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Application of the Finite Element Method to Modeling the Effective Mechanical and Thermomechanical Properties of Metamaterials of the 3D Lattice Structure


Metamaterials are composite materials, the properties of which are determined primarily by their geometric microstructure, and not by the properties of the components included in their composition. Such materials have a cellular structure and are manufactured using a 3D printer. The report describes an approach to a numerical estimation of effective (averaged) properties of metamaterials, based on the calculations on the periodicity cell of the material. Two types of metamaterials were studied. The first kind is metamaterials with negative Poisson's ratio (NPR), which when stretched in one direction expand in the other two. Such materials are used in medicine (stenting), in the manufacture of "smart" filters, to increase the strength of structures experiencing shock loads, in the manufacture of protective and damping materials. For NPR metamaterials, a numerical estimation of effective elastic properties was carried out - in particular, Poisson's ratios. Six boundary value problems of the elasticity theory with various periodic boundary conditions were solved at the periodicity cell of the material. Each problem corresponds to the particular case of effective strain tensor of a cell: three uniaxial tensions and three shear deformations. Distributions of stress tensors obtained as a result of solving problems were averaged over the volume. Effective elastic properties were estimated as a relation between the stress tensor and the strain tensor. We studied a dependence of the effective Poisson's ratios on the geometric parameters of the metamaterial cell. The parameters of cells, in which the Poisson's ratio comes to -1, were found. In addition, metamaterials with a negative thermal expansion coefficient (NTE), which shrink when heated, were studied. Such structures are made of two components: a harder one with a smaller thermal expansion coefficient and a softer one with a larger coefficient. To be precise, the materials that, when heated and cooling, do not change their sizes are of practical interest. They can be used in microchip devices, adhesive fillings, dental fillings and high-precision optical or mechanical devices in environments with varying temperatures. For NTE metamaterials, a numerical estimation of the effective thermal expansion coefficients was carried out. We solved the boundary value problem of thermoelasticity on a periodicity cell with a given heating value and periodic boundary conditions, allowing the cell to deform arbitrarily strongly, while remaining a periodicity cell. The strain tensor field, obtained as a result of solving the problem, was averaged over the volume. The effective thermal expansion was estimated in the form of a linear dependence of the effective deformation tensor on the heating value of the cell. We studied the dependence of the effective thermal expansion coefficients on the geometric parameters of the metamaterial cell. The parameters of the cells, under which the effective thermal expansion coefficient is negative and maximized by the module, are found - as well as the parameters under which it is almost equal to zero. Calculations of the cell stability were performed with the obtained parameters to thermal deformation for deeper analysis. The calculations of effective properties were carried out using the Fidesys Composite software module of CAE Fidesys. The report presents the results of calculations in the form of graphs of dependences of effective properties on the geometric parameters of the cell. The work was supported by the Russian President's grant for young scientists - Ph.Ds MD-208.2021.1.1.

Document Details

AuthorYakovlev. M
Date 28th October 2021
OrganisationFidesys LLC


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