Flat Crack Research Articles

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface

Solution of flat crack problems in shear mode

AbstractSolution of the flat crack problems in shear mode is presented. The least potential energy principle is used to solve the problems. In the solution a family of the crack opening displacements (COD) with some undermined coefficients is assumed. The strain energy stored in body is expressed in the form of a quadratic form with some undetermined coefficients. In the formulation of the quadratic form, the differential–integral equation for the flat crack problem is used. After using the least potential energy principle, the coefficients in the family of COD can be determined and the crack opening displacements can be evaluated immediately. The stress intensity factors along the crack border can be obtained from the known crack opening displacements. A particular feature of the present method is that no singular integral is involved in computation. Several numerical examples are given with the calculated stress intensity factors along the crack border. Copyright © 2005 John Wiley & Sons, Ltd.

Pore fluid effects on shear modulus in a model of heterogeneous rocks, reservoirs, and granular media

To provide quantitative measures of the importance of fluid effects on shear waves in heterogeneous reservoirs, a model material called a “random polycrystal of porous laminates” is introduced. This model poroelastic material has constituent grains that are layered (or laminated), and each layer is an isotropic, microhomogeneous porous medium. All grains are composed of exactly the same porous constituents and have the same relative volume fractions. The order of lamination is not important because the up‐scaling method used to determine the transversely isotropic (hexagonal) properties of the grains is Backus averaging, which, for quasi‐static or long‐wavelength behavior, depends only on the volume fractions and layer properties. Grains are then jumbled together totally at random, filling all space, and producing an overall isotropic poroelastic medium. The poroelastic behavior of this medium is then analyzed using the Peselnick‐Meister‐Watt bounds (of Hashin‐Shtrikman type). We study the dependence of the shear modulus on pore fluid properties and determine the range of behavior to be expected. In particular, we compare and contrast these results to those anticipated from Gassmann's fluid substitution formulas and to the predictions of Mavko and Jizba for very low porosity rocks with flat cracks. This approach also permits the study of arbitrary numbers of constituents, but for simplicity, the numerical examples are restricted here to just two constituents. This restriction also permits the use of some special exact results available for computing the overall effective stress coefficient in any two‐component porous medium. The bounds making use of polycrystalline microstructure are very tight. Results for the shear modulus demonstrate that the ratio of compliance differences R (i.e., shear compliance changes over bulk compliance changes when going from drained to undrained behavior, or vice versa) is usually nonzero and can take a wide range of values, both above and below the value R = 4/15 valid for low porosity, very low aspect ratio flat cracks. Results show that the overall shear modulus in this model can depend relatively strongly on mechanical properties of the pore fluids, sometimes (but rarely) more strongly than the dependence of the overall bulk modulus on the fluids.

Axisymmetric crack problem for a hollow cylinder imbedded in a dissimilar medium

The analytical solution for the linear elastic problem of flat annular crack in a transversely isotropic hollow cylinder imbedded in a transversely isotropic medium is considered. The hollow cylinder is assumed to be perfectly bonded to the surrounding medium. This structure, which can represent a cylindrical coating-substrate system, is subjected to uniform crack surface pressure. Because of the geometry and the loading, the problem is axisymmetric. The z = 0 plane on which the crack lies, is also a plane of symmetry. The composite media consisting of the hollow cylinder and the surrounding medium extends to infinity in z and r directions. The mixed boundary value problem is formulated in terms of the unknown derivative of the crack surface displacement by using Fourier and Hankel transforms. By extending the crack to the inner surface and to the interface, the cases of surface crack and crack terminating at the interface are obtained. Asymptotic analyses are performed to derive the generalized Cauchy kernel and associated stress singularities. The resulting singular integral equation is solved numerically. Stress intensity factors for various crack configurations, crack opening displacements and stresses along the interface and on z = 0 plane are presented for sample material combinations and geometric parameters.

General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method

The three-dimensional equations of transversely isotropic magneto-electro-thermo-elasticity are simplified by the introduction of two displacement functions. A general solution is then rigorously derived by virtue of the operator theory, which is expressed in terms of two functions, satisfying a second-order and a tenth-order homogeneous partial differential equation, respectively. Utilizing the generalized Almansi's theorem, the general solution can be further simplified to the one expressed by six harmonic functions only. This allows us to extend the potential theory method to the mixed boundary value problems of magneto-electro-thermo-elastic materials. A flat crack in an infinite space subjected to symmetric mechanical, electric, magnetic as well as temperature loads at the crack surfaces is considered for instance. One integral equation and three integro-differential equations are derived, which are similar to those reported in the literature. For a penny-shaped crack subjected to a uniform loads, exact three-dimensional expressions for the full-space magneto-electro-thermo-elastic field are obtained in terms of elementary functions.

Impact breakage of semi-brittle spheres

Quasi-static compression cracking and low-velocity impact cracking of 12.7 mm diameter spheres of acrylic resin and polymethyl methacrylate (PMMA), both types regarded as semi-brittle, has been investigated using postfracture optical microscopy observations and high-speed framing photography at framing rates of 200,000 frames per second. It has been shown that under both quasi-static and dynamic loading conditions the initial necessary step is the plastic deformation of the compressed/impacted sphere to an a/ R value of greater than ∼0.5, where a and R are the radii of contact and sphere, respectively. The high-speed photography has shown that under critical conditions one or two primary meridional cracks initiate at the circle of contact and propagate into the bulk, reaching velocities up to 0.8 of the Rayleigh surface wave velocity of the material of the sphere. From the growing primary meridional cracks, secondary meridional cracks initiate and propagate at high speeds towards the free surface of the sphere, leading to the formation of orange segment-shaped fragments. High-speed photography has also revealed that during impact loading, primary and secondary meridional cracks grow around approximately conical zones at the contact regions. High-speed framing photography at a higher framing rate of 500,000 frames per second has also been used to investigate the response of 600–750 μm diameter PMMA spheres when they impacted a smooth silica flat at 200 m s −1. Although under these impact conditions, the spheres did not break up, information about their rebound energies has been obtained and used to analyse the integrity of the spheres. These microspheres are shown to develop at and beyond the edge of the plastically deformed circular flat meridional cracks, which are very similar to those produced in the 12.7 mm diameter spheres.

Hypersingular integral equation method for three‐dimensional crack problem in shear mode

AbstractThis paper presents the use of the hypersingular integral equation method for solving the flat crack problem in shear mode. In the method, the crack opening displacement (COD) functions are assumed to be polynomials with several undetermined coefficients. The involved hypersingular integral can be reduced into a repeat integral in a particular polar co‐ordinate, and further integrated by a known quadrature rule. This technique considerably reduces the effort of derivation and computation to obtain the final solution. The undetermined coefficients in the COD functions are obtained from an algebraic equation. The stress intensity factors (SIF) along the boundary of the flat crack can then be easily calculated. Numerical examples are given to demonstrate the efficiency of the proposed method. Copyright 2004 John Wiley & Sons, Ltd.

On the identification of the flatness of a sound-hard acoustic crack

In this paper, we present results concerning the far field pattern generated by flat and almost flat cracks in 3D and the possibility of identifying these geometrical features from direct inspection of the far field pattern. We address the direct problem using a variational formulation of the boundary integral equation to avoid the hipersingularity in the double layer potential representation. Concerning the inverse problem, some estimates presenting a direct dependence on the far field behavior and the flatness of the crack are derived. From the knowledge of the plane that defines the main directions of the crack it is possible to get a first approximation that may be used as an initial guess for the Newton method. Numerical simulations validate the direct relation between a far field plane having almost null amplitude and the main directions of a plane that defines an almost flat crack.

Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.

Numerical Study of Scale Effects on the Stiffness Modulus of Rock Masses

The purpose of this paper is to contribute to the study of scale effects on the stiffness modulus of discontinuous rock masses from a mechanical point of view. The study has been conducted on the basis of the observation that the length and density of the natural discontinuities (cracks) in rock masses increase with the volume of the mass. Compressive tests on specimens containing open and closed flat cracks have been simulated through a numerical procedure that is based on the displacement discontinuity method and the fictitious stress method. Cracks have been generated, through a random process, with dips in the 0°–180° range. Both open and closed cracks have been considered. The numerical results obtained for specimens containing open cracks are found to approach the values obtained through an analytical solution. Then, the numerical method has been applied to the study of scale effects on the stiffness modulus of specimens with closed cracks. The numerical results show a decrease in the stiffness modu...

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

AbstractThis work contains an analytical study of the asymptotic near‐crack contour behaviour of stresses obtained from the self‐regular traction‐boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The flat crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self‐regular traction‐BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near‐crack contour self‐regular traction‐BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log‐singular stresses are identified and extracted from the BIE. Log‐singular stress terms are revealed for the free integrals written as contour integrals and for the self‐regularized integral with the integration region divided into sub‐regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the final BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identified, and the singular information is all contained in the region near the crack contour. Copyright © 2003 John Wiley & Sons, Ltd.

Three-dimensional stress and deformation fields around flat and slant cracks under remote Mode I loading conditions

This paper investigates the phenomenon of slant fracture observed in stable tearing tests of many ductile materials, where an initially flat crack, loaded under remote Mode I conditions, tends to grow into a slant crack and stay in the slant configuration until final fracture. In an effort to identify potential reasons why cracks prefer to grow in a slant manner, three-dimensional finite element analyses of crack-front stress and deformation fields in Arcan-type specimens containing a flat or slant crack are performed under elastic–plastic and remote Mode-I loading conditions. In particular, the crack-tip opening displacement (COD) at a position behind the crack tip, the mean stress, the effective stress, and a constraint factor (defined as the ratio of the mean stress and effective stress) are studied and compared for the two types of cracks. Analysis results reveal several stress/deformation field variations around flat and slant cracks under identical remote loading conditions. First, close to the crack front, the COD of a slant crack is greater than that of a flat crack. Second, at the specimen’s mid-plane, a flat crack leads to a higher constraint value ahead of the crack than a slant crack. Third, the effective stress ahead of a slant crack is greater than that ahead of a flat crack, especially close to the crack front. The above results seem to suggest that slant fracture may be preferred because a slant crack enhances the driving force in the form of a higher near-tip COD value and because a shearing type of failure is promoted in the case of a slant crack compared to a tensile type of failure in the case of a flat crack.

Bed-limited cracks in effective medium theory

SUMMARY An effective medium theory typically requires the description of a mean crack shape. In general, for simplicity, this is taken to be a flat, circular (‘penny-shaped’) crack. However, this places an unnecessary limitation on the theory, when it is perhaps more realistic to describe a crack in terms of having a bounded width and an otherwise ellipsoidal shape. The generalization of the method of smoothing, as proposed by Hudson (1994, Geophys. J. Int.,117, 555–561) , to extend his original model (Hudson, 1980. Math. proc. Camb. phil. Soc.,88, 371–384), has been used to study the role of the crack width and the ratio of the two larger dimensions in determining the properties of the effective medium. In general, this leads to a description of the medium as having orthorhombic symmetry, and provides a suitable description of a material where the crack dimensions are restricted in one direction owing to, for example, bed-limiting effects, while remaining unconfined in other directions. An elliptical flat crack limit is determined, analoguous to the circular crack description of the original Hudson model. In addition to the isolated crack description, the theory is extended to include the fluid flow mechanism of Tod (2001, Geophys. J. Int.,146, 249–263) that models the flow as being dominated by crack-to-crack flow and is valid for low matrix porosities and over a large range of frequencies, provided that the wavelength is much greater than the crack dimensions.

Solution of flat crack problem by using variational principle and differential–integral equation

Variational principle is used to solve some flat crack problems in three-dimensional elasticity. In the formulation, the strain energy is evaluated by multiplying the crack opening displacement (COD) by the boundary traction. The boundary traction is related to the COD function by a differential–integral representation. By using an integration by part, the portion of the strain energy of the potential functional can be expressed by a repeated integral. In the integral all the integrated functions are non-singular. Letting the functional be minimum, the solution is obtained. In the actual solution, the COD function is represented by a shape function family in which several undetermined coefficients are involved. Using the variational principle, the coefficients are obtained. Several numerical examples are given with the stress intensity factors calculated along the crack border.

Assessment of multiple flat elliptical cracks with interactions

The failure of aging aircraft, pressure vessels and many welded structures is frequently due to elliptical cracks, which arise due to in service operation or the initial fabrication processes. Cracks have the potential to interact. Determination of the residual strength of structure requires an accurate knowledge of the stress intensity factor distribution around the crack front. When interacting cracks prevail, the lack of interaction effect in among multiple cracks may lead to an inadequate evaluation. A hybrid formulation capable of representing stiffness change is developed. This enables the accurate analysis of multiple load path structures containing multiple three dimensional cracks. This procedure complement the standard finite element alternating technique.

Fracture analysis of piezoelectric materials with flat ellipsoidal cracks

The objective of this study is to investigate a three-dimensional, infinitely extended, anisotropic piezoelectric solid containing a flat ellipsoidal crack with emphasis laced on when one of the principal axes of the crack becomes zero. Utilizing the equivalent inclusion method, the electroelastic fields around the crack are obtained explicitly by treating the elastic moduli and piezoelectric coefficients of the flat ellipsoidal inclusion as zero, and neglecting the permittivity. With resulting strain, stress, electric field, and electric displacement, the interaction energy between the crack and electromechanical loads is calculated. Based on the Griffith fracture criterion, the fracture stresses and critical electric displacement of the crack respectively subjected to a simple tension, an in-plane shear, an out-of-plane shear, and an electric displacement are obtained in closed forms by utilizing the obtained interaction energy. It is shown that the resulting fracture stresses can be reduced to those for uncoupled linear elastic fracture mechanics when piezoelectric coupling is absent and the material is isotropic. Numerical results for a three-dimensional PIC-151 piezoelectric solid with a flat ellipsoidal crack are given to demonstrate the application of the proposed formulation. Analysis results also indicate that the electroelastic coupling nature reduces the fracture loads and consequently impedes crack growth.

On the interpretation of different R-curves for soft PZT

From results of literature, it can be concluded that the R-curve of soft piezoelectric ceramics strongly depends on the type of test. In three-point bending tests and measurements with compact tension specimens a pronounced R-curve is found, whereas tests with opposite roller loading show a more flat crack resistance curve. Two possible reasons for this effect are differences in the domain switching state and bridging interactions in the crack wake. Both effects are discussed in terms of fracture mechanics. From the considerations it can be concluded that the dominating effect is the test-dependent change of the orientation state in the crack wake.

On wrinkled penny-shaped cracks

A nominally flat penny-shaped crack is subjected to a static loading. A perturbation method is developed for calculating the stress-intensity factors, based on an asymptotic analysis of the governing hypersingular boundary integral equation for the crack-opening displacement. Comparisons with known exact solutions for an inclined flat elliptical crack and for a crack in the shape of a spherical cap are made.

Numerical solution of three-dimensional crack problem by using hypersingular integral equation

In this paper, a flat crack with the normal boundary loading on the crack face is embedded in an infinite medium. The relevant problem is called the three-dimensional crack problem hereafter. The hypersingular integral equation is used to solve the mentioned problem. Two innovative points are suggested in this paper: (a) the involved hypersingular integrals are evaluated in a polar coordinate system, and are converted to the repeated integrals; (b) one of the repeated integral is hypersingular, and the involved integrated function is approximated by a product of the weight function and Chebyshev polynomials. The hypersingular integrals can be evaluated numerically by using a known result. Therefore, the hypersingular integral equation for the three-dimensional crack problem can be solved immediately. Finally, numerical examples are given to demonstrate the efficiency of the suggested approach.

Equilibrium of a Transversally Isotropic Body with an Elliptic Crack under Thermal Action

A nonuniform thermal effect on the surface of an elliptic crack in a transversally isotropic medium is studied. Use is made of the previous result on the analogy between elastic and thermoelastic problems for an arbitrary smooth flat crack in isotropic and transversally isotropic materials. A numerical analysis is performed to study how the stress intensity factors depend on the parameters and orientation of the crack