Asymptotic Fields Research Articles

On applicability and uniqueness of the correspondence principle to pie-shaped wedge (“wedge paradox”) with various boundary conditions

A new paradigm for re-examination of the concept of the correspondence principle (originally due to Dundurs) in association with the much-studied issue of “wedge paradox” is presented. This study connects for the first time two parallel streams of research: (i) Carothers-Sternberg-Koiter’s wedge paradox study and (ii) Williams' eigenfunction approach. First, a three-dimensional eigenfunction expansion approach for determination of the asymptotic (singular) stress fields in the vicinity of the fronts of infinite wedges of finite thickness, subjected to various wedge-flank boundary conditions, is presented. This analysis is then exploited to establish rigorous rules or conditions, which would depend on the prescribed wedge-flank boundary conditions, the related state of deformation and the elastic property (Poisson’s ratio), for validity (or lack thereof) of the assumptions implicit in the heuristic enunciation of the correspondence principle. More important, a hitherto unavailable analysis for uniqueness (or lack thereof) of the prevalence of such a heuristically postulated correspondence is also developed. Finally, the present study offers both mathematical as well as (heretofore missing) physical explanations behind the working of the correspondence principle.

The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes

This article applies the finite element over-deterministic (FEOD) method to determine the coefficients of the crack tip asymptotic fields in an anisotropic plane. This method relies on finding the best fit of the crack tip displacement equations to the displacements taken from a large number of nodes around the crack tip. We first revisit the asymptotic solution of crack tip fields in an anisotropic plane, followed by giving the exact solution of the crack tip parameters for an infinite anisotropic plane with a central crack. Using the derived displacement field, we then formulate the FEOD method to determine the crack tip parameters from a set of finite element nodal displacements. Numerical examples of cracks in finite and infinite plates are then investigated, where the FEOD results manifest a good agreement with exact or alternative numerical solutions. A convergence study is performed to examine the robustness of the methodology and the reliability of the results. By implementing a least squares error function, we also calculate the corresponding error in approximating the displacement field. The low values of errors demonstrate the high precision of the FEOD method.

Locally exact asymptotic homogenization of periodic materials under anti-plane shear loading

A new implementation of the asymptotic homogenization theory for anti-plane shear loading is constructed based on locally exact elasticity solutions of unit cell problems at different orders in the asymptotic field expansion. The approach is a generalization of the previously developed locally exact homogenization theory for the determination of homogenized elastic moduli under uniform stress fields. The new theory has been developed for periodic materials in the presence of stress gradients and non-vanishing microstructural scale relative to the structural dimensions. The local interior unit cell problems are solved exactly up to the third order using Fourier series representations of the microfluctuation functions. The exterior unit cell periodic boundary value problems at different orders are tackled by an asymptotic extension of the previously introduced balanced variational principle for periodic materials. The effectiveness of the theory is verified by solving a structural problem with varying number of inclusions directly, and comparing the local fields with those reconstructed by the locally exact asymptotic homogenization approach. The theory is also employed to assess the accuracy of an uncoupled homogenization-localization approach in the reconstruction of local stress fields in periodic materials with different levels of microstructural refinement in the presence of stress gradients. The present contribution provides the foundation for further extension of the new locally exact asymptotic homogenization theory to in-plane loading problems of unidirectionally reinforced periodic materials.

Interfacial toughness: dependence on surface roughness and test temperature

Interfacial toughness quantifies resistance to crack growth along an interface and in this investigation the toughness of an aluminum/epoxy interface was measured as a function of surface roughness and test temperature. The large strain response of the relatively ductile epoxy adhesive used in this study was also characterized. This epoxy adhesive exhibits intrinsic strain-softening after initial compressive yield and then deforms plastically at a roughly constant flow stress until it rapidly hardens at large compressive strains. We find that interface toughness scales as the product of the temperature dependent epoxy yield strength and a length scale that characterizes surface roughness. The proposed scaling is based upon dimensional considerations of a model problem that assumes that the characteristic length scale of both the roughness and the crack-tip yield zone is small relative to the region dominated by the linear elastic asymptotic crack-tip stress field. Furthermore, the model assumes that interfacial failure occurs only after the epoxy begins to harden at large strains. The proposed relationship is validated by our interfacial toughness measurements.

Improved crack analysis of two-dimensional functionally graded materials by enriched natural element method

The numerical calculation of stress intensity factors of two-dimensional functionally graded materials is introduced by an enriched Petrov–Galerkin natural element method (enriched PG-NEM). The overall trial displacement field is basically approximated in terms of Laplace interpolation functions and it is enriched by the near-tip asymptotic displacement field. The overall strain and stress fields which were approximated by PG-NEM were smoothened and enhanced by the patch recovery. The modified interaction integral [Formula: see text] is used to evaluate the stress intensity factors of functionally graded materials with the spatially varying elastic modulus. The validity of present method is justified through the evaluation of crack-tip stress distributions and the stress intensity factors of four numerical examples. It has been found that the proposed method effectively and successfully captures the near-tip stress singularity with a remarkably improved accuracy, even with the remarkably coarse grid, when compared with an extremely fine grid and the analytical and numerical reference solutions.

Asymptotic Field Near the Tip of a Debonded Anticrack in an Anisotropic Elastic Material

SummaryWe use the sextic Stroh formalism to study the asymptotic elastic field near the tip of a debonded anticrack in a generally anisotropic elastic material under generalised plane strain deformations. The stresses near the tip of the debonded anticrack exhibit the oscillatory singularities $r^{-3/4\pm i\varepsilon }$ and $r^{-1/4\pm i\varepsilon }$ (where $\varepsilon $ is the oscillatory index) as well as the real power-type singularities $r^{-3/4}$ and $r^{-1/4}$. Two complex-valued stress intensity factors and two real-valued stress intensity factors are introduced to respectively scale the two oscillatory and two real power-type singularities. The corresponding three-dimensional analytic vector function is derived explicitly, and the material force on the debonded anticrack is obtained. Our solution is illustrated using an example involving orthotropic materials.

An Improved Anti-Interference Precoding of Large-Scale Fading System Based on Channel Inversion

Recently, a large-scale fading precoding (LSFP) for the wireless massive multiple-input, multiple-output (MIMO) systems has been proposed. In this precoding, the channel information of all the cells using re-use pilot sequences is processed jointly, and pilot contamination and interference due to a certain number of antennas are effectively eliminated. Additionally, recent studies have found that research in the asymptotic field can be applied to the wireless large-scale MIMO systems. In the LSFP, pilot contamination and signal interference will be completely eliminated when a number of antennas at a base station tend to be unlimited. In this research found that the LSFP method can eliminate most pilot contamination and interference in practical applications only when the number of antennas of the base station reaches hundreds of orders, which greatly increases the equipment construction cost. On the other hand, channel inversion denotes a multi-user channel modulation technology, where a vector signal generated between a user and a base station is used to form an inverse channel matrix so that the channels of each user are balanced during the transmission. In this paper, the channel inversion technology is used in the LSFP. The improved LSFP can effectively reduce the number of antennas required by the base station without affecting the performance of eliminating the pilot contamination and interference. It is shown that when the number of antennas of a base station tends to be unlimited, the improved LSFP can eliminate pilot contamination and signal interference. The simulation results show that in the same practical scenario, when the base station is equipped with the same number of antennas, the improved method can more effectively improve the anti-contamination and anti-interference performance over conventional LSFP.

Poroelastic effects on steady state crack growth in polymer gels under plane stress

Fracture experiments on polymer gels are often conducted with thin specimens, which are close to plane stress in two-dimensional models. However, many of the previous theoretical and numerical studies on fracture of polymer gels have assumed plane strain conditions. The subtle differences between the plane stress and plane strain conditions are elucidated in this paper based on a linear poroelastic formulation for polymer gels, including the asymptotic crack-tip fields and finite element simulations of steady-state crack growth in long strip specimens. Moreover, a poroelastic cohesive zone model is adopted to study the rate-dependent fracture process of polymer gels. It is found that, without the cohesive zone model, the normalized crack-tip energy release rate at the fast crack limit is greater than the slow crack limit, suggesting reduced poroelastic toughening for fast crack growth under plane stress conditions, while the two limits are identical under plane strain conditions. With a solvent-permeable cohesive zone for the case of immersed specimens, solvent diffusion within the cohesive zone enhances the poroelastic toughening significantly as the crack speed increases, leading to a rate-dependent traction-separation relation. On the other hand, with no solvent diffusion in the cohesive zone for the not-immersed case, the poroelastic toughening effect diminishes as the crack speed increases. Based on the present study, the intrinsic steady-state fracture toughness of a poroelastic gel can be determined using long-strip pure-shear specimens, which in general is smaller than the applied energy release rate.

Asymptotic crack-tip fields in perfect plasticity under mixed mode loading

In the paper the refined presentation of the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities is given. This objective of the paper is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions all values of the mixity parameter to within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained by the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher than 0.89 respectively for plane strain conditions and seven sector determine the asymptotic solution for the mixity parameter less than 0.39 and five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

On higher order parameters in cracked composite plates under far‐field pure shear

AbstractThis paper presents the analytical solution of the crack tip fields as well as the crack parameters in an infinitely large composite plate with a central crack subjected to pure shear loading. To this end, the complex variable method is employed to formulate an asymptotic solution for the crack tip fields in an anisotropic plane. Using a stress‐based definition of the crack tip modes of loading, only the mode II crack parameters are found to be non‐zero under pure shear load. Special focus is given to the determination of the higher order parameters of the crack tip asymptotic field, particularly the first non‐singular term, ie, the T‐stress. Unlike the isotropic materials, in which the T‐stress is zero under pure shear, it is found that the T‐stress is non‐zero for the case of anisotropic materials, being the only material‐dependent crack tip stress parameter. The veracity of our exact crack tip fields is assessed and verified through a comparison made with respect to the finite element (FE) solution. Finally, we demonstrate the significance of the T‐stress on stresses near the crack tip in composite plates under pure shear loads.

Empirical model of magnetic field line spreading in isotropic turbulence with zero mean field

In many cases, the random walk of magnetic field line in isotropic turbulence with zero mean field is appropriated to describe the transport of energetic particles in the interstellar medium of our Galaxy. To understand the transport, our previous work determined the asymptotic field line diffusion coefficient by using Corrsin’s hypothesis and presumed models of field line spreading. Two of those models of field line spreading are the diffusive decorrelation (DD) model and the random ballistic decorrelation (RBD) model. The variances of the field line displacement in, say, the x direction for the DD and RBD models are assumed as and , respectively. τ is the field line displacement parameter defined as dτ = ds/B, where s is the arc length along the field line and B is the magnitude of the total magnetic field. D is the asymptotic diffusion coefficient, b is the rms magnetic fluctuation and is the ultrascale. Comparing with simulation results, the DD and RBD models predict the D values with ⩽ 15% error and ⩽ 21% error, respectively. To improve the theoretical model empirically, in this work, we assume that the proper model the variance is , where β is a proportional constant and α is an exponent. We extrapolate β with the proportional constants of DD and RBD models. Comparing with the simulation result of Kolmogorov turbulence, we obtain α = 0.8694. To test validity of the model, we compare the theoretical results of (formulated from Kolmogorov turbulence) with simulation results of two other turbulences: IroshnikovKraichnan turbulence and weak turbulence. The theoretical results of the empirical model match the computer simulation results very well (with ⩽ 0.9% error).

Influence of the vacuum polarization effect on the motion of charged particles in the magnetic field around a Schwarzschild black hole

The consequences of the vacuum polarization effect in magnetic fields around a Schwarzschild Black Hole on the motion of charged particles are investigated in this work. Using the weak electromagnetic field approximation, we discuss the non-minimal coupling between magnetic fields and gravity caused by the vacuum polarization and study the equations of motion for the case of a magnetic field configuration which asymptotically approaches a dipole magnetic field. It is shown that the presence of non-minimal coupling can significantly influence the motion of charged particles around Black Holes. In particular, the vacuum polarization effect, leading to strong amplification or suppression of the magnetic field strength around the event horizon (depending on the sign of the coupling parameter), can affect the scattering angle and minimal distance for the electrons moving in the gravitational field of the Black Hole as well as the dependence of these parameters on the asymptotic magnetic field strength, initial distance and the Black Hole mass. It is further demonstrated that the non-minimal coupling between gravity and astrophysical magnetic fields, caused by the vacuum polarization, can cause significant changes of the parameter space corresponding to bound trajectories around the Black Hole. In certain cases, the bounded or unbounded character of a trajectory is determined solely by the presence of non-minimal coupling and its strength. These effects could in principle be used as observational signatures of the vacuum polarization effect and also to constrain the value of the coupling parameter.

Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms

This paper presents the in-plane asymptotic displacement and stress fields for blunt V-notched components based on Kolosov–Muskhelishvili's approach. In the first part, the displacement and stress components in the polar coordinate system are determined by choosing appropriate complex potential functions. In order to construct the notch geometry, the Neuber's mapping relation is utilized. Then, the notch boundary conditions are imposed to calculate the free parameters of the stress distribution. Eventually, the stress and displacement components are calculated in the Cartesian and polar coordinates in the forms of series expansion. In the second part, the coefficients of series expansions are computed by using the least square method (LSM). The blunt V-notched Brazilian disk (BV-BD) specimen under mixed mode loading is used as an example to verify the proposed procedure. The stress components in arbitrary distances and directions are determined for different blunt V-notches in order to evaluate the accuracy of the calculated stress series solutions and their associated coefficients. The numerical results indicate that a single-term solution can lead to considerable errors, and to achieve good accuracy in the stress field calculation, one should take account of at least three terms in the stress series solution.

Exact charges from heterotic black holes

We derive exact relations to all orders in the α′ expansion for the charges of a bound system of heterotic strings, solitonic 5-branes and, optionally, a Kaluza-Klein monopole. The expressions, which differ from those of the zeroth-order supergravity approximation, coincide with the values obtained when only the corrections of quadratic order in curvature are included. Our computation relies on the consistency of string theory as a quantum theory of gravity; the relations follow from the matching of the Wald entropy with the microscopic degeneracy. In the heterotic frame, the higher-curvature terms behave as delocalized sources that introduce a shift between near-horizon and asymptotic charges. On the other hand, when described in terms of lower-dimensional effective fields, the solution carries constant charges over space which coincide with those of the asymptotic heterotic fields. In addition, we describe why the Gauss-Bonnet term, which only captures a subset of the relevant corrections of quadratic order in curvature, in some cases succeeds to reproduce the correct value for the Wald entropy, while fails in others.

Asymptotic elastic fields near an interface anticrack tip

Using analytic continuation, we derive the asymptotic elastic fields associated with a fixed, semi-infinite anticrack lying on the interface between two dissimilar elastically isotropic half-planes. We derive the two pairs of analytic functions defined in the two half-planes which characterize the elastic fields of stresses and displacements in the bimaterial. The negative anticrack contraction force is expressed in terms of the established complex stress intensity factor.

Astrophysical jets from boosted compact objects

We perform full 3D numerical simulations of compact objects, such as black holes or neutron stars, boosted through an ambient force-free plasma that posses a uniform magnetization. We study jet formation and energy extraction from the resulting stationary late time solutions.The implementation of appropriate boundary conditions has allowed us to explore a wide range of boost velocities, finding the jet power scales as $\gamma v^2$ (being $\gamma$ the Lorentz factor). We also explore other parameters of the problem like the orientation of the motion respect to the asymptotic magnetic field or the inclusion of black hole spin. Additionally, by comparing a black hole with a perfectly conducting sphere in flat spacetime, we manage to disentangle curvature effects from those produced by the perfect conducting surface. It is shown that when the stellar compactness is increased these two effects act in combination, further enhancing the luminosity produced by the neutron star.

A unified method to solve higher order asymptotic crack-tip fields of mode I, mode II and mixed mode I/II crack in power-law creeping solids

A higher order asymptotic analysis with separable variable form is presented in this paper for crack tip mechanics fields under mixed mode creep conditions. By defining a second order mode mixity factor, higher order term solutions are investigated by solving the boundary value problem. The solution procedure is validated first and then the characteristics of various kinds of second order asymptotic solutions for mixed I/II creep crack tip field are presented and discussed. The angular distribution functions of the second order stress and the stress exponent for various cases are given. Detailed relations between the amplitudes of the second order term and the first order term are also discussed. The method and solutions given in this paper encompass the higher order term solutions of pure mode I, pure mode II and mixed mode I/II.

Convergence behaviour of the enriched scaled boundary finite element method

SummaryIn this work, a very efficient numerical solution of three‐dimensional boundary value problems of linear elasticity including stress singularities is discussed, focussing on its convergence behaviour. For the employed scaled boundary finite element method, a discretization is only needed at the boundary, while the solution is considered analytically in a scaling coordinate. This presents a major advantage for two‐dimensional problems, when the scaling center is placed at a stress singularity. Unfortunately, three‐dimensional problems usually do not only include point singularities but also line singularities, which results in singular gradients in the boundary coordinates and thereby diminishes the method's original advantages. To alleviate this drawback, this work discusses an enrichment of the separation of variables representation with analytical asymptotic near fields of the line singularities. In contrast to previous works, besides the near‐field functions with λ=0.5, also those with λ=1.5 were determined and used for enrichment. This leads to a high accuracy and it is shown that this approach is required to recover the convergence properties of smooth boundary value problems without singularities when using quadratic Lagrange shape functions. In order to recover the convergence rates for higher order shape functions, near‐field functions with higher singularity exponent have to be included for enrichment.

The asymptotic solution of the ion-damped acoustic-gravity wave equation

In his classic work Hydrodynamics, Horace Lamb devoted a significant amount of effort to the mathematical analysis of atmospheric waves, i.e. waves in a compressible medium under the influence of a local gravitational field and a variable temperature (and hence sound speed) profile. In so doing, he derived equations for both the divergence and the curl of the velocity field, yielding expressions of considerable mathematical beauty and complexity. Eight decades later, Derek Moore and Edward Spiegel extended Lamb’s analysis to include an arbitrary external applied force in the equations of motion. By a suitable choice of such force, the governing wave equations for a wide variety of Lorenz force/Coriolis force-induced wave motions can be derived. However, they chose to investigate the radiation field resulting from the application of a concentrated vertical force. In so doing, they were able to utilize an important theorem by James Lighthill concerning the asymptotic radiation field from a source with compact support, using the concept of a wavenumber surface. This paper has two main components: the first is to extend the work of Moore and Spiegel to ion-damped acoustic-gravity waves in the ionospheric F-region, based on the seminal work of C. H. Liu and K. C. Yeh. The second part is a consequence of the first: the corresponding wavenumber surface becomes complex, and so Lighthill’s method has to be modified to account for the effects of this. Furthermore, a significant inconsistency in the formulation of the physical problem by Liu and Yeh has been corrected and the corresponding derivations have been reformulated. Some graphical information has also been provided in special cases to illustrate the comparative effects of damping on the asymptotic behaviour of the acoustic-gravity radiation field. A final feature of the paper is that the equations derived are very general and can provide the basis for investigation of more realistic atmospheric temperature profiles in future work.

Improved stress and displacement fields around V-notches with end holes

The complex potential method is used to derive the asymptotic stress field around the V-notches with end holes (VO-notches). Although the classical methods take into account only the singular stress terms, herein the stress distribution is presented in the asymptotic form considering both singular and non-singular terms. In order to calculate the coefficients of the singular and non-singular terms of the stress distribution, an over-deterministic method is employed. Based on the over-deterministic method, the nodal displacement values for a large number of points around the VO-notch are obtained from finite element analysis and fitted to the asymptotic displacement field. The method is then applied to two laboratory test specimens containing VO-notches with three different notch angles and four notch tip radii under mode I loading condition. Moreover, by comparing the results obtained from the truncated stress series with the output stress results of the finite element analysis, the accuracy of the calculated coefficients is evaluated. Finally, by using some numerical examples, it is shown that the proposed asymptotic stress distribution can appropriately describe the stress field around sharp V-, key-hole and U-notches. The results obtained for all mentioned notch types show that neglecting the higher order terms can result in significant errors while considering the first two non-singular stress terms presents very good results.