Rod Of Circular Cross Section Research Articles

A new model for spatial rods incorporating surface energy effects

A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.

Finite element modeling of heat propagation of a complete rod of constant cross-section

In this paper, the definition of the temperature distribution field for a rod made of heat-resistant alloy EI48 is introduced. The authors consider for the study a complete rod of circular cross-section of radius R, of limited length L. Studied body is under the influence of a heat flow q from the surface over the entire cross-sectional area of the left end, and heat exchange with the environment occurs on the cross-sectional area of the right end. The rod is thermally insulated along the side surface. The authors consider two cases: the first is the heat flow with intensity q can be set on the area of a small circle with radius r <R, the second is the heat flow can be set on its part, that is, on the area . During the study, the authors showed that during the thermomechanical process, the strength of each section of the load-bearing structural elements is significantly influenced by the temperature distribution field. The influence of high temperature on the morphology of heat-resistant alloys is also shown. This leads to the fact that in some parts of the structural elements the temperature will be acceptable, and in some — critical. As a result, rapid wear of structural elements and loss of their physical qualities occur. Therefore, mathematical modeling of temperature distribution field for a body of various configurations is an urgent problem. The article presents a method for constructing a mathematical model and a corresponding computational algorithm that allows solving a class of problems to determine the regularities of the temperature distribution field in the elements of rod-shaped structures. To do this, the authors used the energy-variation principle in combination with the finite element method.

Influence of Creep on Longitudinal Fracture of Inhomogeneous Rod Loaded in Torsion and Bending

The present paper analyzes the influence of creep on longitudinal fracture in continuously inhomogeneous rod of circular cross-section loaded in torsion and bending. The rod exhibits continuous material inhomogeneity in both radial and longitudinal directions. The creep is described by using non-linear time-dependent relations between the principle stresses and strains. A time-dependent solution to the strain energy release rate is derived by analyzing the complementary strain energy. The time-dependent strain energy release rate is found also by considering the energy balance for verification. The solutions are applied to perform a parametric study of the strain energy release rate under creep.

Discrete and asymptotic approximations for one stationary radiative–conductive heat transfer problem

AbstractSpecial discrete and asymptotic approximations are proposed for the boundary value problem describing a stationary radiative–conductive heat transfer in a system of absolutely black heat-conducting rods of circular cross-section. Results of numerical experiments are presented to confirm the efficiency of proposed approximations.

Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and Cubic nonlinearities

We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section within the scope of the general weakly-nonlinear elasticity leading to a model with quadratic and cubic nonlinearities. We systematically derive the extended Boussinesq and Korteweg - de Vries - type equations and construct a family of approximate weakly-nonlinear soliton solutions with the help of near-identity transformations. These solutions are compared with the results of direct numerical simulations of the original nonlinear problem formulation, showing excellent agreement within the range of their asymptotic validity (waves of small amplitude) and extending their relevance beyond it (to the waves of moderate amplitude) as a very good initial guess. In particular, we were able to observe a stably propagating "table-top" soliton..

On physical basis of local current overload in vehicle electric mains

Introduction. The data presented in the article indicate that the problem of improving fire safety of vehicles is very topical. The aim of the article is to develop a scientifically based method for studying a copper conductor with signs of local current overload in order to establish the cause of its damage during a fire-technical examination. Materials and methods . The studies were carried out with a JSM-6390LV scanning electron microscope with an add-on device for energy dispersive microanalysis. The fracture surfaces of the copper conductor were analyzed without preliminary sample preparation. Theoretical foundations (theory and calculations). An updated model of ultimate stress-strain state of inelastic pure bending of a copper rod of circular cross section has been developed. The solution has been reduced to simple rating formulas that allow us to evaluate the bearing capacity of flexible single copper conductors. The applicability of the developed mathematical model during the fire-technical examination is shown by a specific example. Results and discussion . Examples of vehicle fires caused by critical bending of the wiring harness are given in the article. Experimental data confirmed that the copper conductor under current overload melts in the critical bend area. The need to clarify the wording of the term “local current overload” is justified. Conclusion. A method for determining the critical bending of a copper conductor at which its melting can occur under electric current has been proposed. The data presented in the article can be used by experts in an expert study of copper conductors from fire sites to establish the mechanism of their damage and, ultimately, the cause of a vehicle fire.

Construction of Constitutive Equations for Orthotropic Materials with Different Properties in Tension and Compression under Creep Conditions

Constitutive steady-state creep equations are proposed for orthotropic materials with different tensile and compressive resistances. The resistances are described using lower functions with different exponents for tension and compression. Equations are written for tension, shear, and plane stress problems. The model is used to solve the problem of torsion by a constant moment at a temperature T = 200°C for annular cross-section rods cut from a plate of AK4-1 transversely isotropic alloy in the normal direction to the plate and in the longitudinal direction. Constitutive equations for torsion are derived. Values of model parameters were obtained in experiments on uniaxial tension and compression of solid circular specimens cut in various directions. An analytical solution for the rate of torsion angle of a circular cross-section rod cut normal to the plate was obtained for the same exponent in tension and compression. For a rod cut in the longitudinal direction, an upper estimate of the torsion angle rate was obtained. The calculated results are in satisfactory agreement with experimental data.

Maxima of the Stresses in the Longitudinal Pochhammer—Chree Waves

The article presents the calculation of stress fields that arise in cylindrical rods of circular cross section in the case of Pochhammer–Chree waves. Examples of calculating the fields for the four lowest wave modes in a steel rod for two cases of phase velocities are considered. The structures of the isostat of the principal stresses in the longitudinal section of the rod are constructed and considered. The fields of the first principal stress and von Mises stress are constructed.

Guided Wave Propagation in Detection of Partial Circumferential Debonding in Concrete Structures.

The following article presents results of investigating the damage detection in reinforced concrete beams with artificially introduced debonding between the rod and cover, using a non-destructive method based on elastic waves propagation. The primary aim of the research was to analyze the possible use of guided waves in partial circumferential debonding detection. Guided waves were excited and registered in reinforced concrete specimens with varying extents of debonding damage by piezoelectric sensors attached at both ends of the beams. Experimental results in the form of time–domain signals registered for variable extent of debonding were compared, and the relationships relating to the damage size and time of flight and average wave velocity were proposed. The experimental results were compared with theoretical predictions based on dispersion curves traced for the free rod of circular cross-section and rectangular reinforced concrete cross-section. The high agreement of theoretical and experimental data proved that the proposed method, taking advantage of average wave velocity, can be efficiently used for assessing debonding size in reinforced concrete structures. It was shown that the development of damage size in circumferential direction has a completely different impact on wave velocity than development of debonding length. The article contains a continuation of work previously conducted on the detection of delamination in concrete structures. The proposed relationship is the next essential step for developing a diagnostics method for detecting debondings of any size and orientation.

Longitudinal wave speed in cylindrical auxetic rods with elastic constraint in radial direction

The combined influence of material auxeticity, strain and lateral boundary condition on the longitudinal wave speed in a prismatic rod of circular cross-section is developed herein. The lateral surface of the rod is elastically restrained in the radial direction such that twisting and bending are prevented while the axial displacement is fully permitted. As such, the imposition of zero and infinite radial spring stiffness reduces this model into longitudinal wave motion in a prismatic rod with free side surface and a plane wave of dilatation, respectively. The wave speed was made dimensionless by two ways, i.e. by normalizing against the elementary wave speed in prismatic rods and against the speed of plane waves of dilatation. Both approaches demonstrate the manner by which the wave speed transits between the two extreme lateral boundary conditions: from being a purely rod (1D stress with 3D strain) to a purely bulk (3D stress with 1D strain) wave motion, with the strain and material properties taken into consideration. Results indicate that the wave speed is increased by (a) greater radial elastic restraint stiffness, (b) more positive longitudinal strain, and (c) the higher Poisson's ratio magnitude of the rod material. These results also suggest that the adjustment of wave speed is more easily controlled in the auxetic region than in the conventional region.

Maximal stresses of the longitudinal Pochhammer – Chree waves

The paper presents a calculation of the stress fields that arise in cylindrical steel rods of circular cross-section, in the case of the axisymmetric Pochhammer – Chree waves for the zero mode. The maximum tension and maximum shear on the surface of the rod and on its axis are calculated. Frequency ranges are obtained for which the maximum stresses on the rod axis are several times higher than on the surface.

Photonic band structure in square and triangular lattices of cylindrical-shell rods

The photonic band structure for transverse electric and transverse magnetic polarization is calculated in a two-dimensional photonic crystal with square and triangular lattices. The crystal comprises cylindrical-shell rods, wherein air rods of circular cross-section are surrounded by a GaAs shell embedded in a background of air. The results showed that increasing the radius of the shell rods reduces the width and number of photonic band gaps for transverse magnetic polarization. In addition, the number of photonic band gaps increases with the decreasing radius of the air rods for both polarizations. Furthermore, increasing the temperature, leads to an increase in the GaAs refractive index, which causes a shift to lower energies in the photonic band structures.

A non-classical Kirchhoff rod model based on the modified couple stress theory

A new non-classical Kirchhoff rod model is developed using the modified couple stress theory, which contains one material length scale parameter and can account for microstructure-dependent size effects. The governing equations and boundary conditions are determined simultaneously by a variational formulation based on the principle of minimum total potential energy. The newly developed model recovers its classical elasticity-based counterpart as a special case when the microstructure effect is not considered. To illustrate the new non-classical Kirchhoff rod model, two sample problems are analytically solved by directly applying the general formulas derived. One problem is the equilibrium analysis of a helical rod of circular cross section deformed from a straight rod, and the other is the buckling of a straight rod of circular cross section induced by an axial compressive force. In the former, the rod undergoes a twisting-dominated deformation, while in the latter the rod deformation is bending dominated. Two closed-form expressions are obtained for the force and couple needed in deforming the helical rod, and an analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, with the microstructure effect incorporated in each case. These formulas reduce to those based on classical elasticity when the microstructure effect is suppressed. For the helical rod problem, the numerical results show that the couple predicted by the current non-classical rod model is significantly larger than that predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase in the rod radius. For the buckling problem, it is found that the critical buckling load based on the new non-classical Kirchhoff rod model is always higher than that given by the classical elasticity-based model, with the difference being significant for a very thin rod.

Polynomial mode approximation for longitudinal wave dispersion in circular rods

The dispersive nature of the longitudinal wave propagation in rods has been studied extensively theoretically as well as experimentally by many authors (see e.g. [1] or [2]). Split Hopkinson Pressure Bars (SHPB) is employed to dynamically characterize materials in the intermediate range of strain rate and the post-processing of the SHPB tests requires the knowledge of the dispersion in the bars [3] [4] [5]. Simplified models of longitudinal wave propagation have been proposed by Love [6] and later by Mindlin et al. [7]. In his paper published in 2006, Anderson [8] developed a rod theory for the propagation of longitudinal waves in slender rods of circular cross section and it is an improvement from the previous works in [7]. In the approximation of the longitudinal and radial displacements he uses only the first two modes of the given basis for each displacement, this leads to the four-mode equation. The four-mode equation is a polynomial whose root is the wave longitudinal velocity. This equation is therefore easier to solve than the Pochhammer equation [1] because the latter is transcendental. However we noticed an error in one of the coefficients of the four-mode equation [8], and this error is not a mere typo. This error was also repeated in [9]. The aim of this paper is threefold. First we want to give the correct four-mode equation and the associated dispersion curve. Secondly, we wanted to examine the convergence of the approximation at higher orders then given by Anderson. Thirdly, this is the occasion to give dimensionless equations. The paper is organised as follows: Section 2 details how to obtain the dimensionless approximation for longitudinal waves. Section 3 recalls the Pochhammer dispersion equation for slender rods -which is later used to compute the reference dispersion curve- and details the way it is solved. Section 4 then gives the correct dispersion curve for the four-mode equation and examines the convergence of the approximate dispersion curves with an increasing number of degrees of freedom (modes). Lastly, Section 5 contains a discussion on the advantages and drawbacks of using either directly the Pochhammer equation or the Jacobi polynomial approximation, we also discuss the computation cost of both methods.

Transient Dynamic Analysis of a Cantilever Rod with Axial Impulse Loading Using Finite Element Method

The behavior of bodies subjected to impulse loading is of prime importance in the study of forces that occur in impulse facilities. Before performing the actual tests, theoretical and numerical simulations are carried out to obtain the response of bodies subjected to impulse loading. The simplest model for this study can be considered as a rod of circular cross section fixed at one end and free at other end. When a transient impulse load is applied on a body, vibrations occur in the body for the brief period of time. In this paper, the effect of a half sine impulse force applied on a cantilever rod in the axial direction has been discussed. The displacements at the tip of the rod were obtained based on two theories, the basic vibration formulae and FEM analysis. Simulations were performed using ANSYS and compared with the displacements obtained from the two theoretical methods.

Mechanical and Micro structural Characterization of Al-Al/B4C Laminated Composites Depending on Manufacturing Parameters

In this study, it is aimed to produce a layered composite structure with Al sheath and Al / B4C cored by extrusion technique and to characterize the mechanical and micro structure of the products. In the production of composite billets, powder in tube (PIT) method was used. In this method, Al7075 (ø 30 mm) tube is filled with Al2124 / B4C powder mixtures. The semi-finished composite billets were sintered at 500 ° C for 1 hour and extruded at the extrusion ratios of R = 14 R = 9 and R = 6.25 at the same temperature to obtain circular cross-section rods. The density changes, hardness distributions and sheath-core section ratios of these bars were determined. The changes of these values depending on the extrusion rate and B4C ratio were investigated. It is also revealed in the micro structure changes depending on these ratios.

Ultrasonic characterization of the complex Young's modulus of polymers produced with micro-stereolithography

Micro-stereolithography is capable of producing polymer parts on the scale of millimeters that contain micron-scale features. Accurate prediction of the dynamic performance of components produced using micro-sterolithography requires knowledge of the as-fabricated material properties, but very little published material property data exist for these materials. This is complicated by the fact that the properties vary as a function of build parameters (i.e., laser exposure time). Designers therefore have limited useful material property information for part design. Frequency-dependent material parameters are often determined by measuring the wave speed and attenuation of an ultrasonic pulse as it propagates through the material. This work employs laser Doppler vibrometry to detect extensional waves in a solid polymer rod of circular cross-section that is excited by a longitudinal ultrasonic contact transducer. Transverse motion associated with extensional waves propagating along the rod axis is measured at m...

Determination of Frictional Forces during Wire Rod Drawing Process by Reverse Method

The article pesents the drawing process of the circular cross-section rod. The mechanical model of plastic deformation in the process of wire drawing is represented by the regularities of yielding of metal at the level of forward flow during the rolling based on theoretical solutions proposed by A. I. Tselikov. The method for determination of drawing forces was probed about by decomposing it into two problems of planar deformation and the superposition of these solutions. The results of the theoretical solution of wire drawing task were used to determine the coefficient of friction during the deformation of low carbon steel in diameter 5.5 mm by means of comparison with the known experimental results of the wire drawing efforts. Satisfactory results were obtained for evaluation of frictional forces during deformation under high plastic deformation zone conditions. The obtained values of friction coefficients can be used to solve the wire drawing tasks and under conditions of a high hearth of plastic deformation as well.

Dynamic characteristics of infinite-length and finite-length rods with high-wave-number periodic parameters

The dynamic characteristics of the infinite-length and finite-length rods with periodic distribution parameters are studied. The differential equation of longitudinal motion of the period-parametric rod is given. The algebraic matrix equation for the wave motion characteristics of the infinite-length periodic rod is derived based on the Bloch theorem and Fourier series. The characteristic frequencies are determined by the matrix eigenvalues which depend on the characteristic wave number and parametric wave number. Then the algebraic matrix equation for the dynamic characteristics of the finite-length periodic rod is derived based on the Galerkin method. The natural frequencies are determined by the matrix eigenvalues which depend on only the parametric wave number. An improving approach algorithm for solving the eigenvalue problem of high degree-of-freedom systems is developed based on the Rayleigh quotient. Finally, the circular cross-section rod with period-varying diameter is considered and numerical results on the dynamic characteristics are given. Large characteristic wave number and parametric wave number are considered for the infinite-length and finite-length periodic rods. The characteristic frequencies varying with the characteristic wave number and parametric wave number are shown, and the band gaps vanishing are revealed for increasing characteristic wave number. The finite-length periodic rod has the dynamic characteristics different from the infinite-length periodic rod. The effect of the term number of the displacement expansion on the natural frequencies and the natural frequencies varying with the parametric wave number and wave amplitude are shown for the finite-length periodic rod. The local resonance and periodical short descent of the natural frequencies with the increase of the parametric wave number and the different changes of the natural frequencies with the parametric wave amplitude are revealed. The above new dynamic characteristics of the infinite-length and finite-length rods with periodic distribution parameters have a potential application to period-structural design and optimization.

Аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе стержней круглого сечения

In applications, it is of great importance to study the heat transfer process in periodic media containing vacuum interlayers or cavities through which the heat transfer is realized by radiation. A numerical solution of such problems requires considerable computational efforts and becomes, in fact, impossible for a large number of heat transferring elements, especially in the case of two-dimensional and three-dimensional structures. Therefore, it is important to construct effective approximation methods. This article continues a series of papers devoted to the construction and substantiation of special semi-discrete and asymptotic approximations of radiation-conductive heat transfer problems in periodic systems of heat-conducting elements separated by a vacuum. In this paper, we consider a stationary problem of radiation-conductive heat transfer in a periodic system consisting of absolutely black heatconducting rods of circular cross section with radius e packed in a square box. We constract two new approximate methods based on special discrete and asymptotic approximations of the original problem. The results of computational experiments are given, which make it possible to draw conclusions about the efficiency of methods and the nature of the dependence of their relative errors on the coefficient of thermal conductivity λ and the radius of the rods e. The seeking value is the absolute temperature. This function is the solution of the boundary-value problem for the stationary heat equation with nonlinear nonlocal intregral conditions describing the heat exchange by a radiation between the rods. Assuming that the temperature on each of the rods is approximately equal to the mean value over the cross section, we comstruct a discrete approximation of the original problem, which is a system of linear algebraic equations with respect to the fourth power of a temperature. Since the matrix of the system is symmetric and positive definite, the method of conjugate gradients can be used to solve it numerically. By its structure, the discrete problem is such that it can be considered as a difference approximation of the homogenized boundary-value problem for the Poisson equation with non-standard boundary conditions. This problem, which we consider as an asymptotic approximation of the original problem, is linear with respect to the fourth power of temperature. A series of computational experiments is carried out, which confirms the operability of the proposed methods for materials with a large coefficient of thermal conductivity. The relative errors of the methods tend to zero as the parameter tends to zero. For a fixed value of radius e, the relative errors of the methods decrease with increasing values of the thermal conductivity, reaching values in the tenths of a percent. As expected, both methods are practically unsuitable for poorly conductive materials and require substantial modification.