Materials Characterization

Figure 1. Constant amplitude cycles at three different mean strains.

Rubber and metal are very different materials that exhibit very different behaviors. Consider the effect of mean strain or stress on the fatigue performance of these materials. Figure 1 illustrates a few typical constant amplitude strain cycles, each at a different level of mean strain. If the stress amplitude is equal to the mean stress, we say that we have pulsating tension or fully relaxing tension. If the mean stress is zero, we say that we have fully reversed tension/compression. If the minimum stress is always positive, then we have nonrelaxing tension (i.e. always under load). Nonrelaxing cycles are quite common in applications. Examples include: pre-loads applied during installation; swaging of a bushing to induce compressive pre-stresses, interference fits, self-stresses occurring due to thermal expansion/contraction; and in tires, shape-memory effects of textile cords.

In metal fatigue analysis, it is customary to define the effect in terms of stress amplitude σ_a and mean stress σ_m, relative to the yield stress σ_y and the ultimate stress σ_u, as shown in Figure 2. Below the fatigue threshold stress σ₀, indefinite life is predicted. The Haigh (or Goodman)

Figure 2. Haigh diagram (left) and Wohler curves (right) showing mean strain effects on fatigue life for a metal.

diagram (left) maps fatigue life as a function of these parameters [1]. Wohler curves (right) provide similar information. For metals, a simple rule may be applied universally: increasing mean strain is detrimental fatigue life. It is also commonly assumed for metals that the critical plane is perpendicular to maximum principal stress direction.

There are many ways that rubber materials differ from metallic materials. At the atomic scale, rubber is composed of long chain molecules experiencing constant thermal motion while interlinked with a permanent network topology. This structure permits large, elastic/reversible straining to occur. Metals could not be more different, existing as individual atoms packed into well-ordered crystals with occasional dislocations or lattice vacancies. This structure permits only vanishingly small strains before inelastic deformation occurs. At the meso scale, rubber is typically a composite material containing fillers such as carbon black, silica or clay, as well as other chemical agents. The mesoscale of a metal is generally described in terms of crystalline grain boundaries and inclusions or voids. Rubber exhibits many “special effects” that are not seen in metals: rate and temperature dependence, ageing, cyclic softening. It is unsurprising that analysis methods for rubber differ substantially from those applied for metals.

Rubber’s fatigue performance has a more complex dependence on mean strain. For amorphous (ie non-crystallizing) rubbers, increasing mean strain reduces the fatigue life, as with metals. But for rubbers that exhibit strain-induced crystallization, mean strain can greatly increase fatigue life, as illustrated in Figure 3. Fatigue simulations therefore must take account of the strain crystallization effect.

Figure 3. Fatigue tests run in simple tension under constant amplitude show a significant increase in life for Natural Rubber (NR), which strain crystallizes, and a decrease of life for Styrene Butadiene Rubber (SBR) which is amorphous [2].

Mean strain effects are specified in the Endurica fatigue code in terms of fracture mechanical behavior, using the concept of an equivalent fully relaxing tearing energy T_eq. The tearing energy for fully relaxing conditions is said to be equivalent when it produces the same rate of crack growth as the nonrelaxing condition. For amorphous rubbers, the equivalent R=0 tearing energy T_eq is simply the range ΔT of the tearing energy cycle, which can be expressed in terms of the min and max tearing energies T_min and T_max, or in terms of R= T_min / T_max. Plugging this rule into the power law crack growth rate function yields the well known Paris law, which predicts faster crack growth for increasing mean strain. For a strain crystallizing rubber, the equivalent fully relaxing tearing energy can be specified using the Mars-Fatemi law. In this case, the equivalent fully relaxing tearing energy depends on a function F(R), which specifies the crystallization effect in terms of its influence on the powerlaw slope of the crack growth rate law. The relationship for amorphous and crystallizing rubbers are summarized in Table 1 [3,4].

Table 1. Models for computing crack growth rate in amorphous and strain-crystallizing rubbers.

Rubber’s fatigue behavior may be plotted in a Haigh diagram, but the contours can be quite different than for metals. In metal fatigue analysis, it is assumed that cracks always develop perpendicular to the max principal stress direction. This is not always true for rubber, especially in cases involving strain crystallization and nonrelaxing loads. For rubber fatigue analysis it is therefore required to use critical plane analysis [5], in which fatigue life is computed for many potential crack orientations, and in which the crack plane with the shortest life is identified as the most critical plane. Figure 4 shows the dependence of the fatigue life and the critical plane orientation on strain amplitude and mean strain. A sphere is plotted for each pair of strain amplitude and mean strain coordinates, on which the colors represent fatigue life, and unit normal vectors indicate critical plane orientations. It can be seen that different combinations of mean strain and strain amplitude can produce a range of crack plane orientations.

Figure 4. Critical plane analysis consists in integrating the crack growth rate law for every possible crack orientation, and identifying the orientation that produces the shortest life (left). Each point in the Haigh diagram (right) is associated with its own critical plane orientation.

The Haigh diagrams for natural rubber (NR) and for styrene butadiene rubber (SBR) are shown in Figure 5. In these images, red represents short fatigue life, and blue long life. For natural rubber (on the left), the long-life region of the Haigh diagram exhibits a notable dome-like shape, indicative of a beneficial effect of mean strain under the influence of strain-induced crystallization. In contrast, SBR always exhibits decreased fatigue life as mean strain increases. Even so, the Haigh diagram for SBR has a nonlinear character associated with the material’s hyperelasticity that is also distinct from a metal.

Figure 5. Haigh diagrams computed for NR (left) and for SBR (right) rubbers.

It should be noted that the strain crystallization effect in rubber depends on temperature. At colder temperatures, the effect is stronger, and at higher temperatures it is weaker. Figure 6 compares experimental Haigh diagrams [6] (top) for a crystallizing rubber to computed results (bottom) for three temperatures.

Figure 6. Experimental Haigh diagram [6] for natural rubber at 3 temperatures (top), compared to computed Haigh diagram (bottom). Increasing temperature tends to reduce the beneficial effect of strain crystallization.

In summary, while tensile mean stresses are always detrimental in metals, in rubber they may be either beneficial or harmful, depending on whether the rubber can strain crystallize. The benefits of mean stresses in rubber can be quite strong – sometimes amounting to more than several orders of magnitude. The beneficial effect is stronger at colder temperatures and is reduced at higher temperatures. Critical Plane Analysis is essential for accurately predicting the effects of strain crystallization in rubber. Wohler curves, commonly used for metal fatigue analysis, incorrectly assume that the worst-case plane is always normal to the max principal stress direction. This is not an accurate approach for strain crystallizing rubber under mean strain. Use the Endurica fatigue solvers to accurately capture these effects when its important to get durability right!

References

[1] Stephens, R. I., Fatemi, A., Stephens, R. R., & Fuchs, H. O. (2000). Metal fatigue in engineering. John Wiley & Sons.

[2] Ramachandran, Anantharaman, Ross P. Wietharn, Sunil I. Mathew, W. V. Mars, and M. A. Bauman. (2017) “Critical plane selection under nonrelaxing simple tension with strain crystallization.” In Fall 192nd technical meeting of the ACS Rubber Division, pp. 10-12.

[3] Mars, W. V. (2009). Computed dependence of rubber’s fatigue behavior on strain crystallization. Rubber Chemistry and Technology, 82(1), 51-61.

[4] Harbour, Ryan J., Ali Fatemi, and Will V. Mars. “Fatigue crack growth of filled rubber under constant and variable amplitude loading conditions.” Fatigue & Fracture of Engineering Materials & Structures 30, no. 7 (2007): 640-652.

[5] Mars, W. V. (2021). Critical Plane Analysis of Rubber. Fatigue Crack Growth in Rubber Materials: Experiments and Modelling, 85-107.

[6] Ruellan, Benoît, J-B. Le Cam, I. Jeanneau, F. Canévet, F. Mortier, and Eric Robin. “Fatigue of natural rubber under different temperatures.” International Journal of Fatigue 124 (2019): 544-557.

2023 marked year 15 for Endurica. If I had to pick one word to describe the past year, that word would be “vector”. Because magnitude and direction. 😊

We updated our core value statement this year. The first one I ever wrote as part of Endurica’s original business plan listed 3 values: technical leadership, customer focus, and trustworthiness. Those values served us well for many years and in many ways shaped who we have become. But it was important this year to take stock again. We’ve grown 8-fold since I wrote those down! So our team spent many hours revisiting our shared values and deliberating over which will best define our culture and steer us right going forward. In the end, we decided to keep the first 3, and we added 3 more: embrace the grit, make an impact, and better every day.

We also completed an exercise to articulate what makes Endurica truly unique in the CAE / durability simulation space. The 3 words we chose are… Accurate, Complete, and Scalable.

Accurate refers to the accurate material models that capture rubber’s many “special effects”, the accurate critical plane analysis method for analyzing multiaxial history, the accurate handling of nonlinear relationships between global input load channels and local crack experiences, and the extensive set of validation cases that have demonstrated our accuracy over the years. Nobody offers a more accurate solution for rubber durability.
Complete refers to our complete coverage of infinite life, safe life and damage tolerant approaches to testing and simulation. It refers to feature completeness that enables users to account for nearly any material behavior under nearly any service conditions. Finally, it refers to the documentation, the materials database, and the examples we distribute with the software and with our webinar series. Nobody offers a more complete solution for rubber durability.
Scalable refers to our capacity to apply our solutions efficiently in all circumstances. Scalability is the training we provide so that users can learn our tools quickly. Scalability is access to powerful, ready-to-use workflows right when you need them. Scalability is the modular approach we take to material testing and modeling so that simple problems can be solved cheaply and complex problems can be solved accurately in the same framework. Scalability is our multi-threading that allows job execution time to be accelerated to complete impactful analysis on tough deadlines. Nobody offers a more scalable solution for rubber durability.

2023 was not all navel-gazing and new marketing. We also had magnitude and direction in other areas.

Top 10 Code Developments:

New Endurica Architecture: After several years of development and a soft launch under the Katana project name, we finally completed our migration to the new architecture. The new architecture provides a huge speed advantage for single thread and now for multithread execution. It uses a new input file format (.json). The json format makes it easier than ever for users to build customized and automated workflows via Python scripting.
Sequence Effects: Sometimes the order of events matters to durability, and sometimes it doesn’t. We introduced Steps and Blocks to our input file, giving users complete control over the specification of multi-block, multi-step scheduling of load cases. There is also a new output request that came out of this work: residual strength.
EIE: 6 channels and support for RPC: Support for 6 channels of load input was one of our most highly requested new features. Fast growing use of this feature led to further enhancements of the workflow (support for rpc file format, studies of map building techniques), and new recommendations on how to implement boundary conditions for specified rotation histories in explicit and implicit finite element models.
Queuing: Design optimization studies need efficient management and execution of multiple jobs. Endurica’s software license manager now supports queueing for licenses. Queuing allows a submitted job to automatically wait to start until a license is available, instead of the prior behavior of exiting with a license error. Now you can submit many jobs without worrying about license availability.
Haigh Diagram Improvements: We implemented an improved discretization of the Haigh diagram, and parallelized its evaluation. Now you get much nicer looking results in a fraction of the time. For details, check out our blog post on Haigh diagrams and also read about other improvements like axis limit setting and smoother contour plots.
Viewer image copy: There is now a button! Its easier than ever to get your images into reports.
Documentation Updates: We have been focusing on improving documentation this year. There are many new sections in the theory manual and user guide, as well as a getting started guide and more examples. Stay tuned for many more examples coming in 2024!
User Defined Planes: It is now possible to define your own set of planes for the critical plane search. One example where you might want to do this would be the situation where you would like to refine the critical plane search on a limited domain of the life sphere.
New Database Materials: We added 7 new carbon black and silica filled EPDM compounds to the database. We are now up to 42 unique rubber compounds in the database.
Uhyper Support: The new architecture now supports user-defined hyperelasticity. If you have a Uhyper subroutine for your finite element analysis, you can use it directly with Endurica.

Testing Hardware

We completed the acquisition and installation at ACE labs of a Coesfeld Instrumented Cut and Chip Analyser (ICCA). The ICCA provides unmatched measurement and control of impact conditions, and provides a way to evaluate rubber compounds for their resistance to cutting and chipping.

Applications, Case Studies, Webinars

Never underestimate the students! We were blown away by the work of undergraduates at the University of Calgary with our tools and Ansys. The students designed an airless tire, completing durability simulations using Endurica software within the scope of a senior design project. They were able to Get Durability Right on a short timeline and a student budget. Check out their multi-objective, high-performance design project here.

Analyzing what happens to tires as they take on the most celebrated testing track in the world might have been the funnest project Endurica’s engineers tackled in 2023. We presented the technical details at The Tire Society annual meeting and more in a followup webinar. An extensive Q&A session followed, and I loved the final question: “So, how long before we have a dashboard display of ‘miles to tire failure’ in our cars?” Bring it. We are ready!

Our Winning on Durability webinar series hit a nerve with the Metal Fatigue DOES NOT EQUAL Rubber Fatigue episodes on mean strain (the tendency of larger mean strains to significantly INCREASE the fatigue life of some rubbers!) and linear superposition (for converting applied load inputs to corresponding stress/strain responses). The great response has lead to our third installment on the differences between rubber and metal fatigue with an upcoming presentation on temperature effects.

Rubber Fatigue ≠ Metal Fatigue Part 1: Mean Strain Effects

2023 – a Year of Magnitude and Direction