論文

Discrete domes are doubly curved structures comprising a network of beam-like elements. We study the mechanics of discrete domes made of ribbons woven in a pentagonal triaxial pattern. Experiments and finite element simulations are performed to characterize the mechanical response of each woven dome under indentation. The observed nonlinear response features force maxima, snap-through inversion, and non-monotonic evolution, leading to additional stable configurations. The dome's rest shape can be tuned continuously by designing the in-plane curvature of the ribbons and is then perturbed by adjustable clamped boundary conditions at their extremities. These control parameters are leveraged to smoothly and selectively modify the nonlinear features of the mechanical response, including multi-stability. Finally, we suggest a simplified model based on an elastica approximation to predict the stability of the inverted state successfully. The strong geometrical constraints imposed by the weaving pattern and the ribbons enable us to rationalize and tune the indentation response of these intriguing discrete structures.

We investigate the load transmission along an elastic rod of finite cross-section in contact with a rigid cylinder, as system often referred to as the generalized capstan problem. In the presence of friction, the idealized classic capstan equation predicts that the tension along a perfectly thin and flexible filament increases exponentially along the contact region. In practical applications, however, the validity of the idealized capstan equation is compromised due to the interplay between finite rod thickness, bending stiffness, and the forces applied at the rod extremities. Here, we combine precision model experiments, finite element simulations, and theoretical modeling to investigate the contact mechanics and the force transmission along an elastic rod in frictional contact with a rigid cylinder. We study two cases when the rod is either static or sliding. First, we focus on the static case, in the absence of friction, by considering equal loads at both extremities of the rod. We show that as the loading force is increased, the nature of contact transitions from a localized region to an extended band at the surface of the cylinder. The latter is characterized by double-peaked contact force distribution. In the sliding case, friction is activated by inducing a relative motion between the rod and the cylinder. We applied a fixed loading force at one rod extremity while pulling the other extremity at a constant velocity. The driving force is monitored during sliding. For increasing loading forces, we find that the force ratio is non-monotonic and displays a local minimum, in contradiction with the constant ideal capstan prediction. This minimum force ratio coincides with the transition from a single contact point to an extended contact region. A theoretical analysis based on Euler's elastica serves to rationalize the results from the physical and numerical experiments. In addition to predicting the nature of the contact region (single point versus extended line), our model provides quantitative predictions for the wrapping angle and the driving-to-loading force ratio. Finally, we leverage our mechanics-based framework to predictively understand the force ratio at the ends of two commercially available engineering belts (spring-steel and polyurethane) in sliding contact with a steel cylinder.

Soft robots are integrated functional structures composed of compliant mechanisms, whose activity spans various industrial applications such as surgery, healthcare, surveillance, and even planetary exploration. One of their promising mobility mechanism is snap-buckling; the instability mode of flexible structures passing from one equilibrium state to another can instantaneously generate large power for its motion. Predicting their performance with even simple geometry requires disentangling material, geometric nonlinearity, and contact, thereby still being an open challenge to date. Here, we study the jumping dynamics of hemispherical elastic shells driven by snap-buckling, as a model system of soft jumping mechanisms, combining experiments, simulations, and analytical theory. We find that the contact transition dynamics trigger the jumping phenomenon upon snap-buckling by constructing the analytical predictions with shell elasticity in excellent agreement with both experiments and simulations. Despite the simple geometry of the shell, its dynamical performance primarily relies on a complex interplay between elasticity, geometry, and contact friction. By elucidating the dynamics of the building blocks of soft robots that undergo large deformations, we can build their predictive experimental and numerical framework. Our research paves the way for designing soft robots suitable for the required loading conditions or structural requirements without empirical methods.

Knitted fabrics exhibit high flexibility due to their periodic loop structures formed by bent yarns. Under compressive loading, they develop three-dimensional (3D) wrinkling patterns that reflect nonlinear interactions between yarn elasticity and local loop deformations, as observed when the sleeves of a sweater are rolled up. Despite their widespread use in garments and medical textiles, the relationship between loop-level geometry and macroscopic buckling remains less understood. Here, we investigate the 3D deformation of knitted fabrics wrapped around a rigid cylinder under uniaxial compression. Circumferential and axial stitch numbers are systematically varied to determine how loop geometry affects the evolution of wrinkle patterns. Samples with a small number of circumferential stitches exhibit sequential wrinkle formation from the compressed end, leading to an accordion-like wrinkle pattern, whereas those with a larger number of stitches form helical wrinkles simultaneously across the surface. Wrinkle morphology changes progressively with stitch geometry, accompanied by systematic variations in compressive force, loop deformation, and helical wrinkle angle. The development of helical wrinkles originates from subtle structural asymmetries introduced during manufacturing processes, including the tension applied during knitting and the direction of sample assembly. These results demonstrate that small variations in local loop deformation can lead to substantial differences in wrinkle morphology, highlighting the sensitivity of macroscopic buckling to microscopic structural features. The study establishes a direct link between loop-level mechanics and global deformation behavior, providing a basis for the predictive design of knitted structures with tailored mechanical responses and complex 3D patterns.

Brownian motion is widely used as a model of diffusion in equilibrium media throughout the physical, chemical and biological sciences. However, many real-world systems are intrinsically out of equilibrium owing to energy-dissipating active processes underlying their mechanical and dynamical features1. The diffusion process followed by a passive tracer in prototypical active media, such as suspensions of active colloids or swimming microorganisms2, differs considerably from Brownian motion, as revealed by a greatly enhanced diffusion coefficient3–10 and non-Gaussian statistics of the tracer displacements6,9,10. Although these characteristic features have been extensively observed experimentally, there is so far no comprehensive theory explaining how they emerge from the microscopic dynamics of the system. Here we develop a theoretical framework to model the hydrodynamic interactions between the tracer and the active swimmers, which shows that the tracer follows a non-Markovian coloured Poisson process that accounts for all empirical observations. The theory predicts a long-lived Lévy flight regime11 of the loopy tracer motion with a non-monotonic crossover between two different power-law exponents. The duration of this regime can be tuned by the swimmer density, suggesting that the optimal foraging strategy of swimming microorganisms might depend crucially on their density in order to exploit the Lévy flights of nutrients12. Our framework can be applied to address important theoretical questions, such as the thermodynamics of active systems13, and practical ones, such as the interaction of swimming microorganisms with nutrients and other small particles14 (for example, degraded plastic) and the design of artificial nanoscale machines15.

Against external loads, shells with ideal symmetries, such as perfect spheres and cylinders, show enhanced rigidities, whose origin is known to be largely geometric and is often termed "geometric rigidity." In contrast, shells with free boundaries (i.e., open shells) may respond quite differently to external loads. They may deflect significantly as free boundaries may allow (almost) inextensional bending deformations that are impossible in the case of closed surfaces. Herein, we investigate the smooth deformation of an open cylindrical shell induced by pinching through a combination of experiment, simulations, and theory. To be more specific, we apply a localized pinch at one end of a long semi-cylindrical shell, and investigate the distance at which the shell recovers its undeformed form; we call this distance the persistence length of a pinch. We establish a new scaling law for the persistence length of shallow shells, which is shown to be valid for an arbitrarily deep shell. We show that the persistence length of a sufficiently deep open cylinder exceeds that of a closed cylinder, thereby highlighting the importance of the subtle interplay between the intrinsic curvature and the free edges on shell mechanics. These findings are rationalized by a detailed asymptotic analysis of the shallow shell equations. Our results reveal how the free boundaries essentially alter the balance between bending and stretching in thin shells.

When a flat elastic strip is compressed along its axis, it is bent in one of two possible directions via spontaneous symmetry breaking, forming a cylindrical arc. This is a phenomenon well known as Euler buckling. When this cylindrical section is pushed in the other direction, the bending direction can suddenly reverse. This instability is called "snap-through buckling" and is one of the elementary shape transitions in a prestressed thin structure. Combining experiments and theory, we study snap-buckling of an elastic strip with one end hinged and the other end clamped. These asymmetric boundary constraints break the intrinsic symmetry of the strip, generating mechanical behaviors, including largely hysteretic but reproducible force responses and switchlike discontinuous shape changes. We establish the set of exact analytical solutions to fully explain all our major experimental and numerical findings. Asymmetric boundary conditions arise naturally in diverse situations when a thin object is in contact with a solid surface at one end. The introduction of asymmetry through boundary conditions yields new insight into complex and programmable functionalities in material and industrial design.

We perform a compare-and-contrast investigation between the equilibrium shapes of physical and ideal trefoil knots, both in closed and open configurations. Ideal knots are purely geometric abstractions for the tightest configuration tied in a perfectly flexible, self-avoiding tube with an inextensible centerline and undeformable cross-sections. Here, we construct physical realizations of tight trefoil knots tied in an elastomeric rod, and use X-ray tomography and 3D finite element simulation for detailed characterization. Specifically, we evaluate the role of elasticity in dictating the physical knot's overall shape, self-contact regions, curvature profile, and cross-section deformation. We compare the shape of our elastic knots to prior computations of the corresponding ideal configurations. Our results on tight physical knots exhibit many similarities to their purely geometric counterparts, but also some striking dissimilarities that we examine in detail. These observations raise the hypothesis that regions of localized elastic deformation, not captured by the geometric models, could act as precursors for the weak spots that compromise the strength of knotted filaments.

Snapping of a slender structure is utilized in a wide range of natural and manmade systems, mostly to achieve rapid movement without relying on musclelike elements. Although several mechanisms for elastic energy storage and rapid release have been studied in detail, a general understanding of the approach to design such a kinetic system is a key challenge in mechanics. Here we study a twist-driven buckling and fast flip dynamics of a geometrically constrained ribbon by combining experiments, numerical simulations, and an analytical theory. We identify two distinct types of shape transitions: A narrow ribbon snaps, and a wide ribbon forms a pair of localized helices. We construct a phase diagram and explain the origin of the boundary, which is determined largely by the geometry. We quantify the effects of gravity and clarify the timescale dictating the rapid flipping. Our study reveals the unique role of geometric twist-bend coupling in the fast dynamics of a thin constrained structure, which has implications for a wide range of biophysical and applied physical problems.

We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper (Kanazawa et al. in Phys Rev Lett 114:090601-090606, 2015). We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.

Efficiency at maximum power (MP) output for an engine with a passive piston without mechanical controls between two reservoirs is studied theoretically. We enclose a hard core gas partitioned by a massive piston in a temperature-controlled container and analyze the efficiency at MP under a heating and cooling protocol without controlling the pressure acting on the piston from outside. We find the following three results: (i) The efficiency at MP for a dilute gas is close to the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency if we can ignore the sidewall friction and the loss of energy between a gas particle and the piston, while (ii) the efficiency for a moderately dense gas becomes smaller than the CNCA efficiency even when the temperature difference of the reservoirs is small. (iii) Introducing the Onsager matrix for an engine with a passive piston, we verify that the tight coupling condition for the matrix of the dilute gas is satisfied, while that of the moderately dense gas is not satisfied because of the inevitable heat leak. We confirm the validity of these results using the molecular dynamics simulation and introducing an effective mean-field-like model which we call the stochastic mean field model.

This study numerically and analytically investigates the dynamics of a rotor under viscous or dry friction as a nonequilibrium probe of a granular gas. In order to demonstrate the role of the rotor as a probe for a nonequilibrium bath, the molecular dynamics (MD) simulation of the rotor is performed under viscous or dry friction surrounded by a steady granular gas under gravity. A one-to-one map between the velocity distribution function (VDF) of the granular gas and the angular distribution function for the rotor is theoretically derived. The MD simulation demonstrates that the one-to-one map accurately infers the local VDF of the granular gas from the angular VDF of the rotor, and vice versa.

The impacts of granular jets for both frictional and frictionless grains in two dimensions are numerically investigated. A dense flow with a dead zone emerges during the impact. From our two-dimensional simulation, we evaluate the equations of state and the constitutive equations of the flow. The asymptotic divergences of pressure and shear stress similar to the situation near the jamming transition appear for the frictionless case, while their exponents are smaller than those of the sheared granular systems, and are close to the extrapolation from the kinetic theoretical regime. In a similar manner to the jamming for frictional grains, the critical density decreases as the friction constant of grains increases. For bi-disperse systems, the effective friction constant, defined as the ratio of shear stress to normal stress, monotonically increases from near zero as the strain rate increases. On the other hand, the effective friction constant has two metastable branches for mono-disperse systems because of the coexistence of a crystallized state and a liquid state.

For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.

The motion of an adiabatic piston under dry friction is investigated to clarify the roles of dry friction in nonequilibrium steady states. We clarify that dry friction can reverse the direction of the piston motion and causes a discontinuity or a cusplike singularity for velocity distribution functions of the piston. We also show that the heat fluctuation relation is modified under dry friction. © 2014 American Physical Society.

We perform three-dimensional simulations of the impact of a granular jet for both frictional and frictionless grains. Small shear stress observed in the experiment is reproduced through our simulation. However, the fluid state after the impact is far from a perfect fluid, and thus the similarity between granular jets and quark gluon plasma is superficial because the observed viscosity is finite and its value is consistent with the prediction of the kinetic theory. © 2012 American Physical Society.

The morphology of an elastic strip subject to vertical compressive stress on a frictional rigid substrate is investigated by a combination of theory and experiment. We find a rich variety of morphologies, which-when the bending elasticity dominates over the effect of gravity-are classified into three distinct types of states: pinned, partially slipped, and completely slipped, depending on the magnitude of the vertical strain and the coefficient of static friction. We develop a theory of elastica under mixed clamped-hinged boundary conditions combined with the Coulomb-Amontons friction law and find excellent quantitative agreement with simulations and controlled physical experiments. We also discuss the effect of gravity in order to bridge the difference in the qualitative behaviors of stiff strips and flexible strings or ropes. Our study thus complements recent work on elastic rope coiling and takes a significant step towards establishing a unified understanding of how a thin elastic object interacts vertically with a solid surface.