Plastic Zone Coalescence and Edge Break Conditions of Internal Cracks in a Semi-infinite Sheet

2006 ◽  
Vol 129 (1) ◽  
pp. 142-147 ◽  
Author(s):  
Toshihiko Nishimura
Keyword(s):  
Plastic Zone ◽  
Integral Equations ◽  
Numerical Results ◽  
Critical Conditions ◽  
Surface Traction ◽  
Crack Face ◽  
Fictitious Crack ◽  
Plastic Zones ◽  
Crack Tip Opening ◽  
Remote Stress

The problem of two cracks in a semi-infinite sheet is analyzed. The critical conditions when adjacent plastic zones just coalesced are obtained. Also, the conditions when a plastic zone just reached the sheet edge are obtained. Assuming the crack and plastic zones as a fictitious crack, the integral equations are formulated in terms of surface traction, nonsingular stress, and zero crack face displacement at the coalescent point or at the sheet edge. By solving the equations, critical remote stress, plastic zone sizes, and crack tip opening displacements are obtained. Numerical results are presented.

Strip Yield Analysis of Plastic Zone Coalescence for Collinear Edge Crack and Internal Crack in a Semi-Infinite Sheet

1999 ◽  
Vol 122 (1) ◽  
pp. 86-89 ◽  
Author(s):  
Toshihiko Nishimura
Keyword(s):  
Plastic Zone ◽  
Integral Equations ◽  
Edge Crack ◽  
Stress Singularity ◽  
Crack Tip ◽  
Internal Crack ◽  
Crack Face ◽  
Plastic Zones ◽  
Crack Tip Opening ◽  
Remote Stress

The coalescence conditions of plastic zones are calculated for an edge crack and an internal crack in a semi-infinite sheet. The cracks and their plastic zones are treated as a fictitious crack, and integral equations are formulated on equilibrium of surface traction, no stress singularity at the crack tip, and zero crack face displacement at the coalesced point of plastic zones. The integral equations are iteratively solved, and critical values of remote stress, plastic zone sizes, and crack tip opening displacements are determined. Numerical results for an edge crack and an internal crack due to remote stress are presented. [S0094-9930(00)00301-2]

Strip Yield Analysis on Coalescence of Plastic Zones for Multiple Cracks in a Riveted Stiffened Sheet

1999 ◽  
Vol 121 (3) ◽  
pp. 352-359 ◽  
Author(s):  
Toshihiko Nishimura
Keyword(s):  
Stress Singularity ◽  
Multiple Cracks ◽  
Algebraic Equations ◽  
Yield Model ◽  
Strip Yield Model ◽  
Crack Tips ◽  
Fictitious Crack ◽  
Plastic Zones ◽  
Crack Tip Opening ◽  
Remote Stress

The coalescence conditions of plastic zones are calculated for multiple cracks in a riveted stiffened sheet using a strip yield model. The multiple cracks and their plastic zones are treated as a fictitious crack, and algebraic equations are formulated on compatibility of displacements, no stress singularity at the fictitious crack tips, and zero displacement at the coalesced points of plastic zones. These equations are iteratively solved, and critical values of remote stress, fastener forces, plastic zone sizes, and crack tip opening displacements are calculated. Some numerical results are presented for two cracks in a sheet with and without stiffeners.

Strip Yield Analysis for Multiple Cracked Sheet With Riveted Stiffeners

1993 ◽  
Vol 115 (4) ◽  
pp. 398-403 ◽  
Author(s):  
T. Nishimura
Keyword(s):  
Plastic Zone ◽  
Stress Singularity ◽  
Fredholm Integral Equations ◽  
Multiple Cracks ◽  
Basic Solution ◽  
Yield Model ◽  
Strip Yield Model ◽  
Crack Tips ◽  
Fictitious Crack ◽  
Crack Tip Opening

An elasto-plastic analysis is conducted based upon a strip yield model for analyzing multiple cracks in a sheet reinforced with riveted stiffeners. Using the basic solution of a single crack and taking unknown fictitious surface tractions and fastener forces, Fredholm integral equations are formulated from the equilibrium condition along multiple cracks in the sheet. In addition compatibility equations of displacements are formulated among the sheet, fasteners and stiffeners. Based upon no stress singularity at the fictitious crack tips, these equations are iteratively solved as a single system of equations. Then the unknown fictitious surface tractions, fastener forces, and plastic zone sizes ahead of the crack tips are determined. The crack tip opening displacements for a multiple cracked sheet with riveted stiffeners are determined from the derived fictitious surface tractions and plastic zone sizes. Four example calculations and predictions are presented.

Elastic-Plastic Analytical Solution for SEMI’s Horizontal Brace with a Circumferential Through Crack under Tension and Bending

2021 ◽  
pp. 1-8
Author(s):  
Fei Wang
Keyword(s):  
Analytical Solution ◽  
Plastic Zone ◽  
Plastic Behavior ◽  
Crack Tip Opening Displacement ◽  
Opening Displacement ◽  
Elastic Plastic ◽  
Bending Load ◽  
Closed Form Analytical Solution ◽  
Plastic Zones ◽  
Crack Tip Opening

The elastic-plastic behavior of semi-submersible’s horizontal brace with a circumferential through crack which lies at its boundary was studied. Both tension and bending were considered to investigate the closed-form analytical solution. The results indicate that the tensile plastic zone and crack tip opening displacement (CTOD) on the cracked section increase sharply after a smoothly increment when loads became larger. The cracked horizontal brace with a greater initial circumferential through crack has a larger tensile plastic zone and earlier compressive plastic zone appearance on the cracked section. Compared with the load of tension, the bending load has larger effect on the plastic zones of the cracked section and CTOD of the 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.

On the Torsion of Elastic Half Spaces With Embedded Penny-Shaped Flaws

1972 ◽  
Vol 39 (3) ◽  
pp. 786-790 ◽  
Author(s):  
R. D. Low
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Rigid Inclusion ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Graphical Displays ◽  
Penny Shaped Crack ◽  
Shaped Crack

The investigation is concerned with some of the effects of embedded flaws in an elastic half space subjected to torsional deformations. Specifically two types of flaws are considered: (a) a penny-shaped rigid inclusion, and (b) a penny-shaped crack. In each case the problem is reduced to a system of Fredholm integral equations. Graphical displays of the numerical results are included.

Plastic Strain Distribution at the Tips of Propagating Fatigue Cracks

1976 ◽  
Vol 98 (1) ◽  
pp. 24-29 ◽  
Author(s):  
D. L. Davidson ◽  
J. Lankford
Keyword(s):  
Aluminum Alloy ◽  
Fatigue Crack ◽  
Plastic Strain ◽  
Plastic Zone ◽  
Strain Distribution ◽  
Dimensionless Parameter ◽  
Fatigue Cracks ◽  
Crack Tip ◽  
Cyclic Plastic Zone ◽  
Plastic Zones

The techniques of selected area electron channeling and positive replica examination have been used to study the plastic zones attending fatigue crack propagation in 304 SS, 6061-T6 aluminum alloy, and Fe-3Si steel. These observations allowed the strain distribution at the crack tip to be determined. The results indicate that the concepts of a monotonic and a cyclic plastic zone are essentially correct, with the strains at demarcation between these two zones being 3 to 6 percent. Strain distribution varies as r−1/2 in the cyclic zone and as ln r in the monotonic plastic zone. The strain distributions for all materials studied may be made approximately coincident by using a dimensionless parameter related to distance from the crack tip.

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

2020 ◽  
Vol 37 (9) ◽  
pp. 3243-3268
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Stochastic Integral ◽  
Bernstein Polynomials ◽  
Numerical Results ◽  
Content Type ◽  
Stochastic Integral Equations

Purpose This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

Elastic Layer Pressed Against a Half Space

1974 ◽  
Vol 41 (3) ◽  
pp. 703-707 ◽  
Author(s):  
K. C. Tsai ◽  
J. Dundurs ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Integral Equations ◽  
Half Space ◽  
Elastic Layer ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Auxiliary Functions ◽  
Receding Contact ◽  
Two Bodies

The paper considers the elastic layer which is pressed against a half space by loads that are not necessarily symmetric about the center of the loaded region. It is shown that the receding contact between the two bodies can be treated by means of superposition, leading to two homogeneous Fredholm integral equations for auxiliary functions that are directly related to the contact tractions. The determination of the extent of contact and the shift between the load and contact intervals can be viewed as an eigenvalue problem of the homogeneous integral equations. Specific numerical results are given for two types of triangular loads, and a comparison is made with certain symmetric loads.

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