On approximating guided waves in plates with thin anisotropic coatings by means of effective boundary conditions

On the Bövik-Benveniste methodology and related approaches for modelling thin layers

This paper reviews several leading approaches for asymptotic modelling of thin layers in elastostatics and wave propagation phenomena. The issues related to applications of the so-called 'equivalent' or 'effective' boundary conditions and their interpretations are highlighted. Comparative analysis of asymptotic models is performed for a two-dimensional elastostatic case using a novel complex variables-based modelling tool. Its implementation allows for straightforward derivations of higher order boundary conditions for problems with layers of arbitrary sufficiently smooth curvatures. Explicit expressions for the conditions up to the third order are provided. All models are tested using available benchmark solutions and the solutions for the limiting cases of the layer parameters. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Predicting bone strength with ultrasonic guided waves

Recent bone quantitative ultrasound approaches exploit the multimode waveguide response of long bones for assessing properties such as cortical thickness and stiffness. Clinical applications remain, however, challenging, as the impact of soft tissue on guided waves characteristics is not fully understood yet. In particular, it must be clarified whether soft tissue must be incorporated in waveguide models needed to infer reliable cortical bone properties. We hypothesize that an inverse procedure using a free plate model can be applied to retrieve the thickness and stiffness of cortical bone from experimental data. This approach is first validated on a series of laboratory-controlled measurements performed on assemblies of bone- and soft tissue mimicking phantoms and then on in vivo measurements. The accuracy of the estimates is evaluated by comparison with reference values. To further support our hypothesis, these estimates are subsequently inserted into a bilayer model to test its accuracy. Our results show that the free plate model allows retrieving reliable waveguide properties, despite the presence of soft tissue. They also suggest that the more sophisticated bilayer model, although it is more precise to predict experimental data in the forward problem, could turn out to be hardly manageable for solving the inverse problem.

A long-wave model for the surface elastic wave in a coated half-space

The paper deals with the three-dimensional problem in linear isotropic elasticity for a coated half-space. The coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long-wave limit. An explicit model is derived for the surface wave using the perturbation technique, along with the theory of harmonic functions and Radon transform. The model consists of three-dimensional ‘quasi-static’ elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two-dimensional hyperbolic equation singularly perturbed by a pseudo-differential (or integro-differential) operator. The latter equation governs dispersive surface wave propagation, whereas the elliptic equations describe spatial decay of displacements and stresses. As an illustration, the dynamic response is calculated for impulse and moving surface loads. The explicit analytical solutions obtained for these cases may be used for the non-destructive testing of the thickness of the coating and the elastic moduli of the substrate.

Dispersion equations for steady waves in an anisotropic elastic plate or a layered plate

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.

Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies

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.

High order approximate low frequency theory of elastic anisotropic lining and coating

A problem of the dynamic behavior of an elastic layer coupled to one or two thick elastic solids is considered. All the materials may possess a general anisotropy and the layer is assumed to be thin enough with respect to the characteristic wavelength. Introducing the asymptotic power series with respect to the thickness-over-wavelength ratio for the main quantities and using the asymptotic integration method the displacements and stresses on the layer surfaces are related. Thus, the so-called impedance boundary conditions (IBC) are deduced for three cases--for a coated substrate with given displacements or with given stresses on the surface and for two substrates with a layer in between. In contrast to previous papers these IBC are obtained for the most general situation with the asymptotic accuracy up to the sixth order, uniform with respect to the representation of the displacements and stresses. Presented theory can be used for studying the surface and interface phenomena as well as for calculating fields and spectra of layered solids. The results are validated numerically and compared with those of other authors.

An O ( h N ) interface model of a three-dimensional curved interphase in conduction phenomena

A thin isotropic three-dimensional curved interphase of thickness h between two isotropic media is considered in the setting of thermal conduction. This interphase is modelled by a surface between the two neighbouring media, and appropriate interface conditions on it are derived for the temperature and normal heat flux fields. The derivation makes use of Taylor expansions for the fields and is correct to O ( h N ), where h denotes the thickness of the interphase. The jumps for the temperature and normal heat flux in the interface model are given in terms of a hierarchy of surface differential forms, which depend on the conductivities of the interphase and surrounding media, and involve surface derivatives of the temperature and normal heat flux along the interface. The analysis is directly transferable to the analogous physical phenomena of electrical conduction, dielectrics, magnetism, diffusion, flow in porous media and anti-plane elasticity.

Ultrasonic wave propagation across a thin nonlinear anisotropic layer between two half-spaces

Boundary conditions and perturbation theory are combined to create a set of equations which, when solved, yield the reflected and transmitted wave forms in the case of a thin layer of material that is perfectly bonded between two isotropic half-spaces. The set of perturbed boundary conditions is created by first using the fully bonded boundary conditions at each of the two interfaces between the thin layer and the half-spaces. Then, by restricting the layer's thickness to be much smaller than an acoustic wavelength, perturbation theory can be used on these two sets of boundary equations, producing a set of equations which effectively treat the thin layer as a single interface via a perturbation term. With this set of equations, the full range of incident and polar angles can be considered, with results general enough to use with a layer that is anisotropic, nonlinear, or both anisotropic and nonlinear. Finally the validity of these equations is discussed, comparing the computer simulation results of this theory to results from standard methods, and looking at cases where the results (or various properties of the results) are known or can be predicted.

Elastic guided waves in a layered plate with rectangular cross section

Guided waves in a layered elastic plate of rectangular cross section (finite width and thickness) has been studied in this paper. A semianalytical finite element method in which the deformation of the cross section is modeled by two-dimensional finite elements and analytical representation of propagating waves along the length of the plate has been used. The method is applicable to arbitrary number of layers and general anisotropic material properties of each layer, and is similar to the stiffness method used earlier to study guided waves in a laminated composite plate of infinite width. Numerical results showing the effect of varying the width of the plate on the dispersion of guided waves are presented and are compared with those for an infinite plate. In addition, effect of thin anisotropic coating or interface layers on the guided waves is investigated.

Theory of elastic wave propagation in anisotropic film on anisotropic substrate: TiN film on single-crystal Si

The delta-function representation of the elastodynamic Green's function is used to derive an expression for the elastic wave forms on the surface of an anisotropic thin film on an anisotropic substrate due to a point or a line source located at the surface of the film. The dispersion relation for surface acoustic waves (SAWs) is obtained from the poles of the Green's function. A computationally efficient algorithm is formulated to obtain the elastic constants and the density of the film from the SAW dispersion data. The theory is used to analyze measured SAW dispersion relations in a titanium nitride film on silicon. The analysis yields values of the elastic constants and the density of the film. Excellent agreement is obtained between the theoretical and experimental dispersion results. Calculated wave forms for the surface wave due to a pulsed line source on the surface of the film are reported.

Off-axis propagation of ultrasonic guided waves in thin orthotropic layers: theoretical analysis and dynamic holographic imaging measurement

The elastic properties of many materials in sheet or plate form can be approximated with orthotropic symmetry. In many sheet material manufacturing industries (e.g., the paper industry), manufacturers desire knowledge of certain anisotropic elastic properties in the sheet for handling and quality issues. Ultrasonic wave propagation in plate materials forms a method to determine the anisotropic elastic properties in a nondestructive manner. This work explores exact and approximate analysis methods of ultrasonic guided wave propagation in thin layers, explicitly dealing with orthotropic symmetry and propagation off-axis with respect to the manufacturing direction. Recent advances in full-field ultrasonic imaging methods, based on dynamic holography, allow simultaneous measurement of the plate wave motion in all planar directions within a single image. Results from this laser ultrasonic imaging approach are presented that record the lowest anti-symmetric (flexural) mode wavefront in a single image without scanning. Specific numerical predictions for flexural wave propagation in two distinctly different types of paper are presented and compared with direct imaging measurements. Very good agreement is obtained for the lowest anti-symmetric plate mode using paper properties independently determined by a third party. Complete determination of the elastic modulus tensor for orthotropic layers requires measurement of other modes in addition to the lowest anti-symmetric. Theoretical predictions are presented for other guided wave modes [extensional (S), flexural (A), and shear-horizontal (SH)] in orthotropic plates with emphasis on propagation in all planar directions. It is shown that there are significant changes in the dispersion characterization of these modes at certain frequencies (including off-axis mode coupling) that can be exploited to measure additional in-plane elastic moduli of thin layers. At present, the sensitivity of the imaging measurement approach limits experimental investigation to relatively large amplitudes easily produced by flexural wave motion (> 0.1 nm). Extension of the measurement range and application to other plate wave modes are in progress and shall be reported in future work.

On ultrasonic guided waves in a thin anisotropic layer lying between two isotropic layers

In this paper, dispersion of guided waves in a three-layered sandwich plate is considered. The focus here is on a configuration consisting of a thin anisotropic layer sandwiched between two identical isotropic layers. This configuration could model, for example, a superconducting tape where the middle layer is a brittle superconductor and the surrounding layers are isotropic and ductile. An approximate dispersion relation correct to O(h) governing the guided waves is obtained by expanding the field inside the thin middle layer in powers of the small thickness h of the layer. Numerical examples are given for two specific systems with superconducting middle layers. Some characteristic features, particularly at low frequencies, are investigated. Comparison between the exact and approximate dispersion relations are made to show that the approximation works well in the frequency interval of interest. The characteristic features may be useful for ultrasonic measurements of the anisotropic elastic constants of the thin layer.