Computational analysis of closed-cells cellular structures.

Hadar, Anton ; Popescu, Diana ; Parpala, Radu 等

Abstract: Cellular solids, due to their specific thermo-plastic and mechanical properties, can be used in many diverse applications such as thermal and acoustic insulations and energy absorbers, scaffolds for tissue engineering, materials for implants and prostheses, light-weight reinforcement, etc. In the attempt to improve the cellular solids mechanical properties, to understand and predict their behavior under certain loadings, and to develop new applications, it is very important to properly choose the type of material, the relative density level and to control the topology. The study presented in the current paper compare the behavior of a cellular material with closed spherical gaps in contrast with the behavior of the base material.

Key words: Cellular material, non-linear, FEM, Rapid Prototyping, mechanical properties

1. INTRODUCTION

A cellular solid is one made up of an interconnected network of solid struts or plates which form the edges of cells (Gibson & Ashby, 2001).

The properties of cellular materials depend on three factors: the material of the cell walls, the cell topology, and the relative density (Stampfl et al., 2004), thus being important to have the possibility to control these parameters in order to optimize the micro--and macro--structural properties for a certain type of application.

In this context, manufacturing solid cellular structures using RP processes presents significant advantages over traditional fabrication processes (Kamrani & Nasr, 2005), allowing building objects with precisely controlled geometry and dimensions. However, not all RP processes can build closed-cell cellular structures, the layered manner of manufacturing and the tool-less approach imposes limitations in obtaining objects with completely enclosed voids, usually a small hole for eliminating the support being necessary.

The possibility offered by FDM process to build closed-cell cellular solids with controlled geometry offers the advantage of being able to comparatively analyze their behavior when subjected to different loadings, for different cells geometries, with impact on developing new applications for these materials. (Popescu et al, 2006)

The mechanical properties of metal foams (and other cellular solids) depend on the properties of the metal that they are made from, on their relative density, and on the cell topology (i.e., cell size, cell shape, open or closed cell morphology, etc.).

The cell size of commercially available metal foams is about 1 to 10 mm. This is on the order of the smallest structural length of specimens in many applications. In such cases, the individual response to a load differs significantly from one cell to another, and the fundamental assumption of the classical continuum theory that the (physical, chemical, mechanical, etc.) properties of a material are uniformly distributed throughout its volume fails. (Tekoglu, 2007).

[FIGURE 1 OMITTED]

As one can see from Fig.1 Stress-strain behavior of cellular materials is different from the one of the base material. (Vesenjak et al, 2005).

2. NON-LINEAR ASPECTS OF CELULAR MATERIALS

Cellular materials have a characteristic stress-strain relationship in compression, which can be divided into four main areas as shown in Fig. 1. The initial response of material is elastic, then buckling is experienced, and plastic deformation and collapse become more evident, which is manifested in large strains at almost constant stress until the cells completely collapse. At this point the cellular material stiffness increase and consequently converges towards the stiffness of base material.

3. FEM MODEL

Because of the complicated geometry of cellular materials and computers capabilities, the realization of a complete detailed FEM model is almost impossible. This is the reason why often cellular materials are modeled by considering a representative volume element or as a simplified structure.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

A regular closed-cell cellular structure was used for the numerical simulation. The test element is a 21.496x21.496x21.496 mm cube with a regular net of gaps; each gap has a radius of 1.1 mm. This corresponds to a relative density of 23.8%. The material was messed entirely with solid 187 which is a high order 3D 10 node element. This tip of element has a quadratic displacement behavior and is well suited for modeling irregular meshes.

The element is defined by 10 nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. The element has plasticity, hyper elasticity, creep, stress stiffening, large deflection, and large strain capabilities. (2).

Because of the 3D model geometrical symmetry and because of the symmetric load the FEM model was reduced at one quarter. This dramatically reduces the number of equations used for analytical computation.

The following loads where used for the FEM simulation:

* Displacement for the base surface

* Distributed force for the top surface (10000 N/m2)

In order to compare the two materials a small load was used in order to ensure that the deformations are in the elastic phase. As we can see from figures 2 and 3 the difference between the maximum displacement of the cellular material and the base material is:

[DELTA][epsilon] = 1.4 x [10.sup.-7] - 0.858 x [10.sup.-7]/1.4 x [10.7] = 38.7%

and the difference between maximum values for the Von Misses stress is (Fig.4 and Fig.5):

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[DELTA][sigma] = 25109 - 15596/25109 = 37.99%

4. CONCLUSION

The development of new applications for cellular solids must rely on a detailed knowledge of their mechanical properties and characteristics, in order to understand how they behave in certain environmental conditions and under certain loadings.

The current paper is focused on the FEM analysis of closed-cell cellular solids manufactured via FDM process. In order to test the behavior of the two analyzed materials, including non-linear characteristics, a stress-strain diagram must be draw by using experimental data.

FEM analyses showed that in case of uniaxial compression, the cell walls at the borders are not under a significant stress.

The initial analyses have proven that a small reduction in weight can dramatically increase the stress of the mechanical structure. The obtained results show that at a relative density of 23.8% the relative displacement of the cellular material is 38.7 % and the relative stress between the two analyzed models is 37.9 %. This must be taken into consideration when using this kind of cellular materials for practical applications.

In further researches, experimental testing will be considered in order to validate the numerical models used for simulating the mechanical behavior of these materials. Also a detailed study of fluid filled close-cell cellular material should be performed in order to see how mechanical properties can be improved.

5. REFERENCES

Gibson, L. & Ashby M. (2001). Cellular Solids: Structure and Properties, Cambridge Solid State Science Series, ISBN 0 521 499119

Kamrani, A.K., & Nasr, E.A, (2005), Rapid Prototyping. Theory and Practice, Springer, ISBN: 0-387-23290-7

Popescu, D., Hadar, A. & Cotet, C., (2006), Fabricarea structurilor celulare solide cu celula inchisa din ABS P400 prin procedeul de fabricare rapida pe straturi Fused Deposition Modeling, Materiale Plastice, Vol. 43, No.2, 175-179, Bucharest, ISSN 0025-5289

Stampfl, J., Fouad, H. & Seidler, S., (2004), Fabrication and moulding of cellular materials by Rapid Prototyping, Materials and Product Technology, Vol. 21, No. 4

Vesenjak, M., Ochsner, A. & Ren, Z., (2005), Influence of Pore Gas in Closed-Cell Cellular Structures under Dynamic Loading, LS-DYNA Anwenderforum, Bamberg 2005, J-1-15-J-1-22

Tekoglu, C., (2007), Size effects in cellular solids, PhD thesis, Rijksuniversiteit Groningen.