Inhomogeneous Beam Research Articles

Uniformly Loaded Logarithmic Beam Mode with Spatially Varying Flexural Rigidity

AbstractThis analysis explores natural leading modes represented by logarithmic functions, achieved by imposing four boundary constraints at the ends of an elastic inhomogeneous beam. The beam possessing constant material inertia, is assumed to be uniformly loaded, and is composed of material with variable stiffness. It is sought analytical expressions for beam deflections in terms of logarithmic functions. Our findings demonstrate that such formulae can be derived for a beam under axially uniform load and with spatially distributed flexural rigidity. Subsequently, the beam shapes and material properties for four specific scenarios are identified: free-free logarithmic beam, cantilevered logarithmic beam, simply-supported logarithmic beam, and simply-supported sliding logarithmic beam. Explicit logarithmic beam responses, governed by a limited number of shape parameters, are illustrated graphically using normalized deflections with respect to the maximum deflection. Highly deflected elastic logarithmic modes emerge as a consequence of high flexural rigidity influenced by the uniformly applied transverse load. These elucidated logarithmic beam modes offer potential practical applications in the structural design of functionally graded materials. They also serve as valuable testing platforms for numerical techniques employed in the analysis of more complex beam problems.

Modal characteristics of functionally graded porous Timoshenko beams with variable cross-sections

The study focuses on the free vibration analysis of beams composed of functionally graded porous materials and characterized by a variable cross-section along their length. A broad range of beams is examined encompassing various tapered configurations, porosity profiles, and porosity content. The equations of motion are derived using Hamilton’s principle within the framework of Timoshenko beam theory. These equations are solved semi-analytically using the differential transform method, which has been adapted to incorporate various boundary conditions such as clamped–clamped, clamped–free, clamped–pinned, and pinned–pinned constraints within a general formulation of the beam eigenvalue problem. To validate the proposed solution technique, computed natural frequencies are compared with existing literature results for tapered inhomogeneous beams and uniform porous beams. Notably, new results are obtained for tapered porous beams. In this regard, a comprehensive parametric study explores the influence of various factors on the natural frequencies and mode shapes of functionally graded porous beams with variable cross-sections. These factors include the type of porosity profiles, a range of porosity parameters, cross-section taper ratios, and specific boundary conditions. The findings deepen our understanding of the modal characteristics of functionally graded porous beams, providing valuable guidance for engineering design and structural optimization in relevant applications. Additionally, they may serve as benchmarks for other researchers.

Dynamic diffraction of thermal neutrons in crystals with continuous deformation field

The purpose of the work is to consider the basics of the formation of a diffraction wave field of thermal neutrons in the case of Laue transmission geometry of a limited beam of neutrons in the crystal, taking into account the peculiarities of the interaction of thermal neutrons with a continuously and slowly changing deformation field of the crystal net. The conditions of both diffraction blurring and diffraction contraction (focusing) up to the practical restoration of the original shape of the packet will be studied.An integral form for determining the quasi-amplitudes of diffracted waves within the framework of the original spatially inhomogeneous beams is formulated, based on the constructed functions of the influence of a point source of thermal neutron waves for crystals with a continues deformation field. It has been shown that neutron focusing is effective, which is revealed by a smaller half-width of the focal spot, an increase in the resolving power of the energy spectrum and focusing efficiency.

Multilayered Inhomogeneous Beams under Torsion and Bending: An Analytical Study of Energy Dissipation

The present theoretical paper is devoted to the problem of energy dissipation in multilayered inhomogeneous beam structures of visoelastic behaviour. In particular, our attention is concentrated on studying the energy dissipation for the case of a beam that is loaded simultaneously in torsion and bending. Both torsion and bending moments which are applied on the beam change with time so that the angles of twist and rotation of the beam end vary continuously with time. The viscoelastic mechanical behaviour of the beam under torsion is treated by a viscoelastic model that has two linear springs and a linear dashpot. The model is under time-dependent shear stress. The viscoelastic behaviour of the beam under bending is modelled in a similar way. An analytical approach of deriving the energy dissipation is applied. The approach treats a multilayered beam that has an arbitrary number of layers of different thicknesses and material properties. The layers are longitudinally inhomogeneous. An investigation of the energy dissipation is carried-out to clarify the effect of the parameters of the time-dependent bending and torsion moments, the material inhomogeneity and the beam size on the energy dissipation.

Indeterminate Continuously Inhomogeneous Beam under End Rotation: A Longitudinal Fracture Study

The present paper describes a theoretical analysis of the effect of the end rotation of a statically indeterminate beam structure on the behaviour of a longitudinal crack. The beam is made of a material that is continuously inhomogeneous along the beam thickness. Besides, the material has non-linear elastic behaviour. The left end of the beam is supported with an axial double rod, while the right end is rigidly fixed. There is no external mechanical loading on the beam. Thus, the only influence on the beam is the rotation of the beam left end. Equations for determination the curvatures and the coordinates of the neutral axes are worked out and solved together with equations for resolving the static indeterminacy. The response of the longitudinal crack to the rotation of the beam left end is studied by deriving the strain energy release rate. For this purpose the balance of the energy in the beam is analyzed. The strain energy release rate is verified by applying the integral, J. A parametric analysis is performed to evaluate the influence of the beam end rotation and other factors on the longitudinal fracture.

Interaction of inhomogeneous warm electron beam with collisional cold plasma

The linear interaction of a warm non-homogeneous electron beam (EB) with a non-homogeneous, unmagnetized, collisional cold plasma is investigated. The inhomogeneity and warmness of beam which leads to electric field amplification have no effect on the dielectric of the unmagnetized plasma. The study revealed that the dielectric constant remains unaffected by the warmness of the beam whereas the amplification of the electric field is affected by the warmness of the beam.

Multilayered inhomogeneous beam under relaxation: an analytical delamination investigation

Delamination in a multilayered inhomogeneous viscoelastic beam structure under relaxation is analyzed. The moduli of elasticity of beam layers change with time. A linear viscoelastic model with two springs and a dashpot is used for describing the relaxation behaviour. The response of the model is obtained by considering the equilibrium of its components. The material properties vary continuously along the width of layers. Besides, the modulus of elasticity of one of the springs changes with time. Time-dependent strain energy release rate (TDSERR) for the delamination is found by using the time-dependent strain energy (TDSE) in the beam. The TDSERR is found also by analyzing the compliance. The results are identical. The influence of different factors and parameters on the delamination is evaluated.

Fracture of inhomogeneous beam with three parallel lengthwise cracks

A lengthwise fracture analysis of an inhomogeneous beam with three parallel cracks is developed. The cracks are located symmetrically with respect to the mid-span. The loading consists of two axial forces applied at the ends of the beam. A vertical notch is cut-out in the mid-span so as the upper three crack arms are free of stresses. Only half of the beam is analyzed due to the symmetry. The cracks are located arbitrary along the beam height. Therefore, the heights of the cross-sections of crack arms are different. The beam exhibits continuous material inhomogeneity in the height direction. The material has non-linear elastic behaviour. Three solutions to the strain energy release rate are derived. For this purpose, the complementary strain energy in the beam is differentiated with respect to the areas of the three cracks. The J-integral approach is applied in order to verify the solutions to the strain energy release rate. The influence of the locations of the three cracks along the beam height on the lengthwise fracture behaviour is studied in detail. The influence of the material inhomogeneity on the lengthwise fracture behaviour of the beam is also analyzed.

Bending and torsion induced stresses in cylindrically orthotropic and inhomogeneous timber beams

The structural design of timber beams subject to bending often relies on the application of the simple Euler–Bernoulli beam theory. However, the simplistic formulas for stress calculations overlook the inherent characteristics of the wood material and the true distribution of the annual rings within the cross-sectional area. This paper introduces a method for determining all six stress components for a cantilever-type beam that is subjected to concentrated end loads. The method considers an inhomogeneous cross-section and employs cylindrically orthotropic material properties. Notably, this approach does not necessitate prior knowledge of elastic and shear centres. It is founded on the formulation of a displacement field incorporating unknown in-plane distortion and warping functions. These are then solved through a straightforward finite element procedure. The efficacy of the method is validated by a series of numerical examples that align with analytical results. Furthermore, a benchmark example for cylindrically orthotropic cross-sections is proposed. As a practical demonstration, an analysis is performed on a real sawn timber cross-section with material parameters representative of Norway spruce. The findings reveal significant disparities in the maximum stresses when compared to conventional engineering approaches. In this specific instance, the maximum longitudinal normal stress resulting from bending is approximately 20 % higher than the outcomes of typical engineering methods. This emphasizes the critical role played by the actual distribution of annual rings across the specific beam cross-section.

Inhomogeneous Beam Brightness Intensity Converter (IBBIC) as support software for cathodoluminescence microscopy studies and images pre-processing

Inhomogeneous Beam Brightness Intensity Converter (IBBIC) as support software for cathodoluminescence microscopy studies and images pre-processing

Light-induced enhanced torque on double-V-type quantum emitters via quantum interference in spontaneous emission

We consider interaction of a pair of counterpropagating spatially inhomogeneous weak vortex beams with a quantum emitter that has a double-V level scheme and closely situated upper levels. We show that in such a situation a quantized torque is exerted on the emitter, which is directly proportional to the topological charge of the vortices and is strongly influenced and even enhanced by the quantum interference of spontaneous emission from the doublet. Depending on the particular initial states of the emitter, absorption, optical transparency, or lasing without inversion can be realized. The interference in spontaneous emission can be modified when the double-V emitter is situated above the surface of a thick slab of Bi2Te3 and the vortex beams propagate parallel to the surface. The light-induced torque rotates the emitters generating a persistent toroidal current flow above the bismuth-chalcogenide surface, whose intensity is remotely controlled by the distance from the surface.

Исследование коэффициентных обратных задач с учетом реологии для функционально-градиентных стержней

The problems of determining the variable rheological properties of functionally gradient beams from some additional information about displacements are considered. Vibrations of a cantilevered inhomogeneous viscoelastic beam are studied in the framework of two models - Euler-Bernoulli and Timoshenko. The oscillations are caused by a moment concentrated at the end of the beam, oscillating in time. Within the framework of the concept of complex modules, operator relations linking the given and desired functions are obtained. Functions reflecting the laws of change of long-term and instantaneous modules have been restored. The results of computational experiments on the restoration of functions of various types are presented.

Lengthwise Fracture Analysis of Inhomogeneous Viscoelastic Cantilever Beam Subjected to Sinusoidal Strains

This paper describes a research of lengthwise fracture in a viscoelastic inhomogeneous cantilever beam under strain that is a sinusoidal function of time. The beam mechanical behavior is investigated by a model having two linear springs and a linear dashpot. The beam material is continuously inhomogeneous along thickness. Therefore, the modules of elasticity of the springs and the coefficient of viscosity of the dashpot vary smoothly in the thickness direction. The compliance method is applied to derive the strain energy release rate (SERR) for the lengthwise crack in the beam structure. The integral J is applied for verification. The stress-strain-time dependence of the viscoelastic model is used for describing the behavior of the beam when obtaining solutions of the SERR and the J-integral. Solutions are derived for both positive and negative rotation angle of the lower crack arm end (when the angle is positive, the upper crack arm is load free, while at negative angle both crack arms are loaded). The effects of various factors including the sign of the angle of rotation on the SERR are analyzed.

Harnessing optical vortices to control current flow via quantized torque in coherently prepared multilevel atoms

This study delves into the interaction between coherently prepared trapped atoms and optical vortices, focusing on the intriguing phenomenon of light-induced torque enhancement and the resultant persistent current flow. Beginning with a three-level atom ensemble in the Λ configuration, we initiate a coherent superposition of lower levels to establish a coherent population trapping medium. The atoms interact with a pair of optical vortices, producing a quantized torque on each trapped atom that is directly proportional to the topological charge of the inhomogeneous coupling beams. This torque imparts a rotation to the entire ensemble, thereby generating a striking ring-shaped atomic current flow. With careful manipulation of the initial internal state of the atoms and the strength of the vortex pair, we can control the magnitude of the torque. Extending the model to include more complex level schemes with additional laser beams offers even greater potential for enhancing light-induced torque and refining our control over the current flow.

Slipping instability of an inhomogeneous relativistic electron beam

The charged particle beams, such as electrons, ions, and plasma compression flow, have received considerable attention due to their applications in science and technology; therefore, studying the stability of these beams is of particular importance. Here, we examine theoretically the stability properties of a cold relativistic electron beam with a transverse velocity shear and non-uniform density profile. We consider a plane-parallel beam propagating along an external magnetic field and evaluate its macroscopic equilibrium state. We derive the dispersion relation of the slipping instability based on the linear electrodynamics of an inhomogeneous plasma and kinetic theory. In this model, the oscillation spectrum and the growth rate are derived by using the eikonal equation and the quasi-classical quantization rule. A linear velocity shear and a non-linear density gradient are assumed. Furthermore, we analyze numerically the dispersion relation of the slipping instability. The impacts of the inhomogeneity parameter and the relativistic factor on the properties of the slipping instability are discussed.

Axial–torsional coupled static behavior of inhomogeneous pretwisted cantilever beams

Axial–torsional coupled static behavior of inhomogeneous pretwisted cantilever beams

Asymptotically Accurate Analytical Solution for Timoshenko-Like Deformation of Functionally Graded Beams

Abstract A closed-form analytical solution is developed for a planar inhomogeneous beam subjected to transverse loading, using a variational asymptotic method (VAM). The VAM decouples the problem into a cross-sectional and an along-the-length analysis, leading to a set of ordinary differential equations. These equations along with associated boundary conditions have been solved to obtain the closed-form analytical solutions. Three distinct gradation models have been used to validate the present formulation against 3D finite element analysis and few prominent results from the literature. Excellent agreement has been obtained for all the test cases. Key contributions of the present work are (a) the solutions have been obtained without any ad hoc and a priori assumptions and (b) the ordered warping solutions result in Euler–Bernoulli type deformation in the zeroth-order, whereas the higher-order solutions provide novel closed-form expressions for transverse shear strain and stress. Finally, the effect of inhomogeneity on various field variables has been analyzed and discussed.

Towards to solution of the fractional Takagi–Taupin equations. The Green function method

Developing the comprehensive theory of the X-ray diffraction by distorted crystals remains to be topical of the mathematical physics. Up to now, the X-ray diffraction theory grounded on the Takagi–Taupin equations with the first-order partial derivatives over the two coordinates within the X-ray scattering plane. In the work, the theoretical approach based on the first-order fractional Takagi–Taupin equations with the ‘quasi-time variable’ of the order $$\alpha \in (0,1]$$ along the crystal depth has been suggested and the corresponding X-ray Cauchy issue is formulated. Accordingly, using the Green function method in the scope of the Cauchy issue, the fractional Takagi–Taupin equations in the integral form have been derived. In the case of the inhomogeneous incident X-ray beam, the solution of the Cauchy issue of the X-ray diffraction by perfect crystal has been obtained and compared with the corresponding one based on the solution of the conventional Takagi–Taupin equations, $$\alpha =1.$$ In turn, notice that the value of order $$\alpha $$ may be adjusted from the experimental X-ray diffraction data.

Polychromatic atom optics for atom interferometry

Coherent manipulation of atoms with atom-optic light pulses is central to atom interferometry. Achieving high pulse efficiency is essential for enhancing fringe contrast and sensitivity, in particular for large-momentum transfer interferometers which use an increased number of pulses. We perform an investigation of optimizing the frequency domain of pulses by using tailored polychromatic light fields, and demonstrate the possibility to deliver high-efficiency and resilient atom-optic pulses even in the situation of inhomogeneous atomic clouds and laser beams. We find that this approach is able to operate over long interrogation times despite spontaneous emission and to achieve experimentally relevant pulse efficiencies for clouds up to 100~mu text{K}. This overcomes some of the most stringent barriers for large-momentum transfer and has the potential to reduce the complexity of atom interferometers. We show that polychromatic light pulses could enhance single-photon-based large-momentum transfer atom interferometry—achieving 850,hbar k of momentum splitting with experimentally accessible parameters, which represents a significant improvement over the state-of-the art. The benefits of the method extend beyond atom interferometry and could enable groundbreaking advances in quantum state manipulation.

Physics-informed neural networks for nonlinear bending of 3D functionally graded beam

This paper proposes a framework for physics-informed neural networks (PINNs) in the nonlinear bending of 3D functionally graded (FG) beams. Utilizing the underlying physical rules governing a 3D FG porous beam resting on a Winkler-Pasternak foundation and motivated by the advancements in the research area of machine learning, this paper develops a PINN framework to predict the nonlinear bending of the beam system. PINNs need much less training data and can achieve high accuracy using a more straightforward network. The powerful tool presented in this work is general enough to handle any class of PDEs. We also take advantage of the recently developed deep learning platform TensorFlow with the company of DeepXDE library to design our network. In this study, the PINN framework takes information from the governing equations and the data from boundary conditions. We developed a mathematical model using the Euler-Bernoulli beam theory and inhomogeneous beam model and validated the results with those extracted from the finite difference method. Furthermore, PINN is presented to accurately predict the nonlinear bending of the system up to 37 times faster than the numerical method.