Two Decades of Critical Plane Analysis

It has been 20 years since Critical Plane Analysis for rubber was first conceived and validated.  There were early signs of its significance.  It won awards wherever I presented it. At the 1999 SAE Fatigue Design and Evaluation meeting, it won the Henry Fuchs award.  At the 2000 Tire Society meeting, it won the Superior Paper award. At the Fall 2000 ACS Rubber Division meeting, it won the Best Paper award.  Upon completing my 2001 doctoral thesis, we applied for and received a US patent (2003) on it.

The strongest early sign was that I soon found myself in company with others pursuing similar thinking.  The earliest was Dr. Nicolas Saintier.  As far as I know, neither of us was aware of the other’s work until 2006.  That was when he published an account similar enough to my own that when it came across my desk and I first started to read it, I felt certain he would cite my 2001 work as a source.  I have to admit to initially feeling let down when I reached the end of his paper and found no mention of my work.  I immediately looked for his other papers and found his 2001 doctoral thesis titled “Fatigue multiaxiale dans un élastomère de type NR chargé: mécanismes d’endommagement et critère local d’amorçage de fissure.” (Multiaxial fatigue life of a natural rubber: crack initiation mechanisms and local fatigue life criterion).  There it was – the same founding principle of Critical Plane Analysis that I had worked so hard to articulate and validate – the idea that cracks develop on a material plane, specifically the most critical material plane, and that their localized experience drives their evolution.  That we both articulated this beautifully simple and powerful principle in the same year with complete independence from each other, when no one else working on elastomers had yet spoken of this approach (there were precedents in the field of metal fatigue analysis), just shows that it was an idea whose time had come.

Although the foundational principle of Critical Plane Analysis was the same, there were also important differences between our accounts.  We differed on 1) how the critical plane is selected, 2) what criterion is used to quantify the severity of loading experience by the critical plane, 3) how damage on the critical plane evolves under solicitation.  The following table summarizes the key differences:

 

Table 1. Comparison of the Mars and Saintier Frameworks for Critical Plane Analysis.

Mars 2001 Saintier 2001
Critical Plane Selection Method Minimize the computed life after evaluation of damage on all planes Maximize the principal stress prior to evaluation of damage
Multiaxial Criterion Energy release rate estimated via cracking energy density on every plane Stress traction on the assumed critical plane
Damage Evolution Law Integration of crack growth rate law Power law Wohler curve
Strain Crystallization Law Treated as R ratio dependence of the crack growth rate law Treated as a modifier of the stress experienced on the critical plane

It may be said that Saintier’s approach followed more closely the precedents for Critical Plane Analysis in metal fatigue, particularly with respect to the method used to select the critical plane.  Selecting the plane is the first step in his method (identify the plane in order to compute the damage), but it is the last step in our method (compute the damage on each plane first and lastly pick the plane with the most damage).  Saintier’s approach also depends on a Wohler curve style characterization of fatigue behavior, where ours is defined via a crack growth rate law.  We have previously discussed the pros and cons of Wohler curves vs. fracture mechanics.  In our approach, we placed a high priority on taking advantage of the very large pre-existing body of knowledge on the fracture mechanical behavior of elastomers, and on the economic and operational advantages that crack growth experiments enjoy.

Since my and Saintier’s first steps, there have now been many others who have contributed in various forms to the overall method, its validation and/or its application.  It is safe to say that Critical Plane Analysis is here to stay, and set to continue expanding for many years (there are now several hundred research papers!).

For our part, Endurica is now in year 12 of delivering commercial grade fatigue analysis solutions built on this method.  Today, Critical Plane Analysis is a production analysis workflow used by many engineering organizations to solve critical durability issues.  It is the heart of the Endurica fatigue solver, and there are hundreds of trained users (look up the #fatigueninjas on twitter!).  It is unrivaled for its reliability, speed and accuracy in computing the impacts of multiaxial loading on durability.

What do the next 20 years hold?  We are going to see a transition in how fatigue analysis is used.  OEM organizations that manage durability and risk across rubber component supply chains will transition away from receiving fatigue simulation results on an optional basis towards requiring fatigue simulations by default on every part at the inception of new programs.  Expectations and achievement of cost-reduction, light weighting and sustainability initiatives will increase as product optimization begins to fully account for actual product use cases.  Critical Plane Analysis has already laid the foundation for these things to happen.  Older fatigue analysis methods that do not compete well against critical plane methods will become obsolete.  On the research side, there will be further development of material models for use in the critical plane framework.  Ageing, inelasticity, rate and anisotropy effects still need further development, for example.  In 20 years, durability will be just one more thing that engineers do well every day, whether or not they know that Critical Plane Analysis was how they did it.

Mars, W. V,  Multiaxial fatigue of rubber. Ph.D. Dissertation, University of Toledo, 2001.

Mars, W. V. “Multiaxial fatigue crack initiation in rubber.” Tire Science and Technology 29, no. 3: 171-185, 2001.

Mars, W. V. “Cracking energy density as a predictor of fatigue life under multiaxial conditions.” Rubber chemistry and technology 75, no. 1: 1-17, 2002.

Mars, W. V., “Method and article of manufacture for estimating material failure due to crack formation and growth.” U.S. Patent No. 6,634,236. 21 Oct. 2003.

Saintier, N, “Fatigue multiaxiale dans un élastomère de type NR chargé: mécanismes d’endommagement et critère local d’amorçage de fissure.” Ph. D Dissertation., Ecole des Mines de Paris, 2001.

Saintier, N, G. Cailletaud, R. Piques. “Crack initiation and propagation under multiaxial fatigue in a natural rubber.” International Journal of Fatigue 28, no. 1: 61-72, (2006).

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Conservatism and Tradition in Fatigue Analysis

Slide Rule

Because Endurica’s Critical Plane Analysis is a relatively new approach to fatigue analysis of elastomers (introduced in 2001), new users often ask whether its predictions are conservative.  Ie, does its predictions reliably lean in favor of safety? And is it more or less conservative than the traditional approaches it supplants?

Fatigue analysis for elastomers follows two distinct traditions.  The earliest tradition traces to Sidney Cadwell’s work in 1940 which followed the even earlier ideas of metal fatigue pioneer August Wohler.  This tradition is based on matching up empirical crack nucleation curves to corresponding in-service operating conditions via convenient parameters such as stress or strain.  It is typically the first approach that engineers encounter in their undergraduate training, as it is often effective and relatively simple to apply.  A later tradition, Fracture Mechanics, traces to the post-WWII work of Ronald Rivlin and Alan Thomas in 1953 which extended Griffith’s seminal 1921 work on rupture to elastomers.  In this tradition, the energy requirements for growing a given crack provide the core organizing principle for analysis.  Combined with empirical crack growth rate curves, this approach can make high accuracy life predictions for a very broad range of application scenarios. This approach is typically first encountered in graduate-level engineering programs, and due to somewhat more complicated mathematics, usually requires specialized calculation software to apply it.

There are a few big holes in the Wohler curve approach.  For elastomers, perhaps the biggest limitation is that this approach assumes a priori that damage is associated with the maximum principal stress or strain.  This is sometimes true for simple cases, but not always: 1) strain crystallization is known to produce off-axis cracking not aligned with the principal stress, 2) compression is known to produce cracks on planes of maximum shearing, and 3) out-of-phase multiaxial loading cases do not even possess a unique, well-defined principal direction – the directions vary in time.  It is also well known that Wohler curves for rubber depend strongly on mode of deformation.  Fatigue experiments in simple tension, biaxial tension, simple shear, and simple compression do not simply resolve to a single universal curve, as the Wohler approach takes for granted.  To use this approach conservatively then requires that the most damaging mode of deformation – simple tension – be used as the baseline.

Perhaps the biggest limitation of the traditional Fracture Mechanics approach is that it typically focuses on only one crack at a time.  In fatigue, structures begin with many microscopic cracks distributed randomly throughout.  Most of the fatigue life of the structure is spent growing many small cracks.  Only towards the very end of life do one or a few large cracks finally emerge as worst cases.  True conservatism would mean tracking the growth of all of possible large cracks, and finding out which one(s) grow the fastest.  But traditional fracture mechanics tools are not well adapted for this task.  They require up front assumptions about the location and shape of the worst case crack.  How can you find a worst case without considering many alternatives?

Critical Plane Analysis is simply the idea that a crack could occur anywhere in a structure, and it could occur in any orientation.  It checks all of the possibilities, and it finds the worst ones.  It looks at the specific loading experiences of each individual crack plane that might occur.  It takes account of material behavior like strain crystallization.  It takes account of crack closure conditions.  It takes account of the fracture mechanical behavior of small cracks.  It does not make unwarranted assumptions about the orientation of cracks.  It correctly predicts the orientation of cracks for all modes of deformation.  It is the most exhaustive and conservative fatigue analysis that you can do.

Don’t mistake traditional approaches with the conservative approach.  Critical Plane Analysis is, by definition, the most conservative approach because it doesn’t make any assumptions about crack location or orientation, and because it checks all of the possible ways a crack might occur.

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