Equilibrium Branches Research Articles

Static behavior and bifurcation of a monosymmetric open cross-section thin-walled beam: Numerical and experimental analysis

The aim of the paper is the numerical and experimental validation of a previously developed nonlinear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section. Nonlinear in-plane and out-of-plane warping and torsional elongation effects are included in the model. To better understand the role of these new contributions a beam with a section with one symmetry axis, undergoing moderately large flexural curvatures and large torsional curvature is taken into account. To obtain a section of a cantilever beam for which the torsional curvature is expected to prevail with respect to the flexural ones, a preliminary study is performed. The attention is focused on the response to static forces and on the stability of the equilibrium branches. Analytical results are compared with results of two different nonlinear finite element models and mainly with experimental results to confirm the validity of the analytical model. Interesting results are obtained for the critical values of the flexural–torsional instability loads.

SPATIALLY CHAOTIC BIFURCATIONS OF AN ELASTIC WEB OF LINKS

Spatially chaotic bifurcations of an elastic web of links are investigated. We numerically construct the global bifurcation diagrams uniquely describing the buckled states, and show that the exponential growth of the number of equilibrium branches with the size of the web indicates spatial chaos. The types of bifurcations from the trivial equilibrium branch are also determined, and we show that cusp catastrophes of any order can appear. This observation relates the buckling of the elastic web of links to the buckling of rods with finite shear and infinite bending and normal stiffness.

The pertinence of jet emitting discs in microquasars. Theory and comparison to observations

AbstractJet Emitting Disc (JED) has dynamical properties quite different from both the standard and advection dominated discs. It also exhibits three different thermal equilibrium branches at a given radius: two stable (cold and hot) and one intermediate unstable. The hot solution has all the characteristics of the so-called “hot corona” generally invoked in XrB sytems in the Low/Hard states. We detail the energetics and radiative expectations of our model and show their good agreement with those observed in Cygnus X-1 in terms of jet power, jet velocity and spectral emission.

TAILORING THE BIFURCATION DIAGRAM OF NONLINEAR DYNAMICAL SYSTEMS: AN OPTIMIZATION BASED APPROACH

Bifurcation tailoring is a method developed to design control laws modifying the bifurcation diagram of a nonlinear dynamical system to a desired one. In its original formulation, this method does not account for the possible presence of constraints on state and/or manipulated inputs. In this paper, a novel formulation of the bifurcation tailoring method overcoming this limitation is presented. In accordance with the proposed approach, a feedforward control law generating an optimal bifurcation diagram is computed by constrained minimization of an objective functional. Then, a feedback control system enforcing stability of the computed equilibrium branch is designed. In this context, bifurcation analysis is exploited to select feedback controller parameters ensuring desired output behavior and, at the same time, preventing the occurrence of multistability. The method is numerically validated on the problem of tailoring the bifurcation diagram of an exothermic chemical reactor.

Stability Loss in Quasilinear DAEs by Divergence of a Pencil Eigenvalue

The divergence through infinity of certain eigenvalues of a linearized differential-algebraic equation (DAE) may result in a stability change along an equilibrium branch. This behavior cannot be exhibited by explicit ODEs, and its analysis in index one contexts has been so far unduly restricted to semiexplicit systems. By means of a geometric reduction framework we extend the characterization of this phenomenon to quasilinear DAEs, which comprise semiexplicit problems as a particular case. Our approach clarifies the nature of the singularities which are responsible for the stability change and also accommodates rank deficiencies in the leading system matrix. We show how to address this problem in a matrix pencil setting, an issue which leads to certain results of independent interest involving the geometric index of a quasilinear DAE and the Kronecker index of its linearization. The results are shown to be of interest in electrical circuit theory, since the differential-algebraic network models actually used in circuit simulation are not in semiexplicit but in quasilinear form.

On the Relationship between Equilibrium Bifurcations and Ideal MHD Instabilities for Line-Tied Coronal Loops

For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated, imposing line-tied boundary conditions. Using a well-studied example based on the Gold – Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad – Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point, because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials, the first bifurcation point of the Grad – Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.

Theory and application of plasma diode equilibria with beam electrons and a fixed particle background of negative charge

The effect of a uniformly distributed, immobile, charged particle background on the stationary states of a plasma diode with beam electrons is studied. Whereas a positively charged particle background results in an increase of the space-charge limit of the electron current across the diode and brings in new branches of equilibria, a negatively charged background decreases the current up to zero current level due to a complete reflection of beam electrons within the diode region. This effect can be used for the design of fast switches.

Influence of residual stresses and geometric imperfections on the elastoplastic collapse of cylindrical tubes under external pressure

The buckling problem of a circular cylindrical shell has long been widely investigated due to its great importance in the design of aerospace and marine structures. Geometric imperfections and residual stresses are inevitable in practice and have been so far frequently considered in analytical and numerical predictions. But little attention has been paid until now on the combined influence of such initial defects on the critical and often unstable response of such elastoplastic structures. In this paper, a shell finite element is designed within the total Lagrangian formulation framework to deal with the elastoplastic buckling and post-buckling of thin cylindrical tubes under external pressure and axial compression. A specific experimental process will be introduced in order to measure residual stresses in the shell very accurately, so as to include them in the numerical calculations. The present formulation will enable us to describe the complete non-linear solutions, namely the critical pressures (bifurcation and limit (collapse) loads), the bifurcation modes and the bifurcated equilibrium branches up to advanced post-critical states. Comparisons will be made between numerical results and the experimental critical value and deformation patterns of a new generation profiler. Furthermore, the combined effects of geometric imperfections, residual stresses and plasticity will be analyzed.

Stability and branching of the relative equilibria of a three-link pendulum in a rapidly rotating frame of reference

The relative equilibria of a plane three-link pendulum with a suspension rotating about a vertical axis in a gravitational force field are investigated. The pendulum is modelled as a system of three point masses, joined in tandem by massless non-elastic rods using cylindrical hinges. All the trivial equilibrium positions, their stability and branching are investigated. Attention is mainly paid to the non-trivial equilibrium positions, that is, to those equilibrium configurations of the pendulum for which not all the links are stretched out along the vertical axis. Questions of the existence, stability and the branching of these non-trivial equilibrium positions are investigated at fairly high angular velocities of rotation. The plane of the geometric parameters of the pendulum is subdivided into domains with a different number of non-trivial relative equilibria. Conditions are presented for each non-trivial equilibrium position which is found, and these are imposed on the system parameters for which an equilibrium position exists, their stability is investigated and a geometric description of the configuration of the pendulum is given.

Bistability and oscillations in chemical reaction networks

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461-3470, 1987; Clarke and Jiang in J Chem Phys 99:4464-4476, 1993; Gatermann et al. in J Symb Comput 40:1361-1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks.

Homogeneous Equilibrium Configurations of a Hyperelastic Compressible Cube under Equitriaxial Dead-Load Tractions

Non-uniqueness, bifurcation and stability of homogeneous solutions to the equilibrium problem of a hyperelastic cube subject to equitriaxial dead-load tractions are investigated. Besides the basic and theoretical questions raised by the analysis, the study is motivated by the somewhat surprising feature of this nonlinear problem for which the symmetric load may give rise to asymmetric stable deformations. In reality, the equilibrium problem, formulated for general homogeneous compressible isotropic materials with polyconvex energy function, may exhibit primary and secondary bifurcations. A primary bifurcation occurs when there exist paths of equilibrium states that bifurcate from the primary path of three equal principal stretches. These bifurcation branches have two coinciding stretches and along them, through secondary bifurcations, other completely asymmetric bifurcation branches, which are characterized by all three stretches different, may risen. In this case, the cube transforms into an oblique parallelepiped. With increasing loads, they are also possible discontinuous paths of equilibria which evince prompt jumps in the deformation process. Of course, the set of asymmetric solutions admitted by the equilibrium problem depends on the specific form of the stored energy function adopted. In this paper, expressions governing the global development of asymmetric equilibrium branches are derived. In particular, conditions to have bifurcation points are individualized. For compressible neo-Hookean and Mooney-Rivlin materials a wide parametric analysis is carried out showing by means of graphs the most interesting branches. Finally, using the energy criterion, a detailed study is performed to assess the stability of the computed solutions.

Fixed contact line helical interfaces in zero gravity

Fluid interfaces supported in microgravity by a helical structure are shown to have a more robust stability than more common structures such as liquid bridges. In particular, helical interfaces can take the form of infinite right circular cylinders over a broad range of configurations. In the case of a single fixed contact line support, the infinite cylinder is stable for all cases in which the pitch to diameter ratio is less than π∕3 (more tightly coiled interfaces). When there are two or more equally spaced fixed contact line supports, the infinite cylinder is stable for all configurations. Furthermore, in the two support case (the double helix), stability persists for all volumes from the cylinder to zero volume, when the pitch to diameter ratio is greater than 2.082 (more loosely coiled interfaces). The equivalent to the axisymmetric Young-Laplace equation is derived for helical interfaces. Interfacial stability is determined from equilibrium branch structure following the application of Maddocks’ method by Lowry and Steen [Proc. R. Soc. London, Ser. A 449, 411 (1995)]. Perturbations to finite wavelength disturbances are considered for the case of a single helical support. Overall stability envelopes are presented for single and multiple support cases. Limited experimental results verify the infinite length stability limit for the single helical support case.

Asymmetric equilibrium configurations of hyperelastic cylindrical bodies under symmetric dead loads

Homogeneous deformations provided by the nonlinear equilibrium problem of symmetrically loaded isotropic hyperelastic cylindrical bodies are investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions, besides the expected symmetric solutions. For general compressible materials, the mathematical condition allowing the assessment of these asymmetric solutions, which describe the global path of equilibrium branches, is given. Explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a Mooney-Rivlin and a neo-Hookean material. A broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. The influence of the constitutive parameters is discussed, and, using the energy criterion, a number of considerations are carried out concerning the stability of the equilibrium solutions.

Post-bifurcation equilibria in the plane-strain test of a hyperelastic rectangular block

Of interest here is the bifurcated equilibrium solution of a homogeneous, hyperelastic, rectangular block under finite, plane-strain tension or compression. A general asymptotic analysis of the bifurcated equilibrium path about the principal solution’s lowest critical load is presented using Lagrangian kinematics. The analysis is valid for any compressible hyperelastic material with axes of orthotropy aligned with the block’s axes of symmetry in the reference (stress-free) configuration. The general theory is subsequently applied to blocks of different constitutive laws. Results are presented in the form of bifurcated equilibrium branch’s curvature at the critical load as function of the block’s aspect ratio, since the sign of this curvature determines the branch’s stability. For small aspect ratios there is agreement with existing structural models, while for relatively higher aspect ratios some rather counter-intuitive stability results appear, which strongly depend on the constitutive law.

Constitutive branching analysis of cylindrical bodies under in-plane equibiaxial dead-load tractions

Finite homogeneous deformations of hyperelastic cylindrical bodies subjected to in-plane equibiaxial dead-load tractions are analyzed. Four basic equilibrium problems are formulated considering incompressible and compressible isotropic bodies under plane stress and plane deformation condition. Depending on the form of the stored energy function, these plane problems, in addition to the obvious symmetric solutions, may admit asymmetric solutions. In other words, the body may assume an equilibrium configuration characterized by two unequal in-plane principal stretches corresponding to equal external forces. In this paper, a mathematical condition, in terms of the principal invariants, governing the global development of the asymmetric deformation branches is obtained and examined in detail with regard to different choices of the stored energy function. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. With reference to neo-Hookean, Mooney–Rivlin and Ogden–Ball materials, a broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. Finally, using the energy criterion, a number of considerations are put forward about the stability of the computed solutions.

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray–Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k−1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as e→ 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

The effects of energy sites on adsorption of Lennard–Jones fluids and phase transition in carbon slit pore of finite length a computer simulation study

A Monte Carlo simulation method is used to study the effects of adsorption strength and topology of sites on adsorption of simple Lennard–Jones fluids in a carbon slit pore of finite length. Argon is used as a model adsorbate, while the adsorbent is modeled as a finite carbon slit pore whose two walls composed of three graphene layers with carbon atoms arranged in a hexagonal pattern. Impurities having well depth of interaction greater than that of carbon atom are assumed to be grafted onto the surface. Different topologies of the impurities; corner, centre, shell and random topologies are studied. Adsorption isotherms of argon at 87.3 K are obtained for pore having widths of 1, 1.5 and 3 nm using a Grand Canonical Monte Carlo simulation (GCMC). These results are compared with isotherms obtained for infinite pores. It is shown that the surface heterogeneity affects significantly the overall adsorption isotherm, particularly the phase transition. Basically it shifts the onset of adsorption to lower pressure and the adsorption isotherms for these four impurity models are generally greater than that for finite pore. The positions of impurities on solid surface also affect the shape of the adsorption isotherm and the phase transition. We have found that the impurities allocated at the centre of pore walls provide the greatest isotherm at low pressures. However when the pressure increases the impurities allocated along the edges of the graphene layers show the most significant effect on the adsorption isotherm. We have investigated the effect of surface heterogeneity on adsorption hysteresis loops of three models of impurity topology, it shows that the adsorption branches of these isotherms are different, while the desorption branches are quite close to each other. This suggests that the desorption branch is either the thermodynamic equilibrium branch or closer to it than the adsorption branch.

Regularization of the Alfvén singularity by the Hall effect

An ideal magnetohydrodynamics (MHD) equilibrium is described by a system of rather complicated singular differential equations when a flow is included. The so-called Alfvén singularity occurs at the place where the Doppler-shifted Alfvén velocity vanishes. It is due to the vanishing of the highest-order derivative in the differential equation. The Hall effect, working as a singular perturbation to the ideal MHD system, yields a new branch of regular solutions that can smoothly connect two regions separated by the Alfvén singularity. The thickness of the transition layer is of the order of the ion skin depth, the intrinsic length scale brought about by the singular perturbation. The regularization mechanism of the nonlinear Hall effect is not as simple as that of the diffusion effect producing an entropy (viscosity) solution in a viscous fluid. The Hall effect removes the restriction binding the magnetic and flow characteristics, and creates the new branch of regularized solutions. One-dimensional analysis loses sight of this new branch of equilibria.

Symmetry breaking for toral actions in simple mechanical systems

For simple mechanical systems, bifurcating branches of relative equilibria with breaking symmetry from a given set of relative equilibria with toral symmetry are found. Lyapunov stability conditions along these branches are given.

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.