Design of Mechanical Properties of Open-Cell Porous Materials Based on μCT Study of Commercial Foams

In the present paper numerical design of mechanical properties of open-cell porous materials is addressed. A detailed knowledge of mechanisms and parameters determining mechanical properties (i.e. Young’s Modulus, Poisson’s Ratio) of foams is essential for applications such as energy absorbers or lightweight construction materials. The foam structures were designed using procedure based on Laguerre-Voronoi tessellations (LVT) with microcomputed tomography of commercial foams used as reference. Foam morphology was studied on post-processed computed tomography images and the parameters of LVT structures were compared with commercial materials. Subsequently finite element method (FEM) calculations were performed on both types of structures to validate the LVT design algorithm. The results show that the described design procedure can be successfully used for modeling mechanical properties of open-cell foam structures.


Introduction
Open porosity foams are multifunctional materials due to their unique properties combining high specific strength and stiffness [1,2] with light weight and excellent energy absorption [3,4].Structural parameters of commercially available foams such as porosity, specific surface, mean strut diameter and mean pore diameter are commonly studied with use of μCT [5][6][7][8][9].Furthermore Open-Cell Porous Materials could imitate various natural materials where porosity is key parameter for structural characteristic [10,11].On the other hand, determining of elastic properties, i.e.Young's modulus and Poisson's ratio, is a more complex process.Extensive research performed in recent years in this area includes experimental [12,13], analytical [14] and numerical [15,16] approaches.Numerical analysis uses both simplified model structures, and calculation models derived from X-ray micro-computed tomography (μCT) data [17].
Most of geometrical models proposed in literature base on the assumption of equal volume of pores of different shapes [18][19][20][21][22][23].In commercial practice foams are very frequently manufactured with narrow pore size distributions, which was confirmed by experimental observations [24].Good approximation of such distribution can be obtained using Laguerre-Voronoi algorithm [25], where tessellations are performed on a set of spheres with pre-determined size distribution.
The aim of present work was to investigate utility of LVT for design of mechanical properties of open-cell foams.To this end, a set of porous structures with homogeneous pore size distribution was generated using Laguerre-Voronoi tessellations.Elastic properties of these structure were calculated using finite element method.Validation of the LVT algorithm was performed by comparing Young's modulus and Poisson's ratio obtained for the modeled structures with values calculated for commercial foam materials.

Materials and methods
The design algorithm developed to model foam structures is based on the Laguerre-Voronoi tessellations method [24].Tessellations are applied to a set of spheres, with specified size distributions, packed in a cube.The foam struts were generated by placing cylinders with defined radius along the cell edges.Utility of LVT for modeling of structure of open-cell foams was described in [26].
The algorithm for the modeling of foam structures employed in the present study consists of the following five steps: 1.Generation of spheres with a defined size distribution (in the computation log-normal distribution of volume was adopted with a pre-selected average value, and the variation coefficient) 2. Packing of the spheres, 3. Performing Laguerre-Voronoi tessellations, 4. Defining the triple edges of the cellular structure, and 5. Generation of struts along the triple edges.
Four random structures with homogeneous pore size distribution were generated using the procedure described above.Each structure was limited by a bounding box of cubic shape and consisted of 200 pores.For each structure strut diameter values were varied from 0.5 to Reference structures used in this study were commercially available aluminum foam specimens with nominal pore densities 10, 20, 30, 40 and 50 pores per inch.Micro-computed tomography was used to investigate structural parameters of reference materials.The three dimensional image acquisition was performed using X-Radia XCT-400 made by SkyScan under acceleration voltage of 150kV and current of 50A, which assured a voxel resolution in the range from 15 to 45 μm/voxel.3D structural analysis was performed using SkyScan cTAN software.First, a median filter was used to remove the noise.
Numerical analysis of elastic properties of both designed structures and commercial materials was performed using finite element method implemented in ANSYS software.Finite element meshes were generated in order to perform numerical simulations corresponding to uniaxial compression tests.Examples of meshes generated on both LVT structures and commercial foam materials are shown on Fig. 1.Ideal elastic material model for matrix has been defined by following sets of isotropic Young's modulus and Poisson's ratio of E=69GPa, ν=0.33 for aluminum.Investigated structures were subjected to numerical simulations of uniaxial compression.Each model was constrained on two nodal components located at the opposite sides -zero displacement at the bottom and displacement equal to 0.1% of initial edge length at the top.Additionally one of the nodes from the bottom was constrained in three directions (ux, uy and uz displacement were fixed).The value of macroscopic engineering stresses has been evaluated as a quotient of the sum of the nodal reaction forces ΣFn divided by initial engineering cross-section area A=a 2 . . ( The engineering strain values were obtained by dividing applied displacement by initial height of the computed structure.

Results and discussion
Complete set of structural parameters of both designed and commercial materials is listed in Table 1.The data for commercial foams does not refer to the entire specimen, because the scanning process could not cover the entire sample.Serious disturbances occurred on the borders of measured specimens.Additionally, the total scanned volume was related to the scanning resolution.10 and 20 ppi foam scans were performed with resolution of 45 μm and resulted in a cubic shaped geometry of 35 mm edge length.Such resolution is sufficient for obtaining precise representation of complex foam geometry, and the obtained sample size seems sufficient to be treated as representative regarding the mean pore size of foams.The 30 ppi foam was scanned at 25 μm resolution (obtained edge length of 25 mm) due to smaller size of struts for this structure.40 and 50 ppi foams required scanning resolution of 12 μm, because scans with lower resolution exhibited distorted foam topology.The obtained edge length of these foams was 15 mm.Obtained structural parameters of commercial foam materials are in good agreement with literature data [8,17,27].One of the most decisive parameters determining Young's modulus of porous materials is porosity, therefore LVT structures with mean strut diameter of 2 mm were chosen as correlated the closest to the reference materials.It has to be noted, that specific surface is significantly lower in case of the designed materials.This discrepancy can be assigned to different nature of triple point connections.Thicker triple point connections were observed in case of commercial materials.Mechanical properties calculated using finite element method are listed in Table 2.Both designed and commercial open-cell foam structures are characterized by randomly distributed cells, therefore calculations were performed in 3 perpendicular directions.Anisotropy was detected for each specimen, however for purpose of comparison, average values from three directions is listed in Table 2, since anisotropy of properties is a subject of separate study.The results obtained for commercial foam materials are in accordance with data reported in literature for similar structures [8,17,28].Mechanical properties of all the designed structures were calculated, however due to similarity of structural parameters explained above, only results for LVT materials with strut diameter of 2 mm will be discussed in comparison with commercial foam materials.Similar values of both Young's modulus and Poisson's ratio were obtained for the designed materials and reference commercial foams with similar structural parameters.Results obtained were considered as a validation of LVT algorithm for design of mechanical properties of open-cell porous materials.

Summary
We used the Laguerre-Voronoi algorithm and the finite element method as a tool for design of open-cell foam structures with defined structural parameters and mechanical properties.The design procedure developed was used to create a set of structures with structural parameters corresponding to commercial aluminum foam materials.Results verified the utility of LVT algorithm for modeling of mechanical properties of foams.The described findings allow for studying the influence of various structural parameters, such as pore size distribution, strut shape or pore diameter, on Young's modulus and Poisson's ratio with use of the described design method.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 2015 mm, which resulted in porosity ranging from 74 to 97%.

Figure 1 .
Figure 1.Example of meshed geometrical models for FEM analysis: a) LVT structures, b) μCT reconstructed commercial material.

Table 1 .
Structural parameters of LVT structures and commercial foam materials.

Table 2 .
Mechanical properties of LVT structures and commercial foam materials.