Wave Transmission Characteristics in Honeycomb Sandwich Structures using the Spectral Finite Element Method

Abstract

Wave propagation is a phenomenon resulting from high transient loadings where the duration of the load is in µ seconds range. In aerospace and space craft industries it is important to gain knowledge about the high frequency characteristics as it aids in structural health monitoring, wave transmission/attenuation for vibration and noise level reduction. The wave propagation problem can be approached by the conventional Finite Element Method(FEM); but at higher frequencies, the wavelengths being small, the size of the finite element is reduced to capture the response behavior accurately and thus increasing the number of equations to be solved, leading to high computational costs. On the other hand such problems are handled in the frequency domain using Fourier transforms and one such method is the Spectral Finite Element Method(SFEM). This method is introduced first by Doyle ,for isotropic case and later popularized in developing specific purpose elements for structural diagnostics for inhomogeneous materials, by Gopalakrishnan. The general approach in this method is that the partial differential wave equations are reduced to a set of ordinary differential equations(ODEs) by transforming these equations to another space(transformed domain, say Fourier domain). The reduced ODEs are usually solved exactly, the solution of which gives the dynamic shape functions. The interpolating functions used here are exact solution of the governing differential equations and hence, the exact elemental dynamic stiffness matrix is derived. Thus, in the absence of any discontinuities, one element is sufficient to model 1-D waveguide of any length. This elemental stiffness matrix can be assembled to obtain the global matrix as in FEM, but in the transformed space. Thus after obtaining the solution, the original domain responses are obtained using the inverse transform. Both the above mentioned manuscripts present the Fourier transform based spectral finite element (FSFE), which has the inherent aliasing problem that is persistent in the application of the Fourier series/Fourier transforms. This is alleviated by using an additional throw-off element and/or introducing slight damping in to the system. More recently wave let transform based spectral finite element(WSFE) has been formulated which alleviated the aliasing problem; but has a limitation in obtaining the frequency characteristics, like the group speeds are accurate only up-to certain fraction of the Nyquist(central frequency). Currently in this thesis Laplace transform based spectral finite elements(LSFE) are developed for sandwich members. The advantages and limitations of the use of different transforms in the spectral finite element framework is presented in detail in Chapter-1. Sandwich structures are used in the space craft industry due to higher stiffness to weight ratio. Many issues considered in the design and analysis of sandwich structures are discussed in the well known books(by Zenkert, Beitzer). Typically the main load bearing structures are modeled as beam sand plates. Plate structures with kh<1 is analysed based on the Kirch off plate theory/Classical Plate Theory(CPT) and when the bending wavelength is small compared to the plate thickness, the effect of shear deformation and rotary inertia needs to be included where, k is the wave number and h is the thickness of the plate. Many works regarding the wave propagation in sandwich structures has been published in the past literature for wave propagation in infinite sandwich structure and giving the complete description of dispersion relation with no restriction on frequency and wavelength. More recently exact analytical solution or simply supported sandwich plate has been derived. Also it is seen by comparison of dispersion curves obtained with exact (3D formulation of theory of elasticity) and simplified theories (2D formulation as generalization of Timoshenko theory) made on infinite domain and concluded that the simplified theory can be reliably used to assess the waveguide properties of sandwich plate in the frequency range of interest. In order to approach the problems with finite domain and their implementation in the use of general purpose code; finite degrees of freedom is enforced. The concept of displacement based theories provides the flexibility in assuming different kinematic deformations to approach these problems. Many of the displacement based theories incorporate the Equivalent Single Layer(ESL) approach and these can capture the global behavior with relative ease. Chapter-2 presents the Laplace spectral finite element for thick beams based on the First order Shear Deformation Theory (FSDT). Here the effect of different choices of the real part of the Laplace variable is demonstrated. It is shown that the real part of the Laplace variable acts as a numerical damping factor. The spectrum and dispersion relations are obtained and the use of these relations are demonstrated by an example. Here, for sandwich members based on FSDT, an appropriate choice of the correction factor ,which arises due to the inconsistency between the kinematic hypothesis and the desired accuracy is presented. Finally the response obtained by the use of the element is validated with experimental results. For high shock loading cases, the core flexibility induces local effects which are very predominant and this can lead to debonding of face sheets. The ESL theories mentioned above cannot capture these effects due to the computation of equivalent through the thickness section properties. Thus, higher order theories such as the layer-wise theories are required to capture the local behaviour. One such theory for sandwich panels is the Higher order Sandwich Plate theory (HSaPT). Here, the in-plane stress in the core has been neglected; but gives a good approximation for sandwich construction with soft cores. Including the axial inertial terms of the core will not yield constant shear stress distribution through the height of the core and hence more recently the Extended Higher order Sandwich Plate theory (EHSaPT) is proposed. The LSFE based on this theory has been formulated and is presented in Chapter-4. Detailed 3D orthotropic properties of typical sandwich construction is considered and the core compressibility effect of local behavior due to high shock loading is clearly brought out. As detailed local behavior is sought the degrees of freedom per element is high and the specific need for such theory as compared with the ESL theories is discussed. Chapter-4 presents the spectral finite element for plates based on FSDT. Here, multi-transform method is used to solve the partial differential equations of the plate. The effect of shear deformation is brought out in the spectrum and dispersion relations plots. Response results obtained by the formulated element is compared and validated with many different experimental results. Generally structures are built-up by connecting many different sub-structures. These connecting members, called joints play a very important role in the wave transmission/attenuation. Usually these joints are modeled as rigid joints; but in reality these are flexible and exhibits non-linear characteristics and offer high damping to the energy flow in the connected structures. Chapter-5 presents the attenuation and transmission of wave energy using the power flow approach for rigid joints for different configurations. Later, flexible spectral joint model is developed and the transmission/attenuation across the flexible joints is studied. The thesis ends with conclusion and highlighting futures cope based on the developments reported in this thesis.