In this paper, we speculate on a possible application of Liquid Crystal Elastomers to the eld of soft robotics. In particular, we study a concept for limbless locomotion that is amenable to miniaturisation. For this purpose, we formulate and solve the evolution equations for a strip of nematic elastomer, subject to directional frictional interactions with a at solid substrate, and cyclically actuated by a spatially uniform, time-periodic stimulus (e.g., temperature change). The presence of frictional forces that are sensitive to the direction of sliding transforms reciprocal,`breathing-like' deformations into directed forward motion. We derive formulas quantifying this motion in the case of distributed friction, by solving a dierential inclusion for the displacement eld. The simpler case of concentrated frictional interactions at the two ends of the strip is also solved, in order to provide a benchmark to compare the continuously distributed case with a nite-dimensional benchmark. We also provide explicit formulas for the axial force along the crawler body.

This paper applies the theory of rate-independent systems to model the locomotion of bio-mimetic soft crawlers. We prove the well-posedness of the approach and illustrate how the various strategies adopted by crawlers to achieve locomotion, such as friction anisotropy, complex shape changes and control on the friction coefficients, can be effectively described in terms of stasis domains.Compared to other rate-independent systems, locomotion models do not present any Dirichlet boundary condition, so that all rigid translations are admissible displacements, resulting in a non-coercivity of the energy term. We prove that existence and uniqueness of solution are guaranteed under suitable assumptions on the dissipation potential. Such results are then extended to the case of time-dependent dissipation.arXiv:1710.08340v2 [math.FA] 30 Dec 2017 p. gidoni: Rate-independent soft crawlers Physically, this equation is a force balance: configurational forces (e.g. tension in an elastic body) are described as the spatial gradient of the internal energy E, while frictional forces are obtained as the subdifferential of a dissipation potential R, assumed to be positively homogeneous of degree one, in order to guarantee rate-independence.The variational structure of the problem has favoured the development of an advanced and extensive mathematical theory of rate-independent systems, assisting and inspired by applications in physics of solids and continuum mechanics, modelling phenomena such as elastoplasticity, fracture, damage, phase transitions.Along with these, systems with dry friction has immediately emerged as one of the most fitting applications of the theory, already from the first major achievements in the 70s, with the introduction of Moreau's Sweeping processes [49]. As today, dry friction still contributes to the development of the theory. We mention the recent results adopting multiscale approaches to study the effective friction in the case of fast-oscillating bodies [32] and of hairy surfaces [28]; or the illustration of the properties of non-convex rate independent systems using using simple, representative dry friction toy models [2].

In this paper we study crawling locomotion based on directional frictional interactions, namely, frictional forces that are sensitive to the sign of the sliding velocity. Surface interactions of this type are common in biology, where they arise from the presence of inclined hairs or scales at the crawler/substrate interface, leading to low resistance when sliding 'along the grain', and high resistance when sliding 'against the grain'. This asymmetry can be exploited for locomotion, in a way analogous to what is done in cross-country skiing �classic style, diagonal stride).We focus on a model system, namely, a continuous one-dimensional crawler and provide a detailed study of the motion resulting from several strategies of shape change. In particular, we provide explicit formulae for the displacements attainable with reciprocal extensions and contractions �breathing), or through the propagation of extension or contraction waves. We believe that our results will prove particularly helpful for the study of biological crawling motility and for the design of bio-mimetic crawling robots.

We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems

We formulate and solve the locomotion problem for a bio-inspired crawler consisting of two active elastic segments (i.e., capable of changing their rest lengths), resting on three supports providing directional frictional interactions. The problem consists in finding the motion produced by a given, slow actuation history.By focusing on the tensions in the elastic segments, we show that the evolution laws for the system are entirely analogous to the flow rules of elasto-plasticity. In particular, sliding of the supports and hence motion cannot occur when the tensions are in the interior of certain convex regions (stasis domains), while support sliding (and hence motion) can only take place when the tensions are on the boundary of such regions (slip surfaces).We solve the locomotion problem explicitly in a few interesting examples. In particular, we show that, for a suitable range of the friction parameters, specific choices of the actuation strategy can lead to net displacements also in the direction of higher friction.

We give a joint normality theorem, which simultaneously generalizes Montel's criterion and Miranda's criterion for normal families.

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