A mixed-order quasicontinuum approach for beam-based architected materials with application to fracture

TL;DR

The paper tackles the challenge of predicting fracture in large-beam architected materials by developing a fully-nonlocal quasicontinuum (QC) framework on multi-lattice networks. A mixed-order interpolation is introduced, combining first-order (linear) elements in fully-resolved regions with second-order (quadratic) elements in coarse regions to overcome stretch locking in bending-dominated lattices, while leveraging repUCs and energy-based sampling for efficiency. The approach is validated in 2D and 3D, showing accurate fracture toughness predictions, substantial reductions in degrees of freedom, and the ability to capture diverse stress distributions across lattice topologies. The results demonstrate significant computational savings without compromising accuracy, enabling robust exploration of fracture in large-scale beam lattices and guiding design of architected materials; future work includes inelasticity and crack propagation.

Abstract

Predicting the mechanics of large structural networks, such as beam-based architected materials, requires a multiscale computational strategy that preserves information about the discrete structure while being applicable to large assemblies of struts. Especially the fracture properties of such beam lattices necessitate a two-scale modeling strategy, since the fracture toughness depends on discrete beam failure events, while the application of remote loads requires large simulation domains. As classical homogenization techniques fail in the absence of a separation of scales at the crack tip, we present a concurrent multiscale technique: a fully-nonlocal quasicontinuum (QC) multi-lattice formulation for beam networks, based on a conforming mesh. Like the original atomistic QC formulation, we maintain discrete resolution where needed (such as around a crack tip) while efficiently coarse-graining in the remaining simulation domain. A key challenge is a suitable model in the coarse-grained domain, where classical QC uses affine interpolations. This formulation fails in bending-dominated lattices, as it overconstrains the lattice by preventing bending without stretching of beams. Therefore, we here present a beam QC formulation based on mixed-order interpolation in the coarse-grained region -- combining the efficiency of linear interpolation where possible with the accuracy advantages of quadratic interpolation where needed. This results in a powerful computational framework, which, as we demonstrate through our validation and benchmark examples, overcomes the deficiencies of previous QC formulations and enables, e.g., the prediction of the fracture toughness and the diverse nature of stress distributions of stretching- and bending-dominated beam lattices in two and three dimensions.

Figures (16)

• Figure 1: Schematic overview of the truss QC method. (a) A finite element mesh is used to bridge between regions of full resolution (such as around a crack), here in dark gray and coarse-grained regions (in light gray). (b) Coarse-graining is based on representative UCs (RepUCs), shown in gray, which carry DOFs whose interpolation approximates the deformation of all other UCs (indicated by the dashed lines). (c) A hexagonal unit cell (UC), shown in gray, is spanned by the basis vectors $\boldsymbol{a}_1$ and $\boldsymbol{a}_2$. Nodes within the unit cell and the beams connecting those (i.e., elements of $\mathcal{E}_u^i$) are colored in green. Beams that connect the UC with neighboring ones (i.e., elements of $\mathcal{E}^n_u$) are colored in red. The DOFs of both nodes within the UC make up the DOFs $\Tilde{\boldsymbol{\varphi}}$ of the UC.
• Figure 2: (a) Square, (b) triangular, (c) hexagonal, (d) star-shaped 2D lattices with their respective UCs highlighted in gray and nodes within the UC shown in green. (e) Tetrakaidecahedral and (f) octet UCs in 3D, highlighting in green again those nodes that belong to the UC.
• Figure 3: Schematic of the selection of repUCs from the full point set $\mathcal{S}_{\text{UC}}$ of UCs (dashed lines) and the creation of simplicial meshes. (a) A possible selection of repUCs (gray) with even Bravais coordinates; (b) the addition of mid-edge repUCs from the set $\mathcal{S}_{\text{mid}}$. (c) Meshing of this set of repUCs leads to either eight first-order or two second-order macroscopic elements, shown in gray.
• Figure 4: Schematic overview of the different elements and the respective choice of sampling UCs for (a) and (c) first-order elements, and (b) and (d) second-order elements in 2D and 3D, respectively. Sampling UCs that are also repUCs are shown in red, additional sampling UCs in gray.
• Figure 5: Schematic showing how the weights of the sampling UCs at a barycenter are calculated in three steps: (a) a macroscopic element (shown in gray) with a box (colored in blue) that fully contains the element; (b) the box is filled with UC locations (shown as dashed lines) as if it was fully resolved; (c) calculation of the distances of each UC to the macroscopic element (UCs with a negative weight are colored in green, others are transparent). UCs on vertices and edges have a distance of zero.