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Growing soft materials which follow a 3D path in space are critical to applications such as search and rescue and minimally invasive surgery. Here, we present a concept for a single-input growing multi-stable soft material, based on a constrained straw-like structure. This class of materials are capable of maneuvering and transforming their configuration by elongation while executing multiple turns. This is achieved by sequenced actuation of bi-stable frusta with predefined constraints. Internal viscous flow and variations in the stability threshold of the individual cells enable sequencing and control of the robot's movement so as to follow a desired 3D path as the structure grows. We derive a theoretical description of the shape and dynamics resulting from a particular set of constraints. To validate the model and demonstrate the suggested concept, we present experiments of maneuvering in models of residential and biological environments. In addition to performing complex 3D maneuvers, the tubular structure of these robots may also be used as a conduit to reach inaccessible regions, which is demonstrated experimentally., Comment: 10 pages and 3 figures

In this paper, we discuss the dynamic modeling of fluid-filled straw-like elements consisting of serially interconnected elastic frusta with both axisymmetric and antisymmetric degrees of freedom, assuming planar motion. Under appropriate conditions each sub-structure has four stable equilibrium states. This gives the system under investigation the ability to remain stable in a large number of complex states, which is a vital ability for myriad of applications, including reconfigurable structures and soft robots. The theoretical model explains the dynamics of a single straw-like element in a discrete manner, considering inertial, damping, and gravitational effects, while taking into account the nonlinear elasticity of the elastic frusta, and assuming hydrostatic behavior of the entrapped fluid. After identifying the geometric and elastic parameters of the theoretical model based on relatively simple experiments, the model is validated compared to numerical simulations and experiments. The numerical simulations validate the theoretical elasticity of the elastic frusta, whereas the overall dynamic behavior of the system and the influence of unmodeled fluidic effects are examined experimentally. It is demonstrated both theoretically and empirically that straw-like elements cannot be adequately modeled using simple uniaxial deformations. In addition, the experimental validation indicates that the suggested model can accurately capture their overall dynamics., Comment: 13 pages, 10 figures

Many marine creatures, gastropods, and earthworms generate continuous traveling waves in their bodies for locomotion within marine environments, complex surfaces, and inside narrow gaps. In this work, we study theoretically and experimentally the use of embedded pneumatic networks as a mechanism to mimic nature and generate bi-directional traveling waves in soft robots. We apply long-wave approximation to theoretically calculate the required distribution of pneumatic network and inlet pressure oscillations needed to create desired moving wave patterns. We then fabricate soft robots with internal pneumatic network geometry based on these analytical results. The experimental results agree well with our model and demonstrate the propagation of moving waves in soft robots, along with locomotion capabilities. The presented results allow fabricating soft robots capable of continuous moving waves via the common approach of embedded pneumatic networks and requiring only two input controls.

Viscous flows in hyperelastic chambers are relevant to many biological phenomena such as inhalation into the lung's acinar region and medical applications such as the inflation of a small chamber in minimally invasive procedures. In this work, we analytically study the viscous flow and elastic deformation created due to inflation of such spherical chambers from one or two inlets. Our investigation considers the shell's constitutive hyperelastic law coupled with the flow dynamics inside the chamber. For the case of a narrow tube filling a larger chamber, the pressure within the chamber involves a large spatially uniform part, and a small order correction. We derive a closed-form expression for the inflation dynamics, accounting for the effect of elastic bi-stability. Interestingly, the obtained pressure distribution shows that the maximal pressure on the chamber's surface is greater than the pressure at the entrance to the chamber. The calculated series solution of the velocity and pressure fields during inflation is verified by using a fully coupled finite element scheme, resulting in excellent agreement. Our results allow estimating the chamber's viscous resistance at different pressures, thus enabling us to model the process of inflation and deflation., Comment: 30 pages, 11 figures

We present a theoretical model and experimental demonstration of thin liquid film deformations due to a dielectric force distribution established by surface electrodes. We model the spatial electric field produced by a pair of parallel electrodes and use it to evaluate the stress on the interface through Maxwell stresses. By coupling this force with the Young-Laplace equation, we obtain the deformation of the interface. To validate our theory, we design an experimental setup which uses microfabricated electrodes to achieve spatial dielectrophoretic actuation of a thin liquid film, while providing measurements of microscale deformations through digital holographic microscopy. We characterize the deformation as a function of the electrode-pair geometry and film thickness, showing very good agreement with the model. Based on the insights from the characterization of the system, we pattern conductive lines of electrode pairs on the surface of a microfluidic chamber and demonstrate the ability to produce complex two-dimensional deformations. The films can remain in liquid form and be dynamically modulated between different configurations or polymerized to create solid structures with high surface quality., Comment: 14 pages and 7 figures in the main manuscript. 4 sections and 5 figures in the SI

Inchworm crawling allows for both quasistatic and dynamic gaits at a wide range of actuation frequencies. This locomotion mechanism is common in nonskeletal animals and exploited extensively in the bio-inspired field of soft robotics. In this work we develop and simulate the hybrid dynamic crawling of a three-link robot, with passive frictional contacts. We fabricate and experimentally test such robot under periodic inputs of joints' angles, with good agreement to the theoretical predictions. This allows to comprehend and exploit the effects of inertia in order to find optimal performance in inputs' parameters. A simple criterion of robustness to uncertainties in friction is proposed. Tuning the inputs according to this criterion improves the robustness of low-frequency actuation, while increasing the frequency allows for gaits with both high advancement velocity and robustness. Finally, the advantages of uneven mass distribution are studied. Time-scaling technique is introduced to shape inputs that achieve similar effect without reassembling the robot. A machine-learning based optimization is applied to these inputs to further improve the robot's performance in traveling distance.

Crawling is a common locomotion mechanism in soft robots and nonskeletal animals. In this work we propose modeling soft-robotic legged locomotion by approximating it with an equivalent articulated robot with elastic joints. For concreteness we study our soft robot with two bending actuators via an articulated three-link model. The solution of statically indeterminate systems with stick-slip contact transitions requires for a novel hybrid-quasitatic analysis. Then, we utilize our analysis to investigate the influence of phase-shifted harmonic inputs on performance of crawling gaits, including sensitivity analysis to friction uncertainties and energetic cost of transport. We achieve optimal values of gait parameters. Finally, we fabricate and test a fluid-driven soft robot. The experiments display remarkable agreement with the theoretical analysis, proving that our simple model correctly captures and explains the fundamental principles of inchworm crawling and can be applied to other soft-robotic legged robots.

We derive a reduced-order model describing the inflation and deflation dynamics of a liquid-filled hyperelastic balloon, focusing on inviscid laminar flow and the extensional motion of the balloon. We initially study the flow and pressure fields for dictated motion of the solid, which throughout deflation are obtained by solving the potential problem. However, during inflation, flow separation creates a jet within the balloon, requiring a different approach. The analyses of both flow regimes lead to a simple piecewise model, describing the fluidic pressure during inflation and deflation, which is then verified by finite element computations. We then use a variational approach to derive the equation governing the balloon's dynamics, yielding a nonlinear hybrid oscillator equation, describing the interaction between the extensional mode of the balloon, and the entrapped fluid. Analytical and graphical investigations of the suggested model are presented, shedding light on its static and dynamic behaviour under different operating conditions. Our suggested model and its underlying assumptions are verified utilizing a fully coupled finite element scheme, showing excellent agreement., Comment: 30 pages, 9 figures

The research fields of microfluidics and soft robotics both involve complex small-scale internal channel networks, embedded within a solid structure. This work examines leveraging viscous peeling as a mechanism to create and activate soft actuators and microchannel networks, including complex elements such as valves, without the need for fabrication of structures with micron-scale internal cavities. We consider configurations composed of an internal slender structure embedded within another elastic solid. Pressurized viscous fluid is introduced into the interface between the two solids, thus peeling the two elastic structures and creating internal cavities. Since the gap between the solids is determined by the externally applied pressure, the characteristic size of the fluidic network may vary in time and be much smaller than the resolution of the fabrication method. This work presents a model for the highly nonlinear elastic-viscous dynamics governing the flow and deformation of such configurations. Fabrication and experimental demonstrations of micron-scale valves and channel-networks created from millimeter scale structures are presented, as well as the transient dynamics of viscous peeling based soft actuators. The experimental data is compared with the suggested model, showing very good agreement., Comment: 8 pages, 6 figures

We demonstrate the existence of a fluid-structure instability arising from the interaction of electro-osmotic flow with an elastic substrate. Considering the case of flow within a soft fluidic chamber, we show that above a certain electric field threshold, negative gauge pressure induced by electro-osmotic flow causes the collapse of its elastic walls. We combine experiments and theoretical analysis to elucidate the underlying mechanism for instability and identify several distinct dynamic regimes. The understanding of this instability is important for the design of electrokinetic systems containing soft elements., Comment: 10 pages, 4 figures

This work analyzes the viscous flow and elastic deformation created by the forced axial motion of a rigid cylinder within an elastic liquid-filled tube. The examined configuration is relevant to various minimally invasive medical procedures in which slender devices are inserted into fluid-filled biological vessels, such as percutaneous revascularization, interventional radiology, endoscopies and catheterization. By applying the lubrication approximation, thin shell elastic model, as well as scaling analysis and regular and singular asymptotic schemes, the problem is examined for small and large deformation limits (relative to the gap between the cylinder and the tube). At the limit of large deformations, forced insertion of the cylinder is shown to involve three distinct regimes and time-scales: (i) initial shear dominant regime, (ii) intermediate regime of dominant fluidic pressure and a propagating viscous-peeling front, (iii) late-time quasi-steady flow regime of the fully peeled tube. A uniform solution for all regimes is presented for a suddenly applied constant force, showing initial deceleration and then acceleration of the inserted cylinder. For the case of forced extraction of the cylinder from the tube, the negative gauge pressure reduces the gap between the cylinder and the tube, increasing viscous resistance or creating friction due to contact of the tube and cylinder. Matched asymptotic schemes are used to calculate the dynamics of the near-contact and contact limits. We find that the cylinder exits the tube in a finite time for sufficiently small or large forces. However, for an intermediate range of forces the radial contact creates a steady locking of the cylinder inside the tube., Comment: 22 pages, 7 figures

A leading concept in soft robotics actuation, as well as in microfluidics applications such as valves in lab-on-a-chip devices, is applying pressurized flow in cavities embedded within elastic bodies. Generating complex deformation patterns typically requires control of several inputs, which greatly complicates the system's operation. In this work, we present a novel method for single-input control of a serial chain of bi-stable elastic chambers connected by thin tubes. Controlling a single flow rate at the chain's inlet, we induce an irreversible sequence of transitions that can reach any desired state combination of all bi-stable elements. Mathematical formulation and analysis of the system's dynamics reveal that these transitions are enabled thanks to bi-stability combined with pressure lag induced by viscous resistance. The results are demonstrated via numerical simulations combined with experiments for chains of up to 5 chambers, using water-diluted glycerol as the injected fluid. The proposed technique has a promising potential for development of sophisticated soft actuators with minimalistic control., Comment: Supplementary document will be added soon

The study of elastic structures embedded with fluid-filled cavities received considerable attention in fields such as autonomous materials, sensors, actuators, and smart systems. This work studies an elastic beam embedded with a set of fluid-filled bladders, similar to a honeycomb structure, which are interconnected via an array of slender tubes. The configuration of the connecting tubes is arbitrary, and each tube may connect any two bladders. Beam deformation both creates, and is induced by, the internal viscous flow- and pressure-fields which deform the bladders and thus the surrounding solid. Applying concepts from poroelasticity, and leveraging Cosserat beam large-deformation models, we obtain a set of three coupled equations relating the fluidic pressure within the bladders to the large transverse and longitudinal displacements of the beam. We show that by changing the viscous resistance of the connecting tubes we are able to modify the amplitude of oscillatory deformation modes created due to external excitations on the structure. In addition, rearranging tube configuration in a given bladder system is shown to add an additional degree of control, and generate varying mode shapes for the same external excitation. The presented modified Cosserat model is applied to analyze a previously suggested energy harvester configuration and estimate the efficiency of such a device. The results of this work are validated by a transient three-dimensional numerical study of the full fluid-structure-interaction problem. The presented model allows for the analysis and design of soft smart-metamaterials with unique mechanical properties. Keywords: autonomous materials, adaptive materials, programmed materials, smart systems, autonomous systems, soft matter, soft robotics, energy harvesting, fluid dynamics, fluid-structure interaction, large deformation., Comment: 49 pages, 16 figures, 2 tables

In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slender linearly elastic cylinder under the creeping flow regime. Considering power-law fluids and applying the thin shell approximation for the elastic cylinder, we obtain a non-homogeneous p-Laplacian equation governing the viscous-elastic dynamics. We present exact solutions for the pressure and deformation fields for various initial and boundary conditions for both shear-thinning and shear-thickening fluids. We show that in contrast to Stokes' problem where a compactly supported front is obtained for shear-thickening fluids, here the role of viscosity is inversed and such fronts are obtained for shear-thinning fluids. Furthermore, we demonstrate that for the case of a step in inlet pressure, the propagation rate of the front has a $t^{\frac{n}{n+1}}$ dependence on time ($t$), suggesting the ability to indirectly measure the power-law index ($n$) of shear-thinning liquids through measurements of elastic deformation., Comment: 27 pages, 5 figures

The aim of this work is to study the dynamics and stability of soft shape-morphing configurations and specifically the modes of interaction between the front and rear airfoil segments. Initially we present several steady-state solutions, such as canceling of deflection due to aerodynamic forces and transition between two predefined cambers via continuous actuation of the airfoil. The steady results are validated by numerical calculations based on commercially available software. We then examine stability and transient dynamics by assuming small deflections and applying multiple-scale analysis to obtain a stability condition. The condition is attained via the compatibility equations of the orthogonal spatial modes of the first-order correction. The results yield the maximal stable speed as a function of elastic damping, fluid density and location of clamping. The results show that the interaction between the front and rear segments is the dominant mechanism for instability for various discrete locations of clamping. Instabilities due to interaction dynamics between the front and rear segments become more significant as the location of clamping approaches the leading edge. Several transient dynamics are presented for stable and unstable configurations, as well as instability dynamics initiated by cyclic actuation at the natural frequency of the airfoil., Comment: 24 pages, 6 figures

We study pressure-driven propagation of gas into a micron-scale gap between two linearly elastic substrates. Applying the lubrication approximation, the flow-field is governed by the interaction between elasticity and viscosity, as well as weak rarefaction and low-Mach-compressibility, characteristic to gaseous microflows. Several physical limits allow simplification of the governing evolution equation and enable solution by self-similarity. These limits correspond to different time-scales and physical regimes which include compressiblity-elasticity-viscosity, compressiblity-viscosity and elasticity-viscosity dominant balances. For a prewetting layer thickness which is similar to the elastic deformation generated by the background pressure, a symmetry between compressibility and elasticity allows to obtain a self-similar solution which includes weak rarefaction effects. The results are validated by numerical solutions of the evolution equation, Comment: 10 pages, 5 figures

We study steady-state oscillations of an elastic Hele-Shaw cell excited by traveling pressure waves over its upper surface. The fluid within the cell is bounded by two asymmetric elastic sheets which are connected to a rigid surface via distributed springs and modeled by the linearized plate theory. Modal analysis yields the frequency response of the configuration as a function of three parameters: the fluidic Womersley number and two ratios of elastic stress to viscous pressure for each of the sheets. These ratios, analogous to the Capillary number, combine the effects of fluid viscosity and the sheets inertia, bending and tension. The resonance frequencies of the configuration include the resonance frequency of the upper sheet, and the resonance frequency of both sheets with a constraint of constant gap. Near the resonance frequency of the upper sheet, the fluid pressure is identical in amplitude and phase to the external excitation. For configurations where both sheets are near resonance, small changes in frequency yield significant modification of the fluidic pressure. In addition, a new resonance frequency is observed, related to the interaction between motion of fluid parallel to the elastic sheets and relative elastic displacements of the sheets. The ratio of the amplitude of the fluidic pressure to the external pressure is presented vs. frequency for several characteristic solid and fluid properties, yielding a bandpass filter behaviour. For configurations with a rigid lower surface the pressure ratio is unity at the passband, while for two elastic sheets pressure amplification or reduction occurs near the resonance frequencies. The results presented here may allow to utilize elastic Hele-Shaw configurations as protective surfaces and mechanical filters.

The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics, and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Foppl-von Karman equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and timescale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Foppl-von Karman equations and present the deformation behavior as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics., Comment: 19 pages, 9 figures

We study the hydrodynamic interaction between two closely-spaced waving elastic cylinders immersed within a viscous liquid, at the creeping flow regime. The cylinders are actuated by a forced oscillation of the slope at their clamped end and are free at the opposite end. We obtain an expression for the interaction force and apply an asymptotic expansion based on a small parameter representing the ratio between the elastic deflections and the distance between the cylinders. The leading-order solution is an asymmetric oscillation pattern at the two frequencies ($\omega_1,\omega_2$) in which the cylinders are actuated. Higher orders oscillate at frequencies which are combinations of the actuation frequencies, where the first-order includes the $2\omega_1,2\omega_2,\omega_1+\omega_2$, and $\omega_1-\omega_2$ harmonics. For in-phase actuation with $\omega_1= \omega_2$, the deflection dynamics are identical to an isolated cylinder with a modified Sperm number. For configurations with $\omega_1\approx \omega_2$, the $\omega_1-\omega_2$ mode represents the dominant first-order interaction effect due to significantly smaller effective Sperm number. Experiments are conducted to verify and illustrate the theoretical predictions.

We analyze flow of non-Newtonian fluids in a Hele-Shaw cell, subjected to spatially non-uniform electroosmotic slip. Motivated by their potential use for increasing the characteristic pressure fields, we specifically focus on power-law fluids with wall depletion properties. We derive a p-Poisson equation governing the pressure field, as well as a set of linearized equations representing its asymptotic approximation for weakly non-Newtonian behavior. To investigate the effect of non-Newtonian properties on the resulting fluidic pressure and velocity, we consider several configurations in one- and two-dimensions, and calculate both exact and approximate solutions. We show that the asymptotic approximation is in good agreement with exact solutions even for fluids with significant non-Newtonian behavior, allowing its use in the analysis and design of microfluidic systems involving electro-kinetic transport of such fluids., Comment: 20 pages, 7 figures

We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure-field. We model the flow-field via the lubrication approximation, and the plate deformation by the Kirchhoff-Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed form expressions. We examine a uniform external pressure acting on the elastic plate over a finite region and during a finite time period, similar to the viscous-elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure, thus suggesting such configurations may be used for impact mitigation.

Elastic deformation due to embedded fluidic networks is currently studied in the context of soft-actuators and soft-robotic applications. In this work, we analyze interaction between the elastic deflection of a slender beam and viscous flow within a long serpentine channel, embedded in the elastic beam. The channel is positioned asymmetrically with regard to the midplane of the beam, and thus pressure within the channel creates a local moment deforming the beam. We focus on creeping flows and small deflections of the elastic beam and obtain, in leading order, a fourth-order partial integro-differential equation governing the time-dependent deflection field. This relation enables the design of complex time-dependent deformation patterns of beams with embedded channel networks, including inertia-like standing and moving wave solutions in configurations with negligible inertia., Comment: 11 pages, 5 Figures

Various microfluidic systems, such as chemical and biological lab-on-a-chip devices, involve motion of multiple droplets within an immersing fluid in shallow micro-channels. Modeling the dynamics of such systems requires calculation of the forces of interaction between the moving droplets. These forces are commonly approximated by superposition of dipoles solutions, which requires an assumption of sufficiently large distance between the droplets. In this work we obtain exact solutions (in the Hele-Shaw limit) for two moving droplets, and a droplet within a droplet, located within a moving immersing fluid, without limitation on the distance between the droplets. This is achieved by solution of the pressure field in a bi-polar coordinate system and calculation of the force in a Cartesian coordinate system. Our results are compared with numerical computations, experimental data and the existing dipole-based models. We utilize the results to calculate the dynamics of a droplet within a droplet, and of two close droplets, located within an immersing fluid with oscillating speed. The obtained results may be used to study the dynamics of dense droplet lattices, common to many microfluidic systems., Comment: 12 pages, 6 figures

We examine transient axial creeping flow in the annular gap between a rigid cylinder and a concentric elastic tube. The gap is initially filled with a thin fluid layer. The study focuses on viscous-elastic time-scales for which the rate of solid deformation is of the same order-of-magnitude as the velocity of the fluid. We employ an elastic shell model and the lubrication approximation to obtain a forced nonlinear diffusion equation governing the viscous-elastic interaction. In the case of an advancing liquid front into a configuration with a negligible film layer (compared with the radial deformation of the elastic tube), the governing equation degenerates into a forced porous medium equation, for which several closed-form solutions are presented. In the case where the initial film layer is non-negligible, self-similarity is used to devise propagation laws for a pressure driven liquid front. When advancing external forces are applied on the tube, the formation of dipole structures is shown to dominate the initial stages of the induced flow and deformation regimes. These are variants of the dipole solution of the porous medium equation. Finally, since the rate of pressure propagation decreases with the height of the liquid film, we show that isolated moving deformation patterns can be created and superimposed to generate a moving wave-like deformation field. The presented interaction between viscosity and elasticity may be applied to fields such as soft-robotics and micro-scale or larger swimmers by allowing for the time-dependent control of an axisymmetric compliant boundary., Comment: 21 pages, 7 figures

We present an analytical study, validated by numerical simulations, of electroosmotic flow in a Hele-Shaw cell with non-uniform surface charge patterning. Applying the lubrication approximation and assuming thin electric double layer, we obtain a pair of uncoupled Poisson equations which relate the pressure and the stream function, respectively, to gradients in the zeta potential distribution parallel and perpendicular to the applied electric field. We solve the governing equations for the fundamental case of a disk with uniform zeta potential and show that the flow-field in the outer region takes the form of a pure dipole. We illustrate the ability to generate complex flow-fields around smooth convex regions by superposition of such disks with uniform zeta potential and a uniform pressure driven flow. This method may be useful for future on-chip devices, allowing flow control without the need for mechanical components., Comment: Accepted to Physics of Fluids

Viscous flows within an elastic structure apply stress on the solid-liquid interface. The stress-field created by the viscous flow can be utilized to counter stress created by external forces and thus may be applied as a tool for delaying the onset of structural failure. To illustrate this concept we study viscous flow within an elastic cylinder under compressive axial force. We obtain a closed-form expression showing an approximately linear relation between the critical buckling load and the liquid inlet pressure. Our results are validated by numerical computations. We discuss future research directions of fluid-solid composite materials which create flow under external stress, yielding enhanced resistance to structural failure.