Mode i energy release rate for extension of a penny shaped crack. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials. N2 a threedimensional penny shaped crack under combined tensile and shear loadings is analyzed. 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 distribution of the normal and tangential components of the contact forces and. Mixedmode fatigue crack propagation of pennyshaped cracks. Moreover, antipov and mkhitaryan 26 studied the plane problem of interaction between a thin rigid inclusion and a. A limiting values of these expressions at the crack plane together with the boundary conditions lead to abeltype integral equations, which admit a closed form solution. It is found that, if a displacement jump crack opening displacement cod takes the form of a 2. The pennyshaped interface crack in a uniform tension field was treated by keer et al. A harmonic potential function representation is used to reduce the problem to a boundary value problem which is solved by an integral equation method.
The surface of the halfspace is assumed to be stressfree. Furthermore, the ps model has also been adopted to study some crack problems in ferro. Abstract closedform solutions are obtained for a pennyshaped crack in a. In view of the symmetry with respect to the cracked plane, this crack problem is formulated by a mixed boundary value problem. The geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks. Threedimensional 3d elastodynamic interaction between a penny shaped crack and a thin elastic interlayer joining two elastic halfspaces is investigated by an improved boundary integral equation method or boundary element method.
The new integration method is based on the continuation approach. Martin i98i has shown how such a green function can be constructed for the penny shaped crack and has derived a fredholm integral equation of the second kind that uniquely determines the c. The interface pennyshaped crack reconsidered 771 the method used is the following. Abstract the assumptions of impermeable and permeable cracks give rise to significant errors in determining electroelastic behavior of a cracked piezoelectric material. Axisymmetric problems of a pennyshaped crack at the interface of a. Effects of electric field on crack growth for a penny. Here, the boundary integrodifferential equations are applied to the numerical calculation of the crack opening displacement of a penny shaped crack in an infinite linear viscoelastic body. The allowance for the contact of the edges of a stationary. In the study, a crack is modeled as distribution of displacement discontinuities, and the governing equation is formulated as singularityreduced integral equations.
The accuracy of the stress intensity factor calculation is satisfactorily examined for rectangular, penny shaped and elliptical planar cracks. Dimensionless axial displacement distribution for solid cylindrical bar with penny shaped crack. A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a pennyshaped crack where displacements are given. A pennyshaped crack in a magneto electroelastic cylinder. The crack tip stress field is obtained by considering the asymptotic behavior of bessel function. Dynamic fracture analysis of a pennyshaped crack in a.
For transversely isotropic media, tsai 1983a, 1983b studied the thermal stress in a transversely isotropic medium containing a penny shaped crack and extended his works tsai 1998, 2000 to the case of a flat toroidal crack using techniques that involve triple integral equations and multiplying factors. The problem was solved by boundary integral equations method using iterative algorithm. Sets of lines is direction of cylindrical coordinates. 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. Deformation due to a pressurized horizontal circular crack. Problem of the plane penny shaped crack edges contact interaction in threedimensional space under action of a normal harmonic tensioncompression wave has been considered. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack opening displacement. Scattering from an elliptic crack by an integral equation. Consider an infinite elastic solid containing a penny shaped crack. Local stress field for torsion of a pennyshaped crack in.
The elastodynamic scattering by a penny shaped crack with spring boundary conditions is investigated. The method of solution is an extension of one recently developed by the writer 1 and involves setting up and solving an integral equation for the radon transform of the relative displacement of the crack faces. We compute the crack opening displacement subject to a plane wave of normal incidence. Journal of mechanics of materials and structures 2. Since many rock types show brittle elastic behaviour under hydrocarbon reservoir. Fundamental solutions of pennyshaped and halfinfinite plane. Thus, the above formula must hold pointwise, and the equation for equilibrium is. The superposed displacement and temperatur fields ar ee also related followin in th conditioe g n of incompressibility. Investigation of crack edges contact interaction in three.
The crack s can be divided into n annular elements. Displacement and stress distributions are calculated in finite circular bars, each containing a penny shaped crack and loaded normal to the crack surface. The equilibrium equation above is valid for any arbitrary volume and thus must hold in the limit that the volume is vanishingly small. The normal to th e crack surfaces which ar located at z 0.
Application of displacement and traction boundary integral equations for fracture mechanics analysis. An analytical solution for the axisymmetric problem of a. The rock mass is assumed to be infinitely extended, homogeneous, and isotropic. On the upper and lower surfaces of a pennyshaped crack s, the axisymmetric distributed extended loadings are applied as 50 p r. The geometry can be found in figure 9, and three mesh models can also be found in figure 10a. Based on the mixed mode dugdale model and the accumulated plastic displacement criterion for crack growth, a fatigue crack growth equation with fourpower effective stress intensity factor dependence is developed for a penny shaped crack under conditions of mixed mode loading and smallscale yielding. Stress intensity factor determination plays a central role in linearly elastic fracture mechanics lefm problems. Hankel transform is used to reduce the problem to solving a fredholm integral equation. The stress intensity factor at the tip of a pennyshaped crack of radius in an infinite domain under uniaxial tension is. Lecture notes elasticity of microscopic structures.
Elastodynamic problem for a layered composite with penny. Extended displacement discontinuity method for nonlinear. Pennyshaped cracks in threedimensional piezoelectric. Abstract the elastodynamic scattering by a penny shaped crack with spring boundary conditions is investigated. The scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method. Scattering by two pennyshaped cracks with spring boundary.
Based on the theory of elasticity, previous analytical solutions concerning a penny shaped interface crack employ the derivative of the crack surface opening displacements as the primary unknowns, thus leading to singular integral equations with cauchytype singularity. Application of ray theory to diffraction of elastic waves. A nominally flat pennyshaped crack is subjected to a static loading. The first integral is over the surface of the material, and the second over its volume. A hypersingular integral equation or a differentialintegral equation is used to solve the pennyshaped crack problem. Thus, the analysis of stresses near the crack tip constitutes an essential part of fracture mechanics. The stress intensity factor is evaluated directly based on displacement discontinuities dd. The formulation of a new displacement discontinuity element the enhanced displacement discontinuity edd element was the second major undertaking of the thesis. Selvadurai 3 performed analysis of the problem pertinent to the complete indentation of a single face of a penny shaped crack by a rigid smooth inclusion. One penny shaped crack to begin with we consider an infinite space which contains one penny shaped crack having the radius of a 0 and the unit normal vector of. The integral representation for the displacement field is given, and an integral equation, which relates crack opening and applied pressure, is derived. 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. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Diffraction of elastic waves by a pennyshaped crack.
Reducing a further such that ba 1 makes a the short dimension of the crack and thus the limiting dimension for crack opening and sif value. Heat extraction from a hydraulically fractured pennyshaped. In this paper, the transient response of a pennyshaped crack embedded in a. 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. Letting the size of the crack approach zero, we obtain greens functions or fundamental solutions corresponding to. The stress intensity factor is evaluated directly based on displacement discontinuities dd using a threedimensional displacement discontinuity, boundary element method based on the equations. According to his results, a simpletensile load would lead to the singularities both in stress and electric displacement at the crack tipas.
T1 mixedmode fatigue crack propagation of penny shaped cracks. The pennyshaped crack on an interface the quarterly. These results are compared with numerically computed exact results. Numerical method for linear analysis of pennyshaped cracks. The stress intensity factors along the front of the penny shaped crack can be found in figure 19. Consider a horizontal penny shaped crack with radius r and depth h in an elastic halfspace. 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. T1 a penny shaped crack in a layer whose upper and lower surfaces are fixed. 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.
Based on the results of these numerical calculations, several conclusions can be made, as follows. Exact expressions for stress and electric displacement intensity factors are also presented. Because this stress field is asymptotic dominant or singular, it is characterized by the stress intensity factor sif. The penny shaped crack is embedded in one of the halfspaces, perpendicular to the interlayer and subjected to a timeharmonic tensile loading on its surfaces. Axial translation of a rigid disc inclusion embedded in a. On solutions of crack surface opening displacement of a penny shaped crack in an elastic cylinder subject to tensile loading.
Stress intensity factor determination for threedimensional crack. It is supposed that the pennyshaped crack is subjected to a pair of normal concentrated forcesp applied in opposite directions at the points. 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. Penny shaped crack in an infinite medium subjected to tension 104. By virtue of the solution to the abel type integral. The indentation of a precompressed pennyshaped crack. 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. A pennyshaped crack has more restricted opening, and has the ratio of 0.
We suppose that timeharmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i. 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. Boundary integral equations in elastodynamics of interface. Dual integral equations involving trigonometric functions are derived and. Asymmetric loading of an externally cracked elastic solid. The stress intensity factor for a pennyshaped crack in an elastic. N2 a vertical, planar pressurized crack is located in a layer with fixed upper and lower surfaces. Briefly, the solution of the displacement equations of equilibrium, for a medium free of body forces, can be represented in. Hypersingular integral equations for the solution of penny. Martin, orthogonal polynomial solutions for pressurized elliptical cracks, quart. Extended displacement discontinuity boundary integral. Therma crack shapl e the finitely deformed medium is assumed to contain a penny shaped crack with zaxi radiu iss s a. Siam journal on applied mathematics siam society for.
Penny shaped crack in an infinite solid the figure shows a circular crack with. 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. For the case of a point force, exact expressions for the fullspace electro. The transition t matrix of the crack is determined and the usefulness of this is illustrated by considering also the scattering by two cracks. Martin, the discontinuity in the elastostatic displacement vector across a penny shaped crack under arbitrary loads, j.
Heat extraction from a penny shaped crack having both inlet and outlet holes is investigated analytically by considering the hydraulic and thermal growth of the crack when fluid is injected at a constant flow rate. The axial displacement of a disc inclusion embedded in a. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an. This new formulation provides information on the inplane confinement. Suppose that is a penny shaped crack, with radius so that the crack occupies the region where and are polar coordinates, and. Stress intensity factor determination for threedimensional. We wish to solve eqn 40, subject to the relation given by eqn 50. Fracture propagation is controlled by the stress field near the crack tip. 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. Axisymmetric displacement boundary value problem for a. Some particular solutions for pennyshaped crack problem by. A hankel transform development of our mixedboundary value problem yields two simultaneous pairs of dual integral equations. Suppose the planar crack is a penny shaped crack centered at the origin of the coordinate system with radius a.
The displacement and stress near the crack tip can be characterized by three. The solution for the penny shaped ligament was given perhaps most accurately by nisitani and noda 8 see murakamis handbook 9, p. With the proposed method several example problems, such as a penny shaped crack, an elliptical crack in an infinite solid and a semielliptical surface crack in an elbow are solved. Application of displacement and traction boundary integral. 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.
Recently, huang 1997 obtained explicit expressions ofstress and electric displacement intensity factors for a pennyshaped crack embedded in an infinitetransversely isotropic piezoelectric solid subjected to three loading conditions, respectively impermeable conditions on crack surfaces were employed. Fabrikant department of mechanical engineering, concordia university, montreal, canada h3g 1m8 received 30 october 1986 and accepted 12 january 1987j abstract closedform solutions are obtained for a penny shaped crack in a transversely. We first present the numerical method of linear analysis. The torsion of a penny shaped crack in a functionally graded strip is considered. By considering the electric field in the crack cavity, the polarization saturation ps model of a penny shaped crack under electrically semipermeable boundary condition in a threedimensional piezoelectric medium is studied via both the extended displacement discontinuity boundary integral equation method and the boundary element method. On solutions of crack surface opening displacement of a. The assumptions of dugdale are applied to estimate the effects of plasticity around the edge of the crack. Fluid flow in a penny swed crack consider a large penny shaped crack having a radius r and width w in the zdirection as shown in fig. The equations for fluid flow are derived and solved to determine the flow pattern in the crack. The stress state and effective elastic moduli of an isotropic solid containing equally oriented penny shaped cracks are evaluated accurately. Read a penny shaped crack in a magneto electroelastic cylinder surrounded by a rigid body and subjected to magnetoelectro mechanical loads, international journal of applied electromagnetics and mechanics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Youn, seungwon, application of displacement and traction boundary integral equations for fracture mechanics analysis 1993. Fluid is injected from the inlet at the center of the crack and removed in part at the outlet, x a, where x is the distance measured in the vertical direction from the center. The penny shaped crack surface is subjected to uniform coupled loadings the solutions and the intensity factors for the isotropic thermoelastic material are given by kassir and sih 1967, 1977.
In this example, an embedded penny shaped crack under nonuniform loading is considered. The somigliana formula is used to reduce an arbitrary elastic crack problem to a system of three integral equations for the components of displacement disc. A formula is derived for the stress intensity factor at the rim of a pennyshaped crack in. The elastic stress field around a crack tip 3 brittle fracture in a solid in the form of crack growth is governed by the stress. Similar results are presented for an annular plate containing internal, tractionfree surface cracks. In this paper, we describe a new method for solving the corresponding linear boundaryvalue problem for u i, which we denote by s. The assumptions of impermeable and permeable cracks give rise to significant errors in determining electroelastic behavior of a cracked piezoelectric material. Results are presented for slits and penny shaped cracks. An expression for the surface displacement of the crack is also given.
By means of the hankel transform and dualintegral equations, the nonlinear response of a pennyshaped dielectric crack with a permittivity kappa0 in a transversely isotropic piezoelectric. The interaction between the disc inclusion and the external crack attention will be focused on the problem of a penny shaped rigid inclusion of radius a which is. For a penny shaped crack subjected to a normal loading, the shear tractions vanish in the plane zo, and eqn. Electric and magnetic polarization saturations for a. Penny shaped crack in an infinite medium subjected to tension. A perturbation method is developed for calculating the stressintensity factors, based on an asymptotic analysis of the governing hypersingular boundary integral equation for the crack opening displacement. If the crack is first loaded in tension and then subjected to a heat flux, it seems reasonable to anticipate that. Numerical results for a pennyshaped interface crack in al2o3pmma. 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. Investigated are the effects of material property parameters and geometry criterion on the stress intensity factor. The crack opening displacement cod is then described by the field. Stress intensity factor and effective stiffness of a solid.
Comninoutype contactzone problem to a singular integral equation to. Stress and displacement fields due to a pennyshaped shear. Additionally are referred to as the edds the planar crack is assumed to be a penny shaped crack centered at the origin of the coordinate system with radius a. Nonlinear solution of the ps model for a semipermeable crack. The energy release rate is defined as the instantaneous loss of total potential energy per unit crack growth area. A generation of special triangular boundary element shape. Asymmetric loading of an externally cracked elastic solid 255 3. The naviercauchy equations of elastic equilibrium are reduced to three sets. Stress intensity factors for cracks in anisotropic. Let qf ff 1 2n and use the gaussian quadrature formula for chebyshev.
1243 615 478 148 1616 1294 938 1479 83 1195 89 653 210 1075 332 463 913 32 1168 1017 1464 439 1475 997 280 1578 1104 1452 1451 1048 468 1259 1297 86 1104