Semi-elliptical Notch Research Articles

Prediction of slip activity of crystal grains around semi-circular and semi-elliptical notches in thin-sheet specimens of pure titanium using formulated macroscopic stress distribution and crystal orientation

Thin metal sheets and wires are important materials for various devices used in electrical, mechanical, and medical fields. With the downsizing of these devices, demand for thinner sheets and wires has increased. Amongst the many metals available, pure titanium has been attracting much attention for use in medical and dental devices because of its good biocompatibility in addition to its light weight and high corrosion resistance. However, thin metal sheets and wires are usually polycrystalline materials and, with the downsizing of materials, there is a loss of homogeneity during deformations. Inhomogeneous deformation becomes significant in thin sheets and wires, owing to the different crystal orientations and geometries of crystal grains. Furthermore, the shapes of such devices are not uniform, unlike, say, a simple rod. Therefore, macroscopic stress and strain concentrations should be taken into consideration when designing these devices as they affect the localization of deformation and the resultant fracture. In this study, semi-circular and semi-elliptical notched specimens made of thin-sheet polycrystalline pure titanium are subjected to tensile testing. Inhomogeneous deformation caused by crystallographic slip is observed near the notch root. Analysis of the crystal orientation and observation of the slip line show that the slip initiation in crystal grains is affected by the macroscopic stress distribution and can be predicted from the slip activity calculated from both the critical resolved shear stress on the slip systems and the resolved shear stress acting on prospective slip planes obtained from the macroscopic multiaxial stress distribution.

High cycle fatigue limit prediction of foreign object damaged aerofoil samples by the theory of critical distance

The fatigue limit at 3 × 107 cycles predicted by the theory of critical distance (TCD) on Ti–6Al–4V aerofoil samples subjected to foreign object damage (FOD) was studied. First-order bending vibration fatigue experiments were carried out with 18 FOD samples after FOD tests. First-order bending vibration mode analysis of each notched sample with various semi-elliptical notch dimensions were carried out by finite element method and the critical distance is deduced by test results and numerical simulation. FOD and HCF tests were carried out on another 9 aerofoil samples, and fatigue limit predictions were carried out using the Peterson formula and the TCD model. The results showed that the TCD model has higher prediction accuracy.

The Crack Growth in the Imitation Model of a GTE Turbine Disk under Operating Loading Conditions

The paper presents the experimental results of growing surface cracks in the turbine disk of a gas turbine engine (GTE) under cyclic tension at room and elevated temperatures. The geometry of the imitation model of the GTE turbine disk with a stress concentration zone in the form of a bolt hole was justified. In order to ensure the similarity of the initial damage of the imitation model and the GTE turbine disc in the plane of symmetry of the stress concentration zone, a semi-elliptical notch was made. The loading conditions of the imitation model were developed based on results of a comparative stress-strain state (SSS) analysis of the stress concentration zone of the imitation model and the GTE turbine disc. As a result of the fatigue test of the imitation model at room and elevated temperatures, the experimental positions and sizes of the crack fronts with respect to the drop potential signal on the crack edges were obtained. The fixed positions and sizes of the crack fronts were used as the basis for the numerical calculation of the fracture resistance parameters. For the numerical studies, ten three-dimensional finite element models with different positions and sizes of the crack fronts were considered. The numerical calculation results based on the finite element method were used to determine the distributions of the elastic stress intensity factors along each crack front. The crack growth rate characteristics both on the free surface and at the deepest point of the crack front were obtained at room and elevated temperature conditions. A technique for the automation tests that simulate the block-type loading of the disk material at elevated temperatures was proposed.

Scattering of SH-waves by oblique semi-elliptical notches in half space

The scattering of steady-state SH waves in a half-space with oblique semi-elliptical notches is presented analytically. Mathieu function addition theorem and multi-elliptical coordinate systems are developed to solve the complex boundary value problem. The existence of local defects has a great influence on the scattering and diffraction of elastic waves. The wave function expansion method which can reveal the physical process of wave scattering and verify the accuracy of numerical methods is a common method to study the influence of defects. The numerical results of the solution are obtained by truncating the infinite equation. Although the truncation of the infinite equation is inevitable, the accuracy of the numerical results has been carefully checked. The proposed solution is verified by comparing the results obtained when the oblique semi-elliptical notch is degraded into a semi-elliptical notch or even a semi-circular notch with previous results. The numerical results for typical cases show the complex effects of the inclination of the notch, the axial ratio of the ellipse, and the incidence angle on the displacement amplitude and dynamic stress concentration near the notch.

Estimation on stress concentration factor of remanufacturing notch for TC4 titanium alloy

Remanufacturing notch would be produced after crack polishing in remanufacturing pre-treatment. In the present study, stress concentration of remanufacturing notch was investigated by a series of semi-elliptical notch models using finite element analysis. Comparing with notch width, notch depth has a critical effect on stress concentration. An equation to estimate stress concentration factor (SCF) has been modified to be suitable for remanufacturing notch. Based on the pre-estimation of stress concentration for remanufacturing notch, references on crack polishing plan can be provided in the remanufacturing field.

Adaptive Neuro-fuzzy Inference System Trained for Sizing Semi-elliptical Notches Scanned by Eddy Currents

The present study explores the capability of COMSOL Multiphysics, as a finite element modelling (FEM) tool, to model the interaction between a split-D differential surface eddy current (ECT) probe and semi-elliptical surface electrical discharge machined (EDM) notches. The effect of the small probe’s lift-off and tilt on its signal is investigated through modelling and subsequently, the simulation outcomes are validated using the probe’s impedance measurements. In the next stage, an adaptive neuro-fuzzy inference system (ANFIS) is designed to take the signal features as inputs and consequently, provide the length of the scanned notch as the system’s output. The system is trained by extracted features of thirty model-generated signals obtained from scanning of the same number of semi-elliptical notches by means of the split-D probe. The trained ANFIS is tested afterwards using the measured signals of 3 calibration EDM notches together with 5 model-based ones. A very low average estimation error is observed with regard to the length estimation of the test notches and the accuracy of the length estimation is found to be quite reasonable.

Numerical Investigation of J-Integral on a Notched Pressure Vessel

Finite elements analysis has been performed to compute a J-integral values in a notched pressure vessel. A semi-elliptical notch was introduced in a wall of cylindrical pressure vessel with aspect ratio of notch depth to wall thickness a/w = 1/6 to 5/6, where the wall thickness was 3 mm. Two cases were examined – inner and outer notch. Numerical examinations were revealed that outer notch has much higher values of J-integral (3-4 times), all other being the same, (i.e. inner pressure and dimensions of the notch). That was explained in terms of triaxality stress state in front of notch. Namely, it was shown that inner notch has positive triaxiality stress ratio and vice-versa, outer notch revealed highly negative hydrostatic stress component ahead the notch.

Fatigue Crack Growth Under Constant and Variable Amplitude Loading at Semi-elliptical and V-notched Steel Specimens

An assessment of fatigue crack propagation in components and structures based on fracture mechanical approaches is fundamental to define periodic intervals for service inspections. This paper focuses on the investigation of flat specimens made of mild steel S355 with V-shaped and semi-elliptical notches under constant and variable amplitude fatigue loading to analyze the influence of load sequence effects on crack propagation in order to obtain information about the remaining service life of components. Depending on the load sequence, crack propagation may be accelerated, delayed or in some cases even stopped, which leads to beach marks within the area of fracture. Fractographic analyses of the tested specimens are carried out by light-optical microscopy to determine different crack propagation stages. Numerical and analytical linear-elastic fracture mechanics (LEFM) calculations based on two- and three-dimensional models are performed for constant and variable amplitude loads. All analytical assessments of the V-notched specimen illustrate conservative results compared to testing, and the numerical results match the experimental investigations well. The maximum deviation of results is observed at variable amplitude loading due to missing retardation effects for the LEFM calculations. Preliminary distortion measurements and application of strain gauges on semi-elliptical notched specimens are performed to investigate the influence of angular deformation due to clamping on the local stress distribution. A simple model accounting for superimposed static bending stresses due to clamping is able to improve the crack growth predictions for semi-elliptical surface cracks significantly. A final comparison of the fractographic analyses and the numerical crack propagation calculations illustrates differences in the results and provides information to assess the fatigue crack growth and service inspection intervals of components under variable amplitude loading more precisely.

A dislocation based method using generative algorithms to model notch geometries: Determination of Mode I SIFs

This Technical note presents a parametric algorithmic tool to generate two-dimensional notches allowing the use of the DDT to solve the problem of a crack emanating from an arbitrarily shaped notch subjected to Mode I loading. The numerical formulation of the DDT for assessing notches of any shape as well as the basic steps implemented within a graphical algorithm editor to create and discretise the notch geometry are described. Finally, the applicability and accuracy of the procedure is assessed by investigating V-notches, U-notches, semi-elliptical notches and dovetail-like notches. The tool proves to be a simple, general and efficient way of assessing Stress Intensity Factors (SIFs) for cracks emanating from notches with low computational cost and numerical modeling expertise in comparison with other numerical methods, as the Finite Element Method.

Stress and Stress Intensity Factor Near Notches in Thick Cylinders

This work investigates the impact of semi-elliptical notches (erosion grooves) at the bore of pressurized and autofrettaged thick cylinders. It provides a robust yet rapid method to determine the rapidly varying stress in the near-notch region prior to crack development and crack tip stress intensity factor after a crack develops at the notch root. The procedure involves a sequence of asymptotic solutions and adjustments. A superposition is presented for the stress concentration factor (SCF) of a small edge notch in a pressurized cylinder with bore pressure infiltrating and acting upon the notch surface. A procedure for adjusting this SCF to account for a varying pre-existing stress field is described. This provides accurate predictions for notch depths of up to 15% of wall thickness. Stress variation beyond the notch root is determined by scaling analytic solutions. Solution accuracy appears to be approximately 5%. Stress profiles were used to calculate stress intensity factor (SIF) for cracks emanating from the notch root and deep into the wall. There are notable differences between SIF behavior in the pressurized tube and in the autofrettaged tube. The main reason for this difference is that compressive reyielding near the notch disrupts the residual compressive stress profile over an extended distance. This leads to the conclusion that a crack originating from a notch in an autofrettaged tube exhibits a much higher cyclic SIF range during pressurization than the same length crack originating from the bore. This will cause higher fatigue crack growth rates and shorter fatigue lifetimes.

A Point Dislocation Interacting with an Elliptical Hole Located at a Bi-Material Interface

The problem of a point dislocation interacting with an elliptical hole at the interface of two bonded half-planes is studied. Complex stress potentials are obtained by applying the methods of complex variables and conformal mapping. A rational mapping function that maps a half plane with a semi-elliptical notch onto a unit circle is used for mapping the bonded half-planes. The solution derived can serve as Green’s function to study internal cracks interacting with an elliptical interfacial cavity.

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.

Heat Conduction and Thermal Stress Analyses of a Strip with Semi-Elliptical Notch Under Electric Current

An electric current is applied in a thin strip conductor with an edge crack or a semi-elliptical notch. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. The Joule heat gives rise to the temperature, and thus the heat flux and also free thermal expansion. Using a rational mapping function, problems of the electrical current, the Joule heat, the temperature, the heat flux and the thermal stress are analyzed and their each solution is obtained as a closed form. The heat conduction problem is solved as a temperature boundary value problem. The distributions of the electric current, the Joule heat, the temperature, and the heat flux are shown in figures. After a free thermal expansion is obtained and a support condition is given, the stress components are calculated. The crack problems are analyzed for the problems of the electric current, joule heat, heat flux and stress. The expressions for intensity of the electric current, the heat flux and the Joule heat at the crack tip are derived. The exact expression of the temperature at the crack tip (notch tip) is also given. The notch problems are analyzed, and the approximate expressions of the concentration are given for each problem, of which error is a few percent. The relations between the electric current density and the melting temperature and between the electric current density and Joule heat are investigated for some crack lengths of aluminum strip. The intensity of the Joule heat relates to the creation of the blow hole.

Experimental and numerical study of stress–strain state of pressurised cylindrical shells with external defects

Work presents the experimental technique with using of strain gauges for determining of strain distribution near longitudinal external defects of semielliptical shape in pressurised cylindrical shells and an appropriate procedure of numerical calculation based on a finite element method (FEM) for assessment of strain state near such defects. Numerical assessment of elastic stress distribution at bottom of external semielliptical notch in pressurised cylindrical shell is in a good agreement with the data received on the base of different analytical models. Here, the FEM results and data based on Glinka-Newport model are the most close for maximal stress. A comparison an experimental and calculation results showed on their acceptable coincidence and this fact gives the ground for using the numerical calculations instead labour-intensive and expensive experimental tests.

Short crack threshold estimates to predict notch sensitivity factors in fatigue

The notch sensitivity factor q can be associated with the presence of non-propagating fatigue cracks at the notch root. Such cracks are present when the nominal stress range Δ σ n is between Δ σ 0/ K t and Δ σ 0/ K f, where Δ σ 0 is the fatigue limit, K t is the geometric and K f is the fatigue stress concentration factors of the notch. Therefore, in principle it is possible to obtain expressions for q if the propagation behavior of small cracks emanating from notches is known. Several expressions have been proposed to model the dependency between the threshold value Δ K th of the stress intensity range and the crack size a for very small cracks. Most of these expressions are based on length parameters, estimated from Δ K th and Δ σ 0, resulting in a modified stress intensity range able to reproduce most of the behavior shown in the Kitagawa–Takahashi plot. Peterson or Topper-like expressions are then calibrated to q based on these crack propagation estimates. However, such q calibration is found to be extremely sensitive to the choice of Δ K th( a) estimate. In this work, a generalization version of El Haddad–Topper–Smith’s equation is used to evaluate the behavior of cracks emanating from circular holes and semi-elliptical notches. For several combinations of notch dimensions, the smallest stress range necessary to both initiate and propagate a crack is calculated, resulting in expressions for K f and therefore for q. It is found that the q estimates obtained from this generalization, besides providing a sound physical basis for the notch sensitivity concept, better correlate with experimental data from the literature.

Pipe-defect assessment based on the limit analysis, failure-assessment diagram, and subcritical crack growth

We analyze the serviceability of a pipe containing a defect by the following three methods: limit analysis, failure-assessment diagram, and subcritical crack growth. The use of the proposed complex approach is explained by the fact that, despite the plasticity of pipe steels, they can suffer quasibrittle fracture (under the conditions of plane deformation) or fail as a result of plastic collapse. Moreover, the final fracture of a pipe with defect is preceded by the stage of stable crack growth. Therefore, in analyzing the serviceability of the pipe, one should also take into account the “leak-before-break” criterion. To take into account all these circumstances, it is necessary to use the indicated three methods, as in the case of evaluation of the serviceability of a pipe of API X52 steel containing a surface semielliptic notch. Defects of this sort reflect the influence of the surface gouges and dents responsible for the failures of pipes under the action of internal pressure.

Green’s functions for a bi-material problem with interfacial elliptical rigid inclusion and applications to crack and thin rigid line problems

The Green’s functions for a point force and dislocation interacting with interfacial elliptical rigid inclusion in a bonded bi-material system are obtained by applying complex variable method and conformal mapping technique. The problem of an internal crack or thin rigid line interacting with the interfacial inclusion is then examined. For mapping the half plane with a semi-elliptic notch a rational mapping function is used. This helps in evaluating certain contour integrals quite easily. The Green’s function solutions are then used to simulate internal cracks or thin rigid lines to study their behavior in the presence of interfacial inclusion. Some interesting observations pertaining to the interaction between rigid inclusion and crack as well as between rigid inclusion and thin rigid line are discussed. In particular, stress intensity factors (SIF) at the tips of internal crack or stress singularity coefficients (SSC) at the tips of thin rigid line exhibit markedly different behavior depending on loading direction and distance between interfacial inclusion and crack (thin rigid line).

Green’s Function of a Bimaterial Problem With a Cavity on the Interface—Part I: Theory

The problem of a point dislocation interacting with an elliptical hole located on a bimaterial interface is examined. Analytical solution is obtained by employing the techniques of complex variables and conformal mapping. A rational mapping function is used to map a half-plane with a semielliptical notch onto a unit circle. In the first part of this paper, complex potentials for the bimaterial system with an elliptical hole on the interface is derived when a point dislocation is present in the upper half-plane without loss of generality. The solution derived can be used as Green’s function to study internal cracks interacting with an elliptical interfacial cavity.

Interaction among a Row of N Semi-Elliptical Notches and Edge Cracks

Interaction among a Row of N Semi-Elliptical Notches and Edge Cracks

Fracture resistance of cracked duplex stainless steel elbows under bending with or without internal pressure

Electricité De France (EDF), in co-operation with Framatome, has conducted a large research programme on the fracture behaviour of aged cast duplex stainless steel elbows. One important task of this programme consisted of testing three large diameter (580 mm O.D.) thermally aged cast elbows containing either one or two semi-elliptical notches on the outer surface of the flank. The first two elbow tests (called SEM1 and SEM2) were conducted under in-plane closure bending at 320°C and the third one (called SEM3) under constant internal pressure and in-plane closure bending at 60°C. Those tests were carried out with 2/3-scale models of PWR primary loop elbows. This paper presents the results of the experiments and the results of the fracture mechanics analyses, based on finite element calculations.