Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

A penny-shaped crack in transversely isotropic magneto-electro-thermo-elastic medium subjected to uniform symmetric heat flux

The problem of a penny-shaped crack subjected to symmetric uniform heat flux in an infinite transversely isotropic magneto-electro-thermo-elastic medium is investigated. The exact solution in the full space is in terms of line integrals and the exact solution in the crack plane also is obtained. Although we start our derivations with magneto-electro-thermo-elastic, the solution presented in this paper is also applicable for linear transversely isotropic thermopiezoelectric, thermomagnetoelastic,thermoelastic materials (see Appendix E). The solution in the crack plane, which shows a great agreement with the solution for a transversely isotropic medium obtained by Tsai (1983), indicates that _σx,_σy,_Dx,_Dy,_Bx${\sigma }_x\text{,}{\sigma }_y\text{,}{D}_x\text{,}{D}_y\text{,}{B}_x$, and _By$B_y$ along the crack rim are of the same singularity of the normal stress or its equivalent quantities. To illustrate how the applied symmetric heat fluxes affect the whole fields, a numerical example is also given.     

Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading

Summary Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems are investigated by means of a so-called deformation map. The deformation map was further used for stability considerations of geometrically nonlinear shells, see Shilkrut [1, 2]. The map reveals the complete picture of the axially symmetric deformations and the stability of the investigated structure. The equilibrium differential equations for the above mentioned circular plate were derived by Timoshenko [3]. The boundary value problem of the investigated structure is transformed to an initial value problem (Cauchy's problem). Then the Runge-Kutta (R. K.) method can be used to solve numerically the equilibrium equations. The geometrically nonlinear, simply supported circular plate subjected to uniform radial force and uniform radial bending moment acting along the supported edge is investigated as example, and some new qualitative and quantitative results are obtained. This approach can be used without essential difficulties for the investigation of axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems in elastic and non-elastic fields.

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Axialsymmetrische Verformung und Stabilität geometrisch nichtlinearer Kreisplatten unter mehrparametrischer statischer Belastung

Übersicht Zur Untersuchung axialsymmetrischer, geometrisch nichtlinearer Verformung von Kreisplatten und ihrer Stabilität bei mehrparametrischer Belastung wird eine sog. Deformationskarte benutzt. Sie wurde auch für Stabilitätsbetrachtungen geometrisch nichtlinearer Schalen benutzt, s. Shilkrut [1,2]. Die Karte zeigt das vollständige Bild der axialsymmetrischen Verformung und die Stabilität der untersuchten Struktur. Das Randwertproblem zu den differentiellen Gleichgewichtsbedingungen, die für die betrachtete Platte von Timoshenko [3] hergeleitet wurden, wird in ein Anfangswertproblem (Caudy-Problem) überführt, welches numerisch nach der Methode von Runge-Kutta gelöst wird. Als Beispiel wird die nichtlineare Kreisplatte unter radialer Zug-und Biegemomentenbelastung am einfach gestützten Umfang untersucht, und man erhält einige neue qualitative und quantitative Ergebnisse. Die Methode läßt sich ohne wesentliche Schwierigkeiten auch auf axialsymmetrische, nichtlineare Verformungen und die Stabilität von Kreisplatten unter anderen mehrparametrischen statischen Belastungen im elastischen und nichtelastischen Bereich anwenden.

     

Axially symmetric long waves on the surface of a varying-depth basin

The axially symmetric Korteweg-de Vries (KdV) equation for the case of a constant-depth basin was obtained in [1]. In the present paper we derive the axially symmetric KdV equation for a varying-depth basin. Conditions are shown for the equation obtained, under which the asymptotic behavior of its solution is described by an equation of the form $$u_t + uu_x + u_{xxx} = 0,$$ whose asymptotic behavior is well known [2].     

The stress state of an elastic orthotropic medium with a penny-shaped crack

The static-equilibrium problem for an elastic orthotropic space with a circular (penny-shaped) crack is solved. The stress state of an elastic medium is represented as a superposition of the principal and perturbed states. To solve the problem, Willis approach is used, which is based on the triple Fourier transform in spatial variables, the Fourier-transformed Greens function for an anisotropic material, and Cauchys residue theorem. The contour integrals obtained are evaluated using Gauss quadrature formulas. The results for particular cases are compared with those obtained by other authors. The influence of anisotropy on the stress intensity factors is studied.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 76–83, December 2004.     

Transonic expansion of a penny-shaped crack

The problem of a penny-shaped crack propagating in an unbounded isotropic linear elastic body is solved. The crack expands from a zero radius in a self-similar manner and is assumed to have speed larger than the S-wave speed and less than the P-wave speed. A tensile load is directed normal to the crack at infinity. The method of self-similar potentials and rotational superposition are applied. Attention is given to the stress singularity at the crack border.     

Small edge crack in a semi-infinite solid subjected to thermal stratification

The mode I stress intensity factor for a small edge crack in an elastic half-space is found when the space is in contact with two stratified fluids of different temperatures, the boundary between the fluids oscillating sinusoidally over the solid surface. The variation in the stress intensity factor, which may lead to thermal fatigue crack growth, is examined as a function of time, crack depth, amplitude and temporal frequency of oscillation, surface heat transfer coefficient and material properties of the half-space. It is shown how this ‘boundary layer’ solution may be applied to problems involving finite geometries.     

On stress analysis for a penny-shaped crack interacting with inclusions and voids

An analytical approach to calculate the stress of an arbitrary located penny-shaped crack interacting with inclusions and voids is presented. First, the interaction between a penny-shaped crack and two spherical inclusions is analyzed by considering the three-dimensional problem of an infinite solid, composed of an elastic matrix, a penny-shaped crack and two spherical inclusions, under tension. Based on Eshelby’s equivalent inclusion method, superposition theory of elasticity and an approximation according to the Saint–Venant principle, the interaction between the crack and the inclusions is systematically analyzed. The stress intensity factor for the crack is evaluated to investigate the effect of the existence of inclusions and the crack–inclusions interaction on the crack propagation. To validate the current framework, the present predictions are compared with a noninteracting solution, an interacting solution for one spherical inclusion, and other theoretical approximations. Finally, the proposed analytical approach is extended to study the interaction of a crack with two voids and the interaction of a crack with an inclusion and a void.     

Penny-shaped crack in a long circular cylinder subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in a long circular cylinder of elastic material containing a penny-shaped crack when it is deformed by the application of a uniform shearing stress. The crack with its center on the axis of the cylinder lies on the plane perpendicular to that axis, and the cylindrical surface is stress-free. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of Fredholm integral equations of second kind. These are solved numerically, and the percentage increase in the stress intensity factor due to the effect of the finite radius of the cylinder is presented in graphical form for various proximity ratios.     

Rapid extension of a penny-shaped crack under torsion

The rapid extension of a penny-shaped crack under torsion is investigated. Both dynamic and quasi-static loading is considered. The wave motion is analyzed through a Green's function technique which leads to an integral equation for the stress field around the crack. Asymptotic expansions for the stress intensity and displacement rate intensity functions which are valid for a small time are obtained for the two types of loading. The propagation of the crack is analyzed through the balance of rates of energy criterion.     

Mode II stress intensity factors for a crack parallel to the surface of an elastic half-space subjected to a moving point load

The direction of propagation of rolling contact fatigue cracks is observed to depend upon the direction of motion of the load. In this paper approximate calculations are described of the variation of Mode II stress intensity factors at each tip of a subsurface crack, which lies parallel to the surface of an elastic half-space, due to a load moving over the surface. In particular the effect of frictional locking of the crack faces under the load is investigated. In consequence of frictional locking the range of SIF at the trailing tip ΔK_T is found to be about 30% greater than that of the leading tip ΔK_L, which is consistent with observations that subsurface cracks propagate predominantly in the direction of motion of the load over the surface. The effects on k_t and k_lof crack length, crack face friction, traction forces at the surface and residual shear stresses are also investigated.     

Thermostressed state of a piezoelectric bodywith a plane crack under symmetric thermal load

The paper addresses a thermoelectroelastic problem for a piezoelectric body with an arbitrarily shaped plane crack in a plane perpendicular to the polarization axis under a symmetric thermal load. A relationship between the intensity factors for stress (SIF) and electric displacement (EDIF) in an infinite piezoceramic body with a crack under a thermal load and the SIF for a purely elastic body with a crack of the same shape under a mechanical load is established. This makes it possible to find the SIF and EDIF for an electroelastic material from the elastic solution without the need to solve specific problems of thermoelasticity. The SIF and EDIF for a piezoceramic body with an elliptic crack and linear distribution of temperature over the crack surface are found as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 96–108, March 2008.     

A long circular cylinder with a circumferential edge crack subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in a long circular cylinder of isotropic elastic material with a circumferential edge crack when it is deformed by the application of a uniform shearing stress. The crack with its center on the axis of the cylinder lies on the plane perpendicular to that axis, and the cylindrical surface is stress-free. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of singular integral equations. This pair of singular integral equations is solved numerically, and the stress intensity factor due to the effect of the crack size is tabulated.     

Axially symmetric problems for a porous elastic solid

This paper is concerned with the linear theory of elastic materials with voids. The Dirichlet and Neumann problems for a half-space are studied by using the technique of integral transforms. The case of a concentrated body load is investigated in detail.