Quasi-static Evolution Research Articles

Experimental & numerical investigations of ultra-high-speed dynamics of optically induced droplet cavitation in soft materials

Perfluorocarbon (PFC) droplets represent a novel class of phase-shift contrast agent with a promise in applications in biomedical and bioengineering fields. PFC droplets undergo a fast liquid-gas transition upon exposure to acoustic or optical triggering, offering a potential adaptable and versatile tool as contrast agent in diagnostics imaging and localized drug delivery vehicles in therapeutics systems. In this paper, we utilize advanced imaging techniques to investigate ultra-high-speed inertial dynamics and rectified quasi-static (low-speed) diffusion evolution of optically induced PFC droplet vaporization within three different hydrogels, each of different concentrations, examining effects such as droplet size and PFC core on bubble dynamics and material viscoelastic properties. Gelatin hydrogels reveal concentration-dependent impacts on bubble expansion and material elasticity. Embedding PFC droplets in gelatin increases internal pressure, resulting in higher equilibrium radius and continuous bubble growth during quasi-static evolution. Similar trends are observed in fibrin and polyacrylamide matrices, with differences in bubble behavior attributed to matrix properties and droplet presence. Interestingly, droplet size exhibits minimal impact on bubble expansion during inertial dynamics but influences quasi-static evolution, with larger droplets leading to continuous growth beyond 60 seconds. Furthermore, the core composition of PFC droplets significantly affects bubble behavior, with higher boiling point droplets exhibiting higher maximum expansion and faster quasi-static dissolution rates. Overall, the study sheds light on the intricate interplay between droplet characteristics, matrix properties, and multi-timescale bubble dynamics, offering valuable insights into their behavior within biomimetic hydrogels.

An explanation for the slow-rise phase of solar eruptions

ABSTRACT Solar eruptions are sudden release of the magnetic free energy accumulated within a quasi-static evolutionary process of the corona. Interestingly, many solar eruptions are preceded by a short-term slow-rise phase, during which the pre-eruption structure rises at a speed significantly larger than that of the quasi-static evolution. Here we suggest an explanation for the slow-rise phase based on a recent high-accuracy magnetohydrodynamic simulation for initiation of solar eruption. The simulation shows that by continuously shearing a bipolar magnetic arcade, an internal current sheet forms gradually, and an eruption begins once magnetic reconnection is triggered at the current sheet. We find in the simulation that the overlying field presents a slow-rise phase before the reconnection sets in. In addition, the rising speed is significantly larger than that of the core field during this phase. This slow rise is a manifestation of the growing expansion of the arcade in the process of approaching a fully open field state, which is inherent to the formation of a current sheet before the eruption. We also show three flare events with slow-rise phases that are highly consistent with these key characteristics in the simulation: an expansion of the overlying coronal loops with speeds much larger than the quasi-static evolution speed, and for those events with filament eruption, the slow rise of filament is much smaller than that of the overlying loops. In this type of events, the eruption might be initiated through the mechanism as shown in the simulation, and the expansion of overlying coronal loops is a better indicator of the slow-rise phase.

Spinodal slowing down and scaling in a holographic model

The dynamics of first-order phase transitions in strongly coupled systems are relevant in a variety of systems, from heavy ion collisions to the early universe. Holographic theories can be used to model these systems, with fluctuations usually suppressed. In this case the system can come close to a spinodal point where theory and experiments indicate that the behaviour should be similar to a critical point of a second-order phase transition. We study this question using a simple holographic model and confirm that there is critical slowing down and scaling behaviour close to the spinodal point, with precise quantitative estimates. In addition, we determine the start of the scaling regime for the breakdown of quasistatic evolution when the temperature of a thermal bath is slowly decreased across the transition. We also extend the analysis to the dynamics of second-order phase transitions and strong crossovers.

Addendum: Thermodynamic quasi-equilibria in high power magnetron discharges: a generalized Poisson–Boltzmann relation (2023 Plasma Sources Sci. Technol. 32 055012)

A recent publication by Köhn et al (2023 Plasma Sources Sci. Technol. 32 055012) studied the quasi-equilibria of high power magnetron discharges through thermodynamic principles. A generalized, magnetic-field aware Poisson–Boltzmann relation for the electric potential and the electron density was established using a non-standard (multi-objective) variational principle. This addendum demonstrates that, assuming slow or quasistatic evolution, the same result can be realized via a standard (single-objective) variational principle, thereby streamlining the theoretical framework while preserving the robustness of the finding.

Direct numerical simulations of microlayer formation during heterogeneous bubble nucleation

In this article, we present direct numerical simulation results for the expansion of spherical cap bubbles attached to a rigid wall due to a sudden drop in the ambient pressure. The critical pressure drop beyond which the bubble growth becomes unstable is found to match well with the predictions from classical theory of heterogeneous nucleation imposing a quasi-static bubble evolution. When the pressure drop is significantly higher than the critical value, a liquid microlayer appears between the bubble and the wall. In this regime, the interface outside the microlayer grows at an asymptotic velocity that can be predicted from the Rayleigh–Plesset equation, while the contact line evolves with another asymptotic velocity that scales with a visco-capillary velocity that obeys the Cox–Voinov law. In general, three distinctive regions can be distinguished: the region very close to the contact line where dynamics is governed by visco-capillary effects, an intermediate region controlled by inertio-viscous effects away from the contact line yet inside the viscous boundary layer, and the region outside the boundary layer dominated by inertial effects. The microlayer forms in a regime where the capillary effects are confined in a region much smaller than the viscous boundary layer thickness. In this regime, the global capillary number takes values much larger then the critical capillary number for bubble nucleation, and the microlayer height is controlled by viscous effects and not surface tension.

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure

Visco-energetic solutions for a model of crack growth in brittle materials

Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introduced by Almgren, Taylor, and Wang in the study of mean curvature flows. With our main result we prove the existence of a visco-energetic solution with a given initial crack. We also show that, if the cracks have a finite number of tips evolving smoothly on a given time interval, visco-energetic solutions comply with Griffith's criterion.

On the determination of the quasi-static evolution of brittle plane cracks via stationarity principle

The crack front evolution in brittle solids is commonly modelled by defining some crack increment criterion, which can be derived from considerations on the stress singularity, from the definition of a dissipative potential, from the introduction of phenomenological concepts such as the crack mobility, and so on. In this work, we faced the problem with no allowance for any crack increment criterion, but using only the elastic properties, the fracture energy, and a stationarity principle. We will show that these ingredients are enough for determining the equilibrium and the quasi-static evolution of generally shaped three-dimensional brittle plane cracks. In doing this, we pointed out some new insight on the crack front motion, a set of new, generalised, domain integrals for measuring the pointwise crack ‘tension’, and a rigorous calculation of the intersection angle between the crack front and the free surface.

Quasi-static evolution of axially and reflection symmetric large-scale configuration

In this paper, we review a recently offered notion of quasi-static evolution of the axial self-gravitating structures at large scales and the criterium to characterize the corresponding evolutionary aspects under the influence of strong curvature regimes. In doing so, we examine the axial source’s dynamic and quasi-static behavior within the parameters of various modified gravity theories. We address the formalism of these notions and their possible implications in studying the dissipative and anisotropic configuration. We initiate by considering higher-order curvature gravity. The Palatini formalism of [Formula: see text] gravity is also taken into consideration to analyze the behavior of the kinematical as well as the dynamical variables of the proposed problem. The set of invariant velocities is defined to comprehend the concept of quasi-static approximation that enhances the stability of the system in contrast to the dynamic mode. It is identified that vorticity and distinct versions of the structure scalars [Formula: see text], [Formula: see text] and [Formula: see text] play an important role in revealing the significant effects of a fluid’s anisotropy. As another example of evolution, we check the influence of Palatini-based factors on the shearing motion of the object. A comparison-based study of the physical nature of distinct curvature factors on the propagation of the axial source is exhibited. This provides the intriguing platform to grasp the notion of quasi-static evolution together with the distinct curvature factors at current time scenario. The importance of slowly evolving axially symmetric regimes will be addressed through the distinct modified gravitational context. Finally, we share a list of queries that, we believe, deserve to be addressed in the near future.

A model of solar magnetic flux rope eruption initiated primarily by magnetic reconnection

ABSTRACT There is a heated debate regarding the specific roles played by ideal magnetohydrodynamic (MHD) instability and magnetic reconnection in triggering solar eruptions. In the context of a pre-existing magnetic flux rope (MFR) before an eruption, it is widely believed that an ideal MHD instability, in particular, the torus instability, is responsible for triggering and driving the eruption, while reconnection, as invoked in the wake of the erupting MFR, plays a secondary role. Here, we present a new numerical MHD model in which the eruption of a pre-existing MFR is primarily triggered and driven by reconnection. In this model, a stable MFR embedded in a strapping field is set as the initial condition. A surface converging flow is then applied at the lower boundary, pushing magnetic flux towards the main polarity inversion line. It drives a quasi-static evolution of the system, during which a current layer is built up below the MFR with decreasing thickness. Once reconnection starts in the current sheet, the eruption commences, which indicates that the reconnection plays a determining role in triggering the eruption. By further analysing the works done by the magnetic flux of the pre-existing MFR and the newly reconnected flux during the acceleration stage of the eruption, we find that the latter plays a major role in driving the eruption. Such a model may explain observed eruptions in which the pre-eruption MFR has not reached the conditions for ideal instability.

Finite element solution of crack-tip fields for an elastic porous solid with density-dependent material moduli and preferential stiffness

In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the classical damped Newton’s method, the sequence of linear problems is then obtained, and the linear PDEs are discretized through a bilinear continuous Galerkin-type finite element method. We perform a series of numerical simulations for material bodies with a single edge-crack subject to a variety of loading types (i.e. the pure mode-I, II, and mixed-mode). Numerical solutions demonstrate that the modeling parameter in our proposed model can control preferential mechanical stiffness with its sign and magnitude along with the change of volumetric strain. This study can provide a mathematical and computational foundation to further model the quasi-static and dynamic evolution of cracks, utilizing the density-dependent moduli model and its modeling framework.

Excitation of Quasiperiodic Fast-propagating Waves in the Early Stage of the Solar Eruption

We propose a mechanism for the excitation of large-scale quasiperiodic fast-propagating magnetoacoustic (QFP) waves observed on both sides of the coronal mass ejection. Through a series of numerical experiments, we successfully simulated the quasi-static evolution of the equilibrium locations of the magnetic flux rope in response to the change of the background magnetic field, as well as the consequent loss of the equilibrium that eventually gives rise to the eruption. During the eruption, we identified QFP waves propagating radially outward of the flux rope, and tracing their origin reveals that they result from the disturbance within the flux rope. Acting as an imperfect waveguide, the flux rope allows the internal disturbance to escape to the outside successively via its surface, invoking the observed QFP waves. Furthermore, we synthesized the images of QFP waves on the basis of the data given by our simulations and found consistency with observations. This indicates that the leakage of the disturbance outside the flux rope could be a reasonable mechanism for QFP waves.

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes

Abstract We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary Γ-convergence.

Perfect plasticity versus damage: an unstable interaction between irreversibility and Γ-convergence through variational evolutions

This paper addresses the question of the interplay between relaxation and irreversibility through quasi-static evolutions in damage mechanics, by inquiring the following question: can the quasistatic evolution of an elastic material undergoing a rate-independent process of plastic deformation be derived as the limit model of a sequence of quasi-static brittle damage evolutions? This question is motivated by the static analysis performed in [J.-F. Babadjian et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854], where the authors have shown how the brittle damage model introduced by Francfort and Marigo (see [G.A. Francfort and J.-J. Marigo, Eur. J. Mech. A Solids 12 (1993) 149–189, G.A. Francfort and J.-J. Marigo, J. Mech. Phys. Solids 46 (1998) 1319–1342]) can lead to a model of Hencky perfect plasticity. Problems of damage mechanics being rather described through evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia is neglected.We consider the case where the medium is subjected to time-dependent boundary conditions, in the one-dimensional setting. The idea is to combine the scaling law considered in [J.-F. Babadjian, et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854] with the quasi-static brittle damage evolution introduced in [G.A. Francfort and A. Garroni, Arch. Rational Mech. Anal. 182 (2006) 125–152] by Francfort and Garroni, and try to understand how the irreversibility of the damage process will be expressed in the limit evolution. Surprisingly, the interplay between relaxation and irreversibility is not stable through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity type.

Nonlocal-to-local limit in linearized viscoelasticity

Abstract We study the quasistatic evolution of a linear peridynamic Kelvin-Voigt viscoelastic material. More specifically, we consider the gradient flow of a nonlocal elastic energy with respect to a nonlocal viscous dissipation. Following an evolutionary Γ-convergence approach, we prove that the solutions of the nonlocal problem converge to the solution of the local problem, when the peridynamic horizon tends to 0, that is, in the nonlocal-to-local limit.

FORMATION OF THE NEAR-FAULT DAMAGE ZONE DURING DYNAMIC RUPTURE IN A CRYSTALLINE ROCK MASS

Many models for analyzing the dynamic propagation of seismogenic ruptures are based on solving classical problems of fracture mechanics. It is assumed that the fault is a shear fracture mode with uniformly distributed friction and stresses at the crack tip. Known fracture mechanics theories do not describe the formation of laterally oriented damage zones, i.e., in the direction perpendicular to the crack plane. Observational data indicate the presence of a fairly extensive zone of damaged material in the vicinity of the fault. This is the zone of dynamic influence where the material has an increased number of fractures, higher permeability and lower elastic wave velocities. A correct assessment of the properties and sizes of zones of dynamic influence is crucial for constructing adequate earthquake preparation models. This paper analyzes the development patterns and quantitative characteristics of the damage zone during dynamic earthquake rupture and quasi-static evolution of the fault. To estimate the size and mechanical characteristics of the near-fault damage zone caused by movement along the slip surface, it is convenient to use the value of the second invariant of the deviatoric stress tensor (shear intensity). Matching of the calculated value with one or another degree of rock mass damage can be done using measurement data from large-scale explosions, by comparing them with the calculation results. It is shown that coseismic movement along the fault leads to insignificant changes in the properties of the host rock mass. Although the longitudinal wave velocity near the fault decreases markedly by 30-35%, the permeability increases only by approximately a factor of three, and the increase in the degree of fracturing is almost unnoticeable. This means that the properties of the rock mass change due to the opening of preexisting cracks. Repeated movements do not radically change the characteristic dimensions and properties of the damage zone. It is concluded that the fault-affected zone is formed mainly at the quasi-static stage of the formation of a single main fault zone through the coalescence of individual macrofractures, and future seismogenic movements add to the already existing amount of fractures.

Quasistatic Evolution of Damage in Thermo-electro-elastic-viscoplastic Materials With Normal Compliance

We have studied contact problem in a quasistatic process with boundary conditions, for which we couple a thermal effect and damage and electrical effect. We assume that the contact is with normal compliance in a form with a gap function and is associated to a slip rate-dependent version of Coulomb’s law of dry friction. We give the variational formulation, then the existence and the uniqueness of the weak solution. The proof is based on arguments of time-dependent variational inequalities, parabolic inequalities, and fixed points.

Level frequency postulate and the dice analog.

A level frequency postulate is proposed in the context of the Onsager regression hypothesis, and is utilized to demonstrate Fourier fluctuation time between levels in an analog system composed of red and white dice. This dice system is shown to be analogous to an isolated composite system of particles through derivation of the level probability distribution. Level fluctuation time is developed as an algebraic expression involving average energy and a Gaussian parameter, with quasistatic evolution demonstrated as an integral over fluctuation time.

Radiative Magnetohydrodynamic Simulation of the Confined Eruption of a Magnetic Flux Rope: Unveiling the Driving and Constraining Forces

We analyze the forces that control the dynamic evolution of a flux rope eruption in a three-dimensional radiative magnetohydrodynamic simulation. The confined eruption of the flux rope gives rise to a C8.5 flare. The flux rope rises slowly with an almost constant velocity of a few kilometers per second in the early stage when the gravity and Lorentz force are nearly counterbalanced. After the flux rope rises to the height at which the decay index of the external poloidal field satisfies the torus instability criterion, the significantly enhanced Lorentz force breaks the force balance and drives the rapid acceleration of the flux rope. Fast magnetic reconnection is immediately induced within the current sheet under the erupting flux rope, which provides strong positive feedback to the eruption. The eruption is eventually confined due to the tension force from the strong external toroidal field. Our results suggest that the gravity of plasma plays an important role in sustaining the quasi-static evolution of the preeruptive flux rope. The Lorentz force, which is contributed from both the ideal magnetohydrodynamic instability and magnetic reconnection, dominates the dynamic evolution during the eruption process.

On an efficient octic order sub-parametric finite element method on curved domains

An efficient finite element method using the sub-parametric transformations of the curved boundaries for the 45-node octic order triangular element is presented in this paper. The one-sided curved and two-sided straight triangle in the universal coordinate system (x,y) is mapped onto an isosceles triangle (ξ,η)∈[0,1] and ξ+η≤1 in the (ξ,η)-coordinate system by using an isoparametric coordinate transformation. Using such a transformation, the curved edge of the triangular element is altered implicitly by a newly defined octic curve. Such an octic curve is obtained by using the topology of the curved edge. With an apparent condition of arc being a parabola, a new transformation is developed to find a unique curve for each edge that passes through the points on the original curved arc. Overall, the new sub-parametric transformations are developed, these include the transformation for the curved edge on the original boundary, in addition to the parameters from the curved triangle's interior. Several numerical experiments are presented to show the efficacy of the proposed method. The computational results obtained for a boundary value problem using octic order basis functions outperform the results obtained using the existing higher order basis functions such as quadratic (6-node), cubic (10-node), quartic (16-node), quintic (21-node), sextic (28-node), and septic (36-node) order triangular elements (TEs) with curves. The method implemented in this paper can be easily extendable to quasi-static and dynamic crack evolution problems within linear and nonlinear elasticity.