Parabolic Notches Research Articles

Recent Advances in the Analytical Stress Field Solutions for Radiused Notches in Orthotropic Solids.

The main aim of this work is to provide a brief overview of the analytical solutions available to describe the in-plane and out-of-plane stress fields in orthotropic solids with radiused notches. To this end, initially, a brief summary on the bases of complex potentials for orthotropic elasticity is presented, with reference to plane stress or strain and antiplane shear problems. Subsequently, the attention is moved to the relevant expressions for the notch stress fields, considering elliptical holes, symmetric hyperbolic notches, parabolic notches (blunt cracks), and radiused V-notches. Eventually, examples of applications are presented, comparing the presented analytical solutions with the results from numerical analyses carried out on relevant cases.

A Parabolic Notch Interacting With a Generalized Antiplane Singularity

Abstract In this study, the interaction of a parabolic notch with a generalized antiplane singularity is studied, and its analytical solution is derived. The singularity may be either an antiplane concentrated force or a screw dislocation, and separate solutions for each of these are found. The driving force present on the screw dislocation due to the notch free boundary is obtained. It is found that a dislocation-free zone may exist beneath the notch root surface when the screw dislocation is placed on the notch geometric symmetry axis, as the driving force will pull dislocations to the free boundary where they will be annihilated. The solutions developed in this study may be used as building blocks to model the damage of material near a parabolic notch under antiplane load conditions and therefore serve as a step in quantifying crack nucleation conditions, which is the novelty of the current study.

Interaction of a parabolic notch with a generalized singularity

In this paper, the interaction of a parabolic notch with a generalized singularity is studied and its analytical solution is derived. The generalized singularity may represent a concentrated force or an edge dislocation. As an example, the parabolic notch interacting with a dislocation is studied in details, and the driving force on the dislocation due to the notch configuration is obtained in terms of the vector J-integral. It is found that a dislocation-free zone may exist beneath the notch root surface. The solutions developed in this study give, on the one hand, basic information about the state of stress induced in the notch neighborhood, and, on the other hand, can be used as building blocks to model damage at the notch root under complex load conditions.

Stress distributions in orthotropic solids with blunt notches under in-plane shear loadings

The present work is devoted to the analysis of the stress distributions in orthotropic solids weakened by blunt notches and loaded under in-plane shear. In particular, using Lekhnitskii's approach, new closed form equations are derived for the stress components, explicitly accounting for the notch geometrical features, namely the notch radius and the opening angle, in conjunction with the elastic material properties.Three main geometries are considered, described using proper conformal mappings: lateral symmetric hyperbolic notches, parabolic notches and hyperbola-like notches in plates loaded under in-plane shear. Appropriate complex potential functions are introduced in the analytical formulation and analytical expressions for the whole stress fields are derived, taking advantage of free-of-stress boundary conditions along the notch edge.Eventually, the very good accuracy of the new solutions is proved against numerical results, as obtained from a number of two-dimensional finite element analyses carried out on plates with rounded notches under shear, also discussing the main features of the mode II stress fields in notched plates.

Design and analysis of a multi-notched flexure hinge for compliant mechanisms

This article presents a new multi-notched flexure hinge, which consists of two right circular and two parabolic notches, for positioning stages based on compliant mechanisms. First, the configuration of the presented multi-notched hinge is obtained using topology optimization, and the final shape is proposed based on post-processing. Second, the dimensionless empirical equations for the stiffness, rotational precision and stress levels of the flexure hinges are developed using finite element analysis (FEA). Third, based on the established equations, the influences of the geometric parameters on the performance of the flexure hinge are investigated. Finally, to further understand the characteristics of this type of flexure hinge, comparisons with flexure hinges of various shapes are performed in terms of stiffness, rotational precision and stress levels.

Stress Concentration Near an Elliptic Hole or a Parabolic Notch in a Quasiorthotropic Plane

We consider the problem of stress distribution in an infinite quasiorthotropic plane containing an elliptic hole whose contour is free of external forces and a homogeneous stressed state is imposed at infinity. The solution of the problem is obtained with the help of the boundary transition from the known analytic solution for an elliptic hole in an orthotropic plane in the case where the roots of the characteristic equation approach each other. In the boundary case where the major semiaxis of the ellipse tends to infinity, these results yield the stress distribution in a plane weakened by a parabolic notch for two main types of deformation (symmetric tension and transverse shear).

Influence of the Anisotropy of Materials on the Distribution of Stresses Near Parabolic Notches

We consider the problems of distribution of stresses in an infinite anisotropic plane containing a parabolic notch for three main types of deformation in the case where the asymptotics of the stress field is specified at infinity (including stress intensity factors at the tip of the corresponding semiinfinite crack). The solutions of these problems are obtained with the help of the limit transition from the known analytic solutions obtained for an elliptic hole in the anisotropic plane for three types of loading at infinity (symmetric tension, transverse and longitudinal shear). These results generalize the well-known data on the distribution of stresses near narrow rounded notches in the isotropic plane and reflect the influence of anisotropy of the material on the stress concentration.

Stress fields due to inclined notches and shoulder fillets in shafts under torsion

Closed-form expressions of the stress fields in notched rounded bars subjected to torsion are obtained. The notch profile is mathematically described according to Neuber’s conformal mapping z = ( u + iv) q, which gives parabolic and hyperbolic profiles depending on q. The notch axis is inclined with respect to the rounded bar axis. This condition results in two eigenvalue functions: the former is associated with the antisymmetric stress field and gives a stress distribution of asymptotic nature when the notch radius is equal to zero. Conversely, the latter is associated with the symmetric stress field and results in a non-singular stress distribution. This specific condition has been noted in the literature only rarely, where only the antisymmetric part of the stress field is discussed in detail. Theoretical results are compared with numerical data as determined from two models weakened by a parabolic notch and a hyperbolic notch, both notches having the local axis inclined to 45° with respect to the bar longitudinal axis. The agreement is found to be satisfactory.

On the determination of stress concentration in a stretched plate with two holes

We present a short survey of studies of the elastic interaction of two holes in a stretched plate. Special attention is paid to the study of the concentration of stresses on the contours of closely positioned holes. For two identical elliptic holes, numerical results are obtained by the method of singular integral equations. With the help of the limit transition, we determined the stress intensity factors at the vertices of semi-infinite parabolic notches. A comparison of the numerical data with known analytic solutions for two circular holes and collinear cracks is performed.

Elastic stress distributions for hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings

Closed-form solutions are developed for the stress fields induced by circumferential hyperbolic and parabolic notches in axisymmetric shafts under torsion and uniform antiplane shear loading. The boundary value problem is formulated by using complex potential functions and two different coordinate systems, providing two classes of solutions. It is also demonstrated that some solutions of linear elastic fracture and notch mechanics reported in the literature can be derived as special cases of the general solutions proposed herein. Finally the analytical frame is used to link the Mode III notch stress intensity factor to the maximum shear stress at the notch tip, as well as to give closed-form expressions for the strain energy averaged over a finite size volume surrounding the notch root.

Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as JVρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of JVρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, JVρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of JVρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.

Stress intensification at crack tips near parabolic notch

Considered in this work is the stress solution of a finite length crack near a traction free parabolic notch. Uses are made from the solution of a crack subjected to arbirary surface tractions and that of concentrated forces in a plane with a parabolic notch. Superposition of the two problems corresponding to stress free conditions on the crack and notch leads to a singular integral equation that can be solved numerically for the crack tip stress intensity factors. Two specific examples are discussed. The first pertains to a pair of concentrated forces in the region across the crack and the second case involves isolated forces applied to the boundary of the parabolic notch. The distances between the crack and notch and location of the load are changed to exhibit the corresponding variations in the stress intensity factors. Discussed are the failure behavior of a notch with a line crack defect ahead.

Stress distribution in a strongly anisotropic elastic plane with a parabolic notch

Stress distribution in a strongly anisotropic elastic plane with a parabolic notch

Asymptotic analysis techniques for stress concentration regions in plane problems

This paper describes how asymptotic techniques can be used in the analysis of data from the stress concentration region at the root of a parabolic notch in a thin plate. Three kinds of data are considered: (i) surface strain values from along the notch boundary, (ii) the sum of principal stresses or strains at interior points and (iii) the difference of principal stress or strains at interior points. The merits of the techniques are critically assessed.

Brittle crack initiation at the elastic-plastic interface

To study fracture initiation near notches experiments were conducted on polycarbonate at room temperature. It could be demonstrated conclusively that cracks are nucleated under the influence of a critical normal stress at the elastic-plastic boundary. Slip line field theory allows the determination of the critical normal fracture stress from the knowledge of the notch root radius and the location of the crack nucleation. For 0.5 in. thick specimens with root radii between 0.001 and 0.01 in. the calculated critical fracture stress of polycarbonate is nearly constant, 21.2 ±0.6 ksi. An analysis of the data in accordance with the theory developed by R. Beeuwkes, Jr. for parabolic notches resulted in critical fracture stress values of approximately 21.5 ksi. Specimens with central holes between 0.04 and 0.05 in. diam also showed crack nucleation below the surface of the hole but the calculated critical fracture stress values were considerably lower, approximately 11.2 ksi. This may be attributed to the loss of plane strain conditions in these 0.5 in. thick specimens. The experimentally observed slip lines in polycarbonate are not orthogonal to each other but intersect at a lower (~80 deg) angle. This may be due to the relief of stresses on unloading and sectioning, and it may indicate that plane strain conditions were not present. Because of its flow characteristics, which are similar to those of metals, and because of its transparency, polycarbonate appears to be a good model material for the study of plastic flow initiated brittle fracture. It could be used to check the theoretical slip line field solutions proposed for a variety of loading conditions.