Isotropic Cylinder Research Articles

Generalized Thermoelasticity of Thermal-Shock Problem in a Non-homogeneous Isotropic Hollow Cylinder with Energy Dissipation

A solution of a thermal-shock problem of generalized thermoelasticity of a non-homogeneous isotropic hollow cylinder using a finite-element method is developed. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory of type II and type III by an appropriate choice of parameters. The problem has been solved numerically using a finite-element method. Numerical results for the distributions of displacement, temperature, radial stress, and hoop stress are represented graphically. The results indicate that the effects of non-homogeneity are very pronounced. The effects of non-homogeneity are presented with the two types of the Green and Naghdi theory.

Fast Compressional Wave Scattering from Cylinder in Porous Media

A mathematical model is presented to describe the scattering of a fast compressional plane wave (fast P-wave) incident at an arbitrary angle on isotropic cylinder embedded in a fluid-saturated porous medium. The expressions of components of solid and filtration displacement, pore fluid pressure and stress tensor in porous medium are obtained by potential function using the Biot’s theory and that of isotropic cylinder are also given. We investigate the properties of the form function of a backscatted fast P-wave for cement and steel cylinder system. It shows the influence of incident angle on form function for steel is more obvious than that for cement cylinder and form function curves are very different between steel and cement cylinder.

Transient elastodynamic response of finite and infinite solid cylinders

The orthogonal eigensolutions for the vibrations of an isotropic finite solid cylinder with a traction-free lateral boundary and rigid-smooth end boundaries are provided. The transient elastodynamic response of this solid cylinder is then constructed using the method of eigenfunction expansion and further extended explicitly and concisely to that of an isotropic infinite solid cylinder. The numerically evaluated analytical solution is shown to compare favorably with that by finite element method (FEM). The effect of external forces on the excitation of each guided wave mode can be quantitatively investigated on the basis of the present solution.

Generalized thermoelasticity of the thermal shock problem in an isotropic hollow cylinder and temperature dependent elastic moduli

In this paper, we construct the equations of generalized thermoelasticity for a non-homogeneous isotropic hollow cylider with a variable modulus of elasticity and thermal conductivity based on the Lord and Shulman theory. The problem has been solved numerically using the finite element method. Numerical results for the displacement, the temperature, the radial stress, and the hoop stress distributions are illustrated graphically. Comparisons are made between the results predicted by the coupled theory and by the theory of generalized thermoelasticity with one relaxation time in the cases of temperature dependent and independent modulus of elasticity.

Accuracy of shell and solid elements in vibration analyses of thin- and thick-walled isotropic cylinders

Three-dimensional (3D) theory of elasticity solutions for vibration analysis of simply supported cylinders are obtained and used to find the natural frequencies. The results are compared with various finite element models using both shell and three-dimensional solid elements to verify the accuracy of these finite element models. In addition, results are obtained for various hollow cylinders for possible verification of shell theories and shell elements based on such theories. A point is made here that although 3D elements can provide accurate results, they are not practical in industrial applications of thin- and thick-walled structures. Instead, shell elements are needed for such applications. However, further improvements to the shell theory used in the finite element formulation are needed to improve the accuracy of these shell elements.

Thermal Stress-Focusing Effect in a Transversely Isotropic Cylinder with Phase Transformation

When a transversely isotropic cylinder is suddenly subjected to an impact cooling, thermal stress waves occur at the surface of a cylinder the moment the thermal impact is applied. The stress waves in a cylinder with a focusing point proceed radially inward to the center of a cylinder. The wave may accumulate at the focusing point and give rise to very large stress magnitudes, even though the initial stresses may be relatively small. This phenomenon is called the stress-focusing effect. When the cooling process passes through the phase transformation point of metal, the phase transformation may be occurred. The eigen strains of phase transformation also cause the stress-focusing effects at the center of a cylinder. In this article, we analyze the stress-focusing effects induced by the thermal and phase transformational stresses caused by the thermal shock in a transversely isotropic cylinder theoretically. The results give a clear indication of the mechanism of stress-focusing effects and clarify the orders of singularities of the stress-focusing effects in a transversely isotropic cylinder.

側面に任意荷重を受ける異方性だ円柱の解析

This paper presents general solutions for anisotropic cylinder using complex stress functions based on Lekhnitskii formalism. Up to now, an analytical solution of isotropic cylinder has been widely used whereas solutions of anisotropic elliptic cylinders subject to arbitrary surface loads have not been derived. The proposed complex stress functions, which are developed by Laurent series, enable to solve conditions under distributed loads like hydro pressures and concentrated loads. The function can also analyze a shear load parallel to longitudinal axis of a cylinder. Analyzed stress distributions are graphically represented.

Intrinsic anisotropy of the effective acoustic properties in metafluids made of two-dimensional cylinder arrays

Within a unified theoretical framework, we extract the omnidirectional effective acoustic parameters for the metafluid consisting of isotropic fluid cylinders embedded in an isotropic fluid background. Besides the analytical formulas for the effective parameters reported previously, i.e., the bulk modulus and the mass density perpendicular to the cylinders, we also derive a simple expression for the effective mass density parallel to the cylinders. As expected, these two effective mass densities are not identical and constitute an anisotropic density tensor. Such intrinsic anisotropy can be engineered much stronger than the pure in-plane anisotropy induced by either anisotropic lattices or anisotropic scatterers.

Wave propagation in a transversely isotropic thermoelastic solid cylinder of arbitrary cross-section

The wave propagation in an infinite, homogeneous, transversely isotropic solid cylinder of arbitrary cross-section is studied using Fourier expansion collocation method, within the frame work of linearized, three-dimensional theory of thermoelasticity. Three displacement potential functions are introduced, to uncouple the equations of motion and the heat conduction. The frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibration and are studied numerically for elliptic and parabolic cross-sectional zinc cylinders. The computed non-dimensional wave numbers are presented in the form of dispersion curves.

On the Modeling of Extension-Torsion Experimental Data for Transversely Isotropic Biological Soft Tissues

For the problem of torsion superimposed on extension of incompressible nonlinearly elastic transversely isotropic circular cylinders, a simple asymptotic analysis is carried out on using a small parameter that reflects the moderate twisting of slender cylinders, which corresponds to a typical testing regime for biological soft tissue. The analysis is carried out for a subclass of strain-energy densities that reflect transversely isotropic material response. On using a four-parameter polynomial expression for the strain-energy density in terms of certain classical invariants, this analysis is shown to be in excellent agreement with experimental data obtained by other authors for rabbit papillary muscles. An explicit condition on the strain-energy density is obtained that determines whether the stretched cylinder tends to elongate or shorten on twisting. For the special case of pure torsion where no extension is allowed, this condition determines whether the classical or reverse Poynting effect occurs. For the rabbit papillary muscles, the theoretical results predict and the experimental results confirm that a reverse Poynting-type effect occurs where the stretched rabbit muscle tends to shorten on twisting.

Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new three-dimensional refined higher-order theory

Closed-form formulation of three-dimensional (3-D) refined higher-order shear deformation theory (RHOST) for the free vibration analysis of simply supported–simply supported and clamped–clamped homogenous isotropic circular cylindrical shells is presented. The proposed RHOST accounts for the effects of in-plane and rotary inertias as well as the effects of transverse normal and shear strains on the dynamic response of cylindrical shells. Also, the present analysis incorporates trapezoidal shape factor of a shell element that arise due to the fact that the stresses over the thickness of the shell are to be integrated on a trapezoidal-like cross-section of a shell element to obtain the accurate stress-resultants. Therefore, the present theory refines other HOSTs established hitherto for free vibration analysis of thick circular cylindrical shells. The equations of motion are obtained using Hamilton's principle. Solutions are obtained using the Galerkin method. Numerical results, not hitherto reported in the literature, are presented for the natural frequencies of isotropic long/short and hollow/solid cylinders. Numerical results indicate that for thick cylinders with large length-to-radius ratios, approximate series truncation methods are inadequate for calculating the stress-resultants and just exact integration yields accurate stress-resultants leading to a reliable prediction of natural frequencies. Natural frequencies associated to higher-mode numbers of thick isotropic cylinders, never published in the literature before, are also considered in this paper. It is shown that stress-resultants calculated using exact integration over the cylinder cross-section are necessary to obtain accurate results of the free vibration analysis, especially for higher-mode numbers of thick isotropic cylinders. The closed-form solutions presented herein are compared with the available exact 3-D elasticity and analytical solutions in the available literature and excellent agreement is obtained. Also, the validity of the results is established with the aid of commercial finite element software.

Modeling of thermoelastic damping and frequency shift of vibrations in a transversely isotropic solid cylinder

PurposeThe purpose of this paper is to analyze the free vibrations in a stress free and thermally insulated (or isothermal), homogeneous, transversely isotropic, solid cylinder based on three‐dimensional coupled thermoelasticity, which is initially undeformed and kept at uniform temperature.Design/methodology/approachThe displacement potential functions have been introduced in the equations of motion and heat conduction in order to decouple the purely shear and longitudinal motions. The system of governing partial differential equations is reduced to four second‐order coupled ordinary differential equations in radial coordinate by using the method of separation of variables. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. The convergence analysis of matrix Frobenius method has been successfully carried out.FindingsThe purely transverse mode is found to be independent of the rest of the motion and temperature change. The natural frequency, dissipation factor, inverse quality factor and frequency shift of vibrations in a stress free solid cylinder get significantly affected due to thermal variations and thermo‐mechanical coupling.Originality/valueA new procedure is used and compared to other methods available in the literature.

A study on dynamic behaviour of functionally graded thick hollow circular cylinders

This article presents the analysis of functionally graded hollow circular cylinders with finite length under axisymmetric dynamic loads. It is assumed that the functionally graded cylinder is comprised of metal-phase and ceramic-phase, whereas material properties are graded in the thickness direction of the cylinder according to power law distribution. Two-dimensional finite element method in conjunction with the Newmark method is used to solve the system of time-dependent coupled equations that govern the dynamic responses. By introducing especial elements, it is possible to distribute the material properties through the thickness of cylinder exactly according to the power law distribution. Dynamic loads applied on the cylinder are axisymmetric in the hoop direction and can vary in the radial and axial directions. As examples, the transient responses of functionally graded cylinders, which are excited in radial direction by increasing and sudden internal pressure and line load and also in axial direction by longitudinal incident wavelet, are calculated, and the characteristics of waves are discussed. The results in the present study are obtained for thick cylinder and cylindrical shell and compared with the results for isotropic cylinders.

Boundary conditions for axisymmetric circular cylinder in one-dimensional hexagonal quasicrystals

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of axisymmetric circular cylinder in one-dimensional (1D) hexagonal quasicrystals (QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, and using the reciprocal theorem and general solution, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the circular cylinder with axisymmetric deformations in 1D hexagonal QCs. The results of the present paper enable us to establish a set of correct boundary conditions, most of which are obtained for the first time. Furthermore, corresponding necessary conditions for the case of axisymmetric circular cylinder with transversely isotropic are derived subsequently, and their isotropic elastic counterparts are also obtained. The accuracy of mixed boundary conditions of isotropic axisymmertic circular cylinder is proved at last.

Asymptotic analysis of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder

The method of asymptotic integration of equations of elasticity [1] is used to study the behavior of the solution of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder of small thickness. Under the assumption that the load is sufficiently smooth, the asymptotic method [1] is used to construct inhomogeneous solutions. An algorithm for constructing exact particular solutions of the equilibrium equations is given for loads of specific types in the case where the cylinder lateral surface is loaded by forces polynomially depending on the axial coordinate. Then the homogeneous solutions are constructed. The asymptotic expansions of homogeneous solutions are obtained, and the above analysis is used to explain the character of the stress-strain state.

Axially symmetric temperature stresses in an elastic isotropic cylinder of finite length

We consider an axially symmetric problem of the thermostressed state of a solid cylinder of finite length with a load-free surface. Using the method of superposition, we have constructed the complete analytical solution of this problem, which is reduced to the solution of a system of linear algebraic equations. We have proposed a method for determining the asymptotic behavior of coefficients in these systems, which enables us to develop an efficient algorithm for the calculation of stresses in the cylinder, including regions near its end-face circles. Typical examples are considered.

Thermal shock problem in a homogeneous isotropic hollow cylinder with energy dissipation

In this work, we constructed the equations of generalized thermoelasticity of a homogeneous isotropic hollow cylinder. The formulation is applied in the context of the Green and Naghdi theory of types II and III. The material of the cylinder is assumed to be homogeneous isotropic both mechanically and thermally. The problem has been solved numerically using a finite-element method. Numerical results for the temperature distribution, displacement, radial stress, and hoop stress are represented graphically. Comparisons are made with the results predicted by the types II and III. The results obtained in this paper can be used to design various homogeneous thermoelastic elements under thermal load to meet special engineering requirements.

Wave Propagation in a Magnetoelectroelastic Rod

In this paper the propagation of axially symmetric compression wave in an infinite transversely isotropic magnetoelectroelastic cylinder is presented. Following Pan(2002), the general boundary conditions on the surface of the magnetoelectroelastic cylinder are written by a simple vector equation that is similar to the purely elastic counterpart. By assuming the possible form of the solution, the generalized Pochhammer frequency equations are presented and the corresponding phase velocity curves in some range are given. Numerical examples are also presented to show phase velocity curves under the first 4 kinds of boundary conditions. Other boundary conditions can be studied under farther research.

Relationships between the Torsions of Anisotropic Circular and of Isotropic Elliptical Elastic Cylinders

Relationships between the solutions to the Saint-Venant torsion of an anisotropic circular cylinder and of an isotropic elliptical cylinder are proven. It is shown that if the geometrical data describing an isotropic elliptical cylinder are certain functions of the shear rigidities of an anisotropic circular cylinder, then the torsional functions and torsional rigidities of the considered cross-sections are the same. The material of both cylinders is homogeneous. The class of anisotropy taken into account has at least one plane of elastic symmetry normal to the axis of the circular cylinder.

Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders

In the context of the theory of non-linear elasticity for rubber-like materials, the problem of finite extension and torsion of a circular bar or tube has been widely investigated. More recently, this problem has attracted considerable attention in studies on the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in non-linear elasticity was concerned specifically with the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic non-linearly elastic materials that undergo severe strain-stiffening in the stress–stretch response. For two specific material models that reflect limiting chain extensibility at the molecular level, it was shown that, in the absence of an additional axial force, a transition value γ=γt of the axial stretch exists such that for γ<γt, the stretched cylinder tends to elongate on twisting whereas for γ>γt, the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models for rubber such as the Mooney–Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. Here we investigate similar issues for fiber-reinforced transversely isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect limited fiber extensibility and in the biomechanics context model the stretch induced strain-stiffening of collagen fibers on loading. They have been shown to model the mechanical behavior of fiber-reinforced rubber and many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The results obtained here have important implications for extension–torsion tests for fiber-reinforced materials, for example in the development of accurate extension–torsion test protocols for determination of material properties of soft tissues.