Contact Interaction of Crack Lips in Shallow Shells in Bending with Tension

In the two-dimensional statement, we consider a problem of bending and tension of a shallow cracked shell. The effect of crack closure caused by bending is described within the framework of the classical theory of shells according to the model of contact of crack lips along a line in the face surface. We study the influence of contact interaction of crack lips and surface curvature on the stressed state and limiting equilibrium of cylindrical and spherical shells. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 4, pp. 45–52, July–August, 2005.     

Stress intensity in the vicinity of a surface flaw in the linear part of a pipeline

Real sharp-edted surface and subsurface flaws detected in a gas pipeline body are modeled by surface semi-elliptical mathematical cracks (cuts) in a closed cylindrical shell. A relationship is proposed that relates the geometrical dimensions of the flaws to the crack aspect ratio. Based on the line spring model, the problem of stress state and boundary equilibrium conditions of a closed cylindrical shell with a surface semi-elliptical crack is reduced to a system of singular integral equations. An algorithm was developed for computational solution of the problem, and numerical analysis was made for the dependence of stress intensity factors on loading conditions and geometrical parameters of shell and crack. For a shell subjected to internal pressure and weakened by a surface longitudinal semi-elliptical crack, a closed approximation formula is proposed that interrelates pressure level, shell/crack dimensions, and material mechanical properties in boundary equilibrium conditions. The maximal error value is indicated for the results obtained using this formula. Lvov Polytechnic State University, Lvov, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 38–47, July–August, 1999.     

Redistribution of residual stresses in a cylindrical piecewise homogeneous shell with crack

We propose a nondestructive theoretical-experimental method for the evaluation of residual welding stresses in piecewise-homogenous cylindrical elements of shell structures based on the solution of inverse problems with the use of the available experimental data. For a structure weakened by a longitudinal crack, the problem of redistribution of stresses near the crack is reduced to a system of singular integral equations whose solution is constructed by the method of mechanical quadratures. As an example, we solve the problem for a piecewise homogeneous cylindrical shell welded from two semiinfinite shells. The dependences of the force and moment intensity factors on the distribution of residual stresses are investigated. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 5, pp. 39–45, September–October, 1999.     

Bilateral bending of a plate with nonsymmetric through-thickness arc crack with allowance for the contact of its edges

The problem of bending of plates with nonsymmetric through-thickness arc crack with distributed moments at infinity with allowance for the contact of the crack edges has been examined on the basis of the classical theory of plate bending. The conditions for the existence of the solution of the problem in this formulation have been established. A numerical analysis of intensity factors of forces and moments, contact force between the crack edges has been performed, the results of which have been graphically illustrated. __________ Translated from Problemy Prochnosti, No. 5, pp. 135–141, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

Limiting equilibrium of a spherical shell nonuniform across the thickness and containing a surface crack

The problem of limiting equilibrium of an elastoplastic spherical shell nonuniform across the thickness and containing a surface crack is reduced to a system of singular integral equations. The algorithm of numerical solution of this system takes into account the conditions of plasticity of thin shells and the conditions of boundedness of stresses in the vicinity of the crack. The influence of internal pressure and the mechanical parameters of functionally inhomogeneous material on the opening displacement of the surface crack and the length of plastic strips on its continuation is investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 43, No. 3, pp. 5–11, May–June, 2007.     

Bending of a protective ring on a cylindrical shell by concentrated forces

An analytical solution of the problem on contact interaction between a cylindrical shell and a ring loaded by a system of radial forces is obtained with the help of the integral Fourier transformation and the method of stressed state synthesis. Simple relations are proposed for the calculation of contact reactions, radial displacements, and internal force factors. The numerical results obtained are in good agreement with analogous data of other authors. Translated from Problemy Prochnosti, No. 2, pp. 89–97, March–April, 1997.     

Contact model of cracks in orthotropic materials

We consider a plane strain problem for an orthotropic half plane loaded at infinity and containing a crack along its fixed edge. To remove a singularity near the right crack tip, we introduce an artificial contact zone. The problem under consideration is reduced to the mixed Dirichlet-Riemann boundaryvalue problem. We present the exact solution of this problem and deduce formulas for stresses in the contact zone and on the continuation of the crack and for the stress intensity factors. By using both analytic and numerical methods, we prove that the energy-release rate is quasiinvariant in the process of crack propagation relative to the size of the contact zone. On the basis of these results, we propose an algorithm for the evaluation of the paramenters of fracture of composites of finite dimensions with interface cracks. As a special case, we develop a model of interface cracks with actual contact zone and establish the dependences of the length of this zone on the external load and elasticity moduli of the material. Dnepropetrovsk State University, Dnepropetrovsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 5, pp. 59–66, September–October, 1999     

Wedging of an edge crack in the elastic wedge

We study the problem of wedging of an elastic wedge realized along the edge crack with the help of a hard wedge. The analytic solution of the problem is obtained by the Wiener–Hopf method. We determine the stress intensity factors, the distributions of stresses on the continuation of the crack and in the contact zone, and the circular displacements of the crack lips.     

Stress intensity near a crack in the composition of a half space and a layer under harmonic loading

We consider a three-dimensional problem for an elastic bimaterial body formed by a half space and a layer and containing a circular crack. The surface of the crack is subjected to the action of stationary torsional forces. The problem is reduced to the solution of a system of boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. The dynamic stress concentration in the vicinity of the crack contour is investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 27–32, March–April, 2008.     

Application of an analog of the δc-model to the analysis of strength of elastoplastic cracked shellsto the analysis of strength of elastoplastic cracked shells

We propose a method for the investigation of the limiting equilibrium of elastoplastic shells with systems of interacting cracks. It can be described as follows: By using an analog of the δ_c-model, the elastoplastic problem is reduced to an elastic problem of the limiting equilibrium of a shell with cracks of unknown length whose lips are subjected to the action of unknown forces and moments satisfying the conditions of plasticity for thin shells. By using equations of the general moment theory of shells and the theory of generalized functions, we reduce the problem to the solution of a system of singular integral equations with unknown limits of integration and singular right-hand sides. We construct an algorithm for the numerical solution of systems of this sort supplemented by the conditions of boundedness of stresses and conditions of plasticity. We investigate crack tip opening displacements in a closed cylindrical shell with a regular system of longitudinal cracks or two transverse cracks. For a cylindrical shell with a single crack, we present an approximate relation for the determination of the critical load or crack length. Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 3, pp. 5–15, May–June, 1996.     

Bending of a plate containing a periodic system of collinear cracks with regard for the contact of crack lips

We study the process of bending of a plate weakened by a periodic system of collinear cracks with regard for the contact of crack lips. The stress-strain state of the plate is found by the method of solution of the problems of linear conjugation from the theory of functions of complex variable. It is shown that if the crack lips are in contact over a part of the thickness of the plate, then the intensity factors of forces decrease and the intensity factors of moments increase as compared with the case of contact of the crack lips along the line of intersection with the surface. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 46–52, March–April, 2008.     

Cracks with bonding between the lips in bushings of friction couples

We study the problem of contact deformation of a bushing of friction couple. The friction surface of the bushing contains rectilinear cracks with end zones characterized by the presence of interaction between the crack lips. In the case of repeated reciprocal motion of a plunger, the material of the bushing finally fails as a result of contact interaction. The boundary-value problem of equilibrium of the bushing of friction couple with cracks whose lips are interacting is reduced to the solution of a system of nonlinear singular integro-differential equations with Cauchy-type kernel. The solution of this system is used to find the normal and tangential forces in the bonds. The condition of limiting equilibrium of a crack with end zone is formulated with regard for the criterion of ultimate length of the bonds in the material. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 2, pp. 53–61, March–April, 2006.     

Periodic system of interface cracks with contact zones in the isotropic bimaterial in the fields of tension and shear

The methods of the theory of functions of complex variable are used to construct the solution of the plane problem of the theory of elasticity in the closed form for an infinite isotropic bimaterial plane (space) with a periodic system of interface cracks in the presence of the zones of smooth contact of crack lips of near the crack tips. As a result of the numerical analysis of the obtained solution, it becomes possible to study the interaction of the cracks and establish, in particular, the regularities of variation of the length of the contact zones and the stress intensity factors as functions of the distance between the cracks. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 4, pp. 101–107, July–August, 2006.     

Interface Edge Crack in a Bimaterial Elastic Half-Plane

The plane strain elastic half-plane problem of an edge crack lying along the interface of two perfectly bonded dissimilar quarter-planes is considered. Moreover, on the boundaries of the two quarter-planes concentrated forces are acting. For the correct formulation of the crack problem at hand, we consider the existence of a small slippage zone near the crack tip where closing stresses act. The mixed boundary value problem is subsequently reduced to a system of two functional equations of the Wiener–Hopf type which are effectively solved. The exact analytical solution of the problem is presented in series form. Numerical results, as well as asymptotic solutions for the most important physical quantities, are also presented. It is shown that there exist certain modes of surface loading of the homogeneous half-space, that result to the formation of two distinct zones at the crack tip region, one where the crack opening occurs and another adjacent to it, where frictionless contact of crack lips takes place. Also, it is demonstrated that in the case of high contrast of Young's moduli of the two quarter-planes, two opening-contact intervals appear consecutively along the crack. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamic stresses in the vicinity of a crack along the cylindrical surface

By using the method of boundary integral equations combined with the method of small parameter, we solve the problem of low-frequency loading of a shallow crack along a cylindrical surface in an infinite elastic body. The distributions of the modes I–III stress intensity factors along the contour of the defect are obtained for different values of crack curvature and wave numbers. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 4, pp. 108–113, July–August, 2006.     

The mechanics and mechanisms of failure of hens’ eggs

The mechanics and mechanisms of failure of hens’ eggs have been examined experimentally under contact loading conditions relevant to industrial conditions by testing eggs of known provenance in compression between stiff platens. Deformation was modelled computationally as a Hertzian contact problem between a thin walled elastic shell and a rigid plate. Contact damage was determined by scanning electron microscopy and by optical examination of transverse sections through the shell. Small stable micro-cracks were found to initiate in the contact area before structural failure, which was characterised by the propagation of one or more macroscopic cracks. Structural failure does not correspond to crack initiation, but crack propagation. Stable micro-cracks were not detectable by routine visual inspection or acoustic resonance, and combined with inconsistent cuticle coverage these could provide a pathway for pathogens to enter the egg contents and thus compromise egg safety.     

Effect of the contact of crack lips under variable loads

We formulate a contact problem of the theory of elasticity with unilateral constraints for cracked bodies under random variable loading and consider a special case of harmonic loads very important for practical applications. The problem is solved by the method of boundary integral equations of the elastodynamic theory of cracked bodies. We also analyze typical singularities of the kernels of potentials of these equations and develop an algorithm for the solution of unilateral contact problems of the dynamics of cracked bodies based on the determination of the saddle point of a subdifferential boundary functional. We solve dynamical problems of propagation of plane harmonic compression-dilatation waves in a plane with cracks of finite lengths under the assumption of unilateral contact of the crack lips. We present numerical results and investigate the effect of contact interaction of the crack lips both quantitatively and qualitatively. Timoshenko Insitutes of Mechanics, Ukrainian Academy of Sciences, Kiev. Kharkov State Technical University of Motor Transport and Roads, Kharkov. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 1, pp. 38–52, January–February, 1996.     

Thermal opening of an initially closed interface crack under conditions of imperfect thermal contact between its lips

We study the thermomechanical behavior of a bimaterial with closed interface crack under the action of compressive forces normal to its surface and a uniform heat flow. The components of the bimaterial have the same mechanical and different thermal properties. The crack lips are in frictionless mechanical and imperfect thermal contact. It is shown that the contact thermal resistance of the crack may be responsible for its opening. The critical intensity of the heat flow at which the crack lips begin to lose their contact depends on the relationship between the thermal parameters of the materials, applied compressive forces, and the thermal resistance of the crack. We pose a problem of thermal opening of a thermally insulated crack and reduce it to the solution, of a singular integral equation for the height of the gap formed between the crack lips. The length of the gap is determined from the condition of its smooth closure. We study the dependences of the height and length of the gap and the stress intensity factors on the force and thermal loads both for the partial and complete opening of the crack. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 5, pp. 14–22, September–October, 1999.     

Stresses Formed Near a Crack in Half Space in Contact with Liquid Under Harmonic Loading

We study a three-dimensional bimaterial object formed by an elastic half space with cracks and a liquid. The crack surfaces are subjected to the action of harmonic loads. The problem is reduced to the solution of a system of two-dimensional boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. We analyze the case of a circular crack whose lips are subjected to the action of stationary loads. The inertial effects in the vicinity of the crack contour are investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 3, pp. 96–100, May–June, 2005.     

Crack closure effect on stress intensity factors of an axially and a circumferentially cracked cylindrical shell

Crack-face closure occurs when a shell or plate containing a through-the-thickness crack is subjected to a bending load, which leads to lower stress-intensity factors than those expected from non-closure assumption. This article presents a theoretical analysis of the effect of crack-face closure on the stress intensity factors of an axially and a circumferentially cracked cylindrical shell subjected to bending moment respectively. The presented analysis extends the shallow shell theories of Delale and Erdogan by incorporating the effect of crack-face closure. In keeping consistent with the shear deformation shell theory, crack-face closure is modeled by a line contact at the compressive edges of the crack face. The unknown contact force is computed by solving a mixed-boundary value problem iteratively, i.e. along the crack length, either the normal displacement of the crack face at the compressive edges is equal to zero or the contact pressure is equal to zero. The results show that the distribution of the contact force along the crack is generally nonuniform. Furthermore, it is found that, similar to the case of spherical shells, crack closure may occur over the full length or only some segments of the crack in cylindrical shells, depending on the geometry of the shell and the nature (direction) of applied bending load. Comparisons of the stress intensity factor ratios between the closure solutions and the non-closure solutions reveal that the crack-face closure influences significantly the magnitude of the stress intensity factors and it tends to reduce the maximum stress intensity factor. The closure effect of crack face on the stress intensity factors is highest when the shell radius becomes very large for a given crack length and shell thickness.