Asymptotic Formulation Research Articles

Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space

Abstract This article is focused on deriving the approximate model for surface wave propagation on an elastic isotropic half-plane under the effects of the rotation and magnetic field along with the prescribed vertical and tangential face loads. The method of study depends on the slow time perturbation of the prevalent demonstration for the Rayleigh wave eigen solutions through harmonic functions. A perturbed pseudo-hyperbolic equation on the interface of the media is subsequently derived, governing the propagation of the surface wave. The established asymptotic formulation is tested by comparison with the exact secular equation. In the absence of the magnetic field, the specific value of Poisson’s ratio, ν = 0.25 \nu =0.25 , is highlighted, where the rotational effect vanishes at the leading order.

Asymptotic solution of electromagnetic heating of skin tissue with lateral heat conduction

We study the temperature evolution in the three-dimensional skin tissue exposed to an electromagnetic beam of millimeter wavelength. The skin absorption coefficient of the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter penetration depth for a 94 GHz wave. In contrast, in the lateral directions perpendicular to the depth, the beam size is usually much larger than the penetration depth. Based on this separation of length scales, we establish an asymptotic formulation in which each term has separable dependences on the depth coordinate and on the lateral coordinates. We solve it analytically to obtain a two-term asymptotic solution of the temperature distribution in the three-dimensional skin tissue. This closed-form analytical solution provides a practical and accurate way of predicting the temperature. When the beam size is moderately larger than the penetration depth (a ratio of 20), the effect of lateral heat conduction is well captured in the asymptotic solution with maximum error less than 0.0017 in the normalized temperature of magnitude well above 1.

Unsteady electroosmotic flow of Carreau–Newtonian fluids through a cylindrical tube

The present study aims to investigate the fully developed electroosmotic flow of Carreau–Newtonian fluids within a circular tube, considering two different boundary conditions of zeta potential. The unsteady behavior of Carreau–Newtonian fluids, is attributed to the pulsatile pressure-driven flow. Studying two distinct formulations, slip and no-slip zeta potentials, illustrates the relative movement of fluid particles and the charged surface. The proposed study’s framework is divided into two regions: a central region containing a non-Newtonian Carreau fluid exhibiting both shear-thinning and thickening behavior, and a peripheral region containing a Newtonian fluid with the effect of an electrical double layer at the solid wall of the cylindrical tube. The Poisson–Boltzmann equation under the assumption of Debye–Hückel approximation governs the potential distribution due to the electric double layer, facilitating the examination of electroosmotic flow through tube. Obtaining analytical solutions for the momentum equations in both regions becomes challenging due to the inclusion of pulsatile pressure-driven flow and a non-Newtonian Carreau fluid. Asymptotic series expansions are employed, utilizing small parameters such as the Weissenberg number (We≪1) and a pulsatile Reynolds number (α≪1), to derive velocity expressions for both regions. Wall shear stress fluctuations, as indicated by time-averaged wall shear stress and oscillatory shear index are assessed to gauge the temporal variation of wall shear stress from the dominant blood flow direction. By utilizing potential and hydrodynamic variables, an asymptotic formulation for the streaming potential is developed to ascertain the asymptotic expression of electrokinetic energy conversion efficiency and scrutinize the impacts of pertinent physical parameters on it. The proposed study prominently shows that the Debye–Hückel parameter, Weissenberg number, pulsatile Reynolds number, and time-dependent pressure gradient significantly influence the fluid velocity, flow rate, and flow resistance. The notable determination of the present study is that the velocity amplitude and electrokinetic energy conversion efficiency are enlarged in the slip-dependent case relative to the no-slip zeta potential. It is noteworthy that as inertial forces become more prominent compared to viscous forces, both time-average wall shear stress and oscillatory shear index experience a slight decrease, although the reduction in oscillatory shear index is minimal owing to the absence of obstruction within the tube wall. The NDSolve numerical scheme in Mathematica software is employed to calculate numerical solutions for the governing equations, which are subsequently verified against results obtained through asymptotic analysis. The findings of this study are expected to deepen our understanding of unsteady electroosmotic flows of Carreau fluid and contribute to the design and development of advanced microfluidic devices with enhanced performance for various applications.

Non-monotonic dynamics (mixed hardening/softening) in nonlinear continuous structures: An asymptotic formulation

Non-monotonic dynamics (mixed hardening/softening) in nonlinear continuous structures: An asymptotic formulation

Describing inherently anisotropic behaviours of natural clay by a hypoplastic model

The purpose of this paper is to model the loading direction‐dependent behaviours of inherently anisotropic intact clay by means of the hypoplastic framework. The basic hypoplastic model for clays proposed by Mašín (2013) is enhanced by incorporating a predefined anisotropic asymptotic state boundary surface formulation which is based on an anisotropic variable by projecting the microstructure tensor onto the normal of the spatially mobilized plane. The capability of the proposed hypoplastic model is first illustrated by simulations of intact Wenzhou soft clay under drained torsional shear tests using the hollow cylinder apparatus. The model is then used to predict undrained shear results on San Francisco Bay mud. Comparisons between the predicted and measured results demonstrate that the proposed hypoplastic model is capable of modelling the combined effect of the principal stress direction and intermediate principal stress on the stress‐strain behaviour of natural soft clay with inherent anisotropy.

Analysis of Torsional Vibration in a Fractured Poroelastic Half-Space Coated with Metal Foam and Sliding Interfaces

Metal foams are highly useful in industries because of their lightweight, energy and vibration absorption properties. This study investigated the propagation of torsional waves in an elastic layer over a fluid-saturated fractured poroelastic half-space with a metal foam coated layer. It is assumed that the interfaces are in sliding contact with two different sliding parameters. The coated layer is closed-cell aluminium foam. We use the separation variable technique and the Bessel function to solve the equation of motion in different layers. The displacement components are written in terms of the second kind Whittaker functions. Using an asymptotic formulation of the Whittaker function and appropriate boundary conditions, the dispersion equation is derived in terms of the determinant. The control of the vibration due to the metal foam-coated layer is one of the important goals of this study. Also, numerical and graphical analyses have been done with the help of Mathematica software to see the effect of different parameters on torsional wave propagation. It is found that the presence of coated metal foam layer decreases the phase velocity of the torsional wave propagation. The work may be helpful in the seismology, automobile, and aerospace industries.

An asymptotic formulation for boundary value problems in a non-local elastic half-space

The primary purpose of this study is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. For the sake of simplicity, the non-locality is limited to the vertical direction, which is represented by a one-dimensional exponential kernel, and the problem is formulated within the framework of Eringen’s theory. The proposed asymptotic approach is based on the assumption that the internal characteristic length is significantly smaller than a typical wavelength. This assumption allows for the development of an asymptotic formulation that expresses the considered boundary value problem in terms of local stresses. Additionally, the formulation includes explicit correction terms to the classical boundary conditions, which arise from the non-local effects. As an example application of the derived formulation, the Rayleigh surface waves in a plane strain problem are considered. Finally, numerical results are presented for certain specific values of elastic parameters to illustrate the effects of non-locality on the analyzed system.

Bayesian forces identification in cable networks with small bending stiffness

The regular monitoring of cable forces is essential for ensuring the safety of cable structures both during construction and throughout their lifetime. This paper aims at developing a vibration-based identification procedure of the axial forces, bending stiffness, and, secondarily, the crossing point position of cable networks. A model constituted by two crossing stays having small bending stiffness and negligible sag effects is considered. The in-plane direct dynamic problem is solved both numerically and through a perturbation approach. The obtained results are compared to the outcomes of a finite element model for verification purposes. The theoretical studies are also supported by experimental tests performed on a real cable-stayed bridge (Haccourt bridge), which provide insights into the dynamics of the system showing that models of cables with small bending stiffness are more appropriate than taut string models. The inverse analysis based on non-linear Bayesian regression is developed and the closed-form asymptotic formulations are used to prove that the bending stiffness, the cable forces, and the crossing point position can be separately identified from a set of observed frequencies. The implemented procedure is then applied to the tested bridge as a proof of concept, showing that the proposed in-plane identification strategy provides satisfactory results.

Seismic metasurface on an orthorhombic elastic half-space.

The article is studying a seismic meta-surface in the case of an oscillatory system arranged on the surface of an orthorhombic elastic half-space. The approach is based on the asymptotic hyperbolic-elliptic formulation for the Rayleigh wave excited by prescribed surface loading. The latter results in hyperbolic equations for surface displacements, with the right-hand sides involving the loading components. The derived model allows a formulation for the meta-surface in the form of a periodic spring-mass system attached to the surface as a hyperbolic equation for the horizontal displacement, with smooth contact stresses emerging from averaging the effect of a regular array of oscillators. The associated dispersion relation is constructed and illustrated numerically for both cases of exponential and oscillatory decay.

Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates

Abstract This article investigates the long-wave anti-plane shear motion in a symmetric three-layered laminate composed of pre-stressed compressible elastic layers. The layers of the laminate are perfectly bonded, while traction-free and fixed boundary conditions are considered on the outer faces of the laminate. In both cases, the dispersion relation is obtained in terms of symmetric and anti-symmetric decompositions. Numerical results and an asymptotic long-wave analysis are presented, corresponding to the three possible vibration modes. It is revealed that a low-frequency mode only exists in respect of symmetric motion with free-faces, while all other cases pose a series of non-zero cut-off frequencies. Comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.

A bootstrapping-based weighted average asymptotic sampling formulation for reliability estimation of highly safe structures

A bootstrapping-based weighted average asymptotic sampling formulation for reliability estimation of highly safe structures

Mixed RF/FSO SWIPT-Based OSLMD Coded AF Cooperative Communication System: Performance Analysis

In this paper, the performance analysis of simultaneous wireless information and power transfer (SWIPT) using time switching (TS) protocol-based amplify-and-forward (AF) dual-hop mixed radio-frequency/free-space optic (RF/FSO) coded and uncoded communication system (CC and UC) is investigated. Specifically, we look at using the One-Step Majority-Logic Decodable (OSMLD) codes at the source and a majority logic decoding algorithm (MLGD) at the destination, jointly with the use of maximal-ratio combining and SWIPT/TS techniques at the relay. Furthermore, the RF link undergoes Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> model, while the second-hop is subject to Máalaga- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {M}$ </tex-math></inline-formula> atmospheric turbulence with the effect of pointing errors. Closed-form and asymptotic formulations for the average bit error probability (ABEP) are obtained in terms of the system parameters for both CC and UC scenarios in addition of the average channel capacity analysis. Computer-based Monte Carlo simulations were used to numerically validate all of the obtained mathematical formulations. Expectedly, it is shown that significant coding gains are achieved using OSMLD codes in comparison with UC. In addition, above a specific threshold of energy efficiency, the system performance gets steady in the high signal-to-noise-ratio regime. Moreover, the decoding method used in this work has been shown to be suitable for the SWIPT/TS-assisted RF/FSO coded AF cooperative communication system where high performance alongside low computational complexity are required.

Asymptotic formulation of the nonlinear bifurcation scenarios in thermomechanically coupled plates

Asymptotic formulation of the nonlinear bifurcation scenarios in thermomechanically coupled plates

Asymptotic Formulation for the Rayleigh Wave on a Nonlocally Elastic Half-Space

This paper deals with the Rayleigh wave, propagating on a nonlocally elastic, linearly isotropic half-space, excited by a prescribed surface loading. The consideration develops the methodology of hyperbolic–elliptic models for Rayleigh and Rayleigh-type waves, and relies on the effective boundary conditions formulated recently, accounting for the crucial contributions of the nonlocal boundary layer. A slow-time perturbation scheme is established, leading to the reduced model for the Rayleigh wave field, comprised of a singularly perturbed hyperbolic equation for the longitudinal wave potential on the surface, acting as a boundary condition for the elliptic equation governing the decay over the interior. An equivalent alternative formulation involving a pseudo-differential operator acting on the loading terms, with parametric dependence on the depth coordinate, is also presented.

A novel asymptotic formulation for partial slip half-plane frictional contact problems

A method of solution and the necessary calibrations are given to permit the steady-state extent of slip to be found in contacts properly described within a half-plane formulation using only two parameters: the contact law and the first-order descriptions of tractions arising at the contact edges. The approach takes the assumption of full stick and corrects for the slip regions using an array of glide dislocations. This is a very versatile approach and is particularly appropriate when studying fretting fatigue, as it permits the region in which cracks nucleate to be defined very simply, and in a form which is transportable from contact to contact, including laboratory tests. The approach has the additional benefit of giving a relatively straightforward expression for the density of dislocations, from which the slip displacement and shear traction within the stick region may readily be calculated. An example implementation is provided in the case of a Hertzian contact in the absence of changes in bulk tension, for which we demonstrate the veracity of the predictions by comparing to previous asymptotic approaches that build upon the traction solution under the assumption of full sliding, as well as the known exact solution.

Elastodynamics of a coated half-space under a sliding contact

The paper deals with elastic wave propagating in a layer on a half-space induced by a vertical force. The focus is on the effect of a sliding contact along the interface and its comparative study with a perfect one. The effective boundary conditions substituting the presence of the layer are derived. The leading order term in these conditions corresponds to vertical inertia of the layer, whereas next order correction involves the effect of plate waves in the coating. Analysis of the associated dispersion relation confirms the existence of a Rayleigh-type wave, along with extensional and shear plate waves. An asymptotic hyperbolic-elliptic formulation for surface wave field is also presented. This includes a hyperbolic equation singularly perturbed by a pseudo-differential operator playing a role of a boundary condition for the elliptic equation governing decay over the interior. The sign of the coefficient at the pseudo-differential operator is demonstrated to be always negative, corresponding to a local maximum of the phase speed at zero wave number, and consequently to a distinct receding type of the Rayleigh-type wave quasi-front induced by an impulse load.

An Investigation of the MIMO Space Time Block Code Based Selective Decode and Forward Relaying Network over n-u Fading Channel Conditions

In this paper, we examine the end-to-end average pairwise error probability (PEP) and output probability (OP) performance of the maximum ratio combining (MRC) based selective decode and forward (S-DF) system over an η–µ scattering environment considering additive white Gaussian noise (AWGN). The probability distribution function (PDF) and cumulative distribution function (CDF) expressions have been derived for the received signal-to-noise (SNR) ratio and the moment generating function (MGF) technique is used to derive the novel closed-form (CF) average PEP and OP expressions. The analytical results have been further simplified and are presented in terms of the Lauricella function for coherent complex modulation schemes. The asymptotic PEP expressions are also derived in terms of the Lauricella function, and a convex optimization (CO) framework has been developed for obtaining optimal power allocation (OPA) factors. Through simulations, it is also proven that, depending on the number of multi-path clusters and the modulation scheme used, the optimized power allocation system was essentially independent of the power relation scattered waves from the source node (SN) to the destination node (DN). The graphs show that asymptotic and accurate formulations are closely matched for moderate and high SNR regimes. PEP performance significantly improves with an increase in the value of η for a fixed value of µ. The analytical and simulation curves are in close agreement for medium-to-high SNR values.

On Rayleigh wave field induced by surface stresses under the effect of gravity

The paper is concerned with development of the asymptotic formulation for surface wave field induced by vertical surface stress under the effect of gravity in the short-wave region. The approach relies on the methodology of hyperbolic-elliptic models for the Rayleigh wave and results in a regularly perturbed hyperbolic equation on the surface acting as a boundary condition for the elliptic equation governing decay over the interior. A special value of the Poisson’s ratio v = 0.25 is pointed out, at which the effect of gravity disappears at leading order.

Nonlinear mode localization in boundary–interior coupled structures by an asymptotic approach

Boundary–interior coupled structures are an important class of complex combined structural systems, with one component kinematically excited at its boundary and the other excited by a coupling force in its interior domain. Nonlinear mode localization of boundary–interior coupled structures is fully investigated in this paper, with a general asymptotic formulation being developed for construction of localized/global/hybrid nonlinear normal modes (NNMs), leading to a nonlinear interpretation of mode localization in boundary–interior​ structures. It is found that localized and global NNMs are induced at the same time as NNMs pairs, if the linear component frequencies are away from each other, while four non-trivial hybrid NNMs associated with two different nonlinear frequencies are produced, if the linear component frequencies are close to each other. Furthermore, the general asymptotic procedure is applied to a beam–beam coupled system with cubic nonlinearity, and various localized/global/hybrid NNMs are explicitly constructed.

Asymptotic Weyl double copy

A characteristic value formulation of the Weyl double copy leads to an asymptotic formulation. We find that the Weyl double copy holds asymptotically in cases where the full solution is algebraically general, using rotating STU supergravity black holes as an example. The asymptotic formulation provides clues regarding the relation between asymptotic symmetries that follows from the double copy. Using the C-metric as an example, we show that a previous interpretation of this gravity solution as a superrotation has a single copy analogue relating the appropriate Liénard-Wiechert potential to a large gauge transformation.