Semi-analytical Formulation for Guided Wave Propagation

Modeling wave propagation in damped waveguides of arbitrary cross-section

This presentation describes methods of using an approximate elasticity formulation to compute natural wave numbers of waves in coated cylindrical shells. The efficient computation of these wave numbers is critical to coating design. A previously developed displacement-based variational formulation for coated shells [J. G. McDaniel and J. H. Ginsberg, J. Appl. Mech. 60, 463–469 (1993)] retains the accuracy of analytical formulations, but avoids the computational burdens associated with special functions of complex argument. This formulation, which was previously applied to two-dimensional problems, is extended to address wave propagation in the axial coordinate for each circumferential harmonic. Because the formulation is energy based, one has ready access to the strain energy distributions of each wave. For a specified real axial wave number, one obtains an easily solvable generalized eigenvalue problem for complex natural frequency. For a specified real frequency, the search for complex axial wave number is iterative. A fast algorithm for finding the complex axial wave numbers, which facilitates the design process, will also be discussed.

This paper gives insight to the ultrasonic wave propagation in arbitrary cross section waveguides such as a rail. Due to the geometrical complexity of the rail cross section, the analytical solution to the wave propagation in the rail is not feasible. A Semi Analytical Finite Element method is described as an alternative yet still robust approach to get the solution of the problem. The free-vibration solution (unforced) and the forced solution to a laser excitation, are shown for the case of an undamped rail up to a frequency of 500 kHz. The effects of different loading patterns are discussed, while experimental results are provided. A mode selection is performed, in accordance to the sensitivity of each mode to the different types of defect that can occur in a rail.

The enormous potential of Guided waves have a significant use for non-destructive evaluation(NDE) and structural health monitoring(SHM). It is common to use different solution approaches to predict dispersion curves depending on the material type, isotropic, transversely isotropic, or orthotropic, of the medium in which the wave propagates, shape of the waveguide, etc. Inability to model waveguides of arbitrary cross-section by existing matrix methods has led to development of Semi-Analytical Finite Element(SAFE) Method. In this study, rapid and accurate prediction of the propagation characteristics for guided waves in a wide range of structures has been performed by implementation of SAFE method. Damped and undamped waveguides of different shapes can be modelled. Wave propagation characteristics in both isotropic and anisotropic composite structures can also be studied. To benchmark the solution, a case study using Transfer Matrix Method (TMM) for composite plate has been evaluated. It was found that the element discretizaton plays an important role in accurate prediction of modes in SAFE method, neglecting which leads to loss of higher order modes.

The existing researches have indicated the conditions that mode pair of fundamental mode and second harmonic mode need to satisfy to produce the cumulative second harmonic. However, due to the multimodal characteristics of ultrasonic guided waves, the conditions are very complex. In this article, a numerical simulation method is proposed to select the optimal mode pair that can generate the cumulative second harmonic mode with the largest amplitude in waveguide of arbitrary cross-section. The optimal mode pair is determined by calculating and comparing the amplitude coefficients of different mode pairs based on mode shape analysis. The CHN60 rail is taken as an example for simulation verification, and the optimal mode pair of nonlinear guided waves in rail is selected.

PROPAGATING WAVES AND END MODES IN PRETWISTED BEAMS

This paper presents a graphical user interface (GUI) for modeling ultrasonic guided wave propagation in elastic solids. The software exploits the semi-analytical finite element (SAFE) method for the calculation of wave-propagation characteristics. The interface allows for the modeling of piezoelectric effects in plate-like and arbitrary cross-sectional waveguides. The isotropic and anisotropic materials with damping effects are also considered. For anisotropic composite material cases, directivity plots can be extracted, containing the phase-velocities, group velocities, and slowness curves. The frequency-dependent mode shapes can also be obtained, including displacement, strain, stress, and other electric components for piezoelectric materials. The corresponding mode shapes for arbitrary cross-sectional waveguides are presented in the form of vivid animations, demonstrating the cross-sectional harmonic motions. All the computational outcomes are compared with commercial finite element (FE) codes via the Bloch-Floquet boundary conditions. The paper finishes with discussion, concluding remarks, and suggestions for future work.

A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides

Our existing Ultrasonic Broken Rail Detection system detects complete breaks and primarily uses a propagating mode with energy concentrated in the head of the rail. Previous experimental studies have demonstrated that a mode with energy concentrated in the head of the rail, is capable of detecting weld reflections at long distances. Exploiting a mode with energy concentrated in the web of the rail would allow us to effectively detect defects in the web of the rail and could also help to distinguish between reflections from welds and cracks. In this paper, we will demonstrate the analysis of a piezoelectric transducer attached to the rail web. The forced response at different frequencies is computed by the Semi-Analytical Finite Element (SAFE) method and compared to a full three-dimensional finite element method using ABAQUS. The SAFE method only requires the rail track cross-section to be meshed using two-dimensional elements. The ABAQUS model in turn requires a full three-dimensional discretisation of the rail track. The SAFE approach can yield poor predictions at cut-on frequencies associated with other modes in the rail. Problematic frequencies are identified and a suitable frequency range identified for transducer design. The forced response results of the two methods were found to be in good agreement with each other. We then use a previously developed SAFE-3D method to analyse a practical transducer over the selected frequency range. The results obtained from the SAFE-3D method are in good agreement with experimental measurements.

Chapter 8 - Guided-wave excitation in aerospace composites

This paper studies the feasibility of using ultrasonic guided waves to rapidly inspect tubes and pipes for possible eccentricity. While guided waves are well established in the long range inspection of structures such as pipes and plates, studies for more complex cross sections are limited and analytical solutions are often difficult to obtain. Recent developments have made the Semi Analytical Finite Element (SAFE) method widely accessible for researchers to study guided wave properties in complex structures. Here the SAFE method is used to study the effect of eccentricity on the modal structures and velocities of lower order guided wave modes in thin pipes of diameters typically of interest to the industry. Results are validated using experiments. The paper demonstrates that even a small eccentricity in the pipe can strongly affect guided wave mode structures and velocities and hence shows potential for pipe eccentricity inspection.

The properties of acoustic surface waveguides are reviewed, with particular reference to topographic structures in which guiding is achieved by drastic deformation of the substrate surface. A numerical technique, capable of computing efficiently and with high accuracy the mode spectrum of an anisotropic piezoelectric heterogeneous waveguide of arbitrary cross section, is described. Characteristics of both the ridge guide and the recently discovered wedge waveguide are discussed in some detail. Techniques for the fabrication of and transduction onto acoustic surface waveguides are discussed, and a preliminary assessment is made of potential linear and nonlinear waveguide applications. A number of experimental devices are described.

Interaction of two counterpropagating coherent transverse-electric electromagnetic waves in the absorbing plate, placed in a waveguide of arbitrary cross section, is considered. It is supposed, that the waves with various initial phases incident on the boundaries of the plate from two sides. The analytical expression for the power transmission interference coefficient is found. Some physical features of tunnel interference in a plate are revealed.

This paper investigates the elastic wave propagation through soft materials that are being subjected to finite deformations. The nonlinear elastic and linearized incremental theories have been exploited to formulate governing wave equations and elastic moduli in Lagrangian space. Semi-analytical finite element (SAFE) method, a numerical approach has been formulated for computing dispersive relations of guided waves in compressible hyper-elastic plates. This framework requires finite element discretization of the cross section of the waveguide and harmonic exponential function assumes the motion along the wave propagation direction. Here, explicit phase velocity results have been shown for soft materials with a prominent stiffening effect by employing the Gent model, and these results are analyzed for elastic wave propagation through compressible materials. It has been noticed that Lamb waves have a strong dependence on the frequency-thickness product, prestretch, and direction of wave propagation. Moreover, with the strain stiffening effect, the dependence becomes stronger, especially for fundamental symmetric and anti-symmetric modes. The numerical results display that at certain prestretch the Gent material encounter snap-through instability resulting from geometrical and material nonlinearities. The influence of material properties like Gent constant and direction of wave propagation on snap-through instability has been discussed. The proposed SAFE framework reveals that finite deformations can affect elastic wave propagation through stiffness and compressibility.

The propagation characteristics of guided waves (GW) are related to both the material, loading and geometry parameters. Correct understanding of GW is the key step towards further GW-based structural health monitoring. A dispersion calculator of GW propagating in the complex geometry with viscoelastic and anisotropic material parameter in the open waveguide is developed based on the semi-analytical finite element (SAFE) method and perfectly matched layer (PML). By assuming the simple harmonic motion along the propagation direction and adopting the mesh discretization in the cross section, SAFE features the capability of GW calculation in the complex waveguide at only a cost of limited calculation resources. Hence, besides the capabilities of the most off-the-shelf tools focusing on the analytical solution of GW in simple plate or pipeline, the developed algorithm further offers the solution of GW in open waveguide with laminates of arbitrary ply stacking sequences and in arbitrarily complex cross section. Quadratic one-dimensional and triangular elements are adopted to automatically discretize the plate structure and complex cross section, respectively. The obtained results of GW in plates are in a good agreement with the results obtained with the off-the shelf-tools that adopt analytical derivations, and show stronger robustness of mode searching and sorting by avoiding the complex root-finding procedure required in the analytical derivation. The most GW features, including phase velocity, group velocity, attenuation, wave structure in terms of displacement, stress, and strain, and animation of wave propagation are directly calculated. All these functions are integrated and updated in the previously developed freeware SAFEDC, whose 1st version SAFEDC v1.0 was published in September 2023.