COMPUTATIONAL ANALYSIS OF CHOSEN MICROSTRUCTURE SAMPLE

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1 COMPUTATIONAL ANALYSIS OF CHOSEN MICROSTRUCTURE SAMPLE Tadeusz Niezgoda, Danuta Miedzinska, Department of Mechanics and Applied Computer Science, Military University of Technology, Gen. Sylwestra Kaliskiego 2, Warsaw, Poland Abstract. One of the possible options as a material for protective layers are aluminium foams which become also very popular due to their lightweight and excellent plastic energy absorbing properties. Such characteristics have been appreciated by the automotive industry with continued research to further understand foam properties. Compressed foaming materials exhibit extensive plastic response, while the initial elastic region is limited in tension by a tensile brittle-failure stress. Aluminium foams have become an attractive material as blast protective layers due to their desirable compressive properties. With different material engineering techniques (as, for example double-layer foam cladding) they can be customized to achieve the most desirable properties. Energy absorption capacity of foams under blast load was analytically confirmed based on a rigid-perfectly plastic-locking foam model. Initial research indicates that energy absorbed by the cladding is much larger than that under quasistatic conditions due to shock wave effect. In this paper the creation process of a real foam microstructure model and its numerical analysis for uniaxial test is presented. Energy absorption of foams strictly depends on the microstructure geometry. From the numerical point of view, the analysed structural domain was described with Lagrangean formulation. Calculations were carried out using the so-called directintegration procedure, colloquially called the explicit integration. Performed research shows that the process selection of the FE model should be based on real foam geometry measurements. In the final part of these investigations the comparison process between numerical and experimental test was performed and presented results shows a good convergence. Keywords: Foam, aluminum, microstructure modeling, experimental and numerical studies 1. INTRODUCTION Porous materials are still quite new and unknown structures. The most popular among them lar are aluminum foams. They are characterized by low density and processing easiness. Also, for example open-cell aluminum foams have very high sound absorbing properties. Foamed materials under the compression load show an extensive plastic response, while the initial elastic region is limited in tension by a tensile brittle-failure stress. The energy absorption ability of foam samples during the crush test was analytically confirmed on the basis of a rigid-perfectly plastic-locking foam model In this paper the development process of a real foam microstructure finite element model and its numerical analysis under an uniaxial compression load is presented. Then the real foam sample model is compared to an idealistic microstructure geometry sample made with the usage of Kelvin tetrakaidecahedrons which have the same shape, size and analysis boundary conditions as the real foam one. The idealistic geometry foam sample numerical model was created with the use of Kelvin tetracaidecahedron structure (Reinelt A., 1996). The model was develop for the comparison of such idealistic, easy to create geometry to the real geometry structure described above. The Kelvin geometry was often used by other authors to estimate the behavior of the foam microstructure (Kırcaa M., 2007; Mills N. J., 2005). The Kelvin structure has a 4 fold rotational symmetry axis at the centre of the square faces (what is assumed in the model) and two mirror planes. When the axis are placed at the hexagonal walls, the structure shows anisotropic behavior. An uniaxial compression test for an aluminum foam sample was carried out. Also the comparison with the experimental results is shown. 2. EXPERIMENTAL RESULTS Quasi static compression test were provided on the INSTRON testing machine with 3000kN head. The friction between the sample and the compression plates were minimized with the grease usage. The force and displacement sensors were used. The results (the load vs. displacement charts) are presented in Fig. 1.

2 Figure 1. The experimental results for the uniaxial compression test 3. NUMERICAL MODELING OF A REAL FOAM SAMPLE For the numerical modeling of the sample a real aluminum closed-cell foam sample was used. The sample was shaped as a cube with the dimension of 35x35x35mm. For preparing a the geometry of the FEM model the thin slices of 0.35mm were removed one by one from a top surface of a foam cube. The process and the used equipment is shown in Fig. 2. Figure 2.The research process and the equipment for the model geometry creation Then the pores were filled with red wax (Fig. 3) and a picture of every slice was taken. Figure 3. A foam sample with pores filled with red wax

3 With the use of the graphic processing software the pictures were converted to 100 bitmaps of 100x100 pixels in black and white (Fig. 4). Figure 4. The conversion of the real foam structure picture to black and white bitmap Thanks to applying solid elements Hex8 (Marc 2007 r1 User Guide) a numerical model based on a position of pixels in each bitmap was built. Each white pixel representing the material was transformed to a finite element. The reconstructed cube had a dimension of 35x35x35mm consisting of 100x100x100 cubes of them represented the material elements (Fig. 5). Porosity of the numerical model was 65% in comparison to porosity of a real foam of 80%. The difference was caused by the limits of the graphical software (e.g. a size of pixels). A dynamic numerical analysis was carried out with the use of LS Dyna computer code. The compression was performed with two rigid plates (LS Dyna rigid walls) a stationary and a moving one. The frictionless contact was applied. A piecewise linear plastic material model was used for aluminum (Young modulus E=71000MPa, Poisson ratio ν=0,33, yield stress R e =250MPa). Dynamic equations were implemented. For the domain, the governing equation is the conservation of momentum. By expressing equilibrium in the current configuration and based on Finite Elements, the set of discrete differential equations is presented as follows: M ü=f ext - e V e B T σdv (1) Here is the tensor representing Cauchy stress in the structure, is a mass matrix, is the vector of nodal accelerations, is the vector of external forces, is the matrix of shape functions derivatives and describes the element e volume. An underlying is related to vector, matrix or tensor quantities. For the solution stability the Couranta Lewy condition were applied. It is described by the following equation: Δt=L e /(Q+(Q 2 +c 2 ) 1/2 ) (2) where Q is the function of the bulk viscosity coefficients, c is the adiabatic sound speed, L e characteristic length of element. The elastic-plastic material model with isotropic hardening was applied to describe the material properties for aluminum (Young modulus E=71000MPa, Poisson ratio ν=0,33, yield stress R e =250MPa). The yield condition is defined in this model as: φ=1/2 s ij s ij -σ y 2 /a=0 (3) where σ y= σ 0 +βe p ε eff p and β is a parameter between 0 and 1 representing kinematic and hardening effects respectively. The penalty function was applied to assess the normal and friction contact force: F nij =ζu nij H(-u nij ) (4) where H( ) is the Heavisidea function, ζ=1/κ, κ penalty coefficient, u n - normal deformation.

4 Figure 5. Numerical model of a real foam structure 4. NUMERICAL MODELING OF AN IDEALISTIC FOAM SAMPLE The sample, shown in Fig. 6, had the dimension of 35x35x35 mm. It was built from shell elements Quad4 (Marc 2007 r1 User Guide) The thickness of the elements was 0.3 mm what resulted in the porosity of 70%. An elastic plastic material model was used for aluminum. A dynamic numerical analysis was carried out with the use of LS Dyna computer code. The same boundary conditions as in the previous model were applied. Figure 6.The numerical model of an idealistic foam structure

5 5. RESULTS During the analysis the relation between the load and the displacement was studied. The results for both numerical analysis and for the experiment are presented in a chart below (Fig. 7). Also, the deformations of the whole structures were presented in Fig. 8. The numerical models were developed in accordance to the experiment. It is noticeable that the real foam numerical structure has a larger stiffness than the real foam. It is caused either by higher porosity of the numerical model comparing with the real foam, or by the inertia of the hitting plate. The same fact is noticeable for an idealistic foam structure what can be caused by the special shape of the pores (Kelvin unit cell) which causes the material surplus at the edges of the model. Figure 7.The load displacement chart for: the experiment, the real foam structure FEM analysis and the idealistic foam structure numerical analysis There are visible differences in the deformations for both FEM models. The real foam structure model shows the behavior similar to the experimental one. Small differences in the transverse cross-section area size are presented. Also the uniform compression process in the whole sample volume was reached. The idealistic foam model shows larger differences in comparison to the real foam compression behavior. The great influence of a shock wave effect is visible. The deformation begins at the bottom of the sample and is irregular in the sample volume. It can be concluded that the modeling based on the real structure geometry is much better to investigate the strength behavior of the foamed material. The value of absorbed energy per mass unit was calculated for the experimental and numerical results what is shown in Tab. 1. The total absolute energy value was calculated as the area under the displacement load curve presented in Fig. 7 for the displacement range from 0 to 10 mm. and then divided by the appropriate sample mass. Table 1. Absorbed energy per mass unit Total absorbed energy per mass unit Experiment Real foam structure FEM analysis Idealistic foam structure FEM Analysis 7374 kj/kg 8186kJ/kg 9225kJ/kg It is shown that the FEM model absorbed more energy than the real foam in both cases, because of the difference between the character of compression processes (static and dynamic). Also a larger difference in the absorbed energy value for an idealistic geometry model is caused by its bigger stiffness described above. 6. REFERENCES Kusner R., Sullivan J. M., Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin s foam, Forma, Vol. 11 (No. 3), pp (1996) Marc 2007 r1 User Guide

6 Reinelt A., Linear Elastic Behavior of Dry Soap, Journal of Colloid and Interface Science 181, Niezgoda T., Małachowski J., Szymczyk W., Modelowanie numeryczne mikrostruktury ceramiki, WNT Hughes T., Hinton E., Finite Element Methods For Plate and Shell Structures, Pineridge Press, UK Mills N. J., The wet Kelvin model for air flow through open-cell polyurethane foams, Journal of Materials Science 40, Kırcaa M., Gülb A., Ekincic E., Yardımc F., Mugana A., Computational modeling of micro-cellular carbon foams, Finite Elements in Analysis and Design 44, Real foam structure Idealistic foam structure 5% compression (displacement = 1,75 mm) 20% compression(displacement = 7 mm) 40% compression (displacement = 14 mm) 60% compression (displacement = 28 mm) Figure 8. The deformations for the real foam structure and the idealistic foam structure numerical analysis 7. ACKNOWLEDGEMENTS The paper was supported by a grant No , financed in the years by Ministry of Science and Higher Education, Poland. 8. RESPONSIBILITY NOTICE The authors are the only responsible for the printed material included in this paper.