A note on the stress singularity at a penny-shaped crack in three-dimensional electrostriction

Zusammenfassung Diese Mitteilung behandelt eine vollständige Bewertung der Spannungs-Singularität am Rande eines flachen kreisförmigen Spaltes unter der Einwirkung von Elektrostriktion. Die Ergebnisse bestätigen die Schlussfolgerungen, welche früher bei zweidimensionalen Problemen gezogen wurden.Die der Elektrostriktion zuzuschreibende Singularität zeitigt eine rein radiale Spannung, welche nicht dazu neigt, die schon bestehenden Spalten ihrer Ausdehnungs-Oberfläche entlang auszubreiten.

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Research supported by the National Science Foundation under grant NSF GP 22567.

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Research supported by the United States Atomic Energy Commission.     

The axial displacement of a disc inclusion embedded in a penny-shaped crack

The present paper examines the axisymmetric problem of the axial translation of a rigid circular disc inclusion of finite thickness which is wedged in smooth contact in a penny-shaped crack. Results for the axial stiffness of the embedded inclusion and the stress intensity factor at the boundary of the penny shaped crack are evaluated in exact closed form.

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Zusammenfassung Die vorliegende Arbeit untersucht das axisymmetrische Problem der axialen Verschiebung einer starren eingebetteten Kreisscheibe mit bestimmter Dicke, die in einer reibungslosen pfennigförmigen Spalte eingekeilt ist. Ergebnisse für die axiale Steifigkeit der eingebetteten Scheibe und der Druckintensitätsfaktor am Rande der pfennigförmigen Spalte sind als eine genaue geschlossene Lösung angegeben.

     

An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.     

Thermal stresses in an elastic sphere containing a penny-shaped crack

Zusammenfassung Es wird der Spannungszustand in einer elastischen Kugel untersucht, die im Zentrum einen münzenförmigen Riss mit konstanter Oberflächentemperatur besitzt, während die sphärische Oberfläche auf der Temperatur null gehalten wird. Dabei wird angenommen, dass an der Oberfläche a) die Schubspannung und die radiale Verschiebung, b) die Normalspannung und die Schubspannung null sind.

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This work was supported by a C.S.I.R. grant No. 3(277)/69-GAU-II.     

The interaction between a penny-shaped crack and a spherical inclusion under torsion

The axisymmetric interaction problem of an elastic spherical inclusion with a penny-shaped crack in an elastic space under torsion is considered. The superposition and reflection methods [3]-[4] are used to solve the mixed boundary value problem in question. With the help of the dual integral equations technique and appropriate re-expansion of the eigenfunction, the problem is reduced to an infinite system of linear algebraic equations of the second kind. The matrix elements of that system decrease exponentially along the rows and the columns. Its unique solution is proved to exist in a proper class of sequences and is shown to be represented by a convergent, in the vicinity of the origin, power series in a geometric parameter, equal to the ratio of the radius of the inclusion to its distance from the crack. This procedure provides an efficient formula for the stress intensity factor.     

Torsional waves in an infinite elastic solid containing a penny-shaped crack

Zusammenfassung In dieser Arbeit wird das axialsymmetrische Problem der Torsionswellen in einem unendlichen, isotropen und homogenen elastischen Körper mit einem kreisförmigen Schlitz betrachtet. Es werden zwei Fredholmsche Integralgleichungen zweiter Art für das gestreute Feld aufgestellt, welche sich bei kleinen Frequenzen zur iterativen Behandlung eignen. Die erste davon ist insofern vorteilhafter, als sie die Verschiebungs- und die Schubspannungsverteilung auf der Schlitzoberfläche direkt liefert und bei iterativer Behandlung viel weniger Arbeit kostet als die zweite. Es werden Ausdrücke für das Verschiebungs- und das Spannungsfeld an der Schlitzoberfläche in Form von Potenzreihen des Frequenzfaktors erhalten. Ebenso wird ein Ausdruck für den Spannungsintensitätsfaktor als Potenzreihe von gewonnen.     

Elastodynamic stress field and bifurcation of a running penny-shaped crack

The elastodynamic stress field around a penny-shaped crack running at a constant speed in an infinite solid is obtained as the sum of the associated static solutions and the wave-effect terms. The results include the stress field in the plane of the running crack and the associated static solutions as special solutions.The crack-tip circumferential stress, , is studied in detail as a function of the crack speed,V, the shear wave speed,c ₂, Poisson's ratio, , and the angle, , between the crack plane and a crack-tip radial line. The theory predicts that the running crack may start to bifurcate at the critical speed of 0.665c ₂ in the direction =41° for =0.25. The bifurcation angle predicted is compared to an experimental observation.

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Zusammenfassung Das elastodynamische Spannungsgebiet um einen pfennigförmigen Bruchriss, der mit konstanter Geschwindigkeit in einem unendlichen Körper läuft, wird als Summe der verbundenen statischen Lösung und der Wellenwirkungsterme aufgefasst. Die Ergebnisse schliessen das Spannungsgebiet im Bereich des laufenden Bruchrisses und die zugehörigen statischen Lösungen als Sonderlösungen ein.Die Ringspannung der Risspitze, wird als Funktion der RissgeschwindigkietV, der Geschwindigkeit der Scherungswellec ₂, der Poissonzahl und des Winkels zwischen der Bruchebene und einer Radiallinie von der Risspitze aus ausführlich studiert. Die Theorie sagt aus, dass der laufende Riss bei der kritischen Geschwindigkeit von 0.665c ₂ in der Richtung =41° für =0.25 zu gabeln anfangen kann. Der vorausgesagte Gabelungswinkel wird mit Versuchserfahrungen verglichen.

     

A contact problem for a smooth rigid disc inclusion in a penny-shaped crack

The present paper examines the problem of the complete indentation of the surface of a penny-shaped crack by a smooth rigid disc inclusion. The integral equation governing the problem is solved numerically to evaluate the axial stiffness of the rigid inclusion and the stress intensity factors at the tip of the penny-shaped crack.     

Hypersingularities in three-dimensional crack configurations in composite laminates

Three-dimensional crack configurations in composite laminates are studied by means of the Scaled Boundary Finite Element Method (SBFEM) particularly regarding stress singularities and their associated deformation modes. The SBFEM is an efficient semi-analytical method that permits solving linear elastic mechanical problems. Only the boundary needs to be discretized while the problem is considered analytically in the direction of the dimensionless radial coordinate pointing from the scaling center to the boundary . An important advantage is that it requires no additional effort for the characterization of existing stress singularities. The situation of two meeting inter-fiber cracks is investigated in detail, considering different materials and fiber / crack orientations. It is shown that in three-dimensional crack configurations in composite laminates so-called hypersingularities can occur, i.e. stress singularities which are even stronger than the classical crack singularity. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theoretical analysis of three-dimensional interface crack

Using the fundamental solution of interface crack and the method of finite-part integral, the problem of three-dimensional interface crack is reduced to solve a set of two-dimensional hypersingular integrodifferential equations with unknown displacement discontinuities of crack surface. Then a systematically theoretical analysis for solving these equations is presented.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.Dedicated to Prof. Dr.-Ing. W. L. Wendland on the occasion of his 50th birthday.A part of this work was done while the first author was a guest at the Georgia Institute of Technology and while the second author was partially supported by the NSF grant DMS-8501797.     

Substructuring FE–XFE approaches applied to three-dimensional crack propagation

Two substructuring methods are investigated in order to allow for the use of the eXtended Finite Element Method (X-FEM) within commercial finite element (FE) codes without need for modifying their kernel. The global FE problem is decomposed into two subdomains, the safe domain and the cracked domain based on the value of the level sets representing the crack. The safe domain is treated by the host FE software while the cracked domain is treated by an independent XFE code. The first substructuring method consists of calculating the Schur matrix of a cracked super-element with the XFE code. The second technique introduces the finite element tearing and interconnecting method (FETI) which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains. The stiffness matrices and nodal forces are provided by the XFE and FE codes for the cracked and safe subdomains, respectively. The solutions obtained with these two techniques are rigorously equivalent to those computed with the stand-alone XFE code. First, the computational efficiency of the two approaches is demonstrated. Second, a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3D cracked geometries. Finally, this contribution presents an application of the FE–XFE–FETI method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure.     

A transversely isotropic medium containing a penny-shaped crack subjected to a non-uniform axisymmetric loading via an anchored smooth rigid disk

A smooth rigid circular anchor disk encapsulated by a penny-shaped crack is embedded in and unbounded transversely isotropic medium. The lamellar rigid disk exerts a nonuniform axisymmetric loading to the upper face of the crack. With the aid of an appropriate stress function and Hankel transform, the governing equations are converted to a set of triple integral equations which in turn are reduced to a Fredholm integral equation of the second kind. For some transversely isotropic materials the normalized stiffness of the system falls well outside of the envelope pertinent to isotropic media. It is shown that mode I stress intensity factor is independent of the material properties and solely depends on the ratio of the radius of the rigid disk to that of the crack; moreover, for the cases where this ratio is less than about 0.9 a simple explicit approximate expression for the mode I stress intensity factor is derived. In contrast, the normalized mode II stress intensity factor is independent of the mentioned geometrical parameters but depends on the elastic properties of the material; depending on the material properties, the normalized mode II stress intensity factor can vary between 0 to ∞ for transversely isotropic materials and between 0 to π/4 for isotropic materials.     

A reciprocal theorem for a first order theory of electrostriction

Résumé On présente un théorème de réciprocité pour l'électrostriction de premier ordre. Ce théorème est employé pour déduire une représentation intégrale pour le déplacement et aussi pour la valeur moyenne des tensions et des déformations.     

The non-linear crack theory

Results obtained in recent years in an attempt to construct a general (physically and geometrically) non-linear crack theory are described.     

The symmetrically branched crack in an infinite elastic medium

Summary The plane elastostatic problem of a symmetrically branched crack in an infinite isotropic body loaded by normal stresses perpendicular to the main crack axis at infinity was studied by using the method of complex potentials. The problem was reduced to a system of three singular integral equations. By means of an approximation of the integrals through the Gauss and Lobatto numerical quadrature procedures, these singular integral equations were transformed into a system of linear equations, which can be readily solved. The stress intensity factors at the tips of the branched crack were computed directly from the solution of the above system of linear equations and were compared with the already existing experimental solutions.

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Zusammenfassung Es wird das ebene elastostatische Problem eines symmetrischen gegabelten Risses für den unendlichen, isotropen und mit senkrecht zur Riss-Hauptachse belasteten Körpers untersucht, und zwar unter Anwendung der Methode der komplexen Potentiale.Das Problem wird auf ein Systen von drei singulären Integralgleichungen reduziert und weiter auf ein System linearer Gleichungen transformiert, vermittelst einer Näherung der Integrale mit Hilfe des leicht lösbaren numerischen Quadraturverfahrens von Gauss und Lobatto. Die Spannungsintensitätsfaktoren in den Spitzen des gegabelten Risses werden rechnerisch ermittelt und mit experimentellen Ergebnissen verglichen.

     

The pressure distribution around a growing crack

Exact solutions of the problem of the pressure distribution around an ideal hydraulic fracture are derived. The crack propagates in a permeable porous medium following a square-root growth law. The case of the penetration of the fracturing fluid into a reservoir is also considered.