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 study we use fullfrequency biots theory of poroelastodynamics to model the. 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. Using electric boundary conditions controlled by the boundaries of an opening crack, the electric displacements at the crack surfaces are determined. 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. Application of ray theory to diffraction of elastic waves. 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. To extend the potential theory method to the crack problem of. Heat extraction from a hydraulically fractured pennyshaped. On the in uence of crack shape on e ective elasticity of.
Siam journal on applied mathematics society for industrial. Dynamic stress intensity factor mode i of a permeable penny. Fracture, mathematical problems of encyclopedia of mathematics. 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. Fundamental solutions of pennyshaped and halfinfinite plane. The indentation of a precompressed pennyshaped crack. Some axially symmetric stress distributions in elastic. Jan 01, 2014 the fracture behavior of a penny shaped crack in a constrained magnetoelectroelastic cylinder of finite radius under magnetoelectromechanical loads is investigated. Dynamic fracture analysis of a pennyshaped crack in a. The gurtinmurdoch continuum theory of elastic material surfaces is adopted, and the hankel integral transform is employed to solve this axisymmetric boundary value problem. Further results are presented for the direct problem of scattering of highfrequency waves by cracks in elastic solids. Results are presented for slits and penny shaped cracks.
Some axially symmetric stress distributions in an infinite elastic solid and in a thick plate containing penny shaped cracks are considered. In this article, a penny shaped crack in the isotropic plane of threedimensional transversely isotropic piezoelectric semiconductors is analyzed via the displacement discontinuity boundary. The pennyshaped crack problem for a finitely deformed. Pennyshaped crack in elastic medium with surface energy. Suppose that is a penny shaped crack, with radius so that the crack occupies the region where and are polar coordinates, and.
Therma crack shapl e the finitely deformed medium is assumed to contain a penny shaped crack with zaxi radiu iss s a. 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. On solutions of crack surface opening displacement of a penny shaped crack in an elastic cylinder subject to tensile loading. The potential theory method has been generalized in this paper to analyze the piezoelectric crackproblem. Pennyshaped cracks in threedimensional piezoelectric. Application of geometrical diffraction theory to qnde analysis. Curves of numerical results are presented for the stress intensity factor and the normal displacement.
All papers iowa state university digital repository. 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. A generation of special triangular boundary element shape. 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 first integral is over the surface of the material, and the second over its volume. Exact expressions for stress and electric displacement intensity factors are also presented. The superposed incremental state of stress corresponds to. 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. 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. Determination of effective elastic properties of microcracked rocks based on asymptotic approximation. 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. 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.
An approximate equivalence of the two ratios implies that, on average. 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. 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. Schapery civil and aerospace engineering departments this work was sponsored by the office of naval research department of the navy contract no. Electric and magnetic polarization saturations for a. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials. 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.
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. Results for a pennyshaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. Due to the fracture size, the relative velocity satis es darcys. 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. Threedimensional brittle shear fracturing by tensile. For simple crack geometries a hybrid method, whereby the crack opening displacement is computed by ray theory, and the scattered field is. Pdf elastic tstress solution for pennyshaped cracks under. By introducing amplitude ratios of relative fluid displacement and solid. Sneddon 1946 solved the problem of an infinitely thin crack. In this approach, special crack border elements with square. The planar crack is assumed to be a pennyshaped crack centered at the origin of the coordinate system with radius a. We will compare the theoretical predictions of the two models and the strengths and weaknesses of each.
A pennyshaped crack in a magneto electroelastic cylinder. I energy release rate for extension of a penny shaped crack with zero displacement on. Finally, we will apply our parallel wall fracture model to the data from tyngsboro. General solutions of a pennyshaped crack in a piezoelectric. The energy release rate of a pressurized crack in soft.
Sudden twisting of a pennyshaped crack journal of applied. For the case of a penny shaped crack situated in an infinite isotropic medium. Pennyshaped cracks in threedimensional piezoelectric semiconductors via greens functions of extended displacement discontinuity. Introduction to fracture mechanics david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029. The paraliel wall fracture model the theory for the parallel wall fracture model has been discussed.
Furthermore, the ps model has also been adopted to study some crack problems in ferro. 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 penny shaped shear crack with nonuniform stress distribution on it in an infinite solid has been researched. In this example, an embedded penny shaped crack under nonuniform loading is considered.
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. T1 a penny shaped crack in a layer whose upper and lower surfaces are fixed. 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. 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. An early attempt in the direction of elasticplastic fracture mechanics was irwins crack extension resistance curve, crack growth resistance curve or rcurve. A hankel transform development of our mixedboundary value problem yields two simultaneous pairs of dual integral equations. Consider a planar crack contained in an infinite space. Model verification to verify the numerical model, we compare its predictions with the available analytical solutions for the penny shaped crack problem. 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. Eight kinds of possible boundary conditions at infinity are considered. 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 paper, the transient response of a pennyshaped crack embedded in a.
The discontinuity in the elastostatic displacement vector. 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. Since the formation of a crack requires the creation of two surfaces, ws is given by 19 where. These results are compared with numerically computed exact results. The normal to th e crack surfaces which ar located at z 0. The geometry can be found in figure 9, and three mesh models can also be found in figure 10a. 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. 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. 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. Threedimensional linear elastic fracture mechanics. 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. The experimental result showed that a buried tensile crack penny. 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.
The governing integrodifferential equation takes the form. Extended displacement discontinuity method for analysis of. 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. In most applications cis a curved 3d domain, with one dimension signi cantly smaller than the dominant two. A complete closed form solution was obtained for a penny shaped crack in an elastic space, subjected to arbitrary internal tractions. Stress and displacement fields due to a pennyshaped shear.
Resolved article pdf available in international journal of fracture 1051. The nonaxisymmetric problem mode i of a permeable pennyshaped crack embedded in an in. As an example, consider an elastic space weakened by a flat crack of general shape, subjected to an arbitrary normal traction. The required cpu time for computing the crack opening displacements was 20,854 sec, and the number of iterations needed for. The extra strain gradient term is calibrated once only on the analytical solution for the penny. Natural frequencies of a pennyshaped crack with spring.
Stress intensity factor for steel hollow pipe with axial crack duration. The superposed displacement and temperatur fields ar ee also related followin in th conditioe g n of incompressibility. 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. 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. 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. 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. Threedimensional poroelastic simulation of hydraulic and. For the case of a penny shaped crack situated in an infinite isotropic medium with the crack faces subjected to arbitrary tractions, the integral equations are solved explicitly. On solutions of crack surface opening displacement of a. 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.
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. The crack is imbedded in a homogeneous medium and on the crack surface the spring boundary conditions are assumed. Deformation due to a pressurized horizontal circular crack in an. 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. On the other hand, more recently, the penny shaped crack in a magnetoelectroelastic material has been considered. Some axially symmetric stress distributions in elastic solids. 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. 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. Mode i energy release rate for extension of a penny shaped crack. Star shaped cracks obtained during directional drying of colloidal suspension in a circular capillary tube. Pennyshaped crack in a transversely isotropic solid. Fracture analysis of a pennyshaped magnetically dielectric. The allowance for the contact of the edges of a stationary. Niraula and wang 2006 derived an exact closedform solution for a penny shaped crack in a magneto.
Extended displacement discontinuity boundary integral. Here e 1 youngs elastic modulus for a continuum approximation. For a pennyshaped crack under axisymmetric loads, the opening crack pro. 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. N2 a vertical, planar pressurized crack is located in a layer with fixed upper and lower surfaces. 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.
Threedimensional static and dynamic stress intensity. 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 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. Results for a penny shaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. This paper considers the electroelastic problem of a threedimensional transversely isotropic piezoelectric material with a penny shaped dielectric crack perpendicular to the poling axis. The axial displacement of a disc inclusion embedded in a. 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. 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. In this study, a penny shaped crack hith a radius of embedded in an infinite elastic medium, as shohn in fig.
A theory of crack growth in viscoelastic media by r. Introduction to fracture mechanics david roylance department of materials science and engineering. The aging material properties are described by the boltzmann volterras linear theory for integral operators with nondifference kernels. The energy release rate is defined as the instantaneous loss of total potential energy per unit crack growth area. A dugdaletype estimation of the plastic zone for a penny. 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. The crack set cis surrounded by the poroelastic domain b nc, where b 0. Fluidsaturated pennyshaped crack in a poroelastic solid. Scaling of strength and lifetime probability distributions. For a circular or penny shaped crack of radius aloaded in mode i by a remote stress, k a. The crack surfaces are assumed to be magnetoelectrically permeable. The potential function theory and hankel transform method are used to obtain a system of.
Siam journal on applied mathematics siam society for. 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 energy release rate of the crack with respect to crack length rather than time, per unit length of crack perimeter, is g a. Quantitative evaluation of microfracture due to disbonding. Analysis of a dielectric crack in a magnetoelectroelastic layer. The stress intensity factors along the front of the penny shaped crack can be found in figure 19.
Application of ray theory to diffraction of elastic waves by. The crack with the radiusa is located in the upper halfspace x 3. Diffraction of elastic waves by a pennyshaped crack. A penny shaped crack is considered under the action of radial shear in a thick transversely isotropic elastic layer. 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. N0001468a03080003 task order nr 064520 technical report no. Natural frequencies of a penny shaped crack are calculated for the threedimensional elastic problem. 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. The discontinuity in the elastostatic displacement vector across a penny shaped crack under arbitrary loads created date. A new potential of a simple layer is introduced to account for the effect of the electric field. This is based on the method outlined in section 11. Fracture analysis of magnetoelectroelastic solid with a penny shaped crack by considering the effects of the opening crack interior. The singular solution is equivalent to that of the sudden appearance of a crack in a body under torsion.