Pressurized Thick-walled Cylinder Research Articles

Implementing a simple 2D constitutive model for rocks into finite element method

Our research group has developed a 2D constitutive model for rocks that incorporates a strain-dependent elastic modulus. This model efficiently represents nonlinear stress–strain curves using just three equations and four parameters. In this study, we have implemented the simple 2D model into finite element codes to tackle engineering problems. We have calculated and compared stresses and displacements around a pressurized thick-walled hollow cylinder, a lined non-circular tunnel, and a rock slope using the elastic, simple 2D, and elasto-plastic models. The results confirmed that the simple 2D model closely matches the stress distribution obtained from the elastic solution at low-stress levels. Additionally, the simple 2D model produces a plastic zone with a larger inward displacement than the elasto-plastic model at higher stress levels. Only the simple 2D model captured sidewall convergence, roof sag, and floor heave in non-circular tunnels, and the toppling failure for 90° slope and toe translational slide for 60° slope in rock slopes with smooth critical slip surface. We did not encounter any convergence difficulties while solving the simple 2D model.

Nonlinear stress and deformation analysis of pressurized thick-walled hyperelastic cylinders with experimental verifications and material identifications

The main purpose of the present research is the investigation of the stress and displacement distributions of the internally pressurized thick-walled hyperelastic cylindrical vessels or pressure pipes. The governing equations are derived by incorporation of the incompressible neo-Hookean and Mooney-Rivlin hyperelastic constitutive models into the direct hyperelasticity theory in the von Karman framework of large deformations. In contrast to all the available researches, no simplifications or linearization are made. The non-linear governing equations are solved by utilizing a second-order point-collocation procedure and implementation of details of the proposed incremental iterative solution scheme in the Matlab code written by the authors. Another novelty of the present research is experimentally identifying the employed polymeric hyperelastic material and verification of the results through experiments conducted by the authors. Due to using both nonlinear constitutive laws and nonlinear stress-displacement relations, through-thickness distributions of the radial displacement and radial and circumferential stresses are extracted for various pressures. The experimental results reveal that while the neo-Hookean hyperelastic model cannot be used for materials with stress-softening, the Mooney-Rivlin model reproduces the experimental results with excellent accuracy. In addition to the experimental verification, the results are verified by the Abaqus finite element analysis code as well.

Investigation of displacement and stress fields in pressurized thick-walled FGM cylinder under uniform magnetic field

In this paper, an analytical solution is presented for computing mechanical displacements and stresses in a thick-walled cylindrical pressurized vessel made of functionally graded materials (FGM). The pressure vessel is subject to axisymmetric mechanical loading within a uniform magnetic field. The analytical model of the pressurized vessel was constructed, where the radial continuous varying of elastic modulus and permeability along the thickness was assumed. It has been assumed that the elastic modulus and permeability are varying through thickness of the FG material according to a nonlinear general distribution along the thickness. Navier’s equation, which is a second-order ordinary differential equation, was derived from the mechanical equilibrium equation with the consideration of the Lorentz force resulting from the magnetic field. The distributions of the displacement and stresses were determined by the exact solution to Navier’s equation. Numerical results clarify the influence of the magnetic field, non-homogeneity parameter and internal pressure on the magneto-elastic response of the functionally graded cylindrical vessel. Thus, these parameters have remarkable effects on the distributions of radial displacement and radial and circumferential stresses.

Morphoelastic fiber remodeling in pressurized thick-walled cylinders with application to soft tissue collagenous tubes

We consider the mechanics of fiber remodeling in a pressurized thick walled cylinder where our focus is on the interaction between inflation and remodeling. The material is taken to consist of an incompressible neo-Hookean ground substance matrix in which the fibers are embedded. The fiber network is assumed to undergo a stretch mediated remodeling process, as suggested by experiments on enzymatic degradation of collagen fibers in biological soft tissues. In our treatment, such a remodeling process is defined in terms of a constant fiber-creation rate, and a fiber-dissolution rate that decreases with the amount of fiber stretch. Here we study how pressure loading of a tube affects this process within the cylindrical wall. The impact of two basic load carrying fibers models on the remodeling of the fiber network are explored, as well as the effect of two different choices for the fiber's natural configuration when replacement fibers are first synthesized.

Displacements and stresses in pressurized thick-walled FGM cylinders: Exact and numerical solutions

In this paper, analytical and numerical solutions are presented for computing the displacements and stresses in a FGM thick-walled cylinder under a uniform internal and external pressure.An analytical and a finite element (FE) model of the pressurized vessel were constructed, where the radial continuous varying of elastic modulus along the thickness was assumed and implemented in a FE code. It has been assumed that the elastic modulus is varying through thickness of the FGM material according to a nonlinear general expression.The analytical solutions of distribution of displacements and stresses were obtained and illustrated in radial direction for pressurized FGM cylinder. The results are checked with FE results. Thus, the inhomogeneity in material properties can be exploited to optimize stress distribution.

Autofrettage of nonlinear strain-hardening cylinders using the proposed analytical solution for stresses

This paper focuses on pressurized thick-walled cylinders with nonlinear strain-hardening behavior. At first, a constitutive model is proposed to capture the test data done on the steel specimen. It is shown that this model is able to cover the various specific behaviors such as the perfect plastic, linear hardening, and nonlinear hardening behaviors. This constitutive model due to high performance in constitutive modeling and simplicity of its mathematical form is applied to pressurized thick-walled cylinders in order to find a closed-form analytical solution for their analysis. For this purpose, elasto-plastic stresses distribution in pressurized thick-walled cylinders made of a nonlinear strain-hardening material is obtained analytically for both loading and unloading phases using Tresca's yield criterion and considering the Bauschinger effect. Then, the solution is compared with experimental data and good agreement between predicted results and experimental data is observed. In the following, the residual stresses distribution are obtained to determine the best level of autofrettage for strengthening and design of these thick-walled cylinders prior to industrial use with the aim of increasing strength-to-weight ratio. It was observed that the parameters such as the material behavior (stress–strain curve) and the Bauschinger effect have a significant influence on the results.

The Bauschinger Effect's Influence on the Stress Intensity Factors of a Semi-Elliptical Crack Emanating From an Erosion at the Bore of a Fully Autofrettaged Pressurized Cylinder

The benefits of autofrettage for thick-walled cylindrical vessels as a means of improving the vessel's durability and sustainability have been addressed in the published literature. However, the presence of the Bauschinger effect (BE) complicates the overall effect of autofrettage, especially when complex three-dimensional crack geometries emanating from erosions at the cylinder bore are considered. In this paper, the BE's impact on the stress intensity factors (SIFs) on such cracks is investigated. The effect of various erosion geometrical configurations on the mode I SIF distribution along the front of a semi-elliptical crack, emanating from the deepest line of the erosion surface (DLES) at the bore of an autofrettaged, pressurized thick-walled cylinder of outer-to-inner radius ratio, Ro/Ri = 2, is investigated. Both autofrettage with BE (BEDA) and Hill's ideal autofrettage residual stress field (BEIA) are considered and simulated by an equivalent thermal load. The SIFs are determined for the semi-elliptical cracks of various crack depths to wall thickness ratio, a/t = 0.05–0.25, and ellipticities, a/c, ranging from 0.5 to 1.5, emanating from the DLES via Ansys software and the nodal displacement method. Three groups of erosion geometries are considered: (a) arc erosions of constant relative depth, d/t, equal to 5% and with relative radii of curvature, r′/t, between 5% and 30%; (b) semi-elliptic erosions of constant relative depth, d/t, of 5% with erosion ellipticity, d/h, varying from 0.3 to 2.0; and (c) semicircular erosions of relative depth, d/t, between 1% and 10% of the wall thickness. KIP, the SIF due to pressurization, is highly dependent on the stress concentration ahead of the DLES which directly relates to the erosion geometry. It is found that the absolute value of KIA, the SIF due to autofrettage, is just slightly reduced by the presence of the erosion. Its change solely depends on, and is directly proportional to, the erosion depth. Thus, the combined SIFs of deep cracks are found to be significantly enhanced by the presence of autofrettage and might result in a shortening of the vessel's fatigue life by up to an order of magnitude. Counteracting this, the combined SIFs are found to be significantly higher for BEDA cases than for BEIA cases. Therefore, the vessel's fatigue life can be profoundly influenced by the presence of the BE.

The Impact of the Bauschinger Effect on Stress Concentrations and Stress Intensity Factors for Eroded Autofrettaged Thick Cylindrical Pressure Vessels

Our previous studies have shown that stress intensity factors (SIFs) are influenced considerably from the presence of the Bauschinger Effect (BE) in thick-walled pressurized cracked cylinders. For some types of pressure vessels, such as gun barrels, working in corrosive environment, in addition to acute temperature gradients and repetitive high-pressure impulses, erosions can be practically induced. Those erosions cause stress concentration at the bore, where cracks can readily initiate and propagate. In this study, the BE on the SIFs will be investigated for a crack emanating from an erosion’s deepest point in a multiply eroded autofrettaged, pressurized thick-walled cylinder. A commercial finite element package, ansys, was employed to perform this type of analysis. A two-dimensional model, analogous to the authors’ previous studies, has been adopted for this new investigation. Autofrettage with and without BE, based on von Mises yield criterion, is simulated by thermal loading and the SIFs are determined by the nodal displacement method. The SIFs are evaluated for a variety of relative crack lengths, a0/t = 0.01–0.45 emanating from the tip of the erosion of different geometries including (a) semicircular erosions of relative depths of 1%–10% of the cylinder’s wall thickness, t; (b) arc erosions for several dimensionless radii of curvature, r′/t = 0.05–0.4; and (c) semi-elliptical erosions with ellipticities of d/h = 0.5–1.5, and erosion span angle, α, from 6 deg to 360 deg. The effective SIFs for relatively short cracks are found to be increased by the presence of the erosion and further increased due to the BE, which may result in a significant decrease in the vessel’s fatigue life. Deep cracks are found to be almost unaffected by the erosion, but are considerably affected by BE.

Creep analysis with a stress range dependent constitutive model

Many materials exhibit the stress range dependent creep behavior. The power law creep observed for a certain stress range changes to the viscous type creep as the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 7–12) may decrease to the low value of approximately unity within the stress range relevant for engineering structures. The aim of this paper is to analyze the influence of the stress range dependent power-law-viscous creep transition on the behavior of structures at elevated temperature. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. To demonstrate basic features of the stress range dependent creep modeling, several elementary examples from structural mechanics are presented. They include a stress relaxation problem, a beam subjected to pure bending and a pressurized thick-walled cylinder. Based on the uni-axial transition stress the transition value of the external load is estimated such that above this value the power law can be applied. For the loading levels below this value the character of the stress distribution as well as the stress values are essentially influenced by the viscous creep.

The Combined Stress Intensity Factors of Multiple Longitudinally Coplanar Cracks in Autofrettaged Pressurized Tubes Influenced by the Bauschinger Effect

The influence of the Bauschinger effect (BE) on the three dimensional, Mode I, combined stress intensity factor (SIF) distributions for arrays of longitudinal coplanar, surface cracks emanating from the bore of a fully or partially autofrettaged thick-walled cylinder is investigated. The combined SIFs, KIN, that depend on pressure effects and the “realistic”—Bauschinger effect dependent Autofrettage (BEDA), or, that depend on pressure effects and the “ideal”—Bauschinger effect independent autofrettage (BEIA), are obtained and compared for crack depth to wall thickness, a∕t=0.01–0.25; crack ellipticity, a∕c=0.5–1.5; crack spacing ratio, 2c∕d=0.25–0.75; and autofrettage level, e=30%, 60%, and 100%. The 3D analysis is performed via the finite element method and the submodeling technique, employing singular elements along the crack front. Both autofrettage residual stress fields, BEDA and BEIA, are simulated using an equivalent temperature field. The combined SIF, KIN, is found to vary along the crack front with the maximum determined by the crack ellipticity, crack depth, and crack spacing ratio. For a partially autofrettaged cylinder, the influence of the BE on the combined SIF, KIN, is substantially reduced as the level of overstrain becomes smaller. For some cases, when comparing like crack distributions, the KIN values obtained from the BEDA model are found to be as much as 100% higher than the KIN values that are computed using the BEIA model. A pressurized thick-walled cylinder with BEDA can be most critical when small cracks are farther apart. As crack depth increases, or when the spacing between cracks is smaller, the SIFs increase. Though the differences in the BEDA SIF, KIA, between e=100% and 60% are small (7–15%, in most cases), the increased level of autofrettage produces a 23–30% decrease in the combined SIF values, KIN. In certain cases, the BEIA model implies an infinite fatigue life, whereas the BEDA model for the same parameters implies a finite life. Therefore, it is important to perform a full 3D analysis to determine the real life cycle of the pressurized cylinder for materials that exhibit the BE.

Steady-state creep of a pressurized thick cylinder in both the linear and the power law ranges

The classical solution of the steady-state creep problem for a pressurized thick-walled cylinder is based on the power law constitutive equation. Several heat resistant steels show, however, the linear dependence of the creep rate on the applied stress within a certain stress range. In this paper we apply an extended constitutive equation which includes both the linear and the power law stress dependencies. The material constants are identified for the 9Cr1MoVNb steel at 600 °C. We recall the boundary value problem of steady-state creep for the thick cylinder under the plane strain condition. We present an approximate solution illustrating the stress redistributions as a result of the creep process. The analysis shows that for the certain range of the internal pressure both the linear and the power law creep must be taken into account. In this case the results according to the extended constitutive model essentially differ from the classical ones. The obtained solution is also applied to verify the developed user-defined creep material subroutine inside a commercial finite element code.

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

A detailed variational formulation is provided for a simplified strain gradient elasticity theory by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the complete boundary conditions of the theory for the first time. To supplement the stress-based formulation, the coordinate-invariant displacement form of the simplified strain gradient elasticity theory is also derived anew. In view of the lack of a consistent and complete formulation, derivation details are included for the tutorial purpose. It is shown that both the stress and displacement forms of the simplified strain gradient elasticity theory obtained reduce to their counterparts in classical elasticity when the strain gradient effect (a measure of the underlying material microstructure) is not considered. As a direct application of the newly obtained displacement form of the theory, the problem of a pressurized thick-walled cylinder is analytically solved. The solution contains a material length scale parameter and can account for microstructural effects, which is qualitatively different from Lamé’s solution in classical elasticity. In the absence of the strain gradient effect, this strain gradient elasticity solution reduces to Lamé’s solution. The numerical results reveal that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.

Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading

Thermoelasticity problem in a thick-walled cylinder is solved analytically using the finite Hankel transform. Time-dependent thermal boundary conditions are assumed to act on the inner surface of the cylinder. For the mechanical boundary conditions two different cases are assumed: Traction–displacement problem (traction is prescribed on the inner surface and the fixed displacement boundary condition on the outer one) and Traction–Traction problem (tractions are prescribed on both the inner and outer surfaces of the hollow cylinder). The quasi-static solution of the thermoelasticity problem is derived analytically, i.e., the transient thermal response of the cylinder is derived and then, quasi-static structural problem is solved and closed form relations are extracted for the thermal stresses in the two problems. The results show to be in accordance with that cited in the literature in the special cases.

Effects of radially varying moduli on stress distribution of nonhomogeneous anisotropic cylindrical bodies

In this study, the stress distribution in a nonhomogeneous anisotropic cylindrical body is investigated. Using equilibrium equations, Hooke’s law and strain–displacement relations, a system of equations is obtained in cylindrical coordinates in terms of stress potentials where elastic properties change in radial direction. Young’s and shear moduli are expressed as power functions of r and Poisson’s ratios are kept constant. Closed-form solutions for stress potentials and stress distribution are obtained for an axisymmetric, orthotropic cylinder. Results are checked with FE results. A pressurized thick walled cylinder example is studied in details. Stresses in radial, tangential and axial directions and Von Mises stresses are plotted for different powers of r.

Analysis of Uniform Arrays of Three-Dimensional Unequal-Depth Cracks in a Thick-Walled Cylindrical Pressure Vessel

The effect of crack depth unevenness on the mode I stress intensity factor (SIF) distributions along the fronts of semi-elliptical surface cracks is studied. These three-dimensional radial cracks pertain to large uniform arrays of unequal-depth cracks emanating from the bore of a pressurized thick-walled cylinder. The analysis is based on the “two crack depth level model,” previously proposed, and is performed via the finite element (FE) method employing singular elements along the crack front. The distribution of KIP-the stress intensity factor due to pressurization, for numerous uneven array configurations bearing n=n1+n2=8 to 128 cracks, a wide range of crack depth to wall thickness ratios, a1/t=0.01 to 0.4, and various crack ellipticities, a1/c1=0.3 to 1.5, are evaluated for a cylinder of radii ratio Ro/Ri=2. To increase the accuracy of the evaluated SIFs an existing improved version of the displacement extrapolation method is used. The results clearly indicate that unevenness, as reflected in KIP distributions, depends on both the number of cracks in the array as well as on the cracks’ depths and ellipticities. The “interaction range” for the various configurations of uneven crack arrays is evaluated. The range of influence between adjacent cracks on the maximal SIF, KPmax, is found to be dependent on the density of the array, as reflected in the inter-crack aspect-ratio, as well as on the cracks’ elipticity.

3-D Stress Intensity Factors for Internal Cracks in an Overstrained Cylindrical Pressure Vessel—Part II: The Combined Effect of Pressure and Autofrettage

K IA and KIP stress intensity factors (SIF) for three-dimensional semi-elliptical, surface, radial cracks prevailing in a pressurized or autofrettaged thick-walled cylinder were evaluated and discussed in Part I of this paper and in Perl et al. 1996, “Three-Dimensional Interaction Effects in an Internally Multicracked Pressurized Thick-Walled Cylinder—Part I: Radial Surface Cracks,” AMSE J. Pressure Vessel Technol. 118, pp. 357–363), respectively. These SIFs were calculated for a wide range of configurations: for cracks pertaining to large arrays of up to 180 cracks, with ellipticities of a/c=0.2, 0.5, 1, 1.5, depth ratios of a/t=0.05−0.6, and for various levels of autofrettage. In Part II of this paper, the effect of the combined SIF KIN=KIP+KIA is considered, which enables the prediction of fracture endurance, crack growth rate, and the total fatigue life for a modern gun barrel. The results reconfirm the impact autofrettage has on delaying crack initiation and propagation. This favorable effect is found to be governed by ψ=σ0/p—the ratio of the vessel’s material yield stress to its internal pressure. The higher ψ is, the more effective autofrettage becomes. While KIA and KIP reach their maximum absolute values, usually, for an array of n=2 cracks, the largest combined SIF-KIN occurs for arrays of 2–16 cracks. Finally, the similarity in the behavior of KIA and KIP along the crack front is studied as well as its relation to the respective stress fields.

Three-Dimensional Analysis of a Semi-Elliptical Crack Emanating From an Erosion at the Bore of an Autofrettaged Pressurized Cylinder

The effect of various erosion configurations on the mode I stress intensity factor (SIF) distribution along the front of a semi-elliptical crack, emanating from the deepest line of the erosion surface (DLES) at the bore of an autofrettaged, pressurized thick-walled cylinder of outer to inner radius ratio, Ro/Ri = 2, is investigated. The three-dimensional (3-D) linear elastic problem is solved via the finite element (FE) method using the ANSYS 5.2 standard code. Hill’s autofrettage residual stress field is simulated by an equivalent thermal load and the SIFs are determined by the nodal displacement method. SIF distribution along the front of semi-elliptical cracks of various crack depths to wall thickness ratios, a/t = 0.05 to 0.25, and ellipticities, a/c, ranging from 0.5 to 1.5, emanating from the DLES, are determined. Three groups of erosion geometries are considered: (a) arc erosions of constant relative depth, d/t, equal to 5 percent and with relative radii of curvature, r′/t, between 5 and 30 percent; (b) semi-elliptic erosions of constant relative depth, d/t, of 5 percent with erosion ellipticity, d/h, varying from 0.3 to 2.0; and (c) semi-circular erosions of relative depth, d/t, between 1 and 10 percent of the wall thickness. The effective SIF along the crack front results from the superposition of KIP—the SIF due to pressurization, and KIA—the negative SIF due to the autofrettage residual stress field. KIP is highly dependent on the stress concentration ahead of the DLES which directly relates to the erosion geometry. The absolute value of KIA is just slightly reduced by the presence of the erosion. Its change solely depends on, and is directly proportional to, the erosion depth. Thus, while deep cracks are almost unaffected by the erosion, the effective SIF for relatively short cracks is found to be significantly enhanced by its presence and might result in a shortening of the vessel’s fatigue life by up to an order of magnitude. Also, it is shown that 2-D analysis may lead to a nonconservative estimate of the vessel’s fatigue life.

The effect of age on residual strain in the rat aorta.

Residual stress is observed in many parts of the cardiovascular system and is thought to reduce transmural stress gradients due to intravascular pressure. Its development is closely associated with normal growth and pathological remodeling, although there appear to be few previous reports of the relationship between aging and residual stress. We have estimated residual strain (an indicator of the magnitude of residual stress) at ten sites along the aorta of rats aged 2.5 to 56 weeks by measuring the degree to which rings of vessel spring open when cut (opening angle). At all ages the opening angle decreased along the aorta, reaching a minimum near the renal arteries and increasing toward the aorto-iliac bifurcation, a result that confirms previous studies. During growth, although the unloaded circumference of the aorta increased steadily, the wall thickness and medial surface area fell to a minimum at the age of 6 weeks before continuing a steady increase. Similarly, the opening angle decreased between the ages of 2.5 and 6 weeks, thereafter increasing with age. In the abdominal aorta, a strong correlation between opening angle and wall thickness relative to midwall radius (h/R) was seen; whereas in the thoracic segment, in which no increase in h/R with age occurred, no such relationship was found. These observations are in keeping with a recently proposed hypothesis that residual stress will change in response to growth-related changes in vessel geometry driven by a tendency to minimize the nonuniform stress distribution inevitably found in pressurized thick-walled cylinders.

Three-Dimensional Interaction Effects in an Internally Multicracked Pressurized Thick-Walled Cylinder— Part I: Radial Crack Arrays

Under certain conditions, numerous internal surface cracks develop in pressurized thick-walled cylinders, both in the radial and longitudinal directions. For fatigue life assessment of such vessels, the 3-D interaction effects among these cracks on the prevailing stress intensity factors (SIFs) need evaluation. In Part I of this paper, radial crack arrays are considered exclusively. The mode I SIF distribution for a wide range of semi-circular and semi-elliptical cracks are evaluated. The 3-D analysis is performed via the finite element method with the submodeling technique, employing singular elements along the crack front. SIFs are evaluated for arrays of up to n = 180 cracks; for a wide range of crack depth to wall thickness ratios, a/t, from 0.05 to 0.6; and, for various ellipticities of the crack, i.e., the ratio of crack depth to semicrack length, a/c, from 0.2 to 2. Using a least-squares fit, two simple expressions for the most critical (n = 2) SIFs are obtained for sparse and dense crack arrays. The formulas, which are functions of a/t and a/c, are of very good engineering accuracy. The results clearly indicate that the SIFs are considerably affected by the interaction among the cracks in the array as well as the three-dimensionality of the problem. In Part II of this paper, the interaction effects between longitudinal coplanar cracks will be analyzed.

Stress intensity factors for internal semi-elliptical surface cracks in pressurized thick-walled cylinders using the hybrid boundary element method

The purpose of this paper is to present a comprehensive range of results of mode I SIFs of three-dimensional surface cracks in internally pressurized thick-walled cylinders. The hybrid boundary element method is summarily reviewed and used to calculate the SIFs of surface cracks in pressurized thick-walled cylinders. The analyzed ratio of crack depth to wall thickness ranges from 0.2 to 0.8; the ratio of crack depth to crack length ranges from 0.25 to 1.0; and the ratio of wall thickness to cylinder radius is 0.5, 1.0 and 2.0. The present normalized SIFs are also compared with other solutions from the literature. The recent results of the body force method and the finite element method agree well ( ca 3%), and the early ones of the boundary integral equation and the finite element method agree fairly well ( ca 10%) with the present results.