Magnetic skyrmion braids

Skyrmions are vortex-like spin textures that form strings in magnetic crystals. Due to the analogy to elastic strings, skyrmion strings are naturally expected to braid and form complex three-dimensional patterns, but this phenomenon has not been explored yet. We found that skyrmion strings can form braids in cubic crystals of chiral magnets. This finding is confirmed by direct observations of skyrmion braids in B20-type FeGe using transmission electron microscopy. The theoretical analysis predicts that the discovered phenomenon is general for a wide family of chiral magnets. These findings have important implications for skyrmionics and propose a solid-state framework for applications of the mathematical theory of braids.

Longitudinal shock waves in a class of semi-infinite stretch-limited elastic strings

In this paper, we initiate the study of wave propagation in a recently proposed mathematical model for stretch-limited elastic strings. We consider the longitudinal motion of a simple class of uniform, semi-infinite, stretch-limited strings under no external force with finite end held fixed and prescribed tension at the infinite end. We study a class of motions such that the string has one inextensible segment, where the local stretch is maximized, and one extensible segment. The equations of motion reduce to a simple and novel shock front problem in one spatial variable for which we prove existence and uniqueness of local-in-time solutions for appropriate initial data. We then prove the orbital asymptotic stability of an explicit two-parameter family of piece-wise constant stretched motions. If the prescribed tension at the infinite end is increasing in time, our results provide an open set of initial data launching motions resulting in the string becoming fully inextensible and tension blowing up in finite time.

On stretch-limited elastic strings

Motivated by the increased interest in modelling non-dissipative materials by constitutive relations more general than those from Cauchy elasticity, we initiate the study of a class of stretch-limited elastic strings : the string cannot be compressed smaller than a certain length less than its natural length nor elongated larger than a certain length greater than its natural length. In particular, we consider equilibrium states for a string suspended between two points under the force of gravity (catenaries). We study the locations of the supports resulting in tensile states containing both extensible and inextensible segments in two situations: the degenerate case when the string is vertical and the non-degenerate case when the supports are at the same height. We then study the existence and multiplicity of equilibrium states in general with multiplicity differing markedly from strings satisfying classical constitutive relations.

Asymptotic analysis of a junction of hyperelastic rods

In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a 1 D-dimensional energy for elastic strings by using techniques from Γ-convergence.

Hyperspherical variables analysis of lattice QCD three-quark potentials: skewed Y-string as the mechanism of confinement?

We have re-analysed the lattice QCD calculations of the 3-quark potentials by: (i) Sakumichi and Suganuma (Phys Rev D 92(3), 034511, 2015); and (ii) Koma and Koma (Phys Rev D 95(9), 094513, 2017) using hyperspherical variables. We find that: (1) the two sets of lattice results have only two common sets of 3-quark geometries: (a) the isosceles, and (b) the right-angled triangles; (2) both sets of results are subject to unaccounted for deviations from smooth curves that are largest near the equilateral triangle geometry and are function of the hyperradius – the deviations being much larger and extending further in the triangle shape space in Sakumichi and Suganuma’s than in Koma and Koma’s data; (3) the variation of Sakumichi and Suganuma’s results brackets, from above and below, the Koma and Koma’s ones; the latter will be used as the benchmark; (4) this benchmark result generally passes between the Y- and the $$\Delta $$ Δ -string predictions, thus excluding both; (5) three pieces of elastic strings joined at a skewed junction, which lies on the Euler line, reproduce such a potential, within the region where the data sets agree, in qualitative agreement with the calculations of colour flux density by Bissey et al. (Phys Rev D 76, 114512, 2007).

Elasticity of tangled magnetic fields

The fundamental difference between incompressible ideal magnetohydrodynamics and the dynamics of a non-conducting fluid is that magnetic fields exert a tension force that opposes their bending; magnetic fields behave like elastic strings threading the fluid. It is natural, therefore, to expect that a magnetic field tangled at small length scales should resist a large-scale shear in an elastic way, much as a ball of tangled elastic strings responds elastically to an impulse. Furthermore, a tangled field should support the propagation of ‘magnetoelastic waves’, the isotropic analogue of Alfvén waves on a straight magnetic field. Here, we study magnetoelasticity in the idealised context of an equilibrium tangled field configuration. In contrast to previous treatments, we explicitly account for intermittency of the Maxwell stress, and show that this intermittency necessarily decreases the frequency of magnetoelastic waves in a stable field configuration. We develop a mean-field formalism to describe magnetoelastic behaviour, retaining leading-order corrections due to the coupling of large- and small-scale motions, and solve the initial-value problem for viscous fluids subjected to a large-scale shear, showing that the development of small-scale motions results in anomalous viscous damping of large-scale waves. Finally, we test these analytic predictions using numerical simulations of standing waves on tangled, linear force-free magnetic-field equilibria.

Asymptotic variational analysis of incompressible elastic strings

Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

Integrated Origami-String System

Origami-inspired mechanical systems are mostly composed of two-dimensional elements, a feature inherited from paper folding. However, do we have to comply with this restriction on our design space? Would it be more approachable to achieve desired performance by integrating elements of different abstract dimensions? In this paper, we propose an integrated structural system consisting of both two-dimensional and one-dimensional elements. We attach elastic strings onto an origami design to modify its mechanical behavior and create new features. We show that, by introducing elastic strings to the recently proposed Morph pattern, we can obtain bistable units with programmable energy landscape. The behavior of this integrated origami-string system can be described by an elegant formulation, which can be used to explore its rich programmability.

Collaborative Control of Multiple Robots Using Genetic Fuzzy Systems Approach

This paper introduces a decentralized approach of collaborative control between multiple robots. A dynamic problem is considered to illustrate the effectiveness of this approach. The objective of this problem is to control three robots that are connected to a ball through elastic strings to bring the ball to a pre-defined target position. Since there is no communication between the robots, each robot does not know how the other robots are going to react at any instant. The only information available to the robots are the current and target positions of the ball. Genetic Fuzzy Systems (GFSs) are used to develop controllers for individual robots to tackle this problem. The nonlinearity of fuzzy logic systems coupled with the search capability of Genetic Algorithm (GA) provides an invaluable tool to design controllers for such tasks. The system is first trained through a set of scenarios and then applied to an extensive test set to test the effectiveness of the approach.