The Effects of Stress Concentration on Crack Propagation
Loads, stress, characteristics of the material and the environment are responsible for crack formation and material failure.
When a structure or a machine is designed, it has to satisfy certain reliability and safety thresholds, based on a number of factors: the loads it is subjected to, the environment it is in and the proposed length of service.
What these thresholds are and how high they are is an evolving matter, and these concepts have changed significantly in the past hundred years or so. Only 50 years ago, we didn’t have any idea why ships would, seemingly out of the blue, break in half, or why commercial airplanes would break down mid-flight.
Since the industrial revolution in the 19th century, we had an ever-growing need to push our engineering designs, which came at a high cost of numerous terrible disasters. Bridges broke down, ships sunk and buildings collapsed, even when things were designed around established safety factors.
It was clear that we were missing something. The development of the field of fracture mechanics and our understanding of how cracks form and propagate has grown greatly since the 1950s, and we can now design and engineer things that would have been considered utterly unsafe and unreliable before.
The Relationship between Cracks and Stress
There are three key factors that are responsible for crack formation and, in turn, material failure: the loads (and stress) the material is subjected to, the characteristics of the material, and the environment it is in.
In engineering terms, stress can be defined as the amount of the load a single material filament is carrying. This is not a real physical representation though, as the stress can be scaled down to the atomic level, where we would observe both the attractive and repulsive forces between molecules and atoms, as well as all the outside forces, or loads, that are acting on them.
We can’t really measure stress, as it is an internal property of a material, and we can only approximate it by knowing the geometry and the applied loads.
Stress is calculated by the formula:
σ = F / A
where F is the given load, and A is a cross-section of the part in question perpendicular to the load vector.
The fact that stress values are an estimate is very important to know, because there are several additional factors at play that make it quite difficult to safely estimate the stresses: internal stresses, residual stresses and stress concentrations.
The aforementioned factors are the reason why seemingly safe structures would break or fall down. If we had a perfectly homogenous material with no changes in geometry, we could calculate stress with absolute accuracy. However, when there are changes in geometry, the areas that are in the immediate vicinity suffer from higher stress than the average value. An example is a the stress concentration at a hole.
Sudden changes increase the value of stress concentration. This must be taken into account when designing structures – even if the whole part has been designed with a high safety factor, there could still be areas where local stress values are higher than the ultimate tensile strength of the material. (Discover the Difference Between Strength and Toughness.)
Even more importantly, no material is truly homogenous. There are always various inclusions, discontinuities, and imperfections that act as local, micro stress concentrators. In addition, residual stresses can be present due to the way the part is produced and assembled, which can also put the stress level over the safety limit.
How does all of this play into crack formation?
Every material has a certain energy threshold before it starts to tear. If we have a combination of external loads, internal stresses, changing geometry, temperature changes and corrosion, it is inevitable that this tearing will occur and a crack will form.
When such a separation appears, it is very thin or sharp (one crack dimension is much greater than the other). This means that stress, due to stress concentration, at the crack tip is theoretically infinite, and it leads to further tearing of the material and crack propagation. Now, this is not as dangerous as it sounds.
Cracks and Failure
Luckily, the same imperfections in material, geometry changes and residual stress can actually be an obstacle to crack propagation. If a crack runs into a discontinuity in the material, it has to have significantly more energy to overcome it and propagate further.
The obvious problem is that the crack reduces the load-bearing surface of the material, which means that the remaining continuous material has to carry more and more of the load. However, this means that we can still use a cracked part, as long as stress levels are below the ultimate tensile strength.
In general, after the crack appears, in the case of ductile materials, there is a period of stable crack growth that we can predict fairly accurately, and we can continue safely using the machine or structure as long as we regularly check up on the crack progression.
This second stage of a crack's life is followed by the final stage of unstable crack propagation, where the crack grows rapidly until the complete failure. Of course, how long each stage is and how the crack behaves is an immensely complicated issue, in no way helped by stress corrosion cracking or corrosion fatigue. (Learn more about how corrosion affects a material's strength in The Effects of Corrosion on the Shear Behavior of Materials.)
The advancements in fracture mechanics and the use of finite element analysis are in large part responsible for our understanding of this crack behavior, and how we can prevent it or combat it.
We can influence crack behavior by introducing intentional residual stresses, or additional geometry changes that even out the stress distribution. For example, we can stop further crack propagation by drilling a hole at the crack tip, or by introducing internal stresses by weld surfacing. These methods are meant to be only a temporary solution as the further degradation of material is inevitable.
The main issue however is that there really are no rules here. Each part is built out of a material that has unique microstructure, its work regime is unique, and its environment and environmental changes are unique. There is no be-all end-all equation that can tell us with certainty where and when a crack will appear.
However, we do have a growing database of parts that suffered failure, which means we can create more and more accurate models that we can use to estimate how a crack will behave when it appears. (See The 3 Stages of Corrosion Failure Analysis for more information.) The new methodology we develop allows us to predict where the chances are highest for such an event, and we can design our machines around that accordingly.
The whole process is far from perfect, but the end result of this engineering paradigm to design around reliability instead of safety factors is that we have more reliable, safer constructions, while using less material with lower production and maintenance costs in the long run.