Bespoke extensional elasticity through helical lattice systems

Abstract

Nonlinear structural behaviour offers a richness of response that cannot be replicated within a traditional linear design paradigm. However, designing robust and reliable nonlinearity remains a challenge, in part, due to the difficulty in describing the behaviour of nonlinear systems in an intuitive manner. Here, we present an approach that overcomes this difficulty by constructing an effectively one-dimensional system that can be tuned to produce bespoke nonlinear responses in a systematic and understandable manner. Specifically, given a continuous energy function E and a tolerance ϵ > 0, we construct a system whose energy is approximately E up to an additive constant, with L∞-error no more that ϵ. The system is composed of helical lattices that act as one-dimensional nonlinear springs in parallel. We demonstrate that the energy of the system can approximate any polynomial and, thus, by Weierstrass approximation theorem, any continuous function. We implement an algorithm to tune the geometry, stiffness and pre-strain of each lattice to obtain the desired system behaviour systematically. Examples are provided to show the richness of the design space and highlight how the system can exhibit increasingly complex behaviours including tailored deformation-dependent stiffness, snap-through buckling and multi-stability.

Exploring Adaptive Behavior of Non-linear Hexagonal Frameworks

“…The energy profile of such elements can be tuned by tailoring their characteristic properties and can even match a target energy profile with an arbitrary level of accuracy (Dixon et al, 2019). However for computational ease we choose l p = l 0 +1…”

“…Furthermore, the helical lattice can be tuned to exploit external fields, such as thermal effects (O'Donnell et al, 2019; McHale et al, 2020a), to modulate the effective stiffness and stability landscape. The potential of helical lattices to obtain truly bespoke elastic responses was considered by Dixon et al (2019). By exploiting the helical lattice as a structural base-unit for nonlinear systems of coupled lattices, an algorithmic tuning process that can match, to an arbitrary precision, any energetic function up to a constant was developed (Dixon et al, 2019).…”

“…The potential of helical lattices to obtain truly bespoke elastic responses was considered by Dixon et al (2019). By exploiting the helical lattice as a structural base-unit for nonlinear systems of coupled lattices, an algorithmic tuning process that can match, to an arbitrary precision, any energetic function up to a constant was developed (Dixon et al, 2019).…”

“…Emergent behavior is not restricted to the helical lattice alone and there exists a range of non-linear units from which emergent behavior is observed as system complexity is increased, such as cylindrical ribbons (Lachenal et al, 2012;Aza et al, 2019;Carey et al, 2019) and snapping arches (Hunt and Dodwell, 2019). This paper seeks to extend the design concepts explored in Dixon et al (2019) beyond their current one-dimensional nature through an exploration of hierarchical space-filling lattices. In doing so, we take one step closer toward the design of truly bespoke (meta-)materials.…”

Visualising Compliance of Composite Shell Mechanisms

“…Refs. [12][13][14][15]. Through careful tailoring of structural form and the composites' anisotropic stiffness, bespoke nonlinear responses can be readily achieved.…”