Rubber Fatigue ≠ Metal Fatigue Part 3: Thermal Effects

All materials are temperature dependent, but some more than others: metals tend to be crystalline solids and will melt at sufficiently high temperatures; in contrast, crosslinked elastomers are always solids. They can be glassy or rubbery, crystalline or amorphous. When heated to extreme temperatures, they burn rather than melt, producing new substances, usually low molecular weight hydrocarbons (i.e. tarry substances and smoke).

Of course, you do not have to melt or burn a material to see the effects of temperature. In fatigue analysis, we are concerned with stress-strain and crack growth behaviour. These can be temperature dependent for both metals and rubbers. However, while metals have a very high thermal conductivity, rubbers have almost the lowest. Therefore, fatigue analyses involving large temperature gradients are much more common in rubber than in metal.

As shown in Fig.1, while a 100°C temperature gradient in a metal can affect the fatigue tensile strength or the fatigue limit by 10% [1], the same 100°C temperature gradient in rubber can reduce the fatigue life by four orders of magnitude [2]!

Fig. 1. Left – Effects of temperature on carbon steel showing tensile strength (+), yield stress (●), and fatigue limit (○), [1]; Right – Effects of temperature on natural rubber (Δ) and styrene butadiene rubber (●) [2].
Temperature and Segmental Mobility

The mechanisms underlying the elasticity of metals and rubbers could hardly be more different.  Under stress, atoms in a metal’s crystal lattice are displaced from their equilibrium positions, and potential energy is stored in strained interatomic bonds.  In rubber, however, the strain energy is not predominantly stored in strained atomic bonds.  Rather, elasticity arises because the constituent long-chain molecules are much more likely to be randomly coiled than to be fully extended.

Thus, provided that the molecules are sufficiently agitated by random thermal fluctuations, an entropic spring effect is created, meaning that potential energy can be stored by working to reduce the entropy of the polymer chain network by increasing the end-to-end distance of individual polymer chains [3].

Polymers in general can exhibit both glassy and rubbery behavior, depending on the temperature.  The rubbery state – in which entropic elasticity dominates – exists above the glass transition temperature Tg, if the molecular motion rate is sufficiently high.  In the rubbery state, very large strains are possible and the rubbery elastic storage modulus E’r determines the stress-strain curve.

Below Tg, however, the lack of thermal molecular mobility prevents molecular reconfiguration, resultng in a glassy stiffness E’g that is several orders of magnitude higher than E’r.  Polymers operating below Tg are thus not capable of large elastic strains and instead exhibit inelastic behavior when strains exceed a few percent.  Figure 2 shows how the storage and loss moduli vary through the glass transition (left), and how the rate of molecular motion rate depends on temperature (right).  The relative rate φ(T)/φ(Tg) of molecular motion as a function of temperature T is described by the WLF equation [4], which has material constants A and B.

Since the fracture mechanical properties of rubber depend on the viscoelastic dissipation in the crack tip process zone, with higher dissipation associated with lower crack growth rates, frequency and temperature effects can be inferred accordingly. Viscoelastic master curves, such as those shown in Fig. 2, can be used as part of the material property rate dependence specification in the Endurica solver.

Fig. 2, Left – Rubber’s elastic and viscous responses depend on temperature relative to the glass transition temperature Tg; Right: The rate of molecular motion depends on temperature relative to the glass transition temperature Tg.

Self-Heating and Thermal Runaway

During a charge cycle, work WL is done on the charge stroke, some of which WU is recovered on the discharge stroke, as shown in Fig. 3.  The unrecovered part of the work H remains in the material as heat energy, increasing the temperature.

 

Fig. 3. Work input WL on the loading stroke is partially recovered as WUon the unloading stroke. A portion H of the energy remains in the material as heat.

The rate of viscoelastic heating of rubber depends on strain amplitude, cycle rate (i.e. frequency) and temperature. The strain amplitude dependence of the viscoelastic storage and loss modulii, G’ and G” respectively, can be specified using the Kraus model [5,6]:

 

where εa is the strain amplitude, and where G’∞, G’0, εa,c, m, G”∞, G”max, and ΔG”U are material parameters. The viscoelastic heat rate per unit volume can be calculated from:

Due to the low thermal conductivity of rubber, small amounts of viscoelastic self-heating can produce large temperature gradients.  Accurately accounting for thermal effects on rubber durability generally requires both structural finite element analysis to calculate stress and strain fields, and a thermal finite element analysis to calculate the temperature field. Endurica fatigue solvers can provide heat rate calculations in a coupled finite element simulation for both transient and steady state thermal analyses.

In cases where the temperature in the rubber exceeds a critical value Tx, an additional heat rate contribution q ̇x occurs due to exothermic chemical reactions.  The effect is illustrated in Fig. 4, for a rubber cylinder subjected to a rotating bending load [7]. The thermal runaway starts after about 250 seconds. Both experimental (dashed line) and Endurica-calculated (solid line) simulation results are plotted for the cylinder centreline (blue) and for the cylinder outer surface (green).  The thermal runaway event typically results in rapid decomposition of the rubber into hydrocarbon gases (i.e. smoke/burning rubber) and low-molecular weight substances (tar).

Fig. 4. When temperature exceeds a critical value Tx, exothermic chemical reactions can produce a thermal runaway failure. Plot (right) shows Endurica calculated transient temperature history (solid lines) for a rotating bending cylinder (structural finite element model shown on left). For comparison, experimentally measured temperature histories are also shown (dashed lines).

Reversible Temperature Effects

The crack growth properties of rubber reversibly depend on temperature.  Higher temperatures tend to reduce the tear strength Tof rubber and increase the crack growth rate, as shown in Fig. 5 [8].  At lower temperatures, the tear strength is increased and crack growth is retarded.  Endurica’s crack growth models can be specified with a temperature dependence via the temperature sensitivity coefficient (see Table 1) or via a table look-up function.

Fig. 6 shows the fatigue life as a function of temperature calculated from the parameters in Table 1 [2].  Over a range of 100°C, natural rubber loses approximately a factor of two in fatigue life, and styrene butadiene rubbers loses four orders of magnitude!

Table 1. Crack growth properties and temperature sensitivity for natural rubber (NR) and styrene butadiene rubber (SBR), estimated from measurements reported in [2].
Fig. 5 – Increasing temperature causes the crack growth rate to increase. Results are shown for natural rubber [8].
Fig. 6. Endurica calculated dependence of fatigue life on temperature for natural rubber (Δ) and for styrene butadiene rubber (●) [2]. Compare to Fig. 1.
Some rubbers undergo strain crystallization, which is beneficial when operating under non-relaxing conditions.  The crystallization effect is strongly temperature dependent and decreases with increasing temperature.

Fig. 7 shows the Haigh diagram calculated by Endurica for three different temperatures: 23, 90 and 110°C.  For example, at a mean strain of 100% and a strain amplitude of 20%, the fatigue life at 23°C exceeds 106 cycles, but at 110°C the fatigue life is approximately 103 cycles.  This effect has been confirmed experimentally in recent work by [9].

Fig. 7. Endurica calculated Haigh diagrams for natural rubber at 23, 90 and 110°C . Increasing temperature tends to reduce strain crystallization, with the result that the mean strain benefit associated with strain crystallization is reduced or even eliminated at high temperatures.

Irreversible Temperature Effects / Ageing

Prolonged exposure to high temperatures can cause permanent changes in the cross link density and mechanical properties of rubber, including stiffness and crack growth properties.  The effect depends on the availability of oxygen [10], as shown in Fig. 8.

Fig. 8 – The evolution of rubber’s properties during ageing depends on the availability of oxygen, and on the temperature [10]. Under aerobic conditions, ageing tends to increase stiffness while strain at break decreases. Under anaerobic conditions, ageing tends to decrease stiffness while strain at break decreases.
When aged under Type I aerobic conditions, rubber becomes brittle as its strain at break λb decreases while its stiffness M100 increases.  When aged under Type II anaerobic conditions, rubber tends to soften while its strain at break decreases.

The rate at which thermochemical ageing of rubber progresses can be specified in Endurica using the Arrhenius law [11] and its activation energy parameter Ea. When following a temperature history θ(t), Endurica integrates the Arrhenius law to determine an equivalent exposure time τ at the reference temperature θ0R is the real gas constant.

The equivalent exposure time controls the evolution of the stiffness and crack growth properties with thermal history. As shown in Fig. 9, the evolution of the crack growth rate law is specified by a tabular function that gives the stiffness E(τ), tensile strength Tc(τ) and the fatigue limit T0(τ).  The material properties are then updated iteratively according to the co-simulation workflow shown in Fig. 10.  This allows the effects of thermal history and ageing on fatigue performance to be considered.

Fig. 9. The crack growth rate law evolves as a function of the equivalent exposure time τ. Crack growth property evolution is specified in Endurica by the dependence of the rubber’s tear strength Tc(τ) and its fatigue limit T0(τ) on exposure time.

 

Fig. 10. Endurica DT’s co-simulation workflow updates the crack length c, exposure time τ, and stiffness E so that stress, strain and temperature fields can be updated during solution.

Conclusion

There are many ways in which metals and rubbers differ in their behaviour, and thermal behaviour is one of the most important.

Rubber more often requires careful attention to thermal effects due to its exceptionally low thermal conductivity, its entropy-elasticity, its visco-elastic properties and tendency to self-heat under cyclic loading, the sensitivity of crack growth properties and strain crystallization to temperature, oxidation, and ageing.

Endurica’s fatigue solvers provide material models and workflows that capture these thermal effects, enabling accurate analysis and “right the first time” engineering.

References

[1] P.G. Forrest, Fatigue of Metals, Pergamon Press: Oxford, New York, 1962.

[2] G.J. Lake and P.B. Lindley, “Cut growth and fatigue of rubbers. II. Experiments on a noncrystallizing rubber”, Journal of Applied Polymer Science, vol. 8(2), pp. 707-721, 1964.

[3] W. V. Mars and T. G. Ebbott, “A Review of Thermal Effects on Elastomer Durability” in Advances in Understanding Thermal Effects in Rubber: Experiments, Modelling, and Practical Relevance, G. Heinrich, R. Kipscholl, J. B.
Le Cam and R. Stoček (eds.), pp. 251–324, Springer Nature: Switzerland, 2024.

[4] M. L. Williams, R. F. Landel and J. D. Ferry, “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids”, Journal of the American Chemical Society, vol. 77 (14), pp. 3701–3707,
1955.

[5] G. Kraus, “Mechanical Losses in Carbon Black Filled Rubbers”, in: Journal of Applied Polymer Science: Applied Polymer Symposium, vol. 39, pp. 75–92, 1984.
[6] J. D. Ulmer, “Strain Dependence of Dynamic Mechanical Properties of Carbon Black-Filled Rubber Compounds”, Rubber Chemistry and Technology, vol. 69, pp. 15–47, 1996.

[7] J. Vaněk, O. Peter et al, “2D Transient Thermal Analytical Solution of the Heat Build-Up in Cyclically Loaded Rubber Cylinder” in Advances in Understanding Thermal Effects in Rubber: Experiments, Modelling, and Practical Relevance,
G. Heinrich, R. Kipscholl, J. B. Le Cam and R. Stoček (eds.), pp. 31–52, Springer Nature: Switzerland, 2023.

[8] D. G. Young, “Fatigue Crack Propagation in Elastomer Compounds: Effects of Strain Rate, Temperature, Strain Level, and Oxidation”, Rubber Chemistry and Technology, vol. 59 (5), pp. 809–825, 1986.

[9] B. Ruellan, J. B. Le Cam et al, “Fatigue of natural rubber under different temperatures”, International Journal of Fatigue, vol. 124, pp. 544–557, 2019.isms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical society77(14), 3701-3707.

[10] A. Ahagon, M. Kida and H. Kaidou, “Aging of Tire Parts during Service. I. Types of Aging in Heavy-Duty Tires”, Rubber Chemistry and Technology, vol. 63 (5), pp. 683–697, 1990. [11] S. Arrhenius, “Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren”, Zeitschrift für Physikalische Chemie, vol. 4 (1), pp. 226–248, 1889.

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4-in-1

Wow – this year has really been one of many firsts for Endurica.  We had our first ever Community Conference in April, we started our first sister company – in Europe, and from September 9 – 13, 2024, we presented 4 technical papers – a new Endurica record for one week!  The other impressive aspect of this latter feat was that the four presentations were on vastly different topics! I’ll just list the venues and titles and then discuss each one.

International Elastomer Conference 2024, Pittsburgh, PA, USA:

  1. “Heat Build-Up and Thermal Runaway in a Rotating Bending Experiment”

44th Annual Meeting and Conference of The Tire Society, Akron OH, USA:

  1. “Coupled Multiphysics Strategy to Monitor the Health of Rubbery Structures Using Endurica Tools”
  2. “Critical Plane Analysis of Surface-proximal Fields for the Simulation of Mechanochemical Wear”
  3. “Models, Materials and the Move Towards Virtual Product Development”

Let’s start with the first presentation on heat build-up. Will Mars presented this paper at the IEC in Pittsburgh on Tuesday the 10th of September. The presentation highlighted a new machine that has been developed by Coesfeld to evaluate the heat build-up behavior of rubber compounds. It uses a hollow rubber tube that is bent to a 60-90 degree arc and then rotated at about 600 rpm to create a tension-compression cycle throughout the tube due to the pre-bending as shown below.

This test offers many advantages over the historical Goodrich Flexometer self-heating test originally developed in 1937.  The Heat Build-Up Analyzer is instrumented to measure internal temperature as well as forces and deformations while the test is progressing.  The recent advances in the Endurica software and workflows are also equipped to predict the transient behavior in this test.  When the rubber reaches a certain high temperature, the rubber starts to break down, often due to the volatilization of low molecular weight additives creating an exothermic reaction, and also due to the reversion of the cross-links.  The exothermic reaction and thermal “runaway” condition can also be predicted by Endurica software.  The animation below shows the elevated temperatures and the internal pressure rise due to the exothermic reactions. The combination of the HBA test and the Endurica FEA-based analysis will add understanding to the heat-rise behavior of compounds for any company.  As with some other Coesfeld machines, Endurica is the sole distributor in the Americas.

The second presentation listed was presented by Mahmoud Assaad, co-authored by others at Endurica and also by Ed Terrill at ARDL.  This work aims to provide the combination of a full oxygen diffusion and oxidation reaction simulation and experimental characterization capability.   The plot here shows the distribution of reacted oxygen in the crown area of a commercial truck tire.  As the oxygen diffuses into the carcass it also reacts with the rubber compounds creating a phenomenon known as Diffusion Limited Oxidation.  Mahmoud, Ed Terrill and I worked on rubber oxidation with Sandia National Laboratories when the three of us worked together at Goodyear. Now we have developed a characterization and simulation capability that should be ready for customers to try in 2025!

For the third presentation listed, Will Mars quickly travelled from the IEC in Pittsburgh to the Tire Society in Akron to give a talk on an evolving capability for wear prediction. This work was co-authored by Lewis Tunnicliff and James Kollar at Birla Carbon as well as others from Endurica. For many years, researchers have been trying to link rubber fracture and tearing behavior to surface wear. One of the early works on this topic is shown in the drawing below from Southern and Thomas in 1979.

This work attempted to explain observations from blade abrader experiments. The Endurica/Birla work broadens this concept to different asperity shapes and a cumulative fatigue process that depends on the depth into the surface.  Temperature distribution near the surface was also calculated and included in the analysis.  Initial results gave similar trends for wear rates as work done by Gent and Pulford in 1983.  This new approach also makes it easy to also incorporate any aging effects that may occur on the surface of a rubber product. Development work on this new capability will continue well into 2025. In the meantime, Endurica does have a more basic FEA-based offering for wear prediction that has been used for multiple customers.

Lastly, on Friday the 13th of September, I had the honor of giving the Plenary Lecture for the Tire Society conference.  Thanks go to Jim McIntyre and the conference organizers for giving me this unique opportunity to address the society.

In April, we conducted the first ever Endurica Community Conference, and we tied in the Solar Eclipse that passed over Findlay, Ohio on April 8th, to produce a very successful event.  I wanted to include the solar eclipse in my Plenary talk and somehow relate it to topics concerning the development of tires.  The two concepts I used to make the connection were:

  • All models are approximations, but some can be very useful, and
  • Some very good physics theories predict singularities. The singularities reveal our ignorance on the topic and show the area where further work and insights are needed.

The first concept comes from the late George E. P. Box, a statistics professor at the University of Wisconsin. The quote is usually stated as: “All models are wrong, but some are useful”. The second concept makes a tie between fracture mechanics and Einstein’s General Theory of Relativity, which was validated by data taken during a solar eclipse in 1919. Both of these theories predict non-physical singularities but remain extremely useful.

The bulk of my talk was on Virtual Tire Development using tire durability as one of the performances to evaluate without building and testing prototypes. It largely followed my experience and contributions to the topic over the 3+ decades I worked on this at Goodyear with many excellent colleagues and partner organizations like Sandia.

All four of these presentations are available on our website at this location: Fatigue Ninja Frontier – Resources from Endurica’s First Annual Meeting.

Please contact us if you have any questions about these presentations or if you would like to chat with us about anything, including possible work together.

One final note: we are working on a revised website. Our Marketing Director, Pauline Glaza, is heading up a project to develop a new website for us that should make navigating our material and interacting with us much easier.  Expect to see our new site in early 2025!

 

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2023 – a Year of Magnitude and Direction

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. Viewer image copy: There is now a button! Its easier than ever to get your images into reports.
  7. 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!
  8. 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.
  9. 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.
  10. 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.

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Crack Growth or Continuum Damage?

The topic of whether to use a crack growth method or a continuum damage method for product fatigue and durability assessment has long been debated. Oftentimes, experts will recommend using a continuum damage approach in the initial phase, when no noticeable cracks are present, and then transition to a crack growth analysis when damage has reached a certain level where cracks are likely to appear.  In other applications, most of the product’s life is consumed in the crack or crack growth initiation phase, so a continuum damage method is deemed most appropriate.  There are also cases where products are in service with known detectable cracks; in this case fracture mechanics and crack growth analysis is employed to predict how fast the crack will propagate and when it will reach a critical size.

The simplest continuum damage analysis uses Wöhler curves, or S-N diagrams and Palmgren-Miner’s rule.  The S-N diagrams are built by running fatigue tests on un-cracked dumbbell specimens at various stress amplitudes, S, and measuring the number of cycles to failure, Nf. Typical S-N diagrams are shown in Figure 1 [1].  The quantity Sf is the Endurance Limit (or Fatigue Limit), below which no failure is predicted to occur.

 

Figure 1. Typicaly S-N Diagrams [1]

A linear damage rule like the Palmgren-Miner rule states that the amount of damage due to a certain number of cycles, ni, at a certain stress amplitude, Si, is a simple linear ratio compared to the number of cycles to cause failure at that stress amplitude, or

(1)

The incremental amount of damage can then be summed over different blocks of cycles at different stress amplitudes to predict failure when

(2)

One of the limitations of this approach is that sequence effects, for example going from a high-to-low stress amplitude vs. going from a low-to-high stress amplitude is not accounted for. Stated another way, the rate of damage accumulation does not depend on the current state of damage. There also tends to be a large amount of scatter in the results.  In finite element implementations, the amount of damage is tracked towards failure, and damage can be included as a state variable in the constitutive law to allow the stiffness to evolve as a function of damage.

The Endurica methods of fatigue analysis combine fracture mechanics, crack growth, and continuum damage methods. In most materials, there are crack precursors on the micron, or sub-micron level that serve as crack growth initiators. Filled elastomers are known to have many discontinuities at the micron level due to, for example, voids filled with air, agglomeration of fillers or clumps of additives.  These are treated as an initial “pre-cursor” crack with the size c0 with typical values between 10 and 100 microns. Crack growth analysis is used to predict the number of cycles, or number of repeats of a block of cycles until the crack reaches a length indicative of the end of life of the product or component.

Rather than using stress as the driver for damage as in the SN diagram, a fracture parameter called Energy Release Rate, or Tearing Energy is used as the driver for crack growth rate.  An example plot is shown in Figure 2.

The analogy to the Endurance Limit in the S-N diagram is the Intrinsic Strength, T0, below which no crack growth is predicted.  The power-law portion of the plot with slope “F” can be expressed as

(3)

 

where rc is the crack growth rate when T = Tc, the Critical Tearing Energy.   In metals, this is termed a Paris Law, in elastomers, it is the Thomas Law [2].

The damage rate in this case is the crack growth rate, dc/dN. Also, the “damage” is tracked as the predicted length of a growing crack.  The summation of the damage over a given set of cycles can be written as

 

(4)

 

The Tearing Energy in a single edge cracked tension specimen is given by

 (5)

 

where W is the strain energy density far from the crack and k is a constant depending on strain level. In a general three-dimensional state of deformation, Endurica uses the Cracking Energy Density, Wc such that,

(6)

In each of these cases, the Tearing Energy, and thus the crack growth rate is predicted to depend on the crack length, c.

Combining equations 6 and 3, we see that the damage rate, dc/dN, in this analysis, will depend on the current state of damage, c, and thus be able to represent sequence effects as part of the analysis.

In the finite element implementation with the Endurica software, there is typically no explicit crack in the FEA model. Thus the calculation of damage in the form of a growing crack is like a continuum damage approach on the macro-scale.  A co-simulation workflow is also available where the stiffness of each element in the FEA model evolves with the calculation of crack length in each element.

The Endurica analysis methods can be viewed as a continuum damage method on the macro-scale, while using fracture mechanics and crack-growth analysis on the micro-scale.  The use of fracture mechanics provides many advantages including a well-developed and validated theory for elastomers, less scatter in fatigue experiments, nonlinear damage evolution and sequence effects, and the easy ability to include many other aspects such as temperature, aging, and strain crystallization.

References

[1]        Stephens, R. I., Fatemi, A., Stephens, R.R., and Fuchs, H.O., Metal Fatigue in Engineering, 2nd edition, John Wiley & Sons, 2001.

[2]        Thomas, A.G., “Rupture of Rubber  IV. Cut Growth in Natural Rubber Vulcanizates,” Journal of Polymer Science, Vol 31, pp 467-480, 1958.

 

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