## Abstract

The paper presents an analytical approach to modelling of Bloch–Floquet waves in structured Mindlin plates. The emphasis is given to a comparative analysis of two simplified plate models: the classical Kirchhoff theory and the Mindlin theory for dynamic response of periodic structures. It is shown that in the case of a doubly periodic array of cavities with clamped boundaries, the structure develops a low-frequency band gap in its dispersion diagram. In the framework of the Kirchhoff model, this band gap persists, even when the radius of the cavities tends to zero. A clear difference is found between the predictions of Kirchhoff and Mindlin theories. In Mindlin theory, the lowest band goes down to *ω* = 0 as the radius of the cavities tends to zero, which is linked with the contrasting behaviour of the corresponding Green functions.

## 1. Introduction

This paper addresses low-frequency asymptotics for Bloch–Floquet waves in structured elastic plates. We will be concerned with the accuracy and consistency of the two principal simplified plate models: classical Kirchhoff theory and Mindlin theory (Mindlin 1951) for dynamic response of periodic structures. The relation between these two theories has been well-summarized by Rose & Wang (2004):
Mindlin plate theory is known to be exceptionally accurate down to wavelengths comparable with the plate thickness

*h*, whereas classical plate theory is of acceptable accuracy only for wavelengths greater than, say, 20 *h*.(Rose & Wang 2004, p. 154)

In fact, as we shall see, this statement can fail in relation to the low-frequency consistency of the two models, for the particular case of a structured plate that develops a complete low-frequency band gap for flexural waves. The low-dimensional plate theories are important for a wide range of practical applications, including those in vibrational control, detection of damage and structural integrity (e.g. Rose & Wang 2010).

The development of Mindlin theory to the cases of a circular cylindrical cavity or a rigid inclusion in an elastic plate, was considered by Pao & Chao (1964) and Lu (1966), respectively. A more comprehensive treatment of circular inhomogeneities in the framework of Mindlin’s theory has been described by Vemula & Norris (1997), following on their comparable work for Kirchhoff theory (Norris & Vemula 1995). The construction of Green’s function by Rose & Wang (2004, 2010) is particularly important in the context of the present paper.

The present authors Movchan *et al.* (2007, 2009) and McPhedran *et al.* (2009), following on the work of Evans & Porter (2007), developed the theory of quasi-periodic flexural waves propagating in a thin elastic plate perforated with a doubly periodic array of circular cavities. A striking conclusion was that, in the case where the cavities had clamped boundaries, the structure develops a low-frequency band gap in its dispersion diagram. Importantly, this band gap persisted, even when the radius of the cavities tended to zero. In this paper, we will develop Mindlin theory for the same problem, and use the asymptotic results for this low-frequency band gap as a decisive test for the consistency of the two models.

The structure of the paper is as follows. Section 2 includes the governing equations for both classical and Mindlin theories, together with the corresponding Green functions. The two Green functions behave differently when the field point approaches the source point, with one remaining finite and the other diverging logarithmically. In §3, we discuss the formulation for Bloch–Floquet waves in structured plates, based on the Rayleigh multi-pole method of Movchan *et al.* (2007). Section 4 contains the derivation of the low-frequency band gap, in the case where the radius of the inclusions tends to zero. We find a clear difference between the predictions of Kirchhoff and Mindlin theories, which we correlate with the contrasting behaviour of the corresponding Green functions. Section 5 includes numerical examples and three-dimensional numerical simulations compared with both Kirchhoff and Mindlin theories.

## 2. Basic representations of displacements and potentials in unstructured plates

We consider a time-harmonic motion, with radian frequency *ω*. The Mindlin framework is used here, and accordingly the displacement is represented in the form (similar to Vemula & Norris 1997)
2.1
where
2.2
and
2.3

where *A*_{1}, *A*_{2} are frequency-dependent coefficients, **e**_{z} is the unit basis vector along the *z*-axis and *W*_{j},*j* = 1,2, and *V* **e**_{z} are scalar and vector potentials, respectively. In the next section, we shall follow the multi-pole method, and there the Bessel function basis will be introduced and the multi-pole expansions will be written for components of equations (2.1) and (2.3).

In the framework of Mindlin’s approximation (Mindlin 1951), a single fourth-order partial differential equation can be derived for *w* by eliminating *Ψ*_{x},*Ψ*_{y}, in the form
2.4
Here,
where *E* is the Young modulus, *h* is the plate thickness, *ν* is the Poisson ratio, *ρ* is the mass density, *G* = *α*^{2}*μ* is the normalized shear modulus, *μ* = *E*/(2(1 + *ν*)) and *α* is Mindlin’s normalization constant (denoted by Mindlin as *κ*). Following Mindlin (1951), we use *α*^{2} = *π*^{2}/12 in the calculations discussed below. The one-dimensional analogue of equation (2.4) represents the Timoshenko beam theory (Timoshenko 1937). The terms corresponding to the derivative (*ρ*/*G*)(∂^{2}/∂*t*^{2}) are referred to as ‘transverse shear-deformation’ terms. If they are omitted, then equation (2.4) is reduced to the form
2.5
The other group of terms corresponds to the derivative (*ρh*^{3}/12)(∂^{2}/∂*t*^{2}). They are referred to as ‘rotatory inertia’ terms, and if they are omitted, then equation (2.4) is reduced to the form
2.6
When both groups (rotatory inertia terms and the transverse shear-deformation terms) are omitted, we arrive at the classical Kirchhoff approximation
2.7

Assuming that both transverse shear and rotational inertia are taken into account, the functions *W*_{1},*W*_{2} and *V* from equations (2.2) and (2.3) satisfy (see Vemula & Norris 1997) the Helmholtz equations
2.8
and
2.9

where
2.10
and
2.11
The constants used in the above equations are defined by
2.12
2.13
The coefficients *A*_{1},*A*_{2} in (2.3) have the representations
2.14

For the set of parameters chosen, the quantities *k*_{2} and *k*_{3} are pure imaginary up to the cut-off frequency where and *k*^{2}_{3} pass through zero. Vemula & Norris (1997) pointed out that the Mindlin model breaks down as *ω* approaches this cut-off frequency. The remaining wavenumber *k*_{1} everywhere takes a real positive value. Hence, equations (2.8) and (2.9) identify three types of waves, which characterize a dynamic response of Mindlin’s plates. Two of these waves are evanescent below the cut-off frequency, whereas the third is a propagating wave. A necessary condition for the Mindlin model to agree with the classical Kirchhoff model is that *k*_{2}∼i*k*_{1} and |*k*_{3}|≫|*k*_{2}|.

In figure 1, we plot the graphs of *k*_{1} and |*k*_{j}|, *j* = 2,3, as functions of *ω*. The data used for the computation is *E* = 2.0 × 10^{11} Pa, *ν* = 0.2, *h* = 0.254 m, , *ρ* = 7.8 × 10^{3} kg m^{−3}, m s^{−1}, m s^{−1}, *D* = *Eh*^{3}/(12(1−*ν*^{2})) N m, and the material in question is steel. As seen in figure 1, for small values of the frequency *ω*, we have |*k*_{2}|∼|*k*_{1}| and |*k*_{3}|≫|*k*_{2}|, which corresponds to the regime where Mindlin and Kirchhoff theories agree for the case of unstructured plates. We will see in later sections that while these conditions are necessary, they are not sufficient for structured systems, where additional lengths are introduced into the problem, entailing the possibility of additional constraints.

Dynamic Green’s functions for Mindlin’s theory were discussed by Rose & Wang (2004), and Evans & Porter (2007) included the discussion of dynamic Green’s functions for the Kirchhoff model. For Mindlin’s plates, a Green’s matrix giving the response to a point force and point moments has been derived by Rose & Wang (2004; see their appendix). In cylindrical coordinates, for the differential operator corresponding to equation (2.4), the relevant (33)-component of this matrix is given by
2.15
This function is singular at the origin, with the leading term in its expansion being
In contrast, Green’s function for the Kirchhoff model is not singular, and in cylindrical coordinates, as given by Evans & Porter (2007), has the form (with appropriate scaling)
2.16
The real part of this expression varies with *r* as

## 3. Boundary conditions and field identities for structured systems

We now consider the case of an elastic plate that has a doubly periodic array of cylindrical inclusions or cavities of circular cross section. We will be interested in finding the dispersion equation of modes obeying the Bloch condition, together with appropriate boundary conditions on the surfaces of the inclusions or cavities. Our previous studies (Movchan *et al.* 2007, 2009; McPhedran *et al.* 2009) have shown that interesting dispersion features arise in the case of inclusions with clamped boundaries, such as a band gap at low frequencies. For this reason, we concentrate on the case of circular cavities of radius *a* with clamped boundaries. In terms of potentials *W*_{1},*W*_{2} and *V* , the boundary conditions are written as follows:
3.1
3.2and
3.3
The functions *W*_{1},*W*_{2} and *V* are represented in terms of the Bessel function series
3.4
3.5and
3.6
These expansions are written for frequencies lying below the cut-off of the second and third wave types (figure 1), i.e. and *k*^{2}_{3} < 0.

The direct substitution of the expansions (3.4)–(3.6) into the boundary conditions (3.1)–(3.3) yields the following matrix relation for the coefficients: 3.7 Here, 3.8 where 3.9and 3.10

The Bloch–Floquet conditions and the periodic arrangement of cylinders allow the multi-pole coefficients on all cylinders to be related to those of the central cylinder using a Bloch factor . Here, **k**_{0} is the Bloch vector and **R**_{p} is the vector linking the origin to the centre of the *p*th cylinder (where *p* is the multi-index of dimension 2).

The application of Graf’s addition theorem (see Abramowitz & Stegun 1965) enables the Rayleigh identities to be derived in the form
3.11
3.12and
3.13
Here, and *S*^{K}_{l−n} are the lattice sums; a brief account of their properties and their numerical evaluation has been given in Movchan *et al.* (2007).

We write equations (3.11)–(3.13) in the matrix form
3.14
Combining equations (3.7) and (3.14), we deduce
3.15
We collect together equations (3.15) for *n* running over a truncated set of integers, and obtain a matrix **S** + **M**. The bands of the doubly periodic system of clamped cylinders, for a given value of *a*, then correspond to the equation
3.16

## 4. Low-frequency dispersion relations for clamped inclusions

For the case of inclusions of small radius *a*, we would like to analyse the low-frequency asymptotics of dispersion relations and hence consider the monopole approximation of the Rayleigh system (3.15). The monopole approximations have proved to be efficient for rigid inclusions of small radii (McPhedran *et al.* 2009), and the related convergence properties in platonic crystal band structures have been analysed in Poulton *et al.* (in press).

The matrix **ℳ**_{0}, characterizing the boundary conditions, is defined by
4.1
where
and
In the monopole approximation, it has the block-diagonal shape, i.e.
4.2
where is a 2 × 2 matrix given by
4.3
with
The dispersion equation is obtained by setting the determinant of the matrix
to zero. The term corresponding to the (33)-element of **ℳ**_{0} is of order *O*(1/*a*^{2}) as while the lattice sum *S*^{K}_{0} is independent of *a* and does not diverge as since |*k*_{3}| does not vanish as (figure 1). It follows that the dispersion equation of the lowest frequency mode is determined by the requirement that for the 2 × 2 matrix
4.4

After straightforward but extensive analytical rearrangements involving expansions of the matrix function and of the Bessel functions therein for small values of *a*, equation (4.4) can be written in the form
4.5

where we have used the relation, which holds for real values of *k*_{1},

In the region of small values of *a*, the dispersion equation (4.5) is inconsistent with the Kirchhoff theory, and it contains additional logarithmic and constant terms. The dispersion properties of the lowest band depend on the radius *a* of small inclusions unless *A*_{1} = *A*_{2} and *k*_{1} = |*k*_{2}|. However, in the framework of Mindlin’s theory,

When both transverse shear and rotational inertia are neglected, as in the classical Kirchhoff theory, the dispersion equation (4.5) reduces to
4.6
as derived in Movchan *et al.* (2007).

## 5. Numerical comparisons and low-frequency band gaps

In figure 2, we compare the asymptotic approximation of formula (4.5) for **k**_{0} = (0.40,0) with the solution of the monopole equation (3.16) and find excellent agreement for radii *a* up to 0.018. We also give in figure 2, the results obtained in the commercial package COMSOL of its Mindlin plates module. These pure numerical results validate the accuracy of both the monopole approximation and its asymptotic development. We note that the monopole approximation is particularly accurate near **k**_{0} = **0**, since in that region, multi-pole coefficients of orders not divisible by four are zero.

In figure 3, we show the *k*_{0y} = 0 section of the lowest band in the Mindlin flexural wave dispersion diagram (obtained in COMSOL) for various values of the radius *a*, which shows reduction of the minimum frequency as *a* decreases. This is consistent with computations, presented in figure 2, showing the tendency of the minimum frequency point on the lowest band to tend towards zero in the inverse logarithmic fashion. The corresponding Bloch mode (for **k** = **0** and the minimum frequency value) is shown in figure 4 for a plate containing a doubly periodic square array of clamped holes of small radius *a* = 10^{−3} m, with a separation of 1 m between neighbouring holes. The computation has been produced using the Mindlin plates module of COMSOL; compared with the classical Kirchhoff model, it is apparent that Mindlin’s model leads to the development of a logarithmic singularity in the displacement when the radius of the clamped hole tends to zero. This feature is directly linked to the statement of closure, at an inverse logarithmic rate, of the low-frequency band gap in the limit configuration of a doubly periodic array of zero radius rigid pins.

Thus, we have seen that in Mindlin theory, the lowest band for a perforated and clamped plate has no radius-independent region, and, as we have seen in figure 3, the lowest band goes down to *ω* = 0 as *a* tends to zero. By contrast, in Kirchhoff theory, the lowest band becomes independent of radius, and it leaves a wide band gap in the low-frequency region.

Finally, we point out that the asymptotics of the lowest band in Mindlin plate problems with clamped cavities is strongly analogous to the behaviour observed in electromagnetism for arrays of perfectly conducting cylinders (see Nicorovici *et al.* 1995 and Pendry *et al.* 1996).

## Acknowledgements

R.C.M. acknowledges the support of the Australian Council through its Discovery Grants Scheme, and of the Liverpool University Research Centre in Mathematics and Modelling.

- Received July 19, 2010.
- Accepted August 31, 2010.

- © 2010 The Royal Society