A meshaindependent method for planar threeadimensional crack. 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. The interface pennyshaped crack reconsidered 771 the method used is the following. A harmonic potential function representation is used to reduce the problem to a boundary value problem which is solved by an integral equation method. N2 a vertical, planar pressurized crack is located in a layer with fixed upper and lower surfaces. 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 thermomagnetoelectroelastic tmee media. The axial displacement of a disc inclusion embedded in a. The formulation of a new displacement discontinuity element the enhanced displacement discontinuity edd element was the second major undertaking of the thesis. Martin, the discontinuity in the elastostatic displacement vector across a penny shaped crack under arbitrary loads, j. The case of a pennyshaped crack with arbitrary normal displacements. 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. References to related problems are also given by kassir and sih 1975, atkinson 1979 and sih and chen 1981. Solution of a flat elliptical crack in an electrostrictive. The elastodynamic scattering by a penny shaped crack with spring boundary conditions is investigated.
An analytical solution for the axisymmetric problem of a. A stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted element. Results are presented for slits and pennyshaped cracks. Consider an infinite elastic solid containing a pennyshaped crack. Stress and displacement fields due to a pennyshaped shear. The continuity of the contour lines verifies that the proper displacement values are. As a special case, the analytical solution for a pennyshaped crack under uniform combined loadings is presented. First, the fundamental green function, point force solution, is taken from achenbach et al. Threedimensional 3d elastodynamic interaction between a pennyshaped crack and a thin elastic interlayer joining two elastic halfspaces is investigated by an improved boundary integral equation method or boundary element method. This is based on the method outlined in section 11. T stress analysis for a griffith crack in magnetoelectroelastic solid. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane.
Using the extended displacement discontinuity boundary element method, penny shaped cracks in transversely isotropic plane of threedimensional piezoelectric semiconductors are studied, and the. If the material is linearlyelastic, the computation of its energy release rate can be much simplified. The pennyshaped crack is embedded in one of the halfspaces, perpendicular to the interlayer and subjected to a timeharmonic tensile loading on its surfaces. When the elliptical crack is degenerated to a penny shaped crack subjected to uniform thermal loadings, the solutions obtained in the present paper are reduced to that given by yang et al for a transversely isotropic medium.
This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic halfspace or full space. Stress intensity factor determination for threedimensional. 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. Williams asymptotic method 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. A closed form fundamental solution is then obtained for a pennyshaped crack subjected to pointforces and point charges symmetrically applied on its upper and lower surfaces. W the crack opening displacement approach to fracture. Natural frequencies of a penny shaped crack are calculated for the threedimensional elastic problem. The pennyshaped crack with heat flux is investigated for the case in which the heat flux is into the material with the lower distortivity. A generation of special triangular boundary element shape. The dugdale plastic zone ahead of a pennyshaped crack in a. 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 present paper examines the problem related to a penny shaped crack which is located at the bonded plane. Electric and magnetic polarization saturations for a. Natural frequencies of a pennyshaped crack with spring.
The stress intensity factors for a periodic array of. Threedimensional static and dynamic stress intensity factor. In particular, the mode i problem of a penny shaped crack in a homogeneous isotropic cylinder under remote tension loading is used as a standard test case. In this approach, special crack border elements with square. The pennyshaped crack at a bonded plane with localized. 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. Since many rock types show brittle elastic behaviour under hydrocarbon reservoir. The stress intensity factor is evaluated directly based on displacement discontinuities. As regards threedimensional 3d crack problems, making use of the displacement discontinuity boundary integral equation method, zhao et al 6 investigated a pennyshaped crack in 3d piezoelectric media and determined the electric yielding size by the ps model. Martin, orthogonal polynomial solutions for pressurized elliptical cracks, quart. Dugdale plastic zone of a pennyshaped crack in a piezoelectric. Utilizing the boundary integral equation method, the singularities of nearcrack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the edd. 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. Axisymmetric displacement boundary value problem for a.
For solutions of crack problems, the dual boundary element method dbem, which is originated from the bem, is a more general and efficient method than other extension methods 6, 1015 in the range of bems, such as the multidomain bem, the symmetric galerkin boundary element method 17, 18, and the displacement discontinuity method. Stress intensity factor determination plays a central role in linearly elastic fracture mechanics lefm problems. The solution for the penny shaped ligament was given perhaps most accurately by nisitani and noda 8 see murakamis handbook 9, p. Results for a pennyshaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data.
Extended displacement discontinuity boundary integral. In this paper, we describe a new method for solving the corresponding linear boundaryvalue problem for u i, which we denote by s. 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. 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. Thermal in an solid an insulated university of michigan. 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. In the study, a crack is modeled as distribution of displacement discontinuities, and the governing equation is formulated as singularityreduced integral equations. For a penny shaped crack subjected to a normal loading, the shear tractions vanish in the plane zo, and eqn. 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. In this paper, values of stress intensity factors sifare obtained for a pennyshaped crack, subjected to either harmonic or impact loads.
The transition t matrix of the crack is determined and the usefulness of this is illustrated by considering also the scattering by two cracks. The method of solution is an extension of one recently developed by the writer 1 and involves setting up and solving an. Piezoelectric semiconductor, greens function, extended displacement discontinuity boundary element method, penny shaped crack, extended stress intensity. Dimensionless axial displacement distribution for solid cylindrical bar with penny shaped crack. Scattering by a horizontal subsurface pennyshaped crack. Mode i energy release rate for extension of a penny shaped crack. Segedin, note on a penny shaped crack under shear, mathematical proceedings of the cambridge philosophical society, vol. Results are presented for slits and penny shaped cracks. The t matrix of a single crack is first determined by a direct integral equation method which gives the crack opening displacement and the integral. The allowance for the contact of the edges of a stationary.
All papers iowa state university digital repository. Barber department of mechanical engineering, university of newcastle upon tyne a solution is given for the steadystate thermal stress and displacement field in an infinite elastic solid containing an insulated penny shaped crack. In fracture mechanics, pennyshaped cracks are often used to model the microstruc. In particular, the mode i problem of a penny shaped crack in a homogeneous isotropic. Siam journal on applied mathematics siam society for. Investigation of crack edges contact interaction in three. Introduction to fracture mechanics david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029.
Pennyshaped cracks in threedimensional piezoelectric. Namely, we consider a penny shaped crack having the radius of a 0 opened by a uniform remote normal tension having the magnitude of p 0. Interaction of a pennyshaped crack and an external circular. In this paper, the boundary integral method is used to solve the scattering problem due to a subsurface pennyshaped crack. We wish to solve eqn 40, subject to the relation given by eqn 50. The extra strain gradient term is calibrated once only on the analytical solution for the penny. Contours of the vertical displacement field in the submodel and the global model are shown for a doubleedged notch specimen in figure 12. Finite element study of a pennyshaped crack along the. In this paper, the transient response of a pennyshaped crack embedded in a. Utilizing the boundary integral equation method, the singularities of near crack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the edd. 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. Numerical method for nonlinear models of pennyshaped cracks.
The required cpu time for computing the crack opening displacements was 20,854 sec, and the number of iterations needed for the convergence of. A selfconsistent method is adopted to determine the real crack opening. Fracture propagation is controlled by the stress field near the crack tip. Jan 01, 20 the effect of crack interactions on stress intensity factors is examined for a periodic array of coplanar pennyshaped cracks. Bui, an integral equations method for solving the problem of a plane crack of arbitrary shape, journal of the mechanics and physics of solids, vol. Integral transform method to the solution of a fredholm integral equation of second kind and numerical approach was implemented by kassir 12, in solving the rectangular. Hence, great efforts 110 have been made in solving the crack problems and it has been widely investigated since the pioneer work by sneddon for a pennyshaped crack. It includes a pennyshaped crack in an infinite piezoelectric plate.
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. To extend the potential theory method to the crack problem of onedimensional. Furthermore, the ps model has also been adopted to study some crack problems in ferro. Application of ray theory to diffraction of elastic waves by. Further results are presented for the direct problem of scattering of highfrequency waves by cracks in elastic solids. Extended displacement discontinuity method for analysis of. Analysis of a pennyshaped crack in a magnetoelectroelastic. 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. These results are compared with numerically computed exact results. Siam journal on applied mathematics volume 17, issue 6 10. Dynamic fracture analysis of a pennyshaped crack in a. Diffraction of elastic waves by a pennyshaped crack. 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. Using the extended displacement discontinuity method, analytical solutions based on the nonlinear ps and db models were derived for a penny shaped crack in 3d piezoelectric medium.
The stress, electric displacement and magnetic induction intensity factors of a pennyshaped crack in batio3cofe2o4 composites are calculated for different volume fractions and different applied combined loadings. This new formulation provides information on the inplane confinement. The extended displacement discontinuity edd method is proposed to analyze cracks in the periodical plane of onedimensional 1d hexagonal quasicrystals with the heat effect. The method allows for the natural introduction of displacement. T1 a penny shaped crack in a layer whose upper and lower surfaces are fixed. The basic unknown in the theory of linear elasticity is the displacement vector. Analysis of mode i conducting crack in piezoelectro. These are the stress intensity factor evaluation by the crack opening displacement method, the strain energy release rate evaluation using the modified crack closure integral method, and the jintegral evaluation using the virtual crack extension technique. Fundamental solutions of pennyshaped and halfinfinite plane. As a special case, the analytical solution for a penny shaped crack under uniform combined loadings is. Scattering by two pennyshaped cracks with spring boundary. The convergence study in crack opening displacement is performed for penny shaped crack and elliptical crack.
The distribution of the normal and tangential components of the contact forces and. 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. The displacement components in terms of this stress function are drf. A system of integraldifferential equations for a displacement jump in the crack plane was derived for generalized loading conditions and solved by this method. In this paper, values of stress intensity factors sifare obtained for a penny shaped crack, subjected to either harmonic or impact loads. Boundary element analysis of nonplanar threedimensional. The present paper examines the problem related to a penny shaped crack which is. This method is based on what is termed a crack green function, and does not depend essentially on the crack geometry. Experimental study of seismic scattering by a penny shaped crack by james francis scheimer submitted to the department of earth and planetary sciences on august 11, 1978 in partial fulfillment of the requirements for the degree of doctor of philosophy abstract ultrasonic model seismology was applied to a study of scattering of. To begin with we consider a simple problem where the exact solution is available. A penny shaped crack at the interface of two bonded dissimilar transversely isotropic elastic halfspaces. 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 first is a linear elastic, plane strain, doubleedged notch specimen under mode i loading, for which bowie 1964 has provided a series solution for the stress intensity factor, k i.
Green functions corresponding to uniformly distributed extended displacement discontinuities on an annular crack element in the isotropic plane of a threedimensional transversely isotropic magnetoelectroelastic medium are derived. For simple crack geometries a hybrid method, whereby the crack opening displacement is computed by ray theory, and the scattered field is. Analysis of arbitrarily shaped planar cracks in three. Boundary integral equations in elastodynamics of interface. Application of ray theory to diffraction of elastic waves.
Extended displacement discontinuity method for nonlinear analysis. The potential theory method has been generalized in this paper to analyze the piezoelectric crackproblem. We suppose that timeharmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i. Threedimensional crack using the displacement discontinuity method with applications to hydraulic fracture height growth and nonplanar. Based on the results of these numerical calculations, several conclusions can be made, as follows. Four examples are presented here for verification purposes. Finally, the mixedmode i and ii problem of a penny shaped crack along the interface in a bimaterial cylinder under three loading conditions is studied in detail. Extended displacement discontinuity method for analysis of penny shaped cracks in threedimensional thermal piezoelectric semiconductors zhao, minghao yang, changhai. Because this stress field is asymptotic dominant or singular, it is characterized by the stress intensity factor sif. This process is experimental and the keywords may be updated as the learning algorithm improves. Stress and displacement distributions are calculated in an axisymmetric, circular bar of finite dimensions containing a penny shaped crack.
Particular attention is devoted to a method by which the crackopeningdisplacement is computed on the basis of ray theory, and the scattered field is subsequently obtained by the use of a representation integral. Then, the general solution for the stress intensity factor was derived, and the corresponding solutions were also presented for a penny shaped crack and a permeable line crack as. However, the integral transform method and the potential theory method are usually limited to some simple cases, such as the case of the pennyshaped crack or uniform loadings. Kachanovs approximate method for crack interactions int. Sets of lines is direction of cylindrical coordinates. Stress intensity factors for pennyshaped cracks citeseerx. The corresponding strain energy area under the curve is equal to. The extended displacement discontinuity boundary element method is developed for these two nonlinear fracture models. The new integration method is based on the continuation approach. 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. Torsion of bonded dissimilar materials containing circular. Let qf ff 1 2n and use the gaussian quadrature formula for chebyshev.
Stress and displacement distributions are calculated in an axisymmetric, circular bar of finite dimensions containing a pennyshaped crack. Stress intensity factors for cracks in anisotropic. Martin i98i has shown how such a green function can be constructed for the pennyshaped crack and has derived a fredholm integral. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bimaterial solid of in. Displacement discontinuity method for modeling axisymmetric cracks in an elastic halfspace. The problem of determining the stresses around a circular crack on the interface between two bonded dissimilar isotropic elastic halfspaces is solved when general polynomial loads are applied to the surfaces of the crack. Each medium is loaded such that the displacements are nonzero only in the circumferential direction.
Using the obtained green functions, an extended displacement discontinuity method is presented to analyze a pennyshaped crack under axisymmetric loadings. Fracture analysis of magnetoelectroelastic solid with a penny shaped crack by considering the effects of the opening crack interior. The problem was solved by boundary integral equations method using iterative algorithm. The pennyshaped crack on an interface the quarterly. 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. Loadpoint displacement curve is linear with a positive slope, and the displacement per unit force applied is defined as the compliance. Using greens functions for th semiirfinite plane, the problem is. The crack is imbedded in a homogeneous medium and on the crack surface the spring boundary conditions are assumed. Results for a penny shaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. A hankel transform development of our mixedboundary value problem yields two.
A novel method for the solution of the three dimensional. The generalized method allows arbitrary placement of the side nodes for. Boundary element analysis of threedimensional crack problems. The use of this method is illustrated by obtaining solutions for two circular geometry and mixed boundary value problems in threedimensional elasticity. The displacement discontinuity method ddm is an efficient method of studying fracture problems. Stress intensity factor determination for threedimensional crack. A hankel transform development of our mixedboundary value problem yields two simultaneous pairs of dual integral equations. Sif for a pennyshaped crack in a finiteradius cylinder submodel method this is a simple threedimensional crack problem in finite domain, a pennyshaped crack in a finiteradius cylinder subjected to remote uniform tension. A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny shaped crack where displacements are given. Threedimensional static and dynamic stress intensity. A new potential of a simple layer is introduced to account for the effect of the electric field.
1540 100 812 170 1237 1421 1081 736 470 892 678 398 200 1212 940 1397 590 245 170 847 13 363 1272 1263 1594 72 240 647 1383 765 1455 344 496 1412 1266