I enjoy discovering the meaning behind some common phrases that we use in everyday English. The term “top notch”, for example, describes something that is of high quality. One of the first references of this term is found in a poem from an 1835 issue of New England Farmer magazine: “But could you rustic rhymer climb/To topnotch of the true sublime.” In New England, where I live, a notch can refer to a mountain pass, so if you are at the “top notch”, you are close to the summit of the mountain.
Notches that appear in engineering structures, on the other hand, are often the location of fatigue failures, and thus represent the opposite of high quality. Notches usually cannot be avoided since they naturally occur at thread roots, transitions, rivet holes, welds, and keyways. Proper consideration of the effect of notches in fatigue can enable long life in these components.
The obvious reason notches decrease the fatigue strength of a part is that they concentrate the stress near the bottom of the notch. A stress concentration, K_{t , }can be calculated that represents the peak stress in the notch divided by the nominal stress in the minimum crosssection as if the notch were not present, as illustrated in Figure 1 above.
Fatigue behavior of notches is not predicted well by smooth specimen fatigue data. In fact, the effect of a notch on fatigue is not as severe as the elastic stress concentration factor would indicate. Referring to Figure 2, which shows fatigue curves for 2024T4 aluminum, the endurance limit of notched fatigue specimen is greater than if the smooth specimen is used and the stress concentration is applied. A fatigue notch factor, K_{f}, is defined as the ratio of smooth to notched fatigue strength. The fatigue notch factor will always be less than the elastic stress concentration factor, but will approach the stress concentration value at high cycle counts and for low ductility materials such as ceramics.
The reasons for the difference between the fatigue notch factor K_{f} and the stress concentration factor K_{t} are related to the stress gradient and localized plastic deformation found at the notch root. Because of the large gradient of stress at a notch, the fatigue life is controlled by the average stress acting over the material at the notch root, which will be lower than the maximum surface stress calculated from K_{t }. When small cracks start to propagate from the notch root, they grow into regions of lower stress due to the stress gradient. Localized plastic deformation also has the effect of blunting cracks in the notch root.
Figure 2: Fatigue Notch Factor

So how does one perform an accurate fatigue assessment of a notched structure? There are three common approaches:
1) Fatigue curves generated from notched specimens.
2) Notch sensitivity and fatigue notch factor approach.
3) Stress gradient methods.
The first method assumes that separate fatigue curves are available for varying stress concentration factor specimens. Figure 3 shows fatigue curves for various mean stresses for aluminum with a K_{t} value of 2.0. This approach has several shortcomings. First, the nominal stress is used with these curves to evaluate life. It is not always easy to determine nominal stress in a complex structure. Second, the stress concentration factor, which depends on the local geometry and mode of loading, may not be well defined in an actual part.
Figure 3: Fatigue Curves for Kt = 2.0

The second approach is to determine the stress concentration factor from the notch using stress analysis, and then determine the fatigue stress concentration factor related to the notch stress concentration factor. A common method to calculate the fatigue stress concentration is by using a notch sensitivity factor, q. Notch sensitivity factors can be found or calculated for classes of material, and are functions of a material length parameter (a) and the notch radius (r). One example is an expression by Peterson:
The material length factor expression is presented below as a function of the ultimate strength of the material (σ_{u}):
The fatigue stress concentration factor can then be calculated from the elastic stress concentration factor and the notch sensitivity factor:
The fatigue life is then estimated by using the nominal stress multiplied by the fatigue stress concentration factor with the smooth specimen fatigue data. As with the previous method, this approach also relies on the ability to determine the K_{t} and nominal stress from a complex part, as well as obtaining an accurate estimate of the material length parameter, a.
The third approach, the stress gradient method, adjusts the stress in the notch based on the stress gradient at the notch. The main advantages of this approach are that it uses the local stress distribution to determine the fatigue response, and it can be used directly with finite element stress analysis so that estimates of nominal stresses and stress concentrations are not required. This approach is incorporated into the nCode DesignLife fatigue code and is performed automatically as part of the fatigue calculation.
The procedure is performed by extracting the six components of stress and their gradients in the direction of the surface normal at the notch. The gradient of the von Mises stress in the surface normal direction is then determined (see Figure 4), and then normalized by the von Mises stress at the surface of the notch. This normalized stress gradient is then used to determine a correction factor based on the type and ultimate strength of the material. In DesignLife, the correction factor is applied to the stress, so fatigue data based on smooth specimens is used. Note that one could alternatively use the notch surface stress and increase the fatigue strength by the same factor.
Figure 4: Determination of Normalized Stress Gradient

It is important to note that if a finite element analysis is performed on a structural component that contains notches, and if the calculated stress in the notch is used with typical smooth specimen fatigue data, the fatigue life will be underestimated unless the analyst accounts for this notch effect in some way.
Hopefully the information in this post will help you perform top notch fatigue assessments! Please comment if you want to share how you handle this effect in your calculations.