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Journal ArticleDOI

Stress intensity factor for an elastic half plane weakened by multiple curved cracks

01 Aug 2018-Applied Mathematical Modelling (Elsevier)-Vol. 60, pp 540-551

Abstract: Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane.

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Stress intensity factor for an elastic half plane weakend by multiple curved cracks
ABSTRACT
Modified complex potential with free traction boundary condition is used to formulate the
curved crack problem in a half plane elasticity into a singular integral equation. The singular
integral equation is solved numerically for the unknown distribution dislocation function.
Numerical examples exhibit the stress intensity factor increases as the cracks getting close to
each other, and close to the boundary of the half plane.
Keyword: Stress intensity factor; Elastic half plane; Curved crack; Singular integral equation
Citations
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ReportDOI
15 Aug 1977
Abstract: : One encounters difficulties in the solution of integral equations, namely, the occurrence of singularities in the kernel, and the occurence of unknown-type singularities in the solution of the integral equation. Standard methods of approximation based on exactness for polynomials up to a certain degree are very poor, and frequently fail for functions having such singularities. The purpose of this contract was to develop methods for solving integral equations which work well in spite of the presence of singularities. This goal has been accomplished, in that the new approximation methods which were developed do work well in the presence of singularities. (Author)

181 citations


Journal ArticleDOI
Abstract: This paper presents the in-plane asymptotic displacement and stress fields for blunt V-notched components based on Kolosov–Muskhelishvili's approach. In the first part, the displacement and stress components in the polar coordinate system are determined by choosing appropriate complex potential functions. In order to construct the notch geometry, the Neuber's mapping relation is utilized. Then, the notch boundary conditions are imposed to calculate the free parameters of the stress distribution. Eventually, the stress and displacement components are calculated in the Cartesian and polar coordinates in the forms of series expansion. In the second part, the coefficients of series expansions are computed by using the least square method (LSM). The blunt V-notched Brazilian disk (BV-BD) specimen under mixed mode loading is used as an example to verify the proposed procedure. The stress components in arbitrary distances and directions are determined for different blunt V-notches in order to evaluate the accuracy of the calculated stress series solutions and their associated coefficients. The numerical results indicate that a single-term solution can lead to considerable errors, and to achieve good accuracy in the stress field calculation, one should take account of at least three terms in the stress series solution.

11 citations


Journal ArticleDOI
Khairum Hamzah1, Khairum Hamzah2, N. M. A. Nik Long2, Norazak Senu2  +1 moreInstitutions (3)
Abstract: This paper deals with the multiple inclined or circular arc cracks in the upper half of bonded dissimilar materials subjected to shear stress. Using the complex variable function method, and with the help of the continuity conditions of the traction and displacement, the problem is formulated into the hypersingular integral equation (HSIE) with the crack opening displacement function as the unknown and the tractions along the crack as the right term. The obtained HSIE are solved numerically by utilising the appropriate quadrature formulas. Numerical results for multiple inclined or circular arc cracks problems in the upper half of bonded dissimilar materials are presented. It is found that the nondimensional stress intensity factors at the crack tips strongly depends on the elastic constants ratio, crack geometries, the distance between each crack and the distance between the crack and boundary.

10 citations


Journal ArticleDOI
Michele Zappalorto1, Marco Salviato2Institutions (2)
Abstract: This contribution investigates the stress fields in orthotropic plates featuring lateral notches under anti-plane shear loading. Four different notch geometries are considered and the relevant analytical expressions for the stress distribution are derived in closed form. For each geometry, the main features of the stress fields and the accuracy of the analytical expressions developed are discussed comparing theoretical results and numerical data from FE analyses carried out on finite plates under longitudinal shear.

4 citations


Journal ArticleDOI
Abstract: The modified complex variable function method with the continuity conditions of the resultant force and displacement function are used to formulate the hypersingular integral equations (HSIE) for an inclined crack and a circular arc crack lies in the upper part of bonded dissimilar materials subjected to various remote stresses. The curve length coordinate method and appropriate quadrature formulas are used to solve numerically the unknown crack opening displacement (COD) function and the traction along the crack as the right hand term of HSIE. The obtained COD is then used to compute the stress intensity factors (SIF), which control the stability behavior of bodies or materials containing cracks or flaws. Numerical results showed the behavior of the nondimensional SIF at the crack tips. It is observed that the nondimensional SIF at the crack tips depend on the various remote stresses, the elastic constants ratio, the crack geometries and the distance between the crack and the boundary.

3 citations


Cites background from "Stress intensity factor for an elas..."

  • ...( ) ( ) 1 2 , L g t dt z t z φ π = − ∫ (6)...

    [...]

  • ...The HSIE for a crack lies in the upper part of bonded dissimilar materials can be obtained by substituted (18) into (5) and applying (6), (7), (21) and (22), then letting point z approaches t0 on the crack and changing d z dz into 0 0 dt dt , yields...

    [...]


References
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Journal ArticleDOI
Brian Cotterell1, James R. Rice2Institutions (2)
Abstract: A solution is presented for the elastic stress intensity factors at the tip of a slightly curved or kinked two-dimensional crack. The solution is accurate to first order in the deviation of the crack surface from a straight line and is carried out by perturbation procedures analogous to those of Banichuk [1] and Goldstein and Salganik [2, 3]. Comparison with exact solutions for circular arc cracks and straight cracks with kinks indicates that the first order solution is numerically accurate for considerable deviations from straightness. The solution is applied to fromulate an equation for the path of crack growth, on the assumption that the path is characterized by pure Mode I conditions (i.e., K II=0) at the advancing tip. This method confirms the dependence of the stability, under Mode I loading, of a straight crack path on the sign of the non-singular stress term, representing tensile stress T acting parallel to the crack, in the Irwin-Williams expansion of the crack tip field. The straight path is shown to be stable under Mode I loading for T 0.

1,608 citations


Book ChapterDOI
01 Jan 1973
Abstract: In this Chapter the numerical methods for the solution of two groups of singular integral equations will be described. These equations arise from the formulation of the mixed boundary value problems in applied physics and engineering. In particular, they play an important role in the solution of a great variety of contact and crack problems in solid mechanics. In the first group of integral equations the kernels have a simple Cauchy-type singularity.

735 citations


ReportDOI
15 Aug 1977
Abstract: : One encounters difficulties in the solution of integral equations, namely, the occurrence of singularities in the kernel, and the occurence of unknown-type singularities in the solution of the integral equation. Standard methods of approximation based on exactness for polynomials up to a certain degree are very poor, and frequently fail for functions having such singularities. The purpose of this contract was to develop methods for solving integral equations which work well in spite of the presence of singularities. This goal has been accomplished, in that the new approximation methods which were developed do work well in the presence of singularities. (Author)

181 citations


Book
04 Jul 2003
Abstract: Chapter 1 Fundamental of plane elasticity crack problem : Introduction Cauchy integrals Some elementary formulae for the calculation of Cauchy integrals Poincare-Bertrand formula Basic equations of complex variable function method in plane elasticity Elementary solutions initiated by point sources Elementary solutions for a single crack problem Modified complex potentials for the case of bonded half-planes Modified complex potentials for the case of circular boundary Eigenfunction expansion form and singular stress field at the vicinity of crack tip Logarithmic singularity at crack tip in plane elasticity. Chapter 2 Multiple crack problems in an infinite plate : Introduction Singular integral equation for the multiple crack problem (type S1) First type of Fredholm integral equation for the multiple crack problem (type R1A) Alternative Fredholm integral equation for the multiple crack problem (type R1B) Alternative singular integral equation for the multiple crack problem (type S2) Third type of Fredholm integral equation for the multiple crack problem (type R2) Hypersingular integral equation for the multiple crack problem (type HS) General case of the multiple crack problem Multiple semi-infinite crack problem in an elastic plane Integral equations for the multiple crack problem in antiplane elasticity T-stress analysis in the multiple e crack problem Interactions between main crack and microcracks. Chapter 3 Multiple crack problems in more complicated cases : Introduction Multiple crack problems or circular regions Multiple cracks in a pressurized cylinder Multiple crack problems of circular region in antiplane elasticity Multiple crack problems for two bonded half-planes in plane and antiplane elasticity Multiple crack problem for an infinite strip Multiple crack problem for a finite plate Multiple crack problem for a rectangular region in antiplane elasticity Periodic crack problem in plane elasticity Doubly periodic crack problem in an infinite plate Multiple branch crack problem in plane elasticity Solution of multiple crack problem of elastic half-plane by using singular integral approach Interaction between a hole edge crack and line crack Interaction between an oblique edge crack and an internal crack in a cracked half-plane Collinear crack problems in an infinite plate An infinite plate containing hypocycloid hole with many cusps Solution of an tiplane elasticity crack problem using conformal mapping Solutions of torsion crack problems of a rectangular bar Solutions of torsion crack problems of a circular cross section bar. (part contents)

79 citations