The applicability of the effective medium theory to the dynamics of cellular beams

https://doi.org/10.1016/j.ijmecsci.2009.06.001Get rights and content

Abstract

The applicability and the limitations of the effective medium assumption for the dynamics of cellular beams are studied. Beams made of uniform triangular and regular hexagonal cells are analysed. The natural frequencies and modal distributions as calculated using the detailed finite element model of the cellular networks are compared with those predicted based on equivalent homogeneous media of the same overall size and shape. It is observed that, for low mode number, a cellular beam behaves as a continuum, provided the cell size is significantly smaller than the external dimensions of the beam. Due to different deformation mechanisms, triangular cells show frequencies independent of area fraction whereas hexagonal cells show this dependence clearly. As the wavelength starts to become of the order of the heterogeneity, the continuum behaviour begins to break down. With the increase in mode number, cellular beams exhibit inherent flexibility with a progressive increase in their modal densities as compared to those of a homogeneous continuum. The modal density increases further when the cell walls start to resonate. During resonance, an abrupt rise in the modal density is observed for the triangular cells as the cell walls start deforming in the flexural mode instead of the axial mode. In contrast, for hexagonal cells, the predominant mode of cell wall deformation is always flexural.

Introduction

Natural and synthetic cellular materials such as wood, cork, bone, polymeric, ceramic and metallic foams have inspired many theoretical, experimental and computational works relating structure to properties. Due to their high stiffness to weight ratio, they are increasingly being used for various industrial applications. Honeycombs and foams are used as core of sandwich structures in aerospace and marine industry. Industrial applications include packaging, thermal insulation, acoustic damping, biomedical scaffolds, etc.

At the mesoscopic scale, i.e. at the length scale of a typical cell, cellular materials are made of complex network of elastic beams or plates. When the overall size of a piece of cellular material is much larger than the length scale of a typical cell, the structure can be treated as homogeneous with some effective properties. The literature shows that the research on cellular structures is primarily devoted in calculating its effective properties, given the geometry, topology, and material properties of the cell wall material.

A generic approach used to analyse periodic cellular structures is to identify a suitable unit cell as a representative of the whole, and then analyse the mechanics of the unit cell for predicting the macroscopic behaviour. A few analytical works are available so far using this sort of unit cell approach [1], [2], [3]. For example, Gibson et al. [1] determined the in-plane elastic constants for infinite lattice made of hexagonal cells. In their unit cell approach, they considered only the bending deformation of the cell walls. Warren and Kraynik [2] used kinematic arguments to analyse the deformations of the cell walls connected at a node. They included both the axial and the bending deformations of the cell walls and evaluated the elastic constants for hexagonal honeycombs for all densities. Gulati [3] derived the expressions for Young's modulus for honeycombs made of triangular cells using strain energy based method.

In case of the numerical approach, the unit cell is modelled as a network of interconnected beams. The bulk behaviour is predicted by analysing the unit cell using the finite element method [4], [5], [6], [7], [8], [9]. For instance, Scarpa et al. [4] used experimental techniques as well as the finite element method to analyse the in-plane elastic behaviour of honeycombs made of inverted hexagons due to uni-axial loading. Silva et al. [5] and Zhu et al. [6] studied the effect of the microstructural variability on the elastic behaviour of two-dimensional cellular solids using the finite element method. Because of the advantage of modelling the intricate geometrical features of the microstructure, finite element method has been used for modelling various defects. For example, Silva et al. [7] analysed the effect of the local defects (e.g. removal of the cell walls) in an irregular topology on the bulk compressive failure behaviour. Grenestedt [8] studied the effect of the wavy nature of the cell walls. Simone and Gibson [9] analysed the dependence of the elastic properties on the thickness variations of the cell walls. Chen et al. [10] studied how these imperfections can affect the yielding of a two-dimensional foam.

Homogenisation theory has been used for calculating the bounds on the bulk properties of the cellular materials. Torquato et al. [11] derived bounds on the effective properties based on the Hashin–Shtrikman bounds and compared their results with the finite element predictions. An extensive discussions on the mechanical behaviour of cellular materials such as elastic, plastic, buckling, failure, thermal conductivity, etc. can be found in the work of Gibson and Ashby [12]. Christensen's review [13] describes the relationships between the elastic properties, cellular topology and the effective density for two- and three-dimensional materials. Mechanical behaviour of metallic foams can be found in Gibson's work [14]. Grenestedt [15] analysed various models for studying the mechanics of perfect cellular materials.

The literature shows that most of the research on cellular materials is devoted to the study of elastic and plastic behaviour under static loads, whereas very few works are available on the dynamics of such materials. Wang and Stronge [16] used a micropolar theory to analyse the behaviour of hexagonal honeycombs under periodic forces. Baker et al. [17] studied the effect of impact and energy absorption behaviour of cellular materials.

The present authors [18], [19] developed a numerical scheme for reducing the computational expense associated with the free vibration and the response calculations of the cellular structures. As opposed to this, here we investigate the applicability and the limitations of the effective medium theory for dynamics of cellular structures. The effective properties based on the statics of infinite lattices are expected to predict the long wave behaviour well. However, it is not clear (1) what happens if a structure is not of infinite extent, because such structure is not strictly periodic, (2) how the dynamic behaviour changes with the increase in the mode number. The first question, in fact, raises the issue of the size of the specimen relative to the cell. In statics, researchers have observed softening effect during compression and stiffening effect in shear when the foam specimen size is not sufficiently large in comparison to the cell size [20], [21]. It is not clear how these effects translate in case of dynamics and what their dependence on mode number is.

Cellular structures are modelled here using the finite element method. This allows modelling of all the geometric features of the structure in detail. In addition, this micromechanics-based approach does not impose any restriction on the size of the model or the nature of the boundary conditions. Periodic boundary conditions are not required in this case as the whole structure itself is analysed. The results from the numerical experiments are chosen as the benchmark for comparisons with the predictions based on the equivalent medium theory. Trends observed are explained using physical reasoning qualitatively and quantitatively.

The paper is organised as follows. An effective medium theory for the dynamics of cellular beams is presented in Section 2, followed by the results and discussions for the low frequency vibration of cellular beams in Section 3. Progressive deviation in the continuum behaviour with increase in mode number and the statistics of modal distribution are described in Section 4. Finally, the concluding remarks are made in Section 5.

Section snippets

Finite element modelling of an elastic network

Free vibration of structures made of uniform triangular and regular hexagonal cells are analysed first. These cell topologies represent two extreme cases: the node connectivity of the cell walls is three for hexagonal cells and six for triangular cells. The dominant strain energy mechanism for the cell walls in a triangular topology is axial stretching, whereas for hexagonal cells bending of the cell walls dominates. Such lattices are isotropic in the plane. Our choice was also guided by the

Numerical results and discussions

Models of cellular beams made of triangular and hexagonal cells were generated for the two orientations in the MATLAB [29] environment. The length l and thickness t of the cell walls for all models were chosen as 10 mm and 0.5 mm, respectively. Therefore, t/l ratio is 1/20, and hence cell walls are thin and shear deformation can be neglected. When the cell walls are modelled using three node elements, the frequencies up to the 25th mode are calculated within 0.1% of the values when only two node

Progressive deviation of the continuum behaviour with the increase in mode number and the statistics of modal distribution

Trends in modal density (number of modes over a band of frequency) for the continuum will be compared with those arising out of cellular matter of the same external dimensions. In order to understand the statistical distribution of modes, a large number of modes need to be evaluated. This raises the issue of convergence in the calculations. When a small fraction of the total number of degree-of-freedom available in the model is used, the calculations are found to be satisfactory and mesh

Conclusions

The applicability and the limitations of the effective medium theory for dynamics of cellular beams were investigated. Beams made of uniform triangular and regular hexagonal cells were analysed. Lattices made of such cells are in-plane isotropic and they represent two extreme examples of topology—nodal connectivity three for hexagons, six for triangles. Natural frequencies calculated using full scale finite element model of the cellular network were used as benchmark for the effective medium

Acknowledgement

This work was supported by the EPSRC Grant GR/R45895/01.

References (31)

  • X.L. Wang et al.

    Micropolar theory for a periodic force on the edge of elastic honeycomb

    International Journal of Engineering Sciences

    (2001)
  • W.E. Baker et al.

    Static and dynamic properties of high-density metal honeycombs

    International Journal of Impact Engineering

    (1998)
  • S. Banerjee et al.

    Free vibration of cellular structures using continuum modes

    Journal of Sound and Vibration

    (2005)
  • S. Banerjee et al.

    The use and limitations of continuum modes for response calculations of cellular structures

    Journal of Sound and Vibration

    (2007)
  • S.M. Han et al.

    Dynamics of transversely vibrating beams using four engineering theories

    Journal of Sound and Vibration

    (1999)
  • Cited by (14)

    • Flexural elasticity of woodpile lattice beams

      2018, European Journal of Mechanics, A/Solids
      Citation Excerpt :

      Slender structures under transverse loading are commonly used as structural components in a variety of non-biomedical contexts. The effective elastic properties of 2D structured beams was studied by Banerjee and Bhaskar (2005, 2007, 2009). They characterised the effective properties of porous media having hexagonal, triangular and irregular cells via the dynamics of cellular beams.

    • Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model

      2014, International Journal of Solids and Structures
      Citation Excerpt :

      Here, we find that the local vibration on the cell level is weak if the H/lE is larger than 12. This trend of the gradually dropping accuracy on eigenfrequencies of the continuum model with increasing mode order has also been summarized by Banerjee and Bhaskar (2009) on the vibration of grid material by a classical continuum model. They also found that the continuum model will finally break down if the order is high enough.

    • Fatigue design of lattice materials via computational mechanics: Application to lattices with smooth transitions in cell geometry

      2013, International Journal of Fatigue
      Citation Excerpt :

      If geometric discontinuities are embedded at either the macro or/and the microscale, then a severe drop in the fatigue resistance is observed. From literature, it appears that the study of fatigue failure in cellular materials, with both random (foam) and periodic arrangement of cells (lattice), has received less attention than their monotonic quasi-static and dynamic properties [8,15–21]. Among the investigations, both experimental and theoretical approaches have been developed for metallic and polymeric cellular materials under fatigue conditions.

    View all citing articles on Scopus
    View full text