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 onedimensional 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 onedimensional 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 prestrain 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 deformationdependent stiffness, snapthrough buckling and multistability.
Original language  English 

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
 multistability
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Dr Isaac V Chenchiah
 School of Mathematics  Senior Lecturer
 Fluids and materials
 Applied Mathematics
Person: Academic , Member