Applied alternating method to analyze the stress concentration around interacting multiple circular holes in an infinite domain

Ting, Kuen; Chen, K.T.; Yang, W.S.

doi:10.1016/s0020-7683(98)00031-6

Cited by 39 publications

(11 citation statements)

References 14 publications

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“…Also, the body force method and conformal mapping technique were developed to study the notch problem (Sih, 1978). An iterative method based on the Airy's stress function was used to 'solve a multiple circular hole problem (Ting et al, 1999). All those investigations are limited to the stress concentration problem for the relevant cases.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Lee

2002

KSME International Journal

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Two-dimensional elastic analysis of doubly periodic circular holes in an infinite plane is given in this paper. Two cases of loading, remote tension and remote shear, are considered. A rectangular cell is cut from the infinite plane. In both cases, the boundary value problem can be reduced to a complex mixed one. It is found that the eigenfunction expansion variational method is efficient to solve the problem. Based on the deformation response under certain loading, the notched medium could be modeled by an orthotropic medium without holes. Elastic properties for the equivalent orthotropic medium are investigated, and the stress concentration along the hole contour is studied. Finally, numerical examples and results are given.

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Section: Introductionmentioning

confidence: 99%

Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Lee

2002

KSME International Journal

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“…In parallel with rapid expansion in computer capabilities, the numerical schemes progressed from a closed-form solution with infinite sums in the bipolar coordinates [Ling 1948] and the alternative iterations [Ting et al 1999] to the advanced FE analysis [Waldman et al 2003] and to the highly accurate solution of the Sherman-type integral equation [Helsing and Jonsson 2000]. Tables 1 and 2 compare literature data against our results [Ting et al 1999 Table 2. Maximum tangential stresses (2.15) for two equal circular holes aligned with the x-axis under biaxial loading B 0 = 1, 0 = 0 as a function of λ.…”

Section: (44)mentioning

confidence: 81%

Shape optimization in an elastic plate under remote shear: from single to interacting holes

Vigdergauz

2008

JOMMS

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An elastic plate with two closely spaced identical holes of fixed area is taken as a two-dimensional sample geometry to find the interface shape which minimizes the energy increment in a homogeneous shear stress field given at infinity. This is a transient model between a single energy-minimizing hole and a regularly perforated plate, both numerically solved by a genetic optimization algorithm together with a fast and accurate fitness evaluation scheme using the complex-valued elastic potentials which are specifically arranged to incorporate a traction-free hole boundary. Here the scheme is further enhanced by a novel shape-encoding procedure through a conformal mapping of a single hole rather than both holes simultaneously as is done in standard practice. The optimized shapes appear to be slightly rounded elongated quadrangles aligned with the principal load axes. Compared to the single (square-like) optimal hole, they induce up to 12% less energy depending on the hole spacing. Qualitatively, it is also shown that the local stresses, computed along the optimal shapes as a less accurate by-product of the optimization, exhibit a tendency to be piecewise constant with no local concentration.

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“…Superÿcially, it can be regarded as a classical Fourier series approach because it deals only with circular boundaries. Even from this viewpoint, however, the method di ers from other Fourier series approaches in that the Fourier coe cients for the boundary displacements and tractions are directly linked together instead of being related indirectly through the coe cients for an Airy's stress function, as in, for example, Reference [12], or a boundary density function, as in Reference [22]. The primary reason for this linkage is that our analysis is based on Somigliana's formula, and this provides the second way of viewing the resulting numerical method: as a special case of the direct boundary integral method in which all boundaries are circular.…”

Section: Discussionmentioning

confidence: 99%

“…More recently, Ting et al [12] described an alternating method for analysing the interactions among multiple circular holes. Their method is based on the analytical solution to the problem of an inÿnite plane containing a single circular hole subjected to arbitrary boundary tractions represented as truncated Fourier series.…”

Section: Introductionmentioning

confidence: 99%

On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries

Crouch

Mogilevskaya

2003

Numerical Meth Engineering

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SUMMARYThis paper considers the problem of an inÿnite, isotropic elastic plane containing an arbitrary number of non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, if desired, be di erent. The analysis is based on the two-dimensional version of Somigliana's formula, which gives the displacements at a point inside a region V in terms of integrals of the tractions and displacements over the boundary S of this region. We take V to be the inÿnite plane, and S to be an arbitrary number of circular holes within this plane. Any (or all) of the holes can contain an elastic inclusion, and we assume for simplicity that all inclusions are perfectly bonded to the material matrix. The displacements and tractions on each circular boundary are represented as truncated Fourier series, and all of the integrals involved in Somigliana's formula are evaluated analytically. An iterative solution algorithm is used to solve the resulting system of linear algebraic equations. Several examples are given to demonstrate the accuracy and e ciency of the numerical method.

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Applied alternating method to analyze the stress concentration around interacting multiple circular holes in an infinite domain

Cited by 39 publications

References 14 publications

Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Shape optimization in an elastic plate under remote shear: from single to interacting holes

On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries

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