William V. Mars, Ph.D., P.E. | Endurica LLC

Virtual vs. Physical in 2021 and Beyond

June 9, 2022August 16, 2021 by William V. Mars, Ph.D., P.E.

An insight to how Endurica stayed connected during Covid-19, by using Microsoft teams to meet virtually. These days everybody’s talking about whether to meet in person or online. There are great tools available for online meetings, and these have helped us navigate Covid-19. Like everyone else, Endurica teammates regularly use online meeting technology. But if there is one thing we learned over the last year, it is that sometimes physical presence really matters. Living and working in isolation is just not healthy in the long run.

The new normal during the pandemic had some benefits that were enabled by the virtual world. Time and energy that used to be consumed by travel were rechanneled into improving our software, testing services, and marketing materials. In our personal lives, we had more opportunities to spend quality time with our immediate families and found more time for fitness activities. We previously talked about our pandemic pivots to bring our training courses online and offer webinars to stay in touch with our existing and potential customers.

But virtual meetings can’t replace the full experience of being together in person. The face-to-face engagement at a trade show, the serendipitous bumping into a client, the spontaneous discussion of ideas with fellow conference-goers with a shared interest, the rapport building that comes from shared experiences. We fundamentally need physical connection. A hug, delivered via Zoom, will never feel the same.

The world of 2021 and beyond is hybrid: part virtual, part in-person. The benefits of the virtual are too great to set aside, and the necessity of the physical is too compelling to neglect. Both are critical to our future, at home and at work.

So, too, with Endurica’s simulation workflows. It was NEVER Simulation OR Build-and-Break. Even the best simulations are not enough to completely skip physical testing. The virtual approach saves significant time and money in product development and design refinement. It allows you to explore a huge space of compound options and of design features before investment in building and testing prototypes. Our simulations enable you to balance difficult trade-offs. Still, before you head into production, you must complete actual physical testing on your rubber part – the physical world is what counts in the end. It is simulation AND build-and-break that are both needed in concert to #GetDurabilityRight.

Just as a Zoom hug will never replace the real thing, software will never replace the role of physical testing. But just as online meetings are creating new opportunities and efficiencies, Endurica’s tools position you for unprecedented success when it’s time to test.

It's a Hybrid World from Here | Endurica's tools position you for unprecedented success when it’s time to test.

Use This One Simple Trick to Ensure Rubber Part Durability

June 9, 2022August 11, 2021 by William V. Mars, Ph.D., P.E.

We’ve just added a new output to the Endurica fatigue solver: Safety Factor. This feature makes it simple to focus your analysis on whether cracks have the minimum energy required to grow. Safety Factor is a quick and inexpensive way to identity potential failure locations. It minimizes the number of assumptions you need to defend, and it is backed by hard science. You don’t need to measure or explain the many influences that together determine how fast cracks grow. You don’t need lengthy materials characterization experiments that take days or weeks. You do need to know your material’s Intrinsic Strength T₀ (ie Fatigue Threshold) and its crack precursor size c₀. The test takes about an hour using the Coesfeld Intrinsic Strength Analyser.

The Safety Factor S is computed as the ratio of T₀ to the driving force T on a potential crack precursor. If the value of the Safety Factor S = T₀/T is greater than 1, it indicates the margin by which crack growth is avoided. If S is less than 1, it indicates that crack growth is inevitable. The calculation of the Safety Factor includes a search for the most critical plane, as we do for our full fatigue life computations.

Although the Safety Factor can’t tell you how long a part will endure, it nevertheless does offer great utility. You can make a contour plot showing the locations in your part where the Safety Factor is the lowest. This is a quick and inexpensive way to identity potential failure locations. You can make statements about the reserve capacity of your design that are easy to communicate and understand with a wide audience.

A vibration isolation grommet operating under small displacement

A vibration isolation grommet operating under large displacement

The images above show a vibration isolation grommet operating under small (Safety Factor 2.6) and large displacements (Safety Factor 0.83). Color contours indicate the Endurica-computed Safety Factor, and use the same scale for both images. Large Safety Factors are shown in blue. Safety Factors approaching 1 are shown in red. Safety Factors smaller than 1 are indicated in black. These results show that the grommet can be expected to operate indefinitely under the small displacements, but that large displacements will produce cracks at some point, in the regions colored black.

Durability by Design on Any Budget

June 9, 2022June 7, 2021 by William V. Mars, Ph.D., P.E.

So, you’ve got a tricky durability problem to solve, a budget, and a deadline. Let’s look at a helpful framework for sorting which Endurica workflows you need. In the grid below, each row represents a potential approach you can take. The approaches are, in order of increasing complexity and cost, the Infinite Life approach, the Safe Life approach, the Damage Tolerant approach, and the Fail Safe approach.

The Infinite Life approach is by far the simplest approach. Here, we say that damage will not be allowed at all. All locations in the part must operate, at all times, below the fatigue limit (ie intrinsic strength) of the rubber. The required material testing is minimal: we need only know the fatigue limit T₀ and the crack precursor size c₀. We avoid the question of how long the part may last, and we focus on whether or not we can expect indefinite life. We report a safety factor S indicating the relative margin (ie S = T₀ / T) by which each potential failure location avoids crack development. When S>1, we predict infinite life. For S<=1, failure occurs in finite time and we must then go on to the next approach…

In the Safe Life approach, the chief concern is whether or not the part’s estimated finite life is adequate relative to the target life. The material characterization now becomes more sophisticated. We must quantify the various “special effects” that govern the crack growth rate law (strain crystallization, temperature, frequency, etc.). We consider the specific load case(s), then compute and report the number of repeats that the part can endure. If the estimated worst-case life is greater than the target life then we may say that the design is safe under the assumptions considered. If not, then we may need to increase the part’s load capacity, or alternatively to decrease the applied loading to a safe level. In critical situations, we may also consider implementing the next level…

The Damage Tolerant approach acknowledges that, whatever the reasons for damage, the risk of failure always exists and therefore should be actively monitored. This approach monitors damage development via inspection and via tracking of accrued damage under actual loading history. A standard nominal load case may be assumed for the purpose of computing a remaining residual life, given the actual loading history to date. Changes in material properties due to cyclic softening or ageing may also be tracked and considered in computing forecasts of remaining life.

The Fail Safe approach takes for granted that failure is going to occur, and obliges the designer to implement measures that allow for this to happen safely. This can take the form of a secondary / redundant load path that carries the load once the primary load path has failed. It can take the form of a sacrificial weak link / “mechanical fuse” that prevents operation beyond safe limits. It can take the form of a Digital Twin that monitors structural health, senses damage, and requests maintenance when critical damage occurs.

The last three columns of the grid show which Endurica fatigue solver workflows align with each design approach. The Endurica solvers give you complete coverage of all approaches. Whether you need a quick Infinite Life analysis of safety factors for a simple part, or deep analysis of Damage Tolerance or Fail Safety, or anything in-between, our solvers have just what you need to get durability right.

Things that went right in 2020 at Endurica

June 9, 2022January 8, 2021 by William V. Mars, Ph.D., P.E.

2020 is burned in all our minds as a chaotic and tough year. Just like the rest of the world, Endurica staff experienced times of isolation and loss due to the pandemic. On a positive note, we invested heavily in making our tools and workflows better than ever so that we’re ready to come back strong in 2021. Here is a list of our top new developments in 2020:

Endurica Software Enhancements

Endurica DT’s new Ageing Feature now enables you to simulate how ageing affects your rubber product. Your compound’s stiffness, strength, and fatigue properties can all evolve with time.
Our new Linux distribution takes our solutions beyond the Windows world.
We’ve added an encryption feature to safeguard your trade secrets.
Viewer Improvements make it easier than ever to visualize your fatigue simulation results.
EIE Enhancements give you blazing-fast compute speed for full road-load signals.
We’ve also planned an aggressive development agenda for 2021. Stay tuned for a new Endurica-based smartphone app for materials engineers, for a new feature that computes fatigue threshold safety margins, for a new block cycle schedule extraction algorithm, and more!

Training

The new Fatigue Ninja Friday webinar series provides step-by-step application training for key the workflows that you need to get durability right. All of the recorded episodes are now available in the online Endurica academy.
The new Winning on Durability webinar series provides high-level overviews of both technical and business topics so you can connect Endurica tools to your strategic imperatives. All of these recorded webinars are available gratis on our website.
We’ve recast our in-person training events as LIVE, ONLINE workshops accessible safely around the world.

Testing Instruments

Coesfeld Tear and Fatigue Analyser – we hosted Coesfeld President Christian Kipscholl for one of the Winning On Durability episodes (#7). We walked through a demo of the T&FA’s fully automated, high-reliability testing workflow.
Intrinsic Strength Analyser – we hosted another Winning On Durability episode (#3) focused on understanding how a rapid cutting experiment tells us about long term fatigue behavior. We also completed a successful validation this year comparing direct measurement of the fatigue threshold with a measurement from the cutting method.

Fatigue Property Mapping Testing Service

We added the Reliability Module to our Fatigue Property Mapping testing service. Use it to quantify crack precursor size statistics when you need to estimate probability of failure.
We also reorganized the Thermal Module and the Ageing Module into Basic and Advanced levels, to offer a lower price-point when a basic option will suffice.

Want to leverage any of these new capabilities in your next durability project? Give us a call and let’s talk!

Does hydrostatic loading cause fatigue damage in rubber?

June 9, 2022January 7, 2021 by William V. Mars, Ph.D., P.E.

A question was recently put to us regarding the effects of cyclic hydrostatic loading on rubber. In hydrostatic loading, no shearing stresses are present, and the 3 principal stresses all have the same value p. For this case, all 3 Mohr’s circles degenerate to a single point on the normal stress axis.

Figure 1. Mohr’s circles degenerate to single point for the case of compressive hydrostatic pressure.

Under dynamic hydrostatic loading, the point may move along the normal stress axis in either of the tensile (p>0) or compressive directions (p<0). When we have pure hydrostatic compression, cracks in all orientations are closed with a tearing energy of zero. We expect infinite fatigue life in this case. On the other hand, when we have hydrostatic tension, growth of a crack will release energy, and so the tearing energy is positive. We then expect crack growth to occur at a rate determined by the tearing energy. Endurica estimates tearing energy T via the following rule:

in which a is the size of the crack, and W_c is the cracking energy density. For a slightly compressible material under hydrostatic loading, the cracking energy density calculation becomes

and, remembering that for volumetric deformation, the linear strain is 1/3 of the volumetric dilatation, we finally obtain

where W is the dilatational strain energy density.

So let’s compute an example using the following material definition:

Let’s compute 8 different fully relaxing hydrostatic loading cases: 4 in hydrostatic compression, 4 in hydrostatic tension. We’ll take these loaded extreme strain levels: -10%, -5%, -2%, -1%, 1%, 2%, 5%, 10%, which correspond to extreme dilatations of -27%, -14%, -6%, -3%, 3%, 6%, 16%, 33%.

As a first check, we plot the hydrostatic pressures computed for each case. The slope of the line is 3000 MPa, which agrees with the assigned bulk modulus.

Figure 2. Computed volume strain – hydrostatic pressure relationship.

Next, we compute the strain energy density and the cracking energy density for each case. As expected, we verify that for p<0, crack closure results in CED=0, and for p>0, CED=SED/3.

Figure 3. Comparison of strain energy density and cracking energy density for hydrostatic compression and tension.

Finally, we compute the fatigue life for each case. In all cases, we see that the damage sphere is uniform over its entire surface, indicating that all possible crack orientations receive equal damage. We also see that for cases involving hydrostatic compression, life is essentially infinite. For cases involving hydrostatic tension, we verify that finite life is predicted, with shorter life at higher hydrostatic tension, as expected.

Figure 4. Predicted life and damage sphere for compressive and tensile hydrostatic loading.

In summary, we have verified that the Endurica fatigue solver behaves as follows with respect to hydrostatic loading:

In hydrostatic compression, no damage accrues, and life is indefinite.
In hydrostatic tension, crack growth is predicted, with shorter fatigue life for higher values of tension. The cracking energy density is 1/3 of the strain energy density for hydrostatic tension.
For all hydrostatic cases, there is no single preferred critical plane. Rather, all planes show equal potential for crack development.

Is It Validated?

June 9, 2022January 5, 2021 by William V. Mars, Ph.D., P.E.

“Is it validated?” – that’s often the first question we hear after introducing our durability simulation capabilities. And for good reason, given the weight that hangs on the hinge of product durability. Endurica takes verification and validation (V&V) very seriously. Let’s look at what that means.

First, it means that our tools are built on well-known, well-established foundations. These foundations include 1) definition of material / crack behavior via fracture mechanics (Rivlin and Thomas, 1953), 2) integration of the crack growth rate law to predict fatigue life (Gent, Lindley and Thomas, 1964), 3) the fact that crack precursors occur naturally in all locations and all orientations in a rubber sample (Choi and Roland 1996, Huneau et al 2016), 4) hyperelastic stress-strain laws compatible with commercial FEA codes (see Muhr 2005 for an excellent review), and 5) rubber’s fatigue threshold (Lake and Thomas 1967). The validation case thus begins with the cumulative authority of thousands of reports that have confirmed these classical results over nearly 70 years.

Critical plane analysis for rubber has now been around for 20+ years, and it has been validated in multiple ways (material level, component level, system level), across multiple experimental programs (industrial and academic), by multiple independent research groups working on multiple applications (see Google/scholar, for example). It has been validated that: 1) it correctly predicts crack plane orientation under uniaxial, proportional and nonproportional loadings (Harbour et al 2008), 2) it correctly predicts fatigue life across different modes of deformation (Mars 2002), 3) it correctly accounts for the effects of crack closure (Mars 2002), 4) it correctly predicts the development of off-axis cracks for nonrelaxing cycles in strain-crystallizing materials (Ramachandran 2017), 5) it correctly predicts the effects of finite straining on crack orientation (Mars and Fatemi 2006).

The literature is full of old experiments that we have used as validation targets. We have validated Endurica’s strain crystallization models by simulating experimental results published by Cadwell et al (1940) and by Fielding et al (1943). We have validated the ability to predict deformation mode effects by simulating experimental results for simple and biaxial tension published by Roberts and Benzies (1977). We have validated Endurica’s temperature dependence against measurements reported by Lake and Lindley (1964). We have validated against multiaxial fatigue experiments reported by Saintier, Cailletaud and Piques (2006).

We’ve done our own validation experiments. My PhD dissertation (University of Toledo, 2001) contains an extensive database of tension/torsion/compression fatigue tests against which our critical plane algorithms were validated. Two additional PhD dissertations that I co-advised generated additional validations. Dr. Malik Ait Bachir’s thesis (2010) validated mathematically that the scaling law we use for small cracks is valid across all multiaxial loading states. Dr. Ryan Harbour’s thesis (2006) contains a database of multiaxial, variable amplitude fatigue experiments against which our rainflow and damage accumulation procedures were extensively validated.

Validation from partners. We partner with several testing labs. We have invested in testing protocols that produce clean, accurate data and we have run validation programs with our partners to verify the effectiveness of our testing protocols. We’ve demonstrated significant improvements to test efficiency and reproducibility (Goosens and Mars 2018) and (Mars and Isasi 2019). We’ve validated techniques for estimating precursor size and size distribution (Robertson et al 2020, Li et al 2015).

Validation from users. Three (3) of the top 12 tire companies and six (6) of the top 10 global non-tire rubber companies now use our solutions. Most of our users have run internal validation programs to show the effectiveness of our solutions for their applications. Most of these studies are unpublished, but the fact that our user base has continued growing at ~20%/year for 12 years (as of this year) says something important both about the technical validation case and the business validation case. Validation studies have been published with the US Army (Mars, Castanier, Ostberg 2017), GM (Barbash and Mars 2016), Tenneco (Goossens et al 2017) and Caterpillar (Ramachandran et al 2017).

Validation from external groups. There are several academic groups that have independently applied and validated components of our approach. There are too many to list completely, but a few recent examples include Zarrin-Ghalami et al (2020), Belkhira et al (2020) and Tobajas et al (2020).

Software verification, benchmarking and unit testing. In addition to the experimental validations mentioned above, each time we build a new version of our software, we execute a series of automated tests. These tests verify every line of code against expected function, and they ensure that as we add new features, we do not introduce unintended changes. The benchmarks include tests that verify things like coordinate frame objectivity (rigid rotations under static load should do no damage and the same strain history written in two different coordinate systems should have the same life), and check known results pertaining to material models and cycle counting rules. You can read more about our software quality processes here.

It is safe to say that no other solution for fatigue life prediction of rubber has been tested and validated against a larger number of applications than Endurica’s.

References

Aıt-Bachir, M. “Prediction of crack initiation in elastomers in the framework of Configurational Mechanics.” PhD diss., Ph. D. thesis, Ecole Centrale de Nantes, Nantes (France), 2010.

Barbash, Kevin P., and William V. Mars. Critical plane analysis of rubber bushing durability under road loads. No. 2016-01-0393. SAE Technical Paper, 2016.

Belkhiria, Salma, Adel Hamdi, and Raouf Fathallah. “Cracking energy density for rubber materials: Computation and implementation in multiaxial fatigue design.” Polymer Engineering & Science (2020).

Cadwell, S. M., R. A. Merrill, C. M. Sloman, and F. L. Yost. “Dynamic fatigue life of rubber.” Rubber Chemistry and Technology 13, no. 2 (1940): 304-315.

Choi, I. S., and C. M. Roland. “Intrinsic defects and the failure properties of cis-1, 4-polyisoprenes.” Rubber chemistry and technology 69, no. 4 (1996): 591-599.

Fielding, J. H. “Flex life and crystallization of synthetic rubber.” Industrial & Engineering Chemistry 35, no. 12 (1943): 1259-1261.

Goossens, J.R., Mars, W., Smith, G., Heil, P., Braddock, S. and Pilarski, J., 2017. Durability Analysis of 3-Axis Input to Elastomeric Front Lower Control Arm Vertical Ride Bushing (No. 2017-01-1857). SAE Technical Paper. https://doi.org/10.4271/2017-01-1857

Goossens, Joshua R., and William V. Mars. “Finitely Scoped, High Reliability Fatigue Crack Growth Measurements.” Rubber Chemistry and Technology 91, no. 4 (2018): 644-650. https://doi.org/10.5254/rct.18.81532

Harbour, Ryan Joseph. Multiaxial deformation and fatigue of rubber under variable amplitude loading. Vol. 67, no. 12. 2006.

Harbour, Ryan J., Ali Fatemi, and Will V. Mars. “Fatigue crack orientation in NR and SBR under variable amplitude and multiaxial loading conditions.” Journal of materials science 43, no. 6 (2008): 1783-1794.

Huneau, Bertrand, Isaure Masquelier, Yann Marco, Vincent Le Saux, Simon Noizet, Clémentine Schiel, and Pierre Charrier. “Fatigue crack initiation in a carbon black–filled natural rubber.” Rubber Chemistry and Technology 89, no. 1 (2016): 126-141.

Lake, G. J., and P. B. Lindley. “Cut growth and fatigue of rubbers. II. Experiments on a noncrystallizing rubber.” Journal of Applied Polymer Science 8, no. 2 (1964): 707-721.

Li, Fanzhu, Jinpeng Liu, W. V. Mars, Tung W. Chan, Yonglai Lu, Haibo Yang, and Liqun Zhang. “Crack precursor size for natural rubber inferred from relaxing and non-relaxing fatigue experiments.” International Journal of Fatigue 80 (2015): 50-57.

Mars, William Vernon. Multiaxial fatigue of rubber. 2001.

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

Mars, W. V., and A. Fatemi. “Analysis of fatigue life under complex loading: Revisiting Cadwell, Merrill, Sloman, and Yost.” Rubber chemistry and technology 79, no. 4 (2006): 589-601.

Mars, W. V., and A. Fatemi. “Nucleation and growth of small fatigue cracks in filled natural rubber under multiaxial loading.” Journal of materials science 41, no. 22 (2006): 7324-7332.

Mars, W. V. “Computed dependence of rubber’s fatigue behavior on strain crystallization.” Rubber Chemistry and Technology 82, no. 1 (2009): 51-61. https://doi.org/10.5254/1.3557006

Mars, William V., Matthew Castanier, David Ostberg, and William Bradford. “Digital Twin for Tank Track Elastomers: Predicting Self-Heating and Durability.” In Proceedings of the 2017 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS). 2017.pdf here

Mars, W. V., and M. Isasi. “Finitely scoped procedure for generating fully relaxing strain-life curves.” In Constitutive Models for Rubber XI: Proceedings of the 11th European Conference on Constitutive Models for Rubber (ECCMR 2019), June 25-27, 2019, Nantes, France, p. 435. CRC Press, 2019.

Muhr, A. H. “Modeling the stress-strain behavior of rubber.” Rubber chemistry and technology 78, no. 3 (2005): 391-425.Lake and Thomas 1967

Ramachandran, Anantharaman, Ross P. Wietharn, Sunil I. Mathew, W. V. Mars, and M. A. Bauman. “Critical Plane Selection Under Nonrelaxing Simple Tension with Strain Crystallization.” In Fall 192nd Technical Meeting of the Rubber Division, pp. 10-12. 2017.

Rivlin, R. S., and A. G. Thomas. “Rupture of rubber. I. Characteristic energy for tearing.” Journal of polymer science 10, no. 3 (1953): 291-318.Gent, Lindley and Thomas, 1964

Roberts, B. J., and J. B. Benzies. “The relationship between uniaxial and equibiaxial fatigue in gum and carbon black filled vulcanizates.” Proceedings of rubbercon 77, no. 2 (1977): 1-13.

Robertson, Christopher G., Lewis B. Tunnicliffe, Lawrence Maciag, Mark A. Bauman, Kurt Miller, Charles R. Herd, and William V. Mars. “Characterizing Distributions of Tensile Strength and Crack Precursor Size to Evaluate Filler Dispersion Effects and Reliability of Rubber.” Polymers 12, no. 1 (2020): 203. Pdf here

Saintier, Nicolas, Georges Cailletaud, and Roland Piques. “Multiaxial fatigue life prediction for a natural rubber.” International Journal of Fatigue 28, no. 5-6 (2006): 530-539.

Tobajas, Rafael, Daniel Elduque, Elena Ibarz, Carlos Javierre, and Luis Gracia. “A New Multiparameter Model for Multiaxial Fatigue Life Prediction of Rubber Materials.” Polymers 12, no. 5 (2020): 1194.

Zarrin-Ghalami, Touhid, Sandip Datta, Robert Bodombo Keinti, and Ravish Chandrashekar. Elastomeric Component Fatigue Analysis: Rubber Fatigue Prediction and Correlation Comparing Crack Initiation and Crack Growth Methodologies. No. 2020-01-0193. SAE Technical Paper, 2020.

Keeping Your Secrets

June 9, 2022August 18, 2020 by William V. Mars, Ph.D., P.E.

The old saying “loose lips sink ships” is as true in product development as it is in war. Maybe more so – while warships are heavily armored, intellectual property is never more secure than the least ethical person’s willingness and ability to misappropriate. And the stakes have never been higher. Simulation software makes it easier than ever to document material properties, geometry, physics and functions of your next product. So, collaborators can communicate design intentions more easily and more fully than ever before. The downside? It’s easier than ever for an adversary (or the next disgruntled employee) to walk out with your crown jewels!

This is why we’ve just implemented an encryption feature in the Endurica fatigue solvers. Now you can password-protect sensitive information. You control which information gets encrypted, and which stays as plain text. You can share material property or load case definitions for use by collaborators without revealing private details in which you are heavily invested.

Here is a quick demo of how the new feature works. Check it out.

One more way we are helping you to win on durability.

Two Decades of Critical Plane Analysis

June 2, 2022April 1, 2020 by William V. Mars, Ph.D., P.E.

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).

Fatigue Property Mapping 2.0

May 27, 2022January 19, 2020 by William V. Mars, Ph.D., P.E.

We have just launched a few updates to our Fatigue Property Mapping service offerings. The changes were:

Addition of the all new Reliability Module for those needing to compute probability of failure in addition to fatigue life. The module gives you Weibull parameters to describe the statistical distribution of crack precursor sizes in your material.
Addition of a pressure-volume test as an optional add-on to the hyperelastic module. Use this add-on when your rubber is loaded under high confinement to the point where its compressibility must be treated more accurately. If the hydrostatic pressure is more than 5% of the bulk modulus, then this option makes sense.
Split of the original Thermal Module in two components: a Basic Thermal Module and an Advanced Thermal Add-on Module. The Basic Thermal Module provides a dynamic strain sweep to quantify dissipation (for use in computing temperature distribution via FEA) and also provides the temperature sensitivity coefficient on the crack growth rate law. The advanced module provides thermal transport properties (conductivity, specific heat), thermal expansion coefficient (for computing thermal pre-stresses), and additional data points for the dissipation and crack growth rate laws.
Split of the original Extended Life (Ageing) Module into two parts: a Basic Ageing Module and a Master Curve Module. The basic module includes characterization of unaged and aged samples for stiffness, critical fracture energy, and intrinsic strength. The oven exposure time and temperature for the aged sample is specified by the client, or can be set by Endurica based upon a client-specified life target. The Full Master Curve Module gives both the Arrhenius law activation energy and a master curve showing how stiffness, critical fracture energy and intrinsic strength depend on exposure time and temperature.

Most prices have remained the same, except for the Thermal and Ageing modules. The Thermal and Ageing modules have now been significantly streamlined, so that we now offer service at a lower price.

The new price list and specifications can be found here.

Conservatism and Tradition in Fatigue Analysis

June 9, 2022January 19, 2020 by William V. Mars, Ph.D., P.E.

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: i.e., 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.