ElasticMatrix: A MATLAB toolbox for anisotropic elastic wave propagation in layered media

Abstract

Simulating the propagation of elastic waves in multi-layered media has many applications. A common approach is to use matrix methods where the elastic wave-field within each material layer is represented by a sum of partial-waves along with boundary conditions imposed at each interface. While these methods are well-known, coding the required matrix formation, inversion, and analysis for general multi-layered systems is non-trivial and time-consuming. Here, a new open-source toolbox called ElasticMatrix is described which solves the problem of acoustic and elastic wave propagation in multi-layered media for isotropic and transverse-isotropic materials where the wave propagation occurs in a material plane of symmetry. The toolbox is implemented in MATLAB using an object oriented programming framework and is designed to be easy to use and extend. Methods are provided for calculating and plotting dispersion curves, displacement and stress fields, reflection and transmission coefficients, and slowness profiles.Keywords partial-wave method · global matrix method · elastic waves 1 Motivation and Significance Matrix models of wave propagation in multi-layered elastic solids have had a significant contribution to research areas such as acoustics, geophysics and electromagnetics. A few examples include: structural health monitoring [1], characterisation of interface bonding [2], detection of debonding in joints [3], measuring material properties [4], designing composite layered structures [5], mode sorting of guided waves [6], the physical interpretation of guided wave structures [7], modelling the directional response of Fabry-Pérot ultrasound sensors [8], reflection and transmission of plane waves [9], elastography of layered soft tissues [10], and ice detection on wind turbines [11].Matrix methods, in particular the partial-wave and global matrix method, represent the stress and displacement fields as a sum of partial-waves for each material of the layered-structure. Each partial-wave represents an upward or downward travelling (quasi-)compressional or (quasi-)shear wave. By invoking boundary conditions at the interfaces of adjacent layers, the partial-wave amplitudes and field properties of the first layer can be related to the last in the form of a 'global' matrix. The resulting matrix equation can be used in two different ways. Firstly, the roots of the equation can be found which give the modal solutions or dispersion curves. Secondly, a subset of partial-wave amplitudes can be defined and the remaining amplitudes solved for. This can be used to calculate the displacement and stress fields within the multi-layered structure when a plane wave is incident. This method will be discussed further in Section 2.Despite its usefulness, there are few available implementations of the partial-wave method. The current state-of-theart implementation is Disperse [12]. This software has been in development since 1990 and is primarily focused on calculating the dispersion solutions for multi-layered structures. The Disperse softwar...

Dichotomy property of dispersion equation of guided waves propagating in anisotropic composite plates

“… 𝑎 (𝑣, 𝜉) ≜ det{𝐃 𝑎 (𝑣, 𝜉)} = 0 for anti-symmetric modes (22) Observing Eqs. ( 19) and ( 21), it is certain that, no matter which SC is considered, 𝐃(𝑣, 𝜉) is a complex-valued matrix due to the existence of complex exponential terms.…”

“…The principal idea of GMM is to assemble all the submatrices representing the continuity conditions of field variables in each layer into a global matrix, along with all layers' wave amplitudes to be determined by the traction free boundary conditions. The pioneering software Disperse [21] and a toolbox implemented in MATLAB called ElasticMatrix [22] are based on GMM. The main merit of GMM rests on its numerical stability even at large 𝑓 𝑑 range but at the cost of heavy computational burden.…”

“…The main merit of GMM rests on its numerical stability even at large 𝑓 𝑑 range but at the cost of heavy computational burden. However, some problems still exist for GMM for some applications such as missing roots or the production of outliers for some mode [23], limited computational capability making that it can only cope with laminates containing no more than 64 layers [24], and the incomplete functionality that it only deals with isotropic and transversely isotropic materials [22], thus restricting the wave propagation direction along materials' principal axes.…”

“…A representative example is the pioneering software Disperse which employed a path-dependent minimal moduli search strategy combined with the convergence criteria of checking the included phase angle of the complex-valued characteristic function that should be greater than 90 • [21,29]. The toolbox ElasticMatrix makes use of MATLAB built-in function fminbnd to achieve the minimal moduli searching purpose [22]. Another complex strategy adopted by the software Dispersion Calculator is to check, in a small interval, both the occurrence of minimum moduli and sign change of the dispersion function (see Table I of [24]).…”

Detection and Imaging of Debonding in Adhesive Joints of Concrete Beams Strengthened with Steel Plates Using Guided Waves and Weighted Root Mean Square

“…The theoretical dispersion curves for the single-layer plate #5 were obtained in the way of solving well-known Rayleigh-Lamb equations in Matlab ® . In the case of the three-layer medium, the theoretical curves were calculated using an open-source toolbox called Elastic-Matrix [59], based on the partial-wave method. Although the proposed algorithm is stated to be inaccurate for leaky cases, the appropriate solution can be obtained by selecting appropriate calculation parameters.…”