An Closed Form Solution for Antiplane Problem of Doubly Periodic Non-Uniform Cracks

2006 ◽  
Vol 324-325 ◽  
pp. 311-314
Author(s):  
Yao Ling Xu ◽  
Wen Feng Tan
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mapping Technique ◽  
Inhomogeneous Materials ◽  
Antiplane Problem ◽  
Microstructure Parameters

Inhomogeneous materials with doubly periodic non-uniform cracks under antiplane shear is dealt with. By using conformal mapping technique and elliptic function theory, the stress field and stress intensity factor at the tip of each crack are derived in closed form. Numerical examples show the influences of some microstructure parameters of crack distribution on stress intensity factor.

Fracture behavior of periodically bonded interface of piezoelectric bi-material under compressive–shear loading

2019 ◽  
Vol 24 (10) ◽  
pp. 3216-3230 ◽  
Author(s):  
S Kozinov ◽  
A Sheveleva ◽  
V Loboda
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Fracture Behavior ◽  
Compressive Loading ◽  
Closed Form Solution ◽  
Industrial Applications ◽  
Shear Loading ◽  
Form Solution ◽  
Applied Loading

A closed-form solution is constructed for a bi-material consisting of two piezoelectric (or piezoelectric and dielectric) half-planes, which are periodically bonded along the interface and can partially contact along the initially unbonded parts. Under compressive loading, the size of the frictionless contact zone is usually quite large; in some cases the interface is completely closed. Such a situation is frequently observed in industrial applications. Since the periodic bonding of two different materials is extremely widespread, it is very important to study the influence of the mutual material properties of the composite and the applied loading on the size and shape of the opened regions, as well as the stress intensity factor at the bonding points. To formulate the problem, the electromechanical factors are presented through piecewise analytic functions, so that the problem in question is reduced to the combined periodic Dirichlet–Riemann problem, which is solved exactly. The obtained solution provides explicit formulas for the mechanical stresses and displacements along the interface and allows one to find the dependence of the contact zones and the stress intensity factor on the ratio of the bonded parts of the interface to the period for the different values of applied loading and materials.

Estimation of Stress Intensity Factor for Circumferential Through-Wall Cracked Elbows Subjected to In-Plane Bending

2015 ◽  
Author(s):  
Han-Bum Surh ◽  
Jong Wook Kim ◽  
Min Kyu Kim ◽  
Min-Gu Won ◽  
Moon Ki Kim ◽  
...  
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Nuclear Power ◽  
Closed Form Solution ◽  
Form Solution ◽  
Accurate Estimation ◽  
Wide Range ◽  
Plane Bending ◽  
Geometric Variables

The stress intensity factor (SIF) is the major fracture mechanics parameter in LEFM concept. Since the SIF can be used for not only calculation of J-integral based on the GE/EPRI and reference stress method but also evaluation of fatigue crack growth, an accurate estimation of the SIF is an important issue for the piping in nuclear power plant. Recently, there is a need to develop the SIF solution which can cover wide geometric variables since there are on-going efforts that are developing next generation reactors in Korea, which is designed to thin-walled structures. For the through-wall cracked straight pipes, many researchers have proposed the SIF solutions which can cover wide range of wall thickness. However, since only limited solutions have been proposed yet for the through-wall cracked elbows, a research related to the SIF estimation for the elbows with wide geometric variables should be performed. In this study, the extended SIF solution for circumferential through-wall cracked elbows subjected to in-plane bending is proposed as the tabulated form through the finite element (FE) analyses. Wide elbow geometries are selected to range between 5 and 50 of Rm/t and range between 2∼20 of Rb/Rm. The existing solutions are then reviewed by comparing with the FE results. Furthermore, effects of geometric variables on the SIF are addressed through systematic investigation of FE based SIF results. These investigated results are expected to contribute to the development of closed form solution for the circumferential through-wall cracked elbows subjected to in-plane bending.

scholarly journals The Part-Through Surface Crack in an Elastic Plate

1972 ◽  
Vol 39 (1) ◽  
pp. 185-194 ◽  
Author(s):  
J. R. Rice ◽  
N. Levy
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Plane Strain ◽  
Surface Crack ◽  
Unit Length ◽  
Mathematical Formulation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Bending Loads

An elastic analysis is presented for the tensile stretching and bending of a plate containing a surface crack penetrating part-through the thickness, Fig. 1. The treatment is approximate, in that the two-dimensional generalized plane stress and Kirchhoff-Poisson plate bending theories are employed, with the part-through cracked section represented as a continuous line spring. The spring has both stretching and bending resistance, its compliance coefficients being chosen to match those of an edge cracked strip in plane strain. The mathematical formulation reduces finally to two-coupled integral equations for the thickness averaged force and moment per unit length along the cracked section. These are solved numerically for the case of a semi-elliptical part-through crack, with results compared to a simple but approximate closed-form solution. Extensive results are given for the stress intensity factor at the midpoint of the part-through crack for both remote tensile and bending loads on the plate. These results indicate that the stress-intensity factor is substantially lower, in general, than for a similarly loaded strip in plane strain with a crack of the same depth.

Closed-Form Stress Intensity Factor Solutions for Surface Cracks With Large Aspect Ratios in Plates

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
Kisaburo Azuma ◽  
Yinsheng Li ◽  
Steven Xu
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Closed Form ◽  
Influence Coefficient ◽  
Maximum Point ◽  
Surface Cracks ◽  
Fourth Degree ◽  
Aspect Ratios ◽  
Influence Coefficients

Abstract Alloy 82/182/600, which is used in light-water reactors, is known to be susceptible to stress-corrosion cracking. The depth of some of these cracks may exceed the value of half-length on the surface. Although the stress intensity factor (SIF) for cracks plays an important role in predicting crack propagation and failure, Section XI of the ASME Boiler and Pressure Vessel Code does not provide SIF solutions for such deep cracks. In this study, closed-form SIF solutions for deep surface cracks in plates are discussed using an influence coefficient approach. The stress distribution at the crack location is represented by a fourth-degree-polynomial equation. Tables for influence coefficients obtained by finite element analysis in the previous studies are used for curve fitting. The closed-form solutions for the influence coefficients were developed at the surface point, the deepest point, and the maximum point of a crack with an aspect ratio a/c ranging from 1.0 to 8.0, where a is the crack depth and c is one-half of the crack length. The maximum point of a crack refers to the location on the crack front where the SIF reaches a maximum value.

Stress Intensity Factor Coefficients for Circumferential ID Surface Flaws in Cylinders for Appendix A of ASME Section XI

2013 ◽  
Author(s):  
Russell C. Cipolla ◽  
Darrell R. Lee
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Flat Plate ◽  
Closed Form ◽  
Fourth Order ◽  
Surface Crack ◽  
Crack Size ◽  
Plate Geometry

The stress intensity factor (KI) equations for a surface crack in ASME Section XI, Appendix A are based on non-dimensional coefficients (Gi) that allow for the calculation of stress intensity factors for a cubic varying stress field. Currently, the coefficients are in tabular format for the case of a surface crack in a flat plate geometry. The tabular form makes the computation of KI tedious when determination of KI for various crack sizes is required and a flat plate geometry is conservative when applied to a cylindrical geometry. In this paper, closed-form equations are developed based on tabular data from API 579 (2007 Edition) [1] for circumferential cracks on the ID surface of cylinders. The equations presented, represent a complete set of Ri/t, a/t, and a/l ratios and include those presented in the 2012 PVP paper [8]. The closed-form equations provide G0 and G1 coefficients while G2 through G4 are obtained using a weight function representation for the KI solutions for a surface crack. These equations permit the calculation of the Gi coefficients without the need to perform tabular interpolation. The equations are complete up to a fourth order polynomial representation of applied stress, so that the procedures in Appendix A have been expanded. The fourth-order representation for stress will allow for more accurate fitting of highly non-linear stress distributions, such as those depicting high thermal gradients and weld residual stress fields. The equations developed in this paper will be added to the Appendix A procedures in the next major revision to ASME Section XI. With the inclusion of equations to represent Gi, the procedures of Appendix A for the determination of KI can be performed more efficiently without the conservatism of using flat plate solutions. This is especially useful when performing flaw growth evaluations where repetitive calculations are required in the computations of crack size versus time. The equations are relatively simple in format so that the KI computations can be performed by either spreadsheet analysis or by simple computer programming. The format of the equations is generic in that KI solutions for other geometries can be added to Appendix A relatively easily.

Closed-Form Calculation of Stress Intensity Factor for an Axial ID Surface Flaw in Cylinder Subjected to Weld Residual Stresses

2013 ◽  
Author(s):  
Douglas A. Scarth ◽  
Steven X. Xu
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Weight Function ◽  
Closed Form ◽  
Function Method ◽  
Current Method ◽  
Surface Flaw ◽  
Weight Function Method ◽  
Influence Coefficients

A method for calculating the stress intensity factor for linear elastic fracture mechanics based flaw evaluation is provided in Appendix A-3000 of ASME Section XI. In the 2010 Edition of ASME Section XI, the calculation of stress intensity factor for a surface crack is based on characterization of stress field with a cubic equation and use of influence coefficients. The influence coefficients are currently only provided for flat plate geometry in tabular format. The ASME Section XI Working Group on Flaw Evaluation is in the process of rewriting Appendix A-3000. Proposed major updates include the implementation of explicit use of Universal Weight Function Method for calculation of the stress intensity factor for a surface flaw and the inclusion of closed-form influence coefficients for cylinder geometry. The explicit use of weight function method eliminates the need for fitting polynomial equations to the actual through-thickness stress distributions at crack location. In this paper, the proposed Appendix A procedure is applied to calculate the stress intensity factors in closed-form for an axial ID surface flaw in a cylinder subjected to a set of nonlinear hoop weld residual stress profiles. The calculated stress intensity factor results are compared with the results calculated based on the current method in Appendix A using cubic equations to represent the stress distribution. Three-dimensional finite element analyses were performed to verify the accuracy of the stress intensity factor results calculated based on the current and proposed Appendix A procedures. The results in this paper support the implementation of the proposed stress intensity factor calculation procedure into ASME Code.

Closed-Form Relations for Stress Intensity Factor Influence Coefficients for Axial Outside Surface Flaws in Cylinders for Appendix A of ASME Section XI

2016 ◽  
Author(s):  
Steven X. Xu ◽  
Darrell R. Lee ◽  
Douglas A. Scarth ◽  
Russell C. Cipolla
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Closed Form ◽  
Inside Surface ◽  
Linear Elastic ◽  
Evaluation Procedures ◽  
Influence Coefficients ◽  
Elastic Fracture ◽  
Flaw Location

Linear elastic fracture mechanics based flaw evaluation procedures in Section XI of the ASME Boiler and Pressure Vessel Code require calculation of the stress intensity factor. Article A-3000 of Appendix A in ASME Section XI prescribes a method to calculate the stress intensity factor for a surface or subsurface flaw by making use of the flaw location stress distribution obtained in the absence of the flaw. The 2015 Edition of ASME Section XI implemented a number of significant improvements in Article A-3000, including closed-form equations for calculating stress intensity factor influence coefficients for circumferential flaws on the inside surface of cylinders. Closed-form equations for stress intensity factor influence coefficients for axial flaws on the inside surface of cylinders have also been developed. Ongoing improvement efforts for Article A-3000 include development of closed-form relations for the stress intensity factor coefficients for flaws on the outside surface of cylinders. The development of closed-form relations for stress intensity factor coefficients for axial flaws on the outside surface of cylinders is described in this paper.

Closed Form Solutions for Stress Intensity Factor Coefficients for Circumferential ID Surface Flaws in Cylinders in ASME Section XI Appendix A

2012 ◽  
Author(s):  
Russell C. Cipolla ◽  
Darrell R. Lee
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Flat Plate ◽  
Closed Form ◽  
Fourth Order ◽  
Surface Crack ◽  
Crack Size ◽  
Plate Geometry

The stress intensity factor (KI) equations for a surface crack in ASME Section XI, Appendix A are based on non-dimensional coefficients (Gi) that allow for the calculation of stress intensity factors for a cubic varying stress field. Currently, the coefficients are in tabular format for the case of a surface crack in a flat plate geometry. The tabular form makes the computation of KI tedious when determination of KI for various crack sizes is pursued and a flat plate geometry is conservative when applied to a cylindrical geometry. In this paper, closed-form equations are developed based on tabular data from API 579 (2007 Edition) [1] for circumferential cracks on the ID surface of cylinders. The closed-form equations provide G0 and G1 coefficients while G2 through G4 are obtained using a weight function representation for the KI solutions for a surface crack. These equations permit the calculation of the Gi coefficients without the need to perform tabular interpolation. The equations are complete up to a fourth order polynomial representation of applied stress, so that the procedures in Appendix A have been expanded. The fourth-order representation for stress will allow for more accurate fitting of highly non-linear stress distributions, such as those depicting high thermal gradients and weld residual stress fields. It is expected that the equations developed in this paper will be added to the Appendix A procedures. With the inclusion of equations to represent Gi, the procedures of Appendix A for the determination of KI can be performed more efficiently without the conservatism of using flat plate solutions. This is especially useful in performing flaw growth calculations where repetitive calculations are required in the computations of crack size versus time. The equations are relatively simple in format so that the KI computations can be performed by either spreadsheet analysis or by simple computer programming. The format of the equations is generic in that KI solutions for other geometries can be added to Appendix A relatively easily.

Sign in / Sign up

Export Citation Format

Share Document