Buckled State Research Articles

Elastic Snap-Through Instabilities Are Governed by Geometric Symmetries.

Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside out. This snap through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological and engineered systems from the Venus flytrap to mechanical metamaterials. Shape transitions are known to be related to the type of bifurcation the system undergoes, however, to date, there is no general understanding of the mechanisms that select these bifurcations. Here we analyze numerically and analytically two systems proposed in recent literature in which an elastic strip, initially in a buckled state, is driven through shape transitions by either rotating or translating its boundaries. We show that the two systems are mathematically equivalent, and identify three cases that illustrate the entire range of transitions described by previous authors. Importantly, using reduction order methods, we establish the nature of the underlying bifurcations and explain how these bifurcations can be predicted from geometric symmetries and symmetry-breaking mechanisms, thus providing universal design rules for elastic shape transitions.

Seismic vibration mitigation of steel storage tanks by metafoundations endowed with linear and bistable columns

AbstractThis paper presents the seismic mitigation of typical storage tanks where extreme loading conditions are considered by safe shutdown earthquakes. To reproduce the main dynamic properties of the superstructure, a standard structural model was considered, where both the presence of the impulsive mode and of the convective mode were considered. Thus, to protect the tank from strong earthquakes, finite locally resonant multiple-degrees-of-freedom (MDoFs) metafoundations were designed and developed; and resonator parameters together with bistable columns were optimized by means of an improved time domain multiobjective optimization procedure. Also, the stochastic nature of the seismic input was taken into account. Therefore, it is proposed: (i) a linear metafoundation endowed with one/two layers and multiple cells, linear springs, and linear viscous dampers; and (ii) a relevant foundation equipped with columns operating in an elastic buckled state. With this arrangement, additional flexibility and dissipation against horizontal seismic loadings are activated. It was shown in both cases, how each metafoundation can be successfully optimized via a sensitivity-based parameter technique. Thus, the performance of the optimized metafoundations was assessed by means of time history analyses; and results were compared with a storage tank endowed with both rigid foundation solutions. Finally, single cells were analysed in the frequency domain while finite lattices and periodic metafoundations in the linear and bistable regime were characterized by means of dispersion relationships.

Omnidirectional wind piezoelectric energy harvesting

A novel energy harvester based on vortex-induced vibration of the sphere and piezoelectric effect is proposed to efficiently gather wind energy from all horizontal directions. An elastically-supported foam sphere is vertically arranged and considerably oscillates in the cross-flow direction when the wind speed is in the lock-in region. A piezoelectric beam is attached to the supported sphere by a spring and deforms periodically around the original buckling state, thereby converting kinetic energy into electrical energy. Experimental studies are performed to assess the power output of the harvester when exposed to wind flows with varying directions and speeds. The omnidirectional wind flow is divided to 12 orientations with the interval angle of 30°, and the wind speeds range from 1.17 m s−1 to 7.87 m s−1. The testing findings indicate that the harvester has excellent consistency in both lock-in region and average power for various wind azimuths. When the wind direction is changed from 0° to 360°, the peak average power changes from 179.8 μW to 247.5 μW, and the wind speed region where the sphere undergoes vibration changes from 2.58 m s−1 7.05 m s−1 to 2.58 m s−1 7.87 m s−1. Following that, the effects of the length of the supporting spring and the diameter of the sphere on the output average power and lock-in region are investigated experimentally. Finally, a demonstration of powering a wireless sensing node is performed to show applications of the designed energy harvester in windy conditions.

Harnessing the nonlinear properties of buckling inflatable tubes for complex robotic behaviors

Imparting complex behaviors to soft robots generally entails complex designs and carefully engineered structures. As such, making soft structures with behaviors such as stiffening, bistability and rapid snapping deformations generally requires specialized solutions. This work introduces the use of short inflatable tubes that buckle under compression to realize behaviors such as self-stiffening, bistability and snap-through induced rapid deformations. These inflatable tubes have a nonlinear behavior resulting from these tubes having two stable states, the buckled and unbuckled states, and the complex transition between states which entails a snap-through buckling instability during the buckling transition, and a range of deformation where the tube is bistable as it is stable in either the buckled or unbuckled states. This work shows how the dimensions of the inflatable tubes affect this behavior and proposes a numerical model that can describe the entire cyclic behavior of buckling and unbuckling inflatable tubes. It is then demonstrated how simply connected multiple inflatable tube segments in series can be used to produce complex robotic behaviors. A soft robotic hand is demonstrated that uses the characteristics of unbuckling inflatable tubes for self-stiffening, and a grasping tool capable of slow or rapid motions by making use of the ability of serially connected inflatable tubes to transition from bistability to monostability.

Multi-blade monolithic Euler springs with optimised stress distribution

Euler springs are used for vertical suspension and vibration isolation as they provide a large static supporting force with a low spring-rate and use minimal spring material. To date, multiple single-width rectangular blades of uniform thickness and stacked flat-face to flat-face have been used in the post buckled state, with half of the blades buckling in each of opposing directions. For ultra-low-noise isolation the ends need to be clamped which results in stick–slip issues at the joints. In this study we investigate the benefits of forming side-by-side oppositely buckling blades from a single monolithic sheet of spring material. Additionally, we study how to distribute the stress evenly along the length of the blade by contouring its width, as well as finding the optimal contour to distribute the stress evenly around the tearing joints between oppositely bending blade sections. We show that this optimal shaping typically improves the inconveniently small spring working range by over 60% compared to an equivalent rectangular blade.

Foreword to the special issue on new developments in structural stability.

Foreword to the special issue on new developments in structural stability.

An M−shaped buckled beam for enhancing nonlinear energy harvesting

Aiming to achieve the high-performance piezoelectric energy harvester under low excitation, a bistable energy harvester with an M−shaped buckled beam is proposed in this paper. When the M−shaped beam is pre-compressed into a buckling state, motion of the M−shaped beam has changed from monostable oscillation to bistable oscillation. Compared with the original monostable M−shaped beam, numerical and experimental results show that the proposed harvester can easily maintain large-amplitude interwell oscillation and offer a wider operation bandwidth in lower frequency range. The harvester can offer an operation bandwidth of 5 Hz or 6 Hz under 3 m/s2 or 5 m/s2 excitation, which is improved by 233.3 % or 140 % in comparison with the monostable configurations. In bistable configurations, the acceleration threshold required for interwell oscillation can be decreased by reducing the buckling level, which can provide higher voltage and wider operation bandwidth in sweep frequency excitation. In monostable configurations, it is observed that the M−shaped beam shows a softening behavior or a hardening behavior with different clamping distance. Numerical and experimental results verify that the performance of monostable configuration with a softening behavior is better than that with a hardening behavior.

Axisymmetric buckling analysis of submerged hemi-elliptic toroidal shells

Toroidal shells have been widely used in various industrial applications for over a decade. Most studies on buckling behavior have focused on circular toroidal shells with uniform loads. In contrast, the existing literature on the shell buckling of a toroid with elliptical cross-sections has received limited attention. This study aims to present numerical solutions for axisymmetric buckling of hemi-elliptic toroidal shells under linearly hydrostatic pressure to append available data and results in this field. The thickness of the hemi-elliptic toroidal shells is assumed to be constant before and after the buckling state. Solution procedures for solving critical buckling loads dealt with eigenvalue problems and were formulated by an energy approach. All relevant parameters such as membrane and bending strain components and in-plane stress resultants were derived based on differential geometry. An in-house FEM program solved the weak formulation derived from a variational method. The buckling results for elliptic toroidal shells with various b/a ratios were established as the critical seawater depth and compared to the previously published results. Additionally, parametric studies on the effect of the variable b/a ratios and the major radius R of elliptic toroidal shells on the axisymmetric buckling loads and mode shapes were examined thoroughly and addressed in more detail herein.

Biomechanical comparison of different prosthetic reconstructions in total en bloc spondylectomy: a finite element study

ObjectiveTo analyse and compare the biomechanical differences between 3D-printed prostheses, titanium mesh cages and poorly matched titanium mesh cages in total en bloc spondylectomy (TES).MethodsThe finite element model of T10-L2 for healthy adults was modified to make three models after T12 total spondylectomy. These models were a 3D-printed prosthesis, titanium mesh cage and prosthesis-endplate mismatched titanium mesh cage for reconstruction. The range of motion (ROM), stress distribution of the endplate and internal fixation system of three models in flexion and extension, lateral bending and axial rotation were simulated and analysed by ABAQUS.ResultIn flexion, due to the support of the anterior prosthesis, the fixation system showed the maximum fixation strength. The fixation strength of the 3D-printed prosthesis model was 26.73 N·m /°, that of the TMC support model was 27.20 N·m /°, and that of the poorly matched TMC model was 24.16 N·m /°. In flexion, the L1 upper endplate stress of the poorly matched TMC model was 35.5% and 49.6% higher than that of the TMC and 3D-printed prosthesis, respectively. It was 17% and 28.1% higher in extension, 39.3% and 42.5% higher in lateral bending, and 82.9% and 91.2% higher in axial rotation, respectively. The lower endplate of T11 showed a similar trend, but the magnitude of the stress change was reduced. In the stress analysis of the 3D-printed prosthesis and TMC, it was found that the maximum stress was in flexion and axial rotation, followed by left and right bending, and the least stress was in extension. However, the mismatched TMC withstood the maximum von Mises stress of 418.7 MPa (almost twice as much as the buckling state) in rotation, 3 times and 5.83 times in extension, and 1.29 and 2.85 times in lateral bending, respectively.ConclusionDifferent prostheses with good endplate matching after total spondylectomy can obtain effective postoperative stable support, and the reduction in contact area caused by mismatch will affect the biomechanical properties and increase the probability of internal fixation failure.

In plane magnetization reversal in nanosized thin truncated conical double-disks of permalloy

In plane magnetization reversal of permalloy thin truncated conical double-disk as a function of the is investigated using micromagnetic simulations with fixed base radius (R) of 100 nm. When external magnetic field is applied along the longer axis of the double-disk, the remanent states change gradually from vortex state to S state and then to buckled magnetization state with reduction of σ. When σ ≈ 1 and the conical nanodisk resembles to a regular cylindrical nanodisk, incoherent magnetization reversal is dominant whereas tapering of conical disk reduces extent of incoherence in magnetization reversal. As tapering of nanodisk goes extreme so that σ ≈ 0.1, the magnetization reversal is governed only by coherent rotation. Correspondingly, coercive field reduces monotonically as σ increases. On the other hand, when a field is applied in plane but perpendicular to the long axis, almost zero coercivity is discovered. These variations are explained using a analytical calculation of demagnetization factors which quantifies shape anisotropy as well as the consideration of incoherence in magnetization reversal.

A hydrogel-based mechanical metamaterial for the interferometric profiling of extracellular vesicles in patient samples.

The utility of mechanical metamaterials for biomedical applications has seldom been explored. Here we show that a metamaterial that is mechanically responsive to antibody-mediated biorecognition can serve as an optical interferometric mask to molecularly profile extracellular vesicles in ascites fluid from patients with cancer. The metamaterial consists of a hydrogel responsive to temperature and redox activity functionalized with antibodies to surface biomarkers on extracellular vesicles, and is patterned into micrometric squares on a gold-coated glass substrate. Through plasmonic heating, the metamaterial is maintained in a transition state between a relaxed form and a buckled state. Binding of extracellular vesicles from the patient samples to the antibodies on the hydrogel causes it to undergo crosslinking, induced by free radicals generated via the activity of horseradish peroxidase conjugated to the antibodies. Hydrogel crosslinking causes the metamaterial to undergo fast chiral re-organization, inducing amplified changes in its mechanical deformation and diffraction patterns, which are detectable by a smartphone camera. The mechanical metamaterial may find broad utility in the sensitive optical immunodetection of biomolecules.

Modelling membrane reshaping by staged polymerization of ESCRT-III filaments.

ESCRT-III filaments are composite cytoskeletal polymers that can constrict and cut cell membranes from the inside of the membrane neck. Membrane-bound ESCRT-III filaments undergo a series of dramatic composition and geometry changes in the presence of an ATP-consuming Vps4 enzyme, which causes stepwise changes in the membrane morphology. We set out to understand the physical mechanisms involved in translating the changes in ESCRT-III polymer composition into membrane deformation. We have built a coarse-grained model in which ESCRT-III polymers of different geometries and mechanical properties are allowed to copolymerise and bind to a deformable membrane. By modelling ATP-driven stepwise depolymerisation of specific polymers, we identify mechanical regimes in which changes in filament composition trigger the associated membrane transition from a flat to a buckled state, and then to a tubule state that eventually undergoes scission to release a small cargo-loaded vesicle. We then characterise how the location and kinetics of polymer loss affects the extent of membrane deformation and the efficiency of membrane neck scission. Our results identify the near-minimal mechanical conditions for the operation of shape-shifting composite polymers that sever membrane necks.

Universal Wrinkling of Supported Elastic Rings.

An exactly solvable family of models describing the wrinkling of substrate-supported inextensible elastic rings under compression is identified. The resulting wrinkle profiles are shown to be related to the buckled states of an unsupported ring and are therefore universal. Closed analytical expressions for the resulting universal shapes are provided, including the one-to-one relations between the pressure and tension at which these emerge. The analytical predictions agree with numerical continuation results to within numerical accuracy, for a large range of parameter values, up to the point of self-contact.

Strongly tunable nonlinear MEMS resonators by electro-thermoelastic buckling

This talk presents the benefit of buckling in strongly tuning nonlinear MEMS resonators. In particular, the talk shows experimental buckling achieving more than 230% tuning in the natural frequency of drumhead MEMS resonators. Moreover, the experiments demonstrate that buckling switches the frequency response between purely stiffening, purely softening, and stiffening-to-softening nonlinearities. We tune the buckling state in these experiments by controlling the electric voltage and the temperature of the resonators. Therefore, these resonators undergo electrostatically-mediated thermoelastic buckling where specific combinations of temperature and voltage are required to access the distinct vibrational responses. We explain the observed linear and nonlinear responses by a reduced-order model (ROM) that lumps the resonator into a 1-degree-of-freedom mass connected to springs reflecting bending, stretching, electrostatic forces, thermal expansions, and residual stresses. The ROM mimics von Mises trusses to model the buckling in the membrane resonators without the need for exact geometry or structure to model the resonators. This developed ROM and the electro-thermoelastic buckling tunability present useful applications for on-chip acoustic devices in different fields such as signal manipulation, filtering, and MEMS waveguides.

Compression-induced buckling of a semiflexible filament in two and three dimensions.

The ability of biomolecules to exert forces on their surroundings or resist compression from the environment is essential in a variety of biologically relevant contexts. For filaments in the low-temperature limit and under a constant compressive force, Euler buckling theory predicts a sudden transition from a compressed state to a bent state in these slender rods. In this paper, we use a mean-field theory to show that if a semiflexible chain is compressed at a finite temperature with a fixed end-to-end distance (permitting fluctuations in the compressive forces), it exhibits a continuous phase transition to a buckled state at a critical level of compression. We determine a quantitatively accurate prediction of the transverse position distribution function of the midpoint of the chain that indicates this transition. We find that the mean compressive forces are non-monotonic as the extension of the filament varies, consistent with the observation that strongly buckled filaments are less able to bear an external load. We also find that for the fixed extension (isometric) ensemble, the buckling transition does not coincide with the local minimum of the mean force (in contrast to Euler buckling). We also show that the theory is highly sensitive to fluctuations in length in two dimensions and the buckling transition can still be accurately recovered by accounting for those fluctuations. These predictions may be useful in understanding the behavior of filamentous biomolecules compressed by fluctuating forces, relevant in a variety of biological contexts.

Tailoring of effective biaxial anisotropy in a 2-D array of thin truncated conical nanodisk

In the present study, the magnetization reversal of 2-D rectangular array of soft magnetic permalloy thin truncated conical disks is simulated by using micromagnetic modeling. The base radius (R) of the conical disk is varied between 20–100nm and upper radius (r) varied between 10nm to R−10nm such that the ratio σ=r/R lies between 0 to 1. The analysis of angular dependence (θ=0° to 90°) of hysteresis loops for the case of R=100nm is done in particular and its comparison with Stoner–Wohlfarth (SW) model reveals the presence of a bi-axial anisotropy which is attributed to the dipolar interaction between the nanodisks in the array. When the nanodisks are close to regular cylindrical shape (for σ∼0.9), the orthogonal axes of the facet of the array acts as magnetic easy axes. As the nanodisks are tapered to σ∼0.5 and 0.1, the same set of axes turns as hard axis of the bi-axial anisotropic system. Corresponding bi-axial anisotropy constant is estimated from the coercivity of the hysteresis loop along the easy axis as Keff=13.508kJ/m3, −5.403kJ/m3 and −4.322kJ/m3 respectively for σ=0.9,0.5 and 0.1 with R=100nm. In addition to this, an analytical expression is also formulated to calculate the total demagnetization energy of the 2-D array as a function of the magnetization orientation. This demagnetization energy is fitted with the effective bi-axial anisotropy energy expression considering an angular dispersion of the bi-axial anisotropy. The fitting suggests that the extent of angular dispersion of bi-axial anisotropy reduces as the conical disks are tapered (σ decreased). Irrespective of the magnetic field orientation, the remanent state exhibits a s-state of spatial magnetization profile when σ is large whereas a buckled magnetization state symmetric to body diagonal is exhibited for lower σ.

Magnetization reversal and ground states in thin truncated conical nanodisks: Analytical and micromagnetic modelling approach

Simulation of magnetization reversal of soft permalloy thin truncated conical disks is done using micromagnetic modelling. The thickness of the disks are fixed at 20nm. Base radius (R) of the conical disk is varied between 50−100nm while the radius of upper surface (r) is varied such that the ratio σ=r/R lies between 0 to 1. As the conical disks are gradually tapered by reducing r, the remanent states change from vortex state to shifted vortex state to buckled magnetization state. In buckled state, the magnetic moments are essentially parallel except at the corners where they curl away from the applied field direction. The extent of curling away of magnetization in buckled state reduce as base radius R is decreased. Thus at smaller base radius, the buckled states converge to homogeneous magnetization state or so called ferromagnetic state. Analytical expressions for the total energy of these three remanent states are calculated using simple models. Analytical calculations reveal that vortex configuration is the ground state for a wide range of geometrical parameters and shifted vortex state is a metastable state which appears as remanent state in the in-plane magnetization reversal due to a possible existence of energy barrier. Other observations from micromagnetic simulations are well corroborated by the results of analytical calculations.

A fluidic relaxation oscillator for reprogrammable sequential actuation in soft robots

A fluidic relaxation oscillator for reprogrammable sequential actuation in soft robots

Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading

Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, the governing equation of vibration of the buckled beam is derived, which is a complex nonlinear partial differential equation with varying coefficients. The first natural frequency of the buckled beam is obtained by linear analysis and effects of material and geometrical parameters on it are numerically investigated. By applying the Galerkin method, the governing equation for nonlinear transverse vibration of the beam in the buckled state is transformed into a series of strongly nonlinear ordinary differential equations. Dynamic characteristics of the hyperelastic beam are investigated by the Runge–Kutta method. Bifurcation diagrams, time traces, phase-plane portraits, and Poincare sections are obtained for different values of the external excitation frequency and amplitude of variation of the axial load. Results show that different dynamic behaviors of the hyperelastic buckled beam such as periodic, multi-periodic, quasiperiodic, and chaotic motions can be found when the amplitude of the axial load is varied. Amplitude–frequency responses of the hyperelastic buckled beam are obtained by the Runge–Kutta method and harmonic balance method. Results from the Runge–Kutta method are in good agreement with those from the harmonic balance method. Lastly, effects of the external mean axial load, amplitude of variation of the axial load, geometrical and material parameters on amplitude–frequency responses of the buckled beam are numerically investigated.

Machine learning driven seismic performance limit state identification for performance-based seismic design of bridge piers

Performance-based earthquake engineering requires the identification of different damage states in structural components. Bridge piers are one of the most critical components in the bridge system that dictate the overall performance of bridges under seismic loading. The ability to predict different damage states of a bridge pier following an earthquake can be useful in restoring service or prescribing appropriate remediation. Machine learning techniques are becoming attractive in earthquake engineering for predicting failure modes of buildings and bridges. This study implements several machine learning techniques to capture different performance limit states such as spalling, core crushing, and bar buckling of reinforced concrete bridge piers under seismic loading. Using experimental data from the large scale testing of bridge piers, this study compares different machine learning techniques for predicting drift capacity performance limit states of circular and rectangular bridge piers. Model tuning using grid search algorithm and 5-fold crossvalidation is performed to optimize model predictions over a wide tuning parameter search space. The efficiency of different methods is compared using a reserved 20% testing dataset to determine the optimal machine learning regression method for each limit state. This study compares the respective bar buckling and concrete spalling drift limit state models against existing physics-based empirical models, where the machine learning models are found to have significantly improved accuracy and precision.