Axisymmetric Stability Research Articles

Enhanced Model for Axisymmetric Stability Analysis of Propagating Circular Defect-Driven Coating Delamination Under Combined Compressive and Diffusion-Induced Stresses.

Enhanced Model for Axisymmetric Stability Analysis of Propagating Circular Defect-Driven Coating Delamination Under Combined Compressive and Diffusion-Induced Stresses.

Axisymmetric vibration and stability of dielectric-elastic tubular bilayer system

Modern transducers and actuators may have functional layers with multi-field coupling and some elastic layers. This paper considers a tubular bilayer system consisting of a thin dielectric tube coated with a thick elastic layer. We study the nonlinear electromechanical response and the linear axisymmetric vibration of the system subject to different applied voltages and inner/outer pressures within the framework of the general nonlinear theory of electro-elasticity, the related linear incremental theory, and by considering the continuity conditions at the interface. We investigate instability behaviour using the same basic formulae. The state-space method provides efficient and accurate free vibration analysis, considering the dynamic response at the lowest frequencies, so we can neglect the viscous and damping effects, which is well suited to this problem. New results indicate that the bilayer system improves its frequency capability and stability compared to the monolayer dielectric tube. The thick outer elastic layer stiffens the bilayer system against axisymmetric bifurcation, bulging and necking instabilities. It also performs better in front of axisymmetric instability, increasing the system’s capability to receive or produce higher voltages, especially for long waves.This work thoroughly explains bilayer functional systems’ behaviour when exposed to extreme environments such as high voltage or pressure.

Finite elements limit analysis formulation using the power type yield criterion for plane strain and axisymmetric stability problems

The current research employs a power type yield criterion which accounts for the dependency of the shear strength parameters, namely, cohesion and internal friction, on stress level which becomes quite effective in obtaining very accurately the solutions of different stability problems in geomechanics involving soil and rock media. The present manuscript introduces plane strain and axisymmetric formulations which deal with the power type yield criterion and are based on the lower and upper bound finite elements limit analysis in conjunction with the power cone programming (PCP). To demonstrate the efficacy of the proposed formulations, the problems of the determination of the bearing capacity of strip and circular foundations in a cohesive-frictional soil medium have been addressed. In the case of a strip footing, the earthquake inertial forces in a horizontal direction have also been incorporated. The adaptive mesh pattern has been employed to improve the accuracy of the solution at the expense of minimal computational effort. The results have been thoroughly validated by making the comparison of the obtained solutions with that reported in literature. The proposed formulations are expected to be quite useful in solving different stability problems in geomechanics.

Nonlinear thermo-mechanical axisymmetric stability of FG-GPLRC spherical shells and circular plates resting on nonlinear elastic medium

ABSTRACT An analytical approach for nonlinear thermo-mechanical buckling of functionally graded graphene platelet reinforced composite (FG-GPLRC) circular plates and shallow spherical shells resting on nonlinear elastic medium using the higher-order shear deformation theory (HSDT) and the nonlinearities of von Kármán is established in the present work. Assuming that the structures are axisymmetrically displaced for the central axis and external pressure and thermal loads are applied. The nonlinear elastic medium is used to model the behaviour of hardening or softening medium depending on the positive and negative values of the nonlinear parameter, respectively. By applying the Ritz energy method, the relation expressions of load-deflection are archived to investigate the postbuckling response and critical buckling load of the structures. Special effects on the nonlinear thermo-mechanical behaviour of plates and shells with different medium stiffnesses, different material parameters, and different geometrical dimensions are explored and discussed in numerical results.

Axisymmetric buckling of annular plate with internal source of heat

In this article, the axisymmetric stability of an elastic annular plate with distributed internal heat source under uniform constant compressive radial loading at the outer edge is investigated. The governing and compatibility equations of the annular plate are taken from modified Karman’s equations, which are solved by the shooting method. The effect of the density of distributed internal heat source on critical load is investigated and discussed in detail. Numerical results of the plate without an internal heat source are in good agreement with the results from the literature, validating the effectiveness and advantages of the current method. The critical buckling load increases with a rise of the density of distributed internal heat sources. Moreover, an increase of wave number increases the critical buckling loads of the plate. The results of this research can be considered as benchmark solutions for buckling of the annular plates with an internal heat source.

Axisymmetric instability of elastic tubes conveying power-law fluids

Instability of collapsible tubes is studied theoretically and experimentally in many papers in the context of biological applications. Up to the present day, only Newtonian fluid flows in elastic tubes have been studied. However, there are circumstances when blood, bile and other biological fluids show essentially non-Newtonian behaviour. In this paper, we re-investigate theoretically axisymmetric stability of elastic tubes conveying power-law fluids. It is shown that for the power-law index$n=1$, i.e. for the Newtonian case, axisymmetric disturbances in infinite-length tubes are damped, which is in accordance with experimental and theoretical observations, where the oscillations always involve non-axisymmetric motion of the tube walls. However, for$n<0.611$, the axisymmetric disturbances can be growing, which predicts a new type of instability of elastic tubes conveying pseudoplastic (shear-thinning) fluids. For$n<1/3$, local instability of axisymmetric perturbations becomes absolute in infinite tubes, while finite-length tubes become globally unstable. The effects of the axial tension, elastic tube length and, if present, lengths of inlet and outlet rigid tubes on the stability of finite-length tubes are analysed.

Lower bound limit analysis for an axisymmetric problem in rock mass using power cone programming

AbstractBy using the lower bound finite elements limit analysis (LB‐FELA) and the power cone programming (PCP), a methodology has been introduced to solve an axisymmetric stability problem with the usage of the generalized Hoek and Brown (GHB) yield criterion‐ applicable for rock mass. The formulation involves the application of the GHB yield criterion in its native form without any smoothing and it does not require any assumption associated with the circumferential stress (σθ). The efficacy of the proposed formulation in a rock mass has been demonstrated by determining (i) the bearing capacity of a circular footing, (ii) the stability numbers for an unsupported vertical cylindrical excavation, and (iii) the vertical uplift capacity of a circular horizontal anchor. The obtained solutions have been compared with that reported in literature. Failure patterns for each problem have also been explored. For the anchor problem, for no solution is reported with the usage of the GHB criterion, the influences of different material yield parameters (GSI, mi, and σci), unit weight of rock mass (γ), and the embedment ratio (H/B) of the anchor on the uplift capacity have been comprehensively examined.

Impact of negative triangularity plasma shaping on the n = 0 resistive wall mode in a tokamak

The axisymmetric (n = 0) resistive wall mode instability is numerically investigated for the negative triangularity plasma shape, which has shown several benefits in terms of improved confinement time and fusion engineering. To evaluate the shape effects, we modified a MHD instability code AVSTAB (Axisymmetric Vertical STABility), which calculates the marginally controllable elongation in a simplified feedback capability parameter. The plasma characteristics (poloidal beta and internal inductance) as well as the geometric effects (wall shape and plasma location) are important to determine the instability. In contrast to positive triangularity, the higher poloidal beta provides more instability drive for the negative triangularity because of the higher Shafranov shift and the higher elongation of the inner flux surface of the MHD equilibrium. Non-conformal wall shapes to the plasmas (positive triangularity wall and negative triangularity plasma) are found to be rather helpful to stabilize the n = 0 mode, unless the plasma is too close to the walls at the nulls for the opposite triangularity.

Temporal instability of a viscoelastic liquid thread surrounded by another viscoelastic fluid in presence of insoluble surfactant and inertia

In recent years, there has been controversy in the literature regarding the role of viscoelasticity in determining the onset of stability for a viscoelastic thread surrounded by another viscoelastic fluid. Whereas a number of authors have found that viscoelasticity serves to destabilize the system, Patrascu and Balan (2018) obtained very different results. They considered a very simple canonical system subject to axisymmetric perturbations that neglected inertia and did not include the effect of surfactant. Their analysis provided two genuinely startling results: firstly, when both fluids have similar elastic properties, the dispersion relation reduces to the Newtonian limit; and secondly, the presence of elasticity in the external medium decreases the value of the unstable growth rate. In this paper, we investigate the axisymmetric temporal stability of an infinitely long viscoelastic thread surrounded by another immiscible viscoelastic fluid in presence of small amount of insoluble surfactant. Our analysis also includes the effects of inertia. The dispersion relations for cases with and without inertia are carefully derived. For the case of zero inertia and no surfactant, our results directly contradict those of Patrascu and Balan. We show that when both fluids have similar elastic properties that the dispersion relation does not reduce to the Newtonian case. We also show that, when one of the fluids is viscoelastic or both fluids are viscoelastic, the system is more unstable than the case in which both fluids are Newtonian. Moreover, our results also reveal that the system with inertia is less unstable than the system without inertia, and when the inner–outer density ratio becomes larger, the system becomes less unstable. We also carry out a parametric study of the effects of surfactant. Our results show that adding surfactant can decrease the growth rate of the instability and increase the wavenumber of the most unstable mode. We also show that the reduction of the unstable growth rate induced by the surfactant is more significant for larger internal–external total viscosity ratios. Finally, an energy budget analysis is performed to investigate the contribution of each physical factor towards the instability. This reveals the roles that the various physical mechanisms play in determining the temporal growth rate. The energy analysis further confirms that the elasticity in either inner or outer fluid increases the growth rate. • Address the controversy regarding the stability of viscoelastic threads. • Dispersion relation for fluids with same elasticity differs from Newtonian case. • Elasticity in the external medium does not decreases the growth rate. • We examine the role played by inertia and surfactant in determining the instability.

On the axisymmetric stability of tokamaks with ferromagnetic walls

Reduced activation ferritic steels are an attractive option for use in large structural components surrounding tokamak plasmas in future fusion power plants, but their ferromagnetic response to the confining magnetic fields must be properly understood. Simultaneously, the advantages of operating at high plasma elongation push tokamak designs toward scenarios that are more vulnerable to vertical displacement events. Passive conducting structures in present tokamaks slow these instabilities such that they may be feedback controlled, but the efficacy of this process is likely to be eroded by ferromagnetic effects. We approach two related analytical models—in cylindrical and spherical geometries—which qualitatively and quantitatively assess the impact of a ferritic steel wall on the vertical instability growth rate for a plasma of certain elongation. Distinct limits for magnetically thick and thin walls give key physical insight, but the dependence on magnetic permeability and wall geometry is, in general, quite complex. Equilibrium considerations, particularly with respect to radial force balance, are also encountered.

Lower bound limit analysis using nonlinear optimization for solving axisymmetric problems using Hoek-Brown yield criterion

ABSTRACT This study presents a computational methodology for solving axisymmetric stability problems in rock mechanics by using the finite element lower bound limit analysis. The generalized Hoek and Brown (GHB) yield criterion, which is often recommended to define failure in the intact and jointed rock mass, has been employed. No assumption(s) are required to make either on the value of circumferential stress (σ θ) or the exponent (α) in the GHB yield expression. It is understood that the assumption of choosing α = 0.5 provides an unsafe solution for the values of GSI ≪ 30. For the purpose of illustration, the bearing capacity factor N σ has been evaluated for a circular footing. The results obtained from the analysis were compared with that reported in the literature and it is concluded that the present formulation by incorporating the true variation of α with GSI significantly improves the existing solution(s) for GSI < 30.

Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.

Gyrokinetic simulations of m = 0 mode in sheared flow Z-pinch plasmas

Axisymmetric stability properties of sheared flow Z-pinch plasmas are studied by making use of the gyrokinetic approximation in the long-wavelength limit. Numerical simulations are carried out with the high-order finite-volume code COntinuum Gyrokinetic Edge New Technology (COGENT) and are analyzed for the parameters characteristic of the FuZE experiment. Reduction of the linear growth rate with increasing shear is observed, and the results are elucidated by making use of a local dispersion relation analysis. In addition, COGENT simulations are compared with fully kinetic particle-in-cell simulations, and with an ideal magnetohydrodynamics (MHD) model. Good agreement between the gyrokinetic and fully kinetic results for the linear stability is found, with the gyrokinetic model requiring much less computational time due to its ability to step over particle gyroperiod. The ideal MHD model is found to be consistent with the kinetic models in the long-wavelength part of the spectra (kρi), while failing to adequately predict short-scale (kρi) stability. Here, k is the axial wavelength vector and ρi is the ion gyroradius.

An Investigation on the Effect of Intermediate Principal Stress on the Solution of Bearing Capacity Problems by the Method of Stress Characteristics

This paper aims to investigate the effect of assuming different values of intermediate principal stress σ2, in the solutions obtained for stability problems in soil engineering. The bearing capacity of strip and circular footings has been selected for this purpose which represent typical plane strain and axisymmetric stability problems. Investigations has been made using the method of stress characteristics as usual; but with yield criteria other than Tresca and Mohr–Coulomb that allow variation in σ2. Variation in σ2 has been considered in the form of usual parameter b which is the ratio of differences of principal stresses. At the outset; the parameter b is set to match these criteria with Tresca or Mohr–Coulomb. This is based on usual assumptions of associativity in plane strain, and Haar–von Karman flow-regime in axial symmetry. The value of b has then been changed from these values to see how the solution is affected by variation in σ2. Results indicate that the usual solutions are not generally on the safe side; and if the material behavior is really different from our assumptions, this should be considered in calculations.

Осесимметричная устойчивость круглой кольцевой плиты, подверженной механическим и термическим нагрузкам

Осесимметричная устойчивость круглой кольцевой плиты, подверженной механическим и термическим нагрузкам

Upper-Bound Finite-Element Limit Analysis of Axisymmetric Problems for Mohr-Coulomb Materials Using Semidefinite Programming

An upper-bound formulation for performing finite-element limit analysis by using semidefinite programming (SDP) for an axisymmetric stability problem involving the Mohr-Coulomb yield criterion has been presented. The SDP technique has an advantage in that it can deal with the yield criterion directly in its native form in terms of principal stresses and strains without any smoothing of the parent yield surface. The associated flow rule and plastic power dissipation are expressed entirely in terms of principal plastic strain rates. Nodal velocities and element plastic strain rates were framed as basic governing variables without involving stresses. The solution was obtained by using the SDP solver MOSEK in MATLAB. The problem of finding the bearing capacity of a circular footing was dealt with by choosing a planar domain that was discretized with (1)three-noded constant strain finite elements and (2)six-noded linear strain finite elements. The results were obtained with and without the provision of the velocity discontinuities, and were compared with those reported in literature. It was learned that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and using velocity discontinuities along all the elements' interfaces.

Note on Electrogravitational Stability of a Double Fluids Interface

This study deals with the axisymmetric stability of the interface between two incompressible Selfgravitating non-conducting fluids in the presence of an electric field. The pressure in the unperturbed state is not constant because the Selfgravitating force is a long-range force. The dispersion relation is derived and discussed. Some previous reported works may be obtained as limiting cases from the presented work with appropriate choices. The study under consideration has strong correlation with instabilities of sun spots and other physical phenomena.

A variational principle for the axisymmetric stability of rotating relativistic stars

It is well known that all rotating perfect fluid stars in general relativity are unstable to certain non-axisymmetric perturbations via the Chandrasekhar–Friedman–Schutz (CFS) instability. However, the mechanism of the CFS instability requires, in an essential way, the loss of angular momentum by gravitational radiation and, in many instances, it acts on too long a timescale to be physically/astrophysically relevant. It is therefore of interest to examine the stability of rotating, relativistic stars to axisymmetric perturbations, where the CFS instability does not occur. In this paper, we provide a Rayleigh–Ritz-type variational principle for testing the stability of perfect fluid stars to axisymmetric perturbations, which generalizes to axisymmetric perturbations of rotating stars a variational principle given by Chandrasekhar for spherical perturbations of static, spherical stars. Our variational principle provides a lower bound to the rate of exponential growth in the case of instability. The derivation closely parallels the derivation of a recently obtained variational principle for analyzing the axisymmetric stability of black holes.

Stabilization of the vertical instability by non-axisymmetric coils

In a published Physical Review Letter (Reiman 2007 Phys. Rev. Lett. 99 135007), it was shown that axisymmetric (or vertical) stability can be improved by placing a set of parallelogram coils above and below the plasma oriented at an angle to the constant toroidal planes. The physics of this stabilization can be understood as providing an effective additional positive stability index. The original work was based on a simplified model of a straight tokamak and is not straightforwardly applicable to a finite aspect ratio, strongly shaped plasma such as in DIII-D. Numerical calculations were performed in a real DIII-D -like configuration to provide a proof of principal that 3-D fields can, in fact raise the elongation limits as predicted. A four field period trapezioid-shaped coil set was developed in toroidal geometry and 3D equilibria were computed using trapezium coil currents of , , and . The ideal magnetohydrodynamics growth rates were computed as a function of the conformal wall position for the symmetry-preserving family. The results show an insignificant relative improvement in the stabilizing wall location for the two lower coil current cases, of the order of 10−3 and less. In contrast, the marginal wall position is increased by 7% as the coil current is increased to , confirming the main prediction from the original study in a real geometry case. In DIII-D the shift in marginal wall position of 7% would correspond to being able to move the existing wall outward by 5 to 10 cm. While the predicted effect on the axisymmetric stability is real, it appears to require higher coil currents than could be provided in an upgrade to existing facilities. Additional optimization over the pitch of the coils, the number of field periods and the coil positions, as well as plasma parameters, such as the internal inductivity , , and would mitigate this but seem unlikely to change the conclusion.

ON THE GRAVITATIONAL STABILITY OF GRAVITO-TURBULENT ACCRETION DISKS

Low mass, self-gravitating accretion disks admit quasi-steady,`gravito-turbulent' states in which cooling balances turbulent viscous heating. However, numerical simulations show that gravito-turbulence cannot be sustained beyond dynamical timescales when the cooling rate or corresponding turbulent viscosity is too large. The result is disk fragmentation. We motivate and quantify an interpretation of disk fragmentation as the inability to maintain gravito-turbulence due to formal secondary instabilities driven by: 1) cooling, which reduces pressure support; and/or 2) viscosity, which reduces rotational support. We analyze the axisymmetric gravitational stability of viscous, non-adiabatic accretion disks with internal heating, external irradiation, and cooling in the shearing box approximation. We consider parameterized cooling functions in 2D and 3D disks, as well as radiative diffusion in 3D. We show that generally there is no critical cooling rate/viscosity below which the disk is formally stable, although interesting limits appear for unstable modes with lengthscales on the order of the disk thickness. We apply this new linear theory to protoplanetary disks subject to gravito-turbulence modeled as an effective viscosity, and cooling regulated by dust opacity. We find that viscosity renders the disk beyond $\sim 60$AU dynamically unstable on radial lengthscales a few times the local disk thickness. This is coincident with the empirical condition for disk fragmentation based on a maximum sustainable stress. We suggest turbulent stresses can play an active role in realistic disk fragmentation by removing rotational stabilization against self-gravity, and that the observed transition in behavior from gravito-turbulent to fragmenting may reflect instability of the gravito-turbulent state itself.