Soft Beam Research Articles

Dynamic modeling and simulation of hard-magnetic soft beams interacting with environment via high-order finite elements of ANCF

Hard-magnetic soft (HMS) beams made of soft polymer matrix embedded with hard-magnetic particles can generate large and fast deformation under magnetic stimulation. Dynamic modeling and simulation of HMS beams interacting with complex environment are challenging in terms of computational accuracy and efficiency. This paper presents a method for high-order modeling and efficient computation of HMS beams. The major contribution of the method is a new three-node HMS beam element of absolute nodal coordinate formulation (ANCF), which applies to two material models of nonlinear and linear elasticities (i.e. neoHookean and St. Venant-Kirchhoff) coupled with magnetic energy. To improve the efficiency of the method, the paper presents how to derive the generalized internal forces and their Jacobians via invariant tensors, and how to determine the generalized external forces to model dynamic loads and interactions including gravity, hydrodynamics in fluids, and frictional contact in pipelines. Afterwards, the paper gives both static and dynamic equations with Rayleigh damping and discusses the numerical algorithms. Finally, the paper makes a comparison of static analysis and the experimental observation to validate the accuracy of the proposed modeling method. The paper also discusses the dynamic simulations, including forced vibration, swimming motion, crawling locomotion, and navigating motion to demonstrate the predictive capability and efficacy of the proposed method for dynamic problems.

Development of a novel fractional magneto-viscoelastic dynamic model for an adaptive beam featuring functional composite magnetoactive elastomers: Simulations and experimental studies

Recently, the rapid emergence of functional composite magnetoactive elastomers with integrated hard magnetic particles (known as hard magnetoactive elastomers) has attracted significant interest in fields such as soft robotics and material science. It is of paramount importance to develop an efficient model that can accurately predict the nonlinear dynamic behavior of adaptive structures for their practical application and the synthesis of effective control strategies. In this study, a novel nonlinear fractional magneto-viscoelastic model of an adaptive cantilevered beam featuring hard magnetoactive elastomers has been developed. This model effectively characterizes the beam’s nonlinear and rate-dependent response behavior under magnetic stimuli of varying frequencies and magnitudes. Considering the fractional Kelvin-Voigt energy dissipation model and large deformation nonlinearity, the governing equations for the cantilevered hard-magnetic soft beam were derived using the Hamilton principle. The finite difference method, combined with the Galerkin modal decomposition scheme, was subsequently utilized to discretize the time and space domains, respectively. A hardware-in-the-loop experimental framework was finally designed to experimentally investigate the beam’s nonlinear response behavior and validate the simulation results. The simulated nonlinear quasi-static responses of the beam, subjected to magnetic loading up to 30 mT, exhibited excellent agreement with the experimental data collected. Moreover, the simulated dynamic responses of the beam, subjected to various amplitudes of the applied magnetic field and magnetic frequencies up to 2 Hz, were thoroughly investigated. These included time-histories, phase-plane, and hysteretic responses, all demonstrating strong alignment with the experimental observations.

Dynamic modeling and analysis of a hard-magneto-viscoelastic soft beam under large amplitude oscillatory motions: simulation and experimental studies

Dynamic modeling and analysis of a hard-magneto-viscoelastic soft beam under large amplitude oscillatory motions: simulation and experimental studies

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.

Machine learning-assisted shape morphing design for soft smart beam

Programming the shape of soft smart materials is a challenging task due to the enormous design space involved. In this study, we propose a novel approach to determine applied stimuli that enable the desired actuated shapes of soft smart materials. Our approach combines finite element methods (FEM), deep neural networks (DNN), and particle swarm optimization (PSO). By employing beams made of dielectric elastomer (DE) as models, we partition a DE beam into multiple actuating units, allowing independent actuation through the application of paired electrical stimuli. FEM is subsequently employed to compute the actuated shapes in response to various applied stimuli. The selection of these stimuli is accomplished through the utilization of Latin hypercube sampling. By utilizing the FEM-derived data, we have developed a machine learning (ML) surrogate model that integrates a long short-term memory (LSTM) network with a fully connected neural network (FCNN). Finally, PSO is employed to determine the optimal applied stimulus that yields the desired actuated shape. The LSTM-FCNN surrogate model is utilized to evaluate the fitness of PSO. The ML-PSO framework exhibits remarkable performance and efficiency in the context of the inverse design of soft smart beams.

Beneficial effects of the precritical nonlinearities on the lateral buckling of extremely flexible beams

The lateral buckling of extremely flexible beams, uniformly or non-uniformly bent in a principal inertia plane, is studied. An exact polar continuum model of an internally constrained 1D straight beam, immersed in the 3D space, is formulated. The model is linearized in the out-of-plane displacement only, so that it is able to capture: (i) the exact planar non-trivial fundamental path, and (ii) the exact critical point along it, at which the beam experiences out-of-plane flexural–torsional buckling. It is proven that the precritical deflections increase the critical load with respect to that evaluated by the classic theory, in which a trivial fundamental path is assumed, thus playing a beneficial role. Moreover, when the beam is extremely flexible, bifurcation disappears, similar to what discussed in literature for axially soft compressed beams, although via a different mechanism of the eigenvalues.

Mechanics of hard-magnetic soft materials: A review

Hard-magnetic soft materials are a class of magnetically responsive composites obtained by embedding hard-magnetic particles into a soft polymeric matrix. They have found widespread applications in shape-morphing systems, soft robotics, biomedical devices, and active metamaterials due to their ability to feature complex, untethered, reversible, and rapid deformations in response to magnetic loads. To guide the rational design of these functional applications, extensive efforts have been devoted to studying the mechanical behavior of hard-magnetic soft materials. In this paper, we review the recent progress in the mechanics of hard-magnetic soft materials. First, we introduce existing constitutive models capable of describing the coupled magneto-elastic deformations of hard-magnetic soft materials. Then, we discuss the mechanical response of structures made of hard-magnetic soft materials, including rods, beams, plates, and shells, under mechanical and magnetic loading. Subsequently, we introduce the design and behavior of magneto-mechanical metamaterials with tunable properties enabled by hard-magnetic soft materials. In addition, optimization-guided inverse design strategies for hard-magnetic soft materials to achieve predefined properties or deformations are also briefly reviewed. Finally, we provide our views on the potential future directions in the field of mechanics of hard-magnetic soft materials. We expect the current review to guide researchers to better understand different theoretical and computational frameworks of mechanics of hard-magnetic soft materials and thus aid with designing functional systems using these materials for various applications.

Tunable topological phase transition in soft Rayleigh beam system with imperfect interfaces

Acoustic metamaterials, especially the topological insulators, have garnered substantial research attention due to their unique wave properties. Recent research emphasizes the crucial role of imperfect interfaces in accurately simulating and understanding wave propagation within these metamaterials. This paper studies the mechanical properties of soft imperfect interfaces and derives the effects of finite deformations on their stiffness. We present a comprehensive model of a soft Rayleigh beam system that incorporates these interfaces, allowing for the observation of the topological phase transition by altering the distance between them. Our investigation into the impacts of finite deformations and interface stiffness adjustment reveals that longitudinal waves' phase transition points are mainly influenced by interface stiffness. In contrast, transverse waves are predominantly affected by finite deformations. The decisive parameters driving these phenomena are also identified theoretically. The transmission studies of supercells further confirm the presence of topologically protected interface modes and their extensive tunability. Overall, this study offers a novel methodology and framework for managing topological phase transition in composite and soft topological insulator systems.

Stable Ultra-thin Silver Films Grown by Soft Ion Beam-Enhanced Sputtering with an Aluminum Cap Layer.

Ultra-thin silver films are susceptible to ambient environments and form grayish layers in the silver mirroring process. The poor wettability together with the high diffusivity of surface atoms in the presence of oxygen accounts for the thermal instability of ultra-thin silver films in the air and at elevated temperatures. This work demonstrates an atomic-scale aluminum cap layer on the silver to enhance the thermal and environmental stabilities of ultra-thin silver films deposited by sputtering with the assistance of a soft ion beam reported in our previous work. The resulted film consists of an ion-beam-treated seed silver layer of ∼1 nm nominal thickness, a subsequent silver layer of ∼6 nm thickness produced by sputtering alone, and an aluminum cap layer of ∼0.2 nm nominal thickness. Although the aluminum cap is only one to two atomic layers and likely non-continuous, it significantly improved the thermal and ambient environmental stability of the ultra-thin silver films (∼7 nm thick) without affecting the film's optical and electrical properties. The improved environmental stability is attributed to the cathodic protection mechanism and reduced diffusivity of surface atoms. The improved thermal stability is attributed to the reduced mobility of surface atoms in the presence of aluminum atoms. Thermal treatment of the duplex film also improves the film's electrical conductivity and optical transmittance by enhancing its crystallinity. The annealed aluminum/silver duplex structure has exhibited the lowest electric resistivity among the reported ultra-thin silver films and high optical transmittance similar to the simulated theoretical results.

A new sandwich beam model with layer-to-layer boundary modified displacements based on higher-order absolute nodal coordinate formulation

The absolute nodal coordinate formulation (ANCF) is one of the efficient modeling methods to solve the dynamics of flexible multibody systems. Zig-zag theory can accurately calculate the stress variation in the sandwich structure with only a few unknown variables, thus providing a good balance between computational efficiency and computational accuracy. Combining ANCF and zig-zag theory, this paper proposes a new sandwich beam model, called HANCF-ZZ, by introducing layer-to-layer boundary modified displacements in a high-order ANCF beam model. The capability of HANCF-ZZ for static deformation analysis and free vibration analysis is verified by comparing experimental, analytical, and numerical results in other literature. The results show that the HANCF-ZZ corrects the global stiffness overestimated by the equivalent single-layer theory, eliminates the locking problem of the low-order ANCF, and captures the nonlinear bifurcation phenomenon in large deformation problems under large loads. The HANCF-ZZ is not only an efficient numerical method for zig-zag theory but also enriches the ANCF element types to enable its application to the dynamics analysis of soft core sandwich beams.

On-demand contactless programming of nonlinear elastic moduli in hard magnetic soft beam based broadband active lattice materials

Engineered honeycomb lattice materials with high specific strength and stiffness along with the advantage of programmable direction-dependent mechanical tailorability are being increasingly adopted for various advanced multifunctional applications. To use these artificial microstructures with unprecedented mechanical properties in the design of different application-specific structures, it is essential to investigate the effective elastic moduli and their dependence on the microstructural geometry and the physics of deformation at the elementary level. While it is possible to have a wide range of effective mechanical properties based on their designed microstructural geometry, most of the recent advancements in this field lead to passive mechanical properties, meaning it is not possible to actively modulate the lattice-level properties after they are manufactured. Thus the on-demand control of mechanical properties is lacking, which is crucial for a range of multi-functional applications in advanced structural systems. To address this issue, we propose a new class of lattice materials wherein the beam-level multi-physical deformation behavior can be exploited as a function of external stimuli like magnetic field by considering hard magnetic soft beams. More interestingly, effective property modulation at the lattice level would be contactless without the necessity of having a complex network of electrical circuits embedded within the microstructure. We have developed a semi-analytical model for the nonlinear effective elastic properties of such programmable lattice materials under large deformation, wherein the mechanical properties can be modulated in an expanded design space of microstructural geometry and magnetic field. The numerical results show that the effective properties can be actively modulated as a function of the magnetic field covering a wide range (including programmable state transition with on-demand positive and negative values), leading to the behavior of soft polymer to stiff metals in a single lattice microstructure according to operational demands.

Tunable Dynamic Walking via Soft Twisted Beam Vibration

We propose a novel mechanism that propagates vibration through soft twisted beams, taking advantage of dynamically-coupled anisotropic stiffness to simplify the actuation of walking robots. Using dynamic simulation and experimental approaches, we show that the coupled stiffness of twisted beams with terrain contact can be controlled to generate a variety of complex trajectories by changing the frequency of the input signal. This work reveals how ground contact influences the system's dynamic behavior, supporting the design of walking robots inspired by this phenomenon. We also show that the proposed twisted beam produces a tunable walking gait from a single vibrational input.

Liquid-Metal-Based Magnetic Controllable Soft Microswitch with Rapid and Reliable Response for Intelligent Soft Systems.

When combined with diverse sensors, soft robots significantly improve their functionalities and intelligence levels. However, most of the existing soft sensors require complex signal analysis devices or algorithms, which severely increase the complexity of soft robot systems. Here, based on the unique fluidic property of liquid metal, we propose a magnet-controllable soft microswitch that can be well-integrated into a soft robot system, e.g., a soft gripper to help it facilely detect and precisely grab objects. The microswitch consists of a flexible soft beam electrode and a fixed electrode, forming a soft microsystem. By tuning the cohesion force of the liquid metal between the electrodes, the microswitch can convert its states between an individual and a self-locking state. The microswitch can achieve a reasonable rapid response (~12 ms) and high switching frequency (~95 Hz). Furthermore, soft microswitches can be customized into logic units and also coupled to control a digital tube showing various numbers. Our work provides a new simple soft sensor unit that may enhance the intelligence of soft systems.

On the dynamics of curved magnetoactive soft beams

Recently, hard-magnetic microparticles were embedded into soft matrix to manufacture a new type of magnetoactive soft materials, i.e. hard-magnetic soft (HMS) materials. Due to the promising applications of HMS materials in the areas of soft robotics, flexible electronics, biomedicine, etc., there is a burgeoning trend in investigating the mechanical responses of HMS structures. Special attention has been placed on the static deformations of initially straight HMS beams. However, to fulfill the potential applications of HMS materials, it is also necessary to examine the dynamical behaviors of HMS beams. The present study aims to develop a large-deformation dynamical model of curved HMS beams actuated by a harmonically rotational magnetic field, which is varying periodically with time. This dynamical curved HMS beam model could be easily reduced to the corresponding static one, which is then solved both analytically and numerically. To tackle the dynamic problem, the derived governing equation for nonlinear vibrations of the curved HMS beam is first discretized by using the Galerkin method and then solved by a fourth-order Runge-Kutta integration algorithm. It is found that the curved HMS beam tends to align with the direction of the applied constant magnetic field as the strength of the applied field increases. The curved HMS beam would vibrate periodically if a periodically rotational magnetic field is applied. Furthermore, when the rotational frequency of the magnetic field is close to the natural frequency of the beam, the system resonates and the vibration amplitude of the curved beam becomes extremely large.

Design of soft and hard active-passive composite beams

The combination of different materials is prevalent in nature and technology. Particularly, smart (active) materials must be combined with other materials to harness their full potential. Three combination cases are explored: Case I (constrained), Case II (combined) and Case III (free) behavior. Based on analytical theories and Finite Element simulations with the Stimulus-Expansion-Model, design approaches for active-passive composites are proposed. These include, choice of Case for an application, bottom-up design of layers with matching stiffness and replacement of materials. The current work substantiates that the stiffnesses are crucial for combining materials, e.g., in smart structures, additive manufacturing and bioelectronic interfaces.

Reprogrammable Mechanical Metamaterials with Heterogeneous Assembly of Soft Shell‐Based Voxels

Mechanical metamaterials are well studied to enable functions that are difficult to achieve in natural materials, in which reprogramming of specific underlying structures targeting different mechanical responses is particularly challenging. Herein, a heterogeneous voxel assembly‐based 2D mechanical metamaterial is purposed, aiming at a reprogrammable design of the mechanical properties and shape‐morphing simultaneously. In particular, each voxel is structured with a bistable soft shell and soft beams, which can be well controlled to achieve various desirable mechanical properties by a parameterized design method. The engineered metamaterial has a periodic arrangement of the voxel structures that can be transformed into an arbitrary aperiodic architecture with low geometrical frustration in the metamaterial plane by manipulating voxels, in which reprogrammable stiffness and Poisson's ratios, as well as deformation sequences and complex surface textures, can be achieved. In addition, the voxelated assembly has two modes allowing for fine or extensive tuning of the stiffness and the optional generation of 1D or 2D textures. Under this design paradigm, the programmability of multiple mechanical responses can support the development of more intelligent mechanical metamaterials with great potentials to soft robotics and active deformable devices.

Submicron 3-D mass spectrometry imaging reveals an asymmetric molecular distribution on chemotaxing cells.

Background: Dictyostelium discoideum is a ~10 µm diameter unicellular eukaryote that lives on soil surfaces. When starved, D. discoideum cells aggregate into streams of cells in a process called chemotaxis. In this report, we studied D. discoideum cells during chemotaxis using 3D - mass spectrometry imaging (3D-MSI). Methods: The 3D-MSI consisted of the sequential generation of 2D molecular maps using burst alignment coupled to delayed extraction time-of flight secondary ion mass spectrometry (TOF-SIMS) combined with a soft sputtering beam to access the different layers. Results: Molecular maps with sub-cellular high spatial resolution (~300 nm) indicated the presence of ions at m/z = 221 and 236 at the front and sides, but reduced levels at the back, of cells moving toward of aggregation streams. The 3D-MSI also detected an ion at m/z = 240 at the edges and back, but reduced levels at the front, of aggregating cells. Other ions showed an even distribution across the cells. Conclusions: Together, these results demonstrate the utility of sub-micron MSI to study eukaryotic chemotaxis.

Generation of soft annular beams with high uniformity, low ring width increment, and a smooth edge.

In this paper, soft-edge toroidal amplitude filter (STAF) and soft-edge toroidal complex amplitude filter (STCAF) are designed according to the principle that soft-edge structures can eliminate diffraction. Based on the Mach-Zehnder interference principle, a double optical path compound interference modulation method that can generate soft annular beams is proposed by using STAF and STCAF. The 1/e2 radius and peak-to-average ratio (PAR) were used to evaluate the ring width and uniformity of the annular beam. Compared with the annular beams generated by STAF and hard-edge toroidal amplitude filter (HTAF), it can be found that the soft annular beam generated by this proposed method has the advantages of high uniformity, small ring width increment, and smooth edges. By analyzing the influence of the number and height of the sawtooth structures on the annular beam propagation performance, the relationship between the PAR and the structure parameters of the STAF was established. Moreover, three kinds of toroidal filters were designed by lithography processing, and an experimental system was built to generate the soft annular beam. In the experiment, the average value of the ring width increment of the soft annular beam is 0.0125, the PAR is less than 1.5, and the root mean square error of the PAR curve is 0.0865, which indicates that the soft annular beam maintains high uniformity during propagation.

Utilization of nonlinear vibrations of soft pipe conveying fluid for driving underwater bio-inspired robot

Creatures with longer bodies in nature like snakes and eels moving in water commonly generate a large swaying of their bodies or tails, with the purpose of producing significant frictions and collisions between body and fluid to provide the power of consecutive forward force. This swaying can be idealized by considering oscillations of a soft beam immersed in water when waves of vibration travel down at a constant speed. The present study employs a kind of large deformations induced by nonlinear vibrations of a soft pipe conveying fluid to design an underwater bio-inspired snake robot that consists of a rigid head and a soft tail. When the head is fixed, experiments show that a second mode vibration of the tail in water occurs as the internal flow velocity is beyond a critical value. Then the corresponding theoretical model based on the absolute nodal coordinate formulation (ANCF) is established to describe nonlinear vibrations of the tail. As the head is free, the theoretical modeling is combined with the computational fluid dynamics (CFD) analysis to construct a fluid-structure interaction (FSI) simulation model. The swimming speed and swaying shape of the snake robot are obtained through the FSI simulation model. They are in good agreement with experimental results. Most importantly, it is demonstrated that the propulsion speed can be improved by 21% for the robot with vibrations of the tail compared with that without oscillations in the pure jet mode. This research provides a new thought to design driving devices by using nonlinear flow-induced vibrations.

Large viscoelastic deformation of hard-magnetic soft beams

This work aims at developing a viscoelastic formulation for the time-dependent finite deformation analysis of beams made of hard-magnetic soft materials (HMSMs) under magnetic loading. After introducing the basic kinematic quantities, a viscoelasticity formulation for the analysis of HMSMs is developed, which is general in the sense that it can be used for 2D and 3D geometries beside the beam, plate, and shell-type structures. Next, the expression for the consistent fourth-order tangent tensors for 3D bodies and beams made of HMSMs are presented. Due to the highly nonlinear nature of the governing equations, a finite element formulation for the numerical solution of beam problems with various loading and boundary conditions is developed. To demonstrate the applicability of the developed formulation, several numerical examples are provided. It is observed that for the case of elastic deformations, the results of the present formulation are very close to those previously reported in the literature. For the case of viscoelastic deformations, the creep response of beams is simulated and the effect of viscoelastic parameters is studied. It is shown that the obtained results are qualitatively in agreement with the basic properties of viscoelastic deformations.