Instability Diagrams Research Articles

Tunnel rockbursts induced by dynamic disturbances: mechanism and mitigation

Rockbursts are characterized by violent rock fractures and pose a significant threat to hard-rock tunnels, potentially resulting in casualties and damage to excavation spaces. Globally recognized as one of the least understood and most feared challenges in underground excavations, rockbursts are often triggered by dynamic disturbances such as engineering activities or nearby vibrations. This study conceptualizes rockbursts as dynamic buckling or instability issues inherent in rock structures. It specifically investigates the mechanism of tunnel rockbursts induced by ambient blasting. The derivation of the governing equation of motion, which incorporates shear deformation and rotary inertia of the rock column, results in coupled Mathieu equations. By employing the proposed numerical method, the conditions triggering rockburst were established using instability diagrams. The study examines the effects of static components, dynamic loading, and frequency on a tunnel example, revealing that the amplitude and frequency of dynamic disturbances are critical in influencing the occurrence of tunnel rockbursts through perturbation effects and parametric resonance mechanisms. These insights offer valuable understanding into the mechanisms, mitigation, and control of tunnel rockbursts.

Hot deformation behavior and microstructure characterization of AlCrFe2Ni2(MoNb)0.2 high entropy alloy

The hot deformation behavior and microstructure evolution of AlCrFe2Ni2(MoNb)0.2 eutectic high entropy alloy at deformation temperatures of 900 °C–1100 °C and strain rates of 0.001 s−1∼1 s−1 were systematically investigated by hot compression tests. The principal structure equations of AlCrFe2Ni2(MoNb)0.2 high entropy alloy were constructed using the Arrhenius equation modified by Zener-Hollomon parameters. The decrease of strain rate and the increase of deformation temperature can enhance the continuous dynamic recrystallization (CDRX) and discontinuous dynamic recrystallization (DDRX) processes simultaneously. The power dissipation and instability diagrams were constructed according to the power dissipation theory and instability criterion, and the hot processing diagrams with strains of 0.3–0.6 were obtained by combining with the Dynamic Material Model (DMM) theory. According to the analysis of power dissipation rate and processing safety zone, the optimal hot processing parameters were found to be ε = 0.6 for the true strain, T = 1050–1100 °C, ε˙ = 0.0037–0.031 s−1.

Hot deformation behavior, dynamic recrystallization mechanism and processing maps of Ti–V microalloyed high strength steel

The hot deformation behavior, dynamic recrystallization mechanism and processing maps of Ti–V complex microalloyed high strength steel under temperatures ranging from 840 to 1040 °C and strain rates varying from 0.01 to 10 s−1 were studied by using thermal-mechanical simulation (Gleeble-3800), optical microscopy (OM), and electron backscatter diffraction (EBSD). The true stress–strain curves of the investigated steel were obtained under the different deformation conditions, the thermal deformation activation energy was calculated, the thermal constitutive equations based on the true stress–strain curves were established, and the 3D power dissipation diagrams, the 3D plastic instability diagrams, and the thermal processing maps under different true strains were drawn. The results show that the flow stress curves are of dynamic recrystallization type at 960–1040 °C and strain rate of 0.01 s−1 and at 1040 °C and strain rate of 0.1 s−1, and the peak stress decreases gradually with the increase of the deformation temperature and the decrease of the strain rate. The hot deformation activation energy of Ti–V microalloyed steel was calculated to be 349.681 kJ/mol. The average grain size of austenite first decreases and then increases with increasing the strain rate under the deformation temperatures of 1000 °C and 1040 °C, and has the smallest value of 15.0 μm and 16.1 μm, respectively, after deformation at the strain rate of 1 s−1. With the true strain increasing from 0.2 to 0.6, the area occupied by the destabilized region increases and the machinable region area decreases. It reveals that the most available hot processing parameter ranges of Ti–V complex microalloyed steel are at the temperature range of 1000∼1040 °C with the strain rate of 0.01–1 s−1.

Numerical simulation of stability and responses of dynamic systems under parametric excitation

The stability and responses of dynamic systems under parametric excitation are often encountered in many fields of science and engineering, such as slender columns and thin plates under axial loadings. This paper proposes a novel numerical simulation method to simultaneously construct stability diagrams and predict the responses of multiple degree-of-freedom (DOF) dynamic systems under arbitrary parametric loadings. The method divides an arbitrary load into discrete segments to approximate the variable excitation function by a step function and then accumulates the system responses of each segment using a matrix method. Dynamic stability and response analysis of undamped and damped 1DOF, 2DOF, 3DOF, and multiple DOF systems are conducted, and numerical examples are compared to conventional methods to show the accuracy and advantage of the proposed numerical simulation method. The instability diagrams are also substantiated by vibration response curves obtained from the same method. The method is applied to calibrate the validity and ranges of applicability of the Hill infinite determinants in the Floquet theory of 1DOF Mathieu-Hill equations. The second order approximation of the first instability region from the Hill determinant is acceptable. However, the fourth order approximation of the second instability region must be used to obtain a relatively accurate result. For 2DOF, 3DOF, and multi-parameter linear periodic systems, the method can simultaneously render four types of instability regions in a single procedure: no resonances, simple parametric resonances, combination additive resonances, and combination differential resonances. These results could provide a better understanding of stability and responses of dynamic systems under parametric excitation.

Simulation on the transition of electrostatic instabilities in EAST steady-state scenario

The parameter dependence of transition between electrostatic instabilities is studied using gyrokinetic simulation based on a real discharge of steady-state scenario in the Experimental Advanced Superconducting Tokamak. The scan of radial locations shows that trapped electron mode (TEM) dominates around the core while the ion temperature gradient mode (ITG) simultaneously dominates outside. The maximum growth rate of TEM appears around ρ = 0.24, where the maximum electron temperature gradient R/L Te locates, ρ is the normalized poloidal flux. Effects of the parameters on the transition between TEM and ITG instability are studied at ρ = 0.24. It is found that TEM dominates in the scanning with individually changing R/L Te from 2.50 to 25.02 or the density gradient R/L n from 1.38 to 13.76. Meanwhile, the electron-ion temperature ratio T e/T i is found to destabilize TEM, the effect of T e is more sensitive than that of T i. The dominant instability diagrams in the (R/L Te, R/L Ti) plane at different T e/T i and R/L n are numerically obtained, which clearly show the parameter range of the dominant TEM or dominant ITG instability region. It is found that the dominant TEM region becomes narrower in the plane by decreasing R/L n when T e/T i > 0.5.

Transfer of Energy from Flexural to Torsional Modes for the Fish-Bone Suspension Bridge Model

We consider a conservative coupled oscillators system which arises as a simplified model of the interaction of flexural and torsional modes of vibration along the deck of the so-called fish-bone (Berchio and Gazzola in Nonlinear Anal 121:54–72, 2015) model of suspension bridges. The elastic response of the cables is supposed to be asymptotically linear under traction, and asymptotically constant when compressed (a generalization of the slackening regime). We show that for vibrations of sufficiently large amplitude, transfer of energy from flexural modes to torsional modes may occur provided a certain condition on the parameters is satisfied. The main result is a non-trivial extension of a theorem in Marchionna and Panizzi (Nonlinear Anal 140:12–28, 2016) to the case when the frequencies of the normal modes are no more supposed to be the same. Several numerical computations of instability diagrams for various slackening models respecting our assumptions are presented.

Dynamic Stability and Responses of Beams on Elastic Foundations Under a Parametric Load

This paper is concerned with numerical simulation of both dynamic stability and responses of beams on elastic foundations under a pulsating axial parametric load in a single matrix method. First, the equation of motion of a beam on an elastic foundation with damping is derived and decoupled into a Mathieu–Hill equation. Three different elastic foundations are considered and compared: Winkler, Pasternak, and Hetenyi models. Then a novel numerical simulation algorithm is proposed to investigate both the dynamic stability and the responses of the beam simultaneously. Accurate instability diagrams are obtained by the numerical simulation and are substantiated by vibration response curves obtained from the same method. These numerically accurate diagrams are used to calibrate the approximate instability boundaries of various orders of Hill infinite determinants from the classical Bolotin method for the first time. A detailed discussion is presented on effects of various aspects including elastic foundation models, damping, and static and dynamic loads. The results provide insights into the efficient and safe application of beams on elastic foundations in engineering. The proposed numerical method can be extended to analyze dynamic stability and vibrations of systems under arbitrary parametric excitations where Mathieu–Hill equations are involved.

Understand and Mitigate Reaction Non-Uniformity in Battery Electrodes

Reaction non-uniformity develops in battery electrodes across a wide range of length scales from the electrode to agglomerate and primary particle level. The resultant non-uniform state of charge distribution increases polarization and reduces energy utilization and rate performance. It generates incompatible volume changes, which triggers microcracks and dislocations. Inhomogeneous intercalation could cause overheating at local spots of high current density and also over-charge/discharge, which leads to irreversible structure decomposition and capacity fading. Understanding the origins of reaction heterogeneity and identifying its mitigation strategy is critical for developing better and safer batteries. In this work, we use mesoscale theory and simulation to elucidate a few important factors in the development of non-uniform reaction distribution at different length scales. At the single particle level, we show that misfit stress generated by lattice mismatch between the lithiated and delithiated phases plays an important role in inducing non-uniform (de)intercalation front through two general mechanisms. The first mechanism, referred to as “surface-mode coherent spinodal decomposition”, occurs in systems that undergo spontaneous phase separation such as olivine LiFePO4. We use phase-field simulation to reveal the morphological evolution of intercalation front in LiFePO4 under stress influence, which satisfactorily explains recent in-situ experimental observations. The second mechanism, which we refer to as “inverted Grinfeld instability”, is analogous to the well-known Asaro-Tiller-Grinfeld instability in epitaxial thin film growth and applies to electrode materials such as Li(Ni0.5Mn1.5)O4 with first-order phase transformations upon cycling. We employ linear stability analysis to establish the criteria of interface instability and construct interface stability diagrams in the space of particle size, applied overpotential / flux and interface location, from which the conditions for achieving uniform intercalation can be identified. At the electrode level, we identify electrolyte depletion as a major factor contributing to reaction inhomogeneity especially at large electrode thickness. By analyzing battery (dis)charge kinetics in the framework of porous electrode theory, two types of non-uniform reaction behavior are discovered for electrode systems that exhibit solid-solution-like (e.g. NMC) or phase-changing (e.g. LiFePO4, Li4Ti5O12) intercalation behavior, respectively. A simplified analytical model was developed to quantitatively describe such behaviors, which shows very good agreement with numerical simulations and provides an efficient tool for predicting reaction distribution within composite electrodes. We further elucidate the coupled effects of ionic and electronic conductivities and surface reaction kinetics on reaction uniformity through a simplified circuit model. Based on the theoretical understanding, we suggest that balancing the kinetic competition between electrolyte transport, electronic conduction and surface reactions represents an effective strategy to promote homogeneous reaction in thick battery electrodes.

A probabilistic description of entrainment instability for cloud‐topped boundary‐layer models

Cloud‐top radiative and evaporative cooling effects on entrainment fluxes are essential to turbulence generation and growth of cloud‐topped boundary layers. While the cloud radiative cooling effects are currently not well understood, their interaction with the cloud‐top evaporation further complicates efforts to parametrize the phenomena due to the random behaviour of the turbulent mixing at the cloud‐top interface. Here we focus on the cloud‐top mixing and treat the typical turbulent mixing eddy in a statistical manner to relate the mean evaporative cooling rate to the distribution of effective mixing fraction (defined as the fraction of air coming from the free atmosphere that enhances the entrainment flux). Existing observations suggest that this effective mixing fraction can be parsimoniously parametrized with beta distributions, the shape parameters of which control the locations of stability lines in the Cloud‐Top Entrainment Instability (CTEI) diagrams used to discriminate between stable and unstable clouds. The probabilistic description of the cloud‐top mixing process allows us to coherently reinterpret various forms of CTEI criteria and generalize them to form new cloud‐top entrainment schemes. We expect that such schemes will help improve cloud dynamic models by embedding realistic distributions of the effective mixing fraction.

Modulational instability and energy localization of twisted DNA with solvent interaction

We investigate the generation of soliton-like pulses along a DNA chain which takes into account both torsional and solvent interaction effects. Interactions between neighboring base pairs are described by a twist angle. Twisting is essential in the model to capture the importance of nonlinear effects for the thermodynamical properties. The nonlinear dynamics of the DNA is then modeled in the Hamiltonian approach by the generalized Dauxois–Peyrard–Bishop model (combination of several models). We introduce the generalized discrete nonlinear Schrödinger equation describing the dynamics of modulated wave through the twisted DNA with solvent interaction. The modulational instability is studied and we present an analytical expression for the MI gain to show the effects of twist angle on MI gain spectra as well as on stability diagram. With the increase of the twist angle the MI gain decreases then increases. Some interesting MI phenomena appear with an additional new MI region as the twist angle increases. The instability and stability diagrams are also affected. Numerical simulations are carried out to show the validity of the analytical approach. The result is that the initial wave breaks into a train of ultrashort pulses with repetition rates, which are trapped in some sites. The impact of the twist angle is investigated and we obtain that the twist angle affects the dynamics of stable patterns generated through the molecule. Thereafter, we study energy localization in the framework of twisted DNA with solvent interaction. While the twist angle leads to a stronger localization of energy, the solvent interaction delocalizes energy along the molecule.

Codimension 2 and 3 situations in a ring cavity with elliptically polarized electromagnetic waves

We study pattern formation on the plane transverse to propagation direction, in a ring cavity filled with a Kerr-like medium, subject to an elliptically polarized incoming field, by means of two coupled Lugiato–Lefever equations. We consider a wide range of possible values for the coupling parameter between different polarizations, B̄, as may happen in composite materials. Positive and also negative refraction index materials are considered. Examples of marginal instability diagrams are shown. It is shown that, within the model, instabilities cannot be of codimension higher than 3. A method for finding parameters for which codimension 2 or 3 takes place is given. The method allows us to choose parameters for which unstable wavenumbers fulfill different relations. Numerical integration results where different instabilities coexist and compete are shown.

Influence of an axial magnetic field on the stability of spherical Couette flows with different gap widths

This paper deals with the linear instability analysis of the spherical Couette flow of an electrically conducting fluid in the presence of an axial magnetic field. The numerical investigations are performed for different ratios, η = 0.5 and η = 0.6, and compared with Hollerbach’s work for η = 0.33 (Hollerbach in Proc R Soc 465:2003, 2009). The corresponding instability diagrams, i.e., the critical values of the Reynolds number Re, the wave number, and the frequency on the Hartmann number Ha, are presented and accompanied by simulation of the transition between three-dimensional flow states of different symmetries. The characteristic subdivision of the linear stability curves into anti-symmetric modes, which is responsible for the instability at small Ha, and symmetric modes occurring at higher Ha is found for larger η, too. However, the extension of the stability corridor between the anti-symmetric and the symmetric modes increases nonlinearly with η. This offers the possibility to stabilize the basic flow up to high Re by appropriately increasing Ha.

Instabilities of a matter wave in a matter grating

We investigate the stability of Bloch waves for a Bose-Einstein condensate moving through a periodic lattice created by another condensate modulated by an optical lattice. We show that the coupling of phonon-antiphonon modes of the two species give rise to a very rich structure of the regimes for dynamical instability, with significant differences with respect to the case of a single condensate in an optical lattice. We characterize the relative weight of each condensate in the mixing and discuss an analytic limit that accounts for the bare structure of the instability diagrams.

Theoretical vs. empirical classification and prediction of congested traffic states

Starting from the instability diagram of a traffic flow model, we derive conditions for the occurrence of congested traffic states, their appearance, their spreading in space and time, and the related increase in travel times. We discuss the terminology of traffic phases and give empirical evidence for the existence of a phase diagram of traffic states. In contrast to previously presented phase diagrams, it is shown that “widening synchronized patterns” are possible, if the maximum flow is located inside of a metastable density regime. Moreover, for various kinds of traffic models with different instability diagrams it is discussed, how the related phase diagrams are expected to approximately look like. Apart from this, it is pointed out that combinations of on- and off-ramps create different patterns than a single, isolated on-ramp.

THE RETURN OF THE MEMBRANE PARADIGM? BLACK HOLES AND STRINGS IN THE WATER TAP

Several general arguments indicate that the event horizon behaves as a stretched membrane. We propose using this relation to understand the gravity and dynamics of black objects in higher dimensions. We provide evidence that: (i) The gravitational Gregory–Laflamme instability has a classical counterpart in the Rayleigh–Plateau instability of fluids. Each known feature of the gravitational instability can be accounted for in the fluid model. These features include threshold mode, dispersion relation, time evolution and critical dimension of certain phase transitions. Thus, we argue that black strings break in much the same way as water from a faucet breaks up into small droplets. (ii) General rotating black holes can also be understood with this analogy. In particular, instability and bifurcation diagrams for black objects can easily be inferred. This correspondence can and should be used as a guiding tool for understanding and exploring the physics of gravity in higher dimensions.

Three-dimensional instabilities of periodic gravity waves in shallow water

A linear stability analysis of finite-amplitude periodic progressive gravity waves on water of finite depth has extended existing results to steeper waves and shallower water. Some new types of instability are found for shallow water. When the water depth decreases, higher-order resonances lead to the dominant instabilities. In contrast with the deep water case, we have found that in shallow water the dominant instabilities are usually associated with resonant interactions between five, six, seven and eight waves. For small steepness, dominant instabilities are quasi two-dimensional. For moderate and large steepness, the dominant instabilities are three-dimensional and phased-locked with the unperturbed nonlinear wave. At the margin of instability diagrams, these results suggest the existence of new bifurcated three-dimensional steady waves.

Monotonic and oscillatory free-convective instability of solution in the space between two plane horizontal electrodes: Solutions containing three types of ions

The conditions for the onset of natural convection (convective instability) in the electrochemical system consisting of two plane horizontal electrodes and solution containing three types of ions between them were analyzed theoretically. The equations of flow of incompressible viscous liquid to the Boussinesq approximation and the equations of material balance with the electroneutrality condition were used as the mathematical model. The set of equations for amplitudes of perturbations was solved using the Galerkin method. The dependences of critical Rayleigh number on the system parameters are obtained for monotonic and oscillatory instabilities in the case that the electrochemical reaction proceeds under the conditions of limiting current. The examples of instability diagrams are presented. In contrast to the binary electrolyte, for a solution containing three types of ions, the critical Rayleigh number depends on the charges of ions and the ratio between the diffusion coefficients of ions in the solution.

On the stability of thermonuclear shell sources in stars

We present a quantitative criterion for the thermal stability of thermonuclear shell sources. We find that the thermal stability of shell sources depends on exactly three factors: they are more stable when they are geometrically thicker, less degenerate and hotter. This confirms and unifies previously obtained results in terms of the geometry, temperature and density of the shell source, by a simplified but quantitative approach to the physics of shell nuclear burning. We present instability diagrams in the temperature-density plane for hydrogen and helium shell burning, which allow a simple evaluation of the stability conditions of such shell sources in stellar models. The performance of our stability criterion is demonstrated in various numerical models: in a 3 MAGB star, in helium accreting CO white dwarfs, in a helium white dwarf which is covered by a thin hydrogen envelope, and in a 1.0 Mgiant.

Kinetic critical thickness for surface wave instability vs. misfit dislocation formation in Ge xSi 1− x/Si (100) heterostructures

The kinetic critical thicknesses for surface wave formation and misfit dislocation generation in UHVCVD-grown GeSi/Si have been quantitatively determined using microscopical techniques. A refined morphological instability theory has been developed using a coupled continuum/atomistic treatment that incorporates a nucleation barrier to the onset of surface wave formation. The theory accurately predicts the onset of surface wave formation as a function of thickness, composition, temperature and deposition rate. The interplay between misfit dislocation generation and surface wave formation can be elucidated from two-dimensional strain relaxation instability diagrams obtained from a 4-D parameter space.

Theory of the kinetics of nucleation in adsorbing layer: the approach based on the relaxation of order parameter field

We use a Ginzburg-Landau type equation for the order parameter field to study the growth kinetics of two-dimensional new phase nuclei, as well as the phenomenon of instability of the azimuthal form of growing nuclei. A connection between the developed approach and phenomenological Zeldovich-Volmer (ZV) theory is established. The onset of various types of conditions imposed in ZV theory on the nucleus boundary is considered. We obtain instability diagrams of the angular perturbations and show that the instability occurs when the main growth mechanism is a diffusive one. Also, we determine the parameters region where the criteria obtained in this paper are the same as the results of phenomenological ZV theory and the continuum DLA model. Additionally, we study the criteria for the growth of angular perturbations in the case of a nonquasistationary growth regime.