Biomimetic Turing machine: A multiscale theoretical framework for the inverse design of target space curves

Abstract

Morphing ribbons and their inverse design are usually confined to plane curves, since in most cases only the curvature is considered. Given that curvature and torsion are equally important geometric characteristics of space curves, it is urgent to propose a systematic theoretical framework for the inverse design. Toward this end, we here present a multiscale theoretical framework named biomimetic Turing machine (BTM) to achieve desired target space curves, which is inspired from two microstructural regulation mechanisms behind the hydration-driven morphing of plant tissues: the graded curvature regulated by matrix volume fraction (c_m) and the helix-like morphology regulated by fibril orientation angle (FOA). By analogizing to Turing machine encoded by binary mapping, the proposed BTM can inversely encode a morphing ribbon with preset microstructural parameters (FOA and c_m) to achieve desired target space curves. The proposed theoretical framework can first bridge the microstructural fiber-matrix swelling and the macroscopic ribbon morphing as a forward problem, in which a twist field is subsequently introduced to create the kinematic map between the target space curve and the ribbon, innovatively posing the inverse design as an initial value problem. To facilitate the experimental implementation of BTM, we further propose an optimization strategy for selecting the twist field and provide design criteria as guidelines for experiments. As a conceptual display, we present a phase diagram in the c_m versus FOA plane to illustrate the complex target morphologies (e.g., hemisphere, hyperboloid, and tendril) characterized by various parameters of curvature and torsion designed rationally by the BTM theory, while in previous studies the morphing morphologies (e.g., helices, arcs, and helicoid ribbons) exhibit only constant curvature or torsion. This work presents a novel inverse design strategy for space curves with both curvature and torsion, broadening the potential for the design and fabrication of morphing materials.