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A displacement function technique has been developed to study plane strain problems in second-order elasticity theory for incompressible elastic materials. The formulation culminates in the development of a single inhomogeneous biharmonic equation of the form ∇ 4 Ψ 2 (x) = f (x) for the second-order displacement function Ψ 2 (x). In the second-order, the isotropic stress associated with constitutive behaviour is governed by an inhomogeneous harmonic equation of the form ∇ 2 p 2 (x) = g (x). Both functions f (x) and g (x) depend only on the solution to the analogous problem in classical elasticity. While there are similarities to the Airy stress function, the displacement function approach enables the evaluation of the first- and second-order displacement fields purely through its derivatives, thus eliminating the introduction of arbitrary rigid body terms normally associated with formulations where the strains derived from a stress function approach need to be integrated. The displacement function approach is applied to develop a second-order elasticity solution to the problem of a Volterra edge dislocation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction of semiclassical spectrum and efficient (simple to implement) explicit semiclassical asymptotic eigenfunctions of the Dirac operator for high-energy bound states in graphene in magnetic field, with considering the effect of trigonal warping [1], [2] to be small. It turns out that the asymptotic spectrum of the operator is invariant under such a perturbation, but due to the symmetry of the problem only, rather than the smallness of this correction. However, the behavior of asymptotic eigenfunctions is quite different; they are significantly affected by trigonal warping that leads to the breaking of certain symmetries. Density plots of asymptotic eigenfunctions can indicate what should be observed using a scanning tunneling microscope. Our approach to constructing asymptotic solutions is based on developments of previous papers [3]–[6], which present a new method for constructing the solution, simplifying practical application. We also provide a rigorous estimate for the tails of the Fourier series of the amplitudes, which permits one to exclude them from consideration. [ABSTRACT FROM AUTHOR]

Maxwell's equations with temporal dispersion and localized initial data are considered. Using the Maslov canonical operator, we construct a uniform asymptotic solution expressed in terms of the derivative of the Airy function of composite argument in a neighborhood of regular points of the leading wave front. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to introduce innovative orthogonal power function series aimed at obtaining accurate nonlinear analytical solutions for axisymmetric circular thin plates. The main features of this paper are as follows: The deflection is expanded by the innovative orthogonal power function series. The Airy stress function, which satisfies the geometric deformation compatibility equation, responds to the nonlinear coupling relationships between the plate deflection and the in-plane force or displacement boundary conditions. The nonlinear algebraic equations are obtained by the energy variational method. Many comparisons are made with the results of related researchers. The present accurate solutions not only allow the problems to be solved perfectly and provide the most reliable basis for engineering design but also set new benchmarks for the verification of various nonlinear numerical and approximate analytical solutions. The developed methodology represents a significant improvement, providing better accuracy and computational efficiency compared to historical approaches. Therefore, the present method is more worthy of promotion. [ABSTRACT FROM AUTHOR]

The two-dimensional deformation of a uniform, elastically isotropic layer of a finite thickness (FT) over a rough-rigid base caused by surface loads has been solved analytically. The stresses and the displacements for an isotropic layer are obtained in the integral form by applying the Airy's stress function approach. Using the appropriate boundary conditions, the problem of surface loads is discussed in detail. The integrals cannot be evaluated analytically due to the complicated expression of denominator. The linear combination of exponential terms occurring in the denominator is expanded by a finite sum of exponential terms (FSET) using the method of least squares and then the integrals are evaluated analytically. The analytical displacements are obtained for normal strip loading, normal line loading and shear line loading. The displacements have been plotted numerically for normal strip loading to find the effect of layer thickness for a Poissonian layer and compared with the half-space. [ABSTRACT FROM AUTHOR]

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi 2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature. • Exact solutions for stress/displacement in gradient elasticity models for anisotropic functionally graded nano-beams. • Analysis focuses on constructing stress fields using "gradient Airy stress function" notation. • Solutions reveal significant strain gradient effects in anisotropic nano-beams. • Literature solutions for CT beams match findings by taking gradient coefficient c to zero. • Gradient stress fields vanish except in y-direction as c approaches infinity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the electro-mechanical buckling analysis of different kinds of smart sandwich shells with functionally graded porous core and piezoelectric sensor–actuator face sheets. Different types of shallow shells with double curvature including convex shells, cylindrical shells and concave shells are analyzed. It is assumed that effective properties of the porous core are functionally graded to vary as a special function of the thickness parameter. The equilibrium equations are established for the doubly-curved smart sandwich shells using the energy method. The stability equations governing the equilibrium position of the smart sandwich shells are obtained as a set of coupled partial differential equations. Closed-form expressions are generated to obtain the critical buckling loads of the smart sandwich shells using the airy stress function. Sandwich shells under different types of electro-mechanical loadings including axial, lateral and hydrostatic pressures are analyzed. These numerical results show the effects of feedback gain, porosity coefficient and piezoelectric layer thickness on the buckling resistance of these smart sandwich structures. [ABSTRACT FROM AUTHOR]

The current work deals with the mechanical buckling behavior of saturated porous toroidal shell segments sandwiched by piezoelectric actuator layers. Material properties of the porous shells with nonuniform distributed porosities are assumed to vary as a specific function of the thickness and porosity parameters. The nonlinear equations of the sandwich toroidal shell having either positive or negative Gaussian curvature are obtained on the basis of Biot's poroelasticity theory. The energy method as the minimum potential energy principle is applied to derive the governing equations of the sandwich shells. The buckling equations of the piezo-porous sandwich shells under various types of mechanical loading are established by employing the adjacent equilibrium criterion. The governing equations as a set of the coupled partial differential equations are solved using an analytical method including the Airy stress function. Closed-form solutions are derived for shallow segments of the sandwich toroidal shell with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loadings. Novel numerical results are presented in this work to show the effects of important factors such as the piezoelectric layer thickness, applied voltage, shell geometrical ratio, and variation of the porosity coefficient on the buckling behavior of these structures. [ABSTRACT FROM AUTHOR]

In the present paper, we have studied the two-dimensional deformation of an elastically orthotropic layer of finite thickness overlying a rough-rigid base due to surface loads. For a two-dimensional approximation, the governing equations are split into two cases: plane strain and anti-plane strain case. We consider plane strain case only. The Airy stress function approach is used to obtain the stresses and displacements in the integral form at any point of the layer. To solve the integrals analytically, the linear combination of exponential terms occurring in the denominator is expanded by binomial expression and then approximated by a finite sum of exponential terms using the least square method. Then, the integrals are evaluated using standard integral tables. The analytical expressions for the displacements are obtained for the normal line loading. Two orthotropic materials, such as Topaz and Barytes, have been considered and graphs showing the displacement field for Topaz, Barytes and the isotropic case have been plotted. [ABSTRACT FROM AUTHOR]

The stress state of a soft interlayer under contact strengthening, when tensile stresses are greater than the yield strength of the interlayer metal and less than the stresses causing a general yield, is considered. The analytical expressions under plain strain and axisymmetric tension are obtained. In the first case, the stresses are determined using the Airy stress function as a corresponding polynomial, in the second case – based on the stress function of the fifth degree, built on the corresponding Legendre polynomial. The stresses satisfy the differential equations of equilibrium and boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we model, classify and investigate the solutions of (normalized) second order ode's with non-constant continuous coefficients. We introduce a generalized frequency function as the solution of a nonlinear integro-differential equation , show its existence and then derive a representation formula for (all) solutions of (normalized) second order ode's with non-constant continuous coefficients. Because this formula specifies the interplay between the coefficients of the ode, the relaxation function ("strongly" decreasing positive function) and the frequency function of the oscillation, it can be applied to design models of dissipative oscillations. As an application, we present and discuss some oscillation models that stop within a finite time period. Via two examples about the Airy function and the Bessel functions, we show that the formula might be useful in analysing certain special functions. Moreover, we demonstrate that a large class of oscillations can be used to design and analyze dissipative waves. In particular, it is easy to model dissipative waves that cause in each point of space an oscillation that stops after a finite time period. Finally, we derive an alternate form of the dissipative wave equation (for a large class of dissipative waves), which is uniquely defined by the sound speed c 0 and the parameter functions in the oscillation equation. [ABSTRACT FROM AUTHOR]

A family of 2D light fields consisting of the product of three Airy functions with linear arguments has been studied theoretically and experimentally. These fields, called three-Airy beams, feature a parameter shift and have a cubic phase and a super-Gaussian circular intensity in the far zone. Transformations of three-Airy beams in the Fresnel zone have been studied using theoretical, numerical, and experimental means. It has been shown that the autofocusing plane of a three-Airy beam is similar to the square root of the shift parameter. We also introduce generalized three-Airy beams containing nine free parameters, and obtain their Fourier transform in a closed form. [ABSTRACT FROM AUTHOR]

In information transfer, the dissipation of a signal is of crucial importance. The feasibility of reconstructing the distorted signal depends on the related permanent loss. Therefore, understanding the quantized dissipative transversal mechanical waves might result in deep insights. In particular, it may be valid on the nanoscale in the case of signal distortion, loss, or even restoration. Based on the description of the damped quantum oscillator, we generalize the canonical quantization procedure for the case of the transversal waves. Then, we deduce the related damped wave equation and the state function. We point out the two possible solutions of the propagating-damping wave equation. One involves the well-known Gaussian spreading solution superposed with the damping oscillation, in which the loss of information is complete. The other is the Airy function solution, which is non-spreading–propagating, so the information loss is only due to oscillation damping. However, the structure of the wave shape remains unchanged for the latter. Consequently, this fact may allow signal reconstruction, resulting in the capability of restoring the lost information. [ABSTRACT FROM AUTHOR]

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫ 0 b x α f (x) Ai (− ω x) d x over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω ≫ 1 . The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [ 0 , 1 ] and [ 1 , b ] . For integrals over the interval [ 0 , 1 ] , we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [ 1 , b ] , we transform the Airy function into the first kind of Bessel function. By applying Cauchy's integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [ 0 , + ∞) , which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. [ABSTRACT FROM AUTHOR]

We consider the Stark operator T = -d²/dx² + x + q (x) on the semiaxis 0 ≤ x < ∞ with the Dirichlet boundary condition at the origin. The asymptotic behavior of the eigenvalues of this operator is studied. By the method of transformation operators, we study the spectral problem. We give a rigorous derivation of the main integral equation for the inverse problem. [ABSTRACT FROM AUTHOR]

In the current research, we delve into the intricate realm of bifurcation analysis for Föppl–von Kármán plates, employing a precise numerical tool. This innovative numerical approach melds the power of spectral discretization with the prowess of a high-order continuation method-based Taylor series development (HODC). It is worth noting that combining the high-order continuation method with such discretization techniques offers an efficient path-following approach, complete with adaptive step lengths, capable of tackling a wide array of nonlinear problems. Despite the extensive applications of nonlinear elasticity, the spectral method remains relatively uncharted territory within this context. However, our deep-rooted understanding and expertise in the field drive us to embrace this method alongside high-order development continuation for bifurcation analysis of Föppl–von Kármán plates. The governing equations governing thin elastic plates experiencing significant elastic deflections manifest as a pair of coupled nonlinear differential equations, famously known as the von Kármán (vK) equations, presented in a strong form with two principal unknowns: deflection (w) and the Airy stress function (F). Leveraging Chebyshev decomposition matrices, we approximate these fourth-order elliptic nonlinear partial differential equations. Subsequently, we harness high-order development continuation techniques to morph these nonlinear systems into linear ones. Our rigorous evaluation and validation of this numerical approach's precision and performance come to fruition through a comprehensive buckling analysis encompassing multiple illustrative examples. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish a connection between Bohr's radius and the analytic and normalized solutions of two differential second-order differential equations, namely y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) . Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F (z) of the above differential equations. We construct several examples by judicious choice of a (z) , b (z) and d (z) . The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. [ABSTRACT FROM AUTHOR]

The magnetotelluric (MT) responses are explicitly solved for a stratiform earth containing multiple transitional layers in which the conductivity varies linearly with depth. In the model under consideration, any one homogeneous layer with constant conductivity or transitional one may be absent in the geometry. The traditional one-dimensional (1D) models with sharp boundaries will be obtained if all the transitional layers are absent in the geometry, while a special 1D model consisting of a sequence of contiguous transitional layers may be obtained if all the homogeneous layers (except the basement layer) are removed from the geometry. The tangential electric and magnetic fields as well as the surface impedance are analytically expressed by Airy functions. The analytical formula is validated in three theoretical examples by comparing with the results from available codes. The apparent resistivity and impedance phase on the surface of three different transitional models are illustrated to analysis the influence of the transitional layers on MT responses. The new formula provides an alternative way to obtain the analytic MT responses for the special layered earth. [ABSTRACT FROM AUTHOR]

The Stark operator with a step-like potential is considered. An explicit form of special solutions to the corresponding Stark equation is found. The scattering problem for the operator L is studied. A formula for expansion in terms of eigenfunctions of a continuous spectrum is obtained. [ABSTRACT FROM AUTHOR]

The goal is to take advantage of the Earth-ionosphere wave guide’s fundamentally two-dimensional wave propagation, utilizing the normalized Airy functions (NAFs) in a complex domain. It is demonstrated that the typical working formula of VLF radio-mode theory may be obtained simply from orthogonality reflections, devoid of the requirement of sophisticated argumentation in the open unit disk. The combination of the expressions is given by considering the symmetry-convex illustration of the NAFs.

Domains with elliptic boundaries are regularly pursued in Elasticity to elucidate the role of circular asymmetry and are associated with many classical closed form solutions. The work under consideration presents a unified semi-analytical approach to solve for stress and displacement field while being subjected to all kinds of plausible boundary conditions on any variant of elliptic geometry i.e. elliptic cylinder to confocal elliptic annulus to elliptic hole in an infinite plane. The generalized representation of Airy stress function in elliptical co-ordinates truncated to finite terms is considered and the associated coefficients are deduced to ensure boundary conditions using collocation technique. The correctness and effectiveness of the method is demonstrated through solution to a variety of problems and its validation via an independent finite element simulation. [ABSTRACT FROM AUTHOR]

The current study investigates the nonlinear vibration characteristic of rotating axially moving conical shell made of shape memory alloy (SMA). For this purpose, the material behavior of SMA is simulated via Boyd-Lagoudas and Brinson models, and three nonlinear governing equations are derived by employing Hamilton principle, Donnell's nonlinear theory assumptions, and SMA constitutive relations. By applying a suitable parametric airy stress function, three nonlinear equations of motion are reduced to one in radial direction, which must be solved with the help of the compatibility equation. By the aid of Jordan conical form and applying the Galerkin method on the equilibrium equation in the radial direction, seven nonlinear nonhomogeneous ODEs are resulted. Then, the set of nonlinear equations is solved using the fourth-order Runge–Kutta method and pseudo-arc length continuation. Furthermore, the bifurcation analysis based on the different parameters especially frequency responses along with the curves of the time histories and phase portraits mention the influence of different phases of the material, axial motion and spinning on the conical shells made of SMA. The results of the present work are validated with available approved data, which shows good agreements. [ABSTRACT FROM AUTHOR]

Thin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflections and their associated Airy stress function, as displayed in the well-known von Kármán equations set. The present study aims at providing explicit expressions for the membrane stresses, the deflections, and the Airy stress function for a general square plate area vertically uniformly loaded to reach large deflection state. This was obtained by using the results of a high-fidelity finite element analysis applied on a lateral loaded simply supported thin square plate, which are then casted to yield approximate Fourier series expressions for the membrane stresses, deflections, and the Airy stress function. The stress map figures provide a good understanding of the critical points on the plate, while the explicit mathematical expressions enabled the calculation of deflections and stresses for the entire plate area. Among other interesting findings, the presence of relatively high tensile and compressive membrane stresses existing near the plate edges was revealed, which might lead to potential failure hazards. The derivatives of the deflections and the Airy stress function enabled the validation of the large deflections von Kármán equations set for the investigated case, and it turned out that the generated expressions for the stresses and the lateral deflection based on a high-fidelity finite element model satisfy the second equation with a good accuracy, while the first one remains to further be investigated. Moreover, using the generated explicit equations, the load influence on the deflections and stresses was also analyzed to yield general novel expressions for the medium and very large deflections states. To generalize the investigated case, the stresses and the deflections were non-dimensionalized so they can be used for any material and plate dimensions. [ABSTRACT FROM AUTHOR]

A new power-exponent-phase vortex beam with nonlinear phase winding has shown flexible control freedom compared with conventional vortex beams. In order to further enrich the modulation freedom and expand the ability of self-healing to meet current application requirements, we conducted a detailed study on the characteristics of the Airy transform of the new power-exponent-phase vortex beam. The influences of the Airy function, the power exponent, and the topological charge on normalized intensity and phase distributions are investigated theoretically and experimentally. More importantly, the self-healing properties of the new power-exponent-phase vortex beam with and without the Airy transform are compared. This shows that the new power-exponent-phase vortex beam with the Airy transform exhibits better self-healing ability when obstructed by obstacles. This study has potential applications in optical trapping and free-space optical communication. [ABSTRACT FROM AUTHOR]

Truss structures composed of members that work exclusively in tension or in compression appear in several problems of science and engineering, e.g., in the study of the resisting mechanisms of masonry structures, as well as in the design of spider web-inspired web structures. This work generalizes previous results on the existence of cable webs that are able to support assigned sets of nodal forces under tension. We extend such a problem to the limit analysis of compression-only "strut nets" subjected to fixed and variable nodal loads. These systems provide discrete element models of masonry bodies, which lie inside the polygon/polyhedron with vertices at the points of application of the given forces ("underlying masonry structures"). It is assumed that fixed nodal forces are combined with variable forces growing proportionally to a scalar multiplier (load multiplier), and that the supporting strut net is subjected to kinematic constraints at given nodal positions. [ABSTRACT FROM AUTHOR]

This paper develops a coupled chemo-mechanical model for stress-assisted diffusion in the framework of micropolar elasticity. The two-way interaction of mechanical and chemical driving forces as well as internal microstructure of the polar media is taken into account. The fundamental governing equations of chemo-mechanics with kinetics driven by diffusion and stress are developed, and mathematical expressions for stress and concentration fields are derived. Using Airy stress functions, a simplified formulation for two-dimensional chemo-elasticity problems under chemical equilibrium is presented. Using a perturbation approach to solve the presented system of equations, closed-form expressions for different field parameters can be obtained. As an illustrative case study, expressions for the stress and solute concentration in an infinite plate with circular hole embedded in a chemical medium have been written. The derived equation and proposed solution approach can be applied to various plane micropolar chemo-elasticity problems for studying the interaction between mechanical and chemical driving forces as well as material length-scale parameters. [ABSTRACT FROM AUTHOR]

A line of contacting hard spheres, placed in a transverse confining potential, buckles under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. This interesting nonlinear problem is enriched by the combined application of both compression and tilt. In a continuous formulation, the profile of transverse sphere displacement is well described by numerical solutions of a second-order differential equation (provided that buckling is not of large amplitude). Here we provide a detailed discussion of these solutions, which are approximated by analytic expressions in terms of Jacobi, Whittaker and Airy functions. The analysis in terms of Whittaker functions yields an exact result for the critical tilt for buckling without compression.

Airy beams based on the Airy function, exhibiting a special curved parabolic trajectory, can effectively alleviate diffraction along its propagation and hold potential in many interesting applications. However, to date, the broadband and high-efficiency generation of non-diffraction Airy beams has remained largely unexplored, especially in the millimeter-wave range that is full of great application potential. In this paper, broadband 1-D and 2-D Airy beam generators utilizing high-efficiency tri-layer metasurfaces operating from 55 to 67 GHz are presented. The metasurfaces consist of elaborately tailored meta-atoms that can independently realize complete amplitude modulation from 1% to 98% and binary phase tuning of 0 and π. The Airy beam generators are engineered and verified by simulations and experiments, which demonstrate the broadband quasi-non-diffraction feature and curved trajectory of the Airy beam. Compared to the 1-D Airy beam generator, the measured quasi-non-diffraction propagation property of the 2-D Airy beam can be maintained by greater than 160 free-space wavelengths across the entire bandwidth. Self-reconstruction property of the 2-D Airy beam is also experimentally validated in the presence of metallic and dielectric obstacles. This work may benefit novel applications of Airy beams in millimeter-wave imaging and detection. [ABSTRACT FROM AUTHOR]

This research examines the uniaxial behavior of core concrete, which is confined by a UHPC (ultra-high-performance concrete) shell, through experimental, numerical, and analytical methods. Various configurations, including different UHPC shell shapes, thicknesses, and core concrete compressive strengths, were tested until failure occurred. Results indicate that the UHPC shell altered the failure modes and enhanced the maximum stress and corresponding strain in uniaxial loading. Additionally, in square and rectangular specimens, debonding between the UHPC and NC (normal concrete) interfaces was observed. Furthermore, we constructed numerical models which integrate the concrete damage plasticity model for NC/UHPC and the coupled adhesive-frictional model implemented in cohesive elements. The model accurately reflects the crack and debonding evolution of the tested specimen. Subsequently, an analytical stress-strain model for uniaxial loading was created based on the experimental results. The confining pressure for square/rectangular specimens was determined using Airy's function. The validity of the analytical model was verified by comparing its predictions with the experimental results. [ABSTRACT FROM AUTHOR]

In this paper, an improved refined shear and normal deformation theory is used in order to investigate the vibration behavior of functionally graded rectangular plates. In this theory, displacements of various points of plate are assumed to be due to in-plane displacements of the middle plane and transverse displacement. Transverse displacement is divided into three parts: bending, shear, and thickness stretching. Using the Airy stress function, corresponding to the compatibility equation, and employing the extended Hamilton's principle, in-plane displacements are omitted from the equations of motions. Thus, the proposed approach uses only three-unknowns in the displacement field. The results of vibration analysis using the proposed approach are in excellent agreement with three-dimensional and quasi-three-dimensional solutions containing a greater number of unknowns to consider the thickness stretching effect. Static and dynamic behavior of wide variety of thin and thick functionally graded plates can be studied using the proposed approach in which not only the number of variables is reduced, but also the contribution of bending, shear, and thickness stretching are completely clarified. [ABSTRACT FROM AUTHOR]

Due to the variable-thickness parallelogram plates in ocean structure are characterized by the presence of strong moment singularity at supported obtuse corners, the wide application has turned the nonlinear mechanical analysis into one of the most important engineering concerns. In the paper, a refined geometrically nonlinear bending model of variable-thickness orthotropic parallelogram plates in curved hull under thermo-mechanical loads is proposed, while influences of linearly or quadratically thickened thickness in symmetrical or unsymmetrical profile on mechanical properties are thoroughly investigated. Three kinds of spatially thermal field in parallelogram domain are formulated on account of the coupled interaction of effects between in-plane distribution and thickness variation corresponding to the Dirichlet, Neumann and Robin thermal boundaries. A novel thickness-dependent Airy stress function is introduced overcoming the failure of traditional Airy stress function in equilibrium of in-plane forces, while the highly coupled and variable-coefficient nonlinear governing partial differential equations are firstly derived. The homotopy-based wavelet method is adopted to investigate the nonlinear thermo-elastic bending behaviors, while convergent process is verified and precision of obtained series solutions has been validated in excellent agreement with published results. The significant conclusion can be made that large-deflection nonlinear bending of such plates can be simplified with little discrepancy by omitting terms involving the derivatives of thickness variation in compatibility equation of deformation, which is generalized to the thermo-mechanical bending and greatly simplifies the analyzing procedures. • Nonlinear thermomechanical model of variable-thickness parallelogram plate is refined. • An Airy function is introduced overcoming the failure in equilibrium of in-plane forces. • Nonlinear thermomechanical bending is simplified by omitting derivatives of thickness. • Accurate bending solutions are obtained in excellent agreement with published results. [ABSTRACT FROM AUTHOR]

An unconstrained global ocean simulation for 2020 supports past observations of diurnal internal tides by acoustic tomography during the 1991–1992 Acoustic Mid‐Ocean Dynamics Experiment in the Western North Atlantic. Explicitly representing the tides, the simulation reproduces the functional form and resonant state of K1 and O1 internal‐tide standing waves, while providing a more realistic physical picture of them. The tomographic data were used to predict the tides in 2020. Not surprisingly, the characteristics of the barotropic and internal tides of the unconstrained simulation deviate from observations. The simulated barotropic tidal currents have excessive, irregular amplitude and lead the acoustic tidal predictions by about 2 hr. While internal‐tide phase coherence is apparent, the simulated internal‐tide variations were irregular in amplitude and phase, unlike the observations. The tomographic tidal measurements therefore provide a quantitative benchmark for improved model representation of tides, internal tides, and dissipation. Plain Language Summary: Oceanic internal waves, supported by stratification and the local rate of rotation of the earth (Coriolis frequency), are ubiquitous, generated by winds, tides, and uneven seafloor. Over the past 30 years, internal‐tide waves of the gravest vertical mode (half‐cosine vertically, 150–500‐km wavelength) were discovered to travel coherently across ocean basins, observed by their signals in temperature or sea‐surface height. Internal tides in a high‐resolution ocean model were compared to acoustical observations made in the western North Atlantic in 1991, with encouraging agreement. Internal waves are refracted back toward the equator at the latitude where their frequency equals twice the earth's local rotation. In the model, internal tides with 24‐hr periods (diurnal) are found to be standing waves with Airy function form, trapped between Puerto Rico and their turning latitude at 30°N, confirming an earlier interpretation of the acoustical observations. The acoustical tidal observations are an excellent benchmark for the development and adjustment of ocean models. Key Points: Past interpretations that signals in acoustic data were caused by standing‐wave, diurnal internal tides are supported by an ocean modelThe historical acoustic data can be used to predict model internal‐tide variability much like ordinary tide predictionThe acoustical observations provide a quantitative benchmark for improved model representation of tides, internal tides, and dissipation [ABSTRACT FROM AUTHOR]

The one-dimensional Schrödinger operator with potential growing at infinity and with a semiclassical small parameter is considered. We obtain estimates via powers of the small parameter for the remainder in the expansion of smooth sufficiently rapidly decaying functions in the exact and asymptotic eigenfunctions. For the asymptotic eigenfunctions, we use a global representation in the form of an Airy function. [ABSTRACT FROM AUTHOR]

Airy and Bessel functions are one of the most popular and important special functions in various branches of physics, mathematics, and engineering. An observation to their behavior for the real argument suggest that they are related. This relation was studied earlier, but were accompanied by a number of assumptions, approximations, and sometimes even misconceptions. This motivated us to develop a fresh and transparent method to establish these relations. As the continuation of our study of the two papers published in resonance already, here we have used the general asymptotic series and the convergent series of these functions and thereby developed two new methods which throw light on the subtle interrelationships between these functions. Numerical evidences of our claims are provided for better clarity and understanding. [ABSTRACT FROM AUTHOR]

The quantum version of the free fall problem is a topic often skipped in undergraduate quantum mechanics courses, because its discussion usually requires wavepackets built on the Airy functions—a difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and for general many-body systems by making use of a gauge transformation that corresponds to a change of reference frame from the laboratory frame to the one comoving with the falling system. Using this approach, the quantum mechanics problem of a particle in an external gravitational potential reduces to a much simpler one where there is no longer any gravitational potential in the Schrödinger equation. It is instructive to see that the same procedure can be used for many-body systems subjected to an external gravitational potential and a two-body interparticle potential that is a function of the distance between the particles. This topic provides a helpful and pedagogical example of a quantum many-body system whose dynamics can be analytically described in simple terms. [ABSTRACT FROM AUTHOR]

The notion of k-symbol special functions has recently been introduced. This new concept offers many interesting geometric properties for these special functions including logarithmic convexity. The aim of the present paper is to exploit essentially two-dimensional wave propagation in the earth-ionosphere wave path using k-symbol Airy functions (KAFs) in the open unit disk. It is shown that the standard wave-mode working formula may be determined by orthogonality considerations without the use of intricate justifications of the complex plane. By taking into account the symmetry-convex depiction of the KAFs, the formula combination is derived. [ABSTRACT FROM AUTHOR]

A class of problems of wave propagation in waveguides consisting of one or several layers that are characterized by linear variation of the squared refractive index along the normal to the interfaces between them is considered in this paper. In various problems arising in practical applications, it is necessary to efficiently solve the dispersion relations for such waveguides in order to compute horizontal wavenumbers for different frequencies. Such relations are transcendental equations written in terms of Airy functions, and their numerical solutions may require significant computational effort. A procedure that allows one to reduce a dispersion relation to an ordinary differential equation written in terms of elementary functions exclusively is proposed. The proposed technique is illustrated on two cases of waveguides with both compact and non-compact cross-sections. The developed reduction method can be used in applications such as geoacoustic inversion. [ABSTRACT FROM AUTHOR]

The nonlinear vibration characteristics of rotating axially moving conical shells are investigated in the current paper. The nonlinear equations of motion and strain compatibility equation based on Donnell's nonlinear shell theory are obtained. Three nonlinear equations of motion are reduced to a radial equation by applying the appropriate Airy stress function, forming a set of equations with the compatibility equation. The compatibility equation is solved by employing the seven degrees of freedom with respect to the system's flexural mode shape. By substituting the flexural mode shape into the equation of motion and applying the Galerkin method, seven nonlinear coupled nonhomogeneous ODEs are achieved, then the set of equations is transformed into the normal form where it has been solved by the numerical method. The effects of the axial and rotational velocity on bifurcation diagrams, frequency response curves, time history, and the phase portraits of the system are discussed. The results of the present paper are validated against available data, and good agreements are achieved. [ABSTRACT FROM AUTHOR]

The theoretical study of the effect of electron beam parameters, in particular, the emittance and its asymmetry on the radiation from relativistic electrons in undulators is conducted both analytically and numerically. The reasons for the odd and even harmonic generation and radiation are explored. The difference in the underlying physical reasons for the spontaneous and stimulated radiation of harmonics in free electron lasers (FELs) is elucidated. The generalized forms of the special functions of the Bessel and Airy type are employed to account analytically for the off-axis and angular effects together with the effect of the beam energy spread. A comparative analysis of the radiation spectra for undulators with different beams is performed. The examples of the radiation at SPARC and LEUTL are given. The effect of the asymmetry of the beam on the radiation properties is analyzed. The alternative theoretical approaches of other authors are also employed for the analytical calculation of the harmonic powers in FELs. The results are compared with existing experimental data. [ABSTRACT FROM AUTHOR]

Model buoyancy frequency distribution and the Wentzel–Kramers–Brillouin method are used to obtain an asymptotic solution to a problem of constructing solutions that describe internal gravity waves in a stratified medium with a background shear flow slowly varying in depth. Dispersion relation asymptotics are expressed in terms of the Airy functions. Asymptotics for various model distributions of background shear flows are used to obtain analytical representations of dispersion relations and eigenfunctions. Exact and asymptotic results are compared for various distributions of background shear flows and generation modes typical of a real ocean. [ABSTRACT FROM AUTHOR]

The stress state from an eccentric ring made of an elastic material symmetrically loaded on the outer boundary by concentrated forces is deduced. The analytical results are obtained using the Airy stress function expressed in bipolar coordinates. The elastic potential corresponding to the same loading but for a compact disk is first written in bipolar coordinates, then expanded in Fourier series, and after that, an auxiliary potential of a convenient form is added to it in order to impose boundary conditions. Since the inner boundary is unloaded, boundary conditions may be applied directly to the total potential. A special focus is on the number of terms from Fourier expansion of the potential in bipolar coordinates corresponding to the compact disk as this number influences the sudden increase if the coefficients from the final form of the total potential. Theoretical results are validated both by using finite element software and experimentally through the photoelastic method, for which a device for sample loading was designed and constructed. Isochromatic fields were considered for the photoelastic method. Six loading cases for two different geometries of the ring were studied. For all the analysed cases, an excellent agreement between the analytical, numerical and experimental results was achieved. Finally, for all the situations considered, the stress concentration effect of the inner hole was analytically determined. It should be mentioned that as the eccentricity of the inner hole decreases, the integrals from the relations of the total elastic potential present a diminishing convergence in the vicinity of the inner boundary. [ABSTRACT FROM AUTHOR]

The ordinary circular and hyperbolic functions of variable x are generalized using a parameter m. so that: (i) they correspond to zero value. m = 0, of the parameter: (ii) for m= 1 are obtained the Airy functions. The generalized circular and hyperbolic cosine and sine are integral functions specified by Maciaurin series valid in the finite complex x-plane of the variable for all complex valties of the parameter m excluding negative integers. Differentiation formulas are obtained for complex variable x and parameter m. and some inequalities are also obtained for real x and m. The extension to the generalized circular and hyperbolic secant. cosecant. tangent and cotangent is made in the usual way. [ABSTRACT FROM AUTHOR]

The Schrödinger equation under the application of a position-dependent mass (PDM) with an exponential form is presented. Several physical models are carried out by choosing different external potential fields including the free field or a confined hard-all potential, the linear potential plus an attractive centrifugal-like term and harmonic oscillator. The eigenfunction of the first case is given by a Bessel function. The calculations of the second case are found to have the Airy function and we are able to get a general form of energy levels based on the zeros of Airy function of the first kind. The last case is found that the eigenfunctions are given by the popular associated Laguerre function. To provide a better physical insight into the solutions, some figures are plotted graphically.

In this article, a 2-D Airy beam with self-bending and non-diffraction characteristics is generated and verified at microwave frequencies by using a metasurface composed of a single-layer of strip resonator. By modifying the rotation angle and structural parameters of the unit cells, the phase and amplitude of the transmission coefficient can be controlled, respectively. In addition, the deflection of the main lobe of the generated 2-D Airy beam can be manipulated by modifying the attenuation coefficients in the 2-D Airy function. The novel excitation and deflection control methods designed in this article propose great application potential in efficient wireless energy transmission at microwave frequencies, particle handling, and integrated beam control systems. [ABSTRACT FROM AUTHOR]

The paper deals with constructive semiclassical asymptotics of eigenfunctions of the Dirac operator describing graphene in a constant magnetic field. Two cases are considered: (a) a strong magnetic field, and (b) a radially symmetric electric field and small mass. Using standard semiclassical methods, we reduce the problem to a pencil of magnetic Schrödinger operators with a correction term. In both cases, the classical system defined by the principal symbol turns out to be integrable, but the correction term destroys the integrability. In case (a), where the correction removes the frequency degeneracy (resonance), we use the averaging method to reduce the problem to an integrable system not only in the leading approximation but also with the correction taken into account. The tori of the resulting system generate a series of asymptotic eigenfunctions of the original operator. In case (b), the system defined by the principal symbol is nondegenerate. Fixing an invariant torus with Diophantine frequencies for this system and looking for a solution of the transport equation for it, we obtain a series of asymptotic eigenfunctions that are in one-to-one correspondence with tori that satisfy the Bohr–Sommerfeld quantization rule and lie in a small neighborhood of the chosen Diophantine torus. In both cases, the construction of the asymptotic eigenfunctions is based on the global representation of the Maslov canonical operator on a two-dimensional torus projected onto the configuration space into an annular domain with two simple caustics via the Airy function and its derivative. A numerical implementation of our formulas in examples shows their efficiency. [ABSTRACT FROM AUTHOR]

We explore an application of the Physics-Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with a minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching the feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs. [ABSTRACT FROM AUTHOR]

This paper presents a new four-node quadrilateral flat shell element, named QFSUQ, for analysis of shell structures. The element is formed by assemblage of a new membrane element and a plate-bending element. The membrane component is an unsymmetric quadrilateral element with drilling degrees of freedom. The trial functions of the membrane element are determined using the element stress fields formulated based on the analytical solutions of the Airy stress function in global Cartesian coordinate system. The corresponding test functions are obtained through the four-node isoparametric-based displacement fields which are enhanced by drilling rotations. The bending component is based on the Hellinger–Reissner variational principle for analysis of Reissner–Mindlin plates. To validate the performance of the proposed element several numerical benchmark problems are employed and the obtained results are compared with other shell elements. The results prove that the QFSUQ element preserves the advantages of the parent element formulation namely explicit stiffness matrix, free of membrane locking, shear locking and singularity problems and is also appropriate in analysis of shell structures with complex geometry, loading and boundary conditions. [ABSTRACT FROM AUTHOR]

Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the bounds is illustrated numerically with an application to Bessel functions of large order. [ABSTRACT FROM AUTHOR]