For the case of 3d planar cracks embedded in a semiinfinite body, there are less available analytical solutions for sif. Some of the results from these references are used to build the present solution for the crack face displacements. Advective transport from a pennyshaped crack in a porous. Thermally loaded penny shaped cracks in thermopiezoelectric materials are investigated in this paper. 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. Threedimensional poroelastic simulation of hydraulic and. Pennyshaped cracks by finite fracture mechanics request pdf. In this paper, in order to decouple these equations we first substitute an analytical approximation for the dry. Selfsimilar solutions of fracture dynamics problems on. The analytical solutions for the stress intensity factors can be found on pages. A 378, 263 1981 has proved that the corresponding linear boundaryvalue problem has precisely one solution. Comparison between analytical tstress solution and fem data, bending load. A simple and direct relationship is established between the limiting values of the. The fundamental solutions for the stress intensity factors.
The lack of general analytical solutions for the problems involving cracks in functionally graded materials is emphasised by eischen 1987. 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 resulting high pressure may form a cylindrical crack around the wellbore. Sneddon 1946 solved the problem of an infinitely thin crack subjected to uniform normal traction p applied to its faces. We show that slip reductions relative to planar faults for 2d line and 3d penny shaped crack models are comparable within 10% when slip is perpendicular to the corrugations. The plane strain version of the problem is also examined. The analytical solutions for the penny shaped cracks subjected to uniform temperature and steady heat flow are discussed. 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 problem of a penny shaped crack embedded in an infinite space of transversely isotropic multiferroic composite medium is investigated. Semianalytical solution for mode i pennyshaped crack in a. Fundamental solutions of pennyshaped and halfinfinite plane. Hypersingular integral equations for the solution of penny. Analytical and numerical analyses for a pennyshaped crack. Sneddon and elliot 1946 presented solutions for semiinfinite, penny shaped, and arbitrarily shaped fractures. An analytical solution for the hydrostaticelastic problem for a wetting fluid inside a 3d penny shaped circular crack in an elastic infinite solid loaded in tension at infinity, when the vapor. 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.
The general representations of the analytical solutions with arbitrary index of selfsimilarity were presented for fracture elastodynamics problems on axially symmetry by the ways of selfsimilarity under the laddershaped loads. The flow of viscous fluid in the fracture is governed by the lubrication equation, while the crack opening and the fluid pressure are. T1 pore pressure cohesive zone modeling of hydraulic fracture in quasibrittle rocks. Using the obtained green functions, the extended displacement discontinuity boundary element method was developed to analyze the ps model and db model for pennyshaped cracks. Pdf elastic tstress solution for pennyshaped cracks under. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of modes i, ii, and iii. A study is carried out of the problem of a penny shaped crack in an infinite body of powerlaw material subject to general remote axisymmetric stressing conditions. Pennyshaped crack, thermopiezoelectric material, analytical solution, finite element method. This paper presents an analytical investigation of a nonclassical fracture mechanics axisymmetrical problem for a nearsurface penny shaped crack at the interface of physical and mechanical properties of a material. In this paper we analyze the problem of a penny shaped hydraulic fracture propagating parallel to the freesurface of an elastic halfspace.
For a penny shaped crack embedded in an infinite 3d elastic body, we need to solve the. In this paper, the general solutions for a penny shaped crack in an infinite solid, subjected to arbitrary tractions on the crack surfaces were derived. Sakamoto 2003 obtained the distribution of normal displacement and stress on the crack plane, located in the center. The analytical solution for a pennyshaped crack subjected to uniform heat flow, in a thermopiezoelectric solid, is obtained in this paper. This paper focuses on tackling the two drawbacks of the dual boundary element method dbem when solving crack problems with a discontinuous triangular element. Abstract closedform solutions are obtained for a penny shaped crack in a transversely isotropic elastic body, with the crack faces subjected to an arbitrary normal and shear loading. Apr 22, 2019 dynamic stress intensification due to the penny. The solution is presented for a crack with constant internal pressure in a homogenous elastic medium.
N2 hydraulic fracturing technology has been widely applied in the petroleum industry for both waste injection and unconventional gas production wells. The crack is assumed to be subjected to uniformly distributed mechanical, electric and magnetic loads applied symmetrically on the upper and lower crack surfaces. Siam journal on applied mathematics siam society for. Asymmetric loading of an externally cracked elastic solid 255 3. By the theory of complex functions, a penny shaped crack on axially symmetric propagating problems for composite materials was studied. Methods of analysis and solutions of crack problems, 368425. Flat, internal, penny shaped crack subjected to remote tension. Interaction of a pennyshaped crack and an external circular. Embedding the appropriate pressure singularities in a family of orthogonal polynomials used for derivation of the solution leads to very rapid convergence of the series, requiring just two terms for an accurate result. The penny shaped crack and the plane strain crack in an infinite body of powerlaw material m.
Here we provide a semianalytical, orthogonal polynomial series solution for a roughwalled radial pennyshaped hydraulic fracture driven by a fully turbulent fluid. The applicability was demonstrated deriving the closedform solutions for a penny shaped circular crack subjected to the lowerorder loading such as constant tension, shear, bending, and torsion. In order to calculate the integrals with higher order singularity, the triangular. 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 problem is compounded by the lack of exact solutions for a propagating pennyshaped hydraulic fracture to benchmark the numerical programs. 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. This paper examines the problem of the advective transport of a contaminant from sources in the shape of either a penny shaped crack or an elongated needle shaped cavity located in a porous medium of infinite extent. An analytical solution was developed by green and sneddon 1950 to calculate the stresses around a flat, elliptical crack.
As a particular case we present explicitly the series expansion for a traction free or clamped penny shaped crack in an axisymmetric or a nonaxisymmetric situation. The solutions presented also apply to powerlaw creeping materials and to a class of strainrate sensitive hardening materials. Results for a plane crack are compared with analytical solutions due to dundurs and gautesen and the method is also used to predict the extent of the contact region for a pennyshaped interface crack in combined shear and tension. Analytical solutions to hyperbolic heat conductive models using greens function method. A large block of steel is loaded to a stress of 34. Analytical solutions for a penny shaped crack, as a special case of the elliptical problem, are given.
Propagation of a pennyshaped hydraulic fracture parallel. We use stress boundary conditions compressive orientated such that frictional contacts shear. In this paper, the main objective is to investigate the interaction of a rigid disc in bonded contact with the surfaces of a penny shaped crack and an in nite transversely isotropic medium. Comparisons are made between the stressintensity factors derived by the analytical solutions and the numerical results using different finite element techniques.
An analytical solution for microannulus cracks developed. Analytical solutions in a closedform of an elliptical crack under the uniform loadings are obtained in section 5. International journal of solids and structures, 3718, 26032619. Penny shaped cra ck in an infinite domain under uniaxial tension. The exact expression of the stressintensity factor is presented.
If the fracture toughness is 44 mpa m12, determine the critical radius of a penny shaped crack. The diffraction of timeharmonic stress waves by a pennyshaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials. The analytical solution for a penny shaped crack subjected to uniform heat flow, in a thermopiezoelectric solid, is obtained in this paper. This paper presents an approximate threedimensional analytical solution to the elastic stress field of a pennyshaped crack and a spherical inhomogeneity. To this end the selfsimilar formulation of the penny shaped model will be analyzed. Penny shaped crack in an infinite domain under uniaxial tension. The problem of a pennyshaped crack embedded in an infinite space of transversely isotropic multiferroic composite medium is investigated. Multiscale tip asymptotics in hydraulic fracture with leak. A 3d pore pressure cohesive zone model was developed to predict nucleation and propagation of a penny shaped fluiddriven fracture.
Further results are presented for the direct problem of scattering of highfrequency waves by cracks in elastic solids. Asymmetric loading of an externally cracked elastic solid. A pennyshaped crack is assumed to be located in the central plane of this layer and. 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. These solutions correspond to an annular crack element applied with uniformly distributed extended displacement discontinuities in the transversely isotropic plane of a 3d piezoelectric medium. 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. This can be used typically to simulate magma overpressure in a sill below a volcano. Critical study of existing solutions for a pennyshaped. International journal for numerical and analytical methods in geomechanics, vol.
The exact solution for a pennyshaped crack is obtained by letting the minor axis of the ellipsoidal cavity approach zero. Exact and complete fundamental solutions for pennyshaped. 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. Suppose that is a penny shaped crack, with radius so that the crack occupies the region where and are polar coordinates, and. Complete and exact solutions of a penny shaped crack in a piezoelectric solid.
Analytical solutions for a pennyshaped crack, as a special case of the elliptical problem, are given. Complete and exact solutions of a penny shaped crack in a. Analytical and numerical solutions of hyperbolic heat. General solutions of a pennyshaped crack in a piezoelectric material under.
We compute the crack opening displacement subject to a plane wave of normal incidence. The geometry analyzed is a pennyshaped crack in an infinite body, subjected to. An analytical solution for stress distribution and displacement along a cylindrical crack formed between the casing and the formation is provided in this paper. Closedform solutions of an elliptical crack subjected to. Closedform solutions of an elliptical crack subjected to coupled. Analytical solutions for two pennyshaped crack problems in thermo. Thermal conductivity of hybrid short fiber composites. The solution is analytical up to the numerical root of the. Read on the penny shaped crack in inhomogeneous elastic materials under normal extension, international journal of solids and structures on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A problem of axisymmetric propagation of a penny shaped crack driven by a thinning fluid is considered. The analytical solutions for the pennyshaped cracks subjected to uniform temperature and steady heat flow are discussed. 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. The geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks. Abstractthe stress intensity factor variation along a semicircular surface law in a three.
Diffraction of elastic waves by a pennyshaped crack. Both numerical and analytical procedures are employed to obtain the main results. The analytical solution for the mode i sif for such a crack is given by sneddon see. 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. Axial translation of a rigid disc inclusion embedded in a. In this brief note, we provide the failure stress of a solid containing a penny shaped crack by means of finite fracture mechanics. This is based on the method outlined in section 11. Citeseerx nearsurface pennyshaped crack at the interface. 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. Guz obtained important analytical results for the stability loss problem for the interface of two bodies. An analytical solution for the axisymmetric problem of a. The fracture opening in the normal direction is given by. When the crack has a special shape, analytical solutions to these equations can be derived.
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. The exact expression of the stressintensity factor is. The interaction between a pennyshaped crack and a spherical. Effective elastic properties of rocks with transversely. A generation of special triangular boundary element shape.
It is demonstrated that the fracture behaviors can be predicted equivalently by both emps and embd models even though they are built on two different physical grounds. Mar 01, 2009 read critical study of existing solutions for a pennyshaped 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. The fracture is driven by an incompressible newtonian fluid injected at a constant rate at the center of the fracture. 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. Exact analytical solution for a penny shaped crack subjected to uniform pressure loading on the crack surfaces was obtained by sneddon 14. The advective transport is induced by darcy flow in the porous medium, where the internal boundary is maintained at a constant potential. Approximate semianalytical solution for a pennyshaped. The problems are governed by integral equations with the webersonin kernel on two segments. 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. The prevailing analytical solutions for hydraulic fracture mainly depend on linear elastic fracture mechanics. The correctness of the derived solution is verified by the finite element results.
This observation raises the question on the legitimacy of commonly used penny shaped crack solutions when applied to fiber reinforced composites. By the mellin convolution theorem the integral equations associated with the models 1 and. Propagation of a pennyshaped fluiddriven fracture in an. Analytical approximations of bulk and shear moduli for dry. Annular and circular rigid inclusions planted into a penny. This approach gives a numerical solution for the entire frequency range and analytical solutions for the low. Penny shaped crack revisited closedform solutions by v.
The stress state and effective elastic moduli of an isotropic solid containing equally oriented penny shaped cracks are evaluated accurately. Pore pressure cohesive zone modeling of hydraulic fracture. Accordingly, investigation of the effects of fracture front profiles on mechanical responses is the thrust of this paper. Pore pressure cohesive zone modeling of hydraulic fracture in quasibrittle rocks.
The rock mass is infinitely extended, homogeneous, and isotropic. For simple crack geometries a hybrid method, whereby the crack opening displacement is computed by ray theory, and the scattered field is. 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. Using the hankel transform method, feng, su, and pan 2007 discussed a penny shaped crack problem, with the crack being idealized to be magnetically dielectric. Siam journal on applied mathematics volume 17, issue 6 10. 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. Particle velocity based hydrofracturing algorithm for a penny. The pennyshaped crack and the plane strain crack in an. Analytical solutions for two pennyshaped crack problems in. The applicability was demonstrated deriving the closedform solutions for a pennyshaped circular crack subjected to the lowerorder loading such as constant tension, shear, bending, and torsion.
Greens functions are defined as a solution to the problem of an elastic, transversely isotropic solid with a penny shaped or an external crack under general axisymmetric loadings acting along a circumference on the plane parallel to the crack plane. 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 solution to the accuracy of four significant digits, at least, is obtained on the basis. Spe 159786 hydraulic fracturing design and optimization. On the general solutions for mixedmode pennyshaped crack. Similar results are presented for an annular plate containing internal, tractionfree surface cracks. The interaction of a penny shaped crack and an external circular crack in a transversely isotropic composite is investigated using the techniques of hankel transform and multiplying factors.
Siam journal on applied mathematics society for industrial. The naviercauchy equations of elastic equilibrium are reduced to three sets of coupled, simultaneous, ordinary differential equations whose solutions are obtained. Pore pressure cohesive zone modeling of hydraulic fracture in. Employing the generalized method of potential theory fabrikant, 1989a, fabrikant, 1991, li et al. Seismic dispersion and attenuation in saturated porous. Furthermore, we consider circular cracks subjected to secondary. 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. The subject of the paper are greens functions for the stress intensity factors of modes i, ii and iii. Thermal shock fracture of a cylinder with a penny shaped crack based on hyperbolic heat conduction. Analytical solutions for two pennyshaped crack problems. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed. Semianalytical solution for mode i pennyshaped crack in a soft. On the pennyshaped crack in inhomogeneous elastic materials. 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 stable growth of a crack created by the hydraulic pressurizing of a penny. Citeseerx document details isaac councill, lee giles, pradeep teregowda. This study partly employs the abovementioned formulations but assumes that the layer is spatially inhomogeneous through its thickness and deals with a mode i penny shaped crack. Results for a penny shaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. These results are important for investigating crack interactions. In their work, the problem was reduced to solutions of fredholm integral equations of the second kind. The crack is located in a nearsurface layer parallel to the surface of a halfspace, which is subjected. Extended displacement discontinuity method for nonlinear. The corresponding solutions of penny shaped crack problems, as a special case of ellipse, are given in section 6. Stress intensity factor and effective stiffness of a solid.
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