18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS MORPHOLOGY EFFECTS ON CONSTITUTIVE PROPERTIES OF FOAMS J. Köll * , S. Hallström Department of Aeronautical and Vehicle Engineering, KTH Royal Institute of Technology, Stockholm, Sweden * Corresponding author (koll@kth.se) Keywords : Rigid foam, morphology, mechanical properties, stochastic modeling 1 Introduction The scope of this work is to investigate if the level Low-density rigid foams are commonly used as core of model sophistication also affects the resulting materials in sandwich structures. Such foams are homogenized mechanical properties. here modeled as stochastic Voronoi partitions of 3D space that are meshed and analyzed with finite 2 Modeling and simulations elements (FE). The constitutive properties and their relation to morphological variations in the cellular The approach is to develop a methodology for structures are studied, especially how the stiffness of generating realistic computer models that are the foam models varies with respect to a number of structurally and mechanically representative of true characteristic measures of the foam structure. foam materials. They should capture the random, Various models of cellular materials have been used amorphous nature of real foam materials for which in the past, ranging from quite simplistic to very the cell shapes and sizes vary significantly. advanced geometrical representations. Strongly The cellular microstructure is modeled as a idealized models of single cells can be used to representative volume element (RVE) containing provide certain scaling laws of foams [1] and there numerous disordered cells. The RVEs should be are a few polyhedral unit-cells that are space-filling. large enough to be globally representative of the However, they all compromise some of the modeled materials. For systematic handling of loads properties that characterize real foam materials, such and boundary conditions spatially periodic RVEs, as connectivity or global isotropy. To increase containing fifty cells each, were built. Bulk material authenticity, computer tomography has also been properties were assigned to the cell walls. The used to depict the true three-dimensional geometry homogenized constitutive properties of the foam of small samples of foams. They have then been models were then determined with FE analysis, digitally recreated and used for FE analysis [2]. Such applying periodic boundary conditions to make the models are amorphous and very representative but models artificially continuous. provide only relatively small samples from a big The RVEs vary in shape but are always of unit variety of cell constellations. These models are volume. The individual cells vary both in shape and neither regular nor periodic, which cause great size. The cells are space fi lling without overlap, and challenges in analysis since adequate loads and define the exact shape of the RVE. An example of a boundary conditions are difficult to apply. Another random disordered polydisperse model with 50 cells way of increasing authenticity is to generate random is shown in fig. 1. and isotropic structures that resemble real foams [3]. 2.1 Voronoi partitioning Previous work has shown not only that input parameters and choice of different methods for The subdivision of the RVE into cells was made as a generating stochastic cellular structures can be Voronoi partitioning. A three dimensional Voronoi related to statistical measures on the morphology, partitioning is based on a set of seed points but also that the methodology needs to be quite distributed in a model space Ω . The Voronoi sophisticated for the models to accurately resemble partition or region V i associated with seed point i is dry foams [4]. given by
2.3 Distribution of solid { } (1) V i = x �� | x � x i � x � x j , i � j Each model was given a prescribed relative density by controlling the wall thickness of the cells. In real where x i and x j are the coordinates of seed points i foams the bulk material is typically somewhat and j respectively. This means that every seed point concentrated along the cell wall edges, and the is surrounded by one cell, and the cell contains all thickness varies over the cell faces. As a first points in space that are closer to this particular seed approximation the material in the models was evenly point than to any other. To obtain spatially periodic distributed, i.e. all cell walls were given equal and models the partitioning is made in a spatially uniform thickness and no additional material was periodic environment, i.e. the model space is added at the cell wall edges. Note however that the surrounded by copies of itself. To create disordered shell elements in the model inherently have small structures, the seed points must be randomly overlaps at the cell edges. distributed in the space Ω . 2.4 Stiffness analysis 2.2 Seed point distributions The stiffness of the models was determined by FE analysis. Bulk material properties were assigned to Generating random points within the model space without any further constraints will result in a cell walls that were modeled with shell elements. Prescribed uniform global strains were applied to the geometry with extreme variation in both cell size and shape, which does not resemble the structure of models together with periodic boundary conditions. real foams. To obtain more foam-like geometries an The reaction forces and the lateral contractions were determined and the stiffness parameters were algorithm preventing seed points to end up too close to each other, was used. Two different methods calculated from principles of elasticity. The full based on packing of hard spheres were tested and orthotropic stiffness matrix was extracted for each model. evaluated with regard to morphological and mechanical properties. The two methods for generating random seed point 3 Results distributions were Random Sequential Adsorption (RSA) and Random Close Packing (RCP). Both 32 models consisting of 50 cells each were methods generate packings of equal-sized hard generated and analyzed. Half of the models were spheres where the sphere centers constitute seed based on RSA distributions and half on RCP points for the Voronoi partitioning. distributions. In addition 5 models based on RCP RSA is a fairly simple way to generate sphere distributions consisting of 350 cells each were packings. Spheres are randomly, sequentially and generated. For these 5 models only the stiffness was irreversibly deposited in the model space where evaluated for comparison. possible without overlap. RSA generates relatively 3.1 Morphology loose packings with φ ≤ 0.36, where φ is defined as the volume fraction of spheres in the model space. The morphology of totally 1600 cells was evaluated. RCP is usually generated using “molecular The normalized cell volume distributions look dynamics” simulations, i.e. based on interaction relatively symmetric. As can be seen in fig. 2 the between particles or spheres. Typically a separating distribution is significantly narrower for the RCP force spreads the particles or spheres in the model models than for the RSA models, which show a space until equilibrium is achieved. RCP enables broader variation in cell volume. This is an expected relatively dense packings with φ ≤ 0.64. result related to the higher packing fraction of All RCP distributions used in the work presented spheres in the RCP distribution. here were generated with a Python-language The normalized face area distributions shown in fig. implementation of Jodrey and Tory’s algorithm 3 reveal interesting characteristics of the Voronoi [5,6]. Several ideas for reducing computational time structures. The RSA models demonstrate a were found in Bargiel and Moscinski’s C-language surprisingly high content of very small faces. As implementation of the same algorithm [7]. might be expected, the RCP models show an
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