Identification of effective sandwich structural properties via an inverse wave approach

Abstract

A structural identification method is presented and applied to one and two-dimensional sandwich structures with honeycomb cores. The method is based on the $k$-space characteristics of measured or simulated data. It uses a harmonic spatial field as primary input. The harmonic spatial velocity field is first correlated to a well defined plane inhomogeneous wave. The resulting algorithm provides a complete $\theta$-dependent dispersion curve. Thus, an equivalent beam or plate model can be identified. The proposed method is successfully applied to sandwich structures with honeycomb cores. The focus is on flexural vibration of symmetric honeycomb beams and panels. The method leads to a pertinent equivalent homogenized model in a wide frequency band.

Introduction

Honeycomb sandwich panels have been increasingly used in every possible engineering field. In fact, a typical laminated structure of great use is the sandwich-type construction. Such structures are commonly included in aerospace, automotive and civil engineering industries among others. Their integration is preferred, as a rule, when a high strength-to-weight ratios and specific dynamic properties are essential. However, whereas a great amount of work exists in order to address some aspects of those sandwich mechanics (damage, static optimization,…), a number of questions arise when strong dynamical constraints are needed. The achievement of a pertinent homogenized model over a wide frequency band for sandwich structures with honeycomb cores is the main subject of discussion of the present paper.

The standard sandwich-type structures are constructed of three layers. Two of them, face sheets, constitute the external layers often treated as thin shells, which are separated by a thick mid-layer playing the role of the core. For the core, non-metallic honeycomb and plastic foam materials are now available and widely used. Arrays of open cells are often glued to the inner surfaces of the face sheets. To achieve high flexural strengths (high flexural natural frequencies) the honeycomb core height is usually about 80%–95% of the total panel thickness. Thus, an important consideration appears for pure numerical modeling of such sandwich structures due to their multi-scale nature. Indeed, the co-existence of multiple dynamical behavior scales makes the direct use, for instance, of finite element techniques “difficult”. One possible solution consists, then, in developing analytical homogenized models which can be used in some dynamical situations with relatively low computational costs.

This paper offers one of those possible alternatives and addresses specifically the question of effective apparent dynamical properties when flexural behavior is essential. The question of parameter characterization is an important task for every material structure [1], [2]. It becomes drastic for some new sandwich materials. For instance, for sandwiches made of recycling thermoplastics, whose properties are not easy to capture. A further major concern the following paper tries to consider is the relevance of structural parameter identification over a wide frequency range. Indeed, structural parameters (Young modulus, Poisson ratio,…) are often given through pure static or quasi-static methods [3] or using low frequency identification techniques based on modal analysis [4], [5], [6].The relevance of such low frequency identified parameters for higher frequencies is also addressed in this paper.

A review of open literature shows that an great deal of interest and work on flexural vibration of honeycomb panels has been carried out [4], [5], [6], [7], [8], [9], [10]. It shows, that the analysis of out-of-plane equivalent material properties (inertia, equivalent stiffness, equivalent density) of honeycomb cores can be obtained from experiments, specific correlations, and structural mechanics modeling. For instance, Saito et al. in [6], used a modal identification technique. A one dimensional beam with honeycomb cores was experimentally tested. Through a Timoshenko beam theory, sandwich honeycomb parameters (precisely Young modulus $E$ and Coulomb modulus $G$) were optimally found in order to match natural frequencies with experiments. The obtained parameters were then included in finite element computations. Randall [5] compared curve-fitting algorithms for the extraction of modal parameters from experiments. Such modal quantities can be used for effective mechanical parameter estimation. Yu et al. [7] and Nilsson et al. [4], [11] conducted interesting investigations on wave propagation in sandwich structures. In Yu et al. [7], a flexural model of a symmetric honeycomb panel was considered. In Nilsson et al. [4], authors employed a modal identification technique with a sandwich beam sample to get the apparent dynamical properties. Such identification was used to validate a dynamical model for multi-layers described in reference [11]. The model orthotropy was assumed and tested using specific oriented beams in the panel. Experimental results and three-dimensional modeling conducted by Lai [8] have indicated that the honeycomb core can be modeled as an equivalent homogeneous orthotropic material in flexural vibration analysis. Hohe [9] investigated a homogenization approach for the determination of the stiffness matrix of sandwich structures. Renji et al. [10] analyzed modal density issues for the composite honeycomb sandwich and pointed out some interesting features specifically when the frequency increases. In short, the question of composite honeycomb modeling and characterization is still an important issue in the literature and it is generally agreed that in using the continuum model for analyzing the free vibration of honeycomb panels it is important that reliable values of the equivalent material properties of honeycomb cores be used.

The idea developed in this paper is original with regards to the above mentioned works. It is fully based on the wavenumber space analysis for effective parameter identification. It is a kind of inverse technique in the wavenumber space ($k$-space). The use of wave properties is also motivated by the frequency band under interest, in particular for mid-high frequency issues [12], [13], [14], [15], [16]. Indeed, the mid and high frequency behavior is relevant for many engineering questions such as pyroshocks and vibroacoustics. Moreover, in this frequency band, modal approaches are of limited interest and most of the predictive tools are based on free wave dispersion curves. In [12], [13], [14], [15] wave properties are used either directly or indirectly to provide mid-high frequency dynamics. Among available predictive tools, we can mention, the deterministic variational theory of complex rays (TVRC) developed by Ladeveze and his colleagues [16], [17] and the statistical energy analysis (SEA) [18], [19]. However, the wave propagation features of complex structures, such as sandwich structures with honeycomb, are not easily described, especially for short wavelengths. Therefore, it seems relevant to propose identification tools of the frequency dispersion equation mainly for complex structures in the medium and high frequency domain.

This contribution uses a method that starts from the spatially distributed fields of a vibrating plane structure, in order to identify the complete $\theta$-dependent dispersion curve. The method uses a harmonic field (say $w(x,y)$) as primary input. The correlation between this field and an inhomogeneous wave is calculated leading to a wavenumber dependent objective function, called here $\mathrm{IWC}$ (for Inhomogeneous Wave Correlation). An inverse technique is then proposed through the optimization of the correlation index. The identification of a complex wavenumber for a given direction $\theta$ leads to the maximization of the function $(k,\gamma )⟶\mathrm{IWC}(k,\gamma ,\theta )$. The introduction of loss factor identification offers the ability of recognizing near-field from far-field. The algorithm, thus, eliminates identified wavenumbers with high apparent loss factors. The method originally proposed for two dimensional structures can be equally applied to one dimensional cases. The method is also successfully compared to another wavenumber identification tool proposed by McDaniel et al. [20], [21]. Application of IWC and McDaniel method to a sandwich beam with aluminum honeycomb cores reveals important conclusions with regards to the apparent structural parameter identification tools. For two dimensional sandwich plates, the identification of the function $k\left(\theta \right)$ allows an equivalent plate model to be identified. This identification is achieved thanks to a simple least mean square algorithm. Application of this method allows understanding of apparent wave propagation in complex structures. A particular sandwich with honeycomb cores is experimentally tested in this paper. It exhibits an orthotropic behavior for short wavelengths. This behavior is successfully compared with an analytical model taken from the literature [4].