Transient analysis of nonlinear locally resonant metamaterials via computational homogenization

Abstract

In this paper, the transient computational homogenization scheme is extended to allow for nonlinear elastodynamic phenomena. The framework is used to analyze wave propagation in a locally resonant metamaterial containing hyperelastic rubber-coated inclusions. The ability to properly simulate realistic nonlinearities in elasto-acoustic metamaterials constitutes a step forward in metamaterial design as, so far, the literature has focused only on academic nonlinear material models and simple lattice structures. The accuracy and efficiency of the framework are assessed by comparing the results with direct numerical simulations for transient dynamic analysis. It is found that the band gap features are adequately captured. The ability of the framework to perform accurate nonlinear transient dynamic analyses of finite-size structures is also demonstrated, along with the significant computational time savings achieved.

A general, implicit, finite-strain FE$$^2$$ framework for the simulation of dynamic problems on two scales

“…These averaging relations can be found in multiple frameworks dealing with homogenization of dynamics, see e.g., [14,31,32,45,51,52,60]. It was shown in [30] that the extended Hill-Mandel averaging relation can be applied in a discretized setting without introducing additional error by scale transition.…”

“…This was extended by [25] to account for matrix cracking at the microscale under impact loading. Other, rather FE 2 -type schemes as [32,33,51,52,56,60,61] calculate the full balance of linear momentum at the microscale. In [32] an explicit, periodic, small-strain framework is presented, which was extended to an implicit time integration method for modeling resonant elastic metamaterials in [33].…”

“…In addition, many require quite elaborate implementations. A recent publication [60] does fulfill the named requirements, however for the derivation of the macroscopic tangent moduli, specific nodes are assumed to experience no microscale fluctuation. This assumption is quite reasonable in the context of metamaterials, whose analysis was primary goal in [60], because there the regions at the unitcell boundaries are particularly stiff.…”

“…A recent publication [60] does fulfill the named requirements, however for the derivation of the macroscopic tangent moduli, specific nodes are assumed to experience no microscale fluctuation. This assumption is quite reasonable in the context of metamaterials, whose analysis was primary goal in [60], because there the regions at the unitcell boundaries are particularly stiff. For microscopic problems not specifically addressing classical metamaterial microstructures, however, more flexibility at the boundaries may be more reasonable which is why we propose an alternative approach here.…”

“…The conducted numerical studies are so far not specifically chosen to highlight this aspect of the framework. To permit a flexible damage evolution in the RVE, which is not dominated by periodic boundary conditions [6] or aforementioned assumptions regarding fixated fluctuations at specific boundary nodes [60], we propose to apply kinematic scale links as constraints on the whole RVE. This allows us to model any type of RVE morphology.…”