Elliptical Notch Research Articles

Non-Propagating Cracks at the Fatigue Limit of Notches: An Analysis of Notch Sensitivity and Size Effect.

The issues of the high-cycle fatigue resistance of notches and the role of non-propagating short cracks in defining the fatigue notch sensitivity and fatigue limit of the configuration are addressed. A fracture mechanics approach is employed to determine the threshold configuration that defines the associated fatigue limit. The influence of notch sharpness, notch size, intrinsic fatigue limit, microstructural dimensions, and the threshold for crack propagation is examined. A simple expression is proposed to estimate the maximum fatigue notch factor, kfMax, which incorporates the influence of these non-propagating cracks. The fatigue limits for both blunt and sharp elliptical notches are analyzed and predicted based on experimental results reported in the literature. Additionally, shallow notches or small defects are analyzed, where it is found that the same hypothesis may not be applicable.

Stress Intensity Factor Solutions for Eccentric Annular External Cracks in Notched Round Bars under Tensile Loading

In this paper, stress intensity factor (SIF) solutions of eccentric annular external cracks in elliptical notched round bars under tension loading have been obtained from 3D finite element analysis, along with their relation to the energy release rate obtained with the J-integral contour. The analysis variables have been the ligament diameter, its eccentricity, and the elliptical notch aspect ratio. The maximum SIF increases with the ligament eccentricity, the presence of the notch (compared to when the bar is smooth), and the elliptical notch axial semi-axis (for the same notch depth); it decreases with the ligament diameter. For external cracks, eccentricity induces bending of the bar subjected to tensile loading, which can produce partial and full contact of the crack surface, relevant phenomena in terms of the SIF value at the different points of the crack front.

Phase-field simulation of fractographic features formation in soda–lime glass beams fractured in bending

The identification of fractographic features, such as the mirror–mist boundary, is central in the fractographic analysis of silicate glasses and other brittle materials. Although phase-field simulations have been previously applied to unstable cracks propagating in amorphous brittle materials, limited efforts have been made to establish the formation of fractographic features. This work proposes two distinct approaches to predict the formation of the mirror–mist boundary in soda–lime glass. Glass beams with embedded corner elliptical notches loaded in bending were considered, and unstable crack propagation was simulated in phase-field. In one approach, the ‘mist’ was expected to form when the crack–front’s speed reached a critical threshold. In the second approach, the thickness of the damaged zone predicted by phase-field was tracked and correlated to the formation of the fractographic features. For the two proposed methods, both the shapes of the estimated mirror–mist boundaries and the magnitudes of the mirror radii were found to be in good agreement with experimental observations gathered from soda–lime glass beams fractured in bending.

Notch Effects on the Stress Intensity Factor and on the Fatigue Crack Path for Eccentric Circular Internal Cracks in Elliptically Notched Round Bars under Tensile Loading.

In this paper, stress intensity factor (SIF) solutions are numerically obtained for notched bars subjected to tensile loading containing an eccentric circular inner crack located in the cross-section corresponding to the notch root. The finite element method and the J-integral have been used to obtain the SIF and to analyze the effect on it of three elliptical notch geometries (of equal radial depth). The results show how the SIF is greater in the notched bars than in the smooth bar and within the former when the axial semi-axis of the notch rises, its effect being greater as the diameter and eccentricity of the inner crack increase. In addition, the fatigue growth of an eccentric crack induces an increase in such eccentricity, greater as the notch axial semi-axis increases. The cause of these phenomena can be attributed to the constraint loss caused by the notch, which also facilitates bending of the specimen due to the asymmetry generated by the crack eccentricity.

2D nonlocal elasticity: Investigation of stress and strain fields in complex shape regions

AbstractThe paper presents the features of solutions to nonlocal elasticity problems in domains with stepped transitions and elliptical notches. A comparative analysis of the solutions obtained in the classical and nonlocal elasticity theories was carried out, as well as an analysis of the influence of the main parameters of the model on the final results. The presented solutions for the region with a step transition are in qualitative agreement with the experimental data and have similar portraits of deformations and resultant stresses along the loading axis. In regions with elliptical notches, similar effects are observed. In addition to qualitative agreement with experiments, solutions in the nonlocal formulation have lower stress values in the concentrators and exhibit edge effects, which are characterized by sharp stress and strain gradients at the domain boundaries. The solutions were carried out numerically by the finite element method.

A semi-analytical quasi-brittle fracture criterion for elliptical notches

The elliptical notch is a widely used model to investigate the failure due to stress concentrations. Although the size effect of the fracture strength of elliptical notches has been studied by many researches, it is still desirable to reveal a clearer physical mechanism. For this purpose, a fracture criterion is proposed in this paper based on an effective stress intensity factor that considers the fracture process zone. This work is greatly facilitated by a semi-analytical stress intensity factor for symmetric cracks around the elliptical notch established with the guidance of the complex potential method. Comparisons with experimental results demonstrate that this criterion has strong predictive power. In addition, the relationship between this criterion and the theory of critical distances is especially highlighted, presenting a clear physical image of this method.

Tuning lamellar structure distribution around elliptical notch in Ti-6Al-4V for fatigue performance

The paper is aimed to investigate the effect of lamellar structure distribution around elliptical notch in bimodal titanium alloys of Ti-6Al-4V. In this regard, within the framework of crystal plasticity based finite element (CPFE), Representative volume element (RVE) model with elliptical notch for bimodal Ti–6Al–4V is developed, to address the effect of lamellar structure on fatigue resistance performance under external fatigue loading. Lamellar structure distribution around elliptical notches with different aspect ratios are investigated. Strain along radial and circular paths are extracted to identify the possible crack initiation points. The numerical results show that, Strain distribution around notches can be adjusted through lamellar structure distribution such that strain uniformity is achieved. Increase of strain oscillation caused by lamellar structure leads to the improvement of fatigue resistance. With the increase of notch aspect ratio, elliptical notch eventually evolves into line crack and tuning performance of lamellar structure appreciably declines. This numerical study provides a new insight into lamellar structure tuning in dual-phase titanium alloys.

Design and Analysis of Compact Diversity Antenna for Wearable Applications

A new compact Ultra-Wideband (UWB) antenna is designed for wearable applications using jeans fabric as the substrate. The proposed antenna has an impedance bandwidth from 3.1 to 11.1 GHz. The antenna footprint is of size 18mm x 26 mm, and the thickness of the substrate is 0.8mm. It is based on a simple patch antenna configuration where the radiator consists of a simple rectangular structure and an ellipse added on top. Circular and elliptical notches are removed from the radiator, and rectangular notches are removed from the feed line for bandwidth enhancement. A partial ground plane of size 18mm x 9mm with a square notch is present on the other side of the antenna. Two such antennas are placed side by side as unit cells, one horizontal and the other vertical, to obtain a 2-port diversity set. The unit cells are placed at a distance such that minimal mutual coupling is present between the cells, and the size is also not compromised. The final diversity set so obtained is of size 44mm x 26 mm. The simulation results for reflection coefficients, SAR, and radiation pattern for both unit cell and diversity port are presented. Diversity parameters like ECC and diversity gain are also simulated for diversity antenna. The results obtained by testing the fabricated prototype are found to be matching reasonably with the simulated results. The antenna is also subjected to bending analysis.

Fatigue life analysis for 6061-T6 aluminum alloy based on surface roughness.

Surface condition is one of the dominant factors affecting fatigue life. Considering the complexity of surface condition, a relatively efficient and economic approach based on surface reconstruction and interpolation method was proposed. The effect of surface roughness on the fatigue life of 6061-T6 aluminum alloy is studied to analyze the fatigue life by surface roughness parameters. Surface topography was simplified into a series of elliptic micro notches, and empirical formula for stress concentration factor is established based on simulation work. Then the extraction method of surface curve is proposed to effectively represent the real surface roughness through 3D model reconstruction. Experiment of surface roughness verified the correctness of the model. The relationship between surface roughness and fatigue life is established and the calculated value of the fatigue life is compared with the test results. The maximum error is 15.65%, indicating that the formula established is reasonable and effective.

Ratcheting Response of SS316 Steel Samples with Different Notch Shapes under Various Loading Spectra

Ratcheting response of notched 316 stainless steel samples was examined under various low-high-high (LHH) and high-low-low (HLL) loading steps and sequences. Plate samples were tested under uniaxial asymmetric stress cycles contained circular or elliptical notches in center. Local ratcheting strain increased over loading steps of steel samples tested with LHH loading sequence, while it dropped in magnitude for samples tested with HLL loading histories. Local ratcheting strains were further evaluated through use of the Ahmadzadeh–Varvani (A–V) and Chaboche (CH) hardening models coupled with Neuber’s rule to predict ratcheting at the vicinity of notch root in steel samples. Finite element (FE) method was employed to numerically assess local ratcheting around the notch root through the use of ABAQUS software. Horizontal X-ellipse, circular and vertical Y-ellipse notch shapes lead to the highest, intermediate and lowest local ratcheting results for steel samples as predicted by means of the A–V and CH frameworks as well as those simulated results through FE method. Simulated ratcheting results through FE analysis positioned on or above the measured values for various loading steps and sequences. Predicted ratcheting results through the CH model positioned slightly above measured data over the first loading step and were turned to higher magnitudes over steps 2 and 3 in LHH samples with different notch shapes. This model underestimated ratcheting over loading steps 2-3 for HLL loading sequences. The predicted results by the A–V model consistently agreed with experimental data over the first step in notched samples tested under LHH and HLL sequences. Last two steps however resulted in lower ratcheting results through this model.

Anti-plane dynamic analysis of a quarter-space piezoelectric material with an elliptic notch in the corner

The aim of this article is to investigate the scattering of anti-plane shear waves by elliptic notches in a quarter-space piezoelectric material. The wave field and electric field expressions in this region are derived using the wave function expansion method. An infinite linear system of equations is established by applying the boundary conditions of no traction and no potential. The results obtained are accurate due to the good convergence of the wave function. The displacement amplitude and dynamic stress in the vicinity of the elliptic arc notch under different characteristic parameters and different wave numbers are computed numerically.

Effect of elliptical notches on mechanical response and progressive damage of FMLs under tensile loading

Abstract This paper aims to explore the residual strength and elucidate the failure mechanisms of elliptical notched FMLs with different design variables through experimental investigations and numerical simulations. Quasi-static tensile experiments are carried out to investigate the effects of off-axis angle, inclination angle and aspect ratio on the tension behaviors with the aid of digital image correlation technique (DIC). Subsequently, a progressive damage model integrated with a VUMAT subroutine is employed to study the progressive damage evolution and failure mechanism, considering thermal residual stress. Meanwhile, damage initiation and evolution as well as the finial damage patterns are explored systemically combining the strain distributions from DIC technique, equivalent plastic strain distributions of aluminum layers and final damage morphologies of specimens. Results show that the notch strength is more sensitive to off-axis angle and inclination angle as compared with the aspect ratio. The failure morphology exhibits a neat and straight crack for on-axis cases, whereas it presents a fracture along the fiber direction of adjacent layer with apparent pull out of fiber for off-axis cases. With the increase of off-axis angle, the ultimate failure mode gradually transfers from fiber-driven to aluminum-driven accompanied by the corresponding critical failure mechanism varying from tension to tension-shear.

Numerical-and-Analytical Method for Creep Investigation in Functionally Graded Complex-Shaped Bodies of Rotation

The spatial axisymmetric creep problem for complex-shaped bodies of rotation from functionally graded materials is considered. For the variational problem statement, the Lagrangian-form functional is defined for kinematically possible displacement rates. For the main unknown creep parameters, namely displacements, stresses, and strains, the Cauchy problem in time scale is formulated for the spatial discretization points. In this case, the initial conditions for the unknown functions are derived by solving the problem of elastic deformation of the body. The numerical-andanalytical method was developed for solving a nonlinear initial boundary-value creep problem based on applying the R-function, Ritz, and Runge–Kutta–Merson methods. The advantages of the proposed method include: the accurate analytical account of the boundary value problem geometry features without their approximation, representation of an approximate solution in the analytical form, and automatic selection of a time step. The creep problem solutions are derived for a hollow straight cylinder and a complex-shaped body of rotation, namely a cylinder with an elliptical notch on the outer surface, loaded by constant internal pressure. The material creep is described by the Norton law. Several variation patterns of the material creep properties along the radial coordinate are considered. The effect of the material gradient properties and geometry on the stress-strain state of bodies of rotation is analyzed. It is shown that the above geometry effect in creep strongly depends on the material properties.

CMOS Analog Filter Design for Very High Frequency Applications

A design strategy for the synthesis of high-selectivity/low-order analog filters in Complementary Metal-Oxide-Semiconductor (CMOS) technology for very high frequency (VHF) applications is presented. The methodology for the reconstitution of a given transfer function by means of Signal Flow Graphs (SFG) manipulation in canonical form is proposed leading to a fully differential g m -C biquad filter. As a practical example, the design of a notch filter intended to suppress interferers in the lower sideband (400 MHz) of the Medical Implant Communication Service (MICS), in single-poly, 6-metal layers; Mixed-Signal/RF 0.18 µm CMOS technology is realized. To compare the performance of the proposal with some other solution, the design of a 7th order elliptic notch filter based on Frequency Dependent Negative Resistors (FDNRs) was also accomplished. The attained simulation results prove that the proposal is competitive compared to the FDNR solution and some other state-of-the-art filters reported in the literature. The most salient features of the proposed notch biquad include: the selectivity, whose value is comparable to that of a 7th order elliptic approach and some other 3rd order filters; a high-frequency operation without resonators; linearity, with a +15 dBm I I P 3 ; a reduced form factor with a total occupied area of 0.004282 mm2 and mostly a low design complexity.

Analytic Solutions to Two-Dimensional Decagonal Quasicrystals with Defects Using Complex Potential Theory

An analytical treatment for two-dimensional point group 10 mm decagonal quasicrystals with defects was suggested based on the complex potential method. On the basis of the assumption of linear elasticity, two new conformal maps were applied to two examples: the first was an arc with an elliptic notch inner surface in a decagonal quasicrystal, where the complex potentials could be exactly obtained; and the second was concerned with a decagonal point group 10 mm quasicrystalline strip weakened by a Griffith crack, which was subjected to a pair of uniform static pressures. Using the basic idea underlying crack theory, the extent of the stress intensity factors was analytically estimated. If the height was allowed to approach infinity, these results can be turned into the known results of an “ordinary” crystal with only phonon elastic parameters when the phason and phonon-phason elastic constants are eliminated.

Double scaling and master curve to predict Kts for elliptically notched orthotropic plates from Kts in circularly notched isotropic plates

The key role played by the so-called stress concentration factors (symbolically usually referred to as Kts) while performing either analysis or design in both mechanical engineering and structural engineering is a well-proven fact; hence, the corresponding accuracy and ease of determination in their estimation have critical implications related to engineering matters. In a previous work, we showed that starting with the Kts for rectangular isotropic plates with circular holes the Kts for rectangular orthotropic plates with elliptic holes can be easily and accurately predicted, using a double scaling procedure (a geometric scaling and a material scaling) together with a basic Kt curve employed as a master curve. In the present work we investigate, in an engineering heuristic manner, whether the stated double scaling procedure exhibits a more general structure so that it can accurately be applied to different plate stress raiser geometries. Specifically, we examine if starting with the known Kts for rectangular isotropic plates with circular notches, used as a master curve, the Kts for orthotropic plates with elliptic notches can be easily and accurately predicted as well. Using a simple hand calculator, we show that the resulting proposed set of predicting mathematical expressions makes virtually unnecessary the use of complex numerical programs (FEM based for example) to determine the corresponding Kts, producing at the same time accurate results with a maximum average recorded error of 2.59%. Additionally, given that a Kt predictive procedure for holed plates was presented in our previous work, and for notched plates in this work, we found necessary first to examine the hypothesis that suggests using Kts from holed plates as good approximations for Kts for notched ones and vice versa. We found this hypothesis not to be valid for either isotropic or orthotropic plates. In the course of this work it was found, rather significantly, that the well-known double square root material contribution to Kts is valid for both holed plates as well as notched ones.

Modeling and design of a two-axis elliptical notch flexure hinge.

As an important part of the joule balance system, the two-axis elliptical notch flexure hinge (TENFH) which typically consists of two single-axis elliptical notch flexure hinges was studied. First, a 6 degrees of freedom (6-DOF) compliance model was established based on the coordinate transformation method. In addition, the maximum stress of the TENFH was derived. The compliance and maximum stress model was verified using finite element analysis simulation. To decouple the attitude of the suspended coil system and reduce the offset between the centroid of the suspended coil mechanism and the mass comparator in the joule balance system, a new mechanical structure of TENFH was designed based on the compliance model and stress model proposed in this paper. The maximum rotation range is up to 10°, and the axial load is more than 5 kg, which meets the requirements of the system. The compliance model was also verified by deformation experimentation with the designed TENFH.

A novel design of Scarecrow-shaped patch antenna for broadband applications

A novel Scarecrow-shaped patch antenna has been presented for broadband applications. The proposed antenna composed of an elliptical shape radiating patch, partial ground plane, and 50 Ω microstrip line which is Boolean added with the radiating patch. To achieve wide impedance bandwidth, an elliptical notch and two rectangular-shaped parasitic slots have incorporated on the ground plane. This antenna has printed on FR-4 substrate (tan(δ) = 0.02, εr = 4.3) with the thickness of 1.6 mm. It is energized by coaxial feed which is connected with a microstrip line through via. The proposed configuration has examined numerically with the help of CST Microwave Studio and experimentally also. The broadband property of this antenna has been achieved by adjusting the dimensions of an elliptical notch. The antenna exhibits the bandwidth of 116.32% from 1.23 to 4.65 GHz for S11 <−10 dB. Current distribution and far-field pattern at resonating frequencies 1.29, 1.77, 2.28, 2.94, and 3.72 GHz have been studied. It has been found that the far-field pattern is distorted at frequency 3.72 GHz due to higher order modes. Effect of parameters on frequency response of the antenna has also been investigated to know the behavior of the antenna.

Numerical solution for multiple confocal elliptic dissimilar cylinders

This paper provides a numerical solution for multiple confocal elliptic dissimilar cylinders. In the problem, the inner elliptic notch is under the traction free condition. The medium is composed of many confocal elliptic dissimilar cylinders. The transfer matrix method is used to study the continuity condition for the stress and displacement along the interfaces. Two cases, or the infinite matrix case and the finite matrix case, are studied in this paper. In the former case, the remote tension is applied in y- direction. In the latter case, the normal loading is applied along the exterior elliptic contour. For two cases, several numerical results are provided.

Crack initiation in elliptically notched plates

AbstractIn this work, crack initiation in elliptically notched plates under uniaxial tension is studied by means of a Finite Fracture Mechanics analysis. The stress concentration factor changes with the aspect ratio of the half axes: from unity for slender ellipses in loading direction to infinity for crack‐like ellipses normal to the loading. The closed‐form solutions of the stress field and an approximate formulae of the stress intensity factor of mode I cracks allows for an efficient Finite Fracture Mechanics analysis of plates with elliptical notches with arbitrary aspect ratios. The results show that the transition from vanishing stress concentration to infinite (crack‐like) stress concentration is rendered continuously by the present approach. A transition from strength of materials to Linear Elastic Fracture Mechanics is obtained that shows distinct size effects. A comparison to experimental results on size effects of notched plates shows good agreement. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)