The sifs which are exact for a penny shaped crack are based on the well known solution for a point load acting normally to such a crack. The study is reduced to the analysis of a hypersingular integral equation with respect to the relative crack face separation over a circular domain occupied by crack. A formula is derived for the stress intensity factor at the rim of a pennyshaped crack in an infinite solid in which there is an axisymmetric distributing of body forces acting in a direction normal to the original crack surfaces. The effective stiffness of a solid containing soarrays of equal pennyshaped cracks was determined by nematnasser et al. T1 stress intensity factor and effective stiffness of a solid containing aligned penny shaped cracks. Dislocation density function and integral transform are used to reduce the problem to the singular integral equation. Problem of the penny shaped crack edges contact interaction in threedimensional space under action of a normal harmonic tensioncompression wave and shear wave has been considered.
The stress intensity factor is generally expressed in the form k y. Basic equations owing to the axial symmetry of the problem posed, it is convenient to adopt a formulation. The paper concerns the contact interaction of the opposite faces of a penny shaped crack under harmonic wave at oblique incidence. The y term allows the effects of finite width, crack shape, position along crack front, bulging, stress concentration factors, local stress concentration. The analysis allows the evaluation of the indenting stress necessary to initiate the. Axial translation of a rigid disc inclusion embedded in a penny shaped crack in a transversely isotropic solid, scientia iranica, 262, pp. This paper considers the transient stress intensity factor mode i of a penny shaped crack in an infinite poroelastic solid. Dynamic stress intensity factor mode i of a pennyshaped.
The singular state associated with the embedded crack with finite, nonaxisymmetric normal loading is that of plane strain. In the former paper the analysis given applies to an axisymmetric distribution of pressure p. An axisymmetric problem for a pennyshaped crack bridged by fibres is considered. Stress intensity factor and effective stiffness of a solid contained aligned penny shaped cracks article in international journal of solids and structures 37. Using laplace and fourier transform, they derived a singular integral equation which was solved numerically to give the dynamic stress intensity factor. Nonnal stress parallel to crack surface in singular region psi n fringe order f material fringe value lbsinorder tfaximum shearing stress in plane perpendicular to crack border psi haximum remote shearing stress in plane perpendicular to crack border at plate surface psi apparent stress intensity factor lbsinj 312. The interface pennyshaped crack reconsidered 771 the method used is the following. The hypersingular boundary integral equations on the crack surface are transformed into new form, where the solution behavior near the crack front is accounted implicitly. 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. An expression for the stress intensity factor has been found by cherepanov 1974, a result which was generalized for a nonhomogeneous body by sankar and fabrikant 1983 a. The solution of some typical examples is given, and the significant effect of the stress intensity factor of the rock on the crack propagation is shown.
The stress intensity factor, is used in fracture mechanics to predict the stress state stress intensity near the tip of a crack caused by a remote load or residual stresses. Hypersingular integral equations for the solution of penny. Hypersingular integral equations for the solution of penny shaped interface crack problems. This integral equation is solved numerically by using gaussian quadrature formulae. Dynamic stress intensity factor mode i of a permeable penny. The paper presents the integral equations governing the pennyshaped crack problem and the numerical results for the stress intensity factor at the tip of the crack.
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. Stress intensity factor and effective stiffness of a solid. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the materials resistance to fracture in modern materials science, fracture mechanics is an important tool used to improve the. When the crack is expanding under a constant flow rate, the classical solution is found to be approximately valid for very large cracks, and nevertheless the crack. Based on the results of these numerical calculations, several conclusions can be made, as follows. The stress intensity factors for a periodic array of. The study is reduced to the analysis of a hypersingular integral equation with respect to the relative crackface separation over a circular domain occupied by crack. Today, it is the critical stress intensity factor k ic, found in the plane strain condition, which is accepted as the defining property in linear elastic fracture mechanics. The problem was solved by boundary integral equation method. The stress intensity factor for a penny shaped crack between two dissimilar materials m.
Let qf ff 1 2n and use the gaussian quadrature formula for chebyshev. The fractallike finite element method ffem is employed to study the interaction of multiple cracks and to demonstrate the efficiency. Stress intensity factor wikimili, the free encyclopedia. The dependence of stress intensity factors on wave numbers was investigated. The influence of layer thickness on the stress intensity factor of a penny shaped crack in a sandwiched viscoelastic bimaterial article in international journal of engineering science 433. Some quantities of physical interest were shown graphically followed by a discussion of the effect of the radii of the inclusion and the crack as well as the thickness medium on the layer deformation. Formulation for the mode i stress intensity factor. The case of a penny shaped crack with arbitrary normal displacements prescribed at its faces inclusion problem was solved exactly and in closed form by. The problem is solved by the method of boundary integral equations using an iterative algorithm. The stress intensity, k i, represents the level of stress at the tip of the crack and the fracture toughness, k ic, is the highest value of stress intensity that a material under very specific planestrain conditions that a material can withstand without fracture. The geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks.
The indentation of a precompressed pennyshaped crack. The pennyshaped crack at a bonded plane with localized. 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 stress intensity factor is evaluated directly based on displacement discontinuities dd using a threedimensional displacement discontinuity, boundary element method based on the equations of proposed in 1. The accuracy of the stress intensity factor calculation is satisfactorily examined for rectangular, pennyshaped and elliptical planar cracks. The stress intensity factors, maximum stress intensity, and strain energy release rate for the epicycloid crack subject to shear load are presented graphically. Fracture mechanics of throughcracked cylindrical pressure. Crack opening displacements and stress intensity factors are also obtained for a throughthickness, center cracked bar with variable thickness. The stress state and effective elastic moduli of an isotropic solid containing equally oriented pennyshaped cracks are evaluated accurately.
The effects of the material anisotropy on the stress intensity factor and on the crack shape are investigated for a penny shaped crack in a transversely isotropic plate of finite thickness. Numerical examples are given to show the effects of the interlayer thickness, distribution parameter, and. Abstract the pennyshaped cracks periodically distributed in infinite elastic body are studied. The assumptions of dugdale are applied to estimate the effects of plasticity around the edge of the crack. Elastodynamic problem for a layered composite with penny. In particular, the effect of the waveinduced fluid flow. A mathematical formulation is presented for the dynamic stress intensity factor mode i of a permeable pennyshaped crack subjected to a timeharmonic propagating longitudinal wave in an infinite poroelastic solid. The torsion of a penny shaped crack in a functionally graded strip is considered. The paper presents the integral equations governing the penny shaped crack problem and the numerical results for the stress intensity factor at the tip of the crack. In this section, we apply the approximate method of kachanov and laures 1989 to the analysis of interacting cracks to obtain the mode i sif of an array of periodic coplanar penny shaped cracks in an infinite medium subjected to a remote normal traction at infinity, figure 1 a. 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. Local stress field for torsion of a pennyshaped crack in. Investigation of the crack faces contact interaction for a. 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 stress intensity factor for a pennyshaped crack in an. In theory the stress at the crack tip where the radius is nearly zero, would tend to infinity. The problem is approximately simplified to that of a single crack embedded in finite length cylinder and the stress intensity factor is obtained by solving a fredholm integral equation. An analytical solution for the axisymmetric problem of a. Stress intensity factors for the pennyshaped crack under.
The stress intensity factor can then be determined for penny shaped cracks in infinite or finite solids subjected to symmetric loading about the plane containing the crack. The stress intensity factor of a concentric pennyshaped mode iii crack in a finite long cylinder of a finite radius is determined by using the basic theorem of the hankel transform and the fourier series. The problem is formulated by using integral transform technique under uniform load and reduced to a singular integral equation. The stress intensity factor is computed using the standard procedure of the fredholm integral. Stress intensity factors are given for various penny. The stressintensity factor for a pennyshaped crack. Fischer institute of mechanics, university for mining and metallurgy, franz josef stra. The representations of the stress intensity factors near the crack edges are obtained. In this closed position only a mode ii type stress intensity factor occurs at the crack tip. The problem of a concentric pennyshaped crack of mode iii. The approximate solution of pennyshaped cracks periodically.
Dynamic stress intensity factor mode i of a permeable penny shaped crack in a fluidsaturated poroelastic solid article in international journal of solids and structures 110111 january 2017. An axisymmetric problem for a penny shaped crack bridged by fibres is considered. Stress intensity factors at any point on the crack front of penny and halfpenny shaped cracks subjected to stress gradients are presented. Stress intensity factor we have modeled a body by using the linear elastic theory. The cracks are located symmetrically and in parallel to one another in the isotropic cylinder. The equations were based on stress intensity factors obtained from threedlmenslonal flnlteelement analyses 8, 9, 18, and 19 that cover a wide range of configuration parameters. The special case of a large crack subjected to modei loading is. This modification allows the direct determination of the stress intensity factors sif in the crack vicinity after solution of equations by the collocation technique. A formula is derived for the stress intensity factor at the rim of a penny shaped crack in an infinite solid in which there is an axisymmetric distributing of body forces acting in a direction normal to the original crack surfaces.
There are various stress intensity factor solutions, particularly for flat plates and pressure vessels with various cracked geometries. Various papers have also studied the stress intensity factor of a pennyshaped crack. This equation can be solved exactly and in an elementary manner for an elliptical crack under polynomial shear loading, and an approximate analytical solution is possible for a general crack under shear loading. Investigated are the effects of material property parameters and geometry criterion on the stress intensity factor. An equilibrium pennyshaped crack in an inhomogeneous elastic medium article in journal of applied mathematics and mechanics 742. As the stress intensity factor reaches the k ic value, unstable fracture. Numerical results for the stress intensity factors are.
The integral equation has been solved numerically in order to determine the variations of the dynamic stress intensity factor at the rim of the penny shaped crack for some particular body force loading cases. The pennyshaped interface crack with heat flow citeseerx. Stress intensity factors at any point on the crack front of penny and half penny shaped cracks subjected to stress gradients are presented. We wish to solve eqn 40, subject to the relation given by eqn 50. Introduction since the first work by sack 1946, the pennyshaped crack. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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 torsional problem of a pennyshaped crack at the interface between two distinctly different materials is investigated by presenting the generalized interlayer model. Stress intensity factors for penny and halfpenny shaped. Stress intensity factor variation of a pennyshaped crack situated close to the free surface of a halfspace k. The crack tip stress field is obtained by considering the asymptotic behavior of bessel function.
Citeseerx an axisymmetric problem of a discshaped rigid. The reported results show a considerable potential for using this. Axial translation of a rigid disc inclusion embedded in a. Timeharmonic body force loading of a modei pennyshaped. It illustrates the manner in which the stress intensity factor can be in. Hankel transform is used to reduce the problem to solving a fredholm integral equation. Axial translation of a rigid disc inclusion embedded in a penny shaped crack in a transversely isotropic. Mixedmode fatigue crack propagation of pennyshaped cracks. The coefficients of the polynomial p x1, x2 are determined using the boundary conditions on the surfaces of the pennyshaped cracks where the stresses on the crack surfaces are zero. Stress intensity factor coplanar pennyshaped crack periodic array interacting cracks abstract the effect of crack interactions on stress intensity factors is examined for a periodic array of coplanar pennyshaped cracks. Fracture mechanics assessment of large diameter wind turbine. Segedin, note on a penny shaped crack under shear, mathematical proceedings of the cambridge philosophical society, vol. The problem of a concentric pennyshaped crack of mode iii in.
T1 mixedmode fatigue crack propagation of penny shaped cracks. The result can be expressed in terms of a fredholm integral equation of the second kind. The torsion of a pennyshaped crack in a functionally graded strip is considered. Oct 01, 2003 in this paper we investigate the stress intensity factors sifs of multiple penny shaped cracks in an elastic solid cylinder under mode i axial tension loading. Kassir department of civil engineering, the city college of the city university of new york, new york, n. The accuracy of the stress intensity factor calculation is satisfactorily examined for rectangular, penny shaped and elliptical planar cracks. The influence of layer thickness on the stress intensity factor of a penny shaped crack in a sandwiched viscoelastic bimaterial.
N2 a mathematical formulation is presented for the dynamic stress intensity factor mode i of a permeable penny shaped crack subjected to a timeharmonic propagating longitudinal wave in an infinite poroelastic solid. An equilibrium pennyshaped crack in an inhomogeneous elastic. The proposed model and the obtained numerical results are in good agreement when compared to gao 40. The stress intensity factor of a concentric penny shaped mode iii crack in a finite long cylinder of a finite radius is determined by using the basic theorem of the hankel transform and the fourier series. In particular, the effect of the waveinduced fluid flow on the dynamic stress intensity factor is analyzed. The use of these formulae is illustrated by a consideration of the special case in which the. 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 solutions to the resulting integral equations permit only the determination of stress intensity factors and energy release rate, and do not directly provide crack opening and sliding displacements. Stress intensity factors for penny and halfpenny shaped cracks subjected to a stress gradient.
Stress intensity factor variation of a pennyshaped crack. The line load solution which is derived from this is different in form to those given by previous workers and is more. Some of these solutions are based on the use of thinshell theory18, which does not take into account the three dimensional nature of the highly localised stresses in the vicinity of the crack front. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a. N2 the stress state and effective elastic moduli of an isotropic solid containing equally oriented penny shaped cracks are evaluated accurately. In some configurations, the range of the equation was extended by using stress intensity factor solutions for a through crack in a similar configuration. A solution is given for the thermal stresses due to a penny shaped crack at the interface between dissimilar materials loaded in tension for the case where the heat flux is into the material with higher distortivity. In this section, we apply the approximate method of kachanov and laures 1989 to the analysis of interacting cracks to obtain the mode i sif of an array of periodic coplanar pennyshaped cracks in an infinite medium subjected to a remote normal traction at infinity, figure 1 a. The stress intensity factors are studied for different values of the wave frequency. An expression for the surface displacement of the crack is also given. Dynamic stress intensity factor mode i of a permeable. Kachanovs approximate method for crack interactions kachanov, m.
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