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Finite Element Analysis of Foam Models Based on Regular and Irregular Arrays of Cubic Open Cells Having Uniform or Normal Distributions

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Advanced Materials Modelling for Mechanical, Medical and Biological Applications

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 155))

Abstract

The paper considers models of foam materials in the form of Gibson–Ashby cell arrays. The method of effective moduli, based on the equality of the potential energies of the porous and homogeneous material, is described. Six boundary value problems of the linear static elasticity theory for a representative volume are given. These problems together allow determining all the coefficients of the effective stiffness matrix for any anisotropy class of the frame material and geometric asymmetry. The finite element package ANSYS and the capabilities of its command language APDL are used to construct representative volumes and to solve homogenization problems numerically. The procedures for creating solid and finite element models of arrays composed of open Gibson–Ashby cells with regular and irregular structure are described in detail. Two different algorithms for regular lattices of low and high porosity are offered. For an irregular lattice, the sizes of the cube frames are randomly generated with a uniform and normal distribution. The results of numerical calculations for stainless steel lattices in a wide range of porosity are presented. The dependencies of the effective elastic moduli on porosity for a single Gibson–Ashby cell and for regular and irregular lattices with uniform and normal distribution are analyzed. It is shown that the applied Gibson–Ashby model predicts the elastic properties of highly porous materials quite well. But the prediction for lattices with porosity less than 75% gives a sufficiently large error. It is noted that regular and irregular lattices with a large number of cells give similar results for effective elastic stiffness moduli. Meanwhile, individual irregular structures with strongly different sizes of cubic cell frames can take extreme values of effective moduli with pronounced anisotropy in different directions. These effects depend on the geometric asymmetry of the irregular lattices and on the stress concentrations. Examples of the stress–strain state in strongly irregular Gibson–Ashby lattices are given. The analysis of the value scatter of various effective elastic moduli, demonstrating the anisotropy of strongly irregular Gibson–Ashby lattices, is given.

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Acknowledgements

This research was supported by the Government of the Russian Federation, contract No. 075-15-2019-1928, and by Russian Foundation for Basic Research, project No. 20-31-90057.

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Authors and Affiliations

  1. Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Milchakova Street 8a, Rostov-on-Don, 344090, Russia

    Alexandr S. Kornievsky & Andrey V. Nasedkin

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  1. Alexandr S. Kornievsky
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  2. Andrey V. Nasedkin
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Correspondence to Andrey V. Nasedkin .

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Editors and Affiliations

  1. Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. DICAAR, Università degli Studi di Cagliari, Cagliari, Italy

    Victor A. Eremeyev

  3. Schmidt Institute of Physics of the Earth, Moscow, Russia

    Alexander Galybin

  4. Don State Technical University, Rostov-on-Don, Russia

    Andrey Vasiliev

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Kornievsky, A.S., Nasedkin, A.V. (2022). Finite Element Analysis of Foam Models Based on Regular and Irregular Arrays of Cubic Open Cells Having Uniform or Normal Distributions. In: Altenbach, H., Eremeyev, V.A., Galybin, A., Vasiliev, A. (eds) Advanced Materials Modelling for Mechanical, Medical and Biological Applications. Advanced Structured Materials, vol 155. Springer, Cham. https://doi.org/10.1007/978-3-030-81705-3_15

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