Seismic dispersion and attenuation in saturated porous. International journal for numerical and analytical methods in geomechanics, vol. Read an integral method for predicting hydraulic fracture propagation driven by gases or liquids, international journal for numerical and analytical methods in geomechanics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at. The problem of a pennyshaped crack embedded in an infinite space of transversely isotropic multiferroic composite medium is investigated. By continuing to use our website, you are agreeing to our use of cookies. Results are discussed and compared with the ones provided by linear elastic fracture mechanics, by theory of critical distances and by cohesive crack. The solution is analytical up to the numerical root of the equation providing the finite crack growth increment. 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 e we use cookies to enhance your experience on our website. Analytical and numerical solutions of hyperbolic heat. Penny shaped crack in elastic halfspace, approximation for ha 1 to compute displacements at surface sun, 1969. As the analytical solutions are proposed for a penny shaped 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 topics discussed in the paper include a crack in layered composites, a crack terminating at and going through a bimaterial interface, a penny shaped crack in a filamentreinforced elastic matrix, and inclusion problems in bonded materials.
The powerlaw rheological model of fluid was assumed and the final scheme was capable of tackling both the viscosity and toughness dominated regimes of crack propagation. Exact analytical solution for a penny shaped crack subjected to uniform pressure loading on the crack surfaces was obtained by sneddon 14. The interaction between a pennyshaped crack and a spherical. Pdf elastic tstress solution for pennyshaped cracks under. The solution of the problem is reduced to that of a system of. The authors start with the solution of a penny shaped crack in a unidirectional, fiber reinforced composite, which demonstrates necessity of considering wavy fracture fronts in. Semianalytical solution for mode i pennyshaped crack in. The closedform solution for the problem with constant pressure applied near the tip of a pennyshaped crack is studied to illustrate the methodology of the analysis and also to provide a fundamental solution for the numerical approach. References to related problems are also given by kassir and sih 1975, atkinson 1979 and sih and chen 1981.
Hypersingular integral equations for the solution of penny. 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 is. Complete and exact solutions of a pennyshaped crack in a piezoelectric solid. The resulting high pressure may form a cylindrical crack around the wellbore. 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 is moving. Analysis of the results shows that the characteristics of the seismic dispersion and attenuation for the slit cracks are similar to known results for penny. The radial model is one of the most important geometries both for benchmarking and as a starter solution for 3d and planar 3d models. Closedform solutions are obtained for a pennyshaped crack in a transversely isotropic elastic body, with the crack faces subjected to an arbitrary normal and shear loading. An analytical solution for stress distribution and displacement along a cylindrical crack formed between the casing and the formation is provided in this paper. 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. Theoretical study of hydraulically fractured penny.
The accuracy of the stress intensity factor calculation is satisfactorily examined for rectangular, penny shaped and elliptical planar cracks. The present paper examines the problem related to a penny shaped crack which is. The penny shaped crack and the plane strain crack in an infinite body of powerlaw material. Pdf herein, we consider the axisymmetric problem of a penny shaped crack in an elastic material sandwiched between other materials. The crack is assumed to be subjected to uniformly distributed mechanical, electric and magnetic loads applied symmetrically on the upper and lower crack surfaces. 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 sun 1969 model calculates analytical solution for surface deformation due to hydrostatic pressure inside a horizontal circular fracture penny shaped in an elastic, homogeneous halfspace. Sun69 penny shaped crack in elastic half space in matlab. It includes a pennyshaped crack in an infinite piezoelectric plate, a piezoelectric strip, a fiber embedded in a matrix, a piezoelectric cylinder with elastic coating, and the. Analytical expressions for deformation from an arbitrarily. This is based on the method outlined in section 11. The stable growth of a crack created by the hydraulic pressurizing of a penny shaped crack in a dry rock mass is investigated.
Diffraction of elastic waves by a pennyshaped crack. Plane elastostatic solution in an infinite functionally. Both numerical and analytical procedures are employed to obtain the main results. This paper develops a closedform approximate solution for a penny shaped hydraulic fracture whose behaviour is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness and fluid leakoff. Propagation of a pennyshaped fluiddriven fracture in an. Analytical and numerical analyses for a pennyshaped crack. International journal of computational mathematics 2014.
Analytic crack solutions for tilt fields around hydraulic. Read critical study of existing solutions for a penny shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact, engineering fracture mechanics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A pennyshaped crack is assumed to be located in the central plane of this layer and. The pennyshaped crack at a bonded plane with localized. An axisymmetric elastostatics problem for a pennyshaped crack placed in the middle of a inhomogeneous fgm elastic layer is considered. The stable growth of a crack created by the hydraulic pressurizing of a penny. If the wall t 1 mm, albb 1mm since a c in this half penny crack. An analytical solution for the axisymmetric problem of a. Semianalytical solution for mode i pennyshaped crack in a soft. In the first part of this paper a universal fluid velocity based algorithm for simulating hydraulic fracture with leakoff was created for a pennyshaped crack. In this brief note, we provide the failure stress of a solid containing a penny shaped crack by means of finite fracture mechanics. Fluid velocity based simulation of hydraulic fracturea. In deriving analytical solution in the present study the elastic parameters. Based on the theory of elasticity, previous analytical solutions concerning a pennyshaped interface crack employ the derivative of the crack surface opening displacements as the primary unknowns, thus leading to singular integral equations with cauchytype singularity.
The problem is compounded by the lack of exact solutions for a propagating pennyshaped hydraulic fracture to benchmark the numerical programs. To validate the present solutions, a numerical code by virtue of finite element method is established for 3d crack problems in the framework of magnetoelectro. A perturbation solution obtained by expanding about the trivial state in which the stress is everywhere parallel to the crack leads to simple formulas which are highly. In this approach, special crack border elements with square. The pennyshaped crack and the plane strain crack in an. Pdf this paper presents the exact close form solutions for elastic tstress of a pennyshaped crack in an infinite body under remote tension. It is assumed that the bimaterial interface contains a pennyshaped crack the surfaces of which are subjected to known tractions. The analytical solutions for the stress intensity factors can be found on pages. The classical eshelbys solution, extended to the thermal conduction problem of an ellipsoidal inclusion embedding in an infinite homogeneous matrix, gives an analytical solution of temperature discontinuity across a nonconducting penny shaped crack. Pdf an analytical solution for the axisymmetric problem. Accordingly, investigation of the effects of fracture front profiles on mechanical responses is the thrust of this paper. Modeling of heat flow and effective thermal conductivity. Thermal shock fracture of a cylinder with a penny shaped crack based on hyperbolic heat conduction.
Matlaboctave codes for analytical deformation models ipgpdeformations matlab. The extra strain gradient term is calibrated once only on the analytical solution for the penny. An improved semianalytical solution for stress at roundtip notches pdf. The following matlab project contains the source code and matlab examples used for sun69 penny shaped crack in elastic half space. Multiscale tip asymptotics in hydraulic fracture with leak. 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. Here we provide a semi analytical, orthogonal polynomial series solution for a roughwalled radial penny shaped hydraulic fracture driven by a fully turbulent fluid. Uniform uniaxial stress edit i f the cra ck is located centrally in a finite plate of width 2 b \displaystyle 2b and height 2 h \displaystyle 2h, an approximate relation for the stress intensity factor is 5. List of research publications civil engineering mcgill. This approach gives a numerical solution for the entire frequency range and analytical solutions for the low. A simple and analytical method of solving the problem of heat extraction from a penny. Arutyunyan, torsion of an elastic ball with a penny shaped crack, mechanics of solids mtt, 1990, 25, no 3, 3240. Approximate semianalytical solution for a pennyshaped. It is assumed that the elastic modulus of the layer varies through the thickness symmetrically with respect to the crack plane.
As a typicalexample, a closedform solution is first obtained for a pennyshaped crack subjected to a pair ofconcentrated forces acting in opposite directions and a pair of point charges on crack surfaces. For a penny shaped crack embedded within an infinite solid and loaded remotely perpendicular to the crack at a stress. Exact expressions for stress and electric displacement intensity factors are also presented. Penny shaped cra ck in an infinite domain under uniaxial tension. The rock mass is infinitely extended, homogeneous, and isotropic. Analytical expressions for deformation from an arbitrarily oriented spheroid in a halfspace. This paper presents an approximate threedimensional analytical solution to the elastic stress field of a pennyshaped crack and a spherical inhomogeneity. The geometry analyzed is a pennyshaped crack in an infinite body, subjected to. When the crack has a special shape, analytical solutions to these equations can be derived. The first objective of this research is to construct rigorous solutions for the problem of a pennyshaped fluiddriven fracture, with a clear statement of their range of applicability. The crack is assumed to be opened by uniform fluid pressure exerted on both sides of the crack. One pennyshaped crack to begin with we consider a simple problem where the exact solution is available. A 378, 263 1981 has proved that the corresponding linear boundaryvalue problem has precisely one solution.
Fundamental solutions of pennyshaped and halfinfinite plane. Several specific distributions of the moduli variations have been analysed. The solution is analytical up to the numerical root of the. These results are important for investigating crack interactions. Abstract closedform solutions are obtained for a pennyshaped crack in a. Later, green and sneddon 12 obtained the exact solution for an embedded elliptic crack in a infinite solid subjected. An approximate solution for a pennyshaped hydraulic. Thus, he has established the existence of a solution to the original boundaryvalue problem which we denote by s for general incident waves. This solution is then validated by the numerical simulation based on the finite elements method. Let 2a 2c, assuming a semicircular throughthickness crack in the appendix wall. Analytical solutions to hyperbolic heat conductive models using greens function method. An analytical solution for microannulus cracks developed. To supplement the dislocation models of davis 1983, okada 1992 and others, this work has extended several elastic crack solutions to provide tilt calculations.
Critical study of existing solutions for a pennyshaped. Sudden twisting of a penny shaped crack in a nonho mogeneous. Diffraction of elastic waves by a penny shaped crack. The solutions include constantpressure 2d, penny shaped, and 3delliptic cracks and a 2dvariablepressure crack. Pennyshaped cracks by finite fracture mechanics request pdf. Analytic and computational micromechanics of clustering.
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