Rubber Fatigue ≠ Metal Fatigue Part 2: Linear Superposition

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Rubber Fatigue DOES NOT EQUAL Metal Fatigue Part 2 Linear Superposition

The load cases to be considered in fatigue analysis can be very lengthy and can involve multiple load axes. Often, load cases are much longer than can be calculated via direct time-domain finite element analysis (FEA).

In metal fatigue analysis, linear superposition is a widely used technique to generate stress-strain history from road loads [1], [2], [3]. When structures behave linearly, this approach is accurate and computationally efficient, allowing the analysis of lengthy load signals. For single axis problems, the finite element (FE) solution for a single unit load case is simply scaled according to the input load history. For multiaxial problems, unit load cases are solved for each of the axes, then scaled and combined according to the input load history.

Due to rubber’s 1) nonlinear material behaviour, 2) nonlinear kinematics, and 3) the possibility of nonlinear contact, linear superposition cannot be applied to rubber fatigue analysis. This article is the second in a series examining how rubber fatigue analysis procedures differ from those used for metal fatigue. Here we present the Endurica EIETM (Endurica Interpolation Engine) solver, which is a tool for the rapid generation of stress-strain histories for fatigue analysis in cases where linear superposition fails.

Nonlinearity figures in the analysis of rubbery materials in several ways including material nonlinearity, kinematic nonlinearity, and contact linearity. Endurica’s EIE solver provides an efficient and accurate method for generating stress-strain history when there is strong nonlinearity.
Fig.1. Nonlinearity figures in the analysis of rubbery materials in several ways including material nonlinearity, kinematic nonlinearity, and contact linearity. Endurica’s EIE solver provides an efficient and accurate method for generating stress-strain history when there is strong nonlinearity.

Brief review of the linear superposition procedure for metals

For linear structures, the relationship between forces [F] and displacements [u] can be written as a matrix multiplication where [k] is the stiffness matrix.

[F] = [k][u]

The associative property of function composition means that multiplying the displacements by a scalar a produces proportionally larger forces.

a[F] = [k](a[u])

The distributive property of addition means that a force system resulting from combined displacements [u] and [v]

[F] = [k][u] +[k][v]

can also be calculated as

[F] = [k]([u] + [v])

Similarly, stress and strain fields can be scaled and combined by linear superposition. Engineers have been using this principle for many years in metal fatigue analysis, particularly for treating multiaxial cases arising from field-recorded load-displacement histories.

The stress and strain fields in a part are assumed to result from a linear combination of unit load cases, where the scale factor for each unit load case is applied to the stress or strain field corresponding to a given input channel.

For example, for the beam shown in Fig.2, if channel 1 is the unit displacement u with magnitude a(t), and channel 2 is another unit displacement v elsewhere in the structure with magnitude β(t) , then the entire history of stress and strain at all points in the beam can be recovered by linear superposition.

Note that the FE solver only needs to produce a single time-independent solution for each unit load case. The time dependence of the solution is obtained entirely through the time variations of the scale factors a(t) and β(t). This extremely efficient method has been used for many years in metal fatigue analysis. It allows rapid analysis of complete road load histories consisting of millions of time steps.

Linear superposition of single load case FE solutions has long been used to generate stress-strain histories from road load histories in metal fatigue analysis.
Fig.2. Linear superposition of single load case FE solutions has long been used to generate stress-strain histories from road load histories in metal fatigue analysis.

Endurica EIETM: load space discretization and interpolation for nonlinear cases

Solving the nonlinear case requires a completely different approach. We wish to retain the advantages of efficiently constructing stress-strain time histories from precomputed FE solutions. Instead of precomputing a single unit load case for each input channel, we precompute a set of load cases from a discretized load space. We call this set a map.

The number of load cases in the map must be sufficient so that we can use interpolation to obtain an reasonable approximation of the nonlinear response at any point within the map. Fig.3 shows a map with two channels defined by x and z displacements. The blue points in the map are precalculated using an FE solver such as Ansys or LS-Dyna following the path traced by the blue line. Once the map is defined, the stress-strain history along the red line can be interpolated from the precomputed solutions in the map.

Endurica EIE discretization map
Fig.3. Two-channel map discretizing a space defined by the x and z displacements. Blue dots represent FE solutions for which the stress-strain fields are precomputed. The blue line represents a solution path, which defines the order in which the solutions are computed and stored in the results database. The red line represents a possible actual displacement history. The stress-strain history for points on the red path is obtained by interpolation from points on the precomputed map.

Endurica EIETM is a general purpose tool for creating and using non-linear maps to generate stress-strain histories for fatigue analysis [4], [5]. EIE is an abbreviation for efficient interpolation engine. EIE provides a simple workflow and powerful utilities for creating and using maps for interpolation. It supports up to six independent input channels.

The entire EIE workflow consists of three main steps. The first step is to create a map. The next step is to specify your history in terms of forces or displacements. Note that any quantity that can be applied as a boundary condition to the FE model can be set up as a channel. The last step is to perform the specified interpolation. The process produces a time history of strain tensor components for each element in your FE model.

The map creation process involves four steps, as shown in Fig.4. First, the number of independent channels that will be used to specify the history must be defined. The map type must also be specified. Several types are available, including a completely customizable map. Grid-based maps are often appropriate for one-, two- and three-dimensional maps. For higher dimensional maps, case vector-based maps are often the most convenient.

Once the map type has been defined, EIE generates solution paths. These consist of enumerated load states that should be applied as boundary conditions to the FE model to generate the map. One or more paths may be generated depending on map type. Each path is called a branch. For each branch, EIE writes a file with the appropriate boundary condition history, which is necessary for the generation of the map. Next, the FE model is set up and executed using EIE’s boundary conditions. Finally, the database of FE results is linked to the corresponding branch in the definition of the map.

At this point the map is complete and ready for interpolation. Note that linear superposition can be implemented as a special case in EIE when unit load case solutions are collected and defined as a map. In general, however, a non-linear map will contain a greater number of solution steps.

 

Steps to specify a map for use by Endurica EIE.
Fig.4. Steps to specify a map for use by Endurica EIETM.

Specifying the load history is as simple as selecting a file containing the time history of each input channel. In the file, each row represents one time step and each column represents an input channel. EIE supports .csv and .rsp formats, both common data formats. Fig.5 shows an example history with  and  displacements. Note that the range of displacements in the history should not exceed the range of the precalculated map. Although interpolated solutions can be quite accurate, extrapolation for non-linear problems can be very risky and inaccurate.

Endurica example of two-channel displacement history for interpolation
Fig.5. Example two-channel displacement history for interpolation.

Once the map and history are specified, interpolation can begin. Endurica EIETM supports multi-threading, meaning that interpolation calculations can be distributed and executed in parallel across available CPUs. This makes interpolating very fast and very scalable to large models and lengthy histories. Note that Endurica EIETM generates large files because it calculates stress and strain tensor components for each time step of each finite element. It is therefore important to ensure that you have sufficient disk space available when running Endurica EIETM.

Comparing linear and non-linear interpolation results for a sway bar under uniaxial loading

As a first example, consider an automotive sway bar link, shown in Fig.7. The sway bar transmits load in a single axial direction. This model uses Ogden’s hyper elastic law, which involves a non-linear relationship between stress and strain. The large deformation solution also involves non-linear kinematics due to the incompressibility of rubber and finite displacements and rotations. To compare the linear and non-linear interpolation methods, we will run the analysis using both: 1) the linear scaling method (where the map consists of a single load case in which we apply one newton of total load in the x-direction to the link and solve for the strain distribution in the part); and 2) the non-linear method (where the map consists of 11 precomputed steps ranging from -10000N to +10000N).

Endurica sway bar analysis area noted by red arrows
Fig.6. Sway bar link under uniaxial loading (left). Axial load history input for strain history interpolation (right).

Figs. 8–10 show the six engineering strain tensor component history results for both the linear superposition procedure (left) and the nonlinear EIE procedure (right). The results are shown for three different locations on the sway bar bushing (highlighted in red). The largest strain component is the 31 shear (orange line). Note that for the linear procedure, a linear increase in the amplitude of the global force results in a linear increase in the strain components. The non-linear procedure produces quite different results. In fact, where the linear solution predicts symmetry of tension and compression loads, the non-linear solution correctly captures asymmetries.

Endurica Sway Bar Analysis linear and nonlinear
Fig.7. Comparison of linear (left) and non-linear (middle) interpolation results for strain tensor components at the location indicated on the right.
Enduria sway bar analysis top area
Fig.8. Comparison of linear (left) and non-linear (middle) interpolation results for strain tensor components at the location indicated on the right.
Endurica sway bar analysis top at edge
Fig.9. Comparison of linear (left) and non-linear (middle) interpolation results for strain tensor components at the location indicated on the right.

As a final comparison, Fig.11 shows the fatigue life calculated using Endurica CLTM. A longer fatigue life is predicted for the non-linearly interpolated case compared to the linearly interpolated case. Note that the fatigue damage is more concentrated in the linear case and more spatially distributed for the non-linear solution.

Endurica sway bar analysis Linear versus Nonlinear
Fig.10. Comparison of fatigue life calculations based on linear (left) and non-linear (right) interpolated strain history.

Endurica EIETMvalidation for a six-channel non-linear interpolation

As a further test of the non-linear interpolation procedure for a six-channel ( forces +  moments) multiaxial load analysis of the gearbox mount shown in Fig.11, the map shown in Fig.12 was defined. This map contained 51 precalculated non-linear FE solutions. The complete loading history to be interpolated is shown in Fig.13. This history was solved in full directly and interpolated from the map using Endurica EIETM.

Endurica Gearbox Mount Analysis
Fig.11. Gearbox mount analysis. All forces and moments (x, y, and z) were applied at the centre of the top rigid mounting plate.
Endurica Six-channel map containing 51 precalculated finite element solutions.
Fig.12. Six-channel map containing 51 precalculated finite element solutions.
Endurica Full six-channel road load history used for validation analysis of gearbox mount.
Fig.13. Full six-channel road load history used for validation analysis of gearbox mount.

The strain tensor histories for the 11, 22 and 12 strain components are compared between the directly solved and interpolated solutions in Fig.14 at the location of the most critical element. A fairly accurate interpolation was obtained with a much shorter run time than the direct finite element analysis of the full history.

Endurica Comparison of EIE-interpolated strain components (blue) v. direct finite element solution (red) at the location of the most critical element.
Fig.14. Comparison of EIE-interpolated strain components (blue) v. direct finite element solution (red) at the location of the most critical element.

The fatigue life of the gearbox mount was calculated with Endurica CLTM using both the EIE-interpolated strain history and the directly solved strain history. The fatigue contours for both cases are shown in Fig.15. The fatigue life for the interpolated history was 7.52E8 and for the directly solve history the fatigue life was 7.87E8. These results indicate a close agreement between the EIE and directly solved cases. Other validation cases were recently published elsewhere (Mars et al 2024).

Endurica comparison of fatigue life calculated from EIE-interpolated strain components (right) and direct finite element solution (left).
Fig.15. Comparison of fatigue life calculated from EIE-interpolated strain components (right) and direct finite element solution (left).

Conclusion

Analysis of rubber components typically involves strong nonlinearities due to material behaviour, finite strain kinematics, and contact. The traditional linear superposition of unit load cases, widely used in metal fatigue analysis, is not effective in such cases. Fortunately, the Endurica EIETM solver can generate strain histories efficiently and accurately in these cases. The EIE tools allow the analysis to precalculate a set of FE solutions for efficient discretization of the load space and accurate interpolation of signals within the load space. With sufficient discretization of the load space, it was shown that quite accurate results can be produced for cases where there are between one and six load input channels.

MORE

This article by Dr. Mars was published in Futurities magazine in Volume 21 No.3 Autumn 2024 issue on pages 34-38 which can be accessed by clicking here. Futurities is published by EnginSoft, a leading technology transfer company, and an Endurica reseller in Italy.

Dr. Mars originally presented this information in Endurica’s Winning on Durability webinar series. To view the webinar click here.

References

[1.] R. W. Landgraf, “Applications of fatigue analyses: transportation”, Fatigue ’87, vol. 3, pp. 1593–1610, 1987

[2.] Moon, Seong-In et al, “Fatigue life evaluation of mechanical components using vibration fatigue analysis technique”, Journal of Mechanical Science and Technology, vol. 25, pp. 631–637, 2011.

[3.] F. A. Conle and C. W. Mousseau, “Using vehicle dynamics simulations and finite-element results to generate fatigue life contours for chassis components”, International Journal of Fatigue, vol. 13(3), pp. 195–205, 1991.

[4.] K. P. Barbash and W. V. Mars, “Critical plane analysis of rubber bushing durability under road loads”, SAE Technical Paper No. 2016-01-0393, 2016.

[5.] W. V. Mars, “Interpolation engine for analysis of time-varying load data signals”. U.S. Patent 9, 645, 041, May 9, 2017.

[6.] W. Mars,  K. Barbash et al, “Durability of Elastomeric Bushings Computed from Track-Recorded Multi-Channel Road Load Input”, SAE Technical Paper No. 2024-01-2253, 2024.

 

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Combine Multiple Load Cases into a Block Cycle Schedule that Executes as a Single Endurica Job

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Our most recent Users Survey garnered two surprising requests:

  • “Very interested in ability to run a single model with increasing load and combine with “Duty cycle” definition to predict/calculate expected lifetime.”
  • “Would like to see more on how to use duty cycles (loads) within one analysis rather than running at one load.”

Endurica already does this! Allow me to break down the process and show how easy it is.

Multiple loading cases for a specific duty cycle is often part of Fatigue analysis. You can piece together a schedule of varying Loads, Displacements, Temperatures, Ozone Exposure, and more with Endurica DT.

I focus on load variability in this example. This duty cycle contains three unique loading conditions for a Simple Tension Strip: (A) 10mm displacement, (B) 20mm displacement, and (C) 35mm displacement.

Each load case is a separate FEA simulation. The strains are all exported separately for use with Endurica DT. Each FEA job is a single cycle of the desired loading.

Figure 1.  Contours of maximum principal engineering strain for each of load cases A, B and C. 

Here is a breakdown of the Duty Cycle for this analysis. One Cycle or “Life” is equivalent to 300 repeats of 10mm, 200 repeats of 20mm, and 100 repeats of 35mm.

Figure 2.  Block cycle schedule consisting of 300 repeats of load case A (displaced of 10mm), followed by 200 repeats of load case B (displaced of 20mm), and by 100 repeats of load case C (displaced of 30mm). 

When setting up the Endurica input file we specify the “schedule” under the “history” header in the input file. The number of “block_repeats” is then specified for each of the loading conditions. Once they are specified you submit the Endurica DT job like you would a single load Endurica CL job. The resulting life you receive will be the total number of cycles till failure.

Figure 3.  Endurica input file json syntax defining the block cycle schedule. 

Once submitted, Endurica provides a minimum life prediction of 2,944 Cycles of the full schedule. That is 883,200 cycles of 10mm, 588,800 cycles of 20mm, and 103,040 cycles of 35mm.

Figure 4.  Contours of fatigue life, reported as repeats of the total block cycle schedule. 

Want more information? Check out more details of Endurica DT’s capabilities.

For tutorials visit Endurica Academy:

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Rubber Fatigue ≠ Metal Fatigue Part 1: Mean Strain Effects

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Rubber Fatigue does not equal Metal Fatigue Part 1 Mean Strain Effects
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 σa and mean stress σm, relative to the yield stress σy and the ultimate stress σu, 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 Teq.  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 Teq is simply the range ΔT of the tearing energy cycle, which can be expressed in terms of the min and max tearing energies Tmin and Tmax, or in terms of R= Tmin / Tmax.  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!

MORE

This article by Dr. Mars was published in Futurities magazine in Volume 21 No.2 Summer 2024 issue on pages 36-38 which can be accessed by clicking here. Futurities is published by EnginSoft, a leading technology transfer company, and an Endurica reseller in Italy.

Dr. Mars originally presented this information in Endurica’s Winning on Durability webinar series. To view the webinar click here.

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.

 

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2023 – a Year of Magnitude and Direction

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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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Latest Addition: the Coesfeld Instrumented Chip & Cut Analyser

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When there is rolling or sliding contact of a rubber surface over a second hard surface of sufficient roughness, localized cutting and damage of the rubber surface sometimes becomes a problem.  It occurs in off-road tires operating on stony surfaces, for example, and it can severely limit the useful life of a tire.  In order to study this “cutting and chipping” failure mode, Endurica last month acquired a new testing instrument: the Coesfeld Instrumented Chip & Cut Analyser (or ICCA).  It has been set up at partner lab ACE Laboratories, commissioned, and it is now ready for running tests.

The ICCA test uses a solid rubber wheel specimen.  These are molded from uncured rubber compound supplied by the client.  Alternatively, the mold can be rented if the client prefers to produce their own specimens.  The ICCA test offers direct control and measurement of the following key parameters

  • Wheel revolution speed
  • Overall impact period, tP
  • Peak impact force, FD
  • Contact duration of impact, tD

Figure 1.  Contact force control signal (left).  Coesfeld ICCA impactor tool actuation (right).

It also records the following measurements

  • Normal force
  • Normal displacement
  • Friction force
  • Friction displacement (i.e. wheel rotation)
  • Abrasion depth

Figure 2.  Normal and friction impact forces and displacements during a single impact.

The Coesfeld ICCA instrument improves on J. R. Beatty’s 1979 “BF Goodrich Cut and Chip” (BFG) test in several ways.  Perhaps the chief improvement is that the force and duration of the impact are accurately controlled and measured.  The BFG test suffers from two major problems: 1) that the impact is passively applied by means of a weighted beam whose natural impact frequency is influenced by the stiffness of the rubber compound, and 2) that the impact forces and displacement are not measured and not easily relatable to applications.  By quantifying the impact conditions of the test, the Coesfeld ICCA offers the opportunity to match those of the actual application.

As the Americas distributor for Coesfeld testing instruments, Endurica is proud and excited to add the Coesfeld ICCA to our portfolio of testing services and testing instruments.  Reach out to us today if you have a need for testing services on the ICCA, or if you would like to bring the Coesfeld ICCA to your lab.

Figure 3.  Endurica President Will Mars and Vice President Tom Ebbott with the newly installed Coesfeld ICCA at partner ACE Laboratories.

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Tolerances in Fatigue Life Prediction with Endurica

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I get this question a lot: how well can the Endurica software predict fatigue life?  Is it as good as a metal fatigue code, where a factor of 2x is often quoted as a target tolerance?

The answer is yes, fatigue life predictions can reach and beat this level of accuracy. But as always, knowledge and control of the problem at hand is key.  We must keep in mind that fatigue behavior varies on a logarithmic scale.  It depends on many variables.  It depends on how failure is defined in the simulation and in the test.  Small variations of an input may lead to large variations of the fatigue life.  So, to achieve the best tolerances, careful specification, measurement, and control are required of both simulation and test.

Analysis tolerances depend on whether the analysis workflow is “open loop” or “closed loop”.  In an open loop workflow, the analyst is typically in the position of having to accept without question the as-given material properties, geometry, boundary conditions and load history.  The analysis is completed and reported.  Decisions are made and life goes on.  In a closed loop workflow, there are additional steps.  These include a careful review of differences between the test and the simulation, as well as identification and correction of any erroneous assumptions (about material properties, geometry, boundary conditions, and load history).

Open loop workflows produce larger tolerances.  Every situation is different, but do not expect tolerances tighter than perhaps a factor of 3x-10x in life, when working in open loop mode.  There is just too much sensitivity, too many variables, and too little control in this mode.  The open loop mode does have a few advantages though.  It takes less work, less time, less cost.  And it is often useful for ranking alternatives (ie A vs. B comparisons).

For high accuracy, a closed loop workflow is required.  It is rarely the case that initial assumptions are sufficiently error-free to support tight tolerances on fatigue life prediction.  Therefore, careful measurement and validation of material property inputs, part dimensions, load-deflection behavior, pre-stresses, etc. should be made.  Where gaps are found between test and simulation, appropriate amendments to the test and/or to the simulation should be adopted.  This approach yields high confidence in the simulation results, and good accuracy in fatigue life predictions.  We have seen users hit life predictions to better than a factor of 1.1x with this approach!  Although this approach requires more effort, it results in more complete mastery of part design, and it yields a much stronger starting position for subsequent products.

While “right the first time” engineering is possible with either open or closed loop, the closed loop approach benefits from progressive refinement of the analysis inputs and it ultimately gives the highest success rate.

 

 

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The New Endurica Architecture – It’s Time to Migrate

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Our transition to a new software architecture is a vital move in navigating the dynamic technological landscape. In a recent webinar, we discussed the aspects of this transition, providing insights into the why and how of adopting a new architectural approach despite having a functional existing one. This post will highlight the motivations behind the shift, the present status of feature migration, alterations in the latest software release, and an overview of projects within this new framework.

The Rationale and Benefits

Why Overhaul?

The complete rewrite of our software’s architecture was not a decision made lightly. The reasoning extends beyond merely wanting a refresh; it was driven by pivotal motivations, primarily surrounding the necessity for speed and efficiency in executing computing processes. Speed is invariably tied to productivity and operational fluency in software and technology. The plot below illustrates a compelling story: the old architecture (represented by the blue line), exhibited a static runtime, regardless of the number of threads engaged, revealing its inability to utilize parallel processing. Contrastingly, the new architecture demonstrates a significant speed-up, even with just a single thread, and scales to allow an increase in speed by many multiples, contingent on thread capacity.

Solving Larger Problems

The pursuit of faster execution isn’t arbitrary; it is intrinsically linked to our objective of solving larger problems. With larger tasks and projects on the horizon, scaling up and utilizing more CPU threads became essential. Exemplified through a job run on a virtual machine with 96 available CPU threads, the linear decrease in runtime with increasing threads (until certain hardware limitations are met) exhibits the new architecture’s adept handling of larger jobs (see plot below). The capability to scale and manage tasks of escalating complexity and size was a crucial driver for our transition.

Enhancing Integrations and Streamlining Workflows

Then, we turned our attention toward improving the user experience in interfacing with our software. Our prior use of the HFI and HFO file formats, while functional, presented numerous challenges regarding modification and integration, particularly when scripted modifications were necessary. The new architecture employs the JSON file format, widely recognized for its robustness and versatility across various industries and applications. With JSON, modifying job inputs and managing data become significantly simplified, as illustrated by a Python script example, wherein the entirety of job modifications, inputs, and submissions can be seamlessly handled with a handful of lines of code.

Improved Usability and Real-Time Error Checking

In an effort to enhance usability and mitigate the common issue of erroneous entries and syntax use, the new architecture, especially when utilized with a text editor like VS Code, offers real-time checking and syntax suggestions. This not only makes job submission more precise but also substantially reduces the trial-and-error cycle, saving valuable time. Additionally, upon job submission, the new architecture performs rigorous error and syntax checks, ensuring smooth execution and user experience.

Comprehensive Feature Migration: A Successful Transition

Reflecting on the past two years, we have accomplished a near-complete feature migration to the new software architecture, with 99% of features now successfully transitioned. This includes all outlined output requests, material models, history types, and various procedures.
Our commitment to supporting multiple interfaces remains, with support for Abaqus, Ansys, and Marc using the new architecture. Furthermore, Endurica Viewer is fully compatible, providing enhanced visualization capabilities under the new system.
The comprehensive migration and the incorporation of new functionalities marks the new architecture as fully operational and ready for use across all undertakings.

Implementation of Directory and Execution Changes in Endurica Software

Refined Directory Structure

In efforts to provide a seamless transition and user experience with the upgraded Endurica software, modifications have been made to the directory structure. The new architecture, once labeled “Katana” during its development phase, has now been ubiquitously integrated into the top-level Endurica directory. With the most recent software installation, users will observe the top-level CL and DT directories contain the new architecture, and the Katana directory has been removed.

Consequently, when we refer to Endurica CL and Endurica DT moving forward, it denotes reference to the new architecture.

Accommodating Transition: The Legacy Folder

Acknowledging that the transition to the new architecture may not be instantaneous for all users, the old architecture will still be available and designated within a “Legacy” folder. Though it requires navigation into subfolders, we ensure its accessibility for users who need more time to transition fully into the new structure.

Executable Naming Conventions

In tandem with the directory adjustments, executable naming conventions have been revised to be more intuitive. Previously, “endurica” was employed to submit fatigue analyses in the old architecture, while “katana” pertained to the new. To streamline, “katana” has been rebranded as “endurica” for submitting the JSON input file, with the legacy version adopting the name “endurica-legacy.” It is crucial to note that users accustomed to utilizing “katana” may continue to do so — “endurica” and “katana” will run the same executable. However, usage of the old architecture requires invoking a new “endurica-legacy” command.

Delivering the Unattainable with Endurica’s New Software Architecture

Embarking upon two recent projects with our new computational architecture, we explored the realms of virtual simulation and data management in tire durability and elastomeric mount durability performance.

Project 1: Tire Durability with Dassault Systems

In collaboration with Dassault Systems, a multi-body dynamic simulation was conducted to compute tire durability at the Nurburgring circuit. Utilizing SIMPACK for generating virtual road load data and employing Endurica EIE and Abaqus to establish a workspace map of driving conditions, the endeavor yielded significant data, processed through 176,000 time steps to evaluate the tire’s fatigue life. After a meticulous analysis, the results spotlighted the fatigue life to be 214 laps, pinpointing the most critical point around the tire bead edge.

Project 2: Durability of an Elastomeric Mount with Ford

Undertaken with Ford, the second project navigated through the durability performance of an elastomeric mount, involving a behemoth of data from 144 load history files, each load file containing tens or hundreds of thousands of time points, accumulating to over 15 million total time points. Utilizing a similar approach as the Nurburgring project, Endurica EIE and Abaqus were used together to generate the strain history data. The analysis focused on membrane elements on the mount’s free surfaces to precisely gauge surface strains. Culminating the analysis, the project succeeded in qualifying the part with a fatigue life of 9.4 repeats of the entire schedule, wherein the requisite was just one repeat.

These projects underscored the capabilities of our new architecture, navigating through large data sets and providing tangible insights in significantly reduced timeframes compared to the old architecture. In essence, the implementation of the new architecture has not only streamlined our processes but also expanded our horizons in handling large data and achieving nuanced analyses in our projects.

Summary

The new Endurica CL and Endurica DT architectures have now fully replaced our old system, maintaining the accuracy our users expect while introducing an easier, more powerful, and scalable solution. Everything has been successfully migrated over to this complete solution. With its enhanced capabilities, it addresses problems that were previously too large or took too long to solve, enabling our customers to tackle challenges they might not have considered before. The ability to solve unprecedented problems is just one more example of our steadfast commitment to providing accurate, complete, and scalable solutions.

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Crack Growth or Continuum Damage?

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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 S–N 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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License Queueing

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Design optimization studies are driving a need to support the efficient management and execution of many jobs.  This is why we are announcing that Endurica’s software license manager now supports queueing for licenses. This allows a submitted job to automatically wait to start until enough licenses are available, instead of the prior behavior of exiting with a license error. Now you can submit many jobs without worrying about license availability.

License queueing is only available for network licenses (not node-locked). It is currently supported for Katana CL/DT jobs and EIE jobs submitted from a command prompt.

To enable queueing, set the environment variable RLM_QUEUE to any value. This environment variable must be set on the client machine (not the license server).

To learn more about license queueing, search for “How to Queue for Licenses” in the RLM License Administration documentation here: https://www.reprisesoftware.com/RLM_License_Administration.pdf

 

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Defining the Temperature Dependence of Strain Crystallization in Endurica

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Crystallization requires the suppression of molecular mobility, which in natural rubber can happen either by reducing the temperature or by increasing the strain.  Crystallization of natural rubber can be extremely beneficial to durability.  Nonrelaxing conditions (ie R>0) can increase life by factors of more than 100!  So, what happens if you have both high mean strain and high temperature?

This was the question studied in 2019 by Ruellan et al.  They constructed Haigh diagrams for a filled natural rubber at 3 temperatures: 23 degC, 90 degC and 110 degC.  They completed a large experimental study using dumbbell shaped specimens with a matrix consisting of approximately 4 R ratios x 4 amplitudes x 3 temperatures = 48 conditions.  Their results show that the increase of fatigue life with increasing mean strain at constant amplitude disappears as temperature is increased.  In particular, notice how at 23 degC each life contour (shown in red) has a strongly defined minimum force amplitude that lies near the R=0 line.  Also notice how, at higher temperatures, the life contours start to reflect a decrease of life with increasing mean strain.

This interesting effect can easily by replicated in the Endurica fatigue solver by letting the strain crystallization effect depend on temperature.  The material definition we have used in this quick demo is given below in both the old hfi format and the new Katana json format.  I have highlighted in yellow those parts of the definition which reflect the temperature dependence.

In the material definition, we have reflected two behaviors:

  1. the increase of crack growth rate with temperature (ie the RC parameter), and
  2. the decrease of strain crystallization with temperature (ie the Mars-Fatemi exponential strain crystallization parameter FEXP).

We have plotted the resulting Haigh diagrams in the Endurica viewer, and directly overlaid Ruellan’s results for comparison.  Although the x and y scales in Ruellan’s results are shown in terms of total specimen force and ours are shown in terms of strain, a quite satisfying match is nonetheless achieved for the interaction of temperature with the mean strain effect.  It is especially satisfying that such rich behavior is so compactly and so accurately described by means of the Mars-Fatemi crystallization parameter.

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