Nonlinear mechanics of non-rigid origami: an efficient computational approach

Abstract

Origami-inspired designs possess attractive applications to science and engineering (e.g. deployable, self-assembling, adaptable systems). The special geometric arrangement of panels and creases gives rise to unique mechanical properties of origami, such as reconfigurability, making origami designs well suited for tunable structures. Although often being ignored, origami structures exhibit additional soft modes beyond rigid folding due to the flexibility of thin sheets that further influence their behaviour. Actual behaviour of origami structures usually involves significant geometric nonlinearity, which amplifies the influence of additional soft modes. To investigate the nonlinear mechanics of origami structures with deformable panels, we present a structural engineering approach for simulating the nonlinear response of non-rigid origami structures. In this paper, we propose a fully nonlinear, displacement-based implicit formulation for performing static/quasi-static analyses of non-rigid origami structures based on 'bar-and-hinge' models. The formulation itself leads to an efficient and robust numerical implementation. Agreement between real models and numerical simulations demonstrates the ability of the proposed approach to capture key features of origami behaviour.

A Theory for the Design of Multi-Stable Morphing Structures

“…The extension of bar k, defined as its length in the infinitesimally deformed configuration minus the length in the target configuration (note that the extension is different from the Green-Lagrange strain derived in Kumar and Pellegrino (2000) and Liu and Paulino (2017)) is given by:…”

“…It has been shown in Liu and Paulino (2017) that, up to second-order infinitesimal quantities, the hinge rotation θ r induced by the displacements d r of nodes i, j, k, l is given by:…”

“…It has size 12 × 12 and consists of 4 × 4 sub-matrices of size 3 × 3. This matrix is symmetric, and hence there are only 10 independent sub-matrices, which are provided in Appendix A of Liu and Paulino (2017) (equations A (10) to A (20)). As an example, the 1,1 block is given by:…”

“…where ⊗ denotes a tensor product. Note that in Liu and Paulino (2017) the initial angle between the triangles, θ, is defined as the dihedral angle of the triangle pair. This is different from the present definition and hence the signs of all entries in C h r and H h r provided in Liu and Paulino (2017) need to be swapped.…”

“…For the second configuration: The calculation of the second-order compatibility matrices for the torsional springs involves a separate calculation for each spring. The Hessian matrices for the rotational springs are obtained from equations A(10) to A(20) in Appendix A of Liu and Paulino (2017). Nodes pinned to the ground are not included in the calculation.…”