A quasicontinuum theory for the nonlinear mechanical response of general periodic truss lattices

Abstract

We present a framework for the efficient, yet accurate description of general periodic truss networks based on concepts of the quasicontinuum (QC) method. Previous research in coarse-grained truss models has focused either on simple bar trusses or on two-dimensional beam lattices undergoing small deformations. Here we extend the truss QC methodology to nonlinear deformations, general periodic beam lattices, and three dimensions. We introduce geometric nonlinearity into the model by using a corotational beam description at the level of individual truss members. Coarse-graining is achieved by the introduction of representative unit cells and an affine interpolation analogous to traditional QC. General periodic lattices defined by the periodic assembly of a single unit cell are modeled by retaining all unique degrees of freedom of the unit cell (identified by a lattice decomposition into simple Bravais lattices) at each macroscopic point in the simulation, and interpolating each degree of freedom individually. We show that this interpolation scheme accurately captures the homogenized properties of periodic truss lattices for uniform deformations. In order to showcase the efficiency and accuracy of the method, we perform simulations to predict the brittle fracture toughness of multiple lattice architectures and compare them to results obtained from significantly more expensive discrete truss simulations. Finally, we demonstrate the applicability of the method for nonlinear elastic truss lattices undergoing finite deformations. Overall, the new technique shows convincing agreement with exact, discrete results for most lattice architectures, and offers opportunities to reduce computational expenses in structural lattice simulations and thus to efficiently extract the effective mechanical performance of discrete networks.

A refinement indicator for adaptive quasicontinuum approaches for structural lattices

“…Essential to realize is that the beam nodes in the cell of Figure 1 are connected to their neighboring beam nodes in different ways. As a distinctive character, the generalized QC method 24 classifies the beam nodes of a cell into different types ‐ based on their connectivity, which is identical to the decomposition of multi‐lattice into several single Bravais lattices in the realm of atomistics 23,25 . The DoFs of each type of beam node are interpolated independently.…”

“…Recently however, Chen et al 24 and Phlipot and Kochmann 25 have proposed generalization in which a more complex connectivity of the lattice is considered (or multi‐lattice in atomistics terminology 23 ). As a result, the method can also be applied to more complex lattice models, in which each lattice node inside a periodic unit cell is connected to its neighboring nodes in different patterns—using interactions of which the mechanical response can vary.…”

“…For this reason, most of the refinement indicators in the realm of structural lattices are based on quantities that are relevant for regular atomistic lattices and metals 3,18,31,32 . For instance, Phlipot and Kochmann 25 have used a refinement indicator based on the (continuum derived) J 2 invariant of the deformation, even though their meso‐structural lattice represents a discrete multi‐lattice 23 . A disadvantage of this indicator is that refinement may also be triggered for homogeneous deformations and rigid body rotations.…”

“…A disadvantage for its use in the generalized variant of the QC method, which effectively superimposes as many finite element shape functions as classes of lattice nodes present in the lattice, is that no macroscopic deformation gradient can be recognized. One possibility would therefore be to combine the Zienkiewicz‐Zhu indicator of Mikeš et al 18 with the approach of Phlipot and Kochmann 25 in which as many macroscale deformation gradient tensors are distinguished as classes of lattice nodes are present.…”

An adaptive quasicontinuum approach for modeling fracture in networked materials: Application to modeling of polymer networks

“…For example, Mikes and Jirasek [24] extended the quasi continuum method to irregular linear elastic lattices with only axial interactions and small deformations. Beex et al [25] extended the method to small deformation uniform linear elastic beam-like lattices for both in plane and out of plane deformation, and Phlipot et al [26] used the method to model linear elastic periodic beam lattices with co-rotational framework. The flexibility of the framework makes it a suitable candidate for applications to irregular polymer networks with both material and geometric nonlinearities, which is the focus of this study.…”