Boundary Behavior Research Articles

A fractal approach to predict the oxidation and corrosion behavior of a grain boundary engineered low SFE high entropy alloy

The current study presents a method to predict the oxidation and corrosion properties in a grain boundary engineered low stacking fault energy (SFE) FCC high entropy alloy (HEA) Ni14Cr21.5Co21.5Fe21.5Mn21.5 through fractal analysis. Two thermo-mechanical processing (TMP) routes namely, unidirectional rolling (UDR) and multistep cross rolling (MSCR) with intermediate annealing, were employed to enhance the percentage of coincident site lattice (CSL) boundaries and break the random boundary network (RBN) in the alloy. Electron backscattered diffraction (EBSD) was utilized to analyze the microstructure and local texture of the samples before and after subjecting the samples to TMP. Fractal analysis (box-method) was carried out to analyze and quantify the RBN in the processed samples. Potentiodynamic polarization experiments were carried out to study the corrosion resistance as a function of CSL boundaries in the processed samples. Thermogravimetric analysis (TGA) was carried out to study the oxidation behavior in the samples. Elemental segregation in the oxide layer was also analyzed using Energy dispersive spectroscopy (EDS) line scan analysis. In the final part of the study, empirical relationships were established between fractal number, oxide layer thickness, and corrosion rate respectively. It was observed that Fractal numbers could be used as a powerful and reliable tool to predict some of the grain boundary character dependent properties like corrosion and oxidation in the alloy system under study.

On the Squeezing Function and Fridman Invariants

For a domain $D \subset \mathbb C^n$, the relationship between the squeezing function and the Fridman invariants is clarified. Furthermore, localization properties of these functions are obtained. As applications, some known results concerning their boundary behavior are extended.

Exploration of the sliding behavior of a Σ11 grain boundary with precipitates in Ni–Al system using molecular dynamics

Molecular dynamics (MD) was used to gain insights into grain boundary (GB) shear mechanisms in alloys that require GB mobility in a field of coherent precipitates. The area fraction of the Σ11 affected by the γ′ was systematically varied to explore how GB shear mechanisms change with increased number of precipitates. Without precipitates, local atomic shuffling accommodates the Σ11 GB shear; above an area-fraction threshold, GB shear required dislocation emission. In addition to precipitate area fraction, the transition in GB shear mechanisms may also depend on precipitate shape/size, temperature, and GB misorientation. These factors will be investigated in future work.

Mechanical behavior of (Ni, Fe)Cr2O4 spinel grain boundaries studied by molecular dynamics simulations.

Using molecular dynamics simulations, we study the fracture initiation of grain boundaries in NiCr2O4 and FeCr2O4 of spinel structure. These compounds are representative of corrosion layers of nickel-chromium-iron austenitic stainless alloys. Uniaxial deformation is applied to several symmetric tilt-, twist-, and random-grain boundaries until complete decohesion is reached, in order to measure the critical cleavage stresses. We find that this mechanical quantity depends on the chemical composition and the structure of the grain boundaries. A correlation between the critical stress and the misorientation angle of the grain boundaries can be established. It is also found than the twist- and the random-grain boundaries exhibit the weakest resistance. Furthermore, these simulations show a localization of the elastic response in the vicinity of the grain boundary plane. Therefore, we were able to compare our simulation results with those provided by the theoretical model based on the universal binding energy relation that relates the critical stress to the cleavage grain boundary energy. It shows that this model provides similar critical stresses than the molecular dynamics simulations if the localization distance parameter is defined as the distance related to the deformation of the grain boundary thickness during the deformation.

The large time behavior of the free boundary for one dimensional compressible Navier-Stokes equations

In this paper, we study the large time behavior of the free boundary for both full Navier-Stokes (NS) equations with heat conduction (FNS) and isentropic NS equations in one dimensional space. We find an interesting phenomenon that the interfaces which separate the fluid and vacuum in FNS or NS will expand outward at exactly the same algebraic rates in time for all γ > 1, except γ = 2, just provided that the initial total entropy of the fluid is positive or the mass of the fluid is conserved, respectively. It should be emphasized that the required condition for FNS equations naturally reduces to the one for isentropic NS equations in which the entropy is constant.

Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.

Dirichlet problem for f-minimal graphs

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with prescribed boundary behavior on a bounded domain $\Omega \subset M$ under suitable assumptions on $f$ and the boundary of $\Omega$. In the second part, we consider the asymptotic Dirichlet problem. Provided that $f$ decays fast enough, we construct solutions to the problem. Our assumption on the decay of $f$ is linked with the sectional curvatures of $M$. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.

Roughness-induced effects on the convection–diffusion–reaction problem in a thin domain

In this paper, we investigate a convection–diffusion–reaction problem in a thin domain endowed with the Robin-type boundary condition describing the reaction catalyzed by the upper wall. Motivated by the microfluidic applications, we allow the oscillating behavior of the upper boundary and analyze the resonant case where the amplitude and period of the oscillation have the same small order as the domain's thickness. Depending on the magnitude of the reaction mechanism, we rigorously derive three different asymptotic models via the unfolding operator method. In particular, we identify the critical case in which the effects of the domain's geometry and all physically relevant processes become balanced.

On the boundary behavior of quasiconformal mappings

We discuss some open questions of the theory of quasiconformal mappings related to the field of studies of Professor G. D. Suvorov. The present work is dedicated to his memory.

Censored symmetric Lévy-type processes

Abstract We examine three equivalent constructions of a censored symmetric purely discontinuous Lévy process on an open set D; via the corresponding Dirichlet form, through the Feynman–Kac transform of the Lévy process killed outside of D and from the same killed process by the Ikeda–Nagasawa–Watanabe piecing together procedure. By applying the trace theorem on n-sets for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions, we analyze the boundary behavior of the corresponding censored Lévy process and determine conditions under which the process approaches the boundary ∂ ⁡ D {\partial D} in finite time. Furthermore, we prove a stronger version of the 3G inequality and its generalized version for Green functions of purely discontinuous Lévy processes on κ-fat open sets. Using this result, we obtain the scale invariant Harnack inequality for the corresponding censored process.

Seepage model for multiple-fractured horizontal wells in coal bed methane reservoirs

ABSTRACT Due to the features of low permeability and porosity, the horizontal well and fracturing treatment are effective ways to develop coalbed methane reservoirs, which have the large drain area leading to high producing rate and recovery. The seepage researches in conventional gas reservoir are sufficient, but there are few researches about the flow behavior and pressure distribution in coal bed methane (CBM) reservoirs, especially for multiple-fractured horizontal wells. In this work, based on the seepage theory, point-source function, and Laplace transforms, the mathematic models of gas flow in multiple-fractured wells are derived considering threshold pressure gradient and unsteady flowing. Subsequently, the type curves of multiple-fractured horizontal well in CBM reservoirs are obtained; there are main eight regimes. Finally, the mainly parameters affecting the dimensionless pressure and dimensionless pressure derivative are analyzed. The percolation characteristics of CBM reservoir are more complex than that of conventional gas reservoir for the presence of adsorption/desorption mainly affecting the seepage features in Middle-later Period and boundary behavior. This method does not only enrich our understanding of multiple-fractured horizontal well transport behaviors in coalbed methane reservoirs, but also has an important guiding significance for determining reasonable development program with multiple-fractured wells in the development of coalbed methane reservoir.

A Markov Chain Approximation Scheme for Option Pricing Under Skew Diffusions

In this paper, we propose a general valuation framework for option pricing problems related to diffusions based on a continuous-time Markov chain approximation to the underlying stochastic process. We obtain an explicit closed-form approximation of the transition density of a general diffusion process, which facilitates the unified valuation of various financial contracts written on assets with natural boundary behaviors, e.g. in foreign exchange market with target zones, and equity markets with psychological barriers. Applications include valuation of European call and put options, barrier and Bermudan options, zero-coupon bonds, and arithmetic Asian options. Motivated by the presence of psychological barriers in the market volatility, we also propose a novel skew stochastic volatility model, in which the latent stochastic variance follows a diffusion process. The framework is shown to be able to also handle the case of jump diffusions. Numerical results demonstrate that our approach is accurate and efficient, and recover various benchmark results in the literature in a unified fashion.

Atomistic simulations of grain boundary migration under recrystallisation conditions

We present details of an approach to simulating the migration of grain boundaries using classical molecular dynamics (MD) in a regime in which the boundary motion is driven by the presence of dislocation defects resulting from plastic deformation. By simulating the shrinkage of spherical grains of deformed material embedded in a perfect crystal, we create conditions in which grain boundary motion takes place on time scales accessible to MD simulations, in which the defect free energy acts as a driving force additional to the capillary force due to excess grain boundary energy. This approach is particularly flexible in allowing arbitrary choices of misorientation axis and angle and providing an interface in which all grain boundary plane orientations are represented. This flexibility is at the cost of additional complexity in the system under study and we establish an approach suitable for analysing the excess grain boundary energy and the components of the force for boundary motion on a local basis over the surface of the shrinking grain. We demonstrate that this approach is able to resolve variation in the grain boundary energy related to the intrinsic grain boundary dislocation network in lower angle boundaries and to detect local maxima in the driving force due to the dislocation structure within the deformed grain. We show that the variation in measured grain boundary velocities is accounted for by the excess energy of the dislocations. We further show that the presence of dislocations in the shrinking grain has a tendency to reduce the anisotropy in grain boundary behaviour. Indicative evidence is presented that the dislocation network may act as a focal point for mechanisms of grain boundary migration.

The Decoupling of Elastic Waves from a Weak Formulation Perspective

Elastic scattering governed by the Lame system associated with the third-type or fourth-type boundary condition is considered. It was shown in [8] by two of the authors that under suitable geometric conditions on the boundary surface of the elastic inclusion, the longitudinal and shear waves can be decoupled. The decoupling result in [8] was derived based on analyzing the local boundary behaviours of the elastic fields. In this article, we provide a different argument from a variational perspective in proving the decoupling result.

On a local version of Jack’s lemma

Abstract The purpose of this paper is to provide a result which concerns with the boundary behavior of analytic functions. It may be a local version of the well known Jack’s lemma when we change the function normalization at the origin.

Grain boundary curvatures in polycrystalline SrTiO3: Dependence on grain size, topology, and crystallography

AbstractBy mapping grain orientations on parallel serial sections of a SrTiO3 ceramic, it was possible to reconstruct three‐dimensional orientation maps containing more than 3000 grains. The grain boundaries were approximated by a continuous mesh of triangles and mean curvatures were determined for each triangle. The integral mean curvatures of grain faces were determined for all grains. Small grains with fewer than 16 neighbors mostly have positive mean curvatures while larger grains with more than 16 neighbors mostly have negative mean curvatures. It is also possible to correlate the mean curvature of individual triangles with the crystallographic characteristics of the grain boundary. The mean curvature is lowest for grain boundaries with (100) orientations and highest for grain boundaries with (111) orientations. This trend is inversely correlated to the relative areas of grain boundaries and directly correlated to the relative grain boundary energy. The direct correlation between the energy and curvature is consistent with the expected behavior of grain boundaries made up of singular orientations. Furthermore, because both the relative energy and curvature of grain boundaries with (100) orientations are minima in the distributions, these boundaries also have the lowest driving force for migration.

Exact boundary behavior of positive large solutions of a nonlinear Dirichlet problem

In this paper, we investigate the exact asymptotic behavior of positive solution to the following singular boundary value problem Δu=b(x)f(u),x∈Ω,u>0 in Ω,u|∂Ω=+∞,where Ω is a C2-bounded domain in RN, (N≥3), f∈C1((0,∞),(0,∞)) is nondecreasing on (0,∞) and b is a function in Clocγ(Ω), (0<γ<1) such that there exist b1,b2>0 satisfying for each x∈Ω, 0<b1=limd(x)⟶0infb(x)h(d(x))≤limd(x)⟶0supb(x)h(d(x))=b2<∞,where d(x)=dist(x,∂Ω) and h(t)≔ct−λexp∫tηz(s)sds, η>diam(Ω), λ≤2 such that z is a continuous function on [0,η] with z(0)=0.

Blow-up solutions for a $p$-Laplacian elliptic equation of logistic type with singular nonlinearity

In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions is for a class of logistic type quasilinear problems in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine comparison principle with our result of existence of solution.

Boundary behaviour of open vesicles in axisymmetric case

A continuous transformation from a closed vesicle to an open vesicle requires that the area of open hole enlarges from zero. Since the shape equation and boundary conditions of lipid open vesicles with free edges have been obtained, we want to know whether this process can be achieved with valid parameters. By studying the boundary conditions in the axisymmetric case, the analytic expression of the boundary edges is obtained generally. It reveals that the radius and line tension of boundary edges are confined strongly by bending moduli. In some cases, there is the minimal nonzero boundary radius and the line tension needs to surmount the maxim following the increase of boundary radius. Without the spontaneous curvature, the line tension will trend to infinite when the boundary radius shrinks to zero. The continuous opening up process requires that the spontaneous curvature is nonzero and the ratio between the bending moduli of Gauss curvature and mean curvature satisfies , which is smaller than the value from experiments and simulations. This result indicates that the opening up process is discontinuous.

Minimal thinness with respect to the Schr\xf6dinger operator and its applications on singular Schr\xf6dinger-type boundary value problems

The application of the new criteria for minimally thin sets with respect to the Schrödinger operator to an approximate solution of singular Schrödinger-type boundary value problems are discussed in this study. The method is based on approximating functions and their derivatives by using the natural and weakened total energies. This study shows that the new criteria are very effective and powerful tools in solving such problems. At the end of the paper, we are also concerned with the boundary behaviors of solutions for a kind of quasilinear Schrödinger equation.