Generalization of Galin's problem to frictional materials and discontinuous stress fields

Tokar, Gabriel

doi:10.1016/0020-7683(90)90047-y

Cited by 16 publications

(14 citation statements)

References 2 publications

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“…In particular, Equations (20) and (21) are suitable for any point on the unit circle in the z-plane, that is, s 2 L 0 .…”

Section: Stress Components On Lmentioning

confidence: 99%

“…For the non-axisymmetric problem, it has always been the difficulty and focus of the research. There are some approximate solutions deduced by the complex variable method [19][20][21] and the perturbation method [22,23]. The complex variable method has played a crucial role in evaluating the shape and size of the plastic zone, and one that was introduced to obtain the exact analytical solution based on the Tresca yield criterion by Galin [24].…”

Section: Introductionmentioning

confidence: 99%

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An analytical method for determining the non-enclosed elastoplastic interface of a circular hole

Cai

2020

Mathematics and Mechanics of Solids

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Based on the Mohr–Coulomb yield criterion, an analytical method is presented to determine the plastic zone in an infinite plate weakened by a circular hole and subjected to non-hydrostatic stresses at infinity. It is worth noting that this paper considers the more complicated case that the plastic zone cannot completely surround the hole, namely the elastoplastic interface is non-enclosed. Initially, the non-circular elastic zone in the physical plane is mapped onto the outer region of a unit circle in the image plane by the conformal transformation in the complex variable method. Thereby, determining the elastoplastic interface is equivalent to solving the mapping function coefficients. The nonlinear equations for solving the coefficients are established by considering both the stress continuity conditions along the elastoplastic interface and the stress boundary conditions along the elastic part of the hole. Naturally, the problem can be further transformed into an optimization problem, which is ultimately achieved by the differential-evolution algorithm; what is more, an analytical solution with high accuracy is obtained. Based on the programmed computing, the influences of various parameters on the shape and size of the plastic zone are given.

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“…In particular, Equations (20) and (21) are suitable for any point on the unit circle in the z-plane, that is, s 2 L 0 .…”

Section: Stress Components On Lmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An analytical method for determining the non-enclosed elastoplastic interface of a circular hole

Cai

2020

Mathematics and Mechanics of Solids

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“…While plastic yielding occurs, various distributions of the plastic zone may appear, depending on the soil strength and boundary conditions (Bradford and Durban, 1998, Tokar, 1990, Yarushina et al, 2010. As an extension of the Galin's (1946) solution to the Mohr-Coulomb material, the major concern of this note is the distribution of the elastic and plastic stresses around the cavity in the states satisfying two prior assumptions (Detournay, 1986, Yarushina et al, 2010: (1) a plastic zone is developed under pressure, and it is statically determinate, and (2) the inner cavity is fully encircled by the formed plastic zone.…”

Section: Elastic and Plastic Stress Analysismentioning

confidence: 99%

“…Analytical solutions for the two-dimensional cylindrical cavity analysis in elastic perfectly-plastic materials was inspired primarily by the ingenious method developed by Galin (1946) in the loading analysis adopting the Tresca yield criterion, for example, the subsequent solutions considering various boundary conditions (Cherepanov, 1963, Parasyuk, 1948, Yarushina et al, 2010 and/or different materials (Detournay, 1986, Tokar, 1990.…”

Section: Introductionmentioning

confidence: 99%

A unified analytical solution for elastic–plastic stress analysis of a cylindrical cavity in Mohr–Coulomb materials under biaxial in situ stresses

Zhuang

2019

Géotechnique

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“…By assuming that the plastic zone is statically determinate and using the complex variable theory in the elastic analysis, both constant and polynomial types of far-field stresses were studied by Galin. 30 Although minor mistakes existed in these solutions as found and improved by Ochensberger et al 31 and Tokar,32 respectively, it is generally believed that this methodology greatly facilitated the development of analytical/semi-analytical solutions for the two-dimensional elastic-plastic analysis of a series of problems with similar boundary conditions. [33][34][35][36][37][38] Comparing with the problem of Galin, 30 uniform shear stresses at the inner cavity wall are additionally considered in this study.…”

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confidence: 99%

Two‐dimensional elastoplastic analysis of cylindrical cavity problems in Tresca materials

Zhuang

2019

Num Anal Meth Geomechanics

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Summary This paper presents analytical elastic‐plastic solutions for static stress loading analysis and quasi‐static expansion analysis of a cylindrical cavity in Tresca materials, considering biaxial far‐field stresses and shear stresses along the inner cavity wall. The two‐dimensional static stress solution is obtained by assuming that the plastic zone is statically determinate and using the complex variable theory in the elastic analysis. A rigorous conformal mapping function is constructed, which predicts that the elastic‐plastic boundary is in an elliptic shape under biaxial in situ stresses, and the range of the plastic zone extends with increasing internal shear stresses. The major axis of the elliptical elastic‐plastic boundary coincides with the direction of the maximum far‐field compression stress. Furthermore, considering the internal shear stresses, an analytical large‐strain displacement solution is derived for continuous cavity expansion analysis in a hydrostatic initial stress filed. Based on the derived analytical stress and displacement solutions, the influence of the internal shear stresses on the quasi‐static cavity expansion process is studied. It is shown that additional shear stresses could reduce the required normal expansion pressure to a certain degree, which partly explains the great reduction of the axial soil resistance due to rotations in rotating cone penetration tests. In addition, through additionally considering the potential influences of biaxial in situ stresses and shear stresses generated around the borehole during drillings, an improved cavity expansion approach for estimating the maximum allowable mud pressure of horizontal directional drillings (HDDs) in undrained clays is proposed and validated.

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Generalization of Galin's problem to frictional materials and discontinuous stress fields

Cited by 16 publications

References 2 publications

An analytical method for determining the non-enclosed elastoplastic interface of a circular hole

An analytical method for determining the non-enclosed elastoplastic interface of a circular hole

A unified analytical solution for elastic–plastic stress analysis of a cylindrical cavity in Mohr–Coulomb materials under biaxial in situ stresses

Two‐dimensional elastoplastic analysis of cylindrical cavity problems in Tresca materials

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