O Ondrej Rokos

Size effects in nonlinear periodic materials exhibiting reversible pattern transformations

Abstract This paper focuses on size effects in periodic mechanical metamaterials driven by reversible pattern transformations due to local elastic buckling instabilities in their microstructure. Two distinct loading cases are studied: compression and bending, in which the material exhibits pattern transformation in the whole structure or only partially. The ratio between the height of the specimen and the size of a unit cell is defined as the scale ratio. A family of shifted microstructures, corresponding to all possible arrangements of the microstructure relative to the external boundary, is considered in order to determine the ensemble averaged solution computed for each scale ratio. In the compression case, the top and the bottom edges of the specimens are fully constrained, which introduces boundary layers with restricted pattern transformation. In the bending case, the top and bottom edges are free boundaries resulting in compliant boundary layers, whereas additional size effects emerge from imposed strain gradient. For comparison, the classical homogenization solution is computed and shown to match well with the ensemble averaged numerical solution only for very large scale ratios. For smaller scale ratios, where a size effect dominates, the classical homogenization no longer applies.

An adaptive variational quasicontinuum methodology for lattice networks with localized damage

Lattice networks with dissipative interactions can be used to describe the mechanics of discrete meso-structures of materials such as 3D-printed structures and foams. This contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture. Lattice networks naturally incorporate non-locality, large deformations, and dissipative mechanisms taking place inside fracture zones. Because the physically relevant length scales are significantly larger than those of individual interactions, discrete models are computationally expensive. The Quasicontinuum (QC) method is a multiscale approach specifically constructed for discrete models. This method reduces the computational cost by fully resolving the underlying lattice only in regions of interest, while coarsening elsewhere. In this contribution, the (variational) QC is applied to damageable lattices for engineering-scale predictions. To deal with the spatially evolving fracture zone, an adaptive scheme is proposed. Implications induced by the adaptive procedure are discussed from the energy-consistency point of view, and theoretical considerations are demonstrated on two examples. The first one serves as a proof of concept, illustrates the consistency of the adaptive schemes, and presents errors in energies. The second one demonstrates the performance of the adaptive QC scheme for a more complex problem.

Human-Induced Loads on Grandstands as Non-Stationary Gaussian Processes

In this contribution, we employ non-stationary filtered Gaussian processes as an enrichment of a periodic mean value in order to approximate crowd loads on grandstands. Our work generalizes previous considerations where the superposition of a mean value and a stationary filtered Gaussian noise was used, and helps therefore to better predict the response of a structure mainly in the transition stages. We specify general theory of stochastic differential equations within the context of grandstands by recalling particular moment equations, and demonstrate its benefits or drawbacks on two simple examples. Overall performance is measured in terms of the second moment evolutions in time and in terms of the total up-crossings of the systems response compared to previously developed stationary approximation and Monte Carlo simulation. Throughout, only an active part of a crowd is considered.

Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields

Abstract This paper presents a homogenization framework for elastomeric metamaterials exhibiting long-range correlated fluctuation fields. Based on full-scale numerical simulations on a class of such materials, an ansatz is proposed that allows to decompose the kinematics into three parts, i.e. a smooth mean displacement field, a long-range correlated fluctuating field, and a local microfluctuation part. With this decomposition, a homogenized solution is defined by ensemble averaging the solutions obtained from a family of translated microstructural realizations. Minimizing the resulting homogenized energy, a micromorphic continuum emerges in terms of the average displacement and the amplitude of the patterning long-range microstructural fluctuation fields. Since full integration of the ensemble averaged global energy (and hence also the corresponding Euler–Lagrange equations) is computationally prohibitive, a more efficient approximative computational framework is developed. The framework relies on local energy density approximations in the neighbourhood of the considered Gauss integration points, while taking into account the smoothness properties of the effective fields and periodicity of the microfluctuation pattern. Finally, the implementation of the proposed methodology is briefly outlined and its performance is demonstrated by comparing its predictions against full scale simulations of a representative example.