Two arc cracks around a circular rigid inclusion

Summary The problem of two symmetric arc cracks lying between a rigid circular inclusion embedded in an infinite matrix under biaxial loading at infinity is considered. By using the complex variable tecnique, the boundary value problem is solved and stress and displacement components are calculated along the inclusion boundary. Moreover, investigating the local stress field, a stress criterion, already proposed by authors, allowing either the crack extension at the interface or its deviation into the matrix to be taken into account, is applied to study the fracture response of the elastic system. The critical applied loads as well as the angle of the incipient crack extension are expressed in terms of the central angle subtended by the crack arcs.Finally the biaxial load effects are graphically shown and discussed.

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Sommario Si considera il problema piano di due «craks» simmetrici all'interfaccia tra una inclusione circolare rigida e la circostante matrice, in regime di carico biassiale. Facendo uso della tecnica delle variabili complesse, viene risolto il problema dei valori al contorno, ricavando altresi le espressioni delle tensioni e degli spostamenti lungo il contorno dell'inclusione. Inoltre, applicando un criterio tensionale, già proposto dagli autori, che consente di studiare sia la estensione del crack all'interfaccia che la sua deviazione nella matrice, si analizza la risposta del sistema considerato. I valori critici dei carichi applicati, nonchè l'angolo di frattura, risultano espressi in funzione dell'angolo al centro sotteso dai cracks.Infine, vengono illustrati graficamente e discussi gli effetti del carico laterale.

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Financial support of the National Research Council (C.N.R.) (research contribution N. 79.01625.07) is gratefully acknowledged.     

Interaction between a rigid line inclusion and an elastic circular inclusion

I.IntroductionManypracticalproblemsinengineering,suchascompositematerial,weldedjointorribbedslab,needustostudytheinteractionproblemoflineinclusionandcircularinclusionasshowninFig.1.Sotheproblemwasdiscussedinthispaper.Proceedingfromthestressfieldofplanecon…     

Stress state of a transversely isotropic spherical shell with a rigid circular inclusion

Analytical expressions are derived for the stresses near a rigid circular inclusion in a transversely isotropic shallow spherical shell under uniform pressure. The form of solution depends on the range of the transverse shear compliance parameter. The influence of the relative radius of a rigid inclusion and transverse shear compliance on stress concentration is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 67–73, December 2005.     

Elastic-plastic analysis of a wheel rolling on a rigid track

A consistent finite element model for a circular wheel is developed based on triangular and quasi-triangular domains and a piecewise linear displacement field. The minimum stress-rate principle of plasticity is used to obtain the solution of this two-dimensional continuum problem with internal unloading. A piecewise approximation of the Tresca yield condition is used. Elastic-plastic solutions of a wheel rolling on a rigid track under its own weight and a hub load are obtained for the first few revolutions until a steady state condition is reached. Shake-down conditions for the wheel are demonstrated.     

Large deformation adhesive contact mechanics of circular membranes with a flat rigid substrate

We study the large deformation mechanics of contact and adhesion between an inflated hyperelastic membrane and a rigid substrate. The initial configuration of the membrane is flat and circular and is clamped at the edge. Two types of friction conditions between the membrane and the substrate are considered: frictionless and no-slip contact. We derive an exact expression for the energy release rate in terms of local variables at the contact edge, thus linking adhesion to the contact angle. Our model can account for the effects of fluid pressure for experiments performed in solution. We also extend our formulation to include surface tension. Numerical simulations for a neo-Hookean membrane are carried out to study the relation between applied pressure and contact area.     

Antiplane shear crack in an infinite plate with a circular inclusion

Summary The effect of an elastic circular inclusion of a different material on the stress state of a single-cracked infinite sheet subjected to anti-plane shear is investigated. The proposed method makes use of complex variables in conjunction with Muskhelishvili's technique and has given interesting results for the stress field in the cracked plate. Numerical results, derived from this technique, investigate the influence of the regional inhomogeneity, produced by the inclusion in the structure, on the variation of stress-intensity factors at the crack tips.

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Nichtebener Schubspannungsriß in einer unendlichen Platte mit einem kreisförmigen Einschluß

Übersicht Es wird der Einfluß untersucht, den ein elastischer, kreisförmiger Einschluß aus verschiedenem Material in einer unendlichen Platte mit einem Riß, welche einer longitudinalen Schubspannung unterworfen ist, auf den Spannungszustand der Platte ausübt. Die vorgeschlagene Methode benutzt komplexe Variablen in Zusammenhang mit der Muskhelishvilischen Technik und gibt für den Spannungszustand in der Platte interessante Resultate. Numerische Ergebnisse wurden gefunden, um den Einfluß der Inhomogenität auf die Spannungsintensitätsfaktoren des Risses zu bestimmen.

     

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.     

Stick-slip analysis of a circular point contact between a rigid sphere and a flat unidirectional composite with cylindrical fibers

The stick-slip contact problem is investigated here when at least one of the contacting bodies behaves as an ideal composite material with long fibers perpendicular to the direction of movement. Cylindrical inhomogeneous inclusions within a homogeneous media and with axes parallel to the contact surface are considered. The Eshelby’s equivalent inclusion method is used to solve the problem numerically. Interactions between close inclusions are taken into account in the numerical procedure, as well as the coupling between the normal and tangential contact problems. It is found that the presence of heterogeneities in the vicinity of the surface contact affects significantly the contact pressure distribution and subsequently the distribution of shear and slip at the interface.     

PLANE ELASTIC PROBLEM ON RIGID LINES ALONG CIRCULAR INCLUSION

The plane elastic problem of circular-arc rigid line inclusions is considered. The model is subjected to remote general loads and concentrated force which is applied at an arbitrary point inside either the matrix or the circular inclusion. Based on complex variable method, the general solutions of the problem were derived. The closed form expressions of the sectionally holomorphic complex potentials and the stress fields were derived for the case of the interface with a single rigid line. The exact expressions of the singular stress fields at the rigid line tips were calculated which show that they possess a pronounced oscillatory character similar to that for the corresponding crack problem under plane loads. The influence of the rigid line geometry, loading conditions and material mismatch on the stress singularity coefficients is evaluated and discussed for the case of remote uniform load.     

Bending of a radially prestressed annular plate by tilting a central rigid inclusion

A thin annular plate contains a rigid, circular, central inclusion. The plate is subjected to a large axisymmetric radial load at its outer edge, where it is also restrained against transverse displacement and rotation. A couple applied to the rigid inclusion causes it to rotate about its diameter out of the plane of the plate. We use the method of matched asymptotic expansions to find an approximate expression for the applied couple as a function of the angle of rotation of the rigid inclusion. If the outer radius of the annulus is very large compared to the inner radius, then the couple required to rotate a truly rigid inclusion is 25% higher than the couple required to rotate an inclusion whose membrane strain stiffness is the same as that of the plate (cf. ref. [3]) through the same small angle.     

Elastic-plastic strain concentrations produced by various skew holes in a flat plate under uniaxial tension

The elastic and elastic-plastic surface strain fields around circular holes drilled at various skew angles to a flat plate have been experimentally evaluated in uniaxial tension. A photoelastic coating and moiré technique were used in the low- and high-strain regions, respectively. The maximum strain-concentration factor is shown to increase markedly with horizontal skew angle and decrease slightly with increasing vertical skew angle. Plastic deformation accentuates the differences between normal and skew holes, so that the angular dependence of the strain-concentration factor increases with nominal strain.     

Thermal stresses in an elastic space with a perfectly rigid flat inclusion under perpendicular heat flow

This paper examines the three-dimensional problem of finding thermal stresses due to an insulated rigid sheet-like inclusion (anticrack) in an elastic space under a uniform perpendicular heat flow. By using appropriate harmonic potentials, a general method of solving this problem is presented. The resulting boundary-value problems are reduced to classical mixed problems of potential theory. For the purpose of illustration, a complete solution in terms of elementary functions for a rigid circularly shaped inclusion is given and discussed from the point of view of material failure.     

Interfacial de-bonding caused by a screw dislocation pile-up at a circular cylindrical rigid inclusion

An array of continuously-distributed screw dislocations piled up against a circular cylindrical rigid inclusion is analyzed by the complex-variable method. Both uniformly applied shearing load at infinity and internal friction stress opposing the movement of dislocations are taken into account. The pile-up tip is away from the matrix-inclusion interface, its distance from the interface being determined by the condition that the stresses should be finite everywhere in the solid. Stress distributions on the interface are determined, and de-bonding of the interface, namely the formation of initial voids or cracks, is discussed. Stress and displacement near the tip of these initial voids are then analyzed. This analysis is combined with the virtual work argument of A.A. Griffith (1920) to yield a criterion for the initial voids to grow along the interface. The critical void-growth load is expressed by the sum of two terms, one proportional to the friction stress and the other inversely proportional to the square-root of the inclusion radius.     

Interaction of a rigid punch and a circular plate with a fixed side and a stress-free face

The axisymmetric contact problem of a rigid punch indentation into an elastic circular plate with a fixed side and a stress-free face is considered. The problem is solved by a method developed for finite bodies which is based on the properties of a biorthogonal system of vector functions. The problem is reduced to a Volterra integral equation (IE) of the first kind for the contract pressure function and to a system of two Volterra IE of the first kind for functions describing the derivative of the displacement of the plate upper surface outside the punch and the normal (or tangential) stress on the plate lower fixed surface. The last two functions are sought as the sum of a trigonometric series and a power-law function with a root singularity. The obtained ill-conditioned systems of linear algebraic equations are regularized by introducing small parameters and have a stable solution. A method for solving the Volterra IE is given. The contact pressure functions, the normal and tangential stresses on the plate fixed surface, and the dimensionless indentation force are found. Several examples of a plane punch computation are given.     

On the thermal stresses of a semi-infinite plate with a circular inclusion under uniform heat flow

Summary The present paper gives a theoretical solution for a semi-infinite plate with a circular hole, which is filled with an elastic inclusion of another material, and subjected to uniform heat flow of the temperature gradient in the direction of the straight edge. The analysis is developed on the basis of the Airy's stress function in the generalized plane stress and by applying the bipolar coordinates. Here the plate is supposed to be insulated and the boundaries are free from the traction. The method of perturbation is adopted for the determination of unknown coefficients involved in the solution.Numerical calculations about the stress distributions along main parts of the semi-infinite plate and the inclusion are worked out in some detail, in order to clarify the effects of an inclusion. The maximum stresses on the common boundary are also calculated and compared with the results available.

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Übersicht Die Aufgabe der Abschätzung des Einflusses eines kreisförmigen Loches, in dieses Loch ist eine clastische Inklusion eingeschweißt, auf die Wärmspannung in einer uniform Wärmeströmebeanspruchten Halbebene wird auf Grund der Airyschen Spannungsfunktion und mittels des bipolaren Koordinatensystems behandelt. Zur Bestimmung der Koeffizienten, die in der Durchspannungsfunktion der Platte enthalten sind, wird die Störungsrechnung angewandt. Numerische Ergebnisse für einige spezielle Fälle werden mitgeteilt.

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The author wishes to express his hearty thanks to Prof. em. S. Higuchi and Prof. O. Tamate of the Tohoku University for their kind advice and interest throughout the progress of the present investigation.     

Photoelastic analysis of an asymmetrically reinforced circular cut-out in a flat plate subjected to uniform unidirectional stress

Three-dimensional photoelastic studies of stresses around an asymmetrically reinforced circular cut-out in a flat plate under uniform unidirectional stress are reported. The frozen-stress technique, with Hysol 4290 material, was used to determine the stress distribution through four critical points on the boundary of the reinforced hole. Included were models with different cross sections of reinforcement, with various interface fillet radii and with different plate widths. For the majority of models, the ratio of volume of reinforcement to volume of hole was unity. It is concluded that, for reducing the stress concentration, there is a limit on the effectiveness of increasing the fillet radius beyond half the plate thickness. It was found that a reinforcement having a thickness of approximately 40 percent of the plate thickness was optimum and that the stress concentration decreases with volume of reinforcement. The authors believe that, with judgment, some of the conclusions reached may be applied, for design purposes, beyond the specific dimensional ranges and loading conditions of the tests.