Linear elastic theory predicts that the stress distribution near the crack tip, in polar coordinates with origin at the crack tip, has the form where is the stress intensity factor with units of stress length 12 and is a dimensionless quantity that varies with the load and geometry. Resolved article pdf available in international journal of fracture 1051. In this paper, a pennyshaped crack in an infinite elastic medium subjected to vertical pressure loading at the crack surface under the influence of surface stress is considered. A penny shaped crack is considered under the action of radial shear in a thick transversely isotropic elastic layer. Relations between different solutions of an interface crack in a neighborhood of the crack tip given by the open model, frictionless and frictional contact models of interface cracks are analyzed numerically for a penny shaped interface crack subjected to remote tension. The stress field around, and the displacement distribution, on a penny shaped shear crack with nonuniform stress distribution on it in an infinite solid has been researched. Quantitative evaluation of microfracture due to disbonding. Curves of numerical results are presented for the stress intensity factor and the normal displacement. Equating eq20 and eq21 solving for fracture stress gives 22 fig 4 a penny shaped circular crack embedded in a solid subjected to a remote tensile stress. Since the formation of a crack requires the creation of two surfaces, ws is given by 19 where. Heat extraction from a hydraulically fractured pennyshaped.
The energy release rate of the crack with respect to crack length rather than time, per unit length of crack perimeter, is g a. The discontinuity in the elastostatic displacement vector. An approximate equivalence of the two ratios implies that, on average. Using the extended displacement discontinuity boundary element method, penny shaped cracks in transversely isotropic plane of threedimensional piezoelectric semiconductors are studied, and the stress, electric displacement, and electric current intensity factors under uniform mechanicalelectriccurrent loads applied on the penny shaped crack. In this paper, the transient response of a pennyshaped crack embedded in a. Finally, we will apply our parallel wall fracture model to the data from tyngsboro. The energy release rate is defined as the instantaneous loss of total potential energy per unit crack growth area. The aging material properties are described by the boltzmann volterras linear theory for integral operators with nondifference kernels. The axial displacement of a disc inclusion embedded in a. The extended displacement discontinuity boundary integral equation eddbie and boundary element method is developed for the analysis of planar cracks of arbitrary shape in the isotropic plane of threedimensional 3d transversely isotropic thermo. Model verification to verify the numerical model, we compare its predictions with the available analytical solutions for the penny shaped crack problem. Threedimensional static and dynamic stress intensity.
Analysis of mode i conducting crack in piezoelectromagneto. Sudden twisting of a pennyshaped crack journal of applied. Furthermore, the ps model has also been adopted to study some crack problems in ferro. All papers iowa state university digital repository.
Here e 1 youngs elastic modulus for a continuum approximation. Extended displacement discontinuity boundary integral. The governing integrodifferential equation takes the form. Therma crack shapl e the finitely deformed medium is assumed to contain a penny shaped crack with zaxi radiu iss s a. Exact expressions for stress and electric displacement intensity factors are also presented.
The superposed displacement and temperatur fields ar ee also related followin in th conditioe g n of incompressibility. A pennyshaped crack in a magneto electroelastic cylinder. Condition for rupture we now consider the problem relating to a finitely deformed incompressible elastic medium containing a penny shaped crack, the surfaces of which are free from surface traction. Some axially symmetric stress distributions in an infinite elastic solid and in a thick plate containing penny shaped cracks are considered. For simple crack geometries a hybrid method, whereby the crack opening displacement is computed by ray theory, and the scattered field is. Sif for a penny shaped crack in a finiteradius cylinder submodel method this is a simple threedimensional crack problem in finite domain, a penny shaped crack in a finiteradius cylinder subjected to remote uniform tension. Siam journal on applied mathematics society for industrial. Particular attention is devoted to a method by which the crack opening displacement is computed on the basis of ray theory, and the scattered field is subsequently obtained by the use of a representation integral. The present paper examines the axisymmetric problem of the axial translation of a rigid circular disc inclusion of finite thickness which is wedged in smooth contact in a penny shaped crack. N2 a vertical, planar pressurized crack is located in a layer with fixed upper and lower surfaces. Jan 01, 2014 the fracture behavior of a penny shaped crack in a constrained magnetoelectroelastic cylinder of finite radius under magnetoelectromechanical loads is investigated. Fracture analysis of a pennyshaped magnetically dielectric.
The first integral is over the surface of the material, and the second over its volume. As regards threedimensional 3d crack problems, making use of the displacement discontinuity boundary integral equation method, zhao et al 6 investigated a penny shaped crack in 3d piezoelectric media and determined the electric yielding size by the ps model. Fracture analysis of magnetoelectroelastic solid with a penny shaped crack by considering the effects of the opening crack interior. A transient stress analysis for the problem of a torque applied suddenly to the surface of a penny shaped crack in an infinite elastic body is carried out. A hankel transform development of our mixedboundary value problem yields two simultaneous pairs of dual integral equations. Pennyshaped cracks in threedimensional piezoelectric semiconductors via greens functions of extended displacement discontinuity. In this example, an embedded penny shaped crack under nonuniform loading is considered. Diffraction of elastic waves by a pennyshaped crack. Mode i energy release rate for extension of a penny shaped crack. Deformation due to a pressurized horizontal circular crack in an. Results for the axial stiffness of the embedded inclusion and the stress intensity factor at the boundary of the penny shaped crack are evaluated in exact closed form. We will compare the theoretical predictions of the two models and the strengths and weaknesses of each.
Pennyshaped cracks in threedimensional piezoelectric. Stress intensity factor for steel hollow pipe with axial crack duration. The potential function theory and hankel transform method are used to obtain a system of. Determination of effective elastic properties of microcracked rocks based on asymptotic approximation. Stress and displacement fields due to a pennyshaped shear. On the other hand, more recently, the penny shaped crack in a magnetoelectroelastic material has been considered. General solutions of a pennyshaped crack in a piezoelectric. Extended displacement discontinuity method for analysis of. To extend the potential theory method to the crack problem of. For a pennyshaped crack under axisymmetric loads, the opening crack pro.
The pennyshaped crack problem for a finitely deformed. Siam journal on applied mathematics siam society for. The effect of a penny shaped crack on the deformation of an infinite piezoelectric material of the hexagonal crystal class 6 mm subjected to mode i electrical and mechanical loading has been studied using the theory of linear piezoelectricity and applying appropriate boundary conditions. On the in uence of crack shape on e ective elasticity of. The stress intensity factors along the front of the penny shaped crack can be found in figure 19. As an example, consider an elastic space weakened by a flat crack of general shape, subjected to an arbitrary normal traction. The gurtinmurdoch continuum theory of elastic material surfaces is adopted, and the hankel integral transform is employed to solve this axisymmetric boundary value problem. A generation of special triangular boundary element shape.
Fundamental solutions of pennyshaped and halfinfinite plane. The discontinuity in the elastostatic displacement vector across a penny shaped crack under arbitrary loads created date. Some axially symmetric stress distributions in elastic solids. Schapery civil and aerospace engineering departments this work was sponsored by the office of naval research department of the navy contract no. Dynamic fracture analysis of a pennyshaped crack in a. Due to the fracture size, the relative velocity satis es darcys. The crack surfaces are assumed to be magnetoelectrically permeable. On solutions of crack surface opening displacement of a penny shaped crack in an elastic cylinder subject to tensile loading. In this study, a penny shaped crack hith a radius of embedded in an infinite elastic medium, as shohn in fig. It is shown that, by use of a representation for the displacement in an infinite elastic solid containing a single crack, representations for the displacements in an infinite solid containing two or more cracks and in a thick plate containing a single.
Surface motions due to a disbonding of a stainless overlay welded on base metal of a pressure vessel have been measured by the use of a commercially available flat. A theory of crack growth in viscoelastic media by r. Natural frequencies of a pennyshaped crack with spring. For a circular or penny shaped crack of radius aloaded in mode i by a remote stress, k a. By introducing amplitude ratios of relative fluid displacement and solid. This paper considers the electroelastic problem of a threedimensional transversely isotropic piezoelectric material with a penny shaped dielectric crack perpendicular to the poling axis. Today, the displacement v at the crack mouth is measured, and the ctod is inferred by assuming the specimen halves are rigid and rotate about a hinge point the crack tip. In this article, a pennyshaped crack in the isotropic plane of threedimensional transversely isotropic piezoelectric semiconductors is analyzed via the displacement discontinuity boundary element method. Fluidsaturated pennyshaped crack in a poroelastic solid. A dugdaletype estimation of the plastic zone for a penny.
Results for a pennyshaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. The stress intensity factor, is used in fracture mechanics to predict the stress state stress intensity near the tip of a crack or notch caused by a remote load or residual stresses. Application of geometrical diffraction theory to qnde analysis. Threedimensional linear elastic fracture mechanics. It is shown that, by use of a representation for the displacement in an infinite elastic solid containing a single crack, representations for the displacements in an infinite solid containing two or more cracks and in a thick plate. An analytical tool using matlab has been developed for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity. Star shaped cracks obtained during directional drying of colloidal suspension in a circular capillary tube. Using the extended displacement discontinuity boundary element method, pennyshaped cracks in transversely isotropic plane of threedimensional piezoelectric semiconductors are studied, and the stress, electric displacement, and electric current intensity factors under uniform mechanicalelectriccurrent loads applied on the pennyshaped crack surface are calculated. N0001468a03080003 task order nr 064520 technical report no. Some axially symmetric stress distributions in elastic. Studying cracks in pscs is beneficial for the design and performance of smart devices, and is important from the perspective of the theory of fracture mechanics for multiplefields. The diffraction of timeharmonic stress waves by a penny shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials. The indentation of a precompressed pennyshaped crack.
The potential theory method has been generalized in this paper to analyze the piezoelectric crackproblem. The geometry can be found in figure 9, and three mesh models can also be found in figure 10a. Application of ray theory to diffraction of elastic waves. The crack closure effect for a penny shaped crack bridged by an arbitrarily located single fibre the crack closure effect for a penny shaped crack bridged by an arbitrarily located single fibre fischer, f. However, the integral transform method and the potential theory method are usually limited to some simple cases, such as the case of the penny shaped crack or uniform loadings. The superposed incremental state of stress corresponds to. In this approach, special crack border elements with square. Using electric boundary conditions controlled by the boundaries of an opening crack, the electric displacements at the crack surfaces are determined. The penny shaped crack with heat flux is investigated for the case in which the heat flux is into the material with the lower distortivity. A new potential of a simple layer is introduced to account for the effect of the electric field. The modified leonov panasyuk dugdales crack model is used with a constant process zone assuming that the critical opening displacement is the fracture criterion.
In this paper, the extended displacement discontinuity edd boundary element method is developed to analyze a penny shaped crack in the isotropic plane of a threedimensional 3d transversely isotropic thermal piezoelectric semiconductor psc. Eight kinds of possible boundary conditions at infinity are considered. The nonaxisymmetric problem mode i of a permeable pennyshaped crack embedded in an in. Introduction to fracture mechanics david roylance department of materials science and engineering. Accurate and fast evaluation of the stress intensity factor for planar cracks shows the proposed procedure is robust for sif calculation and crack propagation purposes. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of. For simple crack geometries a hybrid method, whereby the crackopening displacement is computed by ray theory, and the scattered field is subsequently obtained by the use of a representation theorem, is tested by comparison with exact results. T1 a penny shaped crack in a layer whose upper and lower surfaces are fixed. The required cpu time for computing the crack opening displacements was 20,854 sec, and the number of iterations needed for.
Natural frequencies of a penny shaped crack are calculated for the threedimensional elastic problem. Making use of the displacement discontinuity boundary integral equation method ddbiem, the dimension of the plastic zone at the tip of a pennyshaped crack in a threedimensional elastic medium. We quantify this effect by studying the inflation of a penny shaped crack in an infinite elastic body with applied pressure. The stress field around, and the displacement distribution, on a pennyshaped shear crack with nonuniform stress distribution on it in an infinite solid has been researched. The experimental result showed that a buried tensile crack penny.
Consider a planar crack contained in an infinite space. The planar crack is assumed to be a pennyshaped crack centered at the origin of the coordinate system with radius a. For the case of a penny shaped crack situated in an infinite isotropic medium. The allowance for the contact of the edges of a stationary.
To avoid numerical difficulty caused by singular fields near the crack tip, we derived an expression for the energy release rate which depends on the applied pressure, the surface tension, the inflated crack volume and. As a typicalexample, a closedform solution is first obtained for a penny shaped crack subjected to a pair ofconcentrated forces acting in opposite directions and a pair of point charges on crack surfaces. In particular integral formulae are obtained for the stresses on the plane of the crack beyond the cracktip, and hence for the stress intensity factors. The crack with the radiusa is located in the upper halfspace x 3. I energy release rate for extension of a penny shaped crack with zero displacement on. The paraliel wall fracture model the theory for the parallel wall fracture model has been discussed. The direct problem of the diffraction of timeharmonicwaves by cracks in elastic solids is analyzed for highfrequencies, when the wavelengths are of the same order of magnitude as a characteristic length dimension, a, of the crack. The energy release rate of a pressurized crack in soft. In damage tolerance design and optimization of engineering structures, penny shaped or elliptical cracks are the commonly assumed crack configurations.
Regardless the fracture shape, we nd these ratios to be su ciently close to that of a penny shaped crack imbedded in the same background material. This is based on the method outlined in section 11. Motivated by the current situation, we develop a method of studying arbitrarily shaped planar cracks in the isotropic plane of 3d transversely isotropic tmee media. The crack is imbedded in a homogeneous medium and on the crack surface the spring boundary conditions are assumed. Pennyshaped crack in elastic medium with surface energy. Electric and magnetic polarization saturations for a. On solutions of crack surface opening displacement of a. Threedimensional brittle shear fracturing by tensile. Scaling of strength and lifetime probability distributions.
Threedimensional poroelastic simulation of hydraulic and. Pennyshaped crack in a transversely isotropic solid. Fracture, mathematical problems of encyclopedia of mathematics. The somigliana formula is used to reduce an arbitrary elastic crack problem to a system of three integral equations for the components of displacement discontinuity. Suppose that is a penny shaped crack, with radius so that the crack occupies the region where and are polar coordinates, and. The extra strain gradient term is calibrated once only on the analytical solution for the penny.
To calculate the elastic field around a crack in 3d we assume that the cracks are ellipsoidal voids, and we employ the eshelby 10,11,22 solution for a penny shaped void. These results are compared with numerically computed exact results. Results are presented for slits and penny shaped cracks. Now, let be a simply connected domain in the plane defined as whose boundary has the polar equation, where is bounded and piecewise continuous and is a small positive parameter. Analysis of a dielectric crack in a magnetoelectroelastic layer. A closed form fundamental solution is then obtained for a penny shaped crack subjected to pointforces and point charges symmetrically applied on its upper and lower surfaces. Sneddon 1946 solved the problem of an infinitely thin crack.
The crack set cis surrounded by the poroelastic domain b nc, where b 0. Results for a penny shaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. The threedimensional contact problem for the stationary plane penny shaped crack under arbitrary incident harmonic tensioncompression wave was solved by the method of boundary integral equations with allowance for the crack s edges contact interaction. Niraula and wang 2006 derived an exact closedform solution for a penny shaped crack in a magneto. A complete closed form solution was obtained for a penny shaped crack in an elastic space, subjected to arbitrary internal tractions. Pdf elastic tstress solution for pennyshaped cracks under. An analytical solution is given for the displacement and stress distribution produced in the interior of a transversely isotropie solid containing a pennyshaped crack situated in an elastic symmetry plane and axiallyloaded. Letting the size of the crack approach zero, we obtain greens functions or fundamental solutions corresponding to unit point edds. Employing the dugdale hypothesis and hankel transform theory, the problem of determining the size of the plastic zone is reduced to the numerical solution of a fredholm integral equation of the second kind. An early attempt in the direction of elasticplastic fracture mechanics was irwins crack extension resistance curve, crack growth resistance curve or rcurve. Introduction to fracture mechanics david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029. Dynamic stress intensity factor mode i of a permeable penny. Application of ray theory to diffraction of elastic waves by.
In most applications cis a curved 3d domain, with one dimension signi cantly smaller than the dominant two. Further results are presented for the direct problem of scattering of highfrequency waves by cracks in elastic solids. The normal to th e crack surfaces which ar located at z 0. The singular solution is equivalent to that of the sudden appearance of a crack in a body under torsion. Suppose that is a penny shaped crack, with radius so that the crack occupies the region where and are polar coordinates, and now, let be a simply connected domain in the plane defined as whose boundary has the polar equation, where is bounded and piecewise continuous and is a small positive parameter. In this article, a penny shaped crack in the isotropic plane of threedimensional transversely isotropic piezoelectric semiconductors is analyzed via the displacement discontinuity boundary. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials.
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