Flexural wave dispersion in orthotropic plates with heavy fluid loading Open Access

The bending rigidity of a composite plate may easily vary by an order of magnitude in its principal directions, even for classical composite materials. When the plate is in vacuo, this naturally causes substantial variations in the flexural wavenumber as the direction of wave propagation varies. The present work is concerned with the effects of fluid loading on these variations. Understanding this effect is expected to provide insights into the structural acoustics of finite orthotropic plates (Anderson and Bratos-Anderson, 2005), curved orthotropic structures (Ghinet and Atalla, 2005), and musical instrument models that involve thin orthotropic structures (Derveaux et al., 2003).

Previous work by Fay (1948) on the propagation of flexural waves in fluid-loaded isotropic plates provides a foundation for the present work. Fay found that the dispersion relation for flexural waves involves a tenth-order polynomial in wavenumber. Further analysis at frequencies below coincidence revealed the existence of an unattenuated flexural wave that propagates with a subsonic sound speed. Fahy (1989) provided a simplified dispersion relationship below coincidence in which the effective mass of the fluid-loaded plate is a sum of the plate mass and the mass of a fluid whose volume is determined by the flexural wavenumber.

The present work derives the dispersion relation for a fluid-loaded orthotropic plate, following the derivation for the isotropic plate given by Fahy wherever possible. Next, a convenient method is presented for numerically computing the coefficients of the resulting tenth-order polynomial. The method involves the convolution of various polynomial coefficients, which avoids extensive algebraic manipulation. At frequencies below coincidence, a simplified dispersion relationship is found that is similar to that provided by Fahy. Examples involving classic composite materials indicate that the plate mass competes with the fluid mass for parameters of practical interest.

We begin by considering the equation of motion for an infinite orthotropic plate in contact with an acoustic fluid on one side (Leissa, 1969),

where $w$ is the normal displacement and the coordinates $x$ and $y$ lie in the plane of the plate. This equation assumes that the plate thickness, $h$⁠, is small relative to a flexural wavelength. The variables $Dx$ and $Dy$ represent the flexural stiffness of the plate in the $x$ and $y$ directions, respectively, and $Dxy$ represents the “bending-torsional stiffness” of the plate (Anderson and Bratos-Anderson, 2005). Note that some authors, such as Timoshenko and Woinowsky-Krieger (1959), define the variable $Dxy$ differently.

Derivation of the dispersion relation for the fluid-loaded orthotropic plate will follow a similar derivation by Fahy for isotropic plates, in which wave impedances for the plate and fluid act in mechanical parallel. Substitution of traveling wave representations for displacement and applied traction, $w=R{Wexp[i(ωt−kxx−kyy)]}$ and $f=R{Fexp[i(ωt−kxx−kyy)]}$⁠, into Eq. (1) yields the wave impedance of the plate

Similarly, substitution of a pressure field of the form $p=R{Pexp[i(ωt−kxx−kyy−kzz)]}$ into Euler’s equation yields the acoustic wave impedance of the fluid,

and the acoustic wave equation provides a relationship between the wavenumbers,

Finally, the dispersion relations are obtained by adding the impedances that are in mechanical parallel in the absence of excitation, $Zp+Zfl=0$⁠, or

This polynomial is fifth order in $kx2$ when $ky2$ is specified. Only one of the roots represents a traveling flexural wave that is unattenuated when damping is not present. This wave is the sole focus for the remainder of this paper. For discussions of the other four roots, in the context of isotropic plates, the reader is referred to the paper by Fay.

In numerical simulations, it is convenient to numerically compute the coefficients of this polynomial by convolving vectors of coefficients whenever two polynomials are multiplied. This allows Eq. (5) to be written as $p(kx2)=0$⁠, where $p(kx2)$ is a fifth-order polynomial in $kx2$⁠. Its vector of coefficients, $p$⁠, is given by

where $∗$ denotes a convolution defined by

The vectors are obtained by inspection of Eq. (5),

In this way, the coefficients $p1–p3$ are computed from the above equations and the polynomial coefficients are computed from Eq. (6). This avoids extensive algebraic manipulations to analytically solve for the coefficients of the polynomial.

In this section, we investigate flexural wave dispersion at low frequencies, where the inertia of the fluid dominates its compressibility. For simplicity in the derivation, let a coordinate $x′$ be defined in the propagation direction of a flexural wave in an orthotropic plate. Since the displacement is independent of all other orthogonal coordinates, the equation of motion of the plate with fluid loading on one side assumes the form

where $Dx′$ is algebraically related to $Dx$⁠, $Dy$⁠, and $Dxy$⁠, as well as the angle between $x$ and $x′$⁠. Details of this relationship are given by Leissa (1969) [see Eq. (9.25) and following text].

Assuming a traveling wave of the form

and following the derivation in the previous section results in the dispersion

Introducing the low-frequency approximation

simplifies the dispersion relation in Eq. (13) to

which has the same form as the corresponding result for the isotropic plate given in Fahy.

As in the case of the isotropic plate, the fluid loading represents an added mass that is inversely proportional to the fluid-loaded wavenumber. Larger flexural wavelengths will be increased by smaller amounts. For this reason, fluid loading is expected to make flexural waves less directive in orthotropic plates. The overall magnitude of this effect will depend on the relative values of $ρs$ and $ρ0∕(kx′h)$⁠. Noting that classical plate theory is generally valid when $kx′h<1$ leads to the expectation that the fluid mass will often dominate the plate mass at low frequencies, as noted by Fahy.

The effects of fluid loading were numerically investigated for a plate constructed of graphite-epoxy with thickness $h=0.01m$⁠. Material properties are given in Table I. Note that this material is at least an order of magnitude stiffer in the $x$ direction than in the other directions. The fluid was taken as water with a sound speed $c0=1500m∕s$ and density $ρ0=1kg∕m3$⁠. The wave number $kx$ corresponding to the propagating flexural wave was numerically computed from Eq. (5) for chosen values of $ky$⁠.

Results for a graphite-epoxy plate at frequencies of $1kHz$ and $10kHz$ are shown in Figs. 1 and 2. For comparison, results for the in vacuo case are plotted, as well as results for a similar isotropic plate whose properties are assumed to be those of the orthotropic plate in the $x$ direction. These figures indicate a general magnification of the orthotropic wavenumbers due to fluid loading, however the variation with propagation angle is not significantly distorted when compared to the isotropic plate. This finding was more closely investigated in Table II, which contains the percent increase in the flexural wavenumber due to fluid loading. This table supports the general conclusion that, for the cases studied, fluid loading does not distort the angular dependence of the flexural wave.

An analysis of the propagating flexural wave in a fluid-loaded orthotropic plate has been developed in order to better understand the effects of fluid loading. A method for numerically computing the coefficients of the resulting polynomial allows for direct calculation of the polynomial coefficients and hence the wavenumber. This work is expected to be important in numerically computing dispersion relations for more complicated geometries, such as cylindrical shells, where analytical derivation of the dispersion relation is difficult or impossible. When a single propagation angle is considered, this work shows that the equation of motion has the same form as that of an isotropic plate. This result reveals the underlying physics of flulid loading, particularly in the low-frequency regime where the fluid is primarily inertial and is likely to compete with the plate inertia.

The authors would like to acknowledge funding by ONR under Grant No. N000l4-05-1-0102 and NUWCNPT under the NUWCNPT In-House Laboratory Independent Research (I.L.I.R.) Program.