In the simulated example, fluid is injected at a constant rate into a pennyshaped crack of low initial aperture 10. This particular problem is of interest to the study of indentation testing of brittle ceramic materials fett lo. The two curves for pennyshaped cracks are nearly indistinguishable on the scale of this plot. Ultrasonic nonlinearity evaluation of the cracked interface. In this paper we examine the problem of a pennyshaped crack located in an isotropic elastic solid which is weakened by an external crack situated in the plane of the penny shaped crack fig. Starting with an initial small penny shaped crack, the crack. An extended asymptotic analysis for elastic contact of. Nov 01, 2004 this paper presents the exact close form solutions for elastic t stress of a penny shaped crack in an infinite body under remote tension and bending.
Pramudita and yuji tanabe graduate school of science and technology, niigata university 1. In these equations, the normal stress component on the crack surface e o. Numerical results of the crack opening displacement cod intensity factors are. Assuming that the stress intensity factors for the semiellipse and the semicircular crack are related in the same way as the embedded elliptical and penny shaped crack, he was able to empirically adjust the relations so that the solution for the semiellipse would. In the current work, using the potential method and hankel. The solution is used to calculate the direction of. Pennyshaped crack problems have been analysed extensively in the literature, but all the focus has been on the determination of stress intensity factors. The present work dealt with the epicycloid crack with the application of shear loading in fracture mechanics. 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. The hertz theory of quasistatic impact holds for evaluating. Crack boundary conditions the boundary conditions on the crack plane. Khay pidstryhach institute for applied problems of mechanics and mathematics nasu, 3b naukova str. These exact analytical solutions provide good insight about fracture problems but.
These analytical solutions are compared with known so. One of the important solutions to the penny shaped crack problem was derived by sneddon 6, whose pioneer treatment by means of hankel transform has attracted active contributions to this research field, 710, to name a few. For a rock body weakened by many pennyshaped microcracks, its effective elastic properties such as apparent youngs modulus, shear modulus and poisson s ratio are particularly important to many applications. H 1 v2ne where v is poissons ratio and e the elastic modulus. Thus, he has established the existence of a solution to the original boundaryvalue problem. Boundary integral equations in elastodynamics of interface. 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. In the present paper, the exact tstress solution for a pennyshaped crack. Mishuris 1 introduction hydraulic fracture hf is the phenomenon of a uid driven crack propagating in a solid material. In the current work, using the potential method and hankel transformation technique, the.
Stress intensity factors are also presented for semielliptical surface cracks in internally pressurized cylinders. The rock mass is infinitely extended, homogeneous, and isotropic. Expressions for modes ii and iii stress intensity factors have been given for both the cracks. Threedimensional numerical model of hydraulic fracturing. To begin with we consider a simple problem where the exact solution is available. Particle velocity based hydrofracturing algorithm for a. Much of this work is based upon an analysis of the stress near a circular or pennyshaped crack first discussed by sneddon 3. The results obtained with the numerical method are in good agreement with the analytical solution for uniform loadings. Threedimensional numerical model of hydraulic fracturing in. The discrete problem is solved by taking the solution of the.
Analysis of cracks in isotropic linear elastic halfspace. The penny shaped crack problem we examine the problem of a penny shaped crack which is located at the interface between identical halfspace regions in which the shear modulus varies axially according to the relationships 7. Unit traction is applied to left and right end surfaces. Threedimensional poroelastic simulation of hydraulic and.
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. Since many rock types show brittle elastic behaviour under hydrocarbon reservoir. Due to the axial symmetry of the loading and the spatial. Semianalytical solution for mode i pennyshaped crack in a soft. Fracture toughness of reentrant foam materials with a negative poisson s ratio. Taking numerical inversion of laplace transform, stress intensity factor at. Introduction since the first work by sack 1946, the pennyshaped crack has attracted significant. Figure 9 poisson s ratio fixed point as a function of found numerically for oblate spheroids and pennyshaped cracks, and also for pennyshaped cracks using the analytical expression. Particle velocity based hydrofracturing algorithm for a penny shaped crack d. Stress intensity factor determination plays a central role in linearly elastic fracture mechanics lefm problems. Stress intensity factor determination for threedimensional crack. This paper presents the exact close form solutions for elastic t stress of a pennyshaped crack in an infinite body under remote tension and bending. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials. An analytical approach for an extended asymptotic analysis of 3d wavy surfaces contact was developed on the basis of expansion of a doublesinusoidal surface in fourier series, using cylindrical coordinates.
Since many rock types show brittle elastic behaviour under hydrocarbon. Particle velocity based hydrofracturing algorithm for a pennyshaped crack d. The determination of the distribution of stress in the vicinity of a crack plays a central part in recent theories of fracture 1, 2 and for that reason is of some technical importance. Fracture mechanics assessment of large diameter wind. The proposed matlab script is a literal transcription of the suns paper equations. Elastic fracture mechanics analysis by the boundary. Namely, we consider a pennyshaped crack having the radius of a 0 opened by a uniform remote normal tension having the magnitude of p 0. The contact of elastic regular wavy surfaces revisited. In this paper we examine the problem of a penny shaped crack located in an isotropic elastic solid which is weakened by an external crack situated in the plane of the penny shaped crack fig. Sep, 2019 an analytical approach for an extended asymptotic analysis of 3d wavy surfaces contact was developed on the basis of expansion of a doublesinusoidal surface in fourier series, using cylindrical coordinates. Extended displacement discontinuity boundary integral.
The most relevant is that the solution is shifted in fourier space by one wavenumber. Figure 6 shows the distribution of the normalized temperature discontinuity on a pennyshaped crack centered in the tmee medium under different heat fluxes obtained using the eddbem and the analytical solution. Later, green and sneddon 12 obtained the exact solution for an embedded elliptic crack in a infinite solid subjected. Penny shaped crack problems have been analysed extensively in the literature, but all the focus has been on the determination of stress intensity factors. It is verified on the basis of the equations of fluid dynamics that the fracturing fluid cannot penetrate the entire domain of a crack when the crack. Unit traction is applied to left and right end surfaces as indicated by the red arrows.
Fracture mechanics assessment of large diameter wind turbine. By means of the laplace and hankel transform techniques, the problem is. Analytic crack solutions for tilt fields around hydraulic. An equilibrium pennyshaped crack in an inhomogeneous elastic. The problem reduces to a dual integral equation for which an approximate analytical solution. Exact analytical solution for a pennyshaped crack subjected to uniform pressure. 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. The elastic modulus is 10,000 and poisson s ratio is 0. Segedin, note on a penny shaped crack under shear, mathematical proceedings of the cambridge philosophical society, vol. Segedin, note on a pennyshaped crack under shear, mathematical proceedings of the cambridge philosophical society, vol. The problem reduces to a dual integral equation for which an approximate analytical solution is.
The first results for the solution of the crack problem in an inhomogeneous material using a weighting function were obtained by fett and munz. The crack has zero normal strength, and the insitu stresses are also zero. Abstract in order to develop an analytical method for quantifying the plastic. Semianalytical solution for mode i pennyshaped crack in. Because this stress field is asymptotic dominant or singular, it is characterized by the stress intensity factor sif. An extended asymptotic analysis for elastic contact of three. The penny shaped crack problem has received extensive investigation in the literature of fracture mechanics. An investigation of partial cone cracks in silicon nitride. Interaction between rigiddisc inclusion and pennyshaped. Martin i98i has shown how such a green function can be constructed for the pennyshaped crack and has derived a fredholm integral equation of the second kind that uniquely determines the c. Particle velocity based hydrofracturing algorithm for a penny. This paper derives a novel analytical solution for acoustic nonlinearity evaluation of the cracked interface. In fracture mechanics, pennyshaped cracks are often used to model the.
Pennyshaped cracks have been used extensively to model cracked materials walsh 1965, 1969. Diffraction of elastic waves by a pennyshaped crack. Fracture propagation is controlled by the stress field near the crack tip. Thus, the test conditions approximate those of the analytical solution for the nolag case i. Here, and are the shear modulus and poissons ratio of the material, b is. For a pennyshaped crack the results obtained bv analytical numerical method may be verified as follows. Exact analytical solution for a penny shaped crack subjected to uniform pressure. Crackface displacements for embedded elliptic and semielliptical. An analytical solution for the axisymmetric problem of a penny shaped crack in an elastic layer sandwiched between dissimilar materials kotaro miura, makoto sakamoto, koichi kobayashi, jonas a. Determination of effective elastic properties of microcracked. Analytical expression for the two tangential crack opening displacement potentials have been obtained as series in terms of the crack separation parameter. In these equations, is radius of the circle crack i. The fracture opening in the normal direction is given by. The stable growth of a crack created by the hydraulic pressurizing of a penny.
Elastic t stress solutions for pennyshaped cracks under. The naviercauchy equations of elastic equilibrium are reduced to three sets of coupled, simultaneous, ordinary differential equations whose solutions are obtained. The pennyshaped crack at a bonded plane with localized. It can be encountered in various natural processes, such as subglacial drainage of water or during. Walsh 1969 for oblate spheroids having a small aspect ratio. Dynamic fracture analysis of a pennyshaped crack in a. We will further expand the techniques developed in a number of previous studies aizikovich and alexandrov, 1984, aizikovich, 1995, aizikovich et al.
Deformation due to a pressurized horizontal circular crack. Similar results are presented for an annular plate containing internal, tractionfree surface cracks. To this end, the numerical solution for the stresses in such specimens is derived, based on the solution of the hypersingular integral equation, and the conformal mapping technique is adopted to transform the hypersingular integral equation over a circular region such. Meshes for models 1, 2 and 3 mesh 4 with 12 subdivisions along each quarter of the crack front and 12 subdivisions along the radius is not shown. 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. The correct fixed point for spheres is,and this value is attained in the limit by the. It is clearly shown that the tension and compression elastic asymmetry can result in acoustic nonlinearity. As the analytical solutions are proposed for a pennyshaped fracture with no presence of any obstacle such as natural interfaces, in this work, we presented the results of lattice simulations for hydraulic fracturing in the cement sample, similar to the lab, but with no natural fractures, and compared the results obtained with analytical solutions. The equations are also vectorized for all input parameters. In this paper, the transient response of a pennyshaped crack embedded in a. The problem of a pennyshaped tensile crack in a continuouslyinhomogeneous space is considered. Deformation due to a pressurized horizontal circular crack in. Pdf elastic tstress solution for pennyshaped cracks under. Interaction between rigiddisc inclusion and pennyshaped crack under elastic timeharmonic wave incidence v.
Consider solving the three dimensional laplaces equation, v2u 0, in the half. Exact analytical solution for a pennyshaped crack subjected to uniform pressure loading on the crack surfaces was obtained by sneddon 14. Barenblatt the problem of determining the stresses over the entire surface of an equilibrium crack is formulated as a mixedmixed boundary value problem in the classical theory of elasticity. For an embedded, penny shaped crack in an infinite elastic medium, subject to a remote compression o 0 and internal pressure p, a solution can be found using superposition of solutions in the stress analysis of cracks handbook, by tada. Stress intensity factor determination for threedimensional. Stress distribution at the edge of an equilibrium crack. A pennyshaped crack has more restricted opening, and has the ratio of 0. Sneddon 1946 solved the problem of an infinitely thin crack subjected to uniform normal traction p applied to its faces. When microcracks exist at the interface, the tensile and compressive effective moduli of the cracked interface are considered to be different. An analytical solution for the axisymmetric problem of a. Multiplying this result by a dimensional factor r 3, where r is the crack radius, and recalling that p 0 represents the ratio of the dimensional excess fluid pressure to shear modulus of the medium, we retrieve an analytic solution for a volume of a uniformly pressurized pennyshaped crack in an infinite elastic body e. Impact of torsional load on a pennyshaped crack in an elastic. Interaction of a pennyshaped crack with an elliptic crack. Semianalytical solution for mode i pennyshaped crack in a.
To determine these macroscopic parameters, we first adopted a dilute solution approach. 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 pennyshaped crack in a transversely. Flat, internal, pennyshaped crack subjected to remote tension. Determination of effective elastic properties of microcracked rocks based on asymptotic approximation. An equilibrium pennyshaped crack in an inhomogeneous. The sun 1969 model calculates analytical solution for surface deformation due to hydrostatic pressure inside a horizontal circular fracture pennyshaped in an elastic halfspace. Based on the fact that properties of the elastic fields are superposable, the body force.
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