Modeling of Flexible Beam Networks and Morphing Structures by Geometrically Exact Discrete Beams

Abstract

We demonstrate how a geometrically exact formulation of discrete slender beams can be generalized for the efficient simulation of complex networks of flexible beams by introducing rigid connections through special junction elements. The numerical framework, which is based on discrete differential geometry of framed curves in a time-discrete setting for time- and history-dependent constitutive models, is applicable to elastic and inelastic beams undergoing large rotations with and without natural curvature and actuation. Especially, the latter two aspects make our approach a versatile and efficient alternative to higher-dimensional finite element techniques frequently used, e.g., for the simulation of active, shape-morphing, and reconfigurable structures, as demonstrated by a suite of examples.

Effective continuum models for the buckling of non-periodic architected sheets that display quasi-mechanism behaviors

“…In light of this, engineers have used a variety of reduced order modeling techniques to investigate forward elastic equilibrium and stability problems, as well as to inversely design non-uniform mesostructures at a lesser computational expense. These techniques range from bar-and-hinge [37][38][39] and structural frame [40] models that capture the mechanics of folded sheets, to representations of structural element networks that are based on effective springs [41], equivalent lattices [42], Chebyshev nets [43], discrete elastic rods [43,44] and Kirchhoff rods [45].…”

Adaptation and validation of FFT methods for homogenization of lattice based materials

“…In order to design an optimal lattice for a specific property is fundamental to perform accurate and computationally efficient simulations of the effective response and the local fields developed within the microstructure. Although discrete mechanical models are a good approach for capturing the overall response [5], full-field homogenization approaches allow obtaining a more complete, and often more accurate, result. Under this approach, a micromechanical problem is solved on a representative volume element (RVE) which contains a unit cell or a collection of unit cells in which the lattice geometry is explicitly represented.…”

Numerical modeling of static equilibria and bifurcations in bigons and bigon rings

“…It is not uncommon for an elastic network to exhibit multistable equilibria, separated by unstable energy barriers that normally cannot be obtained in experiments [10,15]. Various numerical frameworks have been proposed to simulate mechanical behaviors of elastic rod and strip networks, such as finite element modeling [15], constrained nonlinear optimization [16], and discrete elastic rods [10,17,18]. The key idea is to simulate elastic networks as coupled rods, which are commonly modeled as elastica for planar branched structures [19], and a Kirchhoff rod or a more general Cosserat rod for spatial elastic networks [10,11,20].…”

“…The orientations of each strip's cross section are implemented through quaternions. Unlike discrete elastic rods that need additional treatment for specifying rotations at coupled joints [18], our formulation allows us to model rigid and flexible nodes by imposing either the continuities of orientations directly or including the constitutive laws of the flexible nodes. The method of formulating a multi-point BVP into a TPBVP is general, which could be applied to elastic networks that are built from general one dimensional structures, such as Cosserat rods [25] and inextensible strips [26].…”