論文

Magneto-rheological elastomers (MREs) are functional materials that can be actuated by applying an external magnetic field. MREs comprise a composite of hard magnetic particles dispersed into a nonmagnetic elastomeric (soft) matrix. Hard MREs have been receiving particular attention because the programmed magnetization remains unchanged upon actuation. Motivated by a new realm of applications, there have been significant theoretical developments in the continuum (three-dimensional) description of hard MREs. In this paper, we derive an effective theory for MRE rods (slender, mono-dimensional structures) under geometrically nonlinear deformation in three dimensions. Our theory is based on reducing the three-dimensional magneto-elastic energy functional for the hard MREs into an one-dimensional Kirchhoff-like description (centerline-based). Restricting the theory to two dimensions, we reproduce previous works on planar deformations. For further validation in the general case of three-dimensional deformation, we perform precision experiments with both naturally straight and curved rods under either constant or constant-gradient magnetic fields. Our theoretical predictions are in excellent agreement with both discrete simulations and precision-model experiments. Finally, we discuss some limitations of our framework, as highlighted by the experiments, where long-range dipole–dipole interactions, which are neglected in the theory, can play a role.

We present a methodology to simulate the mechanics of knots in elastic rods using geometrically nonlinear, full three-dimensional (3D) finite element analysis. We focus on the mechanical behavior of knots in tight configurations, for which the full 3D deformation must be taken into account. To setup the topology of our knotted structures, we apply a sequence of prescribed displacement steps to the centerline of an initially straight rod that is meshed with 3D solid elements. Self-contact is enforced with a normal penalty force combined with Coulomb friction. As test cases, we investigate both overhand and figure-of-eight knots. Our simulations are validated with precision model experiments, combining rod fabrication and X-ray tomography. Even if the focus is given to the methods, our results reveal that 3D deformation of tight elastic knots is central to their mechanical response. These findings contrast to a previous analysis of loose knots, for which 1D centerline-based rod theories sufficed for a predictive understanding. Our method serves as a robust framework to access complex mechanical behavior of tightly knotted structures that are not readily available through experiments nor existing reduced-order theories.

Networks of flexible filaments often involve regions of tight contact. Predictively understanding the equilibrium configurations of these systems is challenging due to intricate couplings between topology, geometry, large nonlinear deformations, and friction. Here, we perform an in-depth study of a simple, yet canonical, problem that captures the essence of contact between filaments. In the orthogonal clasp, two filaments are brought into contact, with each centerline lying in one of a pair of orthogonal planes. Our data from X-ray tomography (µCT) and mechanical testing experiments are in excellent agreement with finite element method (FEM) simulations. Despite the apparent simplicity of the physical system, the data exhibit strikingly unintuitive behavior, even when the contact is frictionless. Specifically, we observe a curvilinear diamond-shaped ridge in the contact-pressure field between the two filaments, sometimes with an inner gap. When a relative displacement is imposed between the filaments, friction is activated, and a highly asymmetric pressure field develops. These findings contrast to the classic capstan analysis of a single filament wrapped around a rigid body. Both the µCT and FEM data indicate that the cross-sections of the filaments can deform significantly. Nonetheless, an idealized geometrical theory assuming undeformable tube cross-sections and neglecting elasticity rationalizes our observations qualitatively and highlights the central role of the small, but nonzero, tube radius of the filaments. We believe that our orthogonal clasp analysis provides a building block for future modeling efforts in frictional contact mechanics of more complex filamentary structures.

Hard magnetic elastomers are composites of soft elastic foundations and magnetic particles with high coercivity. We formulate a theoretical framework to predict the large deformation of a hard magnetic elastomeric rod. In the previous work, the magnetic Kirchhoff rod equations, which constitute a framework for analyzing instabilities for hard magnetic rods, have been developed and validated experimentally for negligible dipole-dipole interactions. Building on previous studies, we derive the magnetic Kirchhoff rod equations with dipole-dipole interactions. The derived equations are integro-differential equations, representing the force and moment balance along the rod centerline that include long-ranged dipole-magnetic force and torque. On the basis of its discrete numerical simulation, we systematically study the effect of the the dipole-dipole interactions strength on the large deformation of hard magnetic rods. In addition, we find that our theory can predict previous experimental results without any adjustable parameters.

We perform a combined experimental and computational investigation of the clove hitch knot. We develop a physical model for the clove hitch by tying an elastic rod onto a rigid cylinder. In the experiments, we characterize the mechanical performance, geometry, and stability conditions of the knot. X-ray tomography allows us to characterize the 3D geometry of the rod centerline. These results also serve to validate our finite element modeling (FEM), which we use to quantify the tension profile, not accessible experimentally, along the knotted rod. We find that the clove hitch comprises alternating segments with two types of contact regions: one where the rod is in single frictional contact with the cylinder, and another with rod self-contact (where a rod segment pinches another against the cylinder). In the first region, the internal tension decays exponentially (akin to the capstan configuration), whereas, in the second, the pinch (nip) regions lead to discontinuous tension drops. We analyze these nip regions with an even simpler model system where an elastic rod is pinched between two rigid cylinders. Despite the complex contact geometry of this pinching experiment, we find that the frictional behavior of our model systems still obeys the classic Amontons–Coulomb law. Ultimately, we can regard the clove hitch knot, if tied correctly, as a functional structure enabling to drop high tension at one extremity of a filament secured onto a rigid post, all the way to zero at the other extremity.

We study the wetting phenomena of a soft viscoelastic solid film on a smooth and flat substrate. A poly-dimethylsiloxane (PDMS) rubber film is suspended from a stage at both ends, and the wetting behavior of the film against a glass substrate is observed while lowering the stage at a constant velocity. We find that the dynamics of the rubber-glass-air contact lines vary with the lowering velocity of the stage. When the stage velocity is sufficiently low, the film wets the substrate smoothly and the contact lines are straight throughout. Consequently, the contact line velocity is proportional to the lowering velocity. As the stage velocity is increased, the contact line velocity reaches a maximum at the critical stage velocity and then subsequently decreases. The contact lines are wavy and sensitive to the defects above the critical velocity, resulting in the trapping of air bubbles at the interface. We reproduce the wetting behavior using a simple numerical model, assuming an upper limit for the contact line velocity. The wetting behavior observed in our experiments is attributed to the transition in the in-plane stress state from tensile to compressive along the film, leading to buckling of the film above the critical stage velocity. Our results suggest the existence and importance of the maximum wetting velocity for viscoelastic solids.

Structures with artificially engineered mechanical properties, often called mechanical metamaterials, are interesting for their tunable functionality. Various types of mechanical metamaterials have been proposed in the literature, designed to harness light or magnetic interactions, structural instabilities in slender or hollow structures, and contact friction. However, most of the designs are ideally engineered without any imperfections, in order to perform deterministically as programmed. Here, we study the mechanical performance of randomly stacked cylindrical shells, which act as a disordered mechanical metamaterial. Combining experiments and simulations, we demonstrate that the stacked shells can absorb and store mechanical energy upon compression by exploiting large deformation and relocation of shells, snap-fits, and friction. Although shells are oriented randomly, the system exhibits statistically robust mechanical performance controlled by friction and geometry. Our results demonstrate that the rearrangement of flexible components could yield versatile and predictive mechanical responses.

We investigate the mechanics of bistable, hard-magnetic, elastic beams, combining experiments, finite-element modelling (FEM) and a reduced-order theory. The beam is made of a hard magneto-rheological elastomer, comprising two segments with antiparallel magnetization along the centreline, and is set into a bistable curved configuration by imposing an end-to-end shortening. Reversible snapping is possible between these two stable states. First, we experimentally characterize the critical field strength for the onset of snapping, at different levels of end-to-end shortening. Second, we perform three-dimensional FEM simulations using the Riks method to analyse high-order deformation modes during snapping. Third, we develop a reduced-order centreline-based beam theory to rationalize the observed magneto-elastic response. The theory and simulations are validated against experiments, with an excellent quantitative agreement. Finally, we consider the case of combined magnetic loading and poking force, examining how the applied field affects the bistability and quantifying the maximum load-bearing capacity. Our work provides a set of predictive tools for the rational design of one-dimensional, bistable, magneto-elastic structural elements. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Snap buckling is a rapid shape transition in slender structures, appearing as a fundamental switching mechanism of natural and man-made systems. Boundary conditions of structures are crucial to predict and control their snap buckling behavior. However, the general framework that relates boundary conditions, geometry, and performance of structures is still absent to date. Here, we study the snap buckling of hemispherical shells in contact with rigid cylinders of different diameters to uncover the roles of boundary conditions in dynamic performance of shells. Specifically, we analyze the jumping dynamics of the pneumatically inverted shells placed on the rigid cylinder by combining experiments and analytical theory. We find the characteristic diameter of the cylinder determined from the shell geometry that can switch between continuous and discontinuous buckling associated with the jumping behavior. The analytical predictions for jumping performance of the shell supplemented with the characteristic diameter are in excellent agreement with our experimental results. Our study clarifies that contact geometry is crucial in predicting the pathway of snap buckling, indicating that dynamic performance of soft robots would be optimized by tuning their surface geometry.

Knitted fabrics are two-dimensional-like structures formed by stitching one-dimensional yarn into three-dimensional curves. Plain stitch or stockinette stitch, one of the most fundamental knitting stitches, consists of periodic lattices of bent yarns, where three-dimensional (3D) curling behavior naturally emerges at the edges. The elasticity and geometry of knitted fabrics have been studied in previous studies, primarily based on 2D modeling. Still, the relation between 3D geometry and the mechanics of knitted fabrics has not been clarified so far. The curling behavior of knits is intricately related to the forces and moments acting on the yarns, geometry of the unit knitted loops, mechanical properties, and contacts, hence requiring a 3D analysis. Here, we show that the curling of plain knits emerges through the elasticity and geometry of the knitted loops, combining desktop-scale experiments and reduced elasticity-based simulations. We find that by changing the horizontal and vertical knitting numbers, three types of curl shapes emerge: side curl and top/bottom curl shapes, which are curled only horizontally and vertically, and double curl shape, in which both curl shapes appear together. We uncover that the knit is side-curled when the knitted loop is vertically elongated, while we observe double curl and then top/bottom curl as the loop becomes horizontally elongated. Furthermore, we find that this characteristic loop shape affects the mechanical properties of knitted fabrics. The fundamental mechanism of intricate shape deformation is clarified through the force and moment balance along yarn whose centerline shape is discretized through the B-spline curves, where elastic stretching, bending, and contact mechanics are taken into account. We reveal that the 3D structure of the single knitted loop plays a critical role in the curling behavior. Our results imply that the change in shape per a single knitted loop has the potential to control the 3D natural overall shape of knitted fabrics. The 3D curling behavior of knitted fabrics is useful for industrial applications such as composite materials, wearable devices, and actuators. Our findings could be applied in predicting or designing more complex 3D shapes made of knitted fabrics.

Fitting two different materials is a versatile methodology in manufacturing complex structures. One of the canonical models for fitting is the snap-fit model, in which flexible materials and rigid structures are assembled by pushing their interlocking components together. The assembly via snap-fit is often accompanied by large deformations of flexible structures and abrupt force drops, highlighting the role of elasticity, geometry, and contact friction. Despite several model studies revealing fundamental mechanics for snap-fit, the current snap-fit design relies on prototyping and empirical rules. In this paper, we analyze a snap-fit model in which a naturally curved beam slips into a rigid hole. We construct an analytical model based on the theory of elastica with contact friction and demonstrate that its predictions are in excellent quantitative agreement with both simulations and experiments. We find three distinct sliding modes: folding, pinning, and unfolding. The classification is systematically organized in a phase diagram based on the geometric parameters of the shells and the hole. Our study complements existing approaches by providing a predictive framework for contact-based structures that involve friction, elasticity, and geometry, and sheds light on a unified understanding of the interactions between an elastic and a rigid body.

Knots across various length scales, from micro to macro-scales, such as polymers, DNA, shoelaces, and surgery, serving their unique mechanical properties. The shapes of ideal knots have been extensively studied in the context of knot theory, while those of physical knots have only been recently discussed in the literature. The complex interplay of elasticity and geometry, such as bending, twisting, and contact, needs to be disentangled to predict their deformation. Still, the unified understanding of the deformation of physical knots is insufficient. Here, we focus on the trefoil knot, a closed knot with a nontrivial topology, and study the relationship between the shapes of the trefoil knot and applied physical twists, combining experiments and simulations. As we twist the elastomeric rod, the knot becomes either tightened or loosened, preserving the original three-fold symmetry, and then buckles and exhibits symmetry breaking at critical angles. The curvature profiles computed through the X-ray tomography (μCT) analysis also exhibit similar symmetry breaking. The transition would be triggered by the mechanical instability, where the imposed twist energy is converted into the bending energy. The phase transition observed here is analogous to the classical buckling phenomena of elastic rings known as the Michell instability. We find that the twist buckling instability of the trefoil knot results from the interplay of bending, twisting, and contact properties of the rod. In other words, the buckling of the knot is predictable based on the elasticity and geometry of rods, which would be useful in avoiding or even utilizing their buckling in practical engineering applications such as surgery and the shipping industry.

The rheology of a three-dimensional granular jet during an impact is investigated numerically. The cone-like scattering pattern and the sheet-like pattern observed in an experiment [X. Cheng, et al. Phys. Rev. Lett. 99, 188001 (2007)] can be reproduced through our calculation. We discuss the constitutive equation for granular jet impact in terms of our simulation. From the analysis of an effective friction constant, which is the ratio between the shear stress and the pressure the assumption of the zero yield stress would be natural in our setup and the shear visocity is not small in contrast to the suggestion by the experiment. © 2013 AIP Publishing LLC.