Dynamics of a bridged crack in a discrete lattice

The paper addresses a problem of partial fracture of a latticeby a propagating fault modelling a crack bridged by elasticfibres. It is assumed that the strength of bonds within thelattice alternates periodically, so that during the dynamiccrack propagation only weaker bonds break, whereas the strongerbonds remain intact. The mathematical problem is reduced tothe functional equation of the Wiener–Hopf type, whichis solved analytically. The load–crack speed dependenceis presented, which also has implications on the stability analysisfor the bridged crack propagating within the lattice. In particular,we address the evaluation of the dissipation rate, which isfound to be strongly dependent on the crack speed. In this latticemodel, our results also cover the case of the supercriticalcrack speed.     

Extension of a piezoceramic bimorph with a crack crossing the interface

We consider the boundary-value problem of electroelasticity for a composite plate weakened by a crack crossing the line joining the media. The initial boundary-value problem, is reduced to a mixed system of singular integral and algebraic equations. We present the calculation results characterizing the variation in the stress intensity factors as a function of the opening angle of a segmented crack for different types of loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 482–488, July–August 1997.     

Invariant energy integrals for the non-linear crack problem with possible contact of the crack surfaces

Generally two-dimensional and three-dimensional formulations of the non-linear crack problem when the crack surfaces do not overlap for a non-uniform anisotropic linearly elastic body are considered. The first derivative of the potential energy function with respect to the perturbation parameter and its representation in the form of an invariant integral over an arbitrary closed contour are obtained for a general form of the differentiable perturbation of a region with a cut, using the method of material derivatives. The sufficient conditions for the existence of an invariant energy integral are derived in general form, and examples of invariant integrals are constructed for different types of perturbations and a different geometry of the cut.     

An interface crack moving between magnetoelectroelastic and functionally graded elastic layers

A constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading is investigated by the integral transform method. Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The singular stress, electric displacement and magnetic induction near the crack tip are obtained asymptotically and the corresponding field intensity factors are defined. Numerical results show that the stress intensity factors are influenced by the crack moving velocity, the material properties, the functionally graded parameter and the geometric size ratios. The propagation of the moving crack may bring about crack kinking, depending on the crack moving velocity and the material properties across the interface.     

Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Problem of thermoelectromagnetoelasticity for a piezoelectric/piezomagnetic bimaterial with interface crack

We consider a plane strain problem for a piezoelectric/piezomagnetic bimaterial space with a crack in the region of the interface of the materials. At infinity, tensile and shear stresses and heat, electric, and magnetic flows are set. Using representations for all mechanical, thermal, and electromagnetic factors in terms of piecewise analytic functions, we formulate problems of linear conjugation that correspond to a model of an open crack and models taking into account the contact zone in the vicinity of a crack tip. Exact analytic solutions of the indicated problems are constructed. Expressions for stresses, the electric and magnetic inductions, jumps of derivatives of displacements, and electric and magnetic potentials on the interface are written. The coefficients of intensities of the indicated factors are presented. We derive a transcendental equation for the determination of the real length of the contact zone. The dependences of this length and the coefficients of intensity on the set external influences are investigated.     

Investigation of the prefracture zone at the tip of a mode I crack reaching a nonsmooth interface of an elastic media

We perform calculation of the initial prefracture zone at the tip of a mode I crack that reaches a nonsmooth interface of two dissimilar elastic media at its corner point by the Wiener–Hopf method. The zone is modeled by a line of normal displacement fracture on a crack continuation. Expressions for the length of the prefracture zone and the potential energy accumulated in it are obtained. Their numerical values are compared with the corresponding values for the prefracture zone in a bonding material on the interface of the media, on the basis of which we draw a conclusion on the possible direction of development of the zone.     

Dynamic stresses in a compound body with circular crack under sliding contact on an interface

We have considered the three-dimensional problem of harmonic loading of a circular crack in an elastic composite consisting of two dissimilar half-spaces under sliding contact on the surface of their bonding. The defect is situated in one of the half-spaces perpendicularly to the interface of materials. Using the representations of solutions in the form of Helmholtz potentials, we have reduced the problem to a boundary integral equation for the function of dynamic defect opening. Based on the numerical solution of this equation, we have obtained the frequency dependences of mode I stress intensity factor near the crack for different relations between the elastic moduli of components of the composite and the depths of crack location with respect to the interface.     

A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.     

An elliptic non-linear equation on a Riemann surface

New existence and uniqueness results for a second order elliptic non-linear equation are obtained by using gauge theory methods on linear holomorphic bundles over an oriented Riemann surface.     

An evolution problem for a non-linear hyperbolic-parabolic system

This work concerns the initial-boundary value problem for a non-linear hyperbolic-parabolic system which governs the motion of a large class of mechanical systems.

     

An inverse problem for a linear diffusion equation with non-linear boundary condition

In this paper we consider the determination of an unknown radiation term in the non-linear boundary condition of a linear diffusion equation from an overspecified condition. It is shown that the solutions of this inverse problem is unique and stable.     

Scattering of a SH-wave by an elastic fiber of nonclassical cross section with an interface crack

The problem of interaction of a plane time-harmonic SH-wave with an elastic fiber of quasi-square or quasi-triangular cross section, when an interface crack is present between an infinite elastic matrix and the fiber, is considered. The modified null-field method taking into account the asymptotic behavior of the solution at crack tips is exploited for obtaining numerical results. The effects of fiber shape, fiber/matrix material combination, debonding (crack size), and direction of wave incidence on the scattering amplitude in the far zone are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 245–254, March–April, 2008.     

The problem of an interface crack with a rigid patch plate on part of its edge

A mixed boundary-value problem is solved for a piecewise-homogeneous elastic body with a rectilinear semi-infinite crack on the line where the materials are joined. A rigid patch plate (a reinforcing plate) of specified shape is attached to the upper edge of the crack on a finite interval adjacent to the crack tip. The edges of the crack are loaded with specified stresses. The body is stretched at infinity by a specified longitudinal stress. External forces with a given principal vector and moment act on the patch plate. The problem reduces to a Riemann-Hilbert boundary-value matrix problem with a piecewise-constant coefficient, the solution of which is explicitly constructed using a Gaussian hypergeometric function. The angle of rotation of the patch plate and the complex potentials describing the stress state of the body are found and the stress state of the body close to the ends of the patch plate, one of which is also simultaneously the crack tip, is investigated. Numerical examples are presented that illustrate the effect of the initial force parameters, the length of the patch plate and other parameters of the body on the angle of rotation of the patch plate and the stress state of the body.     

The interface crack with Coulomb friction between two bonded dissimilar elastic media

We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.