Abstract
One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to H ^1, such that standard nodal H ^1-finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces H ^1 and H (mathrm {curl}), demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.
Highlights
Materials with a pronounced microstructure such as metamaterials, see e.g. [2,3,7,19], porous media, composites etc., activate micro-motions which are not accounted for in classical continuum mechanics, where each material point is equipped with only three translational degrees of freedom
The tests show its inability to find the exact solution for discontinuous microdistortion fields and the corresponding sub-optimal convergence
Comparison between the linear nodal and hybrid element formulations reveals the difference in the arising elemental stiffness matrices, namely K nodal ∈ R12×12 and K hybrid ∈ R8×8, resulting in slower computation times for the nodal element
Summary
Materials with a pronounced microstructure such as metamaterials, see e.g. [2,3,7,19], porous media, composites etc., activate micro-motions which are not accounted for in classical continuum mechanics, where each material point is equipped with only three translational degrees of freedom. In this paper we consider finite element formulations employing either H1 ×[H1]2 or H1 × H(curl) and investigate their validity in correctly approximating results in the relaxed micromorphic continuum. We test both a primal and mixed formulation of the corresponding boundary problem for increasingly large values of the characteristic length Lc. To that end, we consider a planar version of the relaxed micromorphic continuum, namely of antiplane shear [43]. We present several numerical examples to confirm the theoretical results
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