Progressive thermal stress distribution around a crack under Joule heating in orthotropic materials

Stress intensity factor and stress distribution at crack tips are classical problems in solids, which are closely related to the failure and reliability of materials. A crack in a nonlinearly coupled anisotropic medium, on the other hand, is much more difficult to analyze. Using the generalized complex variable method, the thermal stress problem of a crack embedded in an orthotropic medium has been analyzed, and the progressive thermal stress distributions have been obtained in closed-forms. The analysis shows that the thermal stress intensity factors are linear functions of remote thermal flux while are nonlinear functions of remote current; the thermal stress distributions under produced by thermal flux and Joule heating are similar, but not identical; the thermal stress intensity factors are linear functions with respect to the thermal expansion coefficients; with the increase of crack length, the thermal stress intensity factor caused by Joule heat increases rapidly; the thermal stress intensity factors are directly proportional to the temperature difference between the upper and lower crack surfaces and the left and right half crack surfaces divided by the square root of the crack length, and the ratios are only determined by the material parameters. These results provide a powerful tool for the failure and reliability analysis of conductive materials, and suggested that thermal stress analysis may be localized.     

Bifurcations of a plane crack front growing quasistatically in an elastic space

Using variational-asymptotic models of force and energy criteria, situations are found in which bifurcations of the form of the front accompanying the quasistatic propagation of a plane crack in an elastic isotropic space are possible. Two types of bifurcations are revealed for a circular crack in the case of axisymmetric loading: fluctuation of the centre of the crack while preserving its circular form and distortion of the front due to the formation of two or a larger number of “lobes”.     

The pressure distribution in extreme Stokes waves

In this paper we prove that the pressure beneath an extreme Stokes wave over finite depth is strictly increasing with depth. Additionally it is shown that the pressure decreases in moving between a crest-line and trough-line, while it is stationary with respect to the horizontal coordinate along these lines themselves.     

Identification of the material symmetries with anisotropic damage evolution due to a growing mixed-mode crack

A numerical study of a growing mixed-mode internal crack in a unit cell was undertaken with the help of a finite element simulation. The model enables us to measure the components of the elastic compliance tensor modified by damage as the crack grows, showing the evolution of the anisotropic damage and the evolution of the type of material symmetries. The evolution of the elasticity tensor shows that the damage associated with a growing elliptical crack changes the virgin isotropic properties into orthotropic ones and by crack growth the axes of orthotropic symmetry, initially aligned with the local coordinates of the crack, rotate towards the principle loading axes. Crack propagation is simulated using the stepwise method, which consists of the succession of straight segments and crack growth is governed by the principle of maximum driving force which is a direct consequence of the variational principle of a cracked body in equilibrium and considers the effect of all three stress intensity factors. Without any ad hoc assumption, the crack growth rate is calculated using its thermodynamic duality with the local maximum driving force. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stressed state of a transversally isotropic body with inner elliptic crack. A linearly varying pressure on the surface of the crack

An exact solution to the problem of the distribution of stresses in a transversally isotropic elastic body with inner elliptic crack is given. An assumption that the linearly varying pressure acts on the surface of the crack is made. The intensity coefficients of stresses at the vertex of the crack are found. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 117–124. Original     

Capillary-gravity waves produced by a moving pressure distribution

Summary The initial-value problem of surface waves generated by a moving oscillatory pressure distribution is considered and the effect of surface tension on such waves is studied in detail. It is found that the surface tension modifies the critical case in a remarkable way. And also it introduces two extra waves that exist in the upstream side of the pressure distribution.

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Zusammenfassung Das Anfangswertproblem der Oberflächenwellen die durch eine bewegte oszillierende Druckverteilung erzeugt werden, wird behandelt und der Einfluss der Oberflächenspannung wird ausführlich untersucht. Es wird gefunden, dass die Oberflächenspannung den kritischen Fall in bemerkenswerter Weise beeinflusst, und auch zwei besondere Wellen stromaufwärts von der Druckstörung verursacht.

     

The pressure distribution between two elastic bodies in contact

Zusammenfassung Die vorliegende Untersuchung behandelt die Verteilung des Kontaktdruckes von (a) zwei zweidimensionalen, symmetrischen, orthotropen Körpern und (b) zwei transversal-isotropen Rotationskörpern. Man sieht, dass die zwei Probleme mathematisch identisch sind mit den entsprechenden Problemen der undeformierbaren Stempel und dass einfache Lösungen möglich sind. Mehrere Beispiele sind ausgearbeitet.     

The effect of hydrostatic pressure on the crack resistance of particulate reinforced composite materials

The effect of hydrostatic pressure on the fracture toughness of disperse-reinforced composite materials is investigated. It is shown that increased hydrostatic pressure leads to an increase in the critical value of the stress intensity factor and, as a consequence, to an increase in the crack propagation rate. In this case, the pressure-time analogy method can be used to describe the effect of hydrostatic pressure on the characteristics of the crack resistance. This method enables us to represent the experimental data obtained in the form of a generalized dependence of the fracture toughness on the reduced loading rate. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 121–126, January–February, 2000.     

均质油气藏试井分析解的相似结构

针对均质油气藏,建立了考虑井筒储集、表皮效应和三种外边界(无穷大、定压、封闭)条件下的试井分析模型;利用Lap lace变换,在Lap lace空间中得到了储层压力和井底压力分布的精确解;经全面和深入的分析,发现了描述该类油气藏在三种外边界条件下的渗流特征的解式之间具有的相似结构,并作了进一步的讨论.这项研究,极大地方便了试井分析软件的编制,对油气藏渗流规律的理论研究也具有深远的意义.     

Velocity and pressure distribution in the gap of a disk extruder

The axisymmetric two-dimensional flow of a polymer melt in the plane gap of a disk extruder produced by the normal stress effect is considered. The polymer is assumed to be a nonlinear viscoelastic medium, whose strain history is expressed by means of kinematic matrices. A rheological equation of state of the medium, in which all the invariants of the kinematic matrices are function of strain rate intensity, is established. The laws of distribution of the radial and tangential velocity components over the gap are found from the solution of the equations of motion, and expressions are obtained for the radial pressure distribution and the integral thrust.Volgograd Polytechnic Institute. Translated from Mekhanika Polimerov, No. 3, pp. 515–521, May–June, 1971.     

Stress distribution in a transversely isotropic body with a thermally insulating parabolic crack subjected to a uniform heat flux

An explicit static thermoelastic solution is constructed for an infinite transversely isotropic body containing a thermally insulating parabolic crack in the plane of isotropy. The surface of the crack is free of stress. A uniform thermal flux is incident on the crack perpendicular to its surface. Formulas are obtained for the stress intensity factors near the tip of the crack. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Catholic University, Portugal. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 54–66, 1999.     

The Electromagnetic Field around a Subsurface Flaw in a Metal

This work is a theoretical investigation into the detectionand measurement in metals of flaws that do not break the surface,with the use of AC electric fields. The flaw is modelled bya semi-infinite slot, which is perpendicular to the surfaceand whose top edge is at a finite distance below the surface.The electromagnetic field is governed by Maxwell's equationsand the skin depth of the field is assumed to be large comparedto the ligament remaining between the top edge of the crackand the metal surface. Matched asymptotic expansions are usedto describe the field mathematically and numerical results arepresented, illustrating the variation of the potential differenceon the surface of a number of metals with different-size ligamentsin them. Good agreement is obtained from comparisons with someresults using a Fourier-series technique. It is found that themaximum centreline voltage signal is obtained if measurementsof the potential difference are made at a phase of approximately– compared to the upstream signal.     

The non-homogeneous flow of a thixotropic fluid around a sphere

The non-homogeneous flow of a thixotropic fluid around a settling sphere is simulated. A four-parameter Moore model is used for a generic thixotropic fluid and discontinuous Galerkin method is employed to solve the structure-kinetics equation coupled with the conservation equations of mass and momentum. Depending on the normalized falling velocity U*, which compares the time scale of structure formation and destruction, flow solutions are divided into three different regimes, which are attributed to an interplay of three competing factors: Brownian structure recovery, shear-induced structure breakdown, and the convection of microstructures. At small U*( ≪ 1), where the Brownian structure recovery is predominant, the thixotropic effect is negligible and flow solutions are not too dissimilar to that of a Newtonian fluid. As U* increases, a remarkable structural gradient is observed and the structure profile around the settling sphere is determined by the balance of all three competing factors. For large enough U*( ≫ 1), where the Brownian structure recovery becomes negligible, the balance between shear-induced structure breakdown and the convection plays a decisive role in determining flow profile. To quantify the interplay of three factors, the drag coefficient Cs of the sphere is investigated for ranges of U*. With this framework, the effect of the destruction parameter, the confinement ratio, and a possible nonlinearity in the model-form on the non-homogeneous flow of a thixotropy fluid have been addressed.     

The problem of scattering by a partially coated crack

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc in R²$R^2$ as the cross section. We assume that the arc is divided into two parts, and one of the two parts is (possibly) coated on one side by a material with surface impedance λ$\lambda$. Applying potential theory, the problem can be reformulated as a boundary integral system. We obtain the existence and uniqueness of a solution to the system by using Fredholm theory.     

The coupled thermoelastic problem for a plane with a rectilinear crack

We solve the thermoelastic problem for a plane with a rectilinear heat-conducting crack whose conductivity depends on its opening. By modeling the crack as a thin inclusion of variable thickness we reduce the problem to a system of singular integrodifferential equations for the potential densities of the temperature field. We study the behavior of the unknown functions at the ends of the contour of integration and, using a numerical-iteration method, we also determine the solution of the problem. We find an approximate asymptotic solution in the case of a weakly conducting crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 54–58.     

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 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 equilibrium problem for a Timoshenko-type shallow shell containing a through crack

Under study is the nonlinear equilibrium problem for an elastic Timoshenko-type shallow shell containing a through crack. Some boundary conditions in the form of inequalities are imposed on the curve defining the crack. We establish the unique solvability of the variational statement of the nonlinear problem of the equilibrium of a shell. We prove that, for sufficient smoothness of the solution, the initial variational statement is equivalent to the differential formulation of the problem. We deduce the boundary conditions on the inner boundary that describes the crack. In the case of the zero opening of the crack, we prove the local infinite differentiability of the solution function with additional assumptions on the functions defining the curvatures of the shell and the external loads.     

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.