Materials Characterization

Rubber Fatigue ≠ Metal Fatigue Part 2: Linear Superposition

August 29, 2024August 28, 2024 by William V. Mars, Ph.D., P.E.

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 EIE^TM (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.

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.

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 EIE^TM: 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 EIE^TM 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 EIE^TM.

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 EIE^TM 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 EIE^TM 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 EIE^TM.

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 CL^TM. 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 EIE^TMvalidation 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 EIE^TM.

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 CL^TM 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 EIE^TM 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.

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.

Rubber Fatigue ≠ Metal Fatigue Part 1: Mean Strain Effects

April 24, 2024April 19, 2024 by William V. Mars, Ph.D., P.E.

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 σ_a and mean stress σ_m, relative to the yield stress σ_y and the ultimate stress σ_u, as shown in Figure 2. Below the fatigue threshold stress σ₀, indefinite life is predicted. The Haigh (or Goodman)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References

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

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

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

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

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

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

2023 – a Year of Magnitude and Direction

February 8, 2024February 8, 2024 by William V. Mars, Ph.D., P.E.

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

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

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

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

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

Top 10 Code Developments:

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

Testing Hardware

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

Applications, Case Studies, Webinars

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

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

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

Defining the Temperature Dependence of Strain Crystallization in Endurica

February 28, 2023February 28, 2023 by William V. Mars, Ph.D., P.E.

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:

the increase of crack growth rate with temperature (ie the RC parameter), and
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.

Get Durability AND Sustainability Right with Endurica

June 1, 2022February 22, 2022 by William V. Mars, Ph.D., P.E.

Sustainability is all about Precycling with Endurica

How do you respond to the call for sustainable solutions in the rubber industry? Is it via bio-sourced polymers or fillers? elimination of carcinogenic additives from the compound? Inclusion of recycled content in the material? light-weighting aimed at reducing material use or at fuel economy improvements? supply of critical components to EVs?

There are many paths to sustainability, but they are all constrained by these three filters:

Most alternatives risk a reduction in durability (despite the optimistic claims of suppliers).
Your product still must pass its durability qualification requirements.
The number of development iterations is severely limited by the time and cost of durability testing.

Endurica workflows have been driving a “right the first time” engineering culture for the last 14 years. Putting durability characterization and simulation upfront in your development programs means that you find and resolve issues earlier and cheaper than if you depended only on your qualification to discover issues.

With Endurica, you can rapidly evaluate the durability of a series of alternative materials under realistic conditions before you build the first physical prototype. The impacts of polymer alternatives, filler alternatives, additives, recycled content alternatives, etc. can be characterized with a minimal sample of the material and represented accurately with Endurica’s material modeling capabilities. You can see how material property changes play out in your actual part geometry, under actual part loading histories. All without building a single prototype part. The modeling process is simple to automate, enabling much richer explorations of the available design space. Where a purely prototype-based development program may be able to compare two or three alternatives over a six-month period, a simulation-based program can compare several hundred alternatives in the space of a week!

Just because it can be hard to find a sustainable alternative doesn’t mean that they aren’t out there. It is their relative rarity that makes them so valuable. The next crop of winning products will come from those who can quickly and reliably navigate durability.

Towards better rubber compound selection: Introducing Endurica’s new Companion (TM) App

June 9, 2022October 18, 2021 by William V. Mars, Ph.D., P.E.

Endurica Companion App | Fatigue Property Comparator

Rubber can be formulated in a very wide range of properties. For materials developers, this cuts two ways. On one hand, it means that there are almost always excellent options for a given application. On the other, it means that those options are usually hidden among lots of bad options. This is job security for rubber compounders, but it unfortunately also underlies the fact that there are so many instances of sub-optimal materials selection decisions when it comes to rubber. One study found that more than 40% of rubber product failures could have been avoided with better materials selection.

One cause of this statistic is poor visibility into how material properties map into application performance. Too often, the material options are judged based on an over-simplified lab test, or an incomplete specification of application conditions. We made the Companion^TM app to address this gap. Companion makes it easier to find the rubber properties that ensure durability in your application. Companion can compare materials for strain-, stress- and energy control. It can compare applications with different modes of deformation (tension, compression, shear). It can account for fully relaxing and nonrelaxing loading. It can account for temperature effects.

Another cause of too-high rates of poor materials selection is that sometimes different parts of an organization use incompatible approaches to specify, characterize and analyze the material and the application. Gaps between the materials, product and testing silos sometimes create unnecessary confusion, conflict and wasted effort, leading to poor durability. Companion was built with the aim of getting materials engineers and product engineers using a common, validated framework. The material properties and analysis principles in the Companion App are the same as those used in our product simulation software, but the user experience in the app is centered around the materials selection decision. No special knowledge of fatigue theory or simulation technology is needed to start using the app.

With the Companion App you can choose different variables to see how they affect the performance of the rubber

You can use the basic version of Companion for free. Go to companion.endurica.com to set up your account and try it out. The free version lets you define one material and one loading condition. A subscription-based professional version is also available for about $1 USD / day. The subscription version lets you compare 2 materials and 2 load cases side-by-side, it lets you save your material definitions to a local database for future use, and it includes several outputs that give deeper insight into the fatigue behavior of your materials. The workflow is simple: 1) define your material(s), 2) define your load case(s), 3) run the calculation, and 4) review the results and compare performance of the materials for the given load cases.

Give it a try and let us know what you think.

Proper tear testing of elastomers: Why you should tear up the Die C tear test

October 28, 2022February 22, 2021 by Christopher G. Robertson, Ph.D.

Endurica Fatigue Ninja tearing rubber

I spent an interesting and rewarding part of my career helping to lead an elastomer technical college in Yanbu, Saudi Arabia. One of the rubber technology words that was challenging for the Saudis to say in English was ‘tear’. They initially pronounced it like the heteronym related to crying. It might be a stretch to say that tears will come to your eyes if you don’t get tear testing of elastomers right, but proper measurement of critical tearing energy (tear strength) is essential for effective materials development for durability.

The fatigue threshold (intrinsic strength; T₀) is the lower limit of the fatigue crack growth curve shown in the figure below, and we recently reviewed this material parameter including the various measurement options.¹ The upper limit is the tear strength, T_C. If loads in your elastomer component are near or above T_C, then it is not a fatigue problem anymore but rather a critical tearing issue with imminent product failure. It is therefore important to accurately characterize this durability performance characteristic of your materials.

General fatigue crack growth behavior of elastomers

Endurica uses the planar tension (pure shear) geometry for measuring T_C in our Fatigue Property Mapping testing services due to the simple relationship between the strain energy density (W) and the energy release rate (tearing energy, T).^2,3 The T_C is equal to the W at tearing multiplied by the initial specimen height, h. You can see this geometry below along with other tear testing specimens employed in the rubber industry and specified in the ASTM standard.⁴

Comparison of the different durability tests one can conduct: the differences between Crack Nucleation Test and Tear and Crack Growth Tests.

We sometimes get questions from folks with technical backgrounds in metals or plastics about whether rubber tear properties will be different when tested in distinct testing modes (mode I, mode II, etc.). It turns out that the extensibility of rubber causes the deformation to be predominately tension in the tearing region, irrespective of how the crack is opened, such that T_C values are similar for rubber evaluated in different testing modes.^2,3 Therefore, trouser tear testing is an alternative to the planar tension testing, as long as any stretching of the legs is accounted for in the data analysis.^3,5 With no stretching of the legs, T_C is simply given by 2F/t where F is the measured force to propagate the tear and t is the thickness of the specimen. The factor of 2 is surprisingly omitted in the ASTM standard⁴even though it is mentioned in the appendix. The image below shows how to convert the ASTM trouser tear strength to T_C.

Trouser tear strength testing

A proper tear test includes an initial macroscopic cut/crack in the specimen. This is not the case for Die C tear described in the tear testing standard.⁴ Die C is thus not a tear test at all but rather is a crack nucleation experiment akin to normal tensile testing of rubber. Because the strange Die C geometry forces failure in a small region in the center of the specimen, it is actually less useful than tensile strength testing of a dumbbell sample which probes the entire gauge region. The Die C test can also have substantial experimental variability related to the sharpness of the die used to punch out the samples. Unfortunately, the Die C “tear” test is the most popular method in the rubber industry to (incorrectly) assess the tear strength of elastomers, and this reality was a key motivator for writing this post. We look forward to seeing the rubber industry shift away from the Die C test, and we hope that the information provided here will help in that path to #GetDurabilityRight. Click here to learn how intrinsic strength and tear strength can be measured quickly and accurately (0:42 video).

References

Robertson, C.G.; Stoček, R.; Mars, W.V. The Fatigue Threshold of Rubber and its Characterization Using the Cutting Method. Advances in Polymer Science, Springer, Berlin, Heidelberg, 2020, pp. 1-27.
Lake, G.J. Fatigue and Fracture of Elastomers. Rubber Chem. Technol. 1995, 68, 435-460.
Rivlin, R.S.; Thomas, A.G. Rupture of rubber. I. Characteristic energy for tearing. J. Polym. Sci. 1953, 10, 291–318.
Standard Test Method for Tear Strength of Conventional Vulcanized Rubber and Thermoplastic Elastomers. Designation: ASTM D 624-00, ASTM International, West Conshohocken, PA, USA, 2020; pp. 1-9.
Mars, W.V.; Fatemi, A. A literature survey on fatigue analysis approaches for rubber. Int. J. Fatigue 2002, 24, 949–961.

Durability Insights from the ISA for Tire Tread Compound Development

June 9, 2022June 11, 2020 by Christopher G. Robertson, Ph.D.

My last blog post (Getting a Quick Read on Durability with the Intrinsic Strength Analyser) highlighted a one-hour test on the Intrinsic Strength Analyser (ISA) to screen elastomer materials for long-term fatigue performance, with applications in materials R&D and plant mixing quality control. To illustrate the use of this approach for rubber compound development, we recently had the opportunity to collaborate with Dr. Nihat Isitman from Goodyear Tire & Rubber Company in Akron, Ohio and Dr. Radek Stoček from Polymer Research Laboratory in Zlín, Czech Republic.¹ Dr. Isitman led this project and was scheduled to present our research at the Spring 2020 Technical Meeting of the ACS Rubber Division, but the meeting was cancelled due to COVID-19 precautions. Instead, the Rubber Division is offering the content online, and the meeting presentations are available here for a modest fee.

Our study considered model tread compounds based on the well-known green tire formulation, which is a compatible blend of solution styrene-butadiene rubber (SBR) and high-cis butadiene rubber (BR) that is reinforced with a silica-silane system for low rolling resistance (improved fuel economy) passenger tires. Additional production compounds used in actual tire treads were also tested, but the proprietary results for these materials were not included in the public presentation. The SBR/BR ratio, silica loading, and crosslink density were all varied in this investigation. For each rubber formulation, the ISA was used to measure the fatigue threshold (T₀) and critical tearing energy (tear strength; T_c), which bracket the two ends of the fatigue crack growth curve as shown below.

The established cutting method of Lake and Yeoh^2,3is used for assessing T₀ on the ISA, and the one-hour test on this benchtop instrument is concluded with a tearing procedure to measure T_c. The ISA is manufactured by Coesfeld GmbH & Co. in Dortmund, Germany, and distributed in the Americas by Endurica LLC (see photo).

The slide image below summarizes the key findings of this research collaboration. Optimization of T₀ and T_c is possible thanks to different sensitivities to the various compounding variables. It is important to measure both fatigue threshold and tear strength to quantify durability potential of rubber materials, and the ISA is an efficient and effective instrument for these measurements. To learn more about this testing equipment for the rubber lab, please visit our Instruments page and contact us at info@endurica.com with questions.

References

N. Isitman, R. Stoček, and C. G. Robertson, “Influences of compounding attributes on intrinsic strength and tearing behavior of model tread rubber compounds”, paper scheduled to be presented at the 197^th Technical Meeting of the Rubber Division, ACS, Independence, OH, April 28-30, 2020 (online presentation due to meeting cancellation).
G. J. Lake and O. H. Yeoh, “Measurement of Rubber Cutting Resistance in the Absence of Friction”, International Journal of Fracture 14, 509 (1978).
C. G. Robertson, R. Stoček, C. Kipscholl, and W. V. Mars, “Characterizing the Intrinsic Strength (Fatigue Threshold) of Natural Rubber/Butadiene Rubber Blends”, Tire Sci. Technol. 47, 292 (2019).

Getting a Quick Read on Durability with the Intrinsic Strength Analyser

June 9, 2022March 27, 2020 by Christopher G. Robertson, Ph.D.

There is now a one-hour test on a benchtop instrument for the rubber lab to screen materials for long-term fatigue performance. Please continue reading to learn more about this commercialization of a classical elastomer characterization methodology.

Rubber products manufacturers and raw materials suppliers seeking improved materials for next-generation applications depend on lab tests to predict end-use performance. These predictive tests should balance accuracy, relevance, and testing time. The testing time component is particularly challenging when the performance characteristic of interest is fatigue lifetime. The image of traditional fatigue testers chattering along for days or weeks comes to mind for those of us with experience in industrial rubber labs. The time consideration is the reason why tensile stress-strain testing (stretching a material to high strains until failure) is the most common physical test for the fracture behavior of rubber, in clear contrast to the most prevalent application condition for rubber products which is cyclic loading (fatigue) at much lower strains.

Fatigue crack growth is a key element of elastomer behavior that must be determined in order to predict durability, as illustrated below. For example, fatigue crack growth (FCG) testing provides the FCG rate law that is essential for predicting when and where cracks will show up in rubber products using Endurica’s elastomer fatigue software for finite element analysis [https://endurica.com/integrated-durability-solutions-for-elastomers/]. Endurica has developed a finitely scoped, reduced variability measurement approach¹ which is used in our Fatigue Property Mapping testing services and is available on the Coesfeld Tear and Fatigue Analyser (TFA). Our standard FCG measurement protocol takes 20 hours of continuous testing. This testing time is very efficient for characterizing best candidate materials in the development process, but a faster test is needed for narrowing down, for example, 20 initial materials to 5 best candidates or for use in a plant lab to monitor quality of rubber compounding processes.

Key Components of Elastomer Fatigue and Failure

The Intrinsic Strength Analyser (ISA) is a recent addition to the durability testing solutions for elastomers. The ISA was developed through a partnership between Coesfeld GmbH & Co. (Dortmund, Germany) and Endurica LLC (Findlay, OH, USA), and this benchtop instrument employs a testing protocol based on the long-established cutting method of Lake and Yeoh.^3,4 Endurica’s president, Dr. Will Mars, discusses the importance of measuring intrinsic strength (fatigue threshold) in this video on our YouTube channel which also shows some footage of the ISA in operation:

https://www.youtube.com/watch?v=BL92ppsJZfE

The fatigue crack growth curve of rubbery materials is bounded by the fatigue threshold, T₀, on the low tearing energy (T) side and by the critical tearing energy (tear strength), T_c, at the high-T end. This is depicted in the generalized figure below. A streamlined one-hour procedure on the ISA can measure both T₀ and T_c which can then be used to estimate the slope (F) of the intermediate FCG power law response that correlates well with the actual F from rigorous FCG testing using the TFA (see figure). More information about this quick ISA approach to characterizing rubber crack growth behavior for materials development and quality control can be found in the Annual Review 2019 issue of Tire Technology International (open access).²

ISA graph showing Crack Growth Rate compared to tearing energy

The fatigue crack growth slope

The fatigue crack growth slope, F, from the ISA should be considered an approximate value that is useful for comparing the relative FCG behavior of materials. However, the determination of T₀ on the ISA is highly quantitative and the only realistic option for assessing this parameter, since the near-threshold crack growth testing on the TFA needed to define T₀ would take about a month. The implementation areas for the ISA and TFA are compared in the following table. A very conservative approach to product development for elastomer durability is to create a combination of material behavior and component design that places the final operation of the rubber product below the fatigue threshold. If this is your company’s approach to engineering for durability, then the ISA is the testing instrument you need.

Durability Testing Solutions for the Rubber Lab

Crack precursor size is another key characteristic of elastomers that needs to be quantified in order to predict durability. In combination with a standard tensile stress-strain test, the critical tearing energy (T_c) from the ISA can also be used to assess crack precursor size, as we showed recently in an open access publication.⁵

Endurica is the exclusive Americas distributor of the Coesfeld ISA and TFA instruments. Endurica’s efficient and effective testing protocols are provided on these high-quality instruments for the rubber laboratory. To learn more about how to add these testing capabilities to your lab, please contact us at info@endurica.com.

References

J. R. Goossens and W. V. Mars, “Finitely Scoped, High Reliability Fatigue Crack Growth Measurements”, Rubber Chem. Technol. 91, 644 (2018).
C. G. Robertson, R. Stoček, R. Kipscholl, and W. V. Mars, “Characterizing Durability of Rubber for Tires”, Tire Technology International, Annual Review 2019, pp. 78-82.
G. J. Lake and O. H. Yeoh, “Measurement of Rubber Cutting Resistance in the Absence of Friction”, International Journal of Fracture 14, 509 (1978).
C. G. Robertson, R. Stoček, C. Kipscholl, and W. V. Mars, “Characterizing the Intrinsic Strength (Fatigue Threshold) of Natural Rubber/Butadiene Rubber Blends”, Tire Sci. Technol. 47, 292 (2019).
C. G. Robertson, L. B. Tunnicliffe, L. Maciag, M. A. Bauman, K. Miller, C. R. Herd, and W. V. Mars, “Characterizing Distributions of Tensile Strength and Crack Precursor Size to Evaluate Filler Dispersion Effects and Reliability of Rubber”, Polymers 12, 203 (2020).

Calibrating Crack Precursor Size in Endurica CL

June 6, 2022July 22, 2019 by Mark Bauman

Crack precursors exist in all elastomers owing to the heterogeneous microstructure, even before any loads are applied. The size of the typical precursor must be specified as part of the Endurica fatigue analysis workflow. The best practice for finding the precursor size is to leverage both crack growth and crack nucleation experiments to enforce agreement between the nucleation test results and the corresponding simulation-predicted life results. This procedure guarantees that both the crack growth and the crack nucleation experiments add up to an overall consistent story.

Prior to performing the calibration, you will need to have already defined the hyperelastic law, and the fatigue crack growth rate law. Fatigue models used for rubber have the following parameters:

Relationship between tearing energy and crack growth rate
- The parameters needed to define this relationship are obtained through fatigue crack growth experiments. The crack is loaded under a range of tearing energies while tracking growth of the crack. These tests obtain the critical tearing energy, T_c, which is the tearing energy at which the crack reaches end of life failure in one loading. The crack growth rate at critical tearing energy, r_c, and the slope of the curve, F, are determined by fitting a power law to the experimental crack growth and tearing energy.
Threshold
- This is the tearing energy limit T₀ below which cracks do not grow. If you do not specify this parameter, then you will use the Thomas law. If you do specify this parameter, you will use the Lake-Lindley law. The threshold can be measured using an Intrinsic Strength experiment.
Strain Crystallization
- Some rubbers exhibit a strain crystallization behavior that causes an increase of durability under non-relaxing loads. If the duty cycle of your calibration experiment is nonrelaxing, and if you have a strain crystallizing material, then this characterization should be completed before calibrating the precursor size. The strain crystallization effect is measured in the non-relaxing module.
End of life crack size
- This parameter should be set in the material definition prior to calibrating the precursor size. A default value of 1mm is generally adequate, particularly when it turns out that the precursor size is at least 5x smaller than this value. The part is considered to have failed when a crack reaches this size.

The crack nucleation experiment used for the calibration procedure may be made on a material test coupon, or on an actual component. Test coupons are convenient in early development stages as they do not require having a part to test. So long as crack precursor size is controlled by intrinsic features of the compound recipe (and not by the extrinsic features of post-mixing processes), a test coupon is likely to give useful results. There is a risk when using a test coupon: the risk that the precursor size in a manufactured part is actually controlled by some feature of post-mixing process such as factory contamination, part molding, abrasion, etc. This risk can be mitigated by calibrating precursor size on the basis of crack nucleation experiments on the finished part. In the following example, we show the process for calibration based on a finished part. The process for a test coupon is the same, but the model of the part is replaced by a model of the specimen.

To illustrate, take the case of a rubber bumper spring. Its duty cycle consists of compressing the 150 mm long rubber spring by 80 mm. Experiments show a fatigue life of 282,534 cycles for this duty cycle. A finite element analysis of the rubber spring is made to obtain strain history. The rubber spring is shown in the image below at the initial condition, at 50% of the displacement, and at 100% of the displacement during the fatigue duty cycle.

The rubber spring at the initial condition, at 50% of the displacement, and at 100% of the displacement during the fatigue duty cycle

We are now ready to calibrate the as yet unknown precursor size to the known experimental fatigue test result of the spring. The precursor size can be calibrated by calculating the fatigue life for a series of precursor sizes and then interpolating to find the one precursor size that results in the best agreement between fatigue life calculations and the experimental fatigue life. Use the PRECURSORSIZE_CALIBRATION output request in Endurica CL to produce a table of fatigue life vs. crack precursor size. Your output request syntax will look something like this:

**OUTPUT

PRECURSORSIZE_CALIBRATION, NFS=25, FSMIN=1e-2
LIFE

NFS is the number of precursor sizes to evaluate, in this case 25. FSMIN is the smallest precursor size to evaluate, in this case 0.01 mm.

Once you’ve executed the calibration, use the new Endurica Viewer to complete the calibration workflow. It can plot a wide range of Endurica analysis outputs including precursor size calibration. Just open the Endurica output file containing the calibration results and expand the output file contents tree to find the Precursor Size Calibration results. The viewer then plots the computed table of precursor size vs fatigue life.

The viewer plots the computed table of precursor size vs fatigue life

If you click on the plot options in the upper left corner, you can input the target life and the viewer will interpolate the precursor size. In this case, for a life of 282,534 cycles, the corresponding precursor size is 39 microns. Now that the precursor size is calibrated, the spring geometry can be optimized, different loadings analyzed, or entirely different parts can be analyzed using the material model to get fatigue life results that accurately reflect the precursor size that is most representative of the final material in the part. Again, if a part is not available, precursor size can also be calibrated to fatigue results from standard simple tension test specimen.

The calibrated rubber spring FE model with the life result of 282,534 cycles is shown below.