### 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 >0,we construct a system whose energy is approximately E up to an additive constant, with L∞-error no more that . 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.

Original language | English |
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Article number | 0547 |

Number of pages | 28 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 475 |

Issue number | 2232 |

DOIs | |

Publication status | Published - 4 Dec 2019 |

### Keywords

- nonlinear spring
- bespoke stiffness
- lattice
- metamaterials
- anisotropy
- multi-stability

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## Profiles

## Dr Isaac V Chenchiah

- School of Mathematics - Senior Lecturer
- Fluids and materials
- Applied Mathematics

Person: Academic , Member