Elastic–plastic near field solution of an eccentric crack under shear in a finite width plate

The near crack line analysis method is used to investigate an eccentric crack loaded by shear forces in a finite width plate, and the analytical solution is obtained in this paper. The solution includes: the unit normal vector of the elastic–plastic boundary near the crack line, the elastic–plastic stress fields near crack line, variations of the length of the plastic zone along the crack line with an external loads, and the bearing capacity of a finite plate with a centric crack loaded by shear stress in the far field. The results obtained in this paper are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions have been taken. Subsequently, the present results are compared with the traditional line elastic fracture mechanical solutions and elastoplastic near field solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

Flux diffusion upon the compression of a magnetic field by flat strips of variable width

An equation is obtained describing the flux diffusion in flat profiled generators having magnetic cumulation (MC). The critical modes of operation of such generators on active and inductive loads are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 159–162, November–December, 1974.     

Equi-strength design of nonhomogeneous variable thickness high speed rotating disk under steady temperature field

Considering the mutual interaction between the thickness ofthe disk and temperature distribution, and directly using results in [1].[2] and the iteration method stage sied in this paper, we obtain the equi-strength thickness distribution of nonhomogeneous variable thickness high speed rotating disk under axisymmetrieal steady temperature field.Projects Supported by the Science and Technic Fund of the National Education Committee     

Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer

The assumptions of impermeable and permeable cracks give rise to significant errors in determining electro-elastic behavior of a cracked piezoelectric material. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Considering the presence of the dielectric of an opening crack interior and the permeability of the crack surfaces for electric field, this paper analyzes electro-elastic behavior induced by a penny-shaped dielectric crack in a piezoelectric ceramic layer. In the cases of prescribed displacement or prescribed stress at the layer surfaces, the Hankel transform technique is employed to reduce the problem to Fredholm integral equations with a parameter dependent nonlinearly on the unknown functions. For an infinite piezoelectric space, a closed-form solution can be derived explicitly, while for a piezoelectric layer, an iterative technique is suggested to solve the resulting nonlinear equations. Field intensity factors are obtained in terms of the solution of the equations. Numerical results of the crack opening displacement intensity factors are presented for a cracked PZT-5H layer and the effect of applied electric field on crack growth are examined for both cases. The results indicate that the fracture toughness of a piezoelectric ceramic is affected by the direction of applied electric fields, dependent on the elastic boundary conditions.     

Long-term failure of a cylindrically anisotropic viscoelastic material with a crack under a variable load

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 12, pp. 61–68, December, 1994.     

Anti-plane shear crack growth rate of piezoelectric ceramic body with finite width

The theory of linear piezoelectricity is applied to develop an anti-plane crack growth rate equation of a finite crack in a piezoelectric ceramic body with finite width. Plastic zone is assumed to be confined to a sheet ahead of both crack edges similar to the strip model for in-plane loading. The procedure consists of reducing a system of dual integral equations to a Fredholm integral equation of the second kind. The accumulated plastic displacement criterion is used for developing a solution for the crack growth rate. Numerical values of crack growth rate are obtained and the results are displayed graphically to exhibit the electroelastic interactions.     

Growth of a main crack under the influence of gas moving inside it

The motion of a gas or liquid in a growing main crack is examined in connection with the problem of the hydraulic fracture of an oil-bearing bed [1, 2] and evaluation of the quantity of gaseous products escaping from the cavity formed by the underground explosion into the atmosphere by way of the crack [3]. The studies [1, 2] formulated and solved a problem on the quasisteady propagation of an axisymmetric crack in rock under the influence of an incompressible fluid pumped into the crack. An exact solution was obtained in [4] to the problem of the hydraulic fracture of an oil-bearing bed with a constant pressure along the crack. The Biot consolidation theory was used as the basis in [5] for an examination of the growth of a disk-shaped crack associated with hydraulic fracture of a porous bed saturated with fluid. A numerical solution to a similarity problem on the motion of a compressible gas ina plane crack was obtained in [6]. Here we examine the problem of the propagation of a main crack (plane and axisymmetric) under the influence of a gasmoving away from an underground cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 116–122, July–August, 1986.We thank V. M. Entova for his remarks, which helped to improve the investigation.     

Transient crack growth under creep conditions

Transient crack growth in an elastic/power-law creeping material is investigated under antiplane shear loading and small-scale-yielding conditions. At time t = 0 the solid is suddenly loaded far from the crack by tractions that correspond to the elastic crack-tip stress distribution. At that time the crack begins to propagate at a constant velocity. The stress fields evolve in a complex manner as the crack propagates due to the competing effects of stress relaxation due to constrained creep and stress elevation due to the instantaneous elastic material response to crack growth. From detailed finite element calculations it is shown that these fields can be approximated by a simple matching of three asymptotic singular crack-tip solutions. A characteristic stress, distance and time are defined for this problem which provide a normalization that accounts for any crack velocity, loading and all material properties for a given creep exponent n. Results are presented for crack-tip stresses, strains, crack opening displacements and creep zones.     

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.     

Dynamic crack growth under Rayleigh wave

A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed.     

Steady growth of a crack with a rate and temperature sensitive cohesive zone

The steady-state dynamic propagation of a crack in a heat conducting elastic body is numerically simulated. Specifically, a mode III semi-infinite crack with a nonlinear temperature dependent cohesive zone is assumed to be moving in an unbounded homogeneous linear thermoelastic continuum. The numerical results are obtained via a semi-analytical technique based on complex variables and integral transforms. The relation between the thermo-mechanical properties of the failure zone and the resulting crack growth regime are investigated. The results show that temperature dependent solutions are substantially different from purely mechanical ones in that their existence and stability strongly depends on the cohesive zone thermal properties.     

Unsteady free convection flow under the influence of a magnetic field

Summary The unsteady free convection boundary layer at the stagnation point of a two-dimensional body and an axisymmetric body with prescribed surface heat flux or temperature has been studied. The magnetic field is applied parallel to the surface and the effect of induced magnetic field has been considered. It is found that for certain powerlaw distribution of surface heat flux or temperature and magnetic field with time, the governing boundary layer equations admit a self-similar solution locally. The resulting nonlinear ordinary differential equations have been solved using a finite element method and a shooting method with Newton's corrections for missing initial conditions. The results show that the skin friction and heat transfer coefficients, andx-component of the induced magnetic field on the surface increase with the applied magnetic field. In general, the skin friction, heat transfer andx-component of the induced magnetic field for axisymmetric case are more than those of the two-dimensional case. Also they change more when the surface heat flux or temperature decreases with time than when it increases with time. The skin friction, heat transfer andx-component of the induced magnetic field are significantly affected by the magnetic Prandtl number and they increase as the magnetic Prandtl number decreases. The skin friction andx-component of the magnetic field increase with the dissipation parameter, but heat transfer decreases.

---

Instationäre freie Konvektionsströmung unter dem Einfluß eines magnetischen Feldes

Übersicht Untersucht wurde die instationäre freie Konvektionsgrenzschicht am Ruhepunkt eines zweidimensionalen und achsensymmetrischen umströmten Körpers bei vorgegebenem Wärmefluß bzw. bei vorgegebener Temperatur an der Oberfläche. Das magnetische Feld wird parallel zur Oberfläche angelegt, und der Einfluß des induzierten magnetischen Feldes wurde berücksichtigt. Es stellt sich heraus, daß bei bestimmter, zeitlicher Potenzgesetzverteilung des Wärmeflusses bzw. der Temperatur und des magnetischen Feldes an der Oberfläche die geltenden Grenzschichtgleichungen örtlich eine selbstähnliche Lösung erlauben. Die sich ergebenden nichtlinearen gewöhnlichen Differentialgleichungen wurden mittels einer Finite-Element-Methode und einer Shooting-Methode mit Newtonschen Korrekturen für fehlende Anfangsbedingungen gelöst. Die Ergebnisse zeigen, daß die Oberflächenreibung und die Wärmeübergangskoeffizienten sowie diex-Komponente des induzierten magnetischen Feldes an der Oberfläche mit dem angelegten magnetischen Feld zunehmen. Im allgemeinen sind die Oberflächenreibung, der Wärmeübergang und diex-Komponente des induzierten magnetischen Feldes im achsensymmetrischen Fall größer als die entsprechenden Werte im zweidimensionalen Fall. Außerdem verändern sich diese Werte beim zeitlichen Abfallen des Wärmeflusses an der Oberfläche bzw. der Temperatur in höherem Maße als bei der zeitlichen Zunahme dieser Werte. Die Oberflächenreibung, der Wärmeübergang und diex-Komponente des induzierten magnetischen Feldes werden durch die magnetische Prandtl-Zahl erheblich beeinflußt; sie nehmen mit abfallender magnetischer Prandtl-Zahl zu. Die Oberflächenreibung und diex-Komponente des magnetischen Feldes nehmen mit dem Wärmeableitungsparameter zu, der Wärmeübergang jedoch fällt ab.