Curvilinear Crack Research Articles

Numerical solutions for polygonal cracks

An effective numerical scheme capable to deal with polygonal and branching cracks in a plane is proposed. It is suggested to decompose the general singular integral equation, SIE, for curvilinear cracks into a set of SIEs for straight cuts coinciding with straight crack segments. Solutions of SIEs are sought as bounded for all internal ends of the cuts and unbounded for the left end of the left cut and the right end of the right cut. The Gauss-Chebyshev quadrature is applied to each SIE that eventually leads to an over-determined system of linear algebraic equations followed by the application of the least squares method to solve this system. Stress intensity factors are calculated for some crack configurations. This scheme provides satisfactory accuracy although no correct asymptotic behaviour of the solution at internal ends is taken into consideration. The results are verified against the known results for branching cracks.

EFFECT OF A GRADED INTERFACE ON A CRACK APPROACHING AT AN OBLIQUE ANGLE

A model problem is considered in which a crack propagates under ductile conditions through an edge-notched plane-strain tension strip. The strip consists of two monolithic material regions joined transversely by a thin graded layer. The material regions are elastically identical but dissimilar in their plastic and fracture-toughness properties. A range of initial crack configurations is studied, including cracks at various angles to the graded layer and with various initial positions in relation to it. A finite-element-based computational procedure containing a number of novel features is used to study the curvilinear crack growth that occurs in this problem. For the problem parameters under consideration, net-section yielding is induced in the more ductile of the two materials, even when the crack trajectory lies well inside the more brittle material. In general, it is found that the brittle material strongly attracts the crack trajectory, and that the presence of the lower-flow-stress material in conjunction with the constraining effect of the higher-flow-stress material induces a substantial toughening mechanism.

Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks

This paper considers the equations of two-dimensional elasticity theory in nonsmooth domains. The domains contain curvilinear cracks of variable length. On the crack faces, conditions are specified in the form of inequalities describing mutual nonpenetration of the crack faces. It is proved that the solutions of equilibrium problems with a perturbed crack converge to the solution of the equilibrium problem with an unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the length of a curvilinear crack is obtained.

Solutions for curvilinear cracks in bonded dissimilar materials subject to uniform magnetic induction

The magnetoelastic problem of curvilinear cracks lying along the interface between two dissimilar ferromagnetic materials under remote uniform magnetic induction is considered. On the basis of the Hilbert formulation and a special technique of analytical continuation, the solutions of the magnetic and the magnetoelastic stress fields either in the matrix or in the inclusion are expressed in a general form. Just like the pure elastic problem of curved cracks or the magnetoelastic problem of straight cracks in bonded dissimilar media, the trig-log character for the singularity of magnetoelastic stresses is found in this paper. Both the explicit solutions and the numerical results for the illustrating case of a single circular-arc crack in bi-material plate with typical ferromagnetic materials are given to elucidate the use of the present study. It is noted that the stress intensity factors are obtained and verified by comparison with the existing ones under special cases. The validity of the crack opening condition is also discussed.

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.

Crack propagation in a material with random toughness

An efficient Monte Carlo procedure is presented for characterizing the propagation of a crack in a material whose fracture toughness is a random field. The simulations rely on accurate approximate solutions of the integral equations that govern the dislocation densities, stress intensity factors, and energy release rates of curvilinear cracks. For a plate containing an edge crack that propagates towards a subsurface crack representing a traction-free boundary, results for the distributions of crack trajectories, critical applied far-field stresses, and nominal fracture toughness are presented for various parameters that quantify the randomness of the material's critical energy release rate. A demonstrative probabilistic model for crack trajectories is built and size effects are discussed.

Crack path in an elastic material with a random array of small defects

This paper is devoted to the problem of interaction of a semi-infinite crack (propagating in elastic material) with a random array of small defects (e.g. inclusions, voids). This interaction manifests itself in changes of the crack orientation and, overall in randomly varying curvilinear crack trajectories. Making use of the asymptotic results proposed in (Movchan, 1991, 1995) the procedure for configurational averaging is presented which yields the mean crack path and the standard deviation from the mean. The general formulae for the mean crack path can serve as a basis for effective quantitative analysis for specific situations of practical interest. The illustrative examples elaborated in the paper show dependence of the shape of crack trajectory on the probabilistic characteristics of random locations of inclusions and on their elastic parameters. The analysis performed in the paper can be directly used in prediction of failure mechanisms of brittle materials containing small randomly distributed inclusions/voids.

Fracture mechanisms of offset rock joints-A laboratory investigation

To study the failure mechanisms of joints and rock bridges in jointed rock masses a series of uniaxial compression tests were performed on specimens made of rock-like material. Specimens of size 63.5 cm × 27.9 cm × 20.3 cm, made of 72% silica sand, 16% cement (Type I) and 12% water by weight were tested. The joint inclination angle was maintained at 45°, while the offset angle i.e. angle between the plane of the joint and the line that connects the two inner tips of the joints, was changed from 0°–120° with an increment of 15°. The tests were performed using a 2000 kN universal compression machine and a HP data acquisition system. In each sample, five LVDTs were fixed to measure the displacements along and across both the bridge and the joint segment, and the total displacement along the total length of the sample. The failure mechanism was monitored by visual inspection and a magnifier to detect cracks initiation. For each test the failure surfaces were investigated to determine the characteristics of each surface. In all of the tested samples curvilinear cracks called wing cracks initiated at the joints tips due to high tensile stresses concentration. These wing cracks were directed along the direction of the uniaxial load. The coalescence mechanism of two cracks was investigated. Results showed that open cracks could coalesce by shear failure or tensile failure. The coalescence path was found to be mainly dependent on the inclination of the rock bridge between the cracks.

Mixed mode BEM analysis of multiple curvilinear cracks in the general anisotropic solids by the crack tip singular element

We consider multiple curvilinear cracks in the two-dimensional general anisotropic solids and establish a computationally effective technique to determine the stress intensity factors accurately. Each curvilinear crack is represented by a collection of straight crack elements and the crack opening displacement (COD) by the continuous distribution of the dislocation dipoles. The crack element located next to the crack tip is called the crack tip singular element (CTSE), where the known r crack tip opening displacement and the 1/ r stress singularity is mathematically built in the interpolation of the COD using the Chebyshev polynomials. The regular crack elements away from the crack tip use the quadratic polynomial interpolation for the CODs. Simple analytical formulas for the displacement, traction, and the stress intensity factor (SIF) contributions for the CTSE are developed in terms of its own COD coefficients. Since the SIFs are obtained during the main processing no post processing is required; a distinct advantage over the powerful quarter-point element. Numerical results are given for several multiple curvilinear crack problems to demonstrate the accuracy and simplicity of the technique.

Analysis of Shape Variation of a Fatigue Surface Crack in a Pipeline

The authors have elaborated a model of nonthrough surface flaws in a pipeline under variable internal pressure. The model involves a proposed scheme allowing for variation of local fracture stresses along a curvilinear crack front. Verification of the theoretical model is carried out by comparing it with the available experimental data reported elsewhere. The authors have performed a comprehensive parametric study of the influence of initial flaw geometry, pipeline dimensions and variation of properties of steels due to heat treatment and test temperature on the surface crack behavior.

Evaluation of the amount of crack growth in 2D LEFM problems

This work proposes an algorithm for the evaluation of the amount of crack growth in two-dimensional problems of linear elastic fracture mechanics. The method is based on the energetic formulation of the coupled displacement–crack propagation problem defined by Nguyen et al. in 1990 in order to study the stability of crack growth in elastic fracture. To avoid the remeshing usually needed for modelling crack growth, a PUFEM model is utilized. The proposed algorithm is valid for stable rectilinear crack growth in LEFM but it can be extended to evaluate stable curvilinear crack growth in materials showing dissipative deformations.

Shape sensitivity of curvilinear cracks on interface to non-linear perturbations

The 2D-model of an anisotropic, non-homogeneous, bonded elastic solid with a crack on the interface is considered. We state the linear problem with the stress-free boundary condition at the crack faces in addition to the transmission condition at the connected part of the interface. The sensitivity of the model to non-linear perturbations of the curvilinear crack along the interface is investigated. We obtain the asymptotic expansion and the corresponding derivatives of the potential energy functional with respect to the crack length via the material derivatives of the solution. This allows us to describe the growth or stationarity, and the local optimality conditions by the Griffith rupture criterion. The integral expression of the energy release rate for the considered problems is obtained, and the Cherepanov-Rice integral is discussed.

Numerical Analysis of Multiple Discrete Cracks in Concrete Dams Using Extended Fictitious Crack Model

A rigorous analytical procedure for the crack analysis of concrete dams is established using a newly developed numerical technique for analyzing multiple discrete cracks in concrete. The theoretical aspect of the problem solution is addressed first, focusing on the cracking modes of multiple cracks, followed by a brief review on the numerical formulation of the Mode-I type crack propagation for multiple discrete cracks. Next, a simple technique for modeling curvilinear crack propagation is introduced, and the method is examined through numerical studies on the bending tests of simple plain concrete beams with multiple cracks. Then, a well-quoted scale model of a concrete gravity dam is used to study the cracking behaviors in concrete dams; the numerical modeling involves not only the original single-crack problems but also multiple-crack problems. Employing two different types of numerical models and assuming multiple initial notches of different sizes at different locations along the upstream face of the model dams, various kinds of cracking behaviors are obtained and discussed, focusing on the crack interactions.

Simulation of dynamic crack curving and branching under biaxial loading by a time domain boundary integral equation method

Bifurcation and trifurcation of a fast running crack under various biaxial loading conditions is investigated numerically. The solution procedure for the 2D model in the framework of linear elastodynamics employs a time-domain boundary element method and allows for arbitrary curvilinear crack propagation. Branching events are controlled by the criterion of a critical mode I stress intensity factor while the propagation direction and growth rate of each branch are determined from the criterion of maximum circumferential stress. Numerical results are compared with experimental findings and are discussed with respect to macroscopic and microscopic aspects of dynamic fracture.

Prediction of Residual Strength and Curvilinear Crack Growth in Aircraft Fuselages

Two practical engineering approaches to assess the structural integrity of aircraft fuselages are presented. Both approaches combine fracture mechanics with thin-shell finite element analyses to predict structural responses quantitatively. The first approach uses the crack-tip opening angle fracture criterion to predict residual strength of KC-135 fuselages. The second approach uses T-stress and fracture toughness orthotropy to predict crack-growth trajectory in narrow-body fuselages. For residual strength prediction 12 damage scenarios, which might occur in applications, are examined. It is found that the model with small multiple-site cracking and material thinning caused by corrosion damage has the worst load-carrying capacity. For curvilinear crack-growth simulation a directional criterion based on the maximum tangential stress theory accounting for the effect of T-stress and fracture toughness orthotropy is used to predict crack-growth trajectory. Both T-stress and fracture toughness orthotropy are found to be essential to predict the observed crack path, where the trajectory experiences crack turning and flapping phenomena.

The antiplane problem of electroelasticity for a cylinder with a tunnel crack excited by an arbitrary system of surface electrodes

The antiplane mixed boundary-value problem of electroelasticity of the oscillations of an infinite piezoceramic cylinder, weakened by a curvilinear tunnel crack, is considered. Using special integral representations of the solution, the boundary-value problem is reduced to a system of singular integro-differential equations of the second kind with discontinuous kernels. The results of a numerical realization of the algorithm, characterizing the amplitude-frequency characteristics of a piecewise-uniform cylinder and the behaviour of the components of the electroelastic field in the region and on the boundary of the cylinder under conditions of the inverse piezoelectric effect, are presented.

Principle of Comparison and the Method of Equivalence of the Stress Intensity Factors at the Tips of Equitangent Cracks

We propose a principle of estimation and the method of equivalence of equitangent cracks for the evaluation of the stress intensity factors near curvilinear cracks of any convex shape. For their substantiation, we prove two theorems of comparison. The method of equivalence of equitangent cracks is generalized to the case of three-dimensional bodies and approved by analyzing the stressed state near the tips of a hyperboloidal cut as an example. It is shown that the correlation between the calculated and known values of the stress intensity factors near a similar defect is quite good.

Prediction of residual strength and curvilinear crack growth in aircraft fuselages

Prediction of residual strength and curvilinear crack growth in aircraft fuselages

Pitting of the rolling bodies contact surface

A calculational model for investigation of the fracture processes and durability of the surface contact zones of rolling bodies has been proposed within the framework of the fatigue fracture mechanics concepts. Unidirection rolling in conditions of dry friction and boundary lubrication has been investigated. The model basis is the algorithm step-by-step construction of the crack growth paths developed with application of singular integral equations of two-dimensional problem of the elasticity theory for bodies with curvilinear cracks and local fracture criteria for a complex stress-strain state. The paths calculation algorithm also takes into account redistribution of stresses, caused both by crack growth and mutual motion of contacting bodies. In the case of rolling with lubrication, that can fill the crack, its action is modeled by stresses, additionally distributed on the crack faces. Propagation paths of edge cracks have been constructed. The dependence of path form on friction coefficient between contacting bodies, the length and orientation of the initial crack, and in the presence of lubrication, on the intensity of its pressure on the crack faces, has been investigated. In calculations the generalized mode I fracture criterion ( σ θ -criterion) has been used. An attempt has been made to explain the mechanism of development of such crack-like defect as pitting and ‘squat’, often met during rolling contact fatigue. As an example, the residual durability of the ShH-15 bearing steel presurface layer under boundary lubrication has been estimated.

Crack Growth in Rolling Bodies under the Conditions of Dry Friction and Wetting

Within the framework of a two-dimensional mathematical model, we study the development of a surface macrocrack in the course of unidirectional rolling under the conditions of dry friction and wetting. The damaged body in a rolling couple is modeled by an elastic half plane with an edge cut (crack) and the action of the counterbody is simulated by the translational motion of Hertzian contact forces along the boundary of the half plane with taking into account the friction forces. We compute the crack-growth paths depending on the coefficient of contact friction between the bodies and the orientation and length of the crack. The numerical calculations are performed by the step-by-step method based on the use of the singular integral equations of the theory of elasticity for bodies with curvilinear cracks and the criterion of generalized opening. It is shown that, for high coefficients of friction between the bodies of the rolling couple (f = 0.20–40; dry friction), a branch propagates from the original fatigue shear macrocrack into the bulk of the material of the driven body. In the case of low friction coefficients (f < 0.20; wetting), a branch mainly propagates along the boundary of the body in the direction of motion of the counterbody. We propose an explanation of the mechanism of development of burrow-type defects often encountered in railroad rails.