Solid Elastic Cylinder Research Articles

Finite extension of accreting nonlinear elastic solid circular cylinders

Finite extension of accreting nonlinear elastic solid circular cylinders

Analysis of the effects of viscosity on the SH-wave band-gaps of 2D viscoelastic phononic crystals by Dirichlet-to-Neumann map method

The effects of viscosity on the band-baps of two-dimensional (2D) viscoelastic phononic crystals are studied by the Dirichlet-to-Neumann (DtN) map method. The viscoelastic phononic crystals are composed of square or triangular lattices of elastic circular solid cylinders in a viscoelastic solid matrix. Time-harmonic anti-plane transverse (SH) waves are considered and the standard linear solid model is used to describe the viscoelastic behavior of the matrix material, in which the shear modulus depends on the frequency. Based on the DtN map constructed by using the cylindrical wave expansions, a linear eigenvalue problem related to the Bloch wave vector is formulated, which involves relatively small matrices. By using real frequencies and complex-valued wave vectors, the complex band structures are computed for different viscosity parameters, and the wave propagation and attenuation characteristics in dependence of the viscosity are analyzed. Then based on the storage shear modulus, real band structures for the Pb/epoxy phononic crystals with viscosity in the square and triangular lattices are studied, and the effects of the different viscoelastic material parameters and other relevant influencing parameters on the band-gaps are discussed. The results show that the viscosity of the matrix material affects the location and the width of the band-gaps. This fact provides an alternative way to tune the band-gaps of phononic crystals by adjusting the viscoelastic parameters combined with other influencing parameters.

Harmonic Standing-Wave Excitations of Simply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The vibration of a solid elastic cylinder is one of the classical applied problems of elastodynamics. Many fundamental forced-vibration problems involving solid elastic cylinders have not yet been studied or solved using the full three-dimensional (3D) theory of linear elasticity. One such problem is the steady-state forced-vibration response of a simply-supported isotropic solid elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surface. In this paper, we exploit certain recently-obtained particular solutions to the Navier–Lamé equation and exact matrix algebra to construct an exact closed-form 3D elastodynamic solution to the problem. The method of solution is direct and demonstrates a general approach that can be applied to solve other similar forced-vibration problems involving elastic cylinders. The obtained analytical solution is then applied to a specific numerical example, where it is used to determine the frequency response of the displacement field in some low wave number excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a subset of the natural frequencies of the simply-supported cylinder. The analytical solution is also used to compute the resonant mode shapes in some selected asymmetric excitation cases. The studied problem is of general interest both as an exactly-solvable 3D elastodynamics problem and as a benchmark forced-vibration problem involving a solid elastic cylinder.

Implementing Stretch-Based Strain Energy Functions in Large Coupled Axial and Torsional Deformations of Functionally Graded Cylinder

In this paper, coupled axial and torsional large deformation of an incompressible isotropic functionally graded nonlinearly elastic solid cylinder is investigated. Utilizing stretch-based constitutive models, where the deformation tensor is non-diagonal is complex. Hence, an analytical approach is presented for combined extension and torsion of functionally graded hyperelastic cylinder. Also, finite element analysis is carried out to verify the proposed analytical solutions. The Ogden model is employed to predict the mechanical behavior of hyperelastic materials whose material parameters are function of radius in an exponential fashion. Both finite element and analytical results are in good agreement and reveal that for positive values of exponential power in material variation function, stress decreases and the rate of stress variation intensifies near the outer surface. A transition point for the hoop stress is identified, where the distribution plots regardless of the value of stretch or twist, intersect and the hoop stress alters from compressive to tensile. For the Ogden model, the torsion induced force is always compressive which means the total axial force starts from being tensile and then eventually becomes compressive i.e., the cylinder always tends to elongate on twisting.

Closed form solutions for large deformation of cylinders under combined extension-torsion

In this paper, the mechanical response of an incompressible isotropic nonlinear elastic solid circular cylinder is investigated under combined extension and torsion. Since the deformation tensor in extension-torsion is non-diagonal, implementing stretch based energy functions are complex. Hence, in this study, an analytical solution is proposed for both invariant- and stretch-based models. Moreover, finite element analysis of extension-torsion of hyperelastic materials is carried out using both UHYPER and VUMAT user-defined subroutine in ABAQUS to verify the presented analytical methods. Both stretch and invariant-based exponential forms of strain energy function are employed, and its corresponded material parameters are calibrated for silicon-rubber. The finite element results for stress distribution show a good agreement with analytical findings which confirms the validity and accuracy of the proposed method. Results show that the both invariant- and stretch-based exp-exp models are less conservative for a relatively small stretch and twist while for large stretches they yield higher stresses than the other models and the rate of stress variation increases significantly, especially for a stretch-based exp-exp model. For both Invariant- and stretch-based exp-exp models, an alteration axial stretch is identified where the torsion induced axial force alters from tensile to compressive. For stretches smaller than this alteration point, the cylinder always elongates while for the larger stretches, the cylinder tends to shorten for small twists and then elongates on further twisting. Also, as the axial stretch increases the alteration point appears in larger twists.

Static, Transient and Free Vibirational analysis of thermo magneto electric elastic solid cylinder

In this paper the static, transient and free vibration analysis of a thermo- magneto-electric-elastic solid cylinder is analyzed stochastically by using hybrid numerical method (combined finite element and Newmark finite difference method).An infinite solid cylinder made up of 6mm class considered. The constitutive equations containing the mechanical, magnetic, electrical and thermal fields and investigated by free and forced Vibirational boundary conditions. The transient finite element equations are obtained by assumed shape functions. After assembling the Mass, Stiffness and Damping and matrices, the global dynamic equations are in the form of time field. The resulting equations are solved by using the finite difference technique with suitable time instants. By using material constants values the displacement, velocity and acceleration of vibrations are obtained with various time values and the non dimensional frequencies are also obtained by different values of non dimensional wave number. Numerical work is carried out by the electric and magnetic materials Cdse and CoFe2o4. The outcomes are tabulated and represented graphically.

Axisymmetric equilibrium of an isotropic elastic solid circular finite cylinder

Axisymmetric equilibrium of an elastic solid circular finite cylinder is one of the oldest problems in the theory of linear elasticity. Crosswise superposition is a well-known method that is used to solve this boundary value problem (BVP); however, technical realization of its underlying ideas still seems to be vague and the solution obtained by this method suffers from some convergence issues. In this study, we follow two main objectives via analyzing a benchmark problem where an isotropic elastic solid circular cylinder of finite length is subjected to normal lateral loading. The first goal is to add more insight into the method of crosswise superposition by extending the ideas that are used for solving classical BVPs via the superposition principle. For this purpose, a new unified approach that naturally gives rise to the subtle concept of corner conditions (CCs) in the context of crosswise superposition method is introduced. Another goal is to demonstrate the influence of CCs on the convergence of the solution obtained by the method of crosswise superposition. In this avenue, the Love function approach is used to convert the Navier equations for an isotropic elastic material to a single axisymmetric biharmonic equation. Next, a general solution for the axisymmetric biharmonic equation consisting of separable and non-separable solutions is presented in cylindrical coordinates. These two classes of solutions are used to construct the Love function and associated elastic fields through the unified approach. Numerical results reveal that considering the CCs can significantly affect the convergence rate of the solution on the boundaries of the cylinder. Furthermore, it is observed that the solution does not converge to the boundary data at the rims without considering the CCs. Far enough from the boundaries of the cylinder, the solution does not seem to be much different with or without taking the CCs into account.

Contact interaction of thin-walled elements with an elastic layer and an infinite circular cylinder under torsion

We consider a class of contact torsion problems on interaction of thin-walled elements shaped as an elastic thin washer – a flat circular plate of small height – with an elastic layer, in particular, with a half-space, and on interaction of thin cylindrical shells with a solid elastic cylinder, infinite in both directions. The governing equations of the physical models of elastic thin washers and thin circular cylindrical shells under torsion are derived from the exact equations of mathematical theory of elasticity using the Hankel and Fourier transforms. Within the framework of the accepted physical models, the solution of the contact problem between an elastic washer and an elastic layer is reduced to solving the Fredholm integral equation of the first kind with a kernel representable as a sum of the Weber–Sonin integral and some integral regular kernel, while solving the contact problem between a cylindrical shell and solid cylinder is reduced to a singular integral equation (SIE). An effective method for solving the governing integral equations of these problems are specified.

Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families of particular solutions to the Navier-Lame equation in cylindrical coordinates. In this paper, we specifically construct solutions having 2pi-periodic angular parts. These particular solutions can be directly applied to a fundamental set of linear elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis, and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.

Fluid–structure interactions in a cylindrical layered wave guide with application in the spinal column to syringomyelia

Syringomyelia is a disease of the spinal cord in which fluid-filled cavities, called syrinxes, form and expand, compressing the surrounding neural tissue and producing neurological damage. This condition can occur following spinal injury and has limited treatment options in part because of a lack of understanding of its origins. Current pathogenic theories make predictions from localised disturbances in the cerebrospinal fluid dynamics. The poroelastic spinal cord tissues have an effective compressibility as a result of the localised displacement of the intercellular fluid. Also, despite the disease progressing over months and years, the mechanical perturbations from the heart and breathing cycles and from coughs and sneezes operate over much shorter timescales of seconds down to fractions of a second, respectively. To model these pathological features, we solve the harmonic eigenvalue problem for a two-dimensional elastic fluid–solid cylindrically layered waveguide. We analyse the dispersion behaviour and mode shapes over a wide wavelength spectrum and investigate the influence of Poisson's ratio on the wave mode characteristics. The healthy model consists of an elastic solid cylinder (spinal cord) and a surrounding annulus of inviscid fluid (cerebrospinal fluid in the subarachnoid space). To model syringomyelia we then add a cylindrical fluid space (syrinx) within the elastic cylinder. For the geometry with the inner fluid conduit present (diseased spinal cord in situ), the dispersion diagrams show three modes (0, 1, 2) that are propagatory across the wavelength spectrum. At long wavelengths, axial displacement dominates over radial, normal stress dominates over shear, but these become of the same order of magnitude as the wavelength approaches the radius of the elastic cylinder (spinal cord). Wave modes 0 and 1 induce relative motion between the elastic cylinder and the fluid contained within, which supports Williams's so-called slosh mechanism for syrinx lengthening, and all three modes involve stress concentrations (indicated by localised gradient peaks) in the inner wall of the elastic cylinder adjacent to the contained fluid, which would tend to support radial syrinx expansion. Mode 2 is the most sensitive to changes in Poisson's ratio (spinal cord tissue compressibility). We also investigate the progression of post-traumatic syringomyelia by simulating a sequence of discrete states in the gradual occlusion of the subarachnoid space and the subsequent formation and radial expansion of a syrinx. We predict that syrinx-fluid sloshing, hence the stress so induced, diminishes with the continued radial expansion of the syrinx. The largest normal stress concentrations at the syrinx boundary, which appear for modes 0 and 1, initially diminish with radial syrinx expansion; however, they reach a minimum value for intermediate syrinx diameters. On this basis we hypothesize that syrinx development may be part of a homeostatic mechanism to alleviate stress in the spinal cord. Understanding this process may aid in treatment development. The present work also has applications in industrial systems and serves as a platform for more advanced modelling of cylindrical waveguides in biological systems.

Torsion problems of finite cylinders weakened by ring-shaped cracks

Abstract: A finite elastic hollow and solid cylinders are considered. The bottom faces of the cylinders are fixed, the upper faces are free from stress. The tangent axisymmetric loading is applied along their cylindrical surfaces. This leads to the torsion axisymmetric deformation. A system of N ring – shaped cracks is situated inside the cylinders parallel to the cylinder's axis. It is supposed that the branches of the cracks are free from stress. It is necessary to construct the formulas for the stress intensity factor calculation and investigate the stress state of a solid. The initial boundary value problem is reduced with Fourier transformation to a system of integral singular equations with regard to the jumps of the displacements at the cracks' branches. The singularity of the equations kernels is extracted. The system of singular integral equations is solved with the orthogonal polynomial method. The solution of the system is searched as the series by Chebyshev polynomials with the weight function. The realisation of orthogonal polynomial method leads to an infinite system of linear algebraic equations with regard to the unknown coefficients of the series. The formulas for the stresses and displacements of an elastic finite cylinder are presented. The numerical realisation of the proposed method is demonstrated in cases with two and three cracks; the stress state is investigated dependingon the cracks' locations and sizes.

Kinking Instability in the Torsion of Stretched Anisotropic Elastomeric Filaments

The problem of torsion superimposed on extension of incompressible nonlinearly elastic solid circular cylinders is a classic problem of nonlinear elasticity that has been considered by many authors in the context of rubber elasticity particularly for isotropic materials. An interesting aspect of such problems is the instability that arises with sufficiently large twist where a kink or knot suddenly appears. An energy approach to examining this instability when the extension and twist are prescribed was given by Gent and Hua (Int. J. Nonlinear Mech. 39:483–489, 2004) and illustrated there for a neo-Hookean isotropic elastic material. The theoretical results were compared with experimental observations on natural rubber rods. In a recent paper Murphy (Int. J. Nonlinear Mech. 68:96–100, 2015) has shown that the approach in Gent and Hua for isotropic materials can be simplified when the rods are assumed to be thin. Motivated by potential applications to the biomechanics of soft tissues, we are concerned here with the implications of this simplified thin rod instability theory for transversely isotropic materials. The results are illustrated on using three specific strain-energy density functions. These models are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and are consistent with the linear theory. In contrast with the case for isotropic materials, it is shown that the kinking instability arises even in the absence of stretch that is in the case of pure torsion. The results are illustrated on using elastic moduli for muscles based on experimental findings of other authors.

Exact series model of Langevin transducers with internal losses

An exact series method is presented to analyze classical Langevin transducers with arbitrary boundary conditions. The transducers consist of an axially polarized piezoelectric solid cylinder sandwiched between two elastic solid cylinders. All three cylinders are of the same diameter. The length to diameter ratio is arbitrary. Complex piezoelectric and elastic coefficients are used to model internal losses. Solutions to the exact linearized governing equations for each cylinder include four series. Each term in each series is an exact solution to the governing equations. Bessel and trigonometric functions that form complete and orthogonal sets in the radial and axial directions, respectively, are used in the series. Asymmetric transducers and boundary conditions are modeled by using axially symmetric and anti-symmetric sets of functions. All interface and boundary conditions are satisfied in a weighted-average sense. The computed input electrical admittance, displacement, and stress in transducers are presented in tables and figures, and are in very good agreement with those obtained using atila-a finite element package for the analysis of sonar transducers. For all the transducers considered in the analysis, the maximum difference between the first three resonance frequencies calculated using the present method and atila is less than 0.03%.

Correlation between helical surface waves and guided modes of an infinite immersed elastic cylinder

Scattering of obliquely incident plane acoustic waves from immersed infinite solid elastic cylinders is a complex phenomenon that involves generation of various types of surface waves on the body of the cylinder. Mitri [F.G. Mitri, Acoustic backscattering enhancement resulting from the interaction of an obliquely incident plane wave with an infinite cylinder, Ultrasonics 50 (2010) 675–682] recently showed that for a solid aluminum cylinder, there exist acoustic backscattering enhancements at a normalized frequency of ka ⩽ 0.1 . The incidence angle α c at which these enhancements are observed lies between the first (longitudinal) and second (shear) coupling angles of the cylinder. He also confirmed the observations previously reported by the authors that there exist backscattering enhancements of the dipole mode at large angles of incidence where no wave penetration into the cylinder is expected. In this paper, physical explanations are provided for the aforementioned observations by establishing a correlation between helical surface waves generated by oblique insonification of an immersed infinite solid elastic cylinder and the longitudinal and flexural guided modes that can propagate along the cylinder. In particular, it is shown that the backscattering enhancement observed at ka ⩽ 0.1 is due to the excitation of the first longitudinal guided mode travelling at the bar velocity along the cylinder. It is also demonstrated that the dipole resonance mode observed at incidence angles larger than the Rayleigh coupling angle is associated with the first flexural guided mode of the cylinder. The correlation established between the scattering and propagation problems can be used in both numerical and experimental studies of interaction of mechanical waves with cylinders.

OS0106 横等方性弾性円柱の軸対称熱応力(弾性数理解析とその応用,オーガナイズドセッション)

A method of fundamental solution is presented for axisymmetric thermal stress problems of a transversely isotropic elastic circular solid cylinder. We apply fundamental solutions for axisymmetric thermal stress problems and axisymmetric body force problems of transversely isotropic elastic solids. Thermal stresses and temperature distributions of a solid cylinder subjected to a heat distribution at the end surface are considered by numerical calculations.

G0301-2-5 弾性円すい棒と剛体円すい孔接触面角部の応力特異性(材料力学部門一般(2))

G0301-2-5 弾性円すい棒と剛体円すい孔接触面角部の応力特異性(材料力学部門一般(2))

Calculation of the acoustic target strength of elastic objects based on BEM‐BEM‐coupling

Based on a BEM‐BEM‐coupling method, the scattered pressure from elastic objects placed in water and partially buried in the sediment is calculated. For this reason, a special variant of the boundary element method (BEM) is implemented. It contains a pre‐ and a postprocessor with 3D visualization, in order to define the geometry of the scattering objects in the interface layer between fluid and sediment and the parameters needed for characterizing the fluid and the elastic material. The solver is able to perform numerical calculations in a multiple parallel manner. For the solution of the underlying system of linear equations, we use different kinds of approximate and direct solution techniques. Simple acoustical exterior problems, for example, the sound scattering by elastic solid cylinders and spheres placed in a fluid are treated by the BEM‐BEM‐coupling method. The results will be compared with analytical solutions or solutions obtained from other numerical methods.

Axisymmetrical and nonaxisymmetrical guided waves propagating in a solid elastic cylinder embedded in a solid medium

For NDT of rods and pipes, fundamental characteristics of guided waves are to be known, especially dispersion relations between frequency and wave number. A necessary step before detecting defects is to be able to calculate the propagated elastodynamical field in healthy waveguides. Thus, the goal of this work is the calculation of this field propagating in a cylindrical stratified waveguide. The incident field is generated at the end of the cylinder by a force or velocity source which is off‐axis and Gaussian distributed. First, Vector Hankel transform and Fourier series are combined to decompose this field with respect to the angular position. Second, each component is decomposed into an infinite sum of rays, i.e., elementary generalized conical waves. These waves undergo multiple reflections with the guiding surface of the waveguide. Third, we use Generalized Debye Series (GDS) for calculating the global reflection coefficients resulting from these multiple reflections. Finally, the total field is synthesized by the summation of the incident and reflected rays. Many outputs of this code can be exploited like velocity field, stress field, energy field in 2D or 3D spatiotemporal or frequential simulations. Diagrams obtained by this code are compared with results from DISPERSE software.

Analysis of solid elastic cylinders with internal losses using complete sets of functions

A method to determine the response of a solid, elastic, isotropic cylinder with internal losses and arbitrary length to radius ratio to excitations acting on its surface is presented. The method is based on the use of complete sets of functions and the response to arbitrary excitations can be analysed. It is illustrated for axisymmetric excitations that are anti-symmetric with respect to the plane midway between the ends of the cylinder. Excitations that are symmetric have been considered earlier. [D.D. Ebenezer et al., Forced vibrations of solid elastic cylinders, Journal of Sound and Vibration 282 (2005) 991–1007]. All components of displacement and stress are expressed as a sum of two infinite series that contain terms that are complete in the axial and radial direction, respectively. Therefore, arbitrary boundary conditions can be satisfied. In lossless cylinders, for each chosen set, there are difficulties at those frequencies at which certain functions become zero. Some other complete set of functions can be used at these frequencies. However, this difficulty does not arise for cylinders with losses. Two different sets are used to compute the resonance frequencies of lossless cylinders. The responses of cylinders with internal losses to uniform and concentrated excitations on the flat and curved surfaces are also presented to illustrate the approach. They are in good agreement with results obtained using ATILA—a finite element software package.

10402 球状弾性介在物をもつ弾性円柱の引張り(材料(1))

This paper deals with a stress concentration problem of an elastic circular solid cylinder with an elastic spherical inclusion under tension. Effects of Poisson's ratio, especially effects of negative Poisson's ratio, on the stress concentrations are investigated. Lakes has shown that there are materials with negative Poisson's ratios. A method of solution is presented for the problem by applying Green's functions for axisymmetric body force problems of infinite circular solid cylinder and fundamental solutions for axisymmetric problems of elasticity.