An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions

Abdou, M. A.; Jan, Azhar Rashad

doi:10.4236/am.2014.54070

Cited by 4 publications

(1 citation statement)

References 12 publications

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“…Others used complex variable method, Gaursat functions, and this complex method is called Cauchy method. For these types of problems, see Abdou and Khamis [10], Abdou and Aseri [11,12], Jaha and Abdou [13], Sahrama [14], Abdou and Jan [15] and Abdou and Basseem [16]. The problem of focusing stress around a discontinuities shape such as holes or cracks, is of great importance for determining the load of bearing capacity in the structural members, see Exadaktylos et al [17,18].…”

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The singular complex integro differential equation via the boundary value problems of elastic plate weakened by curvilinear hole

Ibrahim

Abdou

2023

Preprint

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In this paper, we presented an integro differential equation (IDE), in compelx plane, with singular kernel. The IDE is investigated from the first and second boundary value problem (BVP) of a weakened infinite plate by a curvilinear hole (C,) in the presence of heat. The hole is conformally mapped inside a unit circle by a special transformation with complex constants coefficients\(z=\omega (\zeta ),\,\zeta =\xi +i\eta ,\,i=\sqrt { - 1} .\) This mapping will conform the curvilinear hole in the infinite elastic plate into a unit circle \(\gamma ,\left| \zeta \right|\;<\,\,1,\) such that inside the circle. The stress and strain compounds were calculated, in the presence of heat effect, and this was clarified by numerical results. Many applications for the problem are discussed. Maple 2019 is used for computations results. MSC (2010): 74B10, 30C20.

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The singular complex integro differential equation via the boundary value problems of elastic plate weakened by curvilinear hole

Ibrahim

Abdou

2023

Preprint

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Stress distribution around waist hole on finite plate based on mapping function

Liu

Wang

2019

AIP Advances

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Holes exist on parts of ships and offshore structures to satisfy functional requirements, reduce weight, and achieve economic benefits. However, stress concentration often occurs around holes and affects the safety of structures. Analytical methods have been proposed for calculating the stress distribution for regular shapes of holes, such as circular, square, and triangular, particularly for the waist hole on the infinite plate. However, the stress distribution of other types of holes on the finite plate is difficult to calculate, although it is more important in practice. In this study, the stress distribution around the waist hole on the finite plate is explored. An improved mapping function for the waist hole, which introduces a new variable based on the rational mapping theory, is proposed, and the range and variation of each parameter in the formula are given according to the structural characteristics of the waist hole. The typical waist hole on the finite plate is calculated using the proposed method, and related numerical simulations are also performed. The results verify the accuracy and reliability of the proposed method. Moreover, the stress distribution of the waist hole under different loads is calculated and discussed. The findings of this study provide a theoretical reference for practical engineering applications.

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An infinite plate with a curvilinear hole having three poles with complex parameters

Bayones

Alharbi

2015

JAM

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This paper covered the study of the boundary value problem for isotropic homogeneous perforated infinite elastic media. For this, we considered the problem of a thin infinite plate of specific thickness with a curvilinear hole where the origin lie outside the hole is conformally mapped outside a unit circle by means of a specific rational mapping . The complex variable method has been applied and it transforms the problem to the integrodifferential equation with Cauchy kernel that can be solved to find two complex potential functions which called Gaursat functions. Many special cases are discussed and established of these functions .Also, many applications and examples are considered. Moreover the components of stress , in each application , are computed.

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An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions

Cited by 4 publications

References 12 publications

The singular complex integro differential equation via the boundary value problems of elastic plate weakened by curvilinear hole

The singular complex integro differential equation via the boundary value problems of elastic plate weakened by curvilinear hole

Stress distribution around waist hole on finite plate based on mapping function

An infinite plate with a curvilinear hole having three poles with complex parameters

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