A rigid inclusion in an elastic space under the action of a uniform heat flow in the inclusion plane

A solution is presented for the three dimensional static thermoelastic problem of an absolutely rigid inclusion (anticrack) in the case when a uniform heat flow is directed along the inclusion plane. By using the potential method and the Fourier transform technique, the problem is reduced to a system of coupled two-dimensional singular integral equations for the shear stress jumps across the inclusion. As an illustration, a typical application to the circular anticrack is presented. Explicit expressions for the thermal stresses in the inclusion plane are obtained and discussed from the point of view of material failure.     

Thermal stresses around a ribbon-like inclusion in a semi-infinite medium under uniform heat flow

On the basis of the two-dimensional theory of thermoelasticity, the present paper deals with the thermal stresses around a thermally insulated and rigid ribbon-like inclusion in a semi-infinite medium under steady-state uniform heat flow. The ribbon-like inclusion is replaced by continuous distributions of sources of temperature discontinuity and concentrated forces. Then we obtain a system of singular integral equations with Cauchy kernels. The solution is given in the form of the product of the series of Chebyshev polynomials of the first kind and their weight function. By means of this method, the singular behavior of the thermal stresses near the tips of the ribbon-like inclusion is estimated exactly and the strength of the thermal stress singularity can be easily evaluated.

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Zusammenfassung Auf dem Grund von der zweidimensionalen Theorie der Thermoelastizität befasst sich der vorliegende Bericht mit der Wärmespannungen um eine thermisch isolierte und starre Bandeinlagerung in einem semi-unendlichen Medium unter stationärzustandlicher gleichmässiger Hitzeausströmung. Die Bandeinlagerung ist durch kontinuierliche Verteilung von Quellen der Temperaturdiskontinuität und konzentrierten Kräften umgestellt. Dann wird ein System von singulären Integralgleichungen mit Cauchyschen Kernen erhalten. Die Lösung kann in der Form des Produkts von der Reihe von Tchebycheffschem Polynomen erster Art und ihrer Gewichtsfunktion angegeben werden. Durch Anwendung von dieser Methode wird das singuläre Benehmen von der Wärmespannungen nahe den Spitzen der Bandein-lagerung richtig geschätzt und kann die Intensität der Wärmespannungsingularität leicht berechnet werden.

     

Spheroidal rigid inclusion in an elastic medium under torsion

The displacement field is determined when a rigid spheroidal inclusion is present in an infinite, isotropic and homogeneous elastic medium under torsion. The values of the force and moment are derived. The solutions for the limiting cases of a sphere and a circular disk are also presented. The analysis is based on suitable distributions of singularities on the axis of symmetry of the inclusion.     

On the load transfer from a rigid cylindrical inclusion into an elastic half space

The present paper examines the problems related to the axial, lateral, and rotational loading of a rigid cylindrical inclusion which is embedded in bonded contact at the boundary of an isotropic elastic half space. The rigid inclusion is modeled as a field of distributed forces which represent the normal and shear tractions that act on the inclusion-elastic-medium interface. The intensities of these distributed tractions are determined by enforcing displacement compatibility conditions at discrete locations of the interface. These compatibility conditions are derived from rigid-body displacement modes appropriate for each loading. The results derived from this numerical scheme are compared with equivalent results derived via analytical techniques which focus on the solution of the governing integral-equation schemes and other approximate-solution schemes. The numerical results presented in the paper illustrate the manner in which the generalized stiffnesses of the embedded inclusion are influenced by its geometry and Poisson's ratio of the half-space region.     

Extension of an elastic space with a rigid bar

An asymptotic model for deformation of an elastic space with a rigid thin reinforcing bar is constructed. The elastic modulus of the fiber far exceeds the elastic modulus of the matrix. The shape optimization problem for the reinforcing bar is solved on the basis of the uniform strength condition. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 120–128, January–February, 2008.     

Interaction between a rigid line inclusion and an elastic circular inclusion

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

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.     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Thermal stresses in a two-dimensional infinite medium containing a rigid inclusion embedded in a line crack

This paper examines the problem of finding thermal stresses, caused by a symmetric indentation of a line crack by an inclusion in an infinite isotropic elastic heat conducting solid. The thermal and elastic problems are reduced to a system of triple integral equations. In each case the solution of the triple integral equations is obtained in a closed form. The expressions for the stress intensity factor at the edge of the line crack, the strain energy density function and the resultant pressure applied to the inclusion are obtained. The expression for the displacement component is also obtained. Finally the results for the physical quantities are displayed graphically.     

Elastic-plastic analysis of a flat plate with a circular rigid inclusion

Summary An elastic-plastic solution is presented for a circular rigid inclusion in a unidirectionally stressed flat plate of linearly strain hardening material. The results are compared to the results obtained for the free hole. It is postulated that for intermediate constraints the plastic flow would be less pronounced than in these limiting cases.     

Anti-plane pull-out of a rigid line inclusion from an elastic medium

The transient and static anti-plane problem of a rigid line inclusion pulled out from an elastic medium is studied. The singular integral equation method is used to solve the stress field. Under the static load, the stress intensity factor(SIF) at the inclusion tips increases with the medium length. The problem becomes equivalent to an inclusion in a medium with an infinite length when the length of the medium is 3.5times longer than that of the inclusion. However, under the transient load, the ...     

Anisotropic thermopiezoelectric solids with an elliptic inclusion or a hole under uniform heat flow

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat flow direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

The torsional oscillations of a rigid spherical inclusion embedded in an elastic stratum

In this paper we investigate the axisymmetric harmonic torsional vibrations of a rigid spherical inclusion, embedded centrally in an elastic stratum of finite thickness. For an infinite solid the problem has a simple closed form solution. Here the stratum problem is formulated as a Fredholm integral equation of the first kind, and a perturbation method employed to solve this equation approximately for large values of the ratio of sphere radius to stratum depth. Approximations are given for quantities of physical interest such as the stress distribution at the inclusion-elastic medium interface, and the stress couple on the sphere.

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Résumé Dans cette étude on recherche les vibrations axisymmétriques harmoniques et torsionales d'une inclusion sphérique et raide, située centralement dans une couche d'une épaisseur bornée. Pour un solide infini le problème a une solution élémentaire.On formule le problème de couche comme une équation intégrale de Fredholm de la première sorte. On se sert d'une méthode de perturbation pour résoudre cette équation d'une façon approximative pour les grandes valeurs de la proportion du rayon de la sphère contre la profondeur de la couche. Les approximations sont calculées pour la distribution d'effort employée sur l'inclusion et pour la couple de torsion employée sur la sphère.

     

Effect of interface stresses on the elastic deformation of an elastic half-plane containing an elastic inclusion

The effect of the interface stresses is studied upon the size-dependent elastic deformation of an elastic half-plane having a cylindrical inclusion with distinct elastic properties. The elastic half-plane is subjected to either a uniaxial loading at infinity or a uniform non-shear eigenstrain in the inclusion. The straight edge of the half-plane is either traction-free, or rigid-slip, or motionless, which represents three practical situations of mechanical structures. Using two-dimensional Papkovich–Neuber potentials and the theory of surface/interface elasticity, the elastic field in the elastic half-plane is obtained. Comparable with classical result, the new formulation renders the significant effect of the interface stresses on the stress distribution in the half-plane when the radius of the inclusion is reduced to the nanometer scale. Numerical results show that the intensity of the influence depends on the surface/interface moduli, the stiffness ratio of the inclusion to the surrounding material, the boundary condition on the edge of the half-plane and the proximity of the inclusion to the edge.     

Thermal stresses in an elastic sphere containing conical cuts

An axisymmetric thermoelastic problem for a sphere with conical cuts in the poles under the action of a quasi-stationary temperature field depending on the meridional angle and on the sphere radius is solved by a method based on the Castigliano variational principle. Orthonormalized systems of polynomials are used as the coordinate functions. Results of numerical calculations of the stress state of a spherical solid are presented.     

Interaction between a rigid inclusion and a line crack under uniform heat flux

The thermoelastic displacement boundary value problem for a rigid inclusion interacting with a line crack in an infinite plane subjected to a uniform heat flux is studied, in which the rigid body rotation of the inclusion is considered. To solve the prescribed problem, we use the principle of superposition to decompose it into two groups of problems, which are further reduced to several basic subproblems including Green’s functions of edge dislocation and heat source couple, as well as the problem of a plane containing the inclusion under uniform heat flux and the problem of the inclusion subjected to a small rotation. The problems are solved using the complex variable method along with the rational mapping function technique. The variations of the stress intensity factors at the crack tips and the rigid body rotation angles with various crack lengths and heat flux angles are shown. The effects of the inclusion shape and size are also investigated.     

Problem of a crack on the boundary of a rigid inclusion in an elastic plate

The problem of equilibrium of a thin elastic plate containing a rigid inclusion is considered. On part of the interface between the elastic plate and the rigid inclusion, there is a vertical crack. It is assumed that, on both crack edges, the boundary conditions are given as inequalities describing the mutual impenetrability of the edges. The solvability of the problem is proven and the character of satisfaction of the boundary conditions is described. It is also shown that the problem is the limit problem for a family of other problems posed for a wider region and describing equilibrium of elastic plates with a vertical crack as the rigidity parameter tends to infinity.