Assessment and Extension of an Analytical Formulation for Prediction of Residual Stress in Autofrettaged Thick Cylinders

2005 ◽  
Author(s):  
Anthony P. Parker
Keyword(s):  
Residual Stress ◽  
Plane Strain ◽  
Bauschinger Effect ◽  
Incompressible Material ◽  
Simple Modification ◽  
Linear Behavior ◽  
Wide Range ◽  
End Conditions ◽  
Von Mises ◽  
Recent Formulation

A recent formulation due to Huang and Cui provides an analytic method for calculating residual stresses in autofrettaged tubes exhibiting non-linear behavior (including Bauschinger effect) during manufacture. The formulation incorporates Von Mises criterion but is limited in its application to plane strain end conditions and an incompressible material (i.e. Poisson’s ratio 0.5) with a fixed unloading behavior. Comparisons indicate that, by selecting unloading behavior typical of that at the bore, the method produces reliable results for both A723 type steels and for steels which exhibit significant strain hardening and/or more dramatic Bauschinger effect. Numerical comparisons show that open-end, compressible conditions produce results which differ from the plane strain, incompressible case, but that these differences may be accurately corrected using a straightforward, pragmatic process. It is concluded that a simple modification of the Huang & Cui procedure may be used to perform extremely accurate straightforward spreadsheet calculations of residual stress in autofrettaged tubes manufactured from a wide range of steels under various end conditions. The Huang & Cui formulation also provides a reliable and highly accurate means for validating numerical formulations including Finite Element methods.

Autofrettage of Open-End Tubes—Pressures, Stresses, Strains, and Code Comparisons

2000 ◽  
Vol 123 (3) ◽  
pp. 271-281 ◽  
Author(s):  
Anthony P. Parker
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Hoop Stress ◽  
Bauschinger Effect ◽  
Diameter Ratio ◽  
Radius Ratio ◽  
Asme Code ◽  
Von Mises ◽  
Numerical Fit ◽  
Autofrettage Pressure

Autofrettage is used to introduce advantageous residual stresses into pressure vessels. The Bauschinger effect can produce less compressive residual hoop stresses near the bore than are predicted by “ideal” autofrettage solutions. A recently developed numerical analysis procedure is adopted and extended. The ratio of calculated autofrettage pressure (numerical)/ideal autofrettage pressure (Tresca criterion and plane stress) is calculated and verified against available solutions. The case of open-end conditions based upon von Mises and engineering plane strain (constant axial strain with zero net axial force) is examined in detail. The ratio in this case varies between unity and 2/3, but exhibits very significant variations from the plane stress case when the diameter ratio of the tube exceeds 1.8. Results are within 0.5 percent of available analytical, numerical, and experimental results. A simple numerical fit allows all autofrettage pressures to be replicated to within 0.5 percent. The true plane strain pressure ratio is examined and shown to be inappropriate in modeling engineering plane strain. A number of residual hoop and axial stress profiles is presented for radius ratio 2.0. Calculated pressures are used to determine residual hoop stress values for tube diameter ratios from 1.1 to 3.0 for the full range of percentage overstrain levels. These comparisons indicate that Bauschinger effect is evident when the ratio autofrettage radius/bore radius exceeds 1.2, irrespective of diameter ratio. To assist designers the important values of residual hoop stress at the bore are summarized in a composite plot and a numerical fit is provided. The accuracy of the current ASME code using pressure criteria is assessed. The code is shown to be generally and modestly conservative. A design procedure is proposed which appears capable of extending code validity beyond 40 percent overstrain (the limit of the current code) and of eliminating the small nonconservatism at very low overstrain. Hoop strain values are calculated at both the bore and outside diameter of a tube of radius ratio 2 at the peak of the autofrettage cycle using von Mises criterion with open-end, closed-end, and plane strain conditions. These are compared with available solutions; general agreement is demonstrated, with agreement within 2 percent with an accepted simple formula in the case of open ends. ASME code predictions of percentage overstrain based upon strains at the peak of the autofrettage cycle are generally within 6 percent of numerical predictions. This is in turn produces an agreement within approximately 3 percent in residual bore hoop stress calculation. This discrepancy is generally conservative, becoming non-conservative only at overstrain levels exceeding 80 percent. Strain during removal of autofrettage pressure, in the presence of Bauschinger effect, is also calculated. This shows that the difference in strain during the unloading phase is up to 8 percent (ID) and 6.3 percent (OD) compared with the predictions of elastic unloading. These latter results show similar agreement with the ASME code as in the peak-strain analysis and permit correction of estimates of percentage overstrain based upon permanent bore enlargement.

Effect of Bauschinger Effect and Yield Criterion on Residual Stress Distribution of Autofrettaged Tube

2005 ◽  
Vol 128 (2) ◽  
pp. 212-216 ◽  
Author(s):  
X. P Huang ◽  
W. C. Cui
Keyword(s):  
Experimental Data ◽  
Residual Stress ◽  
Stress Distribution ◽  
Strain Hardening ◽  
Bauschinger Effect ◽  
Present Model ◽  
Yield Criterion ◽  
Residual Stress Distribution ◽  
Hardening Model ◽  
End Conditions

Many analytical and numerical solutions for determining the residual stress distribution in autofrettaged tube have been reported. The significance of the choice of yield criterion, the Bauschinger effect, strain hardening, and the end conditions on the predicted residual stress distribution has been discussed by many authors. There are some different autofrettage models based on different simplified material strain-hardening behaviors, such as a linear strain-hardening model, power strain-hardening model, etc. Those models give more accurate predictions than that of elastic–perfectly plastic model, and each of them suits different strain-hardening materials. In this paper, an autofrettage model considering the material strain-hardening relationship and the Bauschinger effect, based on the actual tensile-compressive stress-strain curve of material, plane-strain, and modified yield criterion, has been proposed. The predicted residual stress distributions of autofrettaged tubes from the present model are compared to the numerical results and the experimental data. The predicted residual stresses are in good agreement with the experimental data and numerical predictions. The effect of Bauschinger effect and yield criterion on residual stress is discussed based on the present model. To predict residual stress distribution accurately, it is necessary to properly model yield criterion, Bauschinger effect, and appropriate end conditions.

Fully Plastic Crack Problems, Part 1: Solutions by a Penalty Method

1984 ◽  
Vol 51 (1) ◽  
pp. 48-56 ◽  
Author(s):  
C. F. Shih ◽  
A. Needleman
Keyword(s):  
Finite Element ◽  
Numerical Method ◽  
Strain Hardening ◽  
Plane Strain ◽  
Penalty Method ◽  
Incompressible Material ◽  
Material Behavior ◽  
Reduced Integration ◽  
Wide Range ◽  
Crack Problems

We formulate a finite-element reduced integration penalty method applicable to plane-strain problems with incompressible material behavior. This numerical method is employed to generate crack solutions for pure power-hardening solids. For two configurations of interest in applications, an edge cracked panel subject to remote tension and an edge-cracked panel subject to remote bending, we obtain solutions for a wide range of crack lengths and strain-hardening behaviors.

Radial Flow Velocity Field for Predicting Upper-Bound Solutions for Plane Strain Extrusion

1968 ◽  
Vol 90 (1) ◽  
pp. 45-50
Author(s):  
R. G. Fenton
Keyword(s):  
Velocity Field ◽  
Flow Field ◽  
Flow Velocity ◽  
Plane Strain ◽  
Upper Bound ◽  
Radial Flow ◽  
Flow Velocity Field ◽  
Wide Range ◽  
Ram Pressure ◽  
Considerable Range

The upper bound of the average ram pressure, based on an assumed radial flow velocity field, is derived for plane strain extrusion. Ram pressures are calculated for a complete range of reduction ratios and die angles, considering a wide range of frictional conditions. Results are compared with upper-bound ram pressures obtained by considering velocity fields other than the radial flow field, and it is shown that for a considerable range of reduction ratios and die angles, the radial flow field yields better upper bounds for the average ram pressure.

Critical examination of various approaches used for analysing helical cables

1994 ◽  
Vol 29 (1) ◽  
pp. 43-55 ◽  
Author(s):  
M Raoof ◽  
I Kraincanic
Keyword(s):  
Large Diameter ◽  
Numerical Results ◽  
Hertzian Contact ◽  
Critical Examination ◽  
First Order ◽  
Wide Range ◽  
End Conditions ◽  
Parametric Studies ◽  
The Individual ◽  
Fixed End

Using theoretical parametric studies covering a wide range of cable (and wire) diameters and lay angles, the range of validity of various approaches used for analysing helical cables are critically examined. Numerical results strongly suggest that for multi-layered steel strands with small wire/cable diameter ratios, the bending and torsional stiffnesses of the individual wires may safely be ignored when calculating the 2 × 2 matrix for strand axial/torsional stiffnesses. However, such bending and torsional wire stiffnesses are shown to be first order parameters in analysing the overall axial and torsional stiffnesses of, say, seven wire stands, especially under free-fixed end conditions with respect to torsional movements. Interwire contact deformations are shown to be of great importance in evaluating the axial and torsional stiffnesses of large diameter multi-layered steel strands. Their importance diminishes as the number of wires associated with smaller diameter cables decreases. Using a modified version of a previously reported theoretical model for analysing multilayered instrumentation cables, the importance of allowing for the influence of contact deformations in compliant layers on cable overall characteristics such as axial or torsional stiffnesses is demonstrated by theoretical numerical results. In particular, non-Hertzian contact formulations are used to obtain the interlayer compliances in instrumentation cables in preference to a previously reported model employing Hertzian theory with its associated limitations.

scholarly journals Elasto-plastic analysis of the contact region between a rigid ellipsoid and a semi-flat surface

2018 ◽  
Vol 10 (9) ◽  
pp. 168781401879739 ◽  
Author(s):  
Pengyang Li ◽  
Lingxia Zhou ◽  
Fangyuan Cui ◽  
Quandai Wang ◽  
Meiling Guo ◽  
...  
Keyword(s):  
Residual Stress ◽  
Plastic Deformation ◽  
Plastic Strain ◽  
Contact Pressure ◽  
Contact Region ◽  
Three Dimensional ◽  
Von Mises Stress ◽  
Effective Plastic Strain ◽  
Normal Forces ◽  
Von Mises

When the load acting on a mechanical structure is greater than the yield strength of the material, the contact surface will undergo plastic deformation. Cumulative plastic deformation has an important influence on the lifespan of mechanical parts. This article presents a three-dimensional semi-analytical model based on the conjugate gradient method and fast Fourier transform algorithm, with the aim of studying the characteristic parameters of the contact region between a rigid ellipsoid and elasto-plastic half-space. Moreover, normal forces and tangential traction were considered, as well as the contact pressure resulting from various sliding speeds and friction coefficients. The contact pressure, effective plastic strain, von Mises stress, and residual stress were measured and shown to increase with increasing sliding velocity. Finally, when the friction coefficient, contact pressure, and effective plastic strain are increased, the von Mises stress is also shown to increase, whereas the residual stress decreases.

Analytical Solutions of Crack Tip Plasticity Zone Shape for a Semi-Infinite Crack

2003 ◽  
Author(s):  
Peihua Jing ◽  
Tariq Khraishi ◽  
Larissa Gorbatikh
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Analytical Solutions ◽  
Yield Criterion ◽  
Plasticity Zone ◽  
Linear Elastic ◽  
Perfectly Plastic ◽  
Classical Plasticity ◽  
Elastic Perfectly Plastic ◽  
Von Mises

In this work, closed-form analytical solutions for the plasticity zone shape at the lip of a semi-infinite crack are developed. The material is assumed isotropic with a linear elastic-perfectly plastic constitution. The solutions have been developed for the cases of plane stress and plane strain. The three crack modes, mode I, II and III have been considered. Finally, prediction of the plasticity zone extent has been performed for both the Von Mises and Tresca yield criterion. Significant differences have been found between the plane stress and plane strain conditions, as well as between the three crack modes’ solutions. Also, significant differences have been found when compared to classical plasticity zone calculations using the Irwin approach.

Behavior of Lode Dependent Plasticity at Plane Strain Condition and its Implication to Ductile Fracture

2016 ◽  
Vol 258 ◽  
pp. 213-216 ◽  
Author(s):  
František Šebek ◽  
Jindrich Petruška ◽  
Petr Kubík
Keyword(s):  
Ductile Fracture ◽  
Plane Strain ◽  
Uniaxial Stress ◽  
Complex Materials ◽  
Fracture Criteria ◽  
Dependent Plasticity ◽  
Ductile Fracture Criteria ◽  
Stress States ◽  
Aluminum Alloy 2024 ◽  
Von Mises

Variety of metals are complex materials exhibiting various behavior under different loading. Many metallic materials exhibit Tresca-like behavior rather than von Mises. It means different behavior in tension under plane strain and uniaxial stress conditions. This might be described by Lode dependent plasticity which should result in better prediction in force or torque responses of material tests. Good agreement between computation and experiment is also very important when calibrating the ductile fracture criteria. Several tests under plane strain and uniaxial stress states were carried out on aluminum alloy 2024-T351 where the Lode dependency was significant. The Lode dependent plasticity was implemented along with von Mises and Tresca-like yield criteria, which resulted in improvement of force–displacement responses of plane strain tests simulations. But it also caused significant change in the stress state of tensile flat and grooved plates which wrongly approached uniaxial tension condition. This inconvenience prevents plane strain experiments from using for calibration of ductile fracture criteria under these circumstances.

Limit pressure for a spherical vessel with a circumferential slot at its junction with a radial cylindrical protruding branch

1987 ◽  
Vol 22 (4) ◽  
pp. 215-227 ◽  
Author(s):  
M Robinson ◽  
C S Lim ◽  
R Kitching
Keyword(s):  
Spherical Shell ◽  
Lower Bounds ◽  
Yield Criterion ◽  
Spherical Vessel ◽  
Non Linear Programming ◽  
Limit Pressure ◽  
Wide Range ◽  
Cylindrical Branch ◽  
Von Mises ◽  
Plastic Limit Pressure

One of the requirements of the two criteria method of safety assessment of a pressure vessel with a defect is an estimate of the plastic limit pressure. Here the defect is in a spherical shell close to its junction with a protruding radial cylindrical branch. The defect is assumed to be an axisymmetric circumferential slot of uniform depth on the outer surface of the shell. Lower bounds to the limit pressure are calculated for a wide range of geometries. The material is assumed to obey the von Mises yield criterion and a non-linear programming method is used to give optimum lower bounds. Data is supplied for spherical shell radius to thickness ratios from 25 to 100, nozzle radius to vessel radius ratios from 0 to 0.4, nozzle to vessel thickness ratios from 0.25 to 1.0 and ligament thickness to vessel thicknesses (ligament efficiencies) of 0 to 1. Slot widths vary from the significant to the infinitesimal, where it becomes a crack. Vessels of some proportions were shown to have their limit pressures reduced only a little by very low ligament efficiencies.

scholarly journals Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host

Solid Earth ◽  
2021 ◽  
Vol 12 (4) ◽  
pp. 817-833
Author(s):  
Xin Zhong ◽  
Marcin Dabrowski ◽  
Bjørn Jamtveit
Keyword(s):  
Residual Stress ◽  
Aspect Ratio ◽  
Analytical Solution ◽  
Strain State ◽  
Spherical Inclusion ◽  
Raman Shift ◽  
Stress And Strain ◽  
Wide Range ◽  
Inclusion Shape ◽  
Diamond Inclusions

Abstract. Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure and temperature (P–T) conditions of mineral inclusion entrapment. Existing modelling methods in petrology either adopt an assumption of a spherical, isotropic inclusion embedded in an isotropic, infinite host or use numerical techniques such as the finite-element method to simulate the residual stress and strain state preserved in the non-spherical anisotropic inclusions. Here, we use the Eshelby solution to develop an analytical framework for calculating the residual stress and strain state of an elastically anisotropic, ellipsoidal inclusion in an infinite, isotropic host. The analytical solution is applicable to any class of inclusion symmetry and an arbitrary inclusion aspect ratio. Explicit expressions are derived for some symmetry classes, including tetragonal, hexagonal, and trigonal. The effect of changing the aspect ratio on residual stress is investigated, including quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Quartz is demonstrated to be the least affected, while rutile is the most affected. For prolate quartz inclusion (c axis longer than a axis), the effect of varying the aspect ratio on Raman shift is demonstrated to be insignificant. When c/a=5, only ca. 0.3 cm−1 wavenumber variation is induced as compared to the spherical inclusion shape. For oblate quartz inclusions, the effect is more significant, when c/a=0.5, ca. 0.8 cm−1 wavenumber variation for the 464 cm−1 band is induced compared to the reference spherical inclusion case. We also show that it is possible to fit an effective ellipsoid to obtain a proxy for the averaged residual stress or strain within a faceted inclusion. The difference between the volumetrically averaged stress of a faceted inclusion and the analytically calculated stress from the best-fitted effective ellipsoid is calculated to obtain the root-mean-square deviation (RMSD) for quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Based on the results of 500 randomly generated (a wide range of aspect ratio and random crystallographic orientation) faceted inclusions, we show that the volumetrically averaged stress serves as an excellent stress measure and the associated RMSD is less than 2 %, except for diamond, which has a systematically higher RMSD (ca. 8 %). This expands the applicability of the analytical solution for any arbitrary inclusion shape in practical Raman measurements.

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