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.     

Contact of the lips of an interphase crack in the field of concentrated mass forces

We consider a plane problem of the theory of elasticity for a crack located on the boundary of a fixed half plane subjected to the action of an arbitrary concentrated load applied to internal points of the domain. It is assumed that the crack lips contact without friction in the vicinity of the tip. By applying the Fourier integral transformation, we reduce the problem to a system of singular integral equations. In constructing the numerical solution of this system, we take into account the singular behavior of unknown functions near singular points. We establish the dependence of the size of the region of contact of the crack lips on the type of applied load. It is shown that, in this case, the combination of stress intensity factors obtained under the load applied at infinity remains quasiinvariant with respect to the length of the contact region. By using the elastic solution, we approximately determine the boundaries of the plastic domain near the crack tip and analyze their dependence on the intensity of the shearing field.Dnepropetrovsk State University, Dnepropetrovsk. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 2, pp. 35–42, March – April, 1995.     

Fractality of stable cracks in the zones of deformation of glass-like polyarylatesulfone

We present the results of investigation of stable cracks in films of amorphous polyarylatesulfone. We show that a stable crack can be interpreted as a self-similar stochastic fractal, the dimension of which has a clear physical sense, namely, it is inversely proportional to the stress concentration near the tip of a crack. Berbekov Kabardino-Balkar State University, Nalchik, Russia. Dnepropetrovsk State Agrarian University, Dnepropetrovsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 36, No. 1, pp. 51–55, January–February, 2000.     

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.     

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.     

Residual durability of solids contacting under conditions of fretting fatigue

Within the framework of the concepts of fracture mechanics, we propose a numerical-analytic model for the investigation of fracture processes and evaluation of the durability of contacting bodies under the conditions of fretting fatigue. A key element of this model is the step-by-step construction of the paths of propagation of fatigue cracks by using the criteria of local fracture in a complex stressed state, the equations of propagation of a fatigue crack, and the characteristics of cyclic crack resistance of materials. We construct the paths of propagation of edge cracks in the contact zone of one of the bodies subjected to fretting. We assume that the crack growth is controlled by a stress intensity factor of the mixed type K_Iθ, which is determined by the criterial relations of the O_θ- criterion, and that the path of a crack is formed by the maximum_Iθ values in the contact cycle. The dependence of the path shape on the amplitude of the friction factor, the scheme of interaction between the bodies in the contact cycle, the initial lengths of cracks, and their orientation are studied. We show that the obtained theoretical results correlate with experimental data and give an example of calculation of the residual durability of the near-surface layer of VT3-1 titanium alloy in the contact zone. Translated from Fizyko-Khimichna Mekhanika Materialiv,Vol. 36, No. 2, pp. 5–19, March-April, 2000.     

A set of interface cracks with contact zones in a combined tension-shear field

Summary. A set of cracks lying along the interface of two dissimilar isotropic materials under a mixed-mode loading is considered. The interface cracks are assumed to be fully open, partially closed with frictionless contact zones and fully closed. The problem is reduced to a homogeneous combined Dirichlet-Riemann boundary value problem, which is solved in closed form. A set of transcendental equations for the determination of the contact zone lengths for an arbitrary number of cracks and the closed-form expressions for the stresses and the displacement jumps on the material interface are obtained. A single crack with one and two contact zones has been considered in details. An explicit set of two transcendental equations for the relative contact zone length and closed-form expressions for the stress intensity factors at the crack tips are obtained for both cases. The contact zone lengths and the stress intensity factors are investigated numerically for different material pairs under different values of the loading, and a comparison of the results for a crack with one and two contact zones is carried out.     

Analysis of the field of a plane SH-wave scattered by a finite crack on the interface of materials

We solve the problem of diffraction of plane elastic SH-waves on a crack of finite width located in the plane interface of two different perfectly joined isotropic elastic materials. The problem is solved by the Wiener-Hopf method. An approximate solution of the Wiener-Hopf equation is obtained. The numerical results in the far-field zone are presented for a broad frequency band. The specific features of distribution of the radiation field in the zone of critical angles are analyzed. The probability of excitation of lateral waves is investigated and the possibility of detection of defects is discussed. A simple experimental procedure is proposed for the NDT evaluation of crack width in a test bench based on measuring the modulus of the field of displacements in the zone of critical angles. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 5–16, November–December, 2006.     

On the relationship between the stress intensity factors and the far field of SH-waves diffracted on interface cracks

By using the Wiener–Hopf method, we solve the problem of diffraction of plane SH-waves on semiinfinite interface cracks and deduce the relationships between the radiation field in fixed directions and the stress intensity factors. The dependence of the stress intensity factors on the field of displacements scattered in the plane perpendicular to the crack tip is analyzed. It is shown that the ratio of the stress intensity factors can be represented via the ratio of the scattered far fields measured for various angles and frequencies of sounding.     

Contact zone models for an interface crack in a piezoelectric material

Summary A plane strain problem for an interface crack along the fixed edge of a piezoelectric semi-infinite space is examined. Electrically conducting and electrically insulated crack surfaces are considered. By using Fourier transforms the systems of singular integral equations are formulated for both cases. It was found that in the second case for the most commonly used piezoelectric materials instead of oscillating singularity the real singularity of general power type occurs. The dependence of this singularity on the piezoelectric parameters has been investigated. The contact zone model is considered as an alternative one for the case of the oscillating singularity, and the way this model can be used for the investigation of interface cracks in finite size piezoelectrics is suggested.     

Asymmetrical Cracks Parallel to an Interface between Dissimilar Materials

This paper solves a plane strain problem for two bonded dissimilar planes containing a crack parallel to the interface in each layer. The bimaterial system is loaded by tractions distributed along the crack surfaces. Based on the Fourier transform, the problem is reduced to a system of Cauchy type singular integral equations which contain exact and explicit kernel functions. The solution of these equations is obtained easily by utilizing Gauss–Chebyshev integral formulae for various material combinations and geometrical parameters. Several numerical results of stress intensity factors, energy release rate and stress distribution along the interface are presented to exhibit the interaction among cracks and interface. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Plane Problem of a Crack on the Interface of Orthotropic Plates with Friction of Crack Lips

We consider a plane problem of a crack on the interface of different orthotropic arbitrarily oriented semiinfinite spaces subjected to the action of uniform tensile and shear forces applied at infinity. In the vicinity of the crack tip, the crack lips smoothly close up to form a contact zone of a priori unknown length in the presence of constant friction between the lips. A transcendental equation for the length of the contact zone is deduced and analyzed. As a result, we obtain the dependences of the relative length of the contact zone and the stress intensity factors on the orientation of the principal directions of orthotropy for different values of the external load and boundary friction.     

Pressure with friction of a perfectly rigid die upon an elastic half space with cracks

We study the interaction of a rigid die with a base of any shape and the surface of an elastic half space containing cracks in the presence of friction in the contact zone. The solution of the plane contact problem of the theory of elasticity is obtained by the method of singular integral equations. The detailed analysis of the problem is performed for the case where the base of the die is parabolic and a crack is rectilinear and appears on the surface of half space. We also investigate the effects of the friction coefficient, crack length, its orientation, and location on stress intensity factors KI and KII at the crack tip and the distribution of contact stresses under the die.     

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.     

Interaction of parallel cracks across the interface of materials

We reduce a three-dimensional problem of the theory of elasticity about the interaction of cracks in a body to the solution of a system of boundary integral equations for functions that characterize mutual displacements of the opposite surfaces of the cracks in the process of deformation of the body. For the case of parallel cracks, the regular kernels of these equations taking into account the interaction of cracks across the interface of materials are presented in the explicit form. The deduced boundary integral equations are solved by using a numerical-analytic approach. For two disk-shaped parallel cracks in different materials loaded by normal forces, we obtain the dependences of the stress intensity factors on the ratio of elastic constants of the components of the body for various distances from the crack to the interface of materials. State University “L'viv'ska Politekhnika,” L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 2, pp. 12–20, March–April, 1999.     

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.     

Diffraction of SH-waves on interface cracks under the action of concentrated forces

In the mathematically rigorous statement, we solve the problem of diffraction of elastic SH-waves on a finite (semiinfinite) crack located on the plane boundary of two perfectly joined materials. As a sounding field, we consider the field of a point source. The problem is reduced to the Wiener – Hopf functional equation, which is solved by the method of factorization. An approximate solution of this equation is obtained. We numerically study the distribution of the field in the radiation zone and the distinctive features of its formation depending on the location of the source of radiation. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 6, pp. 67–77, November–December, 2008.     

Crack resistance of homogeneous and inhomogeneous butt welded joints

We describe the methods used for the evaluation of strength of butt welded joints according to their crack resistance both by the axial tension of small-sized cylindrical specimens with circular cracks and three-point bending of beam specimens with lateral cracks induced in low-or medium-strength surfacings welded into the high-strength bases of these specimens. We perform the comprehensive investigation of the near-weld zone of butt welded joints, where the lowest crack resistance is exhibited by the metal of the zone of fusion and the highest crack resistance is detected for the source material independently of the type of tested specimens. Metals of the weld and the heat-affected zone are characterized by higher values of crack resistance than the zone of fusion and lower values of crack resistance than the source material. By analyzing, as an example, the case of axial tension of a tubular specimen with internal circular crack in the surfacing, we study the crack resistance of seven types of inhomogeneous welded joints depending on the mode of thermal treatment of 35 and St. 3 steels after normalization and 09G2S, 20Kh, and 30KhGSA steels after hardening, medium tempering, and normalization. __________ Translated from Problemy Prochnosti, No. 4, pp. 62–76, July–August, 2007.