Inhomogeneous Beam Research Articles

Triple longitudinal fracture of inhomogeneous beam under four-point bending an analytical study by using a non-linear elastic material model

The present paper is devoted to fracture analysis of an inhomogeneous non-linear elastic beam configuration with three parallel longitudinal cracks. The material is continuously inhomogeneous along the beam height. The beam is subjected to four-point bending. The longitudinal fracture behavior is studied by applying the J-integral approach. Solutions to the J-integral are obtained for the three cracks. For this purpose, the curvature and the coordinates of neutral axes of the crack arms are determined by using the equations for equilibrium of the elementary forces in the cross-sections of different portions of the beam. The solutions to the J-integral are valid for arbitrary locations of the cracks along the beam height. Thus, the solutions are very useful for evaluating of the effects of locations of cracks on the fracture. The longitudinal fracture behavior is studied also in terms of the strain energy release rate in order to verify the solutions to the J-integral. Solutions to the strain energy release rate are derived for the three cracks by considering the complementary strain energy in the beam. The solutions to the J-integral are applied to evaluate the effects of locations of the cracks, the material inhomogeneity, the sizes of the beam cross-section, and the coordinates of the applications points of the external forces on the longitudinal fracture behavior of the beam.

On-demand multiplexed vortex beams for terahertz polarization detection based on metasurfaces.

The manipulation of polarization states is crucial for tailoring light-matter interactions and has great applications in fundamental science. Nevertheless, conventional polarization measurement approaches are extremely challenging to determine the polarization state of incident terahertz (THz) beams. The combination of metasurfaces and inhomogeneous vector vortex beams (VVBs) provides a new solution for integrated polarization-related functional devices. Herein, a general design strategy for spin-multiplexing all-silicon metasurfaces is presented and demonstrated in THz polarization detection. The employment of basic building blocks with a high aspect ratio (AR) imparts a greater degree of freedom for generating vector beams, and those basic blocks are subsequently utilized to explore the visualized polarization state. With the assistance of a THz near-field scanning system, we evaluate the capability of reconstructing the incident polarization state from the longitudinal polarization component multiplexed by vortex beams with tight focusing characteristics. Not only that, we also utilize the polarization with dynamically varying behavior as the illumination method to elucidate the evolution trend of the polarization state under a single snapshot and establish a visualized parametric model. This work paves the way to realize ultra-compact THz polarization detection-related devices for future applications in remote sensing, high-resolution imaging, and communications.

Fracture study of inhomogeneous viscoelastic beam with longitudinal cracks under axial force and bending moment

Fracture study of inhomogeneous viscoelastic beam with longitudinal cracks under axial force and bending moment

Application of a non-linear viscoelastic model with material properties depending on strains in lenghwise fracture analyses of inhomogeneous beams

Application of a non-linear viscoelastic model with material properties depending on strains in lenghwise fracture analyses of inhomogeneous beams

Градиентная модель изгиба неоднородной пьезоэлектрической балки

The problem of bending a horizontally inhomogeneous piezoelectric cantilever beam is studied taking into account scale effects. The lower and upper surfaces of the beam are electroplated. Three types of loading are considered: load uniformly distributed along the length; transverse force at the other end of the beam; supply of electric potential to the upper electrode. The beam bending is modeled based on the Euler-Bernoulli hypotheses and the quadratic distribution of the electric potential. Based on the application of the variational principle of gradient electroelasticity, a system of differential equations of bending and electrostatics is obtained, as well as an extended range of boundary conditions. To find the bending moment and deflection of the middle line of a homogeneous beam, exact analytical expressions are obtained. In the case of an inhomogeneous beam at large values of the scale parameter, the solution is based on the shooting method. On specific examples, the calculations of moments and deflection are carried out, both in the case of a homogeneous and inhomogeneous beam. The influence of the mechanical and electrostatic gradient parameters, the electromechanical coupling coefficient, and the inhomogeneity parameter on the distribution of deflections has been clarified.

Utilization of viscoelatic models with non-linear springs and dashpots in delamination study of multilayered beams

This paper analyzes delamination of multilayered inhomogeneous beam structure by utilization a non-linear viscoelastic mechanical model. The non-linear time-dependent response is treated by a non-linear spring and a non-linear dashpot connected in series to a system of two linear springs and two linear dashpots. The model is under stress that is a linear function of time. The constitutive law of the model representing a non-linear dependence between stress, strain and time is derived. The main goal of the paper is to obtain a solution of the strain energy release rate for the delamination in the multilayered inhomogeneous beam by applying the non-linear viscoelastic model. Solutions are derived by using the complementary strain energy and by analyzing the balance of the energy with taking into account the non-linear viscoelastic behaviour. For this purpose, the constitutive law of the non-linear viscoelastic mechanical model is used. The solutions are applied to obtain results for multilayered beams with non-linear variation of material properties in longitudinal direction. The influence of different parameters on the time-dependent strain energy release rate is assessed. The study indicates the effectiveness of the viscoelastic mechanical models with non-linear springs and dashpots in time-dependent delamination analyses of multilayered inhomogeneous beam structures.

Precise integration solutions for the static and dynamic responses of axially graded solid beams

By virtue of the scaled boundary finite element method (SBFEM), the precise integration solutions (PIS) to deflections, stresses and transverse vibration parameters of the functionally graded solid beams are provided. It is worth to note that the mechanical parameters of the graded beams are altered across the x-axis according to power-law, exponential, trigonometric or other arbitrary theories. In SBFEM, the in-homogeneous beam is simplified into the one-dimensional model and just two high-order linear spectral elements are exploited to discretize the axially non-homogeneous beam. Nevertheless, the high accuracy can be ensured. It is essential to remind that the displacement-based beam spectral element is characterized by owning only two degrees of freedom. Additionally, the changing patterns of the displacement and stress field through the transverse direction can be expressed as an analytical matrix exponent. Aided by the two-dimensional fundamental equations of elastic materials and the employed scaled boundary coordinate, the governing equation of the axially graded solid beam is derived in the form of second order ordinary differential equation. Furthermore, the stiffness matrix and PIS of displacements and stresses are acquired from evaluating the governing equation by means of the dual vector technology and precise integration methodology (PIM). Applying the kinetic energy theory constructs the mass matrix of the axially heterogeneous beam. Afterwards the PIS to natural frequencies are obtained according to solving the eigenvalue equation. Finally, numerical exercises are presented to portray the excellent agreement and fast convergence ratio of the introduced PIS and discuss influences of supported conditions, gradient exponents, length-to-thickness ratios and formulae of gradation functions on the flexural behaviors and free vibration responses of the axially graded elastic beams.

Inhomogeneous Beam of Circular Cross-Section with Concentric Lengthwise Cracks: A Fracture Analysis with Considering Viscoplastic Behaviour

In this paper, a lengthwise fracture analysis of a viscoplastic inhomogeneous cantilever beam of circular cross-section acted upon by a torsion moment is presented. The beam is with an arbitrary number of concentric lengthwise cracks. The time-dependent mechanical behaviour of the beam is treated by a viscoplastic model representing a series of units which are combinations of springs, dashpots and frictional sliders. Due to the sliders, the model units start to deform one after another with increasing of the external load. Constitutive laws of the viscoplastic model are obtained and applied to treat the beam mechanical behaviour when deriving time-dependent strain energy release rates for the lengthwise cracks.

An Approach in Analyzing Longitudinal Fracture of Inhomogeneous Beams with Using Generalized Viscoplastic Models

This paper introduces a generalized viscoplastic model and describes its application in treating the longitudinal fracture of inhomogeneous beam structures. The model is a combination of an arbitrary number of parallel units. Each unit is obtained by two springs and a dashpot. Besides, a frictional slider is placed parallel with one of the springs in each unit of the model in order to treat the plastic strains. The time-dependent constitutive law of the model obtained by considering strains, stresses and equilibrium of the model components is used in deriving solution of the strain energy release rate for the longitudinal crack in the viscoplastic beam.

Effects of the inhomogeneous fast electron beam on the self-generated magnetic fields in cylindrical targets

Effects of the inhomogeneous fast electron beam on the self-generated magnetic fields in cylindrical targets

Stationary Temperature Field Ensuring the Absence of Thermal Stresses in an Inhomogeneous Rectangular Beam

Stationary Temperature Field Ensuring the Absence of Thermal Stresses in an Inhomogeneous Rectangular Beam

Analytical evaluation of stresses and strains in inhomogeneous non-prismatic beams undergoing large deflections

Analytical evaluation of stresses and strains in inhomogeneous non-prismatic beams undergoing large deflections

Homogeneity of dental curing unit beam profile and its effect on microhardness of dental composites with varying thicknesses

ObjectivesThe dental curing unit (DCU) has been reported to emit inhomogeneous light. However, there are no studies on which elements could affect the inhomogeneity of DCUs. This study aimed to analyze the effect of attenuating factors such as the aperture of the lens or neutral density (ND) filters on the DCU's beam profile and evaluate the effect of DCU's inhomogeneous beam profile on the microhardness distribution on composite resin specimens with different thicknesses. MethodsRadiant emittance and spectrum of eight DCUs were investigated with and without ND filters using different optical density (OD) values. Beam profiles of eight DCUs were photographed while increasing the OD values of the ND filter. The change in the beam profile while changing the f-number of the aperture and the OD value of the ND filters were recorded. The Vickers microhardness of Filtek Z350XT and SDR cured by Bluephase Style 20i with 1, 2, 3, and 4 mm specimens of nine points on each surface was measured. ResultsIrradiance and spectrum of DCUs uniformly decreased after attenuation by the ND filters. The beam profile of the DCUs blurred when the f-number of the aperture was higher. The microhardness of Filtek Z350XT showed differences between the central and peripheral areas, and between the points under violet LED and the center on the bottom surface of the 4 mm specimen. The microhardness of the SDR did not show any differences. SignificanceThe inhomogeneity of the beam profile can be affected by attenuation conditions. DCU's inhomogeneous beam profile may have different effects on the composites depending on the thicknesses and types of composite resin used.

Quadrupole stability in nonuniform fill patterns

In previous publications, the effect of inhomogeneous beam loading of Landau cavities due to a nonuniform fill pattern in a storage ring has been studied with the aim of predicting the resulting longitudinal profiles of the bunches and their time offsets relative to the main rf [T. Olsson et al., Phys. Rev. Accel. Beams 21, 120701 (2018)]. This work was extended to treat dipolar coupled-bunch modes driven by beam-excited higher-order modes in the rf cavities [F. J. Cullinan et al., Phys. Rev. Accel. Beams 23, 074402 (2020)]. These coupled-bunch modes are of interest because they can become unstable, leading to an increase in the energy spread. The theory has now been extended once again to cover the case of coupled-bunch quadrupole modes where it is the lengths of the bunches that are oscillating, not their centroids. The theory is outlined and its predictions are thoroughly benchmarked against predictions from macroparticle tracking. Observations of the effects of nonuniform fill patterns on coupled-bunch quadrupole instabilities are made and interpreted. Results of measurements at the MAX IV 3 GeV ring, interpreted using theoretical calculations, are then presented to continue the investigation.

Analysis of viscoelastic tapered pylons used in transmission lines due to ground vibrations including soil-structure-interaction effects

This work reports on the influence that mass attachments and soft soil support conditions have on the vibratory motion of flexible pylons with non-uniform cross-section, both for axial and flexural vibrations induced by ground motion. The use of such structures is pervasive, both in civil and in mechanical engineering. In particular, the tapered pylon considered here is used in electrified railways to support transmission lines and possibly other types of apparatus. We first present an analytical solution to the governing equations of motion of an inhomogeneous beam, which entails the recovery of the natural frequencies in the presence of an attached top mass and of a compliant base support represented by soil springs. This is followed by the complete solution for the case of harmonic ground motion. Results are presented in term of transmissibility functions, defined as the ratio of the top pylon displacement amplitude to that of the base motion. The aim here is to identify the frequency range for which the presence of a top mass, which may also act as a passive damper, and of compliant soil support, which may also act as a base isolator, is either beneficial or detrimental in terms of minimizing the kinematic field induced in the pylon by ground vibrations.

Natural Frequencies of Beams with Axial Material Gradation Resting on Two Parameter Elastic Foundation

Free vibration analysis is carried out on axially inhomogeneous beams resting on Winkler-Pasternak elastic foundation. The material properties of the beam like Young’s modulus, modulus of rigidity and material density are considered to be varying along the length direction following constant, linear and exponential material models. The beam is subjected to different combinations of clamped and simply supported boundary conditions. The formulation is based on Timoshenko beam theory and energy method along with Hamilton’s principle is used to derive the governing equations. The effect of material gradation and the 2 parameters of elastic foundation on the natural frequencies are studied in detail. The present results are validated by comparing them with established ones and satisfactory matching is observed.
 HIGHLIGHTS
 
 Free vibration behavior of axially functionally graded beams in investigated
 Three different material models and three boundary conditions are considered
 Effect of two parameter elastic foundation is considered
 Timoshenko beam theory and Rayleigh Ritz method are employed
 It is found that the stiffness of elastic foundation significantly affects the natural frequency of the beam
 
 GRAPHICAL ABSTRACT

Viscoelastic inhomogeneous beam subjected to mechanical loading and periodically varying temperature: a longitudinal fracture analysis

This paper deals with analysis of longitudinal fracture of inhomogeneous beam structure that exhibits viscoelastic behaviour under mechanical loading and periodically varying temperature. The analysis is performed assuming that a longitudinal crack is located arbitrarily along the beam thickness. The mechanical loading representing a bending moment applied at the free end of the upper crack arm varies continuously with time. A solution of the strain energy release rate that takes into account the viscoelastic behaviour of the beam is derived. When deriving the solution, the viscoelastic behaviour and the varying temperature are treated by applying a time-dependent modulus that is obtained by using the constitutive law of a viscoelastic model having three components (a spring and two dashpots) under smoothly varying stress with time. The material properties of the viscoelastic model which are involved in the expression for the time-dependent modulus vary continuously along the length and thickness of the beam due to material inhomogeneity. The solution of the strain energy release rate is verified by analyzing the strain energy in the beam structure. The effects of the varying temperature, the loading conditions and the inhomogeneity on the strain energy release rate are investigated.

Effects of static indeterminacy on the lengthwise fracture in non-linear viscoelastic inhomogeneous beams loaded in torsion

Lengthwise fracture of statically undetermined beam structure that has non-linear viscoelastic behaviour is analyzed in this paper. The beam is loaded in pure torsion. The cross-section of the beam is a circle. The beam exhibits continuous material inhomogeneity along the radius of its cross-section. The mechanical behaviour of the beam is treated by a viscoelastic model structured by non-linear springs and non-linear dashpots. The static indeterminacy is resolved first and then the problem of the time-dependent strain energy release rate is solved by considering the complementary strain energy. The energy balance is analyzed for verification of the solution of the strain energy release rate. A parametric investigation of the strain energy release rate is carried-out.

Inhomogeneous beam structures of rectangular cross-section loaded in torsion: a delamination study with considering creep

This paper describes a delamination study of inhomogeneous beam structure with rectangular cross-section subjected to torsion. The beam exhibits linear creep behaviour. The beam is built-up by two longitudinal vertical layers of different width. The layers are continuously inhomogeneous along the beam length. The delamination crack is located inside the beam between layers. The torsion moments in the crack arms are obtained by treating of the beam as internally statically undetermined structure with one degree of indeterminacy. For this purpose, the theorem of Menabrea is applied (the torsion moment in the left-hand crack arm is considered as a redundant unknown). Solution of the strain energy release rate is derived by considering the strain energy in the beam. The strain energy release rate is function of time due to creep behaviour. For verification of the solution, the strain energy release rate is found also by considering the beam compliance. A parametric study is developed.

ИССЛЕДОВАНИЕ КОЛЕБАНИЙ БАЛКИ ИЗ ФУНКЦИОНАЛЬНО-ГРАДИЕНТНОГО МАТЕРИАЛА С УЧЕТОМ ЗАТУХАНИЯ

The problem of oscillations of an inhomogeneous beam made of a functionally gradient material in the framework of various deformation models - the Euler-Bernoulli model and the Timoshenko model in the presence of attenuation, which is described within the framework of the concept of complex modules for a standard viscoelastic body, is considered. The left end of the beam is pinched, and at the right end there is a concentrated moment oscillating with a certain frequency. The solution is built in two ways. In the first of them, on the basis of asymptotic analysis, a solution was constructed in the low-frequency domain, and a complete coincidence of solutions for the models under consideration in the low-frequency domain (up to the first resonance) for any laws of inhomogeneity was shown. In the second case, the influence of rheological factors on the amplitude-frequency characteristics of an inhomogeneous viscoelastic beam in the frequency range up to the third resonance was analyzed using the targeting method. A comparative analysis of the frequency response of two models for different laws of heterogeneity is carried out. Frequency response for various laws of inhomogeneity are presented, the movement of resonant frequencies depending on the dimensionless relaxation time is studied.