Elliptical Crack Research Articles

Microstructural investigation of rolling contact fatigue (RCF) on a failed planetary gear of a windmill gearbox

A systematic investigation of failed planetary gear in a windmill gearbox was carried out. The damaged teeth were investigated in order to determine the damage mechanisms that contributed to the failure. During the visual examination, few representative gear teeth were selected with signatures of spalling, micro pitting, cracks following the elliptical path along with fractured surfaces for further investigation. Subsequently, the SEM fractography showed beach marks indicating the fatigue as a operative mechanism for crack propagation. In addition to this, the surface residual stress analysis by x-ray method showed the tensile nature at the tip, while being compressive at the root for damaged gear teeth. Furthermore, the detailed microstructural investigation of damaged gear teeth at a cross-sectional region revealed the nucleation of microcracks at interphase boundaries, dark etch bands along with severe plastic deformation at the core that further transmitted to the elliptical crack path. Finally, it has been established that the failure was initiated due to micropitting, which aggravated the subsurface microcracks leading to spalling, macro crack growth in the elliptical path across the gear teeth leading to final fracture due to the operation of rolling contact fatigue (RCF).

Crack shape evolution and stress intensity factors for opposite surface cracks in aircraft shaft under tension and combined loading

Stress Intensity Factor (SIF) of opposite surface cracks presented in an un-notched round bar has been determined under tension and combined load conditions. The influence of loading modes on SIF of opposite surface cracks has been attempted using numerical method. Crack depth ratio [(a/d); ‘a’ crack depth and ‘d’ diameter of the bar] ranging 0.1 to 0.4 and crack aspect ratio [(a/c); ‘a’ crack depth and ‘c’ semi major axis of the ellipse] ranging 0.2 to 1.0 were considered. Mixed mode SIF was calculated with consideration of mode-II and mode-III fractures. When the bar was subjected to pure tensile loading, as the crack depth ratio (a/d) increases, a crossover (higher SIF value changes from middle region to surface region) in SIF distribution was observed for an aspect ratio of 0.6. This implies that the growing crack has changed its propagation behavior as the depth ratio increases. In addition to tension, when torque was applied, nonuniform and asymmetric SIF spread was observed for semi elliptic crack [(a/c) = 0.2] compared to semi-circular [(a/c) = 1.0] crack. It was also noted that mode-I fracture is higher at the middle region [P/P0 = 0] of the rack border, whereas mode-II and mode-III fracture is higher at the crack surface region [P/P0 = ±1]. Due to combined loading, more influence of mode-II and mode-III fracture failure was observed and their effect is marginal at the middle region [P/P0 = 0] of the crack border. Using surface plots, SIF of opposite cracks located in un-notched round bar has been determined.

Plane Cracks in a Transversely Isotropic Layer

Problems of plane cracks of normal fracture (mathematical cuts) in the middle plane of a transversely isotropic elastic layer, the outer faces of which are under conditions of a sliding support, are considered. Isotropic planes are parallel or perpendicular to layer faces. Using the Fourier integral transform, the problems are reduced to integro-differential equations for crack opening, from which one can obtain the known equations of the corresponding problems for a transversely isotropic space and an isotropic layer by passing to the limit. A regular asymptotic method is applied for elliptical cracks; this method is effective for a relatively thick layer. It is shown the applicability domain of the method narrows with increasing anisotropy that is characterized by the roots of the characteristic equation (for an isotropic material, all roots are equal to unity). For strip-like cracks, closed solutions are obtained based on special approximations of the kernel symbols of integral equations, the relative errors of which decrease with increasing anisotropy. Calculations are made for known transversely isotropic materials.

Crack tip opening displacement based description of three-dimensional creep crack border stress fields for partially penetrating cracked specimens

The most important ability of fracture mechanics is how to apply the material test data obtained by standard specimens with the through-thickness cracks to practical structures where partially penetrating crack growth usually spends most of the service life, but complicated stress states make it difficult to describe the creep crack border stress states effectively. Based on the creep stress intensity factor Kδ(t)-Tz, a crack tip opening displacement based description is proposed to characterize the creep crack border fields for partially penetrating cracked specimens under out-of-plane constraints described by Tz. Comprehensive three-dimensional finite element results show that in the whole range of elliptical parametrical angle and shape factor, the maximum change in Kδ(t)-Tz is within 19% while C(t) changes over 370%, showing that Kδ(t)-Tz is more stable than C(t). At both primary and steady creep stages, the Kδ(t)-Tz description can characterize the crack border stress fields effectively for specimens with surface and corner elliptic cracks. Consideration of the in-plane constraint coefficient Q* is found to be necessary only for embedded elliptic cracked specimens, where the Kδ(t)-Tz-Q* description can provide an accurate prediction of crack border stress fields. The dominance and stability of Kδ(t)-Tz and Tz make the crack tip opening displacement based description potentially to develop creep fracture assessment approaches for high temperature structures.

Fatigue endurance limit and crack front evolution in metallic glass

Fatigue endurance limit is a threshold stress-amplitude under which a solid subjected to a cyclic loading can sustain infinite life. Such a limit has been confirmed in ferrous materials but remains controversial in many advanced new materials, including bulk metallic glasses with superb strength. By using a combination of ultrasonic fatigue (USF) test and conventional fatigue test, we obtain the stress amplitude vs. loading cycle curve of the Vitreloy 1TM metallic glass, with loading cycles up to 109. There is a clear fatigue endurance limit (FEL) which is about 320 MPa, and is 17% of the strength of Vitreloy 1TM. The residual strength of survivals after 109 cycles is nearly the same as those of intact samples. We demonstrate that all fatigue cracks initiate from spherical pores which are inevitable and intrinsic amid casting and fast cooling, and the fatigue crack front resembles an ellipse arc till final rapid rapture of all tested samples. The size and position of a pore can be utilized to extract the fatigue initiation threshold K0th, and the elliptical fatigue crack front is used to obtain the mode I fracture toughness KIC, which are K0th=2.0±0.5MPam and KIC=18-20MPam for Vitreloy 1TM, respectively. The fatigue endurance limit and its intrinsic origin of Vitreloy 1TM pave the way to understand fatigue in other metallic glasses.

The effects of thermal stress on the crack propagation in AP1000 reactor pressure vessel

In this paper, the effects of thermal stress on AP1000 reactor pressure vessel (RPV) is analyzed, and the elliptical corner crack propagation at the nozzle-cylinder intersection is investigated. The corner crack stress intensity factor (SIF) is computed under static pressure and combination (static pressure plus thermal stress) loadings and crack propagation. The analysis reveals that with an increase degree of the surface corner crack in the longitudinal section of in-let nozzle, SIF will decrease gradually; Due to the influence of thermal stress, the crack growth length is shorter, the growth rate is slower and the SIF is smaller. The numerical results of SIF for a wide range of crack propagation are provided as a reference for the fracture mechanics design of the RPV.

Ellipsoidal inhomogeneity in elliptically orthotropic elastic solid

An unbounded elliptically orthotropic elastic solid containing a single anisotropic ellipsoidal inhomogeneity is considered. The problem statement assumes arbitrary orientation of ellipsoid with respect to the orthotropy axes and non-uniform far loading. By applying the appropriate affine transformations, the boundary value problem is reduced to that one with the isotropic constituents. The complete displacement solution to the transformed problem is obtained using the Papkovich-Neuber representation with the scalar potentials written in terms of ellipsoidal solid harmonics. By accurate fulfilling the interface conditions, the boundary value problem is reduced to the system of the linear algebraic equations for the constants entering the potentials. For the Eshelby problem (uniform remotely applied loading and perfect interface), an explicit analytical solution is written in terms of ellipsoidal harmonics. This result yields solution for the elliptical crack problem in the remarkably simple way. The numerical examples illustrate an effect of the matrix anisotropy on the stress concentration and stress intensity factors. An application of the obtained solution to the micromechanics is discussed. The effective elastic stiffness tensor of a particulate composite with elliptically orthotropic matrix is evaluated using Maxwell homogenization scheme.

A XFEM approach for the three-dimensional cracks in piezoelectric material using interaction integral

Piezoelectric materials are widely used in various engineering applications such as electromechanical actuators, vibrators, energy harvesters, sensors, transducers, and propeller blades. These materials are subjected to various types of cyclic loading and their crack growth analysis is quite useful as it predicts the material’s life. This work proposes an extended finite element method to examine the static three-dimensional cracks in the piezoelectric material. The modeling of a crack is done using crack front elements enriched with the new six folded enrichment functions. These functions have been derived from the analytical solution of semi-infinite crack in a piezoelectric domain and expanded using Laurent-like series. Domain-based interaction integral is obtained to extract the mixed-mode stress intensity factors. Further, the auxiliary mechanical and the electrical fields in the interaction integral are obtained based on Stroh formulation. The XFEM is employed to evaluate SIF and EDIF in a three-dimensional domain is implemented in MATLAB code. The XFEM formulation is validated with static planar penny shaped crack for Mode I SIF and Mode IV EDIF with the standard solutions available in the literature. The error in Mode I and Mode IV is 0.0476 and 0.1444 respectively. The XFEM prediction values matches well with the analytical predictions. Further, simulation of the other cracks like inclined penny shaped crack, lens-shaped crack, and elliptical crack have been carried out with an expectation of usefulness of present methodology to mechanical engineers and designers.

Manchuriophycus-like elliptical cracks in thin mudstones intercalated with lacustrine sandstone: Intrastratal crack formation in water-saturated sediments

Manchuriophycus-like elliptical cracks are found in thin mudstone layers and lenses intercalated with wave-rippled lacustrine sandstone beds of the Lower Cretaceous Jinju Formation, South Korea. The cracks are interpreted as intrastratal shrinkage cracks formed by liquidization and injection of sand, hydroplastic deformation of mud, and concomitant dewatering of the sediments in water-saturated conditions. The cracks are considered to have been formed by earthquake and/or wave-induced loading and initiated where smectite was presumed to be locally concentrated in the mud. It is hypothesized that the cracks were developed into the elliptical outline, reflecting that of an interference ripple trough, due to permeability differences between two thin mud layers deposited over the rippled sand layer.

Fully plastic J-integral and C* equations for small punch test specimen with a surface crack

Small punch (SP) creep test specimen with an elliptical surface crack was suggested for use with creep crack growth testing. Estimation equations for fracture parameters are needed. K-equation was derived previously for the crack in the SP specimen.In this study, estimation equations for J and C* were derived. 3D elastic-plastic finite element analyses were carried out to determine J value along the front of the surface cracks of the same geometries as were employed for the K-estimation. The dimensionless function, h1 was determined for various crack locations according to the EPRI scheme. The plastic limit load solution was also determined. The C*-equation of the cracked SP specimen was derived based on the Jp solution using the analogy between the secondary creep constitutive law and the Ramberg-Osgood plasticity law. The C*-equation may be able to be used to characterize the creep crack growth behavior under the extensive creep condition of the cracked SP specimen.

Extension of Inglis’s elliptic crack solution

We extend Inglis’s two dimensional elliptic tensile crack solution in elastic material after visiting his original work in 1913. Solving static elastic equation of motion, we find a general elliptic crack solution in elliptic coordinate. We transform the solution from elliptic to that in rectangular coordinate system and derive a general singular stress and displacement field in the slit crack model. We then derive extended Irwin’s energy release rate and acceleration associated with the crack just before crack propagation. Pointing out that tensile fracture becomes dominant when the crack velocity approaches zero, we show that splitting is one of the fundamental mode of tensile fracture in confined and compressive stresses in qualitative agreement with experiment.

Fracture mechanics analysis of the effect of clamping stress on the fatigue life of riveted built-up railroad girders under variable amplitude loading

A linear-elastic Fracture Mechanics approach is used to estimate the fatigue life of riveted connections present in railroad bridge built-up steel girders under variable amplitude loading. Special care has been taken to analyze the effect of thermal clamping stresses on fatigue life. The analysis is developed using the variable amplitude stresses generated due to a traversing heavy axle train on a simply supported girder. Stress intensity factors for quarter elliptical corner crack emanating from a through-thickness (rivet) hole under remote tension were used. Thermal clamping stress, typically generated during the installation process, was quantified through a finite element simulation. Superposition principle was utilized to account for localized clamping stress present on the base material along with remote tension. Fatigue life estimations calculated using a linear-elastic Fracture Mechanics approach with and without considering clamping stress are compared. Finally, crack growth proved to be different under the effect of localized clamping stresses when compared to a crack growth emanating from a hole with no clamping stress.

The earthquake arrest zone

SUMMARY Earthquake ruptures are generally considered to be cracks that propagate as fracture or frictional slip on pre-existing faults. Crack models have been used to describe the spatial distribution of fault offset and the associated static stress changes along a fault, and have implications for friction evolution and the underlying physics of rupture processes. However, field measurements that could help refine idealized crack models are rare. Here, we describe large-scale laboratory earthquake experiments, where all rupture processes were contained within a 3-m long saw-cut granite fault, and we propose an analytical crack model that fits our measurements. Similar to natural earthquakes, laboratory measurements show coseismic slip that gradually tapers near the rupture tips. Measured stress changes show roughly constant stress drop in the centre of the ruptured region, a maximum stress increase near the rupture tips and a smooth transition in between, in a region we describe as the earthquake arrest zone. The proposed model generalizes the widely used elliptical crack model by adding gradually tapered slip at the ends of the rupture. Different from the cohesive zone described by fracture mechanics, we propose that the transition in stress changes and the corresponding linear taper observed in the earthquake arrest zone are the result of rupture termination conditions primarily controlled by the initial stress distribution. It is the heterogeneous initial stress distribution that controls the arrest of laboratory earthquakes, and the features of static stress changes. We also performed dynamic rupture simulations that confirm how arrest conditions can affect slip taper and static stress changes. If applicable to larger natural earthquakes, this distinction between an earthquake arrest zone (that depends on stress conditions) and a cohesive zone (that depends primarily on strength evolution) has important implications for how seismic observations of earthquake fracture energy should be interpreted.

A method to compute mixed‐mode stress intensity factors for nonplanar cracks in three dimensions

SummaryMethods to compute the stress intensity factors along a three‐dimensional (3D) crack front often display a tenuous rate of convergence under mesh refinement or, worse, do not converge, particularly when applied on unstructured meshes. In this work, we propose an alternative formulation of the interaction integral functional and a method to compute stress intensity factors along the crack front which can be shown to converge. The novelty of our method is the decoupling of the two discretizations: the bulk mesh for the finite element solution and the mesh along the crack front for the numerical stress intensity factors, and hence we term it the multiple mesh interaction integral (MMII) method. Through analysis of the convergence of the functional and method, we find scalings of these two mesh sizes to guarantee convergence of the computed stress intensity factors in a variety of norms, including maximum pointwise error and total variation. We demonstrate the MMII on four examples: a semiinfinite straight crack with the asymptotic displacement fields, the same geometry with a nonuniform stress intensity factor along the crack front, a spherical cap crack in a cylinder under tension, and the elliptical crack under far‐field tension and shear.

A theoretical analysis of dislocation emission from an elliptical blunt crack tip in nanocrystalline solid

A theoretical model describing edge dislocation emission from an elliptical blunt crack tip in nanocrystalline solid is established to consider the effect of the nanotwin and dislocation pileup at twin boundary. The nanotwin is represented by a wedge disclination quadrupole and some dislocations are deposited on one of twin boundaries. The complex variable function method is used to calculate the force acting on the first dislocation emitted from the tip of blunt crack, and expressions of critical stress intensity factors corresponding to dislocation emission are obtained. Then the influence of the dislocation emission angle, the position and orientation of the twin, the strength of nanotwin, the shape and the surface effects of the elliptical blunt crack on the critical stress intensity factor for dislocation emission is analyze in detail. The results shows that the effect of nanotwin on dislocations emitting from crack tip depends on its position and azimuth. There is an optimal position that dislocations are most easily emitted from the crack tip. The dislocation pileup at the twin boundary will increase the critical stress intensity factors for dislocation emission, making the dislocation emission from crack tip difficult, thus reducing the toughness of the material contributed by dislocation emission. In addition, the surface stress has a significant effect on the critical stress intensity factor.

Uncertainty Modeling of Fatigue Crack Growth and Probabilistic Life Prediction for Welded Joints of Nuclear Stainless Steel.

Welded joints are widely used in the pipeline connection of nuclear power plants. Defects in these joints are an important factor leading to the failure of welded joints. It is critical to study the fatigue crack growth and life prediction methods for the welded joints with defects, to reduce their likelihood. In this paper, we present our study of the uncertainty of fatigue crack propagation and probabilistic life prediction for welded joints of nuclear stainless steel. The standard compact tension (CT) specimens were fabricated according to the American Society for Testing and Materials (ASTM) standard. Fatigue crack propagation tests with different stress ratios were performed on CT specimens, using the Mei Te Si (MTS) fatigue test system. A fatigue crack propagation rate model considering the uncertainty of material parameters, and based on the Paris formula and crack propagation experimental data, was established. A probabilistic life prediction method based on Monte Carlo simulation was developed. The fatigue crack propagation prediction result of a CT specimen was compared with the actual tested result, to verify the effectiveness of the proposed method. Finally, the method was applied to an embedded elliptical crack in welded joints of nuclear stainless steel, to predict the fatigue crack growth life and evaluate the reliability.

Three-dimensional closed-form solution to elliptical crack problem in magneto-electro-elasticity: Electrically and magnetically induced Maxwell stress boundary condition

An elliptical mode-I crack problem in an infinite multi-ferroic composite material with transverse isotropy under remote uniform mechanical, electric and magnetic loadings is analyzed. The electro-magnetically semi-permeable crack boundary condition and Maxwell stress on the crack faces caused by the electric and magnetic fields are considered, which makes the problem nonlinear. On the basis of the general solution for multi-ferroic composite materials and modern potential theory method, the nonlinear integro-differential boundary equations are established and solved analytically. The field variables in the three-dimensional space and some important fracture mechanics quantities, are obtained in explicit forms. From numerical examples, it is found that the combination of the applied electric and magnetic loadings should be in a elliptical region in the electric displacement-magnetic induction coordinate plane to keep the crack open. Furthermore, in the vicinity of the center of this region, the electro-magnetically semi-permeable crack boundary condition without the Maxwell stress could provide relatively accurate results. Otherwise, the Maxwell stress would have a significant influence, and thus should not be neglected.

Effect of crack propagation path on mesh stiffness of spur gears

Gear mesh stiffness is an important indicator for condition monitoring of gears as its value has been found to decrease with initiation of crack. In the current study, the effect of crack propagation path on mesh stiffness has been investigated. Elliptical cracks with different values of semi major and semi minor axes have been considered with same values of crack lengths along these paths. ANSYS 19 software along with FRANC 3D has been used to model the gears with cracks induced. Structural analysis has been performed through Mechanical APDL. The values of mesh stiffness have been found to vary significantly with change in path of crack progression.

Numerical solutions of hypersingular integral equations for stress intensity factors of planar embedded interface cracks and their correlations with bimaterial parameters

This paper studies the stress intensity factors and their correlations with bimaterial parameters for the interface planar cracks by hypersingular integral equations. A new bimaterial parameter γ is defined, and the correlations of the stress intensity factors of interface planar crack with two parameters ε and γ are proved. The relative displacement fundamental function is proposed based on the crack's peripheral equation, thus one kind of element with coupled oscillatory characteristic of mode-1 and mode-2 is established for the interface planar crack. For the interface elliptical cracks with b/a = 0.2, 0.4, 0.6, 0.8, 1.0, the stress intensity factors along the crack front are calculated by solving the hypersingular integral equations. Comparison with the existing exact solutions show that the numerical results have a high accuracy. For the interface planar cracks under tensile load, there exist K1, K2 and K3 at the same time. The numerical results show that K1 is very weakly related to ε and γ, K2 strongly to ε and weakly to γ, and K3 shows a significant correlation with both parameters ε and γ. K3 is much smaller than K1 and K2 and can be ignored in fracture analysis of problems with remote mode-1 loading.

Periodic crack problems for an elastic layer

The problems of periodic mode I 3 D cracks for an elastic layer are reduced to an integro-differential equation (IDE). The principal part of the IDE is the same as in the classical Griffith problem for one crack in an elastic full space. The periodic system of elliptic cracks, featuring equal spaces between neighboring cracks, lies in the middle plane of the layer. Each crack is parallel to the layer’s faces. Four types of boundary conditions are considered on the layer’s faces (sliding or rigid support, stress-free state, sliding support in between elastic layers). The regular asymptotic method has been applied to solving the problems as it proves effective for some fairly big spaces between cracks as well as for relatively thick layers. The stress intensity factor (SIF) has been calculated for a variety of geometric and mechanical parameters. It was compared to that for the well-known case of a single crack in a layer. It has been shown that the SIF for the crack system can be smaller or greater than that for a single crack depending on the boundary conditions on the layer’s faces.