A note on notch shape optimization to minimize stress concentration effects

Highlights

• Stress concentration effects depend on the notch tip profile.

• Circular arcs are not the best notch tip shapes.

• Notch tips with proper variable radii can significantly decrease stress concentration effects.

• Simple optimization schemes are proposed to optimize notches loaded by uniaxial and multiaxial loads.

• Optimized shapes are proposed for several fatigue test specimens.

Abstract

Notches induce localized stress concentration effects that can affect many failure mechanisms, in particular the initiation and growth of short cracks under fatigue loads, significantly reducing the strength of structural components under service loads. To decrease such nocive effects, notches are usually designed with as large as possible circular arc tips, even though it has long been recognized this is not the best solution to minimize such problems. Indeed, notches with properly shaped variable tip radii can have a much smaller deleterious influence on fatigue strength, but such optimized notches still are not routinely used in structural design. In fact, not even standard fatigue specimens specify them. Nevertheless, such improved notches can be a very good design option to augment the strength of structural components, since they barely affect their global dimensions or weight. Moreover, nowadays they can be economically built due to the widespread availability of CNC machine tools. After comparing the improvements achievable by some classic variable radii receipts, two simple and robust numerical routines developed to optimize notch shapes for components that work under general multiaxial loading conditions are presented and evaluated.

Introduction

Most structural components must have brusque geometric transition details such as holes, slots, grooves, corners, shoulders, keyways, splines, threads, welded joints, or similar localized undercuts or even reinforcements, which can be generically called notches. Such notches are usually required for operational, structural, or manufacturing reasons, or else to decrease weight, so they are in fact a practical need. However, if not properly designed they can much perturb the local stress and strain fields around them, locally increasing or concentrating the nominal stresses that would otherwise act at their sites if their effects were negligible. Such localized stress concentration effects depend on the notch geometry and on the loading conditions, and can much decrease the actual component strength. Under higher loads they depend as well on the load level and on the material hardening behavior, since local yielding and other non-linear deformation mechanisms affect the notch tip stresses and the stress gradients around them.

In simple yet very common linear elastic (LE) problems, local effects on notch-tip stresses can be quantified by a material-independent stress concentration factor (SCF) defined by

$$K_t={\sigma }_{\mathit{max}}/{\sigma }_n$$

where σ_max is the maximum stress acting at the notch tip and σ_n is the nominal stresses that would act there if the notch had no effect on the stress field that surrounds it.

Like all LE parameters, K_t are unique values that can be cataloged and then used to solve many important notch problems in structural engineering. They are particularly useful for designing against fatigue crack initiation, for example. However, to properly describe notch effects in elastoplastic analyses, or in multiaxial loading problems, or in anisotropic materials, or even to consider 3D effects in simple uniaxial LE cases (e.g. when the notch tip radius is in the order of or smaller than the component thickness), it is necessary to separate stress from strain concentration effects. In such cases, different stress and strain concentration factors may be defined by K_σ = σ_max/σ_n and K_ε = ε_max/ε_n, respectively, as discussed elsewhere [1], [2].

Pioneer analytical solutions for LE SCF were obtained by Kirsch in 1898 [3], who studied the effect of a circular hole in a tensioned infinite plate, and by Inglis in 1913 [4], who solved the similar elliptical hole problem. Since then, a few analytical and many other numerical and experimental K_t values have been obtained for countless notch geometries. However, most of them by modeling the notches as if they could be properly described by a 2D approximation, solving the stress analysis problem assuming LE plane stress (pl–σ) or eventually plane strain (pl–ε) conditions around their tips. Peterson is a traditional SCF catalog [5], although mostly restricted to plane and axisymmetric LE solutions, whereas Savin [6] is a classical reference for analytical SCF solutions.

A traditional procedure to decrease K_t-effects is to round notch tips using as large as possible circular arcs. This design rule is clearly justified by the classic Inglis’ solution for elliptical notches [4], which in the simplest uniaxial case leads to

$$K_t=1+2a/b=1+2\surd (a/\rho )$$

where a and b are the semi-axes of the elliptical hole in an infinite plate loaded by a normal nominal stress perpendicular to a, and ρ = b²/a is its smallest radius, which occur at the extreme of its 2a axis, so at the points that can be called the elliptical notch tips.

However, although outside the scope of this work, it is important to emphasize that K_t values are not sufficient to quantify all notch-induced stress concentration effects. In fact, both the maximum stresses at the notch tips and the stress gradients around them can significantly affect the actual resistances and consequently the operational lives of structural components. The stress gradient is the main responsible for notch sensitivity under fatigue and under EAC conditions, so very sharp notches are not as bad as it could be anticipated from their very high K_t values because they have very sharp gradients as well, as discussed elsewhere [7], [8]. Anyway, to decrease deleterious effects that can be introduced by sharp notches in structural components, their tips are usually rounded or blunted by circular arcs. The larger such tips radii are the better, meaning the more they tend to alleviate all stress concentration effects induced by the notches. Such facts are well known, and all structural engineers and wise technicians specify generous rounding radii for their notch tips.

Less well known is the fact that circular arcs decrease but do not minimize stress concentration effects around notch tips. Even though this problem has been recognized for a long time, notches with variable radius tips properly optimized to minimize their detrimental influence on the strength of structural components still are not as widely used in engineering designs as they should be. Indeed, albeit efficient receipts for improving notch profiles have been proposed in the early 1930s, the usual practice still is to specify notches with as large as possible constant radius tips, probably because they can be easily fabricated in traditional manually-operated machine tools. To enhance this argument, it can be pointed out that not even standard fatigue crack initiation test specimens are specified with optimized notches to connect their uniform test section to the larger heads required to grip them [9], [10], [11], [12]. Indeed, the generous constant radius notch tips used to significantly alleviate their stress concentration effects do not minimize them. Since such notches locally concentrate stresses and strains around their roots, they may localize the crack initiation point, invalidating in this way the test results, or at least increasing their already intrinsically high dispersion.

On the other hand, natural structural members such as tree branches and bones have learned by evolution to add material where it is needed, so their notches have variable instead of the fixed radii usually specified to smooth engineering notch tips [13], [14], [15], [16], [17]. Since notches with properly specified variable tip radius can have much lower SCF than those obtainable by fixed notch root radii of similar size, such improved notches can be a very good design option to increase fatigue strengths with almost no side effects on the global dimensions or on the weight of most structural components. Moreover, properly optimized notches are now more useful than ever, as nowadays they can be economically specified and manufactured due to the widespread availability of finite elements (FE) codes to calculate and of computer controlled machine tools to fabricate them. These smart design practices can be much cheaper and wiser substitutes for expensive high-performance materials or for major reinforcements in components that tend to fail under service loads.

The aim of this note is first to compare the efficiency of both traditional and modern receipts to design better notch tip profiles, and then to analyze the SCF improvements achievable by optimizing the variable tip radii of notches for uniaxial and multiaxial load applications, using the FE method. To optimize the notches, a simple gradientless optimization method, based on the idea of iteratively adding material where it is needed and removing it where it is superfluous, is proposed and implemented using a self-adaptive remeshing scheme that can be easily adapted to be compatible with most commercial finite element (FE) codes. This technique is used to improve the notches of push–pull, rotary bending, alternated bending, and multiaxial tension–torsion fatigue test specimens, as well as the shape of a tension–torsion load cell, but it can be equally used to optimize any other notch problem. Finally, a more powerful notch-tip optimization method that also considers gradient effects around them is described and evaluated.