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How to Model the Effect of Cracking in Engineering Materials

May 16, 2014 By: Michael Bak

On my first business trip at the start of my engineering career many years ago, I was flying with my supervisor to a meeting to discuss the failure of a particular engineering structure that had catastrophically broken into pieces.  I naively asked how a part could exist that had cracks in it, and was told that all engineering structures have cracks in them, including every part that made up 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 figured I should pay a little more attention to the study of failure due to crack propagation.  There are many examples of cracks causing failure in engineering applications, including some famous ones such as Aloha Airlines Flight 243, where the fuselage fractured in flight in 1988 due to fatigue failure originating from multiple corroded rivet holes, shown in Figure 1.  Since that time, I have worked on many different engineering problems that have involved cracking, and I will briefly summarize the basic approaches to performing failure analysis due to cracking.   For the problems I am referring to, it is assumed that cracks initiate and grow over cycles of loading, as is typical for many engineering structures.
 

Figure 1: Aloha Airlines Failure

The most basic approach to evaluating the effect of cracks in engineering materials, particularly engineering metals such as steel, aluminum and titanium, is fatigue.  A typical fatigue analysis is performed by determining the cyclic stresses in the uncracked body, and then using these stresses to predict life from an S-N fatigue curve.  An S-N curve is obtained by performing testing of the same material,  cycling a constant amplitude loading until failure occurs.  This testing is repeated for different amplitudes of cyclic loading to form the full curve, as shown in Figure 2.  The cyclic stress range can then be used with the S-N curve to predict life in cycles to failure, without ever modeling the crack!
 

Figure 2: Typical S-N Fatigue Curve

Another approach that ignores the presence of the crack in the analysis, is damage modeling.  Damage modeling is typically used to evaluate the behavior of fiber-matrix composite materials, and 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 using degraded material properties.  Figure 3 contains a plot of damage status from modeling damage.
 

Figure 3: Plot of Damage Status In A Composite Specimen

Fracture mechanics provides a more accurate approach to modeling the effect of a crack in a body.  In fracture mechanics, a crack of a given length and location, in a material of known fracture toughness, is modeled to determine if it will propagate to fracture at a given stress level.  Fracture mechanics can be used to determine the rate of crack growth per load cycle, and thus can predict when a specific crack will propagate to fail the material.

There are two general approaches to performing fracture mechanics:

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, shown in Figure 4.  The finite element stresses are used in the expression to determine the stress intensity factor, from which the crack growth per cycle can be found.
 

Figure 4: Theoretical Derivation Of Stress Intensity Factor

The second approach, including the crack directly in the finite element analysis, is by far the most complex and challenging approach.  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 geometry and stress state.  This approach can be very time consuming since the finite element model must be continually updated to model the advancing crack.  Figure 5 shows a cracked finite element model.
 

Figure 5: Modeling A Crack In The Finite Element Model

There are other approaches that have been developed to model the cracking 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.

Which of these approaches do you use to model the effect of cracking in your applications, or do you use something different?  Let us know!