Cracks In Anisotropic Media Research Articles

Stress and Displacement Intensity Factors of Cracks in Anisotropic Media

A relation connecting stress intensity factors (SIF) with displacement intensity factors (DIF) at the crack front is derived by solving a pseudodifferential equation connecting stress and displacement discontinuity fields for a plane crack in an elastic anisotropic medium with arbitrary anisotropy. It is found that at a particular point on the crack front, the vector valued SIF is uniquely determined by the corresponding DIF evaluated at the same point.

Cracks in anisotropic media: pseudodifferential equations, wave fronts, and Irwin’s energy release rate extension

3D problems for plane cracks of arbitrary shape in anisotropic media are considered. The stress intensity factors are determined by the intensity factors of the crack displacement discontinuity field. Strong ellipticity of the constructed pseudodifferential operator is proved. The solution of the pseudodifferential operator of the crack theory is obtained by the specially constructed potentials with densities concentrated at the crack front. Irwin’s relation for the energy release rate is obtained for a plane crack of arbitrary shape in a medium with arbitrary elastic anisotropy.

Stress–displacement intensity factors for cracks in anisotropic media

For a plane crack of arbitrary loading in a medium with arbitrary elastic anisotropy analytic expressions are derived allowing computing stress intensity factors by the corresponding displacement intensity factors. The relation between both intensity factors is derived for media with arbitrary elastic anisotropy. Comparison with the expressions for a plane crack in isotropic medium is outlined.

Through Transmission Characteristics of Ao Mode at Crack Tip in an Isotropic Medium

This paper deals with through transmission characteristics of Ao Lamb mode, which is incident at the tip of a through thickness crack in an isotropic medium. Experiments were conducted transducing Ao Lamb mode at 200kHz frequency in one mm thick aluminium plate. Experimental A-scans captured in time domain, at various configurations, were transformed to frequency domain and variation in normalized amplitude of Ao mode in time and frequency domains with respect to configuration of probe unit, expressed in degree, was evaluated. It was observed that when transmitter and receiver are aligned with the axis of the crack, amplitude of transmitted Ao mode is large. Low amplitude Ao mode was transmitted when the axis of the crack is perpendicular to direction of propagation of Ao mode.

Doubly Periodic Cracks in the Anisotropic Medium with the Account of Contact of Their Faces

Abstract The paper presents complex variable integral formulae and singular boundary integral equations for doubly periodic cracks in anisotropic elastic medium. It utilizes the numerical solution procedure, which accounts for the contact of crack faces and produce accurate results for SIF evaluation. It is shown that the account of contact effects significantly influence the SIF of doubly periodic curvilinear cracks both for isotropic and anisotropic materials.

Nucleation of cracks in an isotropic medium with periodic system of rigid inclusions under transverse shear

A model of nucleation of cracks in composite materials with periodic structures is given. The model is based on the analysis of the cracking zone. It is assumed that the zone cracking is a layer of finite length with weakened bonds between the material particles. The interaction of pre-fracture zone surfaces is modeled by applying cohesive forces caused by bonds to pre-fracture zone surfaces. The analysis of the limit equilibrium of pre-fracture zones under transverse shear involves (i) a definition of relationships between cohesive forces and surfaces pre-fracture zone shear; (ii) the evaluation of displacement and stresses around the pre-fracture zone taking account of external loads and cohesive forces, also disposition of rigid inclusions; (iii) definition of the dependence of critical external loads under which the nucleation of a crack occurs on geometric parameters of the medium. The condition for the appearance of a crack is formulated taking account the criterion of the limit traction of the bonds in the material.

Fracture mechanics analysis of multiple cracks in anisotropic media

AbstractThis paper presents a single‐domain boundary element method (BEM) for linear elastic fracture mechanics analysis in the two‐dimensional anisotropic material. In this formulation, the displacement integral equation is collocated on the un‐cracked boundary only, and the traction integral equation is collocated on one side of the crack surface only. A special crack‐tip element was introduced to capture exactly the crack‐tip behavior. A computer program with the FORTRAN language has been developed to effectively calculate the stress intensity factors of an anisotropic material. This BEM program has been verified having a good accuracy with the previous researches. Furthermore, by analyzing the different anisotropic degree cracks in a finite plate, we found that the stress intensity factors of crack tips had apparent influence by the geometry forms of cracks and media with different anisotropic degrees. Copyright © 2010 John Wiley & Sons, Ltd.

Analysis of Crack-Inclusion Interaction in an Anisotropic Medium by Eshelby Equivalent Inclusion Method

The problem of a semi-infinite crack in anisotropic medium interacting with a near-tip inclusion is analyzed by the Eshelby equivalent inclusion method. The change of mode I stress intensity factor due to crack-inclusion interaction is evaluated using a novel analytical solution for the model I stress intensity factor at the tip of a semi-infinite crack due to near-tip eigenstrains. Numerical results of the mode I stress intensity factor due to the presence of a near-tip circular inclusion are presented to show the influence of the elastic stiffness of an inclusion on the near-tip elastic field. The present scheme can be applied to calculate the stress intensity at a crack-tip in anisotropic media due to the interaction of inclusions with arbitrary shapes.

Laws of crack motion and phase-field models of fracture

Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic surface energy. This force-balance condition can be interpreted physically based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. The obtained law of crack motion reduces in the quasistatic limit to the principle of local symmetry in isotropic media and to the principle of maximum energy-release-rate for smooth curvilinear cracks in anisotropic media. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. Interestingly, for kinked cracks in anisotropic media, force-balance gives significantly different predictions from the principle of maximum energy-release-rate and the difference between the two criteria can be numerically tested. Simulations also show that predictions obtained from force-balance hold even if the phase-field dynamics is modified to make the failure process irreversible. Finally, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed.

Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media

Singularity-reduced integral relations are developed for displacement discontinuities in three-dimensional, anisotropic linearly elastic media. An isolated displacement discontinuity is considered first, and a systematic procedure is followed to develop relations for the displacement and stress fields induced by the discontinuity. The singularity-reduced relation for the stress is particularly important since it is in a form which allows a weakly-singular, weak-form traction integral equation to be readily established. The integral relations obtained for a general displacement discontinuity are then specialized to an isolated crack and to dislocations; the relations for dislocations are introduced to emphasize their direct connection to corresponding results for cracks and to allow earlier independent findings for these two types of discontinuities to be put into proper context. Next, the singularity-reduced integral equations obtained for an isolated crack are extended to allow treatment of cracks in a finite domain, and a pair of weakly-singular, weak-form displacement and traction integral equations is established. These integral equations can be combined to obtain a final formulation which is in a symmetric form, and in this way they serve as the basis for a weakly-singular, symmetric Galerkin boundary element method suitable for analysis of cracks in anisotropic media.

Total induced anisotropy due to tilted cracks in anisotropic medium

This paper is concerned with the relationship between the effective stiffness tensor and the intensity of damage in individual modes for an anisotropic material containing tilted cracks. The predictions are compared favourably with the experimentally measured load-induced changes of the 13 stiffnesses of a 2D C/C-SiC ceramic-matrix composite subjected to off-axis loading. By taking into account the thickness of the cracks, it is possible to understand the change in elastic anisotropy in material and of its inelastic strain.

Hypersingular integral equations for a thermoelastic problem of multiple planar cracks in an anisotropic medium

The problem of calculating the thermoelastic stress around an arbitrary number of arbitrarily located planar cracks in an infinite anisotropic medium is considered. The cracks open up under the action of suitably prescribed heat flux and traction. With the aid of suitable integral solutions, we reduce the problem to solving a system of Hadamard finite-part (hypersingular) integral equations. The hypersingular integral equations are solved for specific cases of the problem.

Cracks in anisotropic elastic media

A numerical method for 3D problems of cracks in anisotropic media is developed, and this is based on the variational approach to the crack opening problem. Properties of the pseudodifferential operator of the crack equilibrium problem are considered. Numerical examples are also presented.

Mode II and mode III stress singularities and intensities at a crack tip terminating on a transversely isotropic-orthotropic bimaterial interface

In a recent paper (referred to as I) we obtained inter alia , the stress and displacement fields at the tips of a transverse crack in an isotropic medium sandwiched between orthotropic media under in-plane loading (mode II). The crack was lying wholly within the isotropic medium so that the singularity at the crack tip was of the usual inverse square root type. In this paper, the analysis is extended to the case when the tip of the crack terminates on the transversely isotropic-orthotropic bimaterial interface and the nature of the singularity at the crack tip depends on the elastic properties of both media. The analysis is performed for both inplane (mode II) and out-of-plane (mode III) shear loading. General solutions are obtained for the crack tip stress singularities and corresponding stress intensity factors, together with the influence of the elastic properties and geometry of the media upon the stress field. These solutions are specialized to the limiting case when the crack terminates on the interface between dissimilar isotropic media in order to demonstrate consistency with published results. As in I, the solutions are used to investigate the influence of ply angle θ upon the stress singularities in [± θ /90°] s fibre-reinforced composite laminates. For this analysis, the outer angle-ply sublaminates are treated macroscopically as homogeneous orthotropic media whose elastic constants are obtained using the classical lamination approximation. Calculations are also carried out to study the variation of stress intensity factors with the ply angle and outer sublaminate thickness.

Plastic relaxation at a crack tip by asymmetric slip

Relaxation at a sharp crack tip by a single slip band is considered. It is shown that for mixed-mode loading of a plane crack in an isotropic medium there is a unique angle between the slip band and the crack for which the energy release rate (or stress intensity factor) of the crack can be reduced to zero. For such a slip-band calculations are made of the slipband length and the crack-opening displacement as a function of the loading, crack length and friction stress acting on dislocations in the slip band. For small-scale yielding, a simple model is discussed that gives a good approximation to the crack-tip opening displacement and slip-band angle.

Mode III Stress-intensity factors in cracked orthotropic plates—An analogy with propagating cracks in isotropic media

In this paper, the method of reflected caustics—which was used to evaluate Mode III SIF's in stationary cracks in isotropic plates—was extended to deal with stationary cracks in orthotropic plates. Furthermore, a correspondence between the anisotropic stationary case and the case of a Mode III dynamic crack, traversing an isotropic plate, is developed by analyzing appropriately the governing equations of the two problems. For this purpose the singulardisplacement field for rectilinearly orthotropic cracked bodies was combined with either Yoffe's model for steady-state, or Broberg's model for transient-crack propagation. Graphs are given where the equivalence between these cases can be established. In this way, the dynamic problem of the propagating crack in an isotropic medium can be readily simulated by considering the experimentally easier anisotropic stationary case.

Interaction between a periodic system of elastic inclusions and rectilinear cracks in an isotropic medium

Interaction between a periodic system of elastic inclusions and rectilinear cracks in an isotropic medium

Thin Inclusions and Cracks in Anisotropic Media

By using the solution of eigenstrain problems in anisotropic media, flat ellipsoidal inclusions and crack problems are investigated. When a dilatational misfit strain is defined in a flat ellipsoidal inclusion (i.e., disk-shaped precipitate), the elastic strain energy associated with the misfit becomes minimum when the plane of the inclusion coincides with one of the crystalline planes, where the matrix and the inclusion are assumed as two different cubic crystals having the same crystalline directions. The minimum value, however surprisingly, depends only on the elastic moduli of the inclusion. When crystalline directions of the inclusion are parallel to the principal axes of the ellipsoid, the elastic strain energy is independent of the orientation of the inclusion with respect to the crystalline directions of the matrix and the constant value is equal to the minimum value in the first case. If an inclusion is simply defined by giving a uniform dilatational misfit strain in an ellipsoidal domain in a homogeneous material (i.e., matrix and inclusion are same material), the condition for the minimum elastic strain energy is the same as that of the first case, namely the minimum occurs when the plane of the inclusion is parallel to one of the crystalline directions. The method employed here can be applied equally to generally anisotropic materials. The method is also applicable to fracture problems by taking the elastic moduli of the inclusion as zero. As examples, an elliptical crack is considered for a simple tension and a pure shear. The interaction energy between the applied stress and a crack is calculated. By using this result, the Griffith criterion for fracture is derived for the penny-shaped crack. Some numerical results for cubic crystals are shown.

Interaction of a doubly-periodic system of rectilinear cracks in an isotropic medium: PMM vol.38, n≗5, 1974, pp. 906–914

The problem of the tension and shear on a plane isotropic medium weakened by a doubly-periodic system of rectilinear cracks is considered. General representations are constructed for the solutions describing a class of problems with doubly-periodic stress distribution outside the cracks. The fundamental singular equation of the problem is reduced to an infinite system of linear algebraic equations without the intermediate step of reducing it to a Fredholm equation. The procedure to determine the stress intensity factors is written out. Questions related to modeling the described lattice by a continuous isotropic medium are examined and the elastic characteristics of this latter are determined (the macroscopic parameters of a medium with cracks). Results of computations are presented. The doubly-periodic problem for a symmetric rhombic lattice has been considered by another method in [1. 2].

An ellipsoidal crack and needle in an anisotropic elastic medium: PMM vol. 37, n≗3, 1973, pp. 524–531

We consider an ellipsoidal crack and needle (of small, but finite thickness) in an anisotropic elastic medium and a homogeneous external field. We obtain and investigate the explicit expressions for the stresses on their surfaces. We show that by decreasing the thickness of the needle the stresses corresponding to any load tend to a finite limit, i.e. they do not contain singularities, while for the ellipsoidal crack a singularity arises if the external field contains a component normal to the plane of the crack. In the case of extension the maximum stress is always attained at the edge of the crack while in the case of pure shear, as a rule, in its small neighborhood. In the latter case there is a sharp peak of stresse and on the edge itself all the components of the stress tensor may vanish. This points to the necessity of investigating the stresses on the entire surface of the crack and not only in its characteristic points. By an ellipsoidal crack (needle) we will understand an ellipsoidal cavity having one small (large) dimension in comparison with the other two dimensions. This allows us, at the computation of the stresses at the surface of the cavity, to restrict ourselves to the principal term of the expansion with respect to a corresponding small but finite parameter. The limiting case when the parameter tend to zero corresponds to the elliptic crack. In this case, only the stresses in the neighborhood of the crack or the limiting values of the nonsingular components of the stress tensor on the crack itself are meaningful. In most cases (see, for example [1] where there are other references) the ellip tic cracks have been studied. The results for an ellipsoidal crack in an isotropic medium can be obtained by a limiting process from the known solutions for the ellipsoidal cavity constructed in [2–4] and which has been done in [5, 6]. How ever, the components of the stress tensor have been studied not on the entire sur face of the crack but only at its edge. The stresses at the vertices of a spheroid; crack and needle in a transversely isotropic medium have been obtained in [7] for external fields which do not have singularities. As opposed to the mentioned papers, here we consider an arbitrary anisotropic medium and we investigate the complete state of stress on the entire surface of an ellipsoidal crack and needle. Such an investigation turns out to be essential since in some cases there is an abrupt increase of stresses near the edge of the crack, although on the edge itself they are equal to zero. In this paper we make use of the general solution of the stress concentration problem on the surface of an ellipsoidal cavity in an anisotropic medium which has been obtained in [8]. In Sect. 1 we give some formulas from [8] which are necessary in what follows and we introduce parameters which are convenient for the limitimg processes. In Sects. 2 and 3 we obtain expressions for the stresses on the surface of the ellipsoidal needle and crack. In Sects 4 and 5 we give a complete study of the stress concentrations at the surface of an ellipsoidal crack in an isotropic and also in an orthotropic medium for all cases of external homogeneous fields.