Modelling of wave propagation in composite plates using the time domain spectral element method (original) (raw)
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Modeling wave propagation in moderately thick rectangular plates using the spectral element method
Applied Mathematical Modelling, 2015
This paper presents development of the spectral element method (SEM) to specify natural frequencies and dynamic response of moderately thick rectangular plates under impact and moving loads. To solve differential equations of moderately thick plate, the displacement field has been expressed in the frequency domain using Fast Fourier Transformation (FFT) algorithm while considering simple boundary conditions for two parallel edges. Closed-form solutions have been derived for the differential equations in frequency domain. Driving exact shape functions of plate in frequency domain, the dynamic solution in time domain has been calculated using Inverse Fast Fourier Transformation (IFFT). In this study, natural frequencies for moderately thick plates with variable and constant thicknesses have been calculated and compared to the past research results. Mode shapes of plates with various boundary conditions have been plotted. Moreover, plate's displacements under impact and moving loads have been calculated using developed SEM. The utilization of a minimum numbers of elements in SEM, consequently leading to a considerable decrease in computational costs, is the main advantage of this method.
A spectrally formulated plate element for wave propagation analysis in anisotropic material
Computer Methods in Applied Mechanics and Engineering, 2005
A new spectrally formulated plate element is developed to study wave propagation in composite structures. The element is based on the classical lamination plate theory. Recently developed method based on singular value decomposition (SVD) is used in the element formulation. Along with this, a new strategy based on the method of solving polynomial eigenvalue problem (PEP) is proposed in this paper, which significantly reduces human intervention (and thus human error), in the element formulation. The developed element has an exact dynamic stiffness matrix, as it uses the exact solution of the governing elastodynamic equation of plate in frequency-wavenumber domain as the interpolating functions. Due to this, the mass distribution is modeled exactly, and as a result, a single element captures the exact frequency response of a regular structure, and it suffices to model a plate of any dimension. Thus, the cost of computation is dramatically reduced compared to the cost of conventional finite element analysis. The fast Fourier transform (FFT) and Fourier series are used for inversion to time-space domain. This element is used to model plate with ply drops and to capture the propagation of Lamb waves.
Waveguides of a Composite Plate by using the Spectral Finite Element Approach
Journal of Vibration and …, 2009
This work presents the extension of an existing procedure for evaluating the waveguides and the dispersion curves of a laminate made up of thin orthotropic composite plates arbitrarily oriented. The adopted approach is based on one-dimensional finite-element mesh throughout the thickness. Stiffness and mass matrices available in the literature for isotropic material are reported in full expanded form for the selected problem. The aim of the work is the development of a tool for the simulation of the most common composite materials. The knowledge of the wave characteristics in a plate allows correct sizing of the numerical mesh for the frequency-dependent analysis. The development of new stiffness matrices and the analysis for different heading angles are detailed to take into account the general anisotropic nature of the composite. The procedure concerns a standard polynomial eigenvalue problem in the wavenumber variable and is focused on the evaluation of the dispersion curves for all the propagating waves within the materials. A comparison with an analytical approach is also shown in the results using the classical laminate plate theory (CLPT). However, limits of CLPT are outlined and spectral finite element method can be successfully used to overcome such limitations.
Engineering Transactions, 2023
Graphite-epoxy composites have been able to meet the multiple requirements of the space industry. However, the radiation from the spatial environment and non-perfect adhesion between the fibers and the matrix can lead to the appearance of imperfections. To handle this, we use non-destructive testing by ultrasonic guided waves known for its high accuracy in detecting defects. In this article, we study the propagation of ultrasonic guided waves in a graphite-epoxy composite plate by the spectral method. First, the mathematical formalism is explained for modeling guided waves in the composite material. Next, we plot the dispersion curves of the composite plate in different orientations of the fibers with a MATLAB program and the results are compared with those of the DISPERSE software. These give us information on the modes that propagate in the structure. We elaborate and explain a technique based on displacement symmetry to distinguish between the different modes. A discussion based on time-saving and accuracy is established to show the advantages of the method. The second part of our paper consists in giving a physical meaning to the spectral displacements normalized in amplitude. We propose to normalize the spectral eigenvectors by the acoustic power. We plot the displacement and stress profiles of the guided modes and we compare our results to the analytical ones. Perfect correspondence is found, indicating the accuracy of the approach developed. In addition, a study of the vibrational state in the composite plate is established for Lamb and horizontal shear modes at a specific frequency.
Composite Structures, 2005
A spectral finite element model for analysis of flexural-shear coupled wave propagation in delaminated, multilayer composite beams is presented. Concept of obtaining the exact spectral element dynamic stiffness matrix for delaminated beam is discussed. Computation is performed in the Fourier domain at FFT sampling points over broad frequency band. Post processing of the response is made in the time domain, which is suitable for structural diagnostics and broad-band wave propagation problems. Implemented numerical examples illustrate the influence of delamination on wave propagation in cantilever multilayer laminated composite beams.
A time domain spectral finite element is developed for improving the efficiency of numerical simulations of guided waves in laminated composite strips. The finite element relies on a new generalized laminate mechanics model formulated to represent symmetric and antisymmetric Lamb waves. The laminate mechanics incorporate third-order polynomial terms for the approximation of axial and transverse displacement fields through the thickness and consider the displacements of the upper and lower surfaces as degrees of freedom. The laminate theory formulation is easily expanded to a high-order layerwise model. Based on the resultant governing equations of the laminate section, a new finite element with 8 nodal degrees of freedom is formulated; its nodes are collocated with Gauss–Lobatto–Legendre integration points in order to improve computational efficiency. Stiffness and mass matrices are assembled and the transient response is predicted using the explicit central differences time integration scheme. The transient response of Aluminum, Carbon Fiber Reinforced Polymer laminated and sandwich strips is investigated. Numerical results are validated against a semi-analytical solution. The accuracy and computational efficiency of the introduced element regarding the prediction of symmetric and anti-symmetric wave propagation is also quantified.
Finite Elements in Analysis and Design, 2004
A frequency domain spectral finite element formulation is presented for the wave propagation analysis of laminated composite curved beams using the first order shear deformation theory (FSDT) and the classical laminate theory (CLT). The elements are derived from the exact solution of the governing equation of motion in frequency domain, obtained through Fourier transformation of the time domain equation. The formulation is validated by comparing the results for natural frequencies with the published results. The new elements are then employed to perform dispersion and wave propagation analyses of curved composite beams. The numerical results reveal that the wavenumbers predicted by the CLT show large deviation from those of the FSDT even for thin beams, and the deviation increases and occurs at lower frequencies with the increase in the thickness to radius ratio. The orthotropy ratio of the composite has a significant effect on the wavenumbers for tangential and mid-surface rotation modes. The wave propagation response predicted by the CLT differs widely from the FSDT prediction, for thin and thick, and shallow and deeply curved beams at both low and high frequencies. Thus, the CLT should not be used for wave propagation analysis of even thin curved laminated beams.
Wave Propagation in Laminated Composite Plates Using Higher Order Theory
Journal of Applied Mechanics, 2000
A higher order displacement based formulation has been developed to investigate wave propagation in fiber-reinforced polymer composite laminated (FRPCL) plates. The formulation has been applied, as an illustration, to a plate made up of transversely isotropic laminae with the axes of symmetry lying in the plane of the lamina. Results for the plane as well as the antiplane strain cases are shown to be in excellent agreement with the exact solutions for isotropic and transversely isotropic single layered plates. Also, numerical results have been obtained for crossply (0 deg/90 deg/0 deg/90 deg) laminated composite plates, which agree very well with the previously published numerical results. The formulation can be employed to expeditiously investigate the dispersion characteristics of waves propagating in a plate with an arbitrary number of anisotropic laminae.
Journal of Sound and Vibration, 2003
A semi-analytical method incorporating various displacement-based formulations has been developed to investigate propagation of time harmonic waves and vibrations in fiber reinforced polymer composite laminated and sandwich plates. Various displacement-based models starting from the first order shear deformation theory to the fourth order theory have been developed using combinations of linear, quadratic, cubic and quartic variation of axial and transverse displacements through the thickness of a lamina or a mathematical sub-layer. These displacement-based formulations have been validated by comparing their results with the analytical solutions available in the literature. Results of all the displacement models have been compared with those obtained by displacement model using quartic variation of in-plane and transverse displacements for vibration problem. Higher order displacement-based theory using cubic variation of in-plane and transverse displacements through the thickness of sub-layer has been found to yield converging results for wave propagation in laminated composite plates as well as for vibration problems. All the investigations performed indicate the importance of higher order theories for analysis of wave propagation and vibrations in composite laminated and sandwich plates.
Dynamic modelling of elastic plates reinforced by strong fibres
IOP Conference Series: Materials Science and Engineering, 2020
Here are considered composite materials reinforced which strong fibres, which have an important property that they are anisotropic, and in many cases this anisotropy may be very strong, in the sense that mechanical properties are strongly dependent on direction. Such materials are highly resistant to deformation by extension in the fibre direction compared to other deformation modes. Thus material has been modelled as a transversely isotropic for which the extensional modulus in the fibre direction is much greater than that in a direction perpendicular to the fibres. Material is modelled as coordinate free and such model is employed for examination of waves propagating in an infinite plate reinforced by strong fibres. Wave propagation in layer reinforced by one family of fibres with wave direction parallel with stress free boundaries, but otherwise with angle arbitrary to fibre direction, are examined here. The dispersion relations, in specific carbon fibres - epoxy resin composites...