Finite Radius Research Articles

Voronoi-Based Discrete Element Analyses to Assess the Influence of the Grain Size and Its Uniformity on the Apparent Fracture Toughness of Notched Rock Specimens

The fracture toughness reflects the rock resistance to crack propagation, and therefore represents an important parameter for rock fracture assessments. From a strict point of view, the real fracture toughness (it {K}_{mathrm{it IC}}) corresponds to a cracked situation in which the notch radius is theoretically equal to zero. However, most of the defects in rocks have a finite radius and, therefore, should be studied as notch-type defects. Here, the notch effect is numerically studied together with the influence of the grain size and the sorting coefficient (grain size uniformity) on the apparent fracture toughness (it {K}_{mathrm{it IN}}). To this end, several four-point bending tests with different U-shaped notch radii, mean grain sizes and degrees of uniformity in grain size and shape have been simulated using the Discrete Element Method. In order to represent the grains of the rocks, the Voronoi tessellation is used to create randomly sized and distributed polygonal blocks. These Voronoi polygons have been defined, on the one hand, by an average edge length of 1, 2 and 3 mm, and, on the other hand, by a different number of iterations (n) in the relaxation process during the generation of the polygons, which defines the grain size uniformity. The numerical analyses performed and the interpretation of the results show a clear notch effect in all the studied cases, as the apparent fracture toughness (it {K}_{mathrm{it IN}}) increases with notch radius. Finally, the obtained stress fields at the notch tip have been compared to those obtained from the traditional finite element method.

A Modified Mean Stress Criterion for Considering Size Effects on Mode I Fracture Estimation of Rounded-Tip V-Notched Polymeric Specimens.

The aim of this paper is to assess the size and geometry effects on the mode I notch fracture toughness of polymeric samples containing rounded-tip V-shaped (RV) notches (V-notch with a finite radius at the notch tip). First, using a large number of fracture tests on an RV-notched Brazilian disk and semi-circular bending polymeric samples with four different sizes, the size-dependent values of the notch fracture toughness are obtained. Then, the mean stress criterion is modified for characterizing the size-dependency of notch fracture toughness in polymeric samples. The resulting modified mean stress criterion considers higher order terms of the stress field when calculating the fracture process zone length around the tip of the defect. Additionally, the critical distance rc is assumed to be associated with the specimen size and a formula containing fitting parameters is utilized for considering this trend of rc. The comparison between the values of notch fracture toughness obtained from experiments and those predicted by the modified mean stress criterion shows that the suggested approach can provide accurate estimations of size-dependent values of notch fracture toughness in polymeric specimens containing RV notches.

A new methodology inspired from the Theory of Critical Distances for determination of inherent tensile strength and fracture toughness of rock materials

Measuring the intrinsic fracture properties of quasi-brittle materials like rocks is of great importance and at the same time a major issue for engineers. In this study, we explore the ability of the Theory of Critical Distances (TCD) to determine accurately both the tensile strength and fracture toughness. To this end, we conduct ring tests and semi-circular bend tests on four rock types including a red sandstone, a white coarse-grained marble, a fine-grained granite and a coarse-grained granite. This selection covers sedimentary, metamorphic and igneous rock types with different grain sizes. The experimental data are analysed using a new methodology developed from the so-called Point Method (PM), a particular form of the TCD, from which we infer the intrinsic tensile strength and the fracture toughness of the studied rock materials. Our results are compared with those obtained from ISRM suggested methodology that is modified to take into account the finite notch root radius used in our experiments. The comparison is successful, supporting that the newly developed methodology is suitable to determine the intrinsic tensile strength and fracture toughness of rock materials.

FULLY 3D FLUID OUTFLOW FROM A SPHERICAL SOURCE

AbstractWe consider fully three-dimensional time-dependent outflow from a source into a surrounding fluid of different density. The source is distributed over a sphere of finite radius. The nonlinear problem is formulated using a spectral approach in which two streamfunctions and the density are represented as a Fourier-type series with time-dependent coefficients that must be calculated. Linearized theories are also discussed and an approximate stability condition for early stages in the outflow is derived. Nonlinear solutions are presented and different outflow shapes adopted by the fluid interface are investigated.

On the Uniform Stability of Recovering Sine-Type Functions with Asymptotically Separated Zeros

We obtain the uniform stability of recovering entire functions of special form from their zeros. To such a form, we can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second orders, including differential systems with asymptotically separated eigenvalues whose characteristic numbers lie on a line containing the origin, as well as the nonlocal perturbations of these operators. We prove that the dependence of such functions on the sequences of their zeros is Lipschitz continuous with respect to natural metrics on each ball of a finite radius. Results of this type can be used for studying the uniform stability of inverse spectral problems. In addition, general theorems on the asymptotics of zeros of functions of this class and on their equivalent representation via an infinite product are obtained, which give the corresponding results for many specific operators.

Pumping-Induced Non-Darcian Flow in a Finite Confined Aquifer

An analytic model for depicting non-Darcian flow caused by pumping in a finite confined aquifer with an outer barrier boundary is established. The model considers the wellbore storage and skin effect. And the semi-analytical solution is derived with the aid of the Izbash’s law-based non-Darcian flow and the method for linearization procedure combined with the Laplace transformation. The new presented solution can reduce to some available solutions for the confined aquifer of unlimited extension. The drawdowns in abstraction well and aquifer are explored. The results suggest that the presence of the no-flux outer boundary makes no difference to the early-time drawdowns, the late-time drawdowns in the unlimited confined aquifer are smaller than those in the finite confined aquifer with a zero-flux outer boundary, and the flow in this finite confined aquifer cannot approach quasi-steady state, especially to a smaller finite outer boundary distance radius. The change of power index n in the non-Darcian flow equation cannot affect the early-time drawdowns in abstraction well, while the drawdowns in aquifer are underestimated for Darcian flow case at early pumping time. The late-time drawdowns in the pumping well and aquifer are significantly overestimated under the assumption of Darcian flow. The early-time drawdowns in abstraction well and aquifer are significantly affected by wellbore storage, and a larger wellbore storage coefficient leads to a smaller drawdown. The skin factor can impact the intermediate-time drawdowns in abstraction well, while the early and intermediate-time drawdowns in aquifer are influenced by skin effect.

Expected signature of stopped Brownian motion on d-dimensional C2,α-domains has finite radius of convergence everywhere: 2 ≤ d ≤ 8

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain Ω, it is only known that the radius of convergence for the expected signature on sufficiently regular Ω is strictly positive everywhere, and that the radius of convergence is finite at some point when Ω is the 2-dimensional unit disc ([2]).In this paper, we prove that on any bounded C2,α-domain Ω⊂Rd with 2≤d≤8, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a “domain-averaging hyperbolic development” (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.

On the Reconstruction of Geodesic Subspaces of ℝN

We consider the topological and geometric reconstruction of a geodesic subspace of [Formula: see text] both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for the reconstruction problem. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and [Formula: see text]–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of [Formula: see text], we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space.

Gyro-average method for global gyrokinetic particle simulation in realistic tokamak geometry

Gyro-average is a crucial operation to capturing the essential finite Larmor radius (FLR) effect in gyrokinetic simulation. In order to simulate strongly shaped plasmas, an innovative multi-point average method based on non-orthogonal coordinates has been developed to improve the accuracy of the original multi-point average method in gyrokinetic particle simulation. This new gyro-average method has been implemented in the gyrokinetic toroidal code (GTC). Benchmarks have been carried out to prove the accuracy of this new method. In the limit of concircular tokamak, ion temperature gradient (ITG) instability is accurately recovered for this new method and consistency is achieved. The new gyro-average method is also used to solve the gyrokinetic Poisson equation, and its correctness is confirmed in the long-wavelength limit for realistically shaped plasmas. The improved GTC code with the new gyro-average method is used to investigate the ITG instability with EAST magnetic geometry. The simulation results show that the correction induced by this new method in the linear growth rate is more significant for short-wavelength modes where the FLR effect becomes important. Due to its simplicity and accuracy, this new gyro-average method can find broader applications in simulating shaped plasmas in realistic tokamaks.

Scattering theory and equation of state of a spherical two-dimensional Bose gas

We analyze the scattering problem of identical bosonic particles confined on a spherical surface. At low scattering energies and for a radius much larger than the healing length, we express the contact interaction strength in terms of the s-wave scattering length. Adopting this relation, we are then able to regularize the zero-point energy of the spherical Bose gas and to obtain its equation of state, which includes the corrections due to the finite radius of the sphere and coincides with the flat-case result in the infinite-radius limit. We also provide a microscopic derivation of the superfluid density of the system, reproducing a result postulated in a previous work. Our results are relevant for modeling the ongoing microgravity experiments with two-dimensional bubble-trapped Bose-Einstein condensates.

Fermionic King model: critical masses resulting from the competition of quantum and evaporation effects

We study the influence of evaporation (the escape of constituents) on the thermodynamics of a self-gravitating non-relativistic gas of fermions in the framework of Newtonian gravitation. For this purpose, it is reconsidered as the so-called fermionic King model introduced by Ruffini and Stella in the context of dark matter halos problems (1983 Astron. Astrophys. 119 3). As the usual King model, the present model predicts density profiles that drop to zero at a certain finite radius R, which is referred to as the tidal radius. Our interest is focused on the thermodynamics of this model at constant total mass M and constant tidal radius R. A special interest is devoted to study the incidence of mass on the collective phenomena of gravitational collapse and evaporation disruption. The underlying physics is driven by the Fermi mass M F ∼ ℏ 6/G 3 m 8 R 3, which corresponds to the total mass of a self-gravitating degenerate Fermi gas of non-relativistic point particles with mass m, whose linear size is equal to the tidal radius R of evaporation. Numerical calculations predict three critical values for the total mass, M 1 > M 2 > M 3, which hallmark the thermodynamic stability. The lower bound is a critical value of the total mass below the which the system gravitation is unable to confine its constituents, so that the system with total mass M < M 3 undergoes an evaporation disruption. The other bounds, M 2 ≃ 90.9M F and M 1 ≃ 8.9 × 106 M F, are critical values that hallmark the continuous (or discontinuous) character of the microcanonical phase transition associated with gravothermal collapse and the presence of states with negative heat capacities.

Gyrofluid analysis of electron β e effects on collisionless reconnection

The linear and nonlinear evolutions of the tearing instability in a collisionless plasma with a strong guide field are analysed on the basis of a two-field Hamiltonian gyrofluid model. The model is valid for a low ion temperature and a finite $\beta _e$ . The finite $\beta _e$ effect implies a magnetic perturbation along the guide field direction, and electron finite Larmor radius effects. A Hamiltonian derivation of the model is presented. A new dispersion relation of the tearing instability is derived for the case $\beta _e=0$ and tested against numerical simulations. For $\beta _e \ll 1$ the equilibrium electron temperature is seen to enhance the linear growth rate, whereas we observe a stabilizing role when electron finite Larmor radius effects become more relevant. In the nonlinear phase, stall phases and faster than exponential phases are observed, similarly to what occurs in the presence of ion finite Larmor radius effects. Energy transfers are analysed and the conservation laws associated with the Casimir invariants of the model are also discussed. Numerical simulations seem to indicate that finite $\beta _e$ effects do not produce qualitative modifications in the structures of the Lagrangian invariants associated with Casimirs of the model.

Critical comparison of collisionless fluid models: Nonlinear simulations of parallel firehose instability

Two different fluid models for collisionless plasmas are compared. One is based on the classical Chew–Goldberger–Low (CGL) model that includes a finite Larmor radius correction and the Landau closure for the longitudinal mode. Another one takes into account the effect of cyclotron resonance in addition to Landau resonance and is referred to as the cyclotron resonance closure (CRC) model [T. Jikei and T. Amano, Phys. Plasmas 28, 042105 (2021)]. While the linear property of the parallel firehose instability is better described by the CGL model, the electromagnetic ion cyclotron instability driven unstable by the cyclotron resonance is reproduced only by the CRC model. Nonlinear simulation results for the parallel firehose instability performed with the two models are also discussed. Although the linear and quasilinear isotropization phases are consistent with theory in both models, long-term behaviors may be substantially different. The final state obtained by the CRC model may be reasonably understood in terms of the marginal stability condition. In contrast, the lack of cyclotron damping in the CGL model makes it rather difficult to predict the long-term behavior with simple physical arguments. This suggests that incorporating collisionless damping both for longitudinal and transverse modes is crucial for a nonlinear fluid simulation model of collisionless plasmas.

Theoretical study of the Alfven eigenmode stability in CFETR steady state discharges

The aim of this study is to analyze the stability of Alfven eigenmodes (AE) in the China Fusion Engineering Test Reactor (CFETR) plasma for steady state operations. The analysis is done using the gyro-fluid code FAR3d including the effect of the acoustic modes, EP finite Larmor radius damping effects and multiple energetic particle populations. Two high poloidal β scenarios are studied with respect to the location of the internal transport barrier (ITB) at r/a ≈ 0.45 (case A) and r/a ≈ 0.6 (case B). Both operation scenarios show a narrow TAE gap between the inner-middle plasma region and a wide EAE gap all along the plasma radius. The AE stability of CFETR plasmas improves if the ITB is located inwards, case A, showing AEs with lower growth rates with respect to the case B. The AEs growth rate is smaller in the case A because the modes are located in the inner-middle plasma region where the stabilizing effect of the magnetic shear is stronger with respect to the case B. Multiple EP populations effects (NBI driven EP + alpha articles) are negligible for the case A, although the simulations for the case B show a stabilizing effect of the NBI EP on the n = 1 BAE caused by α particles during the thermalization process. If the FLR damping effects are included in the simulations, the growth rate of the EAE/NAE decreases up to 70%, particularly for n > 3 toroidal families. Low n AEs (n < 6) show the largest growth rates. On the other hand, high n modes (n = 6 to 15) are triggered in the frequency range of the NAE, strongly damped by the FLR effects.

Analysis of Stability of a Plasma in Porous Medium

The thermal convection of a plasma in porous medium is investigated in the presence of finite Larmor radius (FLR) and Hall effects. Following linear stability theory and normal mode analysis method, the dispersion relation is obtained. It is found that the presence of a magnetic field (and hence the presence of FLR and Hall effects) introduces oscillatory modes in the system which were, otherwise, non-existent in their absence. When the instability sets in as stationary convection, the FLR may have a stabilizing or destabilizing effect, but a completely stabilizing one for a certain wave-number range. Similarly, the Hall currents may have a stabilizing or destabilizing effect but a completely stabilizing one for the same wave-number range under certain condition, whereas the medium permeability always has a destabilizing effect for stationary convection. Also it is found that the system is stable for 𝑔𝛼𝜅 𝜈𝛽 ≤ 27𝜋 4 4 and under the condition 𝑔𝛼𝜅 𝜈𝛽 &gt; 27𝜋 4 4 , the system becomes unstable.

An analytical approach of heat transfer modelling with thermal stresses in circular plate by means of Gaussian heat source and stress function

In the proposed research work we have used the Gaussian circular heat source. This heat source is applied with the heat flux boundary condition along the thickness of a circular plate with a finite radius. The research work also deals with the formulation of unsteady-state heat conduction problems along with homogeneous initial and non-homogeneous boundary condition around the temperature distribution in the circular plate. The mathematical model of thermoelasticity with the determination of thermal stresses and displacement has been studied in the present work. The new analytical method, Reduced Differential Transform has been used to obtain the solution. The numerical results are shown graphically with the help of mathematical software SCILAB and results are carried out for the material copper.

Project Design of a Pulsed D-D Fusion Neutron Source Based on Field Reversed Configuration

The fusion neutron source is of great significance for conducting material testing for future fusion reactors, as it can genuinely reflect the change of material properties under fusion neutron irradiation. The International Fusion Materials Irradiation Facility (IFMIF) is an accelerator-driven neutron source. It still has some differences from the ideal fusion neutron source in terms of fusion neutron energy spectrum, which has led to the reconsideration of the fusion neutron source approach. In this paper, the magnetic field configuration, heating scheme design, and related calculations are carried out regarding the fusion neutron source. The plasma temperature, density evolution process, and the corresponding neutron yield of the field reversed configuration (FRC) plasma after two-stage cascade magnetic compression are analyzed, and the suppression of the magnetic fluid instabilities such as tilted and rotating modes of the FRC plasma after considering the two-fluid effect and the finite Larmor radius effect are studied. The fundamental physical parameters of the fusion neutron source are finally given. The calculation results show that the neutron source is expected to obtain fusion neutrons with an annual average power density higher than 2 MW/m², which can meet the requirements of material testing of the demonstration reactors (DEMO). The power estimation also shows that the scheme has the potential to become an energy source based on the pulsed deuterium-deuterium fusion.

Магнитный резонанс кольцевых спиновых кластеров

Using a quantum-based approach, we modeled the free induction decay signals for ring clusters of finite radius consisting of 6 and 8 particles with spin 1/2. In addition, we modeled a free induction decay signal for a spherical spin cluster by superimposing the ring signals. The current research uses two possible orientations of the spins. The ring plane was assumed to be only parallel or orthogonal to the external magnetic field. The free induction decays curves and their Fourier spectra were calculated both with and without the exchange interaction between the spins. The second and fourth moments of the magnetization response were evaluated by direct theoretical equations. The dependence of relaxation time on the ring radius was found. The exchange interaction was found to cause an increase in the relaxation time of a ring cluster when the exchange is positive, and a decrease in the time when it is negative. At the same time, the results obtained for the spherical cluster model do not show such an effect.

Interaction between long-period ULF waves and charged particle in the magnetosphere: theory and observations (overview)

The paper reviews the current state of the problem of interaction between long-period ultra-low-frequency (ULF) waves and high-energy particles. We consider elements of the theory of energy exchange between waves and particles, particle transport across magnetic shells under the influence of the electromagnetic field of a wave, the acceleration of radiation belt particles by both resonant and non-resonant mechanisms. We examine the mechanisms of generation of azimuthally-small-scale ULF waves due to instabilities arising from the wave–particle resonance. The cases of Alfvén, drift-compressional, and drift-mirror waves are analyzed. It is noted that due to the lack of a detailed theory of drift-mirror modes, the possibility of their existence in the magnetosphere cannot be taken as a proven fact. We summarize experimental data on the poloidal and compression ULF waves generated by unstable populations of high-energy particles. We investigate the mechanisms of modulation of energetic particle fluxes by ULF waves and possible observational manifestations of such modulation. Methods of studying the structure of waves across magnetic shells by recording fluxes of resonant particles with a finite Larmor radius are discussed.

Взаимодействие длиннопериодных УНЧ-волн и заряженных частиц в магнитосфере: теория и наблюдения (обзор)

The paper reviews the current state of the problem of interaction between long-period ultra-low-frequency (ULF) waves and high-energy particles. We consider elements of the theory of energy exchange between waves and particles, particle transport across magnetic shells under the influence of the electromagnetic field of a wave, the acceleration of radiation belt particles by both resonant and non-resonant mechanisms. We examine the mechanisms of generation of azimuthally-small-scale ULF waves due to instabilities arising from the wave–particle resonance. The cases of Alfvén, drift-compressional, and drift-mirror waves are analyzed. It is noted that due to the lack of a detailed theory of drift-mirror modes, the possibility of their existence in the magnetosphere cannot be taken as a proven fact. We summarize experimental data on the poloidal and compression ULF waves generated by unstable populations of high-energy particles. We investigate the mechanisms of modulation of energetic particle fluxes by ULF waves and possible observational manifestations of such modulation. Methods of studying the structure of waves across magnetic shells by recording fluxes of resonant particles with a finite Larmor radius are discussed.