Abstract
Thanks to scale-bridging fabrication techniques, truss-based metamaterials have gained both popularity and complexity, ultimately resulting in structural networks whose description based on classical discrete numerical calculations becomes intractable. We here present a framework for the efficient and accurate simulation of large periodic three-dimensional (3D) truss networks undergoing nonlinear deformation (accounting for linear elastic beams undergoing finite rotations). Although the focus is on elastic beams, the method is sufficiently general to extend to inelastic material behavior. Our approach is based on a continuum representation of the truss (and its numerical implementation via finite elements) whose constitutive behavior is obtained from on-the-fly periodic homogenization at the microstructural unit cell level. We pursue a semi-analytical strategy (previously reported only in two dimensions) which admits the analytical calculation of consistent tangents for convergent implicit solution schemes; the extension to 3D – through the addition of torsional deformation modes and the handling of 3D rotations – results in a powerful tool for the prediction of the complex mechanical response of large structural networks. We validate the small-strain response by comparison to analytical solutions, followed by finite-strain benchmarks that compare simulation results to those of fully-resolved discrete calculations. The homogenization of beam unit cells results in a regularized macroscale model with an intrinsic length scale, which manifests especially when modeling bifurcations or localization. We finally apply our approach to macroscopic boundary value problems involving complex-shaped truss metamaterials (with truss unit cells near the body's boundary mapped onto a conformal surface), which reveal only an insignificant effect of boundary layers on the overall mechanical response, again supporting the applicability of our homogenization approach.
Highlights
To describe the response of large periodic structures such as those exploited in mechanical metamaterials (Greer and Deshpande, 2019), a number of analytical, numerical and semianalytical techniques have been developed (Kochmann et al, 2019) with the aim of replacing expensive discrete calculations by an efficient continuum approximation of the underlying finescale structural deformation mechanisms in macroscopic boundary value problems
We show that our 3D on-the-fly homogenization framework performs well in comparison to exact discrete truss simulations, serving as a computationally tractable alternative for the simulation of periodic truss architectures and truss metamaterials
We focus in the following on linear elastic beams, the overall setup and homogenization approach is sufficiently general to apply to other beam formulations and, in particular, to inelastic behavior through variational constitutive updates (see, e.g., Lestringant et al (2020) for suitable viscous and viscoelastic beam formulations)
Summary
To describe the response of large periodic structures such as those exploited in mechanical metamaterials (Greer and Deshpande, 2019), a number of analytical, numerical and semianalytical techniques have been developed (Kochmann et al, 2019) with the aim of replacing expensive discrete calculations by an efficient continuum approximation of the underlying finescale structural deformation mechanisms in macroscopic boundary value problems. Method (Phlipot and Kochmann, 2019), which localizes discrete calculations at full truss resolution in regions of interest while introducing an efficient continuum approximation elsewhere Such on-the-fly approaches tend to suffer from high computational expenses, which is why much of previous research has been limited to two-dimensional (2D) problems; see, e.g., Vigliotti et al (2014), Vigliotti and Pasini (2012). Vigliotti and Pasini (2012) extended their FE2 approach to 3D, yet computational expenses limited its application to relatively simple benchmarks without significant nonlinearity or localization The latter scenarios benefit from a model with an intrinsic length scale, which has been achieved, e.g., by higher-order homogenization schemes (Kouznetsova et al, 2002; Kouznetsova et al, 2004). We show that our 3D on-the-fly homogenization framework performs well in comparison to exact discrete truss simulations, serving as a computationally tractable alternative for the simulation of periodic truss architectures and truss metamaterials
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