General Analytical Formula Research Articles

Proposed formulas for pillar stress estimation in a regular room-and-pillar pattern

Room-and-pillar mining is one of the oldest and most widely employed underground mining methods worldwide, while in recent years it has also been used for civil engineering projects and applications. The pillar design process is quite straightforward, requiring the assessment of two key elements, namely the pillar's strength and the anticipated loading imposed on it. Lots of research work has been elaborated for the estimation of pillars' strength and the main parameters associated with it, while, on the other hand, in terms of the loading the Tributary Area Theory (TAT) has been traditionally and successfully used as the dominant methodology to assess the post-mining pillar induced stresses. This paper focuses primarily on decoding the pillar loading regime by analyzing the identifying the stresses imposed to them through the use of modern computational tools offered by 2D and 3D numerical analyses, over a series of pillars' type (square, ribs), configurations and initial virgin stress fields. This allows for the development of a benchmarking framework that showcase the differences in the stress conditions and reveal the over-conservative behavior of TAT. More importantly though, it allowed for the expression of two (2) general analytical formulae for the direct estimation of the average vertical elastic stresses on pillars, both for the case of rib & square pillars layout; the most typical patterns applied in this method. They aim at providing an easy to use and more accurate estimation of the pillar stress - as compared to TAT - in alignment with the results obtained from numerical analyses. Thus, with the proposed formulas, the time-consuming numerical process needed for preliminary pillars' dimensioning is tackled in a swift and effective manner.

Analytical formula for signal optimization in stimulated photon-photon scattering setup with three laser pulses

We consider a setup to detect stimulated photon-photon scattering using high-power lasers. Signal photons are emitted from an overlap of the incoming intense laser pulses focused in vacuum from three sides. We derive and justify a general approximate analytical formula for the angular distribution and total yield of such signal photons in terms of the parameters of the incoming pulses, including their intensity, carrier frequencies, durations, focusing, polarizations, mutual orientation and overlap. Using the obtained formula a parametric study of the signal is carried out and optimization is performed. Published by the American Physical Society 2024

Elastic Kink-Meson scattering

In classical field theory, radiation does not reflect off of reflectionless kinks. In quantum field theory, radiation quanta, called mesons, can be reflected. We provide a general analytical formula for the leading order amplitude and probability for the elastic scattering of mesons off of reflectionless quantum kinks. In the case of the Sine-Gordon model we verify that, due to a cancellation of six contributing processes, our general formula yields an amplitude of zero, as is required by integrability.

On the elongational viscosity of viscoelastic slip flows in hyperbolic confined geometries

We study theoretically the elongational viscosity (or Trouton ratio, in dimensionless form) for steady viscoelastic flows in confined and symmetric hyperbolic tubes considering Navier-type slip along the wall(s). Both the planar and the cylindrical axisymmetric geometrical configurations are addressed. Under the classic lubrication approximation, and for a variety of constitutive models such as Phan-Thien and Tanner, Giesekus, and Finite Extensibility Nonlinear Elastic with the Peterlin approximation models, the same general analytical formula for the Trouton ratio is derived as for the Oldroyd-B model, in terms of the velocity at the midplane/axis of symmetry and the Deborah number only. Assuming that the velocity field is approximated by the Newtonian lubrication profile, based on our previous study in the absence of slip, we show that a constant extensional strain rate can be achieved in the limits of zero or infinite slip. For finite slip, a slight modification of the geometry is required to achieve a constant strain rate. In these cases, the formula for the steady state Trouton ratio reduces to that for transient homogeneous elongation. We also provide analytical formulae for the modification (decrease) for both the extensional strain rate and the Hencky strain achieved in the confined geometries because of introducing wall slip.

Light curves of BSM-induced neutrino echoes in the optically thin limit

High-energy neutrinos from astrophysical transients serve as a probe of neutrino physics beyond the Standard Model. In particular, nonstandard interaction of neutrinos with the cosmic neutrino background or dark matter (DM) may have imprints on not only their spectra but also the arrival and time-delay distributions. Assuming that the interaction occurs at most once during the neutrino propagation, we provide general analytic formulas for light curves of the neutrino echoes induced by BSM. The formulas can be used for constraining neutrino-neutrino scattering, neutrino-DM scattering, and other scattering processes experienced by relativistic particles.

Diffusiophoresis of a Weakly Charged Liquid Metal Droplet.

Diffusiophoresis of a weakly charged liquid metal droplet (LMD) is investigated theoretically, motivated by its potential application in drug delivery. A general analytical formula valid for weakly charged condition is adopted to explore the droplet phoretic behavior. We determined that a liquid metal droplet, which is a special category of the conducting droplet in general, always moves up along the chemical gradient in sole chemiphoresis, contrary to a dielectric droplet where the droplet tends to move down the chemical gradient most of the time. This suggests a therapeutic nanomedicine such as a gallium LMD is inherently superior to a corresponding dielectric liposome droplet in drug delivery in terms of self-guiding to its desired destination. The droplet moving direction can still be manipulated via the polarity dependence; however, there should be an induced diffusion potential present in the electrolyte solution under consideration, which spontaneously generates an extra electrophoresis component. Moreover, the smaller the conducting liquid metal droplet is, the faster it moves in general, which means a smaller LMD nanomedicine is preferred. These findings demonstrate the superior features of an LMD nanomedicine in drug delivery.

Analytical and Statistical Modelling of a Fast Ion Source Formed by Injection of a Neutral Beam into Magnetically Confined Plasma

Mathematical modelling of heating and current drive as well as yields and distributions of fusion products in a magnetically confined plasma subject to neutral beam injection requires, in turn, modelling of distributions of fast ions, which is a complex task including calculations of the source of suprathermal particles, i.e., the number of fast ions occurring in unit volume during unit time owing to the injection of fast atoms. The knowledge of the magnetohydrodynamic equilibrium, beam injection geometry and spatial distribution of the magnetic field are the necessary prerequisites. Explicit general analytical formulae for the source of fast ions have been obtained by two different methods. In addition, a method of statistical modelling is presented. Calculations of spatial and angular distributions of the fast ion source for a tokamak and verifications of the obtained results have been performed by a number of methods.

A FRACTAL ELECTRICAL CONDUCTIVITY MODEL FOR WATER-SATURATED TREE-LIKE BRANCHING NETWORK

Electrical conductivity is an important physical property of porous media, and has great significance to rock physics and reservoir engineering. In this work, a conductivity model including pore water conductivity and surface conductivity is derived for water-saturated tree-like branching network. In addition, combined with Archie’s law, a general analytical formula for the formation factor is presented. Through the numerical simulation of the analytical formula above, we discuss the impact of some structural parameters ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in tree-like branching network on the resistance, conductivity and formation factor. The results show that the total resistance [Formula: see text] is proportional to [Formula: see text], [Formula: see text], and inversely proportional to [Formula: see text], [Formula: see text]. The relation between conductivity and porosity in this model is contrasted with previous models and experimental data, and the results show considerable consistency at lower porosity. It is worth noting that when [Formula: see text], the conductivity and porosity curve of this model overlap exactly with those plotted by the parallel model. The fractal conductance model proposed in this work reveals the operation of the current in the tree-like branching network more comprehensively.

Autofocusing field constructed by ring-arrayed Pearcey Gaussian chirp beams.

In this work, we propose and demonstrate the ring-arrayed Pearcey Gaussian chirp beams (RAPGCBs) synthesized by multiple two-dimensional Pearcey beams. The general analytical formula for the propagation of RAPGCBs is presented. We find that, depending on synthesized number n, the profiles of the beams present different polygonal shapes, and the autofocusing properties can be controlled by chirp factor β. Furthermore, we study the properties of the RAPGCBs carrying optical vortices (OVs). It shows that a single OV or two positive OVs form an autofocusing hollow field, and opposite OVs will annihilate, which results in greatly increased autofocusing ability. Our experimental results agree with the simulations. Such beams have potential applications in particle trapping and biology medical fields.

Neuroendocrine pancreatic tumors: about 13 cases

This article intends to revisit the electro-diffusional theory for the wall shear stress measurement from mass transfer probes of rectangular shape by considering the existence of two components of the wall shear rate (i.e., axial and transversal). General analytical formulas for the effective transfer length and the dimensionless mass transport coefficient were derived as a function of two parameters: a dimensionless angle of the flow direction, relative to the leading edge of the probe, and the aspect ratio between the width and the length of the strip probe. The correctness of the analytical relations for arbitrary flow direction and the aspect ratio was confirmed by numerical solutions of the transport equation in the convective-diffusive regime. It has also been proved that the differences between the Lévêque solution and the general analytical formula exhibit a significant deviation for a specific range of parameters. In the case of the three-dimensional boundary layers, in addition to the magnitude of the wall shear stress, the direction of the fluid flow in the vicinity of the probe’s surface is of paramount importance. Accordingly, a measurement methodology is proposed using two strip probes with different aspect ratios. The resulting equations required to quantify the magnitude of the wall shear rate vector and the dimensionless angle are also derived.

A new spherical indentation approach to determine fracture toughness of high strength steels

Fracture toughness indicates the resistance to unstable crack growth that is very important in the assessment of structural integrity. As the Instrumented Indentation Test (IIT) technique is relatively simple to perform, it has been suggested as an alternative way to determine fracture toughness of engineering materials. This paper presents a new spherical IIT approach to predicting fracture toughness of engineering materials using the critical contact pressure. Firstly, according to Johnson’s maximum strain energy theory under indentation, the formula of critical contact pressure under IIT is derived by considering strain hardening and fundamental tensile properties. Secondly, by using an automatic iterative procedure of translating the input and output data between MATLAB and ANSYS, finite element analysis is carried out to build a dummy database of critical contact pressure versus strain hardening and tensile properties for a broad scale of engineering materials. Finally, for the sake of engineering applications, a general analytical formula is put forward according to the dummy database, by which the critical contact pressure and thus the critical indentation depth can be read directly. Fracture toughness of engineering materials is then determined using the concept of indentation energy to fracture at the critical indentation depth. Validation of predicted fracture toughness and tensile properties of four high strength steels is taken out using compact tension (CT) and standard tensile tests, respectively. Good agreements are found between the standard tests and IIT.

Resummation for quantum propagators in bounded spaces.

We outline an approach to calculating the quantum mechanical propagator in the presence of geometrically nontrivial Dirichlet boundary conditions. The method is based on a generalization of an integral transform of the propagator studied in previous work (the so-called "hit function") and a convergent sequence of Padé approximants that exposes the limit of perfectly reflecting boundaries. In this paper the generalized hit function is defined as a many-point propagator, and we describe its relation to the sum over trajectories in the Feynman path integral. We then show how it can be used to calculate the Feynman propagator. We calculate analytically all such hit functions in D=1 and D=3 dimensions, giving recursion relations between them in the same or different dimensions and apply the results to the simple cases of propagation in the presence of perfectly conducting planar and spherical plates. We use these results to conjecture a general analytical formula for the propagator when Dirichlet boundary conditions are present in a given geometry, also explaining how it can be extended for application for more general, nonlocalized potentials. Our work has resonance with previous results obtained by Grosche in the study of path integrals in the presence of delta potentials. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.

SU(2) hyper-clocks: Quantum engineering of spinor interferences for time and frequency metrology

In 1949, Ramsey's method of separated oscillating fields was elaborated boosting over many decades metrological performances of atomic clocks and becoming the standard technique for very high precision spectroscopic measurements. A generalization of this interferometric method is presented replacing the two single coherent excitations by arbitrary composite laser pulses. The rotation of the state vector of a two-level system under the effect of a single pulse is described using the Pauli matrices basis of the SU(2) group. It is then generalized to multiple excitation pulses by a recursive Euler-Rodrigues-Gibbs algorithm describing a composition of rotations with different rotation axes. A general analytical formula for the phase-shift associated with the clock's interferometric signal is derived. As illustrations, hyper-clocks based on three-pulse and five-pulse interrogation protocols are studied and shown to exhibit nonlinear cubic and quintic sensitivities to residual probe-induced light-shifts. The presented formalism is well suited to optimize composite phase-shifts produced by tailored quantum algorithms in order to design a new generation of optical frequency standards and robust engineering control of atomic interferences in AMO physics with cold matter and anti-matter.

Analytic formula for quasinormal modes in the near-extreme Kerr-Newman–de Sitter spacetime governed by a non-Pöschl-Teller potential

Quasinormal modes of scalar, electromagnetic, and gravitational fields in the extreme Schwarzschild-de Sitter background are known to be expressed in analytic form as eigenvalues of the P\"oschl-Teller wavelike equation. We show that perturbations of fermionic fields (given by Dirac and Rarita-Schwinger equations) do not lead to the P\"oschl-Teller effective potential. Nevertheless, using the Frobenius method we find quasinormal modes analytically in this case as well. We write down the analytical formula for quasinormal frequencies of the near-extreme Schwarzschild-de Sitter black holes, which is valid for both bosonic and fermionic fields. We further extend the analysis to the case of charged rotating black holes and find a general analytical formula for quasinormal modes of the fields of various spin for the near extreme Kerr-Newman-de Sitter spacetime.

Tie Rod-Equivalent Non-Linear Constitutive Law for Uniformly Loaded Cables.

Cables are typically used in engineering applications as tensile members. Relevant examples are the main cables of suspension bridges, the stays of cable-stayed bridges, the load-bearing and stabilizing cables of tensile structures, the anchor cables of floating mooring structures, the guy-ropes for ship masts, towers, and wind turbines, the copper cables of electrical power lines. Since cables are characterized by non-linear behavior, analysis of cable structures often requires advanced techniques, like non-linear FEM, able to consider geometric non-linearity. Nevertheless, a traditional simplified approach consists in replacing the cable with an equivalent tie rod, characterized by a suitable non-linear constitutive law. Currently used equivalent constitutive laws have been derived by Dischinger, Ernst and Irvine. Since the equivalence is restricted to taut cables, characterized by small sag to chord ratios, these traditional formulae are not appropriate for uniformly loaded sagging cables: the main cables of suspension bridges are a particularly emblematic case. Despite some recent attempts to find more refined solutions, the problem is still open, since closed form solutions of general validity are not available. In the paper, general analytical formulae of the non-linear constitutive law of the equivalent tie rod are proposed, distinguishing two relevant cases, according as the length of the cable can vary or not. The expressions, derived by applying the general form of the theorem of virtual work, can be applied independently on the material, on the sag to chord ratio, on the load intensity and on the stress level, so allowing the replacement of the whole cable with a single equivalent tie rod. The expressions are critically discussed referring to a wide parametric study also in comparison with the existing formulae, stressing the influence of the most relevant parameters.

Analytical solution to the problem of ice ring formation in underground cryogenic gas storage

Storage of liquefied gas in an underground cavity created in hard rocks is a promising new technology that can significantly reduce the dimensions of a storage facility. The intense heat transfer between the cryogenic fluid and the surrounding rocks leads to the formation of an ice ring in water-saturated rocks, which acts as a natural protective barrier. In this paper, we provide a general theoretical analysis of the existence and evolution of the freezing ring. The analysis is based on solving the problem of heat transport in heterogeneous domain with two thermal waves initiated inside the domain and propagating into two opposite directions. The analytical solution was obtained by the modified Karman–Pohlhausen integral method. We have developed the extension of this method to a heterogeneous medium, in which the effect of the temperature semi-stabilization occurs.It has been shown that the propagation of the freezing front is non-monotonic and changes direction, so that the lifetime of the ice ring is finite. General analytical formulae have been obtained that describe all process parameters depending on the thickness of the insulating layer, in particular: the maximum thickness of the ice ring, its lifetime, conditions of existence, the critical thickness of the insulation above which the ice ring does not arise, as well as the global evolution of the temperature field. These results enable to select an optimal thickness of the insulation.

Extreme Poisson’s Ratios of Honeycomb, Re-Entrant, and Zig-Zag Crystals of Binary Hard Discs

Two-dimensional (2D) crystalline structures based on a honeycomb geometry are analyzed by computer simulations using the Monte Carlo method in the isobaric-isothermal ensemble. The considered crystals are formed by hard discs (HD) of two different diameters which are very close to each other. In contrast to equidiameter HD, which crystallize into a homogeneous solid which is elastically isotropic due to its six-fold symmetry axis, the systems studied in this work contain artificial patterns and can be either isotropic or anisotropic. It turns out that the symmetry of the patterns obtained by the appropriate arrangement of two types of discs strongly influences their elastic properties. The Poisson’s ratio (PR) of each of the considered structures was studied in two aspects: (a) its dependence on the external isotropic pressure and (b) in the function of the direction angle, in which the deformation of the system takes place, since some of the structures are anisotropic. In order to accomplish the latter, the general analytic formula for the orientational dependence of PR in 2D systems was used. The PR analysis at extremely high pressures has shown that for the vast majority of the considered structures it is approximately direction independent (isotropic) and tends to the upper limit for isotropic 2D systems, which is equal to +1. This is in contrast to systems of equidiameter discs for which it tends to 0.13, i.e., a value almost eight times smaller.

Dipole radiation and beyond from axion stars in electromagnetic fields

We investigate the production of photons from coherently oscillating, spatially localized clumps of axionic fields (oscillons and axion stars) in the presence of external electromagnetic fields. We delineate different qualitative behaviour of the photon luminosity in terms of an effective dimensionless coupling parameter constructed out of the axion-photon coupling, and field amplitude, oscillation frequency and radius of the axion star. For small values of this dimensionless coupling, we provide a general analytic formula for the dipole radiation field and the photon luminosity per solid angle, including a strong dependence on the radius of the configuration. For moderate to large coupling, we report on a non-monotonic behavior of the luminosity with the coupling strength in the presence of external magnetic fields. After an initial rise in luminosity with the coupling strength, we see a suppression (by an order of magnitude or more compared to the dipole radiation approximation) at moderately large coupling. At sufficiently large coupling, we find a transition to a regime of exponential growth of the luminosity due to parametric resonance. We carry out 3+1 dimensional lattice simulations of axion electrodynamics, at small and large coupling, including non-perturbative effects of parametric resonance as well as backreaction effects when necessary. We also discuss medium (plasma) effects that lead to resonant axion to photon conversion, relevance of the coherence of the soliton, and implications of our results in astrophysical and cosmological settings.

Broadband dynamic elastic moduli of honeycomb lattice materials: A generalized analytical approach

A generic analytical framework is proposed to obtain the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The dynamic deformation behaviour of the individual beam elements of a lattice is distinct from the behaviour under a static condition. This leads to a completely different global deformation pattern of the lattice material and subsequently opens up a tremendous opportunity to modulate amplitude and phase of the elastic properties of lattices as a function of the ambient vibration. The dynamic stiffness approach proposed in this article precisely captures the sub-wavelength scale dynamics of the periodic network of beams in a lattice material using a single beam-like member. Here the dynamic stiffness matrix of a damped beam element based on the Timoshenko beam theory along with axial stretching is coupled with the unit cell-based approach to derive the most general closed-form analytical formulae for the elastic moduli of lattice materials across the whole frequency range. It is systematically shown how the general expressions of dynamic elastic moduli can be reduced to different special cases by neglecting axial and shear deformations under dynamic as well as classical static conditions. The significance of developing the dynamic stiffness approach compared to conventional dynamic finite element approach is highlighted by presenting detailed analytical derivations and representative numerical results. Further, it is shown how the analytical framework can be readily extended to lattices with non-prismatic beam elements with any spatial variation in geometry and intrinsic material properties. In general, research activities in the field of lattice metamaterials dealing with elastic properties revolve around intuitively designing the microstructural geometry of the lattice structure. Here we propose to couple the physics of deformation as a function of vibrating frequency along with the conventional approach of designing microstructural geometry to expand the effective design space significantly. The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex-valued elastic moduli due to the presence of damping in the constituent material. The amplitude, as well as the phase of effective elastic properties of lattice materials, can be quantified using the proposed approach. The dependence of Poisson's ratio on the intrinsic material physics in case of a geometrically regular lattice is found to be in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only the function of microstructural geometry. The generic analytical approach for analysing the elastic moduli is applicable to any form of two- or three-dimensional lattices, and any profile of the constituent beam-like elements (different cross-sections as well as spatially varying geometry and intrinsic material properties) through a wide range of frequency band. The closed-form expressions of elastic moduli derived in this article can be viewed as the broadband dynamic generalisation of the well-established classical expressions of elastic moduli under static loading, essentially adding a new exploitable dimension in the metamaterials research in terms of dynamics of the intrinsic material.

A revisit of the electro-diffusional theory for the wall shear stress measurement

This article intends to revisit the electro-diffusional theory for the wall shear stress measurement from mass transfer probes of rectangular shape by considering the existence of two components of the wall shear rate (i.e., axial and transversal). General analytical formulas for the effective transfer length and the dimensionless mass transport coefficient were derived as a function of two parameters: a dimensionless angle of the flow direction, relative to the leading edge of the probe, and the aspect ratio between the width and the length of the strip probe. The correctness of the analytical relations for arbitrary flow direction and the aspect ratio was confirmed by numerical solutions of the transport equation in the convective-diffusive regime. It has also been proved that the differences between the Lévêque solution and the general analytical formula exhibit a significant deviation for a specific range of parameters. In the case of the three-dimensional boundary layers, in addition to the magnitude of the wall shear stress, the direction of the fluid flow in the vicinity of the probe’s surface is of paramount importance. Accordingly, a measurement methodology is proposed using two strip probes with different aspect ratios. The resulting equations required to quantify the magnitude of the wall shear rate vector and the dimensionless angle are also derived.