Assessment of component lifespan - How to read a Wöhler curve correctly

Components fail under certain loads. For engineers, therefore, a deep understanding of the relationships between material, geometry, manufacturing process, and expected stress is essential. Metallic materials fail either through a so-called sudden fracture or a fatigue fracture. To ensure that machines, vehicles, or other technical systems can be operated safely without harming people and to prevent unforeseen downtime and associated costs, designers and manufacturing engineers must understand the relationship between stress and material fatigue. 

Sudden fracture is very easy to explain. If the stress exceeds a material’s strength either once or after only a few load cycles and the component breaks, this is referred to as sudden fracture. 

Material fatigue, or fatigue fracture, can be divided into three phases: 

1. Crack formation or crack initiation
   At the point of the highest local stress, usually on the surface of the component, the first plastic deformations occur at the crystal plane level. As a result, crystal planes slide past one another, creating an initial crack—or a point from which the crack can then grow.
    
2. Stable crack propagation or stable crack growth
   Once a crack has formed, it continues to grow with each load amplitude as the load persists. The crack propagates across the cross-section of the component. In the fracture pattern, it appears finely structured.
    
3. Residual fracture
   If the crack is too large, the “load-bearing” cross-sectional area of the component is reduced. If this area is too small, a sudden brittle fracture occurs and the component breaks completely. The material area is also clearly identifiable in the fracture surface as a rough brittle fracture surface.

The Structural Integrity Analysis

When designing a component mechanically, it is essential to address the issue of service life verification. The goal is to predict the component’s service life as accurately as possible. A wide variety of tools and methods are available for this purpose, and certain boundary conditions must be taken into account. For example, the material used, the component geometry, and the manufacturing process must be considered. 

Furthermore, the load on the component is, of course, fundamental. If the load in actual operation is a torsional load, it makes no sense to perform the service life estimation based on a design for a tensile-compressive load. At the core of the structural integrity verification is the evaluation of fatigue strength or the fatigue analysis. It serves as the experimental basis for the estimation and forms the foundation for all further analyses. 

If the component is subjected to dynamic loading, it experiences a defined cyclic load. As a rule, the loading is regular and uniform. This means that the individual load cycles are of equal duration and have the same amplitude values (loading with constant amplitude). A more complex test is, of course, also possible, but this will not be discussed further here. 

The Wöhler Curve

The result of the fatigue analysis is usually a so-called Wöhler curve or Wöhler line. It shows the relationship between the expected number of load cycles (the fatigue life N) and the allowable load (constant load amplitude s_a ). 

August Wöhler (born in 1819 in Soltau, less than 50 km from the current headquarters of ECOROLL AG in Celle) was a railway engineer. He discovered that many railway components failed even though the permissible stress of the material, as defined at the time, had not been exceeded. At that time, components were still designed based on static strength. According to his observations, the allowable stress of a component decreased as the number of load cycles increased. Based on his observations, the results of fatigue strength analysis are still plotted today in the form of a Wöhler curve or Wöhler line. 

A Wöhler line shows the number of cycles N on the X-axis and the load stress at which the component failed on the Y-axis. If a large number of tests with their respective failure results are plotted, a three-part curve is obtained (Figure 1). At low cycle counts, the curve initially runs very flat before then dropping off as the cycle count increases. At the end of this second region, the curve then approaches a constant value again. Since fatigue behavior involves a superproportional relationship, the Wöhler line is usually plotted on a double-logarithmic scale.

The three sections of the Wöhler curve can be divided into three regions: 

1. Short-term strength (up to approx. 10³^…10^⁴load cycles)
2. Time-dependent strength (up to approx.^{10⁶…10⁷}load cycles)
3. Fatigue strength 

In the short-term strength range, the loads are so high that the strength depends heavily on the static properties. Consequently, this range is correspondingly flat and is characterized by a nearly constant curve. 

In the long-term fatigue range, a disproportionate relationship is observed with the constant C (no physical significance) and the slope of the Wöhler line k. 

The value k already provides information about how severe the fatigue behavior of the component is here. Based on this value, different Wöhler curves can already be compared. The larger the value k, the greater the strength decline at high cycle counts. 

At the end of the Wöhler curve, an inflection point can be ideally represented. This inflection point marks the transition to the ultimate strength region. Starting from the stress s_Dat this point, the component can theoretically be loaded for an arbitrarily long time; it will never fail. 

When examining a Wöhler curve, one often finds the representation shown in Figure 2. However, it is important to note that the line shown does not indicate a clear separation. Not all combinations above the line imply “failure,” and not all below it imply “no failure.” 

The Wöhler line is the result of a statistical analysis. Essentially, behind every point on the line lies a histogram with a Gaussian distribution. This means that a certain percentage of components operated at this stress will fail before reaching the indicated number of load cycles. The rest survive this number of load cycles and fail at a later point in time. This is illustrated in Figure 3. The 10% Wöhler line or the 50% Wöhler line is frequently used in this context. 

Prerequisites for working with a Wöhler curve

Anyone working with a Wöhler curve must meet or observe certain basic requirements: 

• A Wöhler curve always applies only to a specific component. The geometry and the material must be the same. If, for example, even just one notch radius changes, the Wöhler curve can no longer be used without further ado for the fatigue strength verification.
• Only components that have undergone the same manufacturing process can be grouped together in a Wöhler curve. This is because manufacturing processes also influence mechanical properties.
• A Wöhler curve also applies only to a defined stress condition, meaning a specific stress ratio and type of loading.
• The Wöhler curve is not a binary curve, but rather the result of a static analysis. Therefore, a specific safety factor must always be factored into the component design. 

Influence of surface and subsurface properties on the Wöhler curve

The shape of a Wöhler curve for a component can be influenced by various parameters. The effects of sample size or geometry are easy to understand. If a component is thicker, it can withstand more load. And if notches are introduced into the component, the service life is reduced due to stress concentrations at the notch root. 

Less straightforward are the influences of so-called surface and edge zone properties. These properties are significantly influenced by the manufacturing process and are present precisely where the highest stresses act, namely at the surface. 

Subsurface properties, in particular, are increasingly being utilized in design processes to further increase the power density of components. Important properties in this context include the presence of compressive residual stresses and an increase in surface hardness. 

Special manufacturing processes, such as smooth rolling or hard rolling, can be used to positively influence component properties after machining. In this process, a rolling roller or rolling ball is pressed against the surface with a defined force. This results in local plastic deformation in the edge region. On the one hand, the surface is smoothed; on the other hand, deep-seated compressive residual stresses are introduced, and an increase in hardness is achieved through work hardening. 

With regard to the Wöhler curve, a smooth surface generally increases service life and shifts the Wöhler curve toward higher load stresses. The roughness valleys on the surface act as micro-notches under load. This leads to a local increase in stress. If the notch is too large—that is, if the roughness is too high—the micro-notch becomes the starting point of a crack, from which fatigue then progresses. Accordingly, smoothing out the roughness peaks can already lead to a significantly increased fatigue strength. 

In particular, the presence of compressive residual stresses has a positive effect on the slope of the Wöhler curve and the fatigue strength s_D. When comparing a conventionally turned specimen with a stress-relieved specimen, it is noticeable in the Wöhler curves that, on the one hand, the fatigue strength is significantly shifted upward. On the other hand, the curve is flatter. What is not affected by cold rolling, however, is the static strength, i.e., the short-term strength range. Here, both curves are virtually identical, even if statistical analysis sometimes makes it appear as though the short-term strength of cold-rolled specimens is low (Figure 4).