Strain-Induced Crystallization in High Cis Butadiene Rubber: Fact or Fiction?

There is an ongoing drive to create synthetic rubber that can give mechanical performance matching the properties of natural rubber (NR) – excellent strength, tear resistance, and fatigue crack growth resistance – which are attributed to the ability of NR to strain crystallize.  I have attended several technical conferences in the tire and rubber field where I have witnessed presentations about new grades of polybutadiene (butadiene rubber (BR)) with high cis-1,4 structure, wherein a claim is made about the improved ability of the BR to undergo strain-induced crystallization (SIC) for better mechanical properties in tires and other rubber applications.  Similar statements can be found in technical marketing materials and in patents.1,2  To what extent is this true?  This post takes a closer look at this topic.

It is first useful to show why strain-induced crystallization is key to the unique mechanical properties of natural rubber.  One of the most conclusive studies on the function of strain-induced crystals to slow the growth of cracks in cyclic fatigue is the work by Brüning et al.3  The movie below from this research shows an edge crack region in NR reinforced with 40 phr of N234 carbon black that is experiencing cyclic deformation from 0% to 70% planar tension at a frequency of 1 Hz.  Each pixel at each time in the video represents a wide-angle X-ray diffraction pattern collected in real time while stretching in a synchrotron X-ray beam, with red color used to indicate crystallinity and black for purely amorphous regions.  You can see the crystals form during stretching which self-reinforces the rubber at the crack tip region to resist further growth of the crack.  These experiments are very revealing, but they are extremely difficult and expensive to perform, as they require scattering analysis expertise, specially designed stretching devices with precise spatial control, and access to one of the few national laboratory sites where synchrotron X-ray studies can be performed.

Stereoregular polymers can crystallize in a window bounded by the glass transition temperature (Tg) and the melting temperature (Tm). When stretched, a polymer can crystallize at temperatures above its normal melting temperature due to melting temperature elevation caused by the well-known chain orientation/entropy effect.4  Chain orientation results in an increase in melting temperature from Tm to Tm,SIC.  This is shown schematically in the figure below for NR and BR.  The Tg and Tm for BR are significantly lower than the values for NR.  Therefore, even after perfecting the structure of polybutadiene to achieve >98% cis-1,4 structure though catalyst and process innovations, BR still has an intrinsic disadvantage compared to NR when it comes to SIC in the common temperature range where durable rubber components are employed.

 

Gent and Zhang5 recognized the lower Tm,SIC for BR compared to NR for samples crystallized in uniaxially strained conditions.  Ultra high cis BR with 98% cis content did not show any evidence of strain-induced crystallization at 23 °C whereas NR clearly exhibited SIC in a study by Kang and coworkers.6  In an investigation by Toki et al.,7 is was necessary to reduce the temperature to 0°C before high cis BR showed a crystalline X-ray scattering pattern at a strain of 5 (500% elongation).

As an aside, I will caution the use of the entropy-driven melting temperature increase as the only explanation for strain-induced crystallization above the normal Tm.  Natural rubber is quite a slow crystallizing polymer in the unstretched state, even when annealed in the prime crystallization regime midway between Tg and Tm.  In stark contrast, strain-induced crystallization occurs very fast in NR which highlights the complexity of behavior beyond the over-simplification given in the figure above.

So, why is there not more focus on synthetic high cis-1,4-polyisoprene which has the same polymer microstructure and strain crystallization window as NR?  The short answer is that the butadiene monomer is much more commercially available than isoprene at many synthetic rubber plants around the world.  I remember the early years in my industrial R&D career when I was working with some very talented polymer chemists at Bridgestone / Firestone.   I was developing new molecular architectures for improved synthetic rubber to use in various tire compounds.  The chemists told me that they could make almost any polymer design I wanted, as long as it could be made from styrene and/or butadiene.  This restriction reflected the fact that these two monomers were the commonly available feedstocks at the commercial plants where the polymers would eventually be produced.

In closing, it is a fact that high cis polybutadiene can strain crystallize at sub-ambient temperatures, but it is fiction that it will strain crystallize in the same manner as natural rubber at room temperature and above.

Do you want to see if your rubber will really exhibit strain-induced crystallization in a practical way that is relevant to end-use applications?  Contact me to learn more about the Non-Relaxing Module in Endurica’s portfolio of testing services that was specifically developed to efficiently highlight SIC effects.8-10

References

1 http://arlanxeo.de/uploads/tx_lxsmatrix/sustainable_mobility_01.pdf

2  S. Luo, K. McKauley, and J. T. Poulton, “Bulk polymerization process for producing polydienes”, U.S. Patent 8,188,201, granted May 29, 2012 to Bridgestone Corporation.

3 K. Brüning, K. Schneider, S. V. Roth, and G. Heinrich, “Strain-induced crystallization around a crack tip in natural rubber under dynamic load”, Polymer 54, 6200 (2013).; movie provided to Endurica LLC by the authors (movie to be used only with permission from Karsten Brüning).

4 P. J. Flory, “Thermodynamics of crystallization in high polymers. I. Crystallization induced by stretching”, Journal of Chemical Physics 15, 397 (1947).

5 A. N. Gent and L.-Q. Zhang, “Strain-induced crystallization and strength of elastomers. I. cis-1,4-polybutadiene”, Journal of Polymer Science Part B: Polymer Physics 39, 811 (2001)

6 M. K. Kang, H.-J. Jeon, H. H. Song, and G. Kwag, Strain-induced crystallization of blends of natural rubber and ultra high cis polybutadiene as studied by synchrotron X-ray diffraction”, Macromolecular Research 24, 31 (2016).

7 S. Toki, I. Sics, B. S. Hsiao, S. Murakami, M. Tosaka, S. Poompradub, S. Kohjiya, and Y. Ikeda, “Structural developments in synthetic rubbers during uniaxial deformation by in situ synchrotron X-ray diffraction”, Journal of Polymer Science Part B: Polymer Physics 42, 956 (2004).

8 K. Barbash and W. V. Mars, “Critical Plane Analysis of Rubber Bushing Durability under Road Loads”, SAE Technical Paper 2016-01-0393, 2016, https://doi.org/10.4271/2016-01-0393.

9 W. V. Mars and A. Fatemi. “A phenomenological model for the effect of R ratio on fatigue of strain crystallizing rubbers”, Rubber Chemistry and Technology 76, 1241 (2003).

10 http://endurica.com/specifying-strain-crystallization-effects-for-fatigue-analysis/

 

 

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Specifying Strain Crystallization Effects for Fatigue Analysis

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 Tmax and R. For purposes of calculation, it is convenient to define an “equivalent” energy release rate Teq 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 Teq.  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

or

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 Tc (x=1), at a given R.

This can be rearranged into the desired Teq (Tmax,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 Teq(Tmax,R), we show 2-D contour plots of Teq with R on the x-axis and ∆T on the y-axis.  ∆T is used instead of Tmax 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 Tc=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 Teq as R increases.  This effect is most pronounced when Tmax is much smaller than the critical energy release rate Tc.  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

  1. 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.

 

 

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Tire Society 2017 – Best Question

Every year, the top minds from academia, government and industry gather in Akron to share their work at the Tire Society annual meeting, and to enjoy a few moments of professional camaraderie.  Then we all return to fight for another year in the trenches of the technology wars of our employers.

This year, the meeting offered the latest on perennial themes: modal analysis, traction, materials science, noise, simulation, wear, experimental techniques for material characterization and for model validation.  Too much to summarize with any depth in a blog post.  If you are interested, you should definitely resolve to go next year.  Endurica presented two papers this year.

I presented a demonstration of how the Endurica CL fatigue solver can account for the effects of self-heating on durability in a rolling tire.  Endurica CL computes dissipation using a simple microsphere model that is compatible, in terms of discretization of the shared microsphere search/integration domain, with the critical plane search used for fatigue analysis.  In addition to defining dissipative properties of the rubber, the user defines the temperature sensitivity of the fatigue crack growth rate law when setting up the tire analysis.  In the case considered, a 57 degC temperature rise was estimated, which decreased the fatigue life of the belt edge by a factor of nearly two, relative to the life at 23 degC.  The failure mode was predicted at the belt edges.  For 100% rated load, straight ahead rolling, the tire was computed to have a life of 131000 km.

The best audience question was theoretical in nature: are the dissipation rates and fatigue lives computed by Endurica objective under a coordinate system change?  And how do we know?  The short answer is that the microsphere / critical plane algorithm, properly implemented, guarantees objectivity.  It is a simple matter to test: we can compute the dissipation and fatigue life for the same strain history reported in two different coordinate systems.  The dissipation rate and the fatigue life should not depend on which coordinate system is used to give the strain history.

For the record, I give here the full Endurica input (PCO.hfi) and output (PCO.hfo) files for our objectivity benchmark.  In this benchmark, histories 11 and 12 give the same simple tension loading history in two different coordinate systems.  Likewise, 21 and 22 give a planar tension history in two coordinate systems.  Finally, 31 and 32 give a biaxial tension history in two coordinate systems.  Note that all of the strain histories are defined in the **HISTORY section of the .hfi file.  In all cases, the strains are given as 6 components of the nominal strain tensor, in the order 11, 22, 33, 12, 23, 31.  The shear strains are given as engineering shear components, not tensor (2*tensor shear = engineering shear).

The objectivity test is successful in all cases because, as shown in the output file PCO.hfo, both the fatigue life, and the hysteresis, show the same values under a coordinate system change.  Quod Erat Demonstrondum.

ObjectivityTable

 

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

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.

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Welcome to the Endurica Blog

Welcome to the Endurica blog, written by founder William Mars, Ph.D. These earlier posts (AND MANY MORE) are available on Will’s LinkedIn page:

 

Tire Society 2016: Notes on Advances in Computing Durability –  September 22, 2016

 

 

Strain Crystallization and Durability of Elastomers – August 26, 2016

 

Maximum Principal Stress Damage – August 3, 2016

 

 

Microstructure in Elastomers: Flaw or Feature? – March 26, 2016

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