Auxetic orthotropic materials: Numerical determination of a phenomenological spline-based stored density energy and its implementation for finite element analysis

Abstract Auxetic materials, which have negative Poisson’s ratio, show potential to be used in many interesting applications. Finite element analysis (FEA) is an important phase in implementing auxetic materials, but may become computationally expensive because simulation often needs microscale details and a fine mesh. It is also necessary to check that topological aspects of the microscale reflects not only micro but macromechanical behavior. This work presents a phenomenological approach to the problem using data-driven spline-based techniques to properly characterize orthotropic auxetic material requiring neither analytical constraints nor micromechanics, expanding on previous methods for isotropic materials. Hyperelastic energies of auxetic orthotropic material are determined from experimental data by solving the equilibrium differential functional equations directly, so no fitting or analytical estimation is necessary. This offers two advantages; (i) it allows the FEA study of orthotropic auxetic materials without requiring micromechanics considerations, reducing modeling and computational time costs by two to three orders of magnitude; (ii) it adapts the hyperelastic energies to the nature of the material with precision, which could be critical in scenarios where accuracy is essential (e.g. robotic surgery)

Numerical homogenization of foam-like structures based on the FE2- approach

The computation of foamlike structures is still a topic of research. There are two basic approaches: the microscopic model where the foamlike structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE2-approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point. In this contribution, we present a twodimensional geometrically nonlinear FE2-framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p-version. The p-version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuumbased approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beamlike structures. In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed

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