Thick-walled Cylindrical Tube Research Articles

On the helical buckling of anisotropic tubes with application to arteries

The helical stability experienced by elastic cylinders is investigated using numerical methods. A doubly fiber-reinforced incompressible nonlinear elastic tube subject to axial loading, internal pressure and twist is examined using a numerical scheme based on the modified Riks (quasi-static) procedure. Under the application of such loadings (qualitatively physiological for arteries) vessels form tortuous shapes and a range of topologically and geometrically complex morphologies. These configurations can be highly unstable owing to the nonlinear interaction (geometry, material and self contact) among the multiple bifurcation modes. The present work attempts to model such complex configurations using a thick-walled cylindrical tube. These complex morphologies are most commonly termed as helical coiling, looping and winding in the biomechanics community. We show that these bifurcations are very sensitive to the applied pressure, axial stretch and the amount of twist. Illustration of the helical buckling is then provided by considering an anisotropic constitutive model that includes both fiber stretching and fiber shearing, as opposed to previous analyses that only consider fiber stretching. The numerical implementation is achieved via user routine in the finite element code Abaqus.

Modelling of a production process of irreversible strains in a material of a thick-walled cylindrical tube under the influence of inner pressure

This article considers the processes of the accumulation of irreversible strains in an elastoviscoplastic material of a thick-walled cylindrical tube in the framework of small strains. The material is deformed under the action of uniform inner pressure changing with time. Irreversible strains accumulate in the material by means of two different mechanisms, namely creep and plastic flow. Initially, whereas stresses increase in the deformed body due to mechanical influence, irreversible strains are produced as creep strains because of the viscous material properties. When the stress state reaches a loading surface, the production mechanism of irreversible strains becomes plastic. When unloading, this sequence in the production mechanisms is opposite, i.e. the fast plastic mechanism is replaced by the slow viscous one. The continuity in the growth of irreversible strains is provided by the corresponding setting of creep and plastic potentials. The possibility of the use of these potentials in the form of piecewise linear functions is investigated. It makes possible to use the developed mathematical formalism of the ideal plasticity theory. The creep and plastic flow processes under increasing and constant inner pressures are considered. The unloading process and repeated plastic flow under decreasing inner pressure, the stress relaxation after the full removal of a loading force are investigated. The reversible and irreversible strains, stresses and displacements in the medium are calculated. The laws of the motion of elastoplastic boundaries in the cylindrical layer are determined.

Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

The application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure (P) – inflation (lambda or v) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

Computational simulations of the helical buckling behavior of blood vessels.

Tortuous vessels are often observed in vivo and could hinder or even disrupt blood flow to distal organs. Besides genetic and biological factors, the in vivo mechanical loading seems to play a role in the formation of tortuous vessels, but the mechanism for formation of helical vessel shape remains unclear. Accordingly, the aim of this study was to investigate the biomechanical loads that trigger the occurrence of helical buckling in blood vessels using finite element analysis. Porcine carotid arteries were modeled as thick-walled cylindrical tubes using generalized Fung and Holzapfel-Gasser-Ogden constitutive models. Physiological loadings, including axial tension, lumen pressure, and axial torque, were applied. Simulations of various geometric dimensions, different constitutive models and at various levels ofaxial stretch ratios, lumen pressures, and twist angles were performed to identify the mechanical factors that determine the helical stability. Our results demonstrated that axial torsion can cause wringing (twist buckling) that leads to kinking or helical coiling and even looping and winding. The specific buckling patterns depend on the combination of lumen pressure, axial torque, axial tension, and the dimensions of the vessels. This study elucidates the mechanism of how blood vessels buckle under various mechanical loads and how complex mechanical loads yield helical buckling.

Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field

This paper is concerned with the bifurcation analysis of a pressurized electroelastic circular cylindrical tube with closed ends and compliant electrodes on its curved boundaries. The theory of small incremental electroelastic deformations superimposed on a finitely deformed electroelastic tube is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect cylindrical shape of the tube. First, prismatic bifurcations are examined and solutions are obtained which show that for a neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result contrasts with that for the purely elastic case for which prismatic bifurcation modes were found only for an externally pressurized tube. Second, axisymmetric bifurcations are analyzed, and results for both neo-Hookean and Mooney–Rivlin electroelastic energy functions are obtained. The solutions show that in the presence of a moderate electric field the electroelastic tube becomes more susceptible to bifurcation, i.e., for fixed values of the axial stretch axisymmetric bifurcations become possible at lower values of the circumferential stretches than in the corresponding problems in the absence of an electric field. As the magnitude of the electric field increases, however, the possibility of bifurcation under internal pressure becomes restricted to a limited range of values of the axial stretch and is phased out completely for sufficiently large electric fields. Then, axisymmetric bifurcation is only possible under external pressure.

On the effect of temperature gradient on the stability of circular tubes made of hyperelastic entropic material

Rubber-like materials are very applicable in almost all fields of industries, but due to their large deformation characteristic, they can exhibit a variety of instabilities. Accordingly, many researchers have been motivated to investigate the effects of different parameters on the stability of hyperelastic cylindrical tubes under finite deformation, while the effects of temperature gradient have not been considered. In this paper, the effects of temperature variation on the stability and thermo-mechanical behavior of the cylindrical tubes made of the entropic materials such as rubber-like materials and elastomers are investigated via an effective strain energy density function. To this purpose, an Ogden-type strain energy density with only integer powers is applied in order to determine an analytical solution, not involving the integral form, for the stress distribution through the wall thickness of cylindrical tubes at finite deformation thermoelasticity. This problem is examined in two cases including (i) a thick-walled cylindrical tube under internal pressure and uniform variation of temperature and (ii) a thick-walled cylindrical tube under internal pressure and temperature gradient, simultaneously. It was observed that the positive temperature gradients in comparison with environment temperatures improve the stability of the circular tubes made of the entropic materials.

Analytical solutions for yield onset achievement in FGM thick walled cylindrical tubes undergoing thermomechanical loads

Abstract A thick-walled cylindrical tube made of a Functionally Graded Material (FGM) and undergoing thermomechanical loads has been investigated. The tube material properties have been considered to be isotropic and vary as a power function of the radius. A yield variable parameter has been defined according to von Mises criterion, has been introduced into the stress governing equation. The value of the radius where yielding may appear has been assessed in relation to internal pressure, temperature difference and heat generation. An analytical solution has been derived, allowing the prediction of radial position of yeld onset as related to the elastic limit domain configuration according to FGM parameter settings and thermomechanical loads. Moreover, elastic limit region changes have been investigated for two different designs of FGM materials. Stress and yield parameter variations along the thickness have been obtained. The effects of FGM parameters and of the shape factor of the cylinder cross-section on initial yielding due to thermomechanical loads have been studied.

Finite deformations of an electroelastic circular cylindrical tube

In this paper the theory of nonlinear electroelasticity is used to examine deformations of a pressurized thick-walled circular cylindrical tube of soft dielectric material with closed ends and compliant electrodes on its curved boundaries. Expressions for the dependence of the pressure and reduced axial load on the deformation and a potential difference between, or uniform surface charge distributions on, the electrodes are obtained in respect of a general isotropic electroelastic energy function. To illustrate the behaviour of the tube, specific forms of energy functions accounting for different mechanical properties coupled with a deformation independent quadratic dependence on the electric field are used for numerical purposes, for a given potential difference and separately for a given charge distribution. Numerical dependences of the non-dimensional pressure and reduced axial load on the deformation are obtained for the considered energy functions. Results are then given for the thin-walled approximation as a limiting case of a thick-walled cylindrical tube without restriction on the energy function. The theory described herein provides a general basis for the detailed analysis of the electroelastic response of tubular dielectric elastomer actuators, which is illustrated for a fixed axial load in the absence of internal pressure and fixed internal pressure in the absence of an applied axial load.

Analytical solution for the general mechanochemical corrosion of an ideal elastic–plastic thick-walled tube under pressure

The paper presents analytical solutions for the equal-rate mechanochemical wear of an ideal elastic–plastic thick-walled cylindrical tube subjected to any combination of internal and external pressure. The rates of corrosion at the inner and outer surfaces are supposed to be proportional to the equivalent tensile stress at the surface involved when it exceeds a given threshold. Furthermore, the corrosion rate can decay exponentially with time. The obtained solutions allow to assess the time of the initial yielding at the bore of the tube and the time of fully plastic yielding. Calculations showed that the time of plastic-zone propagation throughout the tube wall can be much greater than the length of the pure elastic stage. The proposed analytical solutions are to be used for design purposes and as benchmark solutions for numerical analysis.

Experimental Parameter Estimation Method for Nonlinear Viscoelastic Composite Material Models: An Application on Arterial Tissue

This study is aimed at setting a method of experimental parameter estimation for large-deforming nonlinear viscoelastic continuous fibre-reinforced composite material model. Specifically, arterial tissue was investigated during experimental research and parameter estimation studies, due to medical, scientific and socio-economic importance of soft tissue research. Using analytical formulations for specimens under combined inflation/extension/torsion on thick-walled cylindrical tubes, in vitro experiments were carried out with fresh sheep arterial segments, and parameter estimation procedures were carried out on experimental data. Model restrictions were pointed out using outcomes from parameter estimation. Needs for further studies that can be developed are discussed.

Arterial remodeling in response to increased blood flow using a constituent-based model

Previous theoretical models of arterial remodeling in response to changes in blood flow were based on the assumption that material properties of the arterial wall remain unchanged during the remodeling process. According to experimental findings, however, remodeling due to increased flow is accompanied by alteration in the structural properties of elastin, which results in a decrease in its effective elastic stiffness. To account for these effects, we propose a predictive model of arterial remodeling hypothesizing that the variation in mechanical properties of elastin is initiated and driven by the deviation of the intimal shear stress from its baseline value. Geometrical remodeling restores the wall stress distribution as it was under normal flow conditions. A constrained mixture approach is followed. Artery is modeled as a thick-walled cylindrical tube made of non-linear, elastic, anisotropic and incompressible material. Data for a rabbit thoracic aorta have been employed. At the final adapted state, the model predicts a non-monotonic dependence of arterial compliance on the magnitude of flow. This result is in agreement with available experimental data in the literature.

Simple shakedown of structures under variable multi-loadings

ABSTRACT The shakedown analysis of structures under variable multi-loadings is considered, and the corresponding simple shakedown condition is presented in this paper. Distribution of fixed stresses field is given, and the self-equilibrium of fixed stresses field is analyzed. Elastic shakedown and plastic shakedown conditions are presented based on the fixed stresses field. The theorem is convenient to evaluate the shakedown limit of structures under cyclical variable multi- loadings through solving positive scalar fields and fixed stresses field factors at a series of dangerous positions of the structure, and tedious computations are avoided. Finally the theorem is applied to a thick-walled cylindrical tube under variable pressure and temperature, and the rolling contact problem. The results are in good agreement with some computational results.

Combined implementing the hole-drilling method and reflection hologram interferometry for residual stresses determination in cylindrical shells and tubes

Main features inherent in simplified approach to residual stresses determination in cylindrical shells and tubes, external diameter of which is not less than 60 mm, by combing the hole-drilling method and reflection hologram interferometry are discussed in detail. Initial experimental information in a form of hole diameter increments in principal stress directions is derived from high-quality reflection holograms recorded near cylindrical objects of intermediate curvature value. Converting measured parameters into required stress values is based on the transition model that corresponds to plane stress conditions of pure membrane type. The technique developed is capable of determining residual stress component values within 5% accuracy in an absence of stress gradients over the probe hole diameter when a type of residual stress field corresponds to the transition model adopted. The accuracy analysis involved is based on matrix formulation of conventionally direct problem and an assumption on a pure membrane character of residual stress field under study for thin-walled shell. Required error estimations in a case of inspecting thick-walled cylindrical tube are obtained by combining the above-mentioned approach and an analogy of reconstructed fringe patterns with actual and artificial interferograms, which follow from drilling blind hole of the same geometrical parameters in thick-walled plates. Experimental verification of the developed approach is founded upon a determination of actual stresses in thin-walled cylindrical shell and obtaining residual stress distributions at the proximity of welded joint in thick-walled cylindrical tube.

A Model for Diffusion in White Matter in the Brain

Diffusion of molecules in brain and other tissues is important in a wide range of biological processes and measurements ranging from the delivery of drugs to diffusion-weighted magnetic resonance imaging. Diffusion tensor imaging is a powerful noninvasive method to characterize neuronal tissue in the human brain in vivo. As a first step toward understanding the relationship between the measured macroscopic apparent diffusion tensor and underlying microscopic compartmental geometry and physical properties, we treat a white matter fascicle as an array of identical thick-walled cylindrical tubes arranged periodically in a regular lattice and immersed in an outer medium. Both square and hexagonal arrays are considered. The diffusing molecules may have different diffusion coefficients and concentrations (or densities) in different domains, namely within the tubes’ inner core, membrane, myelin sheath, and within the outer medium. Analytical results are used to explore the effects of a large range of microstructural and compositional parameters on the apparent diffusion tensor and the degree of diffusion anisotropy, allowing the characterization of diffusion in normal physiological conditions as well as changes occurring in development, disease, and aging. Implications for diffusion tensor imaging and for the possible in situ estimation of microstructural parameters from diffusion-weighted MR data are discussed in the context of this modeling framework.

Modeling diffusion in white matter in the brain: A composite porous medium

We model diffusion in white matter fascicles as a problem of diffusion in an array of identical thick-walled cylindrical tubes immersed in an outer medium and arranged periodically in a regular lattice. The diffusing molecules have different diffusion coefficients and concentrations (or densities) within the tubes' inner core, membrane, myelin sheath, and within the outer medium. For an impermeable myelin sheath, diffusing molecules within the inner core are completely restricted, while molecules in the outer medium are hindered due to the tortuosity of the array of impenetrable tubes.

Remarks on the Instability of an Incompressible and Isotropic Hyperelastic, Thick-Walled Cylindrical Tube

The problem of instability of a hyperelastic, thick-walled cylindrical tube was first studied by Wilkes [1] in 1955. The solution was formulated within the framework of the theory of small deformations superimposed on large homogeneous deformations for the general class of incompressible, isotropic materials; and results for axially symmetrical buckling were obtained for the neo-Hookean material. The solution involves a certain quadratic equation whose characteristic roots depend on the material response functions. For the neo-Hookean material these roots always are positive. In fact, here we show for the more general Mooney–Rivlin material that these roots always are positive, provided the empirical inequalities hold. In a recent study [2] of this problem for a class of internally constrained compressible materials, it is observed that these characteristic roots may be real-valued, pure imaginary, or complex-valued. The similarity of the analytical structure of the two problems, however, is most striking; and this similarity leads one to question possible complex-valued solutions for the incompressible case. Some remarks on this issue will be presented and some new results will be reported, including additional results for both the neo-Hookean and Mooney–Rivlin materials.

Damage and shakedown analysis of structures with strain-hardening

In this article, the ductile damage of materials is introduced into the shakedown theory of strain-hardening structures. A mathematical programming method is developed to calculate an upper bound of the damage factor in a structure subjected to varying loads, which is suggested as the criterion of structural failure. Based on it, a lower bound of the safe load factor can be obtained for a strain-hardening structure via a mathematical programming. The application of this theory is demonstrated by analysing the thick-walled cylindrical tube.

A General Solution for the Pressurized Elastoplastic Tube

The problem of a thick-walled cylindrical tube subjected to internal pressure is investigated within the framework of continuum plasticity. Material behavior is modeled by a finite strain elastoplastic flow theory based on the Tresca yield function. The deformation pattern is restricted by the plane-strain condition but arbitrary hardening and elastic compressibility are accounted for. A general solution is given in terms of quadratures. The analysis also includes treatment of a second plastic phase, characterized by corner relations, that may develop at the inner boundary. It is shown that the interface between the two plastic regions moves initially outwards and then, beyond a certain strain level, it moves back inwards. Some useful and simple results are given for thin-walled tubes of hardening materials and for thick-walled elastic/perfectly plastic tubes.

Effect of non-homogeneity un yield stress in a thick-walled cylindrical tube under pressure

The material of the cylindrical tube has been assumed to be non-homogeneous both elastically and plastically. The expression for the critical pressure for yielding has been found, when the tube is under the action of internal and external pressures. The effect of non-homogeneity on yielding has been exhibited graphically.

Extrapolation of Strain and Stress From Holographically Measured Surface Displacement

For isotropic elastic materials, following Hooke’s law, the strain and stress field below the surface is uniquely determined by the knowledge of the displacement of the surface itself. From the holographically measured surface displacement u and the boundary conditions for the surface-stresses one can determine all 9 components of the vector-gradient grad u, which describes the strains as well as the tilt and the rotation of a volume element adjacent to the surface. It is shown that strain and stress can be uniquely calculated in a cone-shaped zone below the observed part of the surface by stepwise linear extrapolation. The depth of this cone depends on the density of the sample points and on the accuracy of the displacement measurement on the surface, as well as on the required accuracy of the extrapolated strain and stress values. The suggested extrapolation method has been tested numerically for a thick-walled cylindrical tube under internal pressure using simulated input data. The limitations and the accuracy are discussed.