18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS APPLICABILITY OF AN INCLUSION-BASED HOMOGENISATION APPROACH TO MODELLING OF BALSA-LIKE POROUS MATERIALS O. Shishkina*, Y. Zhu, L. Gorbatikh, S.V. Lomov, I. Verpoest Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, Leuven, Belgium *Corresponding author (oksana.shishkina@mtm.kuleuven.be) Keywords : Balsa wood, polymeric foams, inclusion-based homogenisation method, micro-compression tests of a scanning electron microscope (SEM) for the 1 Introduction same materials. Balsa wood and polyvinyl chloride (PVC) foam are popular core materials for lightweight sandwich structures used, for example, in shipbuilding and 2 Materials Description windmill industry. Due to the limited availability, materials from natural sources are usually expensive. 2.1 Structure of balsa wood Their annual supply and quality can vary with On the meso-level balsa wood consists of three climatic conditions. These disadvantages explain an distinct types of cells (Fig. 1). Two of them have an increased interest and efforts towards development important mechanical function: 1) tracheids are cells of artificial cellular materials that can substitute that have a bean pod shape and are highly elongated balsa wood in the above applications. Rigid PVC and aligned in the axial direction of a tree stem. foams have comparable to wood properties and Their average length/diameter ratio varies from 16 replace timber in some applications. On a up to 25 [2]. Tracheids are responsible for the cost/volume basis PVC foams are more expensive stiffness and axial strength of a tree trunk; 2) rays than wood [1], therefore, engineers are still looking are smaller, more rectangular cells that are organized for cheaper alternatives. in radial arrays, which penetrate tracheids. The main Promising candidates to replace balsa and PVC functions of ray cells are to transport nutrients foams are low-cost polymeric foams, such as radially and contribute to the radial strength of the foamed polypropylene (PP) and polyurethane (PU). wood [3]. They are especially relevant nowadays, in the era of 2.2 Structure of nano-reinforced polymeric foams nano-materials, as used in a combination with nano- reinforcements (such as nanotubes, graphene sheets The nano-reinforced polymeric foams have a closed- etc), they have a potential to deliver unprecedented cell structure, which is more random than balsa’s material performance. Achieving the exceptional organization (Fig. 2) and highly influenced by the mechanical properties of balsa in these foams would production technique. Addition of nano-fillers to the be difficult without the help of nano-reinforcements polymer increases nucleation efficiency but usually and strong built-in anisotropy in the foam leads to an inhomogeneous structure with cells of microstructure. different sizes and shapes. Most cells have spherical To date, understanding of nano-scale materials and or ellipsoidal shape with an average length/diameter their properties has been achieved primarily through ratio equal to 2. empirical or discovery-based research. While this approach will continue to make important contributions, a systematic understanding of physics 3 Modelling approach fundamentals through modelling approaches is also Macroscopic properties of heterogeneous materials needed. based on their micro-structural parameters can be In this work, we discuss the applicability of the predicted using many available methods of the inclusion-based methods to predict elastic properties micromechanics. They are based on volume (or of balsa-like cellular materials and results of the ensemble) averaging operations (homogenisation micro-compression tests performed inside a chamber procedure) within a representative volume element
(RVE). The inclusion-based models [4] (for the non-uniform loading can result in early failure at example, Mori-Tanaka homogenisation scheme, the corners or edges of the specimen. The surface self-consistent method) are a popular class of parallelism is especially relevant for tests on the methods for the prediction of the elastic constants of micro-scale. materials containing inclusions. Examples of such Samples were compressed using a stage shown in materials are composites reinforced by particles of Fig. 4. Applied displacements were controlled different sizes and shapes, unidirectional fibre- manually by means of a screw. The stage was placed reinforced composites, porous materials (pores also inside of the SEM chamber where images were can be treated as inclusions with zero stiffness) etc. taken. The inclusion-based homogenisation approach was When balsa is compressed axially (Fig. 5a-f), no chosen in the present work. bending moments are applied to the length of the cell walls. Instead, the walls suffer axial compression. Failure occurs in the ends of tracheids 4 Results and discussion and by forming of bands in the collapsed cell walls (Fig. 5e). The inclusion-based homogenisation approach was When wood is loaded in the radial or tangential applied to the RVE of balsa wood (Fig. 3) and directions (Fig. 6a-f), the cells deform by the polymeric foams in order to calculate effective progressive bending of the cell walls [2]. Even in the elastic constants of these materials. elastic mode (2% of strains) balsa’s honeycomb Porosity, aspect ratio and orientation of cells, the structure loses stability and local buckling of cell cell wall properties and the presence of different walls occurs, which is clearly seen in Fig. 6b. families of pores were used as an input data for the Similar behaviour can be observed in polymeric model. foams (see Fig. 7 for a PU foam and Fig. 8 for a The cell wall properties are important parameters PVC foam). The dominant mechanisms of that significantly influence predictions of the model. deformation are wrinkling of the cell faces (PU Elastic constants of cell walls are difficult to foam) or bending of the cell walls (PU and PVC measure because of their small sizes. Therefore in foam). Cell faces are the weakest parts of the PU the literature there is a large scatter of these foam. Their thickness is much less than one of the properties. In this work typical values of 35 GPa for struts. The deformation tends to localize thus modulus in the axial direction of the cell and 10 GPa causing rupture of the cell faces (Fig. 6e). In PVC [2] in the radial and tangential directions was used foams cells have hexagonal shapes and polymer is for balsa. Solid polymers are isotropic materials homogeneously distributed within the cellular frame. therefore only one value of Young’s modulus was Cell walls tend to bend (Fig. 8c-d) due to the low used: 2 GPa [5], 0.5 GPa [6] and 3 GPa [7] for PP, elastic modulus of the cell wall material in PU and PVC foams respectively. comparison with balsa and defects in the structure For balsa wood the predicted values of the Young’s caused by a production process. modulus of the RVE in the axial direction were in The inclusion-based methods follow from a dilute good agreement with experimental data of [2]. But solution of a single inclusion problem in an infinite for balsa in the transversal directions (radial or matrix [4]. Because of the latter these approaches do tangential) and for foams the predicted values of the not account for the bending/wrinkling effects and elastic constants were overestimated (approximately cannot predict the elastic properties of highly porous 3 times higher than experimental ones). materials accurately. They seem to give accurate The reason for this discrepancy was found in a predictions of the elastic moduli of wood [8], metal different response of the material microstructure to and ceramic porous materials [9, 10]. For balsa the applied load. In order to prove or disprove this wood in the axial direction, the methods work even statement, compression tests of different cellular for high volume fractions of pores (up to 0.95)! The materials were performed on the micro-scale. reason is that internal microstructures of these Cubic samples with the edge of 5 mm were cut from materials are stable to bending deformations. For balsa and foam panels using a razor blade. Great cellular polymeric materials the inclusion-based attention was paid to the parallelism of the top and method, however, needs to be advanced in order to bottom surfaces of the samples. This issue is account for bending deformation mechanisms that important in compression tests because the uniform are typical for foams. loading conditions should be obtained. If the specimen ends are not parallel to the load surfaces,
Acknowledgments We thank the European Commission for financing this work within the FP7 program through the NanCore project (“Microcellular nanocomposite for substitution of Balsa wood and PVC core material”). Sap channel Tracheids Rays Fig.1. Micro-CT image of balsa wood. Cell collapse Fig. 5. SEM images of balsa under axial loading: a – 0%, b – 2%, c – 4%, d – 8%, e – 10% of strains; f – typical stress-strain curve. Wall bending Fig.2. Structural hierarchy of nano-reinforced PP foams. Fig. 3. Schematic of the balsa wood model. Sample Fig. 6. SEM images of balsa under loading in the radial Screw direction: a – 0%, b – 2%, c – 4%, d – 6%, e – 8% of strains; f – typical stress-strain curve. Fig. 4. Schematic of the stage for micro-compression.
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