Structural cracks are everywhere. What can be done about it, and how do we identify them? We will learn how to determine the potential for failure from crack propagation due to fatigue in this article.

Early in my engineering career, I was traveling by air with an experienced aerospace engineer to attend a meeting to discuss the failure of an engineering structure that had catastrophically fractured. I naively asked how it was possible for a critical part to be put into service that contained cracks. The experienced engineer told me that all engineering structures have cracks in them, including every single part in the aircraft in which we were currently flying!

Once we safely landed, and after kissing the ground to express my happiness for getting out of the cracked-filled flying machine, I decided I should pay more attention to the study of failure due to crack propagation. Preventable failures due to cracking in engineering materials have caused many disastrous crashes, which have resulted in the loss of many lives. One infamous example is the Aloha Airline Flight 243 in 1988. The fuselage fractured in-flight due to fatigue failure originating from multiple corroded rivet holes, resulting in the disastrous crack propagation shown in Figure 1.

Since my early flight, I have worked on many different engineering problems that have involved predicting the life of engineering components due to crack propagation using computer-based numerical-modeling approaches. In this article, we assume that cracks exist in the structures, and the cracks potentially grow from repeated cycles of loading, commonly referred to as fatigue loading. Three core analysis approaches used to determine the potential for failure from crack propagation due to fatigue will be discussed, with emphasis on when and how to adopt these approaches to analyze the life prediction of engineering structures. These approaches are:

1. Fatigue analysis of bodies without modeling the defects
2. Damage modeling
3. Fracture mechanics-based fatigue analysis of cracked bodies

#### Fatigue Analysis of Bodies Without Modeling the Defects

The most basic approach (and most prevalent method) to evaluate the effect of cracks in engineering materials and predicting the life of a component – particularly ones made of engineering metals such as steel, aluminum and titanium – is fatigue analysis. A typical fatigue analysis is performed by first determining the cyclic stresses occurring in the part due to the applied loading. Although the failure mechanism is assumed to be the initiation and propagation of cracks, a general fatigue analysis is performed on the “uncracked” geometry.

Fatigue analyses predict the life by comparing to known fatigue test data, and the data is plotted on what is known as S-N curves, where S is the cyclic stress magnitude and N is the number of cycles.

To generate fatigue data, one would run a physical test where the part is cyclically loaded at a given stress range until it fails. The cyclic stress, S, is plotted against the number of cycles to failure, N. An identical part could then be loaded with a different cyclic load amplitude. The cycles to failure for that stress range are plotted, and the process is continued until the S-N curve for that part is fully described. There is usually a lot of scatter in the data due to variations in the material tested, so it is commonly represented with a curve fit on a logarithmic scale. An example of an S-N curve is shown in Figure 2.

A structural-analysis simulation, typically using a computer-based technique such as finite-element analysis, can now be performed by applying the cyclic loading (pressure, temperature, acceleration, etc.) on the part. Based on the predicted stresses, the fatigue data is used to determine the cumulative cycles required to induce failure. And even though the cause of failure is the propagation of cracks in the part, this approach is performed without ever explicitly modeling a crack.

Unfortunately, fatigue data for specific part geometry and components typically does not exist and can be expensive to generate. Reasonably accurate life assessments can be made using specimen testing, however, where standard test specimens for given materials are used to generate the fatigue data. Specimen-based fatigue data for many engineering materials is readily available.

Differences between the fatigue specimen and actual components, including surface finish, manufacturing process and size effects, are accounted for by applying various safety factors to the life calculation. Most finite-element codes have built-in fatigue modules to assist in these calculations. A finite-element model with a color contour of cycles to failure generated from a fatigue analysis is shown in Figure 3.

#### Damage Modeling

Another approach that ignores the geometric presence of the crack in failure analysis is damage modeling, which is most commonly used to evaluate the failure behavior of fiber-matrix composite materials. It consists of reducing the elastic material constants gradually to model the accumulation and progression of damage from cracking that can occur in the fibers, the matrix and the interface between lamina. Again, no actual crack is modeled, but the effect of the crack is accounted for, in this case by using degraded material properties.

When modeling damage, the cyclic loading is continually reapplied in the finite-element model, the loading is redistributed in the composite structure based on the current state of damage, and the updated structural response is calculated. This procedure can be continued until the load-carrying capability falls below a certain level or the analysis stops converging due to excessive “numerical” damage. As with general fatigue analyses, most finite-element analysis codes have the ability to model damage. Figure 4 contains a plot of finite-element-based damage simulation.

#### Fatigue and Fracture Mechanics Analysis

Fracture mechanics is an analysis approach where a crack of a given shape, length and location in a material of known fracture toughness is modeled to determine if the crack will propagate to failure under a given stress level. The derivation of the equations of stress near a crack tip – obtained directly from the theory of elasticity – show that if the stress intensity is greater than the material fracture toughness, the crack will propagate. Fracture toughness is a measurable material property found from testing.

Fracture mechanics can be combined with fatigue such that the rate of crack growth per load cycle can be determined for a given crack under cyclic loading. Therefore, the cyclic loading is applied, the current rate of crack growth is calculated, the crack is extended a small amount and the procedure is repeated.

Since one is explicitly modeling the actual crack geometry, this procedure of combining fracture mechanics with fatigue is considered a more accurate approach than using S-N data. This is true because the actual crack direction and depth is included in the analysis, and the entire crack-propagation procedure is modeled. This approach is often used for critical engineering parts to set inspection intervals. For example, an inspection of an engineering part can be used to find the current largest flaw existing in the part. An analysis can be done to determine, for this particular flaw, the number of load cycles until it grows to a specified size. A future inspection can be scheduled for the predicted interval (with a healthy safety factor included), and the inspection/analysis procedure can be repeated.

There are two different ways of performing the combined fatigue/fracture-mechanics method:

1. Obtaining the stresses in the uncracked body and using these stresses in stand-alone fracture-mechanics calculations
2. Including the crack directly in the finite-element analysis

In the first case, the type of crack is selected from a library of known fracture-mechanics solutions. An example is the through-thickness crack under tension (Fig. 5). The finite-element stresses are used in the expression to determine the stress-intensity factor from which the crack growth per cycle can be identified.

The second approach, including the crack directly in the finite-element analysis, is by far the most complex and challenging. The procedure consists of building the crack into the model, performing the analysis to obtain the stress-intensity factor at the crack tip, then extending the crack and repeating the analysis for the full crack-propagation path. The path that the crack will propagate along must be determined based on the changing crack geometry and corresponding stress state.

This approach can be very time-consuming since the finite-element model must be continually updated to model the advancing crack front. In addition, a new fine mesh is required near the changing crack tip to accurately predict the stress-intensity factor. Figure 6 illustrates a finite-element model with explicit modeling of a crack.

There are other more recently developed approaches to simulate the potential crack-based failure of structures. The XFEM method includes the effect of a crack in the finite-element model by changing the element formulations for the elements that lie along the crack path.

Cohesive zone modeling (CZM) is a technique for modeling delamination cracks in layered media. Some special-purpose codes or user-defined routines in standard finite-element codes have been developed that automatically update the mesh to account for a propagating crack. All of these techniques can be applied to a fatigue analysis used to predict the cycles to failure of a propagating crack.

#### Selecting the Right Approach

As with most engineering analysis decisions, the right approach to discover and mitigate disastrous fatigue-driven crack propagation is based on the application and the overall goal of the analysis. Many engineering structures continue to be successfully designed and analyzed using S-N data with appropriate safety factors.

The more complex and accurate approaches will take longer and be more expensive. They are focused on engineering structures where material optimization, cost and loss-of-life potential are present. Regardless of the approach, failure and fatigue analysis is a vital tool for designing safe engineering structures.

For more information:  Contact Dr. Michael Bak, senior engineering manager, CAE Associates, 1579 Straits Turnpike Suite 2B, Middlebury, CT 06762; tel: 203-758-2914; e-mail: bak@caeai.com; web: https://caeai.com