Hyperbolic 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.

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.

Study of stress–strain state of elastic body with hyperbolic notch

The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two-dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems are presented.

State‐of‐the‐art review on extended stress intensity factor concepts

ABSTRACTThe stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K1, K2 and K3 are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K1ρ, K2ρ and K3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K1p, K2p and K3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.

Numerical evaluation of the original “Neuber’s rule” for pure out-of-plane shear loading

Neuber’s rule has become a popular analytical tool for evaluating notch root stress and strain for fatigue life estimation. The original study was based on the geometry of prismatic bodies with hyperbolic notches under pure out-of-plane shear with a nonlinear material model. From this study, the author observed that “The geometric mean of stress and strain concentration factors is equal to theoretical elastic stress concentration factor, Kt.” Although this conclusion was based on a “special deformation law,” Neuber suggested that it can be approximately extended to any stress–strain law and to tensile and bending loads replacing shear stress and strain with equivalent von Mises stress and strain and Neuber’s rule evolved. Extensive research has been conducted to evaluate and generalize Neuber’s rule for fatigue analysis. However, the applicability of Neuber’s rule to a specific situation requires consideration of the material model utilized, in addition to the notch geometry and loading conditions. This article presents a detailed finite element analysis of Neuber’s rule using out-of-plane shear loading on prismatic bodies with hyperbolic notch geometries identical to those considered by Neuber. Neuber’s nonlinear material model (special deformation law) is compared to the Ramberg–Osgood material model, and discernable differences were observed. Neuber’s observations were approximately valid for conditions congruent with his constitutive model. For the Ramberg–Osgood model, this happens to occur only for a strain hardening exponent of about 0.2. For other values of the strain hardening exponent, Neuber’s model cannot emulate the shape of the Ramberg–Osgood curve, and notch strain predictions from Neuber’s rule are less accurate.

Torsion of a round shaft of variable diameter

Problems for a round shaft of variable diameter subjected to torsion are studied. By transforming the governing equation into oblate spheroidal coordinates, a general solution is obtained in terms of the associated Legendre functions of order 2. Two illustrative problems, one being a shaft with a hyperbolic notch, the other a cylindrical shaft containing a small oblate spheroidal cavity located on its central axis, are solved. On the basis of the ensuing stress concentration, the important connection between the deep notch and crack is exploited. Several previously known solutions can be recovered. Results are extended to the cases of composites and transversely isotropic materials.

Distribution of stresses over the contour of a rounded V-shaped notch under antiplane deformation

The solution of the antiplane problem of the theory of elasticity for a plane with semiinfinite rounded V-shaped notch is obtained by the method of singular integral equations. On this basis, we deduce the relationships between the stress concentration and stress intensity factors for sharp and rounded notches. The obtained solution is compared with the well-known solution of a similar problem for a hyperbolic notch. It is shown that both the radius of rounding of the notch tip and the shape of its neighborhood strongly affect the distribution of stresses on the boundary contour.

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.

Two-dimensional fracture mechanics problems for solids with sharp and rounded V-notches

The survey of research on the problems of fracture mechanics of solids with V-shaped notches is discussed. The methods of solution of plane problems of the elasticity theory for domains with angular points on boundary contours are considered. Special attention is paid to the results of the research which authors obtained applying the unified approach to the solution of stress concentration problems in the neighborhood of the vertices of sharp and rounded V-notches. The solutions of elastostatics problems for domains with rounded V-shaped notches were obtained by singular integral equation method. Based on the relation between stress intensity factor at the sharp notch vertex and the stress concentration factor at the vertex of rounded V-notch (notch with straight edges and vertex rounded with circular arc), the stress intensity factors for V-notches in various domains were obtained. The presented examples include notch stress intensity factors for periodically placed notches in the edge of half-plane, diamond and lens-like holes in infinite plane. Through limit transition from the system of periodically placed diamond-shaped ovals to infinite double-sided notch, the stress concentration factors for rounded deep V-notch and the notch stress intensity factors for sharp notch were obtained. The comparison between calculated data and known solutions for the hyperbolic notch documents strong influence of the notch shape in the neighborhood of the rounded vertex on the stress concentration values for small vertex radii. The numerical results are presented not only in graphical form but also the numerous approximation formulas are given. Based on the solution of the nonlinear problem of rounded V-notch with plastic strip at the vertex, the new deformation fracture criterion is proposed. This criterion contains only standard strength parameters and it is intended to determine fracture toughness of the elements with rounded notches made of quasi-brittle materials.

A non-local stress failure condition for structural elements under multiaxial loading

A non-local stress condition for crack initiation and propagation is proposed and applied to several particular cases, such as plate with wedge-shaped notch, elliptical hole and hyperbolic notch. Brittle failure initiation for notched elements under complex loading (Modes I and II) is studied in detail. A value of critical load and crack orientation is predicted from the non-local condition, which is applicable to both regular and singular stress concentrations.

Closed-form functions for elastic stress concentration factors in notched bars

The closed-form equations given are based on the results of finite element analyses of double-edge-notched plates subject to tension or in-plane bending. The notch dimensions were varied in a parametric survey from shallow, part-circular forms to deep, sharp, slits with semi-circular ends, giving stress concentration factors varying from 1.2 to 13 (net stress basis). The concept of a configuration factor for notches, similar to that used to calculate crack-tip stress field intensity factors, is introduced. It is shown in the first instance that the analogous crack configuration factor can be used directly to modify the elastic stress concentration factor for an elliptical hole, giving closed-form functions that do not involve empirical fitting constants and have acceptable practical accuracy. Reasons for the effectiveness of this form are given, together with an analysis of the points where the notch stress concentration factors diverge from the simple closed form. Further refinements that improve accuracy are given and comparisons are also made with stress concentration factors for hyperbolic edge notches.

Stress concentration of a strip with V- or U-shaped notches under transverse bending

This paper deals with the stress concentration analyses of a 60°V-shaped single edge notch, a U-shaped single edge notch, 60°V-shaped double edge notches and U-shaped double edge notches in an infinite strip under transverse bending. The moment field induced by a moment in a semi-infinite plate is used as a fundamental solution to solve those problems. The results show that the Neuber formula gives underestimated stress concentration factors for the wide range of the notch depth. However, in the case of blunt notches, the Neuber solution of deep hyperbolic notches still gives a sufficient accuracy. The stress concentration factors of the 60°V-shaped notches and of the U-shaped notches are represented by diagrams. In addition, the stress concentration factors of a V-shaped notch in a semi-infinite plate and the effect of the notch angle on the stress concentration factor are also discussed in the appendix.

Stress concentration of a strip with double edge notches under tension or in-plane bending

The stress concentration analysis of 60° V-shaped or partially-circular double edge notches in an infinite strip under tension or in-plane bending is discussed. The stress field induced by a point force in an infinite plate is used to solve these problems. The present results for semicircular notch are in close agreement with other reports. The results calculated on the 60° V-shaped notches show that the Neuber formula gives an underestimated stress concentration factor of about 11% for tension case and in about 9% for bending case. These errors exist for a wide range of notch depth. However, in the case of blunt notches, the Neuber solution of deep hyperbolic notches still gives a sufficient accuracy in engineering use. In addition, the stress concentration factors of 60° V-shaped notches are also represented by diagrams for wide use.

Application of body force method to the analysis of stress concentration of an axi-symmetric body under bending: I. Basic theory and application to several simple problems

The general theory is established for the application of the body force method to the stress concentration analysis of an axi-symmetrical body under bending. Determination of fundamental solutions in bending are not self-evident as in tension or torsion problems. However, comparing the boundary conditions to be satisfied along circumferential position of imaginary boundary and stress fields due to the body force distributed trigonometrically along a ring around the axis, it is found that three kinds of fundamental solutions are necessary and sufficient. Thus, the axi-symmetrical problem could be treated in the similar manner as two-dimensional problems.For example, problems of a spheroidal cavity and a troidal cavity in an infinite body under bending are solved numerically. The error in the former problem is less than 0.07%. The results of the latter problem agree with the exact solution in the two limiting cases; a deep hyperbolic notch and two dimensional elliptic hole.

In-plane bending of a strip with symmetric 60.DEG.V-shaped or partially-circular notches.

This paper deals with the stress concentration analysis of 60°V-shaped or partially-circular double edge notches in an infinite strip under remote in-plane bending. The stress field induced by a point force in a semi-infinite plate is used to solve these problems. The present results for semi-circular notch are in close agreement with other reports. The results calculated on the 60°V-shaped notches show that the Neuber formula gives an underestimated stress concentration factor in about 9% for a wide range of notch depth. However, in case of blunt notches, the Neuber solution of deep hyperbolic notches still gives a sufficient accuracy in engineering use. In addition, the stress concentration factors of 60°V-shaped notches are also represented by diagrams for wide use.

Stress concentration of a fatigue test specimen having a partially-circular circumferential notch under torsion, tension or bending.

The body force method is applied to calculate stress concentration factors of partially-circular circumferential notched cylindrical specimens subjected to torsion, tension of bending. This type specimen is used frequently for fatigue tests. The stress field induced by a ring force in an infinite body is used to solve these problems. The fundamental solutions and calculation method developed here have a wide applicability for various kinds of axisymmetric notch problems under torsion, tension or bending. Through the comparison of the present result with Neuber's analytical solution for a deep hyperbolic notch, the Neuber's solution is found to give a sufficient accuracy for a blunt notch in engineering use.

Tension of a strip with symmetric 60.DEG. V-shaped or partially-circular notches.

This paper deals with the stress concentration analysis of 60°V-shaped or a partially-circular double edge notches in an infinite strip under remote tension. The stress field induced by a point force in a semi-infinite plate is used to solve these problems. The present results for semi-circular notch are in close agreement with other reports. The results calculated on the 60°V-shaped notches show that the Neuber formula gives an underestimated stress concentration factor in about 11% for a wide range of notch depth. However, in case of blunt notches, the Neuber solution of deep hyperbolic notches still gives a sufficient accuracy in engineering use. In addition, the stress concentration factors of 60°V-shaped notches are also represented by diagrams for wide use.

Determination of notch-tip plasticity by X-ray diffraction and comparison to continuum mechanics analysis

Dislocation-free silicon crystals of ({\bar 2 \bar 1 \bar 1}) orientation with a hyperbolic notch, subjected to tensile deformation at 1073 K, were used as model material for the analysis of the induced plastic zone. The results obtained by X-ray topography and X-ray rocking-curve measurements were compared to theoretical calculations and predictions based on continuum mechanics. Good agreement between experiment and theory was obtained regarding the shape of the plastic zone, the contribution of the active slip systems to the size of the plastic zone and the direction of the maximum plastic strain trajectory in the zone. Discrepancies between experiment and theory regarding the symmetry relation of the plastic zone lobes and the dislocation density near the notch tip were attributed to the interactions and resulting work-hardening. These aspects were not taken into account in calculations of continuum mechanics.

A Mechanics Approach to the Creep Rupture Strength of Notched Bodies under Longitudinal Shear

This paper deals with a mechanical analysis of the notch creep rupture behavior under the longitudinal or mode III shear stress condition. The crack initiation, propagation and rupture lives (nondimensional) of prismatic bodies with double-edged hyperbolic notches were predicted using the results of stress and strain analysis of these bodies under the steady-state creep condition, as combined with a fracture criterion, in which a crack was assumed to grow by a characteristic distance, ρs, when the strain at the distance, ρs, ahead of the crack reached a critical value.The effects of several geometrical and material parameters on the crack initiation life of the notched prismatic bodies and also on the rupture life and the crack propagation rate of a deep-cracked prismatic body were predicted. It was found that “notch-weakening” in creep rupture strength prevails under the longitudinal shear stress condition, and that the crack propagation rate changes approximately linearly in a log-log diagram against the net section stress or the elastic stress intensity factor of the prismatic body. The physical interpretation and the extension of the fracture criterion used were discussed.The prediction was extended to the tensile or mode I notch creep rupture behavior, using an analogue between the modes I and III, which is empirically accepted in fracture mechanics. Qualitative agreement was observed between the predicted values and the experimental mode I results available in the literature.

A Hypothesis of Non-Propagating Fatigue Crack

A hypothesis for the mechanism of non-propagating fatigue crack related with the stress amplitude and the critical strength of a material in a region ahead of the fatigue crack tip below notch root was proposed. It was shown that the calculated results based on the hypothesis well explained the experimental results of a rotating bending fatigue test of a round bar specimen with a circumeferential deep hyperbolic notch (i.e.Neuber type notch) in each case of two sample (fine grained steel and coarse grained steel) which had been obtained by the heat-treatment of a mild steel.