p* is a Lagrange multiplier which is introduced to allow and to vary independently. The counterpart of (2.5) is(3.4)We note that, for incompressible materials, strong ellipticity is not sufficient to guarantee the positive definiteness of *T* (Chadwick 1997), so that (3.4) cannot be solved to find explicitly. To get around this problem, we multiply (3.2) by an arbitrary constant *λ* and add the resulting equation to (3.4), obtaining(3.5)where(3.6)Provided *λ* is positive and sufficiently large, the defined above will be positive definite and we may then write(3.7)Equations (2.6)_{1}, (2.7) and (2.10) are still valid, but (2.11) is now replaced by(3.8)(3.9)where(3.10)The Hamiltonian/Stroh formulation is completed if we can eliminate the unknown multiplier *p* from (3.8) and (3.9). This is achieved by taking the dot product of (3.7) with *e*_{2} and making use of (3.1). We have(3.11)where(3.12)On substituting (3.11), (3.8) and (3.9) into the Hamiltonian equations (2.10), we obtain the Stroh formulation (1.7) with *N* taking the form(3.13)where(3.14)(3.15)(3.16)We expect that the three matrices *N*_{1}, *N*_{2}, *N*_{3} should be independent of the arbitrary constant *λ*. This is indeed the case, since it can be shown thatFor instance, to show d*N*_{2}/d*λ*=0 we may simply differentiate (3.15) with respect to *λ* and then simplify the resulting expression usingobtained from , (3.6)_{2} and (3.12), respectively.

The above results can easily be modified to describe a prestressed incompressible elastic material which is initially isotropic and for which the equilibrium equations are(3.17)where(3.18)The _{jilk} are the first-order instantaneous elastic moduli (Chadwick & Ogden 1971), whereas and *p* are the pressure associated with the prestressed state and the incremental pressure, respectively; see Dowaikh & Ogden (1990). The *p* here plays the same role as the Lagrange multiplier introduced earlier in (3.3). It is immediately seen that, to apply the results obtained in the earlier part of this section, we need only to make the substitution(3.19)Correspondingly, the matrices *Q*, *R* and *T* are now calculated from(3.20)We also note that for surface waves with speed *v*, the *Q* throughout this section should be replaced by *Q*−*ρv*^{2}*I*. With these substitutions made, the expressions (3.14)–(3.16) are equivalent to the expressions (2.26)–(2.28) by Fu (2005*b*). When comparing with Chadwick (1997), we find that the eqn (3.24) there is in error; the missing term should be added to its right-hand side. For another derivation of the Stroh formulation for incompressible materials, we refer to Destrade *et al.* (2002) in which the authors first expressed the Stroh formulation for compressible materials in terms of elastic compliances and then took an appropriate limit corresponding to incompressibility.

## 4. Hamiltonian formulation for anisotropic elastic plates

In this section, we consider edge waves propagating along the edge of a generally anisotropic elastic plate defined by(4.1)where *h* is a constant. Under the Kirchhoff hypothesis, the leading-order displacement field takes the form(4.2)where *u*_{1} and *u*_{2} are the in-plane displacement components at the mid-plane, *w*(*x*_{1}, *x*_{2}, *t*) is the deflection of the mid-plane and *θ*_{1}, *θ*_{2} are related to *w* through(4.3)In the rest of this paper, Greek subscripts range from 1 to 2 only. As a result of the assumption *σ*_{33}≡0 which is used to express in terms of , the constitutive relations become(4.4)where the reduced elastic moduli are related to the elastic moduli *c*_{ijkl} by

Under an appropriate averaging procedure (see Christensen (1991), Reddy (1997) or Fu & Brookes (2006)), the governing equations are replaced by(4.5)where ,(4.6)(4.7)(4.8)(4.9)the *k* in (4.9) being the wavenumber and the symbol 〈 〉 denoting averaging across the plate thickness, e.g.In (4.5) we have neglected terms involving the wave speed *v* which are of higher order than those that have been retained; see Fu & Brookes (2006). For static problems, we set *v*=0.

On substituting (4.6) and (4.7) into (4.5)_{1,2}, we obtain(4.10)(4.11)which may be rewritten as(4.12)(4.13)where(4.14)To obtain the corresponding variational principle for the current problem, we first considerwhich suggests that the variational principle should bewith *L* given by(4.15)where the Lagrange multiplier *p* is introduced to allow *θ*_{1} and *θ*_{2} to vary independently. From *δ*=0, we obtain (4.10) and, instead of (4.11),(4.16)The consistency of (4.16) with (4.11) requires(4.17)and elimination of *p* from (4.17) by cross-differentiation then yields the third equilibrium equation (4.5)_{3}. The above connection between the shear force *q*_{2} and the multiplier *p* suggests that the shear force *q*_{2} should play only a passive role in our Hamiltonian formulation (as the multiplier *p* in §3); there is no need to expand/express it in terms of ** u** and

**, as has been done by Fu & Brookes (2006). Indeed, this realization will significantly simplify our derivations.**

*θ*We note that the constraint may also be written as(4.18)We denote by ** t** and

**the variables conjugate to**

*s***and**

*u***, respectively. As in §2 they are computed according toand are given by(4.19)(4.20)We note that in Fu & Brookes (2006) the ‘generalized coordinates’ are chosen to be**

*θ**u*

_{1},

*u*

_{2},

*w*, −

*w*

_{,2}, whereas here we have found it more convenient to choose them to be

*u*

_{1},

*u*

_{2}, −

*w*

_{,1}, −

*w*

_{,2}instead. It is known from Fu & Brookes (2006) that the variable conjugate to

*w*is

*M*

_{12,1}+

*q*

_{2}. It then follows, through integrating by parts, that the variable conjugate to

*θ*

_{1}=−

*w*

_{,1}must necessarily bewhich, in view of the connection (4.17)

_{2}, is consistent with (4.20)

_{1}.

In a manner similar to that in §2, the functional is defined by(4.21)and the Hamiltonian equations are given by(4.22)While this Hamiltonian structure guarantees that the fundamental matrix in our final Stroh formulation will have the desired property, we find it easier to derive the Stroh formulation directly with the aid of (4.12), (4.13), (4.19) and (4.20).

First, the two equations (4.19) and (4.20) may be written as(4.23)With the inverse of the coefficient matrix above defined by(4.24)equation (4.23) gives us the first two sets of equations in the Stroh formulation. For instance, we have(4.25)The other two sets of equations are obtained by differentiating (4.19) and (4.20), followed by the use of (4.12) and (4.13). We have(4.26)(4.27)where the derivatives with respect to *x*_{2} on the right-hand sides can be eliminated with the aid of (4.23). The Stroh formulation is completed if the multiplier *p* is eliminated from these expressions. This last task is accomplished by taking the dot product of (4.25) with *e*_{1} and then using (4.18) to eliminate . We have(4.28)where(4.29)Finally, on substituting (4.28) into equations (4.23), (4.26), (4.27) and manipulating the resulting equations into a matrix form, we obtain(4.30)where(4.31)(4.32)*I*_{33}, *I*_{34}, *I*_{43}, *I*_{44} being the 4×4 matrices whose *IJ*-elements are *δ*_{3I}*δ*_{3J,} *δ*_{3I}*δ*_{4J,} *δ*_{4I}*δ*_{3J,} *δ*_{4I}*δ*_{4J,} respectively. In (4.30), the new variables ** ϕ** and

**are defined by(4.33)and are introduced so that all the derivative terms in (4.30) have the same order. They in fact have physical interpretations as follows. First, the**

*ψ***is a ‘stream function’ for the equilibrium equationwhere, as indicated earlier, (**

*ϕ**N*

_{α1}) denotes the vector whose

*α*-th component is

*N*

_{α1}. Likewise, the second expression in (4.33) suggests that

**should also be a ‘stream function’ for another equilibrium equation. The latter equation can only be the third equilibrium equation (4.5)**

*ψ*_{3}; this becomes transparent if we note from (4.5)

_{2}and (4.17) that the third equilibrium equation (4.5)

_{3}is equivalent to(4.34)where we have made use of (4.20)

_{1}. It then follows that(4.35)From this and , obtained from (4.20)

_{1}and (4.33)

_{2}, we obtain(4.36)and hence(4.37)The effective shear forces , are then found to be given by(4.38)Thus, all the bending moments and effective shear forces are expressed in terms of a single vector function

**. The expressions (4.37) and (4.38) have previously been given by Cheng & Reddy (2002).**

*ψ*Once the fundamental elasticity matrix *N* is known, we may define the edge-impedance matrix *M* through(4.39)On substituting (4.39) into the Stroh formulation (4.30), we may deduce that *M* satisfies the matrix equation(4.40)Such a matrix equation for the surface-impedance matrix associated with compressible materials has previously been derived by Biryukov (1985), Mielke & Sprenger (1998) and Fu & Mielke (2002).

As in Mielke & Fu (2004) and Fu & Brookes (2006), the following integral representation for *M* can be derived with the aid of the matrix equation (4.40)(4.41)where(4.42)The *N*_{1}(*θ*) and *N*_{2}(*θ*) above are obtained from the expressions (4.31) for *N*_{1} and *N*_{2} by replacingbyrespectively, whereandThe integral representation for the surface-impedance matrix was pioneered by Barnett & Lothe (1973), and the counterparts of the above *H* and *S* in the context of generalized plane strain elasticity are called the Barnett–Lothe tensors. It was shown by Fu & Brookes (2006) that a combination of the matrix equation (4.40) and the integral representation (4.41) provides a robust numerical method for computing the edge wave speed.

We have checked to confirm that our expressions (4.31) are equivalent to their counterparts given by Cheng & Reddy (2005) and Fu & Brookes (2006). Our integral representation for the edge-impedance matrix is checked by substituting the expression (4.41) into the matrix equation (4.40) for a random selection of numerical examples.

In the special case when the mid-plane *x*_{3}=0 is a plane of material symmetry, we have *B*_{αβγω}≡0 and the problem for *w* decouples from the plane stress problem for *u*_{α}. It was shown by Fu (2005*a*) that in this case the flexural edge-wave problem associated *w* is mathematically identical to the plane strain surface-wave problem associated with an incompressible elastic material and that more explicit results can be obtained.

## 5. Conclusion

The original Stroh formulation was written down in the context of general compressible anisotropic elasticity. Extension of this formulation to other elastic problems with different forms of governing equations is known to be non-trivial. The difficulty is not only to obtain an *N* with the desired structure but also to express the *N* in terms of matrices whose corresponding forms relative to a rotated coordinate system can be written down immediately using the tensor transformation rule. The latter is required, as we have demonstrated at the end of §4, in the construction of the Barnett–Lothe tensors and the surface/edge impedance matrix which feature in a wide range of applications (see Chadwick & Smith 1977; Ting 1996). In this paper, we have shown that the Hamiltonian interpretation provides a systematic and fail-proof framework for deriving such a matrix *N*. In particular, the procedure given in §3 for incompressible materials is also applicable for elastic materials suffering from other types of constraints such as inextensibility.

## Acknowledgments

The author wishes to thank the referees for their constructive comments and suggestions.

## Footnotes

- Received June 19, 2007.
- Accepted August 14, 2007.

- © 2007 The Royal Society