Notch Problems Research Articles

Simetrik V Şekilli Plakadaki Gerilme Yığılma Faktörünün Yapay Sinir Ağı ile Modellenmesi

Machine parts are exposed to stress accumulation due to geometric differences. Determining the stress accumulation locations is crucial to the design procedures. Studies on stress concentrations have been conducted in the past using a variety of theoretical and experimental methodologies, and distinct interpretations have been offered depending on the geometry of the machine part to be produced. The ability to complete activities with the least amount of effort and in the shortest amount of time has emerged as a result of the new computer technologies and software that have impacted many aspects of our everyday lives. One of these methods is the artificial neural networks (ANN) model, which is a branch of artificial intelligence. It is argued as a thesis in this study that fast and low-cost solutions can be found to problems in the field of solid mechanics by using the ANN model. For this purpose, a model has been developed to determine the SCF value with the ANN model of a plate with symmetrical V-shaped notch. The graphs obtained from previous experimental studies were converted to digital format and the Kt values obtained for the V-shaped notch problem with different parameters were converted into a data file. In this file, the SCF values to be obtained according to the strength upper limit safety factor value of the machine part, depending on the dimensional dimensions and material type required for the design, are calculated numerically in the form of an Excel file. An ANN-based code was created in MATLAB software and a new solution method was presented for parts containing a V-shaped notch.

On the effects of higher order stress terms in pure mode III loading of bi‐material notches

AbstractThe effects of higher order terms (HOTs) on the stress distribution at the vicinity of the sharp corner of two bonded dissimilar materials were investigated under pure out‐of‐plane (mode III) loading. To achieve this aim, finite element (FE) analysis in conjunction with the over‐deterministic technique were performed for two specimens, each containing four different notch opening angles. The coefficients of singular and non‐singular stress terms were then computed from the displacement field determined from the FE analysis. The elastic stresses obtained directly from the FE analysis were compared with those reconstructed from the truncated series expansions, in order to validate the accuracy of the calculated coefficients. The numerical results show that although the contribution of HOTs in the out‐of‐plane stress field is less than that of the in‐plane notch problems, ignoring their effects can lead to considerable errors in the stress field estimation. The magnitude of numerical error depends on the geometry and loading conditions of the bi‐material notch.

The revisited phase-field approach to brittle fracture: application to indentation and notch problems

In a recent contribution, Kumar et al. (J Mech Phys Solids 142:104027, 2020) have introduced a comprehensive macroscopic phase-field theory for the nucleation and propagation of fracture in linear elastic brittle materials under arbitrary quasistatic loading conditions. The theory can be viewed as a natural generalization of the phase-field approximation of the variational theory of brittle fracture of Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998) to account for the material strength at large. This is accomplished by the addition of an external driving force—which physically represents the macroscopic manifestation of the presence of inherent microscopic defects in the material—in the equation governing the evolution of the phase field. The main purpose of this paper is to continue providing validation results for the theory by confronting its predictions with direct measurements from three representative types of experimentally common yet technically challenging problems: (i) the indentation of glass plates with flat-ended cylindrical indenters and the three-point bending of (ii) U-notched and (iii) V-notched PMMA beams.

Higher order stress terms in sharp notch problems under pure‐out‐of‐plane loading

AbstractCurrent paper studies the effects of higher order terms (HOTs) on the stress and displacement fields around sharp V‐notched components under pure antiplane shear (mode III) loading. For this aim, two V‐notched test samples each containing four different notch angles are simulated in a finite element (FE) commercial software. Subsequently, the over‐deterministic technique is used as an approach to calculate the coefficients of singular and nonsingular terms out of the nodal displacement values. To verify the accuracy of the values calculated as the coefficients, the stress field reconstructed from the truncated series expansions is compared with the direct nodal outputs of FE analysis. The results demonstrate that the use of nonsingular terms of the asymptotic series expansion is necessary only in the case that the distribution of stress at far distances from the notch tip is sought and the notch angle is small. To check the stability of the results, some of the parameters which affect accurate determination of unknown coefficients and convergence of the numerical results are studied.

Solution for square notch problem with small round corners in plane elasticity under remote loading (technical note)

In the paper, the solution for square notch problem with small round corners in plane elasticity under remote loading is studied. Based on the two obtained particular solutions and the extrapolation technique, the general relation between the stress concentration and the small radius at the corner point is obtainable. It is emphasized here that this relation can be used to the arbitrary small radius case.

A plane strain analysis of the elastostatic fields near the notch-tip of a Blatz-Ko material

An asymptotic plane strain analysis is carried out in order to study the notch problem in a homogeneous isotropic compressible hyperelastic material whose behavior is governed by the Blatz-Ko constitutive law. The work is realized within the framework of the fully nonlinear fracture mechanics at large strain. Close to the notch tip, a stress-free state is assumed in the notch edges, while at infinity (sufficiently far to the notch tip) fracture modes I and II are considered. The results emphasizes many discrepancies with linear theory predictions like the emergence of a logarithmic singularity and show that solution depends on the notch initial opening angle.

A singular element based on dual interpolation BFM for V-shaped notches

• A new singular element based on dual interpolation boundary face method is proposed. • The new element takes into account the variable singularity orders at the notch tip . • Special basis function is introduced to build the shape functions of the proposed element. • The dual interpolation method can provide much higher accuracy than traditional interpolation method. • The numerical results for stress intensity factors obtained by the new element are convergent. A singular element based on dual interpolation boundary face method (DiBFM) is presented for solving V-shaped notch problems in this paper. The stress field around sharp notches is singular, and the singularity orders vary with the notch angle. Thus an element with usual shape functions or traditional singular element cannot lead to high accurate results. To accurately model the distribution of displacement around the notch tip, a new displacement singular element based on DiBFM is proposed. The new element takes into account the variable singularity orders at the notch tip. The dual interpolation method which combines conventional polynomial element interpolation and moving least-squares approximation can provide much higher accuracy than traditional interpolation method. With the proposed singular element based on DiBFM, more accurate displacement results in the vicinity of the notch tip can be evaluated, thus more accurate stress intensity factor (SIF) of the V-shaped notches can be obtained. Numerical examples have demonstrated the validity and accuracy of our method.

The singular elastostatic fields at the notch-tip of a compressible Ciarlet-Geymonat material

This paper is devoted to the asymptotic analysis of the plane strain notch problem in a compressible Ciarlet-Geymonat material. The far field conditions correspond to mode I and mode II loadings, while close to the notch tip, a traction-free state is assumed. The analysis shows that solution depends on the notch opening angle and that it is necessary to refine asymptotic approximation to obtain coherent solution. The stress field analysis reveals that there exists more than one singular term and that the dominant term is stronger than that predicted by the linearized theory. It was also shown that the notch opens close to the notch tip even if the applied loading is antisymmetric, which is also in contradiction with linear theory.

Multi-parameter average strain energy density factor criterion applied on the bi-material notch problem

The stress field near a bi-material notch tip has the singular character. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress term exponent. The terms can be either singular or non-singular depending on the value of stress term exponent. The exponents are determined as solution of eigenvalue problem and depend on material properties and local geometry. The factors are calculated by analytical-numerical approaches and depend on global geometry and applied loading. The singular terms become unbounded when approaching the bi-material notch tip while the non-singular terms vanish. The non-singular terms increase precision in stress description on larger distances from the notch tip. The crack initiation conditions from the notch tip can be assessed by the average strain energy density factor (SEDF) criterion. In the case of bi-material notch problem, the crack can initiate in the direction of global minimum, the local minimum or the interface. The actual crack initiation load is calculated for each of abovementioned directions considering the individual fracture toughnesses. Contrary to the case of a crack, the direction of minimum of SEDF changes in dependence on the distance from the tip, therefore the average value over specific distance is used. The multi-parameter criterion considers singular and non-singular stress terms. Therefore in cases, when the specific distance is large, the non-singular terms provide means to improve precision of prediction of crack initiation parameters.

Numerical fatigue life estimation of aluminium 6201-T81 wires containing geometric discontinuities.

Previous works have shown that fretting fatigue can be modelled as a notch problem due to the mild superficial damage produced in the partial slip regime during fretting. In this context, the aim of this numerical and experimental study is to estimate the fatigue life of aluminium 6201-T81 wires, used in overhead conductors, containing geometric discontinuities. These discontinuities induce a stress field analogous to the fretting problem. To perform the fatigue life assessments theLM-Nfrelationship, which is an extension of the Theory of Critical Distance (TCD), was applied. TheLM-Nfrelationship was calibrated using twoS-Ncurves, one of a plain wire and other of a notched wire. The validation of the numerical approach estimations was performed using a wire containing a through hole. The estimations were fallen almost entirely among factor 3 bands when compared with the experimental results. This factor 3 represents the biggest scatter in theS-Ncurves used to calibrate theLM-Nfrelation. The TCD approach applied, in terms of point method, provide accurate life estimations in the range 105and 106cycles.

A unified solution approach for a large variety of antiplane shear and torsion notch problems: Theory and examples

In this work, a unified solution approach is proposed for the analytical evaluation of the stress fields close to notches under antiplane shear and torsion loadings, which allows a large variety of notch problems to be tackled. The method is based on the complex potential approach for antiplane elasticity combined with the use of proper conformal mappings. In particular, it is shown that a well defined analytical link does exist between the complex potential to be used to determine stresses and the first derivative of the conformal mapping used to mathematically describe the notch profile. This makes some methodologies such as Schwarz-Christoffel transformation, which allows describing any polygonal domain automatically, very attractive for the direct solution of notch problems. A bulk of solutions are provided to support this finding, from cracks and pointed notches, to radiused notches. In addition, the accuracy of each proposed solution is discussed in detail taking advantage of a bulk of results from FE analyses.

Inclined Hole Under Different Loading Conditions: A Review of Recent Results

Three-dimensional (3D) elastic stress distributions in the vicinity of the sharp corners of an inclined diamond hole in a plate are investigated. A detailed 3D finite element model under different loading conditions is analyzed to study the intensity of different fracture modes due to the thickness effect. The stress results are compared with those provided by a recent theory which reduces the 3D governing equations of elasticity to a differential equation system, which includes a biharmonic equation and a harmonic equation. They provide the solution of the corresponding in-plane and out-of-plane notch problem, respectively, and have to be concurrently satisfied. Comparing numerical results and theoretical stress distributions, a good agreement is found.

Analytical solution for the three-dimensional stress fields in anisotropic composite bimaterial corners

The paper presents a theoretical treatise for the three-dimensional elastic stress distributions close to anisotropic bimaterial corners according to which the 3D governing equations are successfully reduced to two uncoupled equations in the two-dimensional space. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. With the new theory an analytical solution for the stress fields for bimaterial corners in thick anisotropic composite plates is presented and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE analyses carried out on bimaterial composite plates and joints.

Effect of Machining the Load Capacity Notched Components

Nowadays is issue investigation the notch problems and their influence on component durability. For many manufactured components, we find various types of notches, such as grooves, step and holes. They have a tend to be the place where is concentrated stress, so called the macroscopic stress concentrator. In this area is higher risk of part destruction. Nowadays, there are hypotheses that are based on the assumption that the higher the roughness, the lower the durability. In many cases the designers prescribe unnecessarily high surface quality. It is necessary to maintain adequate quality of the surface, and also necessary that the component has attained a high durability. The paper deals with the influence of machining technology carrying capacity of notched components. As the test material was used steel Fe510 according to EN ISO (11523, according to CSN 42 0002).

On fracture initiation angle near bi-material notches – Effects of first non-singular stress term

The effect of first non-singular stress term of elastic stress field near bi-material notches (I-stress) on the fracture initiation angle is investigated. A modified maximum tangential stress (MMTS) criterion is suggested for predicting the fracture initiation angles which takes into account the effect of I-stress as well as the singular terms. It is shown that the I-stress can play an important role in fracture initiation behavior adjacent the interface corners and neglecting this term may introduce significant errors in predicting the fracture initiation angle. Then, to evaluate the proposed criterion, a finite element (FE) simulation is done on a typical test specimen and the presented criterion is applied for predicting the fracture initiation direction. It was observed that the MMTS predictions are very close to the results obtained from the FE analysis (which takes into account all stress terms) showing the reasonable accuracy of the MMTS criterion for bi-material notch problems.

Three‐dimensional stress fields due to notches in plates under linear elastic and elastic–plastic conditions

AbstractThe paper deals with a 3D multi‐parametric stress field representation ahead of notches in plates of finite thickness under different loading conditions. Under certain hypotheses, the 3D governing equations of elasticity can be reduced to a system where a bi‐harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, and the latter provides the solution of the corresponding out‐of‐plane shear notch problem. The solution is valid close the notch edge, through the plate thickness, with exclusion of a very limited zone close to the free surface of the plates. Also, under elastic–plastic conditions there are circumstances where it is possible to separate the in‐plane problem from the out‐of‐plane problem. A new analytical frame is proposed, and the results are compared with those obtained from 3D finite element models of a plate with a square notch. In the presence of remote applied tensile stress, the slope of the induced, out‐of‐plane, shear stress component matches that of the mode III plastic problem.

Application of maximum tangential stress criterion in determination of fracture initiation angles of silicon/ glass anodic bonds

Article history: Received January 25, 2014 Received in Revised form May, 10, 2014 Accepted 7 June 2014 Available online 9 June 2014 Silicon/glass bi-materials are used in micro-assembly and packaging of microelectromechanical systems (MEMS) and micro-electronics devices. In this paper, maximum tangential stress (MTS) concept is used for determination of the fracture initiation angles of silicon/ glass bi-material notches. First, the MTS criterion is analytically formulated for a bimaterial notch problem. Then, the criterion used for prediction of fracture initiation angles of some experimental data given in literature for silicon/ glass bi-material notches. In addition, the modified MTS (MMTS) criterion, which considers the effect of I-stress, was compared with the MTS criterion and the experimental data. It was shown that MMTS criterion provides more accurate results than the MTS criterion for estimation of the fracture initiation angle. © 2014 Growing Science Ltd. All rights reserved.

An element‐free method for in‐plane notch problems with anisotropic materials

SUMMARYThis study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.

ANALYSIS OF GENERALIZED STRESS INTENSITY FACTORS OF V-SHAPED NOTCH PROBLEMS BY FEM

This paper deals with a-finite element method (FEM) based on a V-shaped notch corner tip stresses to solve generalized stress intensity factors (GSIFs) in 2D elastic bodies. The method does not need extremely refined meshes and special elements accounting for the analytical form of singularities around the V-shaped notch corner tip. The generalized stress intensity factors of the V-shaped notch problems are evaluated from the ratios of FEM stress values at the notch corner tip for a given problem and a reference one. Several numerical examples show that present method is effective and applicable to dealing with the V-shaped notch problems.

Recent developments in multi-parametric three-dimensional stress field representation in plates weakened by cracks and notches

The paper deals with the three-dimensional nature and the multi-parametric representation of the stress field ahead of cracks and notches of different shape. Finite thickness plates are considered, under different loading conditions. Under certain hypotheses, the three-dimensional governing equations of elasticity can be reduced to a system where a bi-harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. The analytical frame is applied to some notched and cracked geometries and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE models.