A Newton Solver for Micromorphic Computational Homogenization Enabling Multiscale Buckling Analysis of Pattern-Transforming Metamaterials

Abstract

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour. Such a behaviour is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical macroscale predictions, a micromorphic computational homogenization scheme has recently been developed. Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. To address them, a full Newton method is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge.

34 pages, 17 figures, 1 table, 1 algorithm, abstract shortened to fulfill 1920 character limit

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Cited by in corpus (7)

• A Comparative Study of Enriched Computational Homogenization Schemes Applied to Two-Dimensional Pattern-Transforming Elastomeric Mechanical Metamaterials
• Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterials
• Wang tiles enable combinatorial design and robot-assisted manufacturing of modular mechanical metamaterials
• Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials
• Computational frameworks for homogenization and multiscale stability analyses of nonlinear periodic metamaterials
• Harvesting Deformation Modes for Micromorphic Homogenization from Experiments on Mechanical Metamaterials
• Towards active stiffness control in pattern-forming pneumatic metamaterials