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.     

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.     

The penny-shaped crack in three-dimensional electrostriction

Zusammenfassung Zur Untersuchung der Spannungs- und Verschiebungsfelder in einem unendlichen elastischen Dielektrikum verwenden wir die dreidimensionale lineare ungekoppelte Theorie der Elektrostriktion.Von besonderem Interesse ist das Wesen der Spannungs-Singularität am Rande eines flachen kreisförmigen Spaltes innerhalb der begrenzenden Geometrie. Ein zweifacher Sachverhalt wurde festgestellt: 1. diese Spannungs-Singularität kann von höherer Ordnung sein als diejenige, welche mit den klassischen Umgebungen mechanischer und thermischer Art verknüpft ist;2. diese Singularität neigt nicht dazu, den Spalt in bezug auf eine spannungslose Spaltoberfläche auszubreiten. Die obigen Ergebnisse stimmen mit unseren früheren zweidimensionalen Analysen überein.

---

---

This work was supported by the United States Atomic Energy Commission.     

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.

---

---

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

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.

---

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.

     

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.     

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.     

Elastostatic bounds for the stiffness of an elliptical disc inclusion embedded at a transversely isotropic bi-material interface

The present paper examines the problem related to the axial translation of a rigid elliptical disc inclusion which is embedded in bonded contact at the interface of a transversely isotropic bi-material elastic medium. The result for the stiffness of the elliptical disc inclusion is presented in the form of a set of bounds. These bounds are developed, in exact closed form, by imposing displacement and traction constraints at the bi-material interface.

---

Resumé On examine le problème d'une inclusion rigide de form elliptique soumise à une translation axiale. Cette inclusion est située à l'interface d'un milieu élastique bi-matériel à isotropie transversale et elle est en contact lié avec se dernier. La rigidité de l'inclusion elliptique est exprimée par un ensemble de limites et la forme exacte de ces limites à été obtenue en imposant des contraintes de déplacement et traction à l'interface du bi-matériau.

     

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.

---

---

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

---

Research supported by the United States Atomic Energy Commission.     

Nonlinear control of a variable displacement axial piston pump

In this contribution a mathematical model of a variable displacement axial piston pump controlled by a solenoid valve is derived. For the purpose of a controller design the mathematical model is simplified using singular perturbation arguments. The goal of the controller design is to track prescribed trajectories in the load pressure for arbitrary unknown load conditions. The control concept being proposed comprises a feedforward controller and a nonlinear backstepping controller combined with a load estimator for the trajectory error system. The feasibility of this control concept is shown by means of measurements on a test-stand. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the problem of a thin rigid inclusion embedded in a Maxwell material

We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established.     

In-plane loading of a cracked elastic solid by a disc inclusion with a Mindlin-type constraint

The paper examines the problem related to the axisymmetric interaction between an external circular crack and a centrally placed penny-shaped rigid inclusion located in the plane of the crack. The interface between the inclusion and the elastic medium exhibits a Mindlin-type imperfect bi-lateral contact. Analytical results presented in the paper illustrate the manner in which the lateral translational stiffness of the inclusion and the stress intensity factor at the boundary of the external circular crack are influenced by the inclusion/crack radii ratio.     

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.     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

Out-of-plane displacement measurement near the crack tip during a crack propagation: Validation of a 3D formulation for a specimen in PMMA loaded in mode I.

The fundamental aim of this study is the determination zone of the 3D effects and the transient one at the vicinity of the crack tip during a crack propagation in brittle materials ( PMMA ) using an optical method (Michelson interferometer). With the obtained interferograms, we can extract the phase (thus the relief) by using a new numerical approach based on the principle of images correlation between real fringes and virtual fringes. Different dynamic tests are realized by a plate loaded in mode I under a constant loading. We compare the obtained data with the two-dimensional theory of Westergaard (plane stress hypothesis) [1]. With the divergence is established, we propose a new 3D formulation, based on a formulation employed for static crack, which takes into account 3D and transient effects. For the static cracks, the 3D effects relate to a presence of the state of three-dimensional stresses. However in dynamics, the transient effects appear and are related to the crack propagation velocity. The 3D effects and transient effects lead to results equivalent to experimental ones in terms of displacement but are completely different to results given by the two-dimensional theory near the crack tip. It is possible to quantify the zone when the plane stress hypothesis is not valid according to the crack propagation speed V. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Helmholtz equation in a non-smooth inclusion

In this paper, we study the Helmholtz equation in a non-smooth inclusion, i.e., in a doubly connected bounded domain B$B$ in R²$R^2$ with boundary ∂B$\partial B$ that consists of two disjoint closed curves Γ$\Gamma$ and _Γ0${\Gamma }_0$. The existence and uniqueness of a solution to the Helmholtz equation for mixed boundary conditions on Γ$\Gamma$ are obtained by using Riesz–Fredholm theory.     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

The inclusion of storage in a queueing system

Service operations involving an intermediate store are modelled using an extended definition of the system state which is a conventional definition of state supplemented with an indicator of the live capacity in the store. Additional to the parameters of mean service rate and mean arrival rate, a mean discharge rate of the store is introduced and the steady state probabilities are calculated in terms of these three quantities. Both finite source and infinite source cases are considered under the assumption of exponential distributions. The models permit production comparisons for systems with and for systems without storage devices.