Penny-shaped Crack Research Articles

Three-Dimensional Elastic Analysis of Rock Excavations by Using the g2 Constant Displacement Discontinuity Method

A fast computational code is presented that is dedicated for the elastic analysis of three-dimensional excavations and cracks in rocks. The problem is solved on the boundaries that are discretized with a new triangular leaf constant displacement discontinuity element with one collocation point. The creation of the new triangular element was inspired from Mindlin's special version of grade-2 or strain-gradient elasticity theory (second gradient of displacement, g2). This element is characterized by a much better measure of the average stress at the center of gravity of the triangular element compared to that of the classical elasticity element close to regions with stress or strain gradients (e.g. notches, cracks etc). In a verification stage, the accuracy of the computational algorithm for the pressurized penny-shaped and mixed-mode elliptical crack problems that have analytical solutions is demonstrated. More specifically, it is shown that the average error of the crack tip Stress Intensity Factor predicted by the gradient modified method for nine discretizations of varying density is around 3.5% with a máximum error of 5%, while the constant displacement discontinuity element displays errors varying around 14%. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity with only one collocation point per element, but it is far more efficient compared to it, especially close to the crack tips and corners of excavations where the displacement and stress gradients are highest.

Quantitative Evaluation of the Flaking Strength of Rolling Bearings with Small Defects: (Part 1: FEM Analyses of the Stress Intensity Factor, KII, under Rolling Contact)

It has been demonstrated that the rolling contact fatigue (RCF) test, using a specimen with a small drilled hole, is a useful means of evaluating the influence of a minor defect on the flaking strength of steels. In this study, RCF tests were conducted on rolling bearings with small drilled holes. Flaking failure was determined to be caused by the shear-mode fatigue cracks that emanated from the small defects. As a first step to quantifying the crack-growth threshold according to fracture mechanics principles, using the finite element method (FEM), it was necessary to analyze the Mode II stress intensity factor (SIF) range, ΔKII, of a ring-shaped crack, as emanated around the edge of a drilled hole after the passage of a rolling element. Subsequently, the derived values were correlated with the ΔKII values of penny-shaped cracks in an infinite body under uniform shear via a correlation factor, fdrill. The SIF of the ring-shaped crack was also uniformly correlated with that of the penny-shaped crack, using the single factor, fdrill, irrespective of the hole diameter, d, the depth of the hole-edge, h′, and the maximum contact pressure, qmax, within the following ranges: d = 0.05 ~ 0.2 mm, h′ = 0.05 ~ 0.345 mm and qmax = 2.0 ~ 3.0 GPa. The obtained results were later applied towards the quantification of the RCF test results, as detailed in Part 2 of the report on this research.

転がり軸受のはく離強度に及ぼす微小欠陥寸法の影響（第1報:転がり接触下のモードII応力拡大係数のFEM解析）

It has been shown that rolling contact fatigue (RCF) test using specimen having a small drilled hole is a useful method for evaluating the effect of small defect on flaking strength of steels. In this study, RCF tests of rolling bearings having a small drilled hole were carried out. The flaking failure was considered as a problem of shear-mode fatigue crack emanating from the small defect. As a first step to quantify the crack-growth threshold on the basis of fracture mechanics, mode II stress intensity factor range, ΔKII, of a ring-shaped crack emanated around the edge of a drilled hole under the passing of a rolling element was analyzed by using finite element method. And then, the obtained values were correlated with the ΔKII values of penny-shaped cracks in an infinite body under uniform shear through the intermediary of a correlation factor, fdrill. The stress intensity factor of the ring-shaped crack was uniformly correlated with that of the penny-shaped crack by the single factor fdrill irrespective of hole diameter, d, depth of hole edge, h’, and maximum contact pressure, qmax, within the ranges: d = 0.05 ~ 0.2 mm, h’ = 0.05 ~ 0.345 mm and qmax = 2.0 ~ 3.0 GPa. The obtained results will be applied for the quantification of RCF test results shown in the subsequent paper.

Stress intensity factor solutions for several crack problems using the proportional crack opening displacements

A general finite element procedure based on the proportional crack opening displacements for obtaining the stress intensity factors is presented. The procedure is applied to the nonsingular 3-node linear, 4-node linear, 8-node parabolic, 8-node axisymmetric elements and 8-node hexahedral solid elements for a test. It is found that the current method exhibits good element type adaptability and significantly less mesh dependency, and accurate results can be obtained effectively using rather coarse meshes. The accuracy of the current procedure is evaluated by applying it to two-dimensional interface cracks, three-dimensional penny-shaped cracks as well as circumferential surface cracks. Comparison with the published data from the literature shows that the current procedure gives accurate stress intensity factors. Furthermore, the current method is fairly efficient and less computational resource consuming and can be used as an effective tool in the reliability analysis of the bonded multi-layers.

Investigation of stress intensity factor for internal cracks in FG cylinders under static and dynamic loading

This paper investigates the variations of mode I stress intensity factor (KI) for inner penny-shaped and circumferential cracks in functionally graded solid and hollow thick walled cylinders, respectively with the changes of crack geometry, material gradation and loading conditions. The functionally graded material of cylinders consists of epoxy and glass. It is assumed that the mechanical properties vary with a power law in the radial direction of cylinders. Micromechanical models for conventional composites are used to estimate the material properties of functionally graded cylinders. The equations of motion obtained from the extended finite element discretization are solved by the Newmark method in the time domain. The interaction integral method is employed to calculate the mode I stress intensity factor (KI). The MATLAB programming environment was implemented to solve the problem.

Penny-shaped cracks in three-dimensional piezoelectric semiconductors via Green’s functions of extended displacement discontinuity

In this article, a penny-shaped crack in the isotropic plane of three-dimensional transversely isotropic piezoelectric semiconductors is analyzed via the displacement discontinuity boundary element method. The general solutions are derived based on the generalized Almansi’s theorem and the operator theory. Green’s functions are derived using the Hankel transformation under uniformly distributed extended displacement discontinuities (including displacement discontinuities, potential discontinuity, and electron density discontinuity) on a penny-shaped crack in the isotropic plane of three-dimensional transversely isotropic piezoelectric semiconductors. Using the extended displacement discontinuity boundary element method, penny-shaped cracks in transversely isotropic plane of three-dimensional piezoelectric semiconductors are studied, and the stress, electric displacement, and electric current intensity factors under uniform mechanical-electric-current loads applied on the penny-shaped crack surface are calculated. The fracture behaviors of piezoelectric semiconductors are studied.

Micromechanics-based simulations of compressive and tensile testing on lime-based mortars

The purpose of this paper is to propose a continuum micromechanics model for the simulation of uniaxial compressive and tensile tests on lime-based mortars, in order to predict their stiffness, compressive and tensile strengths, and tensile fracture energy. In tension, we adopt an incremental strain-controlled form of the Mori–Tanaka scheme with a damageable matrix phase, while a simple J2 yield criterion is employed in compression. To reproduce the behavior of lime-based mortars correctly, the scheme must take into account shrinkage cracking among aggregates. This phenomenon is introduced into the model via penny-shaped cracks, whose density is estimated on the basis of particle size distribution combined with the results of finite element analyses of a single crack formation between two spherical inclusions. Our predictions show a good agreement with experimental data and explain the advantages of compliant crushed brick fragments, often encountered in ancient mortars, over stiff sand particles. The validated model provides a reliable tool for optimizing the composition of modern lime-based mortars with applications in conservation and restoration of architectural heritage.

Frictionless contact of a rigid disk with the face of a penny-shaped crack in a transversely isotropic solid

In the framework of linear elastic continuum mechanics, an analytical formulation is presented for the axisymmetric axial interaction of a rigid disk in frictionless contact with the face of a penny-shaped crack in a transversely isotropic solid. The problem is reduced to an integral equation and is shown to be degenerated to the formulation of isotropic materials in the literature. As the closed-form solution is not possible, by means of a numerical procedure, the obtained integral equation is solved and the accuracy of numerical procedure and mathematical formulation is verified through comparison with the available results for the relevant analysis in isotropic solids. Through several numerical displays, the effect of material anisotropy on the stiffness of the interacting system and the stress intensity factors at the tip of penny-shaped crack is surveyed.

Discrete model of hydraulic fracture crack propagation in homogeneous isotropic elastic media

An infinite homogeneous and isotropic elastic medium with a penny shape crack is considered. The crack is subjected to the pressure of fluid injected in the crack center. Description of the crack growth is based on the lubrication equation (balance of the injected fluid and the crack volume), equation for crack opening caused by fluid pressure on the crack surface, the Poiseullie equation related local fluid flux with the crack opening and pressure gradient, and classical criterion of crack propagation of linear fracture mechanics. The crack growth is simulated by a discrete process consisting of three basic stages: increasing the crack volume by a constant crack size, crack jump to a new size defined by the fracture criterion, and filling the appeared crack volume by the fluid. It is shown that the model results a reasonable dependence of the crack size on the time as well as the pressure distribution of fluid on the crack surface. Comparisons with the solutions of hydraulic fracture problems existing in the literature are presented.

Elastic-wave velocity characterization of gas hydrate-bearing fractured reservoirs in a permafrost area of the Qilian Mountain, Northwest China

There are two types of gas hydrate-bearing reservoirs in the permafrost area of Qilian Mountain. Most of the gas hydrates occur mainly in the fractured mudstone reservoirs and rarely in the pores of the sandstone reservoirs. In this study, for the acoustic velocity characterization of the fractured gas hydrate reservoirs of the Qilian Mountain permafrost area, some mudstone core samples were collected for physical rock experiments, such as the acoustic experiment and the porosity and permeability experiment. An acoustic velocity numerical simulation of gas hydrate reservoirs was performed according to the Biot theory and the differential effective medium theory, with the conditions of multiple gas hydrate occurrence models, including the suspension model, the semi-cementation model and the cementation model, and considering both infinite and penny-shaped cracks. Fracture porosity was added to the core samples that only contain matrix porosity. With fracture porosity ranging from 0.01% to 5%, the variation laws between acoustic velocity with fractured porosity and hydrate saturation are obtained: (1) In the case of an infinite crack, if the fractured porosity is 0.01%–1%, the P-wave velocity decreases rapidly in the case of the three occurrence models. If the fractured porosity is higher than 1%, the acoustic velocity decreases gradually. If the crack shape is a penny-shaped crack, the P-wave velocity decreases almost linearly with increasing fracture porosity. (2) If the hydrate occurrence model is the suspension model, the P-wave velocity increases slightly with increasing hydrate saturation. If the occurrence model is the semi-cementation model or the cementation model, when the gas hydrate saturation of the infinite crack ranges from 0 to 80%, the acoustic velocity increases approximately linearly, whereas when the gas hydrate saturation ranges from 80% to 100%, the velocity increases rapidly. If the crack is a penny-shaped crack, the velocity increases almost linearly with increasing gas hydrate saturation from 0 to 100%. (3) It is found that the fractured gas hydrate reservoirs of the Qilian Mountain permafrost area contain both penny-shaped and infinite cracks, of which the infinite crack is the main crack shape. The gas hydrate occurrence in the Qilian Mountain permafrost area mainly follows the suspension model. This has significance for the seismic exploration and log evaluation of gas hydrate-bearing fractured reservoirs in the permafrost area of the Qilian Mountain in studying the acoustic velocity characterization, the crack shapes and occurrence models of gas hydrate reservoirs in the study area.

Penny-shaped crack in elastic medium with surface energy effects

In this paper, a penny-shaped crack in an infinite elastic medium subjected to vertical pressure loading at the crack surface under the influence of surface stress is considered. The Gurtin–Murdoch continuum theory of elastic material surfaces is adopted, and the Hankel integral transform is employed to solve this axisymmetric boundary value problem. A set of simultaneous dual integral equations is solved by employing an appropriate numerical solution scheme. Selected numerical results are presented to portray the influence of the surface stress on the elastic field. Numerical results reveal that the surface stress has a significant influence on both stress and displacement fields. It is also found that the material becomes tougher with the presence of surface stress. In addition, the elastic field also shows size-dependent behavior deviating from the classical crack solution. The solutions presented in this study provide fundamental understanding of the influence of surface stress on fracture mechanics problems. It can also be used as a benchmark solution for the development of numerical techniques, such as FEM and BEM, for analysis of more complicated problems associated with cracks under the influence of surface stress.

Development of stress boundary conditions in smoothed particle hydrodynamics (SPH) for the modeling of solids deformation

This paper develops a method for imposing stress boundary conditions in smoothed particle hydrodynamics (SPH) with and without the need for dummy particles. SPH has been used for simulating phenomena in a number of fields, such as astrophysics and fluid mechanics. More recently, the method has gained traction as a technique for simulation of deformation and fracture in solids, where the meshless property of SPH can be leveraged to represent arbitrary crack paths. Despite this interest, application of boundary conditions within the SPH framework is typically limited to imposed velocity or displacement using fictitious dummy particles to compensate for the lack of particles beyond the boundary interface. While this is enough for a large variety of problems, especially in the case of fluid flow, for problems in solid mechanics there is a clear need to impose stresses upon boundaries. In addition to this, the use of dummy particles to impose a boundary condition is not always suitable or even feasibly, especially for those problems which include internal boundaries. In order to overcome these difficulties, this paper first presents an improved method for applying stress boundary conditions in SPH with dummy particles. This is then followed by a proposal of a formulation which does not require dummy particles. These techniques are then validated against analytical solutions to two common problems in rock mechanics, the Brazilian test and the penny-shaped crack problem both in 2D and 3D. This study highlights the fact that SPH offers a good level of accuracy to solve these problems and that results are reliable. This validation work serves as a foundation for addressing more complex problems involving plasticity and fracture propagation.

Dielectric Breakdown Model for an Electrically Semi-Permeable Penny-Shaped Crack in Three-Dimensional Piezoelectric Media

The dielectric breakdown (DB) model for a penny-shaped crack under a semi-permeable boundary condition in a three-dimensional piezoelectric medium is studied. An approximate analytical solution is derived by using the boundary integral equation with extended displacement discontinuity, and the corresponding boundary element method with double iterative approaches is developed to analyze the semi-permeable crack. The effect of electric boundary conditions on crack faces is discussed on the basis of DB model. By comparing the DB model with the polarization saturation (PS) model for different piezoelectric materials, some interesting phenomena related to the electric yielding zone and local J-integral are observed.

Normal impact response of a saturated porous cylinder with a penny-shaped crack

The transient response of a saturated porous cylinder containing a penny-shaped crack and subjected to a suddenly applied normal loading is investigated. The general field equations of Biot’s theory are applied in the formulation in conjunction with the Laplace and integral transforms to reduce the mixed boundary-value problem to the solution of Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used, and the time-dependent magnitudes of the local stress field near the crack border measured in terms of the dynamic stress-intensity factor are computed for selected material property values. These results are presented and compared to the results of dry medium to find out the influence of pore fluid and its extent on the dynamic stress-intensity factor.

Finite element analysis of internal penny-shaped crack problem in an unbounded thermoelastic medium

ABSTRACTIn this article, the generalized thermoelastic theory with one relaxation time is used to investigate the internal penny-shaped crack problem in an unbounded thermoelastic material. The boundary of the crack is subjected to a prescribed stress distribution and temperature. The problem has been solved numerically using a finite element method. The results of the temperature, displacement components, and stress components have been verified numerically and are represented graphically. Comparisons are made with the results predicted by the presence and absence of the relaxation times.

Transient thermal fracture analysis of a penny-shaped magnetic and dielectric crack in a magnetoelectroelastic cylinder

ABSTRACTThe problem of penny-shaped magnetic and dielectric crack in a magnetoelectroelastic cylinder is investigated under thermal shock load. The problem is reduced to solve three coupled Fredholm integral equations. The field intensity factors are derived. Numerical results of crack opening displacement intensity factors are presented, and the effects of thermal shock time, crack configuration, and magnetoelectrical crack surface conditions on crack propagation and growth are evaluated. Among others, the larger cylinder's radius, the easier to propagate the crack is. For a fixed crack configuration, magnetoelectrical crack surface conditions have different effects on crack propagation as well.

Theoretical estimates for flat voids coalescence by internal necking

Coalescence of voids by internal necking is in most cases the last microscopic event related to ductile fracture and corresponds to a localized plastic flow between adjacent voids. Macroscopic load associated to the onset of coalescence is classically estimated based on limit analysis. However, a rigorous upper-bound mathematical expression for the limitload required for flat voids coalescence that remains finite for penny-shaped voids/cracks is still unavailable. Therefore, based on limit analysis, theoretical upper-bound estimates - both integral expression and closed-form formula - are obtained for the limit-load of cylindrical flat voids in cylindrical unit-cell subjected to boundary conditions allowing the assessment of coalescence, for axisymmetric stress state. These estimates, leading to finite limit-loads for pennyshaped cracks, are shown to be in very good agreement with numerical limit analysis, for both cylindrical and spheroidal voids. Approximate formula is also given for coalescence under combined tension and shear loading. These coalescence criteria can thus be used to predict onset of coalescence of voids by internal necking in ductile fracture modelling.

Axisymmetric thermo-elastic field in an infinite space containing a penny-shaped crack under a pair of symmetric uniform heat fluxes and its applications

This paper deals with the problem of an infinite isotropic solid weakened by a penny-shaped crack, which is subjected to a pair of uniform heat flux loadings symmetrically exerted on the upper and lower crack surfaces. Three-dimensional thermo-elastic field variables are obtained by means of potential theory method in company with the newly developed general solution. Some physical quantities on the cracked plane, such as temperature, radial displacement, crack surface displacement, normal stress at the crack front and stress intensity factor, which are important parameters in fracture mechanics, are presented in closed-forms. In the framework of ideal thermo-elasto-plasticity, the extent of the plastic zone at the crack front is determined by the Dugdale crack model, which linearizes the original non-linear problem. The refined crack surface displacement and normal stress outside the crack are also obtained via the principle of superposition. Numerical calculations are carried out to validate the present solutions and to show the distributions of the axisymmetric thermo-mechanical fields. The present solution could be served as a reference for the infrared-thermography technique.

Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth

We present a novel numerical method to simulate crack growth in 3D, directly from the Computer-Aided Design (CAD) geometry of the component, without any mesh generation. The method is an isogeometric boundary element method (IGABEM) based on non-uniform rational B-splines (NURBS). NURBS basis functions are used for the domain and crack representation as well as to approximate the physical quantities involved in the simulations. A stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted elements. Convergence studies in the crack opening displacement is performed for a penny-shaped crack and an elliptical crack. Two approaches to extract stress intensity factors (SIFs): the contour M integral and the virtual crack closure integral are compared using dual integral equations. The results show remarkable accuracy in the computed SIFs, leading to smooth crack paths and reliable fatigue lives, without requiring the generation of any mesh from the CAD model of the component under consideration.

Strip yield zone of a penny-shaped crack in a magnetoelectroelastic material under axisymmetric loadings

The strip yield zone ahead of a penny-shaped crack in a magnetoelectroelastic material, subjected to electric, magnetic and axisymmetric mechanical loadings, is evaluated analytically. Hankel transform is employed to reduce the mixed boundary value problem of the penny-shaped crack to dual integral equations, which are solved exactly under the assumption of electrically and magnetically permeable crack face conditions and the plastic strip yield zone crack model. An analytic solution to the mixed boundary problem has been obtained to predict the relationship between the length of the strip yield zone and the applied loadings. The distribution of mechanical, electric and magnetic fields in and outside of the strip yield zone in the cracked magnetoelectroelastic material has been derived analytically, and the crack opening displacement has been investigated. The effects of the mechanical, electric and magnetic loadings on the size of the yield zone are discussed in detail.