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Endurica CL and fe-safe/Rubber provide several material models for defining cyclic crack growth under nonrelaxing conditions. Nonrelaxing cycles occur when the ratio R is greater than zero. R is defined as

where T is the energy release rate (note that T will always be greater than or equal to zero).

The crack growth rate under nonrelaxing conditions is, in general, a function of both T_max and R. For purposes of calculation, it is convenient to define an “equivalent” energy release rate T_eq that gives the same steady state rate of crack growth as the operating condition on the nonrelaxing crack growth curve, but which is instead on the fully relaxing crack growth curve. In other words,

Using this scheme, you can set up models for both amorphous and strain-crystallizing rubbers, depending on your definition of T_eq. Amorphous rubbers follow the well-known Paris model, and strain-crystallizing rubbers follow the Mars-Fatemi model (or you can define a lookup table).

Paris Model (Amorphous):

The Paris model is the simplest to derive, as it does not involve any material parameters. It defines the equivalent energy release rate as

This definition is only suitable for rubbers that do not strain-crystallize.

For strain-crystallizing rubbers, one of the other two models should be used.

Mars-Fatemi Model (Strain-crystallizing):

The Mars-Fatemi model accounts for strain crystallization by treating the power-law slope, F, of the Thomas fatigue crack growth rate law as a function of R, where

The exponential version is more compact, but the polynomial version is more flexible.

By substituting F(R) into the fatigue crack growth rate equations for relaxing and nonrelaxing cases, and doing a bit of algebra, the following relationship is obtained

Lookup Table (Strain-crystallizing):

The most flexible and accurate way to define strain crystallization is via a lookup table. The lookup table takes R as an input and returns x(R) as an output. This function can be defined as the fraction x(R) by which the nonrelaxing crack growth curve is shifted between the fully relaxing crack growth curve (x=0), and the vertical asymptote at T_c (x=1), at a given R.

This can be rearranged into the desired T_eq (T_max,R) form, as follows

Comparisons:

Visualizing the differences between the models helps gain a better understanding of how strain crystallization can affect fatigue performance. Since all of these models can be represented in the same form of T_eq(T_max,R), we show 2-D contour plots of T_eq with R on the x-axis and ∆T on the y-axis. ∆T is used instead of T_max to make it easier to compare back to the simple Paris model.

From the figures above, we see that for the Paris model, the equivalent energy release rate depends only on ∆T. When using this model, changes in R will have no effect on fatigue performance (when ∆T is also held constant).

For strain-crystallizing rubbers, changes in R should influence fatigue performance. This is seen in the figures for the Mars-Fatemi and lookup table models.

The Mars-Fatemi example uses the following parameters:

The lookup table example uses T_c=10.0 kJ/m² and Lindley’s data for unfilled natural rubber (P. B. Lindley, Int. J. Fracture 9, 449 (1973)).

For these models, there is a significant decline in T_eq as R increases. This effect is most pronounced when T_max is much smaller than the critical energy release rate T_c. Also, there is a point where the effect is reversed (around R=0.8 in these examples) and the high R-ratio starts to have a negative effect on fatigue performance.

Implications:

A material’s strain crystallization properties’ impact on fatigue performance under non-relaxing conditions should not be ignored. Whether you are seeking to take advantage of strain-crystallization effects or simply comparing the results of different materials/geometries/loadings, strain-crystallization should be accurately represented in your simulations.

Follow these tips to take advantage of strain crystallization and help ensure your fatigue performance is the best it can be.

Take advantage of Endurica’s material characterization service (the FPM-NR Nonrelaxing Module generates the strain crystallization curve) or use your own in-house testing to create an accurate strain crystallization model of your material (the nonrelaxing procedure is available for the Coesfeld Tear and Fatigue Analyser).
Use output requests like DAMAGE_SPHERE, CEDMINMAX and CEDRAINFLOW to observe R-ratios for your duty cycles.

References

B. Lindley, Int. J. Fracture 9, 449 (1973)

Mars, W. V. “Fatigue life prediction for elastomeric structures.” Rubber chemistry and technology 80, no. 3 (2007): 481-503.

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

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.

Two graphs depicting the relationship between cycles and tearing energy. Through these graphs they show a relationship between a facture mechanics experiment and a crack nucleation experiment.

The relationship between crack nucleation and fracture mechanics experiments for polymers was first documented in 1964 by Gent, Lindley and Thomas (Journal of Applied Polymer Science, 8, 455, 1964.)

Some weeks ago, I attended the WCX 2017 SAE World Congress and Exhibition, where a Technical Expert Panel Discussion on the topic of Durability Analysis in CAE was held. The panel was moderated by Yung-Li Lee (FCA US LLC), and included topic experts Abolhassan Khosrovaneh (General Motors LLC), Xuming Su (Ford Motor Co., Ltd.), and Efthimio Duni (FCA EMEA). The discussion was excellent and wide ranging, owing both to the panelists, and also to the audience, which (judging by the high engagement) was very well versed with the core of the topic, as well as its frontiers. I will not attempt to give a complete summary of the event, but I do want to highlight a memorable discussion thread, and to offer a few thoughts.

I do not know who raised the topic. It could have been a doctoral student or young professional. Clearly, it was a person wanting to align his own efforts well relative to larger industry trends. He started out with the observation that the classical crack nucleation methods (in which fatigue behavior is defined by a stress-life or strain-life curve) are quite popular in the automotive sector for analyzing fatigue of metals. He also observed that modern tools for rubber take a different approach based upon a fracture mechanics method (in which fatigue behavior is defined by a crack growth rate curve). He then asked (I’m paraphrasing from memory here):

Which method (nucleation vs. fracture mechanics) is preferred for analysis of polymers?
Should we try to unify all testing and analysis efforts for metals and polymers under the same method?

The panelists made several points in responding to this prompt. They started with the point that differences in methodology may be hard to avoid, if only because metals and polymers are so different in composition, molecular structure, and microstructure. Of course, it is possible to use fracture mechanical methods with metals, although there are some limitations implied by the granular crystalline structure of metals when cracks are very small. Likewise, it is also possible to use stress-life methods with polymers, although certain aspects of the material behavior may be incompatible with the usual procedures, leading to questionable results. From a practical standpoint, it would be quite difficult to change the methods used by the industry for metal fatigue analysis – the methods are quite mature at this point, and they have been implemented and validated across so many codes and projects that it is hard to imagine what could be gained by making a change. For polymers, CAE durability methods are newer, and we should use what works.

There is a final point that I believe will ultimately define how this all plays out. It is that 1) fatigue analysis for polymers is usually driven by multiple “special effects”, and that 2) the economics of the testing required to characterize these effects scales very differently between the two approaches.

Let me illustrate with a typical example: we have a Natural Rubber compound used in a high temperature application, for an extended time, under nonrelaxing loads. Let’s compare our options:

Option 1

Stress-Life Method

Option 2

Fracture Mechanics + Critical Plane Method

To use the stress-life method, we will need to develop curves that give the effect of 4 parameters on the fatigue life: 1) strain amplitude, 2) mean strain, 3) temperature, and 4) ageing. The experiment is a simple cycle-until-rupture procedure, with one test specimen consumed per operating condition tested.

Let’s assume that we measure each of the four parameters at only 3 levels, and that we will require 3 replicates of each experiment. The total number of fatigue experiments we need is therefore:

N = 3 amplitudes x 3 means x 3 temperatures x 3 ageing conditions x 3 replicates = 3⁵ = 243 fatigue to failure tests

With the fracture mechanics method, a single run of the experiment solicits the crack at many different operating conditions, enabling observation of the crack growth rate at each condition. Using Endurica’s standard testing modules, the example testing program (including replication) would require the following procedures:

Core module: 9 experiments (amplitude effect)

Nonrelaxing module: 3 experiments (mean effect)

Thermal module: 12 experiments (temperature effect)

Ageing module: 30 experiments (ageing effect)

243 tests required

54 tests required

In this example, the fracture mechanics method is almost 243/54 = 4.5x more efficient than the stress-life method! If you need more than 3 levels, or if you have more than 4 key operating parameters, the experimental cost for the stress-life method quickly becomes completely impractical, relative to the fracture mechanics method. Based on these scaling rules, and on the fact that polymers exhibit so many special effects, you can now appreciate why the fracture mechanics method must prevail for polymers. For metals, the case is less compelling: there aren’t as so many special effects, and the industry testing norms are already well established.

Bottom line: for fatigue of polymers, the economics of testing for ‘special effects’ strongly favors a fracture mechanics approach. This fact is certain to shape the future development of fatigue life prediction methods for polymers.

Specifying Strain Crystallization Effects for Fatigue Analysis

Durability Analysis in CAE: panel discussion of metals vs. polymers at the SAE World Congress