In-plane buckling of circular arches and rings with shear deformations

A finite strain formulation is developed for elastic circular arches and rings in which the effects of shear deformations are included. Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for stress and finite strain are based on a hyperelastic constitutive model. Virtual work and equilibrium equations are derived. Closed-form in-plane buckling solutions are developed for circular rings and high arches under hydrostatic pressure. The effects of axial deformation prior to buckling as well as shear deformations are included in the buckling analysis. The formulation developed is compared with solutions in the literature and to the predictions of the finite element package ANSYS. The importance of including the effects of shear deformations for deep arches is investigated.     

Uniqueness and stability of rigid-plastic spherical shells subjected to finite deformation under hydrostatic pressure

The rate problem for rigid-plastic strain-hardening deformations in structures subjected to prescribed hydrostatic pressure surface load is stated rigorously with due account of finite deformations. From the basic theory, a complete solution for admissible stress and velocity fields occurring at bifurcation is obtained for the problem of a spherical shell under arbitrary combinations of internal and external pressures. An earlier proven, sufficient condition for the uniqueness of continuing quasi-static deformation of a spherical shell is shown to be one of necessity. In the case of solely external pressure, it is shown that buckling modes are excluded by attention to an isotropically strain-hardening material with a non-singular yield surface. For preponderant internal pressure, however, it is possible for the predicted bifurcation mode to occur under increasing pressure.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials

The classical problem of simple shear in nonlinear elasticity has played an important role as a basic pilot problem involving a homogeneous deformation that is rich enough to illustrate several key features of the nonlinear theory, most notably the presence of normal stress effects. Here our focus is on certain ambiguities in the formulation of simple shear arising from the determination of the arbitrary hydrostatic pressure term in the normal stresses for the case of an incompressible isotropic hyperelastic material. A new formulation in terms of the principal stretches is given. An alternative approach to the determination of the hydrostatic pressure is proposed here: it will be required that the stress distribution for a perfectly incompressible material be the same as that for a slightly compressible counterpart. The form of slight compressibility adopted here is that usually assumed in the finite element simulation of rubbers. For the particular case of a neo-Hookean material, the different stress distributions are compared and contrasted.     

Lateral buckling of beams with shear deformations – A hyperelastic formulation

The equilibrium and buckling equations are derived for the lateral buckling of a prismatic straight beam. A consistent finite strain constitutive law is used, which is based on a hyperelastic model for an isotropic material. The kinematics of the cross-sectional deformations are based on a Timoshenko type beam displacement of the cross-sectional plane using Euler angles and two shear finite rotations coupled with warping taken normal to the displaced plane. Also derived are the second order approximations to the displacements, curvatures, twist and internal actions. The constitutive relationships for the internal actions reveal new coupling terms between the bending moments, torsion and bimoment, which are functions of the cross-sectional warping and shear deformations. New Wagner type nonlinear torsion terms are derived which are functions of the warping of the cross-sectional plane, and are coupled to the twisting and shear deformations of the cross-section. Solutions are determined for the lateral buckling of a prismatic monosymmetric beam under pure bending and the flexural–torsional buckling under axial compression. For the flexural–torsional buckling problem it is found that the Euler type column buckling formula is consistent with Haringx’s column buckling formula while the torsional buckling formula is different to conventional equations. The second variation of the total potential is also derived. The effects of shear deformations are explored by examining the non-dimensional lateral buckling equation for a simply supported beam.     

Finite deformation of an ideal rigid-plastic membrane

We consider finite deformation of a membrane which is a cylinder in the initial state under a pressure uniformly distributed over the inner surface. The problem is solved using the model of deformation of an incompressible rigid-plastic material with full plasticity. Exact analytical relations are obtained for the pressure and kinematic characteristics as functions of the rotation angle of the normal on the contour of the shell. Elongations and displacements are found as functions of the radial coordinate. The time of reaching the maximum value of the external pressure is determined. It is shown that the change in the membrane thickness along the radial coordinate is constant.     

Steady diffusion of an ideal fluid through a pre-stressed and reinforced hollow conduit subjected to combined finite deformations

A problem dealing with the radial steady diffusion of an ideal fluid through a pre-stressed fibrous hollow cylinder subjected to finite deformations is investigated. Such a problem has relevance to several technical problems such as: (a) improving the method for performing prosthesis conduit for use with living tissue, (b) more understanding the problem of atherogenesis, and (c) ultra filtration process. The numerical results show that the presence of a pre-stress reduces considerably the sensitivity of the dimensionless trans-mural pressure to imposed dimensionless flux. This effect is confirmed with respect to the variation of the circular and axial shear.     

In-plane buckling of prismatic funicular arches with shear deformations

The in-plane buckling behavior of funicular arches is investigated numerically in this paper. A finite strain Timoshenko beam-type formulation that incorporates shear deformations is developed for generic funicular arches. The elastic constitutive relationships for the internal beam actions are based on a hyperelastic constitutive model, and the funicular arch equilibrium equations are derived. The problems of a submerged arch under hydrostatic pressure, a parabolic arch under gravity load and a catenary arch loaded by overburden are investigated. Buckling solutions are derived for the parabolic and catenary arch. Subsequent investigation addresses the effects of axial deformation prior to buckling and shear deformation during buckling. An approximate buckling solution is then obtained based on the maximum axial force in the arch. The obtained buckling solutions are compared with the numerical solutions of Dinnik (Stability of arches, 1946) [1] and the finite element package ANSYS. The effects of shear deformation are also evaluated.     

Creep of thick-walled cylinders subjected to internal pressure and axial load

A creep theory is presented to predict deformations at any specified time for a thick-walled cylinder subjected to internal pressure and axial load. The theory is based on the usual assumptions that the deformations are infinitestimal, that the material is incompressible and that the total strain theory is valid. The stress-strain-time relation for the material is assumed to be represented by an isochronous stress-strain diagram which is approximated by an arc hyperbolic sine function. The experimental part of the investigation included tests of thick-walled cylinders made of high-density poly-ethylene whose ratio of outside to inside radii were either 1.5 or 2.0. The test cylinders were either tested as closed-ented cylinders with internal pressure or subjected to a combination of internal pressure and axial load. Also, the application of the theory for varying load conditions was studied. Good agreement was found between theory and experiment.     

Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model. This revised version was published online in August 2006 with corrections to the Cover Date.     

The torsion of a composite,nonlinear-elastic cylinder with an inclusion having initial large strains

This article considers a static problem of torsion of a cylinder composed of incompressible, nonlinear-elastic materials at large deformations. The cylinder contains a central, round, cylindrical inclusion that was initially twisted and stretched (or compressed) along the axis and fastened to a strainless, external, hollow cylinder. The problem statement and solution are based on the theory of superimposed large strains. An accurate analytical solution of this problem based on the universal solution for the incompressible material is obtained for arbitrary nonlinear-elastic isotropic incompressible materials. The detailed investigation of the obtained solution is performed for the case in which the cylinders are composed of Mooney-type materials. The Poynting effect is considered, and it is revealed that composite cylinder torsion can involve both its stretching along the axis and compression in this direction without axial force, depending on the initial deformation.     

Axisymmetric deformations of circular rings made of linear and Neo-Hookean materials under internal and external pressure: A benchmark for finite element codes

The axisymmetric deformations of thick circular rings are investigated. Four materials are explored: linear material, incompressible Neo-Hookean material and Ogden's and Bower's forms of compressible Neo-Hookean material. Radial distributed forces and a displacement-dependent pressure are the external loads. This problem is relatively simple and allows analytical, or semi-analytical, solution; therefore it has been chosen as a benchmark to test commercial finite element software for various material laws at large strains. The solutions obtained with commercial finite element software are almost identical to the present semi-analytical ones, except for the linear material, for which commercial finite element programs give incorrect results.     

Finite strain––beam theory

An appropriate strain energy density for an isotropic hyperelastic Hookean material is proposed for finite strain from which a constitutive relationship is derived and applied to problems involving beam theory approximations. The physical Lagrangian stress normal to the surfaces of a element in the deformed state is a function of the normal component of stretch while the shear is a function of the shear component of stretch. This paper attempts to make a contribution to the controversy about who is correct, Engesser or Haringx with regard to the buckling formula for a linear elastic straight prismatic column with Timoshenko beam-type shear deformations. The derived buckling formula for a straight prismatic column including shear and axial deformations agrees with Haringx’s formula. Elastica-type equations are also derived for a three-dimensional Timoshenko beam with warping excluded. When the formulation is applied to the problem of pure torsion of a cylinder no second-order axial shortening associated with the Wagner effect is predicted which differs from conventional beam theory. When warping is included, axial shortening is predicted but the formula differs from conventional beam theory.     

Analysis of functionally graded thick truncated cone with finite length under hydrostatic internal pressure

Finite Element Method based on Rayleigh–Ritz energy formulation is applied to obtain the elastic behavior of functionally graded thick truncated cone. The cone has finite length, and it is subjected to axisymmetric hydrostatic internal pressure. The inner surface of the cone is pure ceramic and the outer surface is pure metal, and the material composition varying continuously along its thickness. Using this method, the effects of semi-vertex angle of the cone and the power law exponent on distribution of different types of displacements and stresses are considered.     

Inelastic Deformation of Flexible Spherical Shells with Two Circular Openings

Elastoplastic analysis of thin-walled spherical shells with two identical circular openings is carried out with allowance for finite deflections. The shells are made of an isotropic homogeneous material and subjected to internal pressure of known intensity. The distributions of stresses (strains or displacements) along the contours of the openings and in the zone of their concentration are studied by solving doubly nonlinear boundary-value problems. The solution obtained is compared with the solutions that account for only physical nonlinearity (plastic deformations) and only geometrical nonlinearity (finite deflections) and with a numerical solution of the linearly elastic problem. The stress–strain state near the two openings is analyzed depending on the distance between the openings and the nonlinear factors accounted for     

On the Stability of Incompressible Elastic Cylinders in Uniaxial Extension

Consider a cylinder (not necessarily of circular cross-section) that is composed of a hyperelastic material and which is stretched parallel to its axis of symmetry. Suppose that the elastic material that constitutes the cylinder is homogeneous, transversely isotropic, and incompressible and that the deformed length of the cylinder is prescribed, the ends of the cylinder are free of shear, and the sides are left completely free. In this paper it is shown that mild additional constitutive hypotheses on the stored-energy function imply that the unique absolute minimizer of the elastic energy for this problem is a homogeneous, isoaxial deformation. This extends recent results that show the same result is valid in 2-dimensions. Prior work on this problem had been restricted to a local analysis: in particular, it was previously known that homogeneous deformations are strict (weak) relative minimizers of the elastic energy as long as the underlying linearized equations are strongly elliptic and provided that the load/displacement curve in this class of deformations does not possess a maximum.     

Axisymmetric smooth contact for an elastic isotropic infinite hollow cylinder compressed by an outer rigid ring with circular profile

A contact problem for an infinitely long hollow cylinder is considered. The cylinder is compressed by an outer rigid ring with a circular profile. The material of the cylinder is linearly elastic and isotropic. The extent of the contact region and the pressure distribution are sought. Governing equations of the elasticity theory for the axisymmetric problem in cylindrical coordinates are solved by Fourier transforms and general expressions for the displacements are obtained. Using the boundary conditions, the formulation is reduced to a singular integral equation. This equation is solved by using the Gaussian quadrature. Then the pressure distribution on the contact region is determined. Numerical results for the contact pressure and the distance characterizing the contact area are given in graphical form. The English text was polished by Yunming Chen     

Control of vortex shedding behind circular cylinder for flows at low Reynolds numbers

It has been observed by researchers in the past that vortex shedding behind circular cylinders can be altered, and in some cases suppressed, over a limited range of Reynolds numbers by proper placement of a second, much smaller, ‘control’ cylinder in the near wake of the main cylinder. Results are presented for numerical computations of some such situations. A stabilized finite element method is employed to solve the incompressible Navier–Stokes equations in the primitive variables formulation. At low Reynolds numbers, for certain relative positions of the main and control cylinder, the vortex shedding from the main cylinder is completely suppressed. Excellent agreement is observed between the present computations and experimental findings of other researchers. In an effort to explain the mechanism of control of vortex shedding, the streamwise variation of the pressure coefficient close to the shear layer of the main cylinder is compared for various cases, with and without the control cylinder. In the cases where the vortex shedding is suppressed, it is observed that the control cylinder provides a local favorable pressure gradient in the wake region, thereby stabilizing the shear layer locally. Copyright © 2001 John Wiley & Sons, Ltd.     

Fluid-pressure eigenstates and bifurcation in tension specimens under lateral pressure

An analysis is presented of an eigenstate that may be significant in deformation processes where part of the surface of a body is subjected to loading by uniform fluid pressure. The ‘fluid-pressure eigenstate’ is a configuration in which quasi-static incremental deformation is possible under surface traction-rates that are related to the instantaneous velocity field in a certain way, the fluid pressure being momentarily stationary. Deformation processes exist such that, given certain rate boundary-conditions, uniqueness of the incremental deformation is guaranteed at every instant up to a fluid-pressure eigenstate. For a cylindrical specimen, of arbitrary cross-section, of elastic/plastic or incompressible, finite elastic material it is shown that the first fluid-pressure eigenstate to be reached on a path of uniform stretching corresponds to the instant at which the ‘effective load’ reaches a maximum. No fluid-pressure eigenstates are reached in isotropic Cauchy-elastic solids under all-round fluid pressure loading provided the physically reasonable conditions that the instantaneous bulk and shear moduli remain positive are satisfied.     

Extension,torsion and expansion of an incompressible,hemitropic Cosserat circular cylinder

Constitutive equations for the stress and couple stres on an incompressible, hemitropic, constrained Cosserat material are derived, and the theory is applied to study the problem of finite extension, torsion and expansion of a circular cylinder. As in the theory of isotropic simple elastic materials, it is found that the deformation is controllable by application of only a normal force and a tosional moment at the cylinder ends. It is shown that in general the well known universal relation between the torsional stiffness and the axial force for incompressible, isotropic simple materials in the limit as the twist goes to zero does not exist for incompressible, hemitropic Cosserat materials. However, for a special and unusual class of hemitropic materials, the same universal formula is found to hold for a certain reduced torsional stiffness. The main problem is solved completely for incompressible, hemitropic, linearly elastic, Cosserat materials; and certain additional special features of the Kelvin-Poynting type, which here appear to the first order in the amount of twist of the cylinder, are derived and discussed in relation to experimentally observed composite material behavior.