On the expansion of a penny-shaped crack by a rigid circular disc inclusion

Selvadurai, A. P. S.; Singh, Brij Mohan

doi:10.1007/bf01152750

Cited by 46 publications

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“…Since such a problem for a purely elastic isotropic medium has been solved in [19], we will use its solution and its relationship to the electroelastic problem. Note that the paper [19] reduces the problem to triple integral equations effectively solved by expanding the unknown function into a series in powers of the small parameter e = a b / .…”

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Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

Kirilyuk

2008

Int Appl Mech

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The paper establishes a relationship between the solutions for cracks located in the isotropy plane of a transversely isotropic piezoceramic medium and opened (without friction) by rigid inclusions and the solutions for cracks in a purely elastic medium. This makes it possible to calculate the stress intensity factor (SIF) for cracks in an electroelastic medium from the SIF for an elastic isotropic material, without the need to solve the electroelastic problem. The use of the approach is exemplified by a penny-shaped crack opened by either a disk-shaped rigid inclusion of constant thickness or a rigid oblate spheroidal inclusion in an electroelastic medium Keywords: transversely isotropic piezoceramic medium, plane crack, rigid inclusion, isotropy plane, penny-shaped crack, disk-shaped rigid inclusion of constant thickness, oblate spheroidal inclusion, stress intensity factorIntroduction. Methods to solve three-dimensional problems of elasticity for bodies with cracks are well developed. Results on stress intensity factors (SIFs) for cracks in an elastic medium are presented in numerous publications [3, 7, 8, 15, 19, 21, etc.]. The brittle fracture of prestressed bodies is studied in [2]. The extensive use of piezoceramic materials, which are distinguished by considerable brittleness, necessitates studying the mechanical and electric fields around stress concentrators such as cavities, inclusions, and cracks in electroelastic bodies [1, 4-6, 9-14, 6-18, 20, 22-24]. However, solving electroelastic problems involves severe mathematical difficulties compared with elastic problems because the original equations for electric and strain states constitute a more complicated system of differential equations [1,4]. The homogeneous equations of electroelasticity for piezoceramic bodies are addressed in [5,22].The present paper establishes a relationship between the solutions of the three-dimensional problem for plane cracks opened by rigid inclusions in an elastic isotropic space and in a transversely isotropic electroelastic medium. It is assumed that the crack is in the isotropy plane of the transversely isotropic electroelastic material and there is no friction between the crack and the inclusion. Thus, we can calculate the stress intensity factor (SIF) K I in a piezoceramic space from the expressions for the SIF in an elastic isotropic medium with a plane crack and a rigid inclusion of the same shape, without the need to solve the electroelastic problem. As examples of using such a relationship, we will consider a penny-shaped crack opened by either a rigid disk-shaped inclusion of constant thickness or a rigid inclusion in the form of an oblate spheroid in an electroelastic space. Problem Formulation and Basic Equations.Consider a three-dimensional transversely isotropic electroelastic space with a system of plane cracks (occupying a domain S ) located in a plane perpendicular to the polarization axis and opened by rigid inclusions (occupying a domain S 1 , i.e., S S 1 Ì ). The free portions of the cracks are not sub...

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“…Note that the paper [19] reduces the problem to triple integral equations effectively solved by expanding the unknown function into a series in powers of the small parameter e = a b / . As a result, we obtain …”

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Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

Kirilyuk

2008

Int Appl Mech

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“…For both problems the crack-opening mode stress-intensity factors at the boundary r = b are evaluated; these are denoted by (K,) Al and (K/) ao . These auxiliary problems have been examined in the literature (17,21,22) and the stress-intensity factors can only be evaluated either in a power-series form in terms of a/b or numerically. The radius of the zone of separation can be evaluated by invoking the conditions (K 7 ) Al = (K/) Al = (K,)^, A[ + A 2 = A and (K,) A + (Kj) ao = 0.…”

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A Unilateral Contact Problem for a Rigid Disc Inclusion Embedded Between Two Dissimilar Elastic Half-Spaces

Selvadurai¹

1994

Q J Mechanics Appl Math

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The present paper examines the axisymmetnc elasto-static problem where the contact between two smoothly compressed dissimilar elastic half-spaces is perturbed by a smooth disc inclusion of finite thickness. The paper develops the governing integral equation and the additional constraint which is associated with the separation condition. The analysis focuses on the estimation of the influence of the precompression and the elasticity mismatch between the bimatenal half-space regions on the radius of the separation zone. The paper also records a useful analytical approximation to the unilateral contact problem which is derived by prescribing a priori the displacements in the contact region beyond the separation zone.

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“…This three-part mixed boundary value problem can be reduced to a set of triple integral equations, which has been examined by a number of authors (see e.g., [109][110][111][112][113][114][115]), and an approximate solution to the system (8) to (10) can be obtained as a series expansion in terms of the parameter c(= a/b) < 1. The result of interest to the interpretation of the surface permeability tests conducted on the cuboidal regions of both Stanstead and Lac du Bonnet granite is the steady flow rate Q 0 (units L 3 /T) that will be established from the interior region of the impermeable annulus, when the interior region is subjected to a constant fluid pressure of p 0 , and the exterior region is maintained at zero pressure.…”

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Estimates for the Effective Permeability of Intact Granite Obtained from the Eastern and Western Flanks of the Canadian Shield

2020

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Granitic rock from the western part of the Canadian Shield is considered as a potential host rock for the siting of a deep geological repository for the storage of heat-emitting high-level nuclear fuel waste. The research program focused on the use of surface permeability measurements conducted at 54 locations on a 300 mm cuboid of granite, obtained from the Lac du Bonnet region in Manitoba, to obtain an estimate for the effective permeability of the cuboid. Companion experiments are conducted on a 280 mm cuboid of granite obtained from Stanstead, Quebec, located in the eastern part of the Canadian Shield. The surface permeabilities for the cuboids of granite are developed from theoretical relationships applicable to experimental situations where steady flow is initiated at a sealed annular surface region with a pressurized central domain. The experimental values for the surface permeability are used with a kriging procedure to estimate the permeability variations within the cuboidal region. The spatial variations of permeability are implemented in computational models of the cuboidal regions to determine the one-dimensional permeabilities in three orthogonal directions. The effective permeability of the granite cuboids is estimated by appeal to the geometric mean. The research provides a non-destructive methodology for estimating the effective permeability of large specimens of rock and the experiments performed give estimates for the effective permeability of the two types of granitic rock obtained from the western and eastern flanks of the Canadian Shield.

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On the expansion of a penny-shaped crack by a rigid circular disc inclusion

Cited by 46 publications

References 5 publications

Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

A Unilateral Contact Problem for a Rigid Disc Inclusion Embedded Between Two Dissimilar Elastic Half-Spaces

Estimates for the Effective Permeability of Intact Granite Obtained from the Eastern and Western Flanks of the Canadian Shield

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