Analysis of the derivative of the energy functional with respect to the length of a curvilinear crack in an elastic body with a possible crack-edge contact

A homogeneous two-dimensional body with a crack of variable length is considered. At the crack edges, conditions are formulated in the form of inequalities that describe mutual nonpenetration of the edges. The derivative of the elastic-energy functional with respect to the length of the curvilinear crack is analyzed. It is shown that the derivative is independent of the crack path, provided that the curve along which the crack propagates is reasonably smooth. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 138–145, September–October, 2007.     

Use of the energy criterion of fracture to determine the shape of a slightly curved crack

An asymptotic formula is obtained for the total-energy increment during quasistatic growth of a semiin finite crack in an anisotropic elastic plane under complex loading. It is assumed that the shear loads are much larger than the tearing loads. The shape of the slightly curved crack was determined using the Griffith criterion in two versions: global and local. It is shown, in particular, that the first version leads to an improbable result. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 119–130, September–October, 2006.     

Crack stabilization in a brittle body using stiffeners

The behavior of a crack under uniaxial tension in the presence of reinforcement is studied. The reinforcing members (riveted stiffeners) are modeled by point loads. Only four members nearest to the crack are taken into account. It is shown that stiffeners allow one to arrest a crack and prevent its catastrophic growth. Relations between the geometrical and force characteristics for which the crack is stabilized are obtained. The stabilization mechanism is discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 137–143, November–December, 2006.     

Limiting velocity of crack propagation in elastic materials

A crack is represented as a continuous set of linear dislocations. Simple analytical expressions are obtained for the potential and kinetic energies of the environment of moving cracks and the attached mass of cracks for an arbitrary form of the stress applied to the crack P(x). It is shown that the indicated analytical expressions are bilinear integrals of the functions P(x) and ∂P(x)/∂x. These integrals are calculated in explicit form for a constant stress over the entire crack length and the stress due to the action of molecular adhesion forces in a narrow region near the crack openings. It is shown that the calculation results does not depend on the form of molecular adhesion forces. The proposed approach to describing cracks and calculations based on it has made it possible for the first time to obtain a complete analytical expression for the limiting crack propagation velocity in elastic materials as a function of the main mechanical characteristics of such materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 158–166, July–August, 2009.     

Stress state in the plastic zone on the continuation of an elliptic mode I crack under bi-and tri-axial loading

The spatial stress state in a circular plastic zone near an elliptic crack under bi-and tri-axial asymmetric loading at infinity is studied. It is shown that the plastic constraint factor peaks at the points with maximum external tensile tangential stresses on the crack boundary. The use of a two-parameter failure criterion leads to the conclusion that the limit state can first be reached at the ends of the major axis of the elliptic crack depending on the relation between the external stresses __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 24–30, November 2007.     

Asymptotics of stresses and strain rates near the tip of a transverse shear crack in a material whose behavior is described by a fractional-linear law

An approximate solution of the problem of determining the fields of stresses and strain rates due to creep near the tip of a transverse shear crack in a material whose behavior is described by a fractional-linear law of the theory of steady-state creep is given. It is shown that the strain rates have a singularity of the type ∼ r^−α near the crack tip; the order of singularity α changes discretely, depending on the polar angle, and takes the values 1, 2/3, and 1/2. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 165–176, January–February, 2009.     

Eigenvalues of the antiplane-shear crack problem for a power-law material

This paper discusses the problem of finding the eigenvalue spectrum in determining the stress and strain fields at the tip of an antiplane-shear crack in a power-law material. It is shown that the perturbation method provides an analytical dependence of the eigenvalue on the material nonlinearity parameter and the eigenvalue of the linear problem. Thus, it is possible to find the entire spectrum of eigenvalues and not only the eigenvalue of the Hutchinson-Rice-Rosengren problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 173–180, January–February, 2008.     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

Optimization formulation of the evolutionary problem of crack propagation under quasibrittle fracture

For the evolutionary problem describing crack propagation in a solid with allowance for the irreversible work of plastic deformation due to the crack propagation, a general optimization formulation is proposed and investigated. For the optimum crack, data on the H²-smoothnesses of the displacement field in the solid and, hence, on the finiteness of the stress at the crack tip, are obtained. The solvability of the optimization problem (i.e., the existence of an optimum crack) is proved for a curvilinear crack propagation path specified a priori. For the particular case of a straight path, a generalized criterion of crack growth is proposed. The question of the choice of a crack propagation path is discussed and a comparison with existing fracture criteria is made. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 107–118, September–October, 2006.     

Subcritical crack growth in an aging plate with variable elastic modulus

The paper addresses subcritical growth of a crack in a thin isotropic plate made of an aging viscoelastic material with time-dependent elastic modulus. The behavior of the material is described by Arutyunyan’s creep theory. To simulate fracture, a modified Leonov–Panasyuk–Dugdale model and a critical crack opening displacement criterion are used. An equation describing the subcritical growth of the crack is derived assuming that Poisson’s ratio is constant. As an example, the critical loads are determined, and curves of subcritical crack growth are plotted for a specific material. The results are compared with the case of constant elastic modulus     

Effect of contact interaction of the crack faces for a crack under harmonic loading

This paper is concerned with dynamic problems in fracture mechanics for elastic solids having cracks with contacting faces. The contact problem for a penny-shaped crack with a nonzero initial opening under normally incident harmonic wave is solved by the method of boundary integral equations. The solutions are compared with those that neglect the contact interaction of the crack faces. Results are presented for different values of the initial crack opening Presented at the 6th International Conference on Modern Practice in Stress and Vibration Analysis (Bath, United Kingdom, September 5–7, 2006). Published in Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 125–131, July 2007.     

Effect of scattering of elastic harmonic waves in the far zone by a spatial crack

A three-dimensional wave field formed owing to diffraction of low-frequency waves on a curved crack in an infinite elastic solid at a large distance from the defect is studied by the method of boundary integral equations. Direction diagrams of the scattered field versus the excentricity of the crack surface and wavenumber are obtained for different directions of incidence of planar longitudinal waves onto a gently sloping spheroidal crack. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 115–123, July–August, 2006.     

One model of fatigue crack growth

A model for crack growth is proposed based on studies of the variation in the curvature radius at the crack tip during cyclic loading. Relations are obtained between mechanical material characteristics, crack geometry, and the rate of crack growth in a structure under cyclic loading. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 167–175, July–August, 2009.     

Passages to the limit in the dynamic problem for a crack at the interface between elastic media

The paper analyzes numerically the passages to the limit in the dynamic problem for a penny-shaped crack at the interface between dissimilar linear elastic, homogeneous, isotropic materials as either the frequency of harmonic load or the difference between the properties of the materials decreases. It is shown that as the frequency decreases, the solution of the dynamic problem tends to that of the static problem, and as the physical and mechanical properties of the materials become less different, the original problem goes into the dynamic problem for a crack in a homogeneous body __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 26–34, July 2008.     

Integral equation for stress concentration at the edge of a plane crack of arbitrary contour

Equations are derived for stress concentration near a crack of closed contour lying in a plane. A system of one-dimensional integral equations for the concentration factor is obtained. The right sides of the equations contain the initial approximation—a solution of the problem of a circular crack whose sides are acted upon by nonaxisymmetric loading. Mining Institute, Siberian Division, Russian Academy of Sciences, Novosibirsk 630091. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 143–148, September–October, 1999.     

The use of the “trident” model in the analysis of plastic zones near crack tips and corner points

The paper deals with calculation of a plastic zone near a crack tip in a homogeneous elastoplastic solid and near a corner point of the boundary of this solid. The calculations are conducted for a solid subject to plane strain and within the framework of models with plastic strips. It is shown that in comparison with the widely used model with two straight slip-lines, the process of plastic deformation is described by the “trident” model more accurately. The results of calculations of the plastic zone by the “trident” model that correspond to different stages of the development of plastic deformation are given for a crack of normal separation in a quasibrittle material. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 95–100, March, 2000.     

Effect of the parameters of the biaxial field of rock pressure on the shape of the fracture zone formed by an explosion of a cylindrical charge in a brittle medium

The effect of the biaxial field of external rock pressure on the deformation of the fracture zone formed by radial cracks in an elastic-brittle medium is studied. We consider cylindrical charges that are rather thin compared to the diameter of the explosive borehole. This allows one to exclude the grinding zone from consideration. At the initial moment of time, the system of emerging cracks originating at the boundary of the circular hole is assumed to be symmetric. To solve the problem, we use singular integral equations and the fracture criterion σ_ϑ. The propagation trajectories of the system cracks are calculated in the quasistatic approximation in a step-by-step manner in relation to the parameters of the external compressive stress field. Two ideal variants of loading of the crack system are analyzed. In the first variant, gaseous detonation products penetrate into cracks, and the pressure in the explosion cavity and the cracks instantaneously equalizes. In the second variant, gases do not penetrate into the cracks of the system. The fracture zone is shown to become an ellipse whose long axis is oriented in the direction of the largest compressive stress in magnitude acting at infinity. The effect of the variants of loading of the crack system on the shape and dimensions of the deformed fracture zone is evaluated. Mining Institute, Siberian Division, Russian Academy of Sciences, Novosibirsk 630091. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 129–137, January–February, 1998.     

A phenomenological model of fatigue crack growth in perfectly plastic infinite plates under completely reversed uniaxial loading

The fatigue failure of a thin infinite center-cracked plate under completely reversed uniaxial loading is considered. A two-stage fatigue crack model including the incubation and crack propagation stages is constructed. The stress distribution in the vicinity of the crack tip is described using the concept of a conventional elastic crack. The crack-tip plastic zone is simulated by a Dugdale thin plastic zone, and the condition for the movement of the failure front is given by criteria of damage mechanics. It is shown that the fatigue crack growth rate in perfectly plastic materials with a plastic zone of constant length is a power-law function of the stress intensity factor range. This relationship is quadratic when the length of the plastic zone is not constant Published in Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 116–127, December 2005.     

Control of the Strain Field Near a Crack in Elastic Layer

In the present paper, we consider the strain analysis near the crack parallel to boundary surfaces of a linear isotropic elastic layer of constant thickness. We pose the problem of minimizing the opening of crack faces when a normal force is applied to the boundary of the layer. The control secondary loading is a normal stress uniformly distributed over the boundary surfaces as to oppose the opening caused by the external applied force. We show that the most dangerous are tangential deformations, and the normal ones are always by many times less dangerous. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 265–277, April–June, 2005.     

Unsteady problem of crack propagation in the beam approximation

Unsteady crack propagation is studied in the beam approximation. The Euler and Timoshenko models of beam theory are used. Crack propagation is described using an energy balance equation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 177–189, September–October, 2008.