Nonlinear axisymmetric deformations of an elastic tube under external pressure

The problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the C++ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the “corner bulging” of short tubes, and multiple modes of deformation for longer tubes.     

Numerical and experimental study of elastoplastic tension-torsion processes in axisymmetric bodies under large deformations

We consider a numerical solution technique for generalized axisymmetric problems with torsion for elastoplastic bodies of revolution of arbitrary shape under large strains, as well as simple or complex loading, and the conditions of inhomogeneous stress-strain state. The processes of elastoplastic deformation, strain localization, and fracture of solid axisymmetric steel samples of variable thickness are studied experimentally and numerically for the cases of proportional and nonproportional kinematic torsional and/or tensile loading until failure. The mutual influence of torsion and tension on the deformation and failure under large strains is estimated.     

A semi-analytic analysis of shape memory alloy thick-walled cylinders under internal pressure

In this paper, a new method for analysis of the pseudoelastic response of shape memory alloy thick-walled cylinders subjected to internal pressure is proposed. Two cases of short and long cylinders are considered by assuming the plane stress and plane strain conditions. In each case, a three-dimensional phenomenological SMA constitutive model is simplified to obtain the corresponding two-dimensional constitutive relations. The cylinder is partitioned into a finite number of narrow annular regions, and appropriate assumptions are made in order to find a closed-form solution for the equilibrium equations in each annular region. The global solution is obtained by enforcing the stress continuity condition at the interface of the annular regions and imposing the boundary conditions. Several numerical examples are presented to demonstrate the efficiency of the proposed method, and the results are compared with three-dimensional finite element simulations.     

Analysis of dynamic stress intensity factors of thick-walled cylinder under internal impulsive pressure

Dynamic stress intensity factors are evaluated for thick-walled cylinder with a radial edge crack under internal impulsive pressure. Firstly, the equation for stress intensity factors under static uniform pressure is used as the reference case, and then the weight function for a thick-walled cylinder containing a radial edge crack can be worked out. Secondly, the dynamic stresses in uncracked thick-walled cylinders are solved under internal impulsive pressure by using mode shape function method. The solution consists of a quasi-static solution satisfying inhomogeneous boundary conditions and a dynamic solution satisfying homogeneous boundary condi- tions, and the history and distribution of dynamic stresses in thick-walled cylinders are derived in terms of Fourier-Bessel series. Finally, the dynamic stress intensity factor equations for thick-walled cylinder containing a radial edge crack sub- jected to internal impulsive pressure are given by dynamic weight function method. The finite element method is utilized to verify the results of numerical examples, showing the validity and feasibility of the proposed method.     

Gravity waves under nonuniform pressure over a free surface. Exact solutions

Plane periodic oscillations of an infinitely deep fluid are studied in the case of a nonuniform pressure distribution over its free surface. The fluid flow is governed by an exact solution of the Euler equations in the Lagrangian variables. The dynamics of an oscillating standing soliton are described, together with the scenario of the soliton evolution and the birth of a wave of an anomalously large amplitude against the background of the homogeneous Gerstner undulation (freak wave model). All the flows are nonuniformly vortical.     

Dynamic neck development in a polymer tube under internal pressure loading

The initiation and growth of necks in polymer tubes subjected to rapidly increasing internal pressure is analyzed numerically. Plane strain conditions are assumed to prevail in the axial direction. The polymer is characterized by a finite strain elastic–viscoplastic constitutive relation and the calculations are carried out using a dynamic finite element program. Numerical results for neck development are illustrated and discussed for tubes of various thicknesses. The sensitivity to the wave number of the thickness imperfections is studied with a focus on comparing a long wave length imperfection and a short wave length imperfection. After some thinning down at the necks, the mode of deformation switches to neck propagation along the circumference of the tube. A case is shown in which the necks have propagated along the entire tube wall, so that network locking in the polymer results in high stiffness against further expansion of the tube. The rate dependence of the necking behavior gives noticeable differences in neck development for slow loading versus fast loading.     

Collapse of thick-walled cylinders under external pressure

Hydrostatic collapse tests performed on thick-walled capped cylinders are described. Finite-element predictions which incluce the effects of end-cap stiffening, cross-section ovalities and material strain hardening are compared to experimental results. The analyses correctly predict the sequence of events leading to collapse, but experimental failure pressures are significantly below predictions. It is concluded that the von Mises yield criterion used in the analysis did not accurately represent the yield behavior of the 1018 steel tubing material of the test-cylinders for the triaxial-stress conditions of interest.     

Finite creep deformations of thick-walled tubes

As an application of the theory of finite deformations to creep problems, the steady-state creep of pressurized thick-walled tubes is analysed. Constitutive equations of steady-state creep in case of finite deformations are first derived by assuming the Prager-Drucker potential and Norton's law. Closed form solutions are obtained, and compared with the corresponding experiment as well as the analysis disregarding the effect of J₃ and that of infinitesimal deformations. The relation between the present analysis and that of Rimrott, performed by modifying the procedure of infinitesimal deformations, is also discussed.     

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.     

Limit analysis of viscoplastic thick-walled cylinder and spherical shell under internal pressure using a strain gradient plasticity theory

Plastic limit load of viscoplastic thick-walled cylinder and spherical shell subjected to internal pressure is investigated analytically using a strain gradient plasticity theory. As a result, the current solutions can capture the size effect at the micron scale. Numerical results show that the smaller the inner radius of the cylinder or spherical shell, the more significant the scale effects. Results also show that the size effect is more evident with increasing strain or strain-rate sensitivity index. The classical plastic-based solutions of the same problems are shown to be a special case of the present solution.     

Electromechanical phase transition of a dielectric elastomer tube under internal pressure of constant mass

The electromechanical phase transition for a dielectric elastomer(DE) tube has been demonstrated in recent experiments, where it is found that the unbulged phase gradually changed into bulged phase.Previous theoretical works only studied the transition process under pressure control condition, which is not consistent with the real experimental condition. This paper focuses on more complex features of the electromechanical phase transition under internal pressure of constant mass. We derive the equilibrium equations and the condition for coexistent states for a DE tube under an internal pressure, a voltage through the thickness and an axial force. We find that under mass control condition the voltage needed to maintain the phase transition increases as the process proceeds. We analyze the entire process of electromechanical phase transition and find that the evolution of configurations is also different from that for pressure control condition.     

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.     

Exact solutions for functionally graded pressure vessels in a uniform magnetic field

Analytical studies for magnetoelastic behavior of functionally graded material (FGM) cylindrical and spherical vessels placed in a uniform magnetic field, subjected to internal pressure are presented. Exact solutions for displacement, stress and perturbation of magnetic field vector in FGM cylindrical and spherical vessels are determined by using the infinitesimal theory of magnetoelasticity. The material stiffness and magnetic permeability obeying a simple power law are assumed to vary through the wall thickness and Poisson’s ratio is assumed constant. Stresses and perturbation of magnetic field vector distributions depending on an inhomogeneous constant are compared with those of the homogeneous case and presented in the form of graphs. The inhomogeneous constant, which includes continuously varying volume fraction of the constituents, is empirically determined. The values used in this study are arbitrary chosen to demonstrate the effect of inhomogeneity on stresses and perturbation of magnetic field vector distributions.     

Stability of an elastoplastic sphere under external pressure

The stability of ideally plastic, elastoplastic, and reinforced elastoviscoplastic bodies subjected to large subcritical strains was investigated in [1–4], The problems solved in these papers were related to the stability of systems in which homogeneous stress and strain fields arise in the initial state. The stability of an elastic thick-walled spherical shell subjected to external pressure leading to large subcritical strains was investigated in [5]. The stability of an axisymmetric sphere of elastoplastic material subjected to large plastic strains is examined below.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 155–159, September–October, 1977.     

A model for elastic-plastic pressure sensitive materials subjected to large deformations

Development of a finite deformation elasto-plastic model for pressure sensitive materials is presented. The chosen model, which has its roots in the MRS-Lade material model is influenced by recent developments. The thermodynamic consequences of introducing non-associative yielding (both deviatoric and volumetric) and hardening⧸softening characteristics are assessed. The consistently linearized Algorithmic Tangent Stiffness (ATS) tensor is presented. This tensor is used in the constitutive driver as a key feature of the efficient iterative procedure for satisfying equilibrium in the case of stress (or mixed) control.The chosen model is calibrated using data from experiments conducted in a Directional Shear Cell (DSC) , which has been used extensively at the University of Colorado at Boulderto investigate the behavior of pressure sensitive materials under deformations of large magnitude.     

Crystal plasticity based analysis of localized necking in aluminum tube under internal pressure

The finite element method is used to numerically simulate localized necking in aluminum alloy tube under internal pressure. The measured electron backscatter diffraction (EBSD) data are directly incorporated into the finite element model and the constitutive response at an integration point is described by the single crystal plasticity theory. The tube is assumed sufficiently long, so that length changes as well as the end effects can be ignored and a plane strain analysis can be performed. Localized necking is assumed to be associated with surface instability, the onset of unstable thinning. It is demonstrated that such a surface instability/necking is the natural outcome of the present approach, and an artificial initial imperfection required by other approaches is not necessary in the present analysis. The effects of spatial grain orientation distribution, material strain rate sensitivity, work hardening, and initial surface topography on necking are discussed. It is found that localized necking depends strongly on both the initial texture and its spatial orientation distribution, while the initial surface topography has a negligible effect on necking.