Closed-form Weight Function for an Internal Elliptical Crack in an Infinite Body

1989 ◽  
Vol 69 (3) ◽  
pp. 155-160 ◽  
Author(s):  
Z.-G. Zhang ◽  
Y.-W. Mai
Keyword(s):  
Weight Function ◽  
Closed Form ◽  
Elliptical Crack ◽  
Infinite Body ◽  
Internal Elliptical Crack

Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

2018 ◽  
Vol 24 (6) ◽  
pp. 1821-1848 ◽  
Author(s):  
Yuan Li ◽  
CuiYing Fan ◽  
Qing-Hua Qin ◽  
MingHao Zhao
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Analytical Solutions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Elliptical Crack ◽  
Two Dimensional ◽  
Crack Face ◽  
Penny Shaped Crack ◽  
Crack Problems

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

Explicit Formulas for the Associated Jacobi Polynomials and Some Applications

1987 ◽  
Vol 39 (4) ◽  
pp. 983-1000 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Weight Function ◽  
Closed Form ◽  
Recurrence Relation ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Jacobi Polynomials ◽  
Main Diagonal ◽  
Discrete Point ◽  
Explicit Formulas ◽  
Explicit Representations

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

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.

scholarly journals Weight Function Approach for Semi Elliptical Crack at Blade Mounting Locations in Steam Turbine Rotor System

2018 ◽  
Vol 62 (3) ◽  
pp. 203-208
Author(s):  
Damarla Kiran Prasad ◽  
Kavuluri Venkata Ramana ◽  
Nalluri Mohan Rao
Keyword(s):  
Stress Intensity ◽  
Weight Function ◽  
Steam Turbine ◽  
Stress Intensity Factors ◽  
Turbine Rotor ◽  
Rotor System ◽  
Elliptical Crack ◽  
Influence Coefficient ◽  
Intensity Factors ◽  
Steam Turbine Rotor

This paper presents analysis of stress intensity factors at blade mounting locations of steam turbine rotor system. General expressions for the stresses induced in a rotating disc are derived and these equations are applied to steam turbine rotor disc. It is observed that the radial stress increases instantly at blade mounting location which indicates the probability of crack initiation and growth. A semi elliptical crack is considered at that location and weight function approach is used to determine the stress intensity factors. The results are validated with the influence coefficient approach. The differences of present approach with influence coefficient approach are less than 3 %. Hence the present approach is suitable for determination of stress intensity factors in a semi elliptical crack at blade mounting locations of a steam turbine rotor disc.

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

2014 ◽  
Author(s):  
Steven X. Xu ◽  
Darrell R. Lee ◽  
Douglas A. Scarth ◽  
Russell C. Cipolla
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Weight Function ◽  
Closed Form ◽  
Nuclear Power ◽  
Cubic Equation ◽  
Evaluation Procedures ◽  
Influence Coefficients ◽  
Surface Flaws

Analytical evaluation procedures for determining the acceptability of flaws detected during in-service inspection of nuclear power plant components are provided in Section XI of the ASME Boiler and Pressure Vessel Code. Linear elastic fracture mechanics based evaluation procedures in ASME Section XI require calculation of the stress intensity factor. In Article A-3000 of Appendix A of the 2013 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 stress intensity factor influence coefficients. The influence coefficients are only provided for a flat plate geometry. The ASME Section XI Working Group on Flaw Evaluation is in the process of rewriting Article A-3000 of Appendix A. Major updates include the implementation of an alternate method for calculation of the stress intensity factor for a surface flaw that makes explicit use of the Universal Weight Function Method and does not require a polynomial fit to the actual stress distribution, and the inclusion of stress intensity factor influence coefficients for the cylinder geometry. Tabular data of influence coefficients for the cylinder geometry are available in API 579-1/ASME FFS-1 2007. Effort has been made to develop closed-form relations for the stress intensity factor influence coefficients for the cylinder geometry based on API data. With the inclusion of the explicit weight function approach and the closed-form relations for influence coefficients, the procedures of Appendix A for the calculation of stress intensity factors can be used more efficiently. The development of closed-form relations for stress intensity factor influence coefficients for axial ID surface flaws in cylinders is described in this paper.

Calculation of Stress Intensity Factor for Surface Flaws Using Universal Weight Functions With Piece-Wise Cubic Stress Interpolation

2012 ◽  
Author(s):  
Steven X. Xu ◽  
Douglas A. Scarth ◽  
Russell C. Cipolla
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Stress Distribution ◽  
Intensity Factor ◽  
Weight Function ◽  
Closed Form ◽  
Function Method ◽  
Linear Interpolation ◽  
Surface Flaw ◽  
Weight Function Method

Linear elastic fracture mechanics based flaw evaluation procedures in ASME Section XI require calculation of the stress intensity factor. The method to calculate the stress intensity factor that is provided in the 2010 Edition of Appendix A of ASME Section XI is to fit the stress distribution ahead of the crack tip to a polynomial equation, and then use standardized influence coefficients. In PVP2011-57911, an alternate method for calculation of the stress intensity factor for a surface flaw that makes explicit use of the Universal Weight Function Method and does not require a polynomial fit to the actual stress distribution was proposed for implementation into Appendix A of Section XI. The alternate method provides closed-form solutions for the stress intensity factor. A numerical approximation is the assumed piece-wise linear variation of stress between discrete locations where stresses are known. For highly nonlinear stress distributions, piece-wise cubic interpolation of stress over intervals between discrete locations where stresses are known is an improvement over piece-wise linear interpolation. Investigation of a cubic interpolation of stress between discrete locations where stresses are known has been conducted. Closed-form equations for calculation of the stress intensity factor for a surface flaw were developed using the Universal Weight Function Method and generic piece-wise cubic interpolation of stress over intervals. Example calculations are provided to compare the results of stress intensity factors using piece-wise cubic interpolation with piece-wise linear interpolation.

scholarly journals On the Polyconvolution with the Weight Function for the Fourier Cosine, Fourier Sine, and the Kontorovich-Lebedev Integral Transforms

2010 ◽  
Vol 2010 ◽  
pp. 1-16
Author(s):  
Nguyen Xuan Thao
Keyword(s):  
Weight Function ◽  
Closed Form ◽  
Integral Equations ◽  
Integral Transforms ◽  
Convolution Product ◽  
Factorization Property ◽  
Fourier Cosine ◽  
Fourier Sine ◽  
Systems Of Integral Equations

The polyconvolution with the weight functionγof three functionsf,g, andhfor the integral transforms Fourier sine(Fs), Fourier cosine(Fc), and Kontorovich-Lebedev(Kiy), which is denoted by∗γ(f,g,h)(x), has been constructed. This polyconvolution satisfies the following factorization propertyFc(∗γ(f,g,h))(y)=sin y(Fsf)(y)⋅(Fcg)(y)⋅(Kiyh)(y), for ally>0. The relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has been obtained. Also, the relations between the polyconvolution product and others convolution product have been established. In application, we consider a class of integral equations with Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the new polyconvolution. An application on solving systems of integral equations is also obtained.

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