Limit Point Instability Research Articles

Structural instabilities in soft electro-magneto-elastic cylindrical membranes

In the present work, a nonlinear coupled electro-magneto-elastic membrane formulation is developed for soft functional materials starting from the variational form of 3D governing equations. The resulting 2D model is applied to an internally pressurized cylindrical membrane in an azimuthal magnetic field and radial electric field. The results of our model are verified with existing literature for some special cases. The model is subsequently used to analyze mechanical and electrical limit-point instabilities and the effect of external fields on the onset of these instabilities. It is observed that the onset of mechanical limit-point instability, defined by a loss of monotonicity in the pressure versus deformation plots, is dependent on the material properties, geometrical parameters, and applied electromagnetic fields. Magnetic field-induced instability results in an initial dip in the pressure versus stretch plots, subsequently converging to the purely hyperelastic membrane behavior at larger stretches. On the other hand, applying an electric field results in an early onset of limit-point instability (i.e., at smaller stretches) compared to the hyperelastic case. In this work, we additionally observe the presence of an electrical limit-point instability, characterized by a loss of monotonicity in voltage versus stretch plots. The dependence of electrical-limit point on magnetic field and force inputs is studied. To the best of our knowledge, this effect has not been discussed in prior literature. Finally, we study the effect of Maxwell stress due to electromagnetic fields. It is observed that ignoring the Maxwell stress related traction boundary term results in an error of up to 10%, depending on the force and magnetic field inputs. In summary, our membrane model describes the interactions between the electromagnetic fields. The electric, magnetic and elastic limit points observed in these membranes under different set of loading conditions can be used towards improving the design of soft actuator and energy-harvesting devices.

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

Non-Linear Instability of Pin-Ended Functionally Graded Material Arches Under Locally Distributed Radial Loads

The arch is a common structural form in bridge engineering; its collapse is often caused by instability. In this article, in-plane nonlinear instability of pin-ended functionally graded material (FGM) arches with two cross-sectional types under local radial loads is studied. New analytical solutions to nonlinear equilibrium paths, limit point instability, bifurcation instability, and multiple limit point instability of pin-ended FGM arches under local radial load are obtained. Modified slenderness corresponding to different instability patterns of FGM arches is also derived. Comparison with the numerical results of ANSYS demonstrates that the analytical solution is accurate. The results show that cross-sectional types of FGM arches have a great influence on limit-point instability and bifurcation instability. Localized parameters increase lead-to-limit point instability load and bifurcation instability load increases, while increasing the modified slenderness ratio results in decreased limit point instability load and bifurcation instability load. In addition, a material proportion coefficient and power law index increase can also lead to limit point instability load and bifurcation instability load decrease.

Extended Gent models for residually stressed thick spheres and cylinders

We develop some Gent-like limited extensible models for initial stress symmetry. The proposed models exhibit sharp strain-stiffening characteristics when the stiffening parameter approaches the maximum allowable limit. When this maximum allowable limit tends to infinity, these models reduce to some other existing models for initial stress. We perform a micro-mechanical investigation which agree with the proposed constitutive behaviour. This investigation also affirms that the residual stress influences the limited elastic behaviour as exhibited by the proposed models. We apply the proposed extended Gent models and the extended Mooney–Rivlin model to investigate the inflation and instability characteristics of residually stressed thick spheres and cylinders. The influence of residual stress on the stability characteristics is examined. We observe that the stationary inflection point is obtained for a single value of residual stress, in the inflation characteristics for extended Mooney–Rivlin spheres. On the other hand, in the inflation characteristics of extended Gent spheres, stationary inflection points are obtained for two residual stress limits. These residually stressed spheres exhibit stable inflation characteristics when residual stress is beyond the limits of stationary inflection points. Within these limits, however, the inflation characteristics show inflation-jump instability. Limit point instability is also exhibited by some of the proposed models in absence of residual stress. We next study the inflation of residually stressed thick cylinders for both constrained and unconstrained axial stretch. The thick cylinder inflation characteristics with unconstrained axial stretch show both stability and inflation-jump instability based on the magnitude of residual stress.

Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

The application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure (P) – inflation (lambda or v) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

Effects of movement and rotation of supports on nonlinear instability of fixed shallow arches

Abstract In practice, the foundations of arch bridges or other arch structures may have non-uniform settlements leading to uneven movements or rotations of the supports of the arch. The uneven movements and rotations of the supports influence the deformations and internal forces produced by the external load as well as the in-plane non-linear instability behaviour of arches significantly. Little research is reported about effects of support movements and rotations on the nonlinear in-plane instability of shallow arches in the opening literature insofar. This paper focuses on effects of uneven movements and rotations of the supports on the in-plane nonlinear instability of fixed shallow arches. New analytical solutions for the limit point instability load of the fixed shallow arches under a uniformly localized radial load incorporating the influences of uneven movements and rotations of the supports are derived. Comparisons against the finite element results show the analytical solutions are adequately accurate. It is found that the uneven support movements and rotations have significant effects on the nonlinear in-plane instability behaviour of fixed shallow arches. It is also found that the in-plane nonlinear instability load of a fixed shallow arch decreases with an increase of the magnitudes of uneven movements and rotations of the support.

A new analytical model of wellbore collapse pressure based on stability theory

In theory, the wellbore collapse for the petroleum drilling belongs to the limit point instability, and should be treated as the stability problem, but instead of the strength problem. For this purpose, a new analytical model was established to determine the collapse pressure based on the stability theory. Using the elastoplastic mechanics, the stress and displacement around the wellbore were solved, and the equilibrium path (i.e., the relation between the inner pressure and the radial displacement) was proposed. According to the equilibrium path, it can be found that the wellbore collapse occurs at the end of the plastic softening stage and the beginning of the residual stage. Additionally, a comprehensive numerical validation was performed based on the numerical methods, to examine the proposed analytical model. The stress and displacement around the wellbore by the proposed method agree well with those from the numerical method, with a deviation of less than 8.9%. Finally, the parametric analysis was performed to investigate the effects of the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength on the equilibrium path. The results show that the collapse pressure decreases as the plastic hardening modulus, the plastic softening modulus, the yield limit and the residual strength increasing. On the contrary, an increasing of the strength limit can increase the wellbore collapse pressure, which is different to the common understanding.

Instabilities in the free inflation of a nonlinear hyperelastic toroidal membrane

Study on an incompressible nonlinear hyperelastic thin-walled toroidal mem- brane of circular cross-section subjected to inflation due to a uniform pressure is conducted in this work. Comparisons are made for three elastic constitutive mod- els (neo-Hookean, Mooney–Rivlin, and Ogden) and for different geometric aspect ratios (ratio of the radius of cross-section to the radius of revolution). A variational approach is used to derive the equations of equilibrium and bifurcation. An analysis of the pressure–deformation plots shows occurrence of the well-known limit point (snap through) instabilities in membrane. Calculations are performed to study the elastic buckling point to predict bifurcation of solution corresponding to loss of symmetry. Tension field theory is employed to study the wrinkling instability that, in this case, typically occurs near the inner regions of tori with large aspect ratios.

Localized loading and nonlinear instability and post-instability of fixed arches

In the engineering practice, arches, such as arch bridges, are often subjected to localized loading. It is known that the nonlinear in-plane instability of shallow arches is caused by the significant in-plane bending action. The different localized loading produces very different in-plane bending distributions and so the instability of arches under localized loading would be quite different from that under a central point load or under a radial load uniformly distributed over the entire arch. However, there are no studies about the nonlinear instability and post-instability of arches under localized loading reported in the literature. This paper presents nonlinear in-plane analyses for fixed shallow circular arches under a localized uniform radial load. Novel theoretical solutions for nonlinear equilibrium, limit point instability, and bifurcation instability for arches under a localized uniform load are obtained, and new theoretical solutions of the certain modified slenderness switching the instability patterns of an arch under localized uniform loading are also obtained. It is found that the length of localized loading segment significantly influences the in-plane instability and post-instability responses of an arch. The limit point and bifurcation instability loads decrease with a decrease of the length of the localized loading segment. However, the modified slenderness switching instability patterns is found to increase with a decrease of the length of the localized loading segment. Comparisons against the finite element results show the theoretical solutions derived in this paper are accurate. These theoretical solutions provide comprehensive solutions for nonlinear instability and post-instability of fixed arches and they can reduce to those under a central point load or to extend to those under a uniform load over the entire arch.

Truss-based nonlinear mechanical analysis for origami structures exhibiting bifurcation and limit point instabilities

Origami, the art of paper folding, is a technologically transformative art with applications at many length scales from material microstructure design to space deployable structures. The transformation of a two-dimensional fold pattern to a three-dimensional structure makes origami a practical basis for deployability, light-weight materials, and self-actuation. In order to predict the complex motions of origami structures including path traversal of instabilities, accurate and efficient modeling techniques are required. While many approaches consider rigid facets and/or linear approximations to rotations, a truss and hinge finite element method is presented here within a global coordinate system framework to accurately capture the geometric nonlinearities while allowing for small to moderate facet deformation. Accurately capturing these modes of deformation is critical toward understanding the elastic energetic states required for design and analysis of multistable origami structures and mechanical metamaterials. Particular formulation developments that we address to solve origami mechanics problems include formulation of a continuous and differentiable fold angle, strategy for selecting fold path off the flat state, and integration of common numerical approaches important for origami (large rotation formulation, enforcement of periodic boundary conditions for tessellations, arc-length continuation method for solving highly nonlinear loading paths, and a perturbation strategy for handling bifurcations). After formulating this truss finite element method, we verify our modified finite element method on well-studied structures. The waterbomb fold pattern is studied, and the interplay of the stretching and folding energies are considered toward design of a bistable structure that exhibits a fold based snap-through instability. This study is followed by analysis of a cylindrical network of waterbombs (axial periodicity) that exhibits a stretch driven instability. Lastly, a tessellated square twist pattern, exhibiting multiple bifurcations off the flat state is presented that employs all aspects of our numerical method.

Nonlinear behaviour of short elastic cylindrical shells under global bending

A recent computational study identified four distinct domains of stability behaviour at different lengths in thin elastic cylindrical shells under global bending. Cylinders of sufficient length suffer from fully-developed cross-sectional ovalisation and fail by local buckling at a moment very close to the Brazier prediction. Progressively shorter cylinders experience less ovalisation owing to the increasingly strong restraint provided by the boundary at the edges. Very short thin cylinders, however, restrain the formation of even a local buckle and fail through a limit point instability at moments and curvatures significantly in excess of the classical elastic prediction. This limit point behaviour is not caused by ovalisation but by the growth of a destabilising fold on the compressed meridian.The nonlinear behaviour of very short cylinders under global bending is investigated in detail herein, covering a wide range of lengths, radius to thickness ratios and boundary conditions with both restrained and unrestrained meridional rotations corresponding to ‘clamped’ and ‘simply-supported’ conditions respectively. Two types of imperfections are investigated, the critical buckling eigenmode and a realistic manufacturing-related ‘weld depression’. A complex insensitivity to these imperfections is revealed owing to a pre-buckling stress state dominated by local compatibility bending, and the cylinder length is confirmed as playing a crucial role in governing this behaviour. The study contributes to the characterisation of multi-segment shells with very short individual cylindrical segments, often found in the aerospace and marine industries as well as in specialised civil engineering applications such as LIPP® silos.

Bifurcations and limit cycles in cytosolic yeast calcium

Calcium homeostasis is a fundamental cellular process in yeast. The regulation of the cytosolic calcium concentration is required for volume preservation and to regulate many vital calcium dependent processes such as mating and response to stress. The homeostatic mechanism is often studied by applying calcium pulses: sharply changing the calcium concentration in the yeast environment and observing the cellular response. To address these experimental investigations, several mathematical models have been proposed to describe this response. In this article we demonstrate that a previously studied model for this response predicts the presence of limit point instabilities and limit cycles in the dynamics of the calcium homeostasis system. We discuss the ways in which such dynamic characteristics can be observed with luminometric techniques. We contrast these predictions with experimentally observed responses and find that the experiments reveal a number of features that are consistent with modeling predictions. In particular, we find that equilibrium cytosolic concentrations have a sharp change in behavior as pulse size changes in the micromolar range. We show that such change is consistent with the presence of limit point instabilities. Additionally, we find that the response of synchronized yeast cells to millimolar range pulses is non-monotonic in its late stages. This response has characteristics similar to those associated with limit cycles.

Nonlinear Equilibrium and Buckling of Fixed Shallow Arches Subjected to an Arbitrary Radial Concentrated Load

This paper is concerned with an analytical study of the nonlinear in-plane equilibrium and buckling of fixed shallow circular arches that are subjected to an arbitrary radial concentrated load. The structural behavior of an arch under an arbitrary radial concentrated load is quite different from that of an arch under a central concentrated load. It is shown that a fixed arch under an arbitrary radial concentrated load can buckle in a limit point instability mode, but cannot buckle in a bifurcation mode, which is different from that of a fixed arch under a central concentrated load that can buckle in a bifurcation mode or in a limit point instability mode. Analytical solutions for the nonlinear equilibrium path and limit point buckling load of shallow circular arches under an arbitrary radial concentrated load are derived. It is found that the load position influences the buckling load significantly and the influence is much related to the modified slenderness of the arch defined in the paper. It is also found that when the modified slenderness of an arch is smaller than a specific value, the arch has no typical buckling behavior. The analytical solution for the relationship of the specific modified slenderness with the load position is also derived. Comparisons with finite element (FE) results show that the analytical solutions can accurately predict the nonlinear equilibrium and buckling load of shallow fixed arches under an arbitrary radial concentrated load.

Field-Controlled Electrical Switch with Liquid Metal.

When immersed in an electrolyte, droplets of Ga‐based liquid metal (LM) alloy can be manipulated in ways not possible with conventional electrocapillarity or electrowetting. This study demonstrates how LM electrochemistry can be exploited to coalesce and separate droplets under moderate voltages of ~1–10 V. This novel approach to droplet interaction can be explained with a theory that accounts for oxidation and reduction as well as fluidic instabilities. Based on simulations and experimental analysis, this study finds that droplet separation is governed by a unique limit‐point instability that arises from gradients in bipolar electrochemical reactions that lead to gradients in interfacial tension. The LM coalescence and separation are used to create a field‐programmable electrical switch. As with conventional relays or flip‐flop latch circuits, the system can transition between bistable (separated or coalesced) states, making it useful for memory storage, logic, and shape‐programmable circuitry using entirely liquids instead of solid‐state materials.

04.01: Buckling of very short elastic cylinders with weld imperfections under uniform bending

ABSTRACTThe length‐dependent domains of behaviour of thin elastic cylindrical shells under uniform bending have recently received significant research attention. Ovalisation is known to affect very long cylinders which undergo significant cross‐sectional flattening before failing by local buckling. This effect is restrained by the end boundary condition in shorter cylinders, which fail instead by local buckling at moments close to the classical analytical prediction. In very short cylinders, however, even this local buckling is restrained by the end boundary, and failure occurs instead through the development of a destabilising meridional fold on the compressed side. Although this is a limit point instability under bending, ovalisation does not play any role at all. This ‘very short’ length domain has only recently been explored for the first time with the aid of finite element modelling.A brief overview of the nonlinear buckling behaviour of very short elastic cylinders under uniform bending is presented in this paper. Two types of edge rotational restraint are used to illustrate the influence of a varying support condition on the stability in this short length range. It is shown that short cylinders under bending do not suffer at all from local short‐wave buckling. Additionally, when the meridional dimension of such cylinders becomes particularly short the resulting numerical models may predict indefinite stiffening without a limit point, even when the shell is modelled using more complete 3D solid continuum finite elements. Idealised weld depressions, which are realistic representations of a systemic manufacturing defect, are used to demonstrate only a very mild sensitivity to geometric imperfections at such short lengths owing to a pre‐buckling stress state dominated by local compatibility bending. The topic should be of interest to researchers studying shell problems dominated by local bending with computational tools and designers of multi‐segment shells with very close segment spacing.

Buckling of very short elastic cylinders with weld imperfections under uniform bending

The length-dependent behaviour domains of thin elastic cylindrical shells under uniform bending have recently received significant research attention. Ovalization is known to affect very long cylinders that undergo significant cross-sectional flattening before failing by local buckling. This effect is restrained by the end boundary conditions in shorter cylinders, which instead fail by local buckling at moments close to the classical analytical prediction. In very short cylinders, however, even this local buckling is restrained by the end boundary, and failure occurs instead through the development of a destabilizing meridional fold on the compressed side. Although this is a limit point instability under bending, ovalization does not play any role at all. This ‘very short’ length domain has only recently been explored for the first time with the aid of finite element modelling. A brief overview of the non-linear buckling behaviour of very short elastic cylinders under uniform bending is presented in this paper. Two types of edge rotational restraint are used to illustrate the influence of a varying support condition on the stability in this short length range. It is shown that short cylinders under bending do not suffer at all from local short-wave buckling. Additionally, when the meridional dimension of such cylinders becomes particularly short, the resulting numerical models may predict indefinite stiffening without a limit point, even when the shell is modelled using more complete 3D solid continuum finite elements. Idealized weld depressions, which are realistic representations of a systemic manufacturing defect, are used to demonstrate only a very mild sensitivity to geometric imperfections at such short lengths owing to a pre-buckling stress state dominated by local compatibility bending. The topic should be of interest to researchers studying shell problems dominated by local bending with computational tools and designers of multi-segment shells with very close segment spacing.

Limit points in the free inflation of a magnetoelastic toroidal membrane

One common phenomenon native to inflation of membranes is the elastic limit-point instability—a bifurcation point at which the membrane begins to deform enormously at the slightest increase of pressure. In the case of magnetoelastic materials, there is another possible phenomenon which we call magnetic limit-point instability, a state referring to the non-existence of an equilibrium state – either stable or unstable. In this work, we are concerned with such instabilities in an incompressible isotropic magnetoelastic toroidal membrane with an initial circular cross-section. A non-uniform magnetic field is generated using a circular current carrying loop placed inside the membrane in addition to inflation by a uniform hydrostatic pressure. An energy formulation based on magnetization is used to model the magneto-mechanical coupling along with a Mooney–Rivlin constitutive model for the elastic strain energy density. Computations show that the magnetic field strongly influences the location of elastic limit points and in some cases can cause them to vanish. Multiple equilibrium states are obtained as solutions of the governing equations and a criterion based on second variation is employed to determine their stability. Existence and dependence of magnetic limit point on the magnetic field is demonstrated. While the quantitative results obtained here are specific to the toroidal geometry, the deformation behaviour can be generalized to any magnetoelastic membrane.

Bifurcation instabilities in finite bending of circular cylindrical shells

This work re-visits the finite pure bending problem for circular cylindrical shells within the elastic range. The interest here is primarily directed towards the bifurcation instabilities of such configurations when the progressive flattening of the cylindrical cross-section is explicitly taken into account (the so-called Brazier effect). By coupling Reissner’s axisymmetric solution to the buckling equations for a quasi-shallow toroidal shell we formulate a novel boundary-value problem able to capture such bifurcations. Numerical simulations of this problem confirm that buckling occurs before the usual limit-point instability is reached, while singular perturbation methods allow us to obtain simple asymptotic approximations for the critical curvature and bending moment associated with the bifurcations.

Increasing Practical Safety of Von Mises Truss via Control of Dynamic Escape

This paper investigates the nonlinear dynamics and stability of the shallow von Mises truss, which is a prototype for buckling analysis of several planar and spatial truss systems and shallow lattice shell structures, including the geodesic dome, and which has a theoretical and practical interest in many engineering fields. These structural systems are liable to limit point instability and may fail at load levels well below the theoretical limit point load due to complex nonlinear phenomena that reduce the safety and the dynamic integrity of the structure; this often occurs as a consequence of the erosion of the basins of attraction of the safe pre-buckling solutions. So, it is vital to increase the safety of the structure in a dynamic environment. In this paper a method for controlling the global nonlinear dynamics of mechanical systems is applied to the shallow von Mises truss. The method consists of the (optimal) elimination of homoclinic intersection by properly adding superharmonic terms to a given harmonic excitation. By means of the solution of an appropriate optimization problem, it is possible to select the amplitudes and the phases of the added superharmonics in such a way that the manifolds distance is as large as possible. This methodology is here applied to increase the integrity of the basins of attraction of the system and consequently its practical safety.

Limit point instability in pressurization of anisotropic finitely extensible hyperelastic thin-walled tube

Mechanical responses of materials undergoing large elastic deformations can exhibit a loss of stability in several ways. Such a situation can occur when a thin-walled cylinder is inflated by an internal pressure. The loss of stability is manifested by a non-monotonic relationship between the inflating pressure and internal volume of the tube. This is often called limit point instability. The results, known from the literature, show that isotropic hyperelastic materials with limiting chain extensibility property always exhibit a stable response if the extensibility parameter of the Gent model satisfies Jm<18.2. Our study investigates the same phenomenon but for tubes with anisotropic form of the Gent model (finite extensibility of fibers). Anisotropy, used in our study, increases the number of material parameters the consequence of which is to increase degree of freedom of the problem. It will be shown that, in stark contrast to isotropic material, the unstable response is predicted not only for large values of Jm but also for Jm≈1 and smaller, and that the existence of limit point instability significantly depends on the orientation of preferred directions and on the ratio of linear parameters in the strain energy density function (this ratio can be interpreted as the ratio of weights by which fibers and matrix contribute to the strain energy density). Especially tubes reinforced with fibers oriented closely to the longitudinal direction are susceptible to a loss of monotony during pressurization.