Fatigue Analysis | Endurica LLC

Rubber Fatigue ≠ Metal Fatigue Part 2: Linear Superposition

October 18, 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.

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.

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

June 14, 2024June 14, 2024 by Ethan Steiner

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:

Fatigue Ninja Friday (FNF) #4 presentation on Damage Extrapolation with Endurica DT
FNF #9 Simulating Ageing Effects with Endurica DT
FNF 11 Simulating Tires and Ageing with Endurica DT;
Fatigue Ninja Friday (#18) Sequence Effects in Rubber Fatigue.

Rubber Fatigue ≠ Metal Fatigue Part 1: Mean Strain Effects

September 25, 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!

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.

My SAE WCX 2022 Top Takeaway

June 9, 2022April 20, 2022 by William V. Mars, Ph.D., P.E.

SAE WCX | Detroit, Michigan | April 5-7, 2022

There were several papers on fatigue life prediction for elastomers at SAE WCX 2022, but the highlight for us was this one from Automotive OEM Stellantis: “Fatigue Life Prediction and Correlation for Powertrain Torque Strut Mount Elastomeric Bushing Application” by Dr. Touhid Zarrin-Ghalami, Durability Technical Specialist at FCA US LLC with coauthors C Elango, Sathish Kumar Pandi, and Roshan N. Mahadule from FCA Engineering India Pvt, Ltd. Check out the abstract or buy the paper here…

The study shows that very accurate fatigue life prediction results are possible for elastomeric components under block cycle loading using Critical Plane Analysis. A key feature of the analysis is the characterization and modeling of rubber’s hyperelastic properties, fatigue crack growth properties, crack precursor size, and strain crystallization behavior. Careful measurement of these analysis ingredients led to a nearly perfect correlation of the predicted life (520 blocks) with the tested life (523 blocks, average of 4 replicate tests), and of predicted failure mode with observed failure mode.

Endurica users like Stellantis are developing a solid track record of routine and successful fatigue life prediction. We soon expect to see the day when CAE fatigue life prediction for rubber components is regarded as obligatory, given the risk and cost avoided with “right the first time” engineering.

Congratulations to the Stellantis team on this impressive success!

Fatigue Life (block) demonstrating the accuracy of the CAE Virtual Simulation compared to a physical test

Citation: Elango, C., Pandi, S.K., Mahadule, R.N., and Zarrin-Ghalami, T., “Fatigue Life Prediction and Correlation of Engine Mount Elastomeric Bushing using A Crack Growth Approach,” SAE Technical Paper 2022-01-0760, 2022, doi:10.4271/2022-01-0760.

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.

Behind the Scenes Tour of Endurica Software Development and QA Practices

June 15, 2022November 21, 2018 by Jesse D. Suter

Ever wonder what it takes to consistently deliver quality and reliability in our software releases? Here’s a brief overview of the systems and disciplines we use to ensure that our users receive timely, trouble-free updates of Endurica software.

Automation:

Throughout the life of our software, changes are made to our source code for a variety of reasons. Most commonly, we are adding new features and capabilities to our software. We also make updates to the code to improve performance and to squash the inevitable bugs that occasionally occur.

With each change committed to the code repository, the software needs to be built, tested, and released. Endurica’s workflow automates these steps so that any change to the source repository triggers a clean build of the software. A successful build is automatically followed by a testing phase where our suite of benchmarks is executed and compared to known results. Finally, the build is automatically packaged and stored so that it is ready to be delivered. At each step along the way, a build error or failed test will cancel the workflow and send an alert warning that the release has been rejected, so that the issue can be addressed, and the workflow restarted.

Endurica's build and testing process ensures that high quality standards are met for every new release. Black arrow: normal flow, Red arrow: an error or failed test — Figure 1: Endurica’s build and testing process ensures that high quality standards are met for every new release. Black arrow: normal flow, Red arrow: an error or failed test.

Reliability:

The automated testing phase that every release goes through helps ensure the reliability of our software. For example, every Endurica CL release must pass all 70 benchmarks. Each benchmark is a separate Endurica CL analysis made up of different materials, histories, and output requests. Results from a new build are compared to known results from the previous successful build. If results do not agree, or if there are any errors, the benchmark does not pass and the build is rejected.

The testing phase prevents “would-be” bugs from making it into a release and makes sure that any issues get resolved.

Repeatability:

The automated nature of our development workflow naturally helps with repeatability in our releases. Each build flows through the same pipeline, creating consistent releases every time. There is less worry, for example, that a component will be forgotten to be included. It also allows us to recreate previous versions if comparisons need to be made.

Traceability:

Our version control system enables us to easily pinpoint where and when prior changes were introduced into the software. Each release is tied to a commit in the repository. This allows any future issues to be easily traced back and isolated to a small set of changes in the source for quick resolution.

Responsiveness:

Automating the build and release pipelines greatly increases our responsiveness. If an issue is discovered in a release, the problem can be resolved, and a fully corrected and tested release can be made available the same day. We can also quickly respond to user feedback and suggestions by making small and frequent updates.

The systems and disciplines we use in our development process make us very efficient, and they protect against many errors. This means we can spend more of our time on what matters: delivering and improving software that meets high standards and helps you to get durability right.

Top 10 Reasons to Celebrate Endurica’s 10-Year Anniversary

June 6, 2022September 19, 2018 by Christopher G. Robertson, Ph.D.

Endurica | Get Durability Right | 10 Years In considering ways to capture the contributions and essence of Endurica LLC to celebrate its tenth year of existence – and educating myself some more about the company I joined a little more than a year ago – I decided to put together the following top 10 list. Enjoy this informative snapshot of Endurica.

10 years of providing software and testing solutions for elastomer applications to #GetDurabilityRight in automotive, tire, aerospace, sealing, defense, consumer products, energy, and medical industries.

9 countries are using Endurica’s elastomer fatigue analysis software products (Endurica CL™, fe-safe/Rubber™, Endurica DT™, and Endurica EIE™) for finite element analysis (FEA).

8 specialized elastomer characterization modules are available in our Fatigue Property Mapping™ testing services.

7 years ago, the first training course was offered by Endurica. Today there are three courses that are each taught multiple times around the world every year.

6 is the number of full-time teammates working at Endurica LLC.

5 types of integrated durability solutions are offered by Endurica: FEA software, material characterization services, testing instruments, training, and consulting.

4 patents for Endurica’s innovative technology (3 granted plus 1 pending application).

3 testing instruments are available in the Americas region through our partnership with Coesfeld GmbH & Co. KG (Germany).

2 members of the Endurica team received the Sparks-Thomas Award from the Rubber Division of the American Chemical Society for outstanding contributions and innovations in the field of elastomers.

1^st (and only) commercial FEA software to predict when and where cracks will show up in an elastomer product with complex loading and geometry for users of Abaqus™, ANSYS™, and MSC Marc™.

Integrated Durability Solutions for Elastomers

June 7, 2022March 16, 2018 by Christopher G. Robertson, Ph.D.

Will the durability of your new rubber product meet the expectations of your customers?

Do you have a comprehensive capability that fully integrates all of the disciplines required to efficiently achieve a targeted durability spec?

Your engineers use finite element analysis (FEA) to model the elastomer component in the complex geometry and loading cycle for the desired product application. One traditional approach to predicting durability is to develop a rough estimate of lifetime by looking at maximum principal strain or stress in relation to strain-life or stress-life fatigue curves obtained for the material using lab specimens in simple tension. The difficulties and uncertainties with this method were discussed in a recent blog post.

a rough estimate of lifetime by looking at maximum principal strain or stress in relation to strain-life or stress-life fatigue curves obtained for the material using lab specimens in simple tension

A modern approach to elastomer durability is to use the Endurica CL™ durability solver for FEA. This software uses rubber fracture mechanics principles and critical plane analysis to calculate the fatigue lifetime – which is the number of times the complex deformation cycle can be repeated before failure – for every element of the model. This provides engineers with the ability to view lifetime throughout the FEA mesh, allowing them to modify design features or make material changes as needed to resolve short-lifetime areas.

view lifetime throughout the FEA mesh, allowing them to modify design features or make material changes as needed to resolve short-lifetime areas.

A sound finite element model of the elastomer product in the specified loading situation and fundamental fatigue material parameters from our Fatigue Property Mapping™ testing methods are the two essential inputs to the Endurica CL software. This is illustrated in the figure below.

The requisite elastomer characterization methods can be conducted by us through our testing services or by you in your laboratory with our testing instruments. For some companies, consulting projects are a route to taking advantage of the software before deciding to license the unique predictive capabilities. The following diagram shows how our products and services are integrated.

Durability Solutions for Elastomers

For companies that are just getting started with implementing our durability solutions, the following is a typical testing services and consulting project:

We use our Fatigue Property Mapping™ testing methods, through our collaboration with Axel Products Physical Testing Services, to characterize the properties of cured sheets of rubber compounds sent to us by the client. The minimum requirements for fatigue modeling are crack precursor size and crack growth rate law, and these are quantified within our Core Fatigue Module. Special effects like strain-induced crystallization and aging/degradation are accounted for using other testing modules when applicable.
The client sends us the output files from their finite element analysis (FEA) of their elastomer part design for the deformation of their complex loading cycle. It is common for the goal to be a comparison of either two designs, two distinct loading profiles, two different rubber compounds, or combinations of these variations. Our software is fully compatible with Abaqus™, ANSYS™, and MSC Marc™, so the simulations can be conducted on any of these FEA platforms. In some situations where a client does not have their own FEA capabilities, one of Endurica’s engineers will set up the models and perform the analyses instead.
The fatigue parameters and FEA model are inputted to Endurica CL fatigue solver to calculate values of the fatigue lifetime for every element of the model. The lifetime results are then mapped back onto the finite element mesh in Abaqus, ANSYS, or MSC Marc so that the problem areas (short lifetime regions) within the geometry can be highlighted.
We review the results with the client and discuss any opportunities for improving the fatigue performance through design and material changes.

Advanced implementors of our durability solutions have licensed the Endurica CL software and are using our rubber characterization methods in their laboratories on a routine basis, with instruments provided through our partnership with Coesfeld GmbH & Co. (Germany). One recently publicized example of a company using the Endurica approach to a very high degree is Tenneco Inc., which you can read about here.

We want to help you #GetDurabilityRight, so please contact us at info@endurica.com if you would like to know more about how Endurica’s modern integrated durability solutions for elastomers can help enable a product development path that is faster, less expensive, and more confident.

Wohler Curves or Fracture Mechanics?

May 4, 2022February 22, 2018 by William V. Mars, Ph.D., P.E.

Endurica uses a fracture mechanics based description of rubber’s fatigue behavior, rather than the classical Wohler curve (ie S-N curve) approach. This is why:

1) Wohler curves in rubber show the combined effects of several nonlinear processes, but they do not easily deconvolve into useful information about the individual processes. This means that Wohler curve users struggle to trace the causes of fatigue failures any deeper than the single monolithic empirical SN curve. When the customer or the boss asks why the part is failing, Wohler curve users end up falling back on the old “rubber is mysterious” defense. Meanwhile, users of critical plane analysis + fracture mechanics are hypothesis testing. They can check what events and what loading directions are most damaging, and what material parameters (crack precursor size, strain crystallization, threshold, crack growth rate law, thermal effects, etc.) can be exploited to gain leverage and solve the issue.

2) Fatigue failure in rubber is often dominated by “special effects”: dependence on strain level, dependence on R ratio, dependence on temperature, dependence on rate, dependence on ageing, etc. The Wohler curve crowd must choose between ignoring/oversimplifying these special effects, or running an experimental matrix that rapidly scales to an infeasibly huge size as more variables are added. While fracture mechanics users obtain a wealth of information from a single test specimen (one test can probe many different strain levels, temperatures, rates, etc), Wohler curve users obtain 1 data point per tested specimen. Look in the rubber technical literature and count the number of S-N-curves that are given, relative to the number of fatigue crack growth rate curves. Google/scholar returns less than 2000 results for “rubber Wohler curve”, and 78700 results for “rubber crack growth curve”. There is a reason that crack growth rate curves outnumber Wohler curves.

3) SN based methods are not conservative. Wohler curve users end up assuming that a crack will show up perpendicular to a max principal stress or strain direction. This assumption only works when you have the very simplest loading cases, no compression, and no strain crystallization. Users of fracture mechanics + critical plane analysis don’t worry about whether they have simple loading, finite straining, out-of-phase loading, compressive loading, changing principal directions, and/or strain crystallization. Critical plane analysis checks every possible way a crack might develop and is therefore assured to always find the worst case regardless of detailed mechanisms.

4) Wohler curves are messy. They depend strongly on crack precursor size, which naturally varies specimen-to-specimen, batch-to-batch, and between lab mix and factory processes. During SN curve testing, the size of the crack is neither measured nor controlled. This accounts for the extra scatter that is typical in these tests. In fracture mechanics testing, on the other hand, the crack is measured and controlled, leading to more repeatable and reliable results. Noisy data means that the Wohler curve crowd has trouble differentiating between material or design options. Users of fracture mechanics benefit from cleaner results that allow more accurate discrimination with less replication.

A Wohler curve does have one valuable use. The Wohler curve can be used to calibrate the crack precursor size for a fracture mechanics analysis. It only takes a few data points – not the entire curve, since the crack precursor size does not depend on strain level, or other “special effects” variables. Our recommended practice is to run a small number of nucleation style tests for this purpose only, then leverage fracture mechanics to characterize the special effects.

The bottom line is that, for purposes of general fatigue life prediction in rubber, the Wohler curve method loses technically and economically to the fracture mechanics + critical plane analysis based method that is used in modern fatigue solvers.

Fatigue Life Analysis of Free Surfaces

February 8, 2018February 1, 2018 by Mark Bauman

Free surfaces are critical in fatigue analysis because cracks in a physical part tend to form and grow fastest on such surfaces. Extra care is required when analyzing free surfaces because typical 3D solid finite elements have their worst accuracy at the free surface (gauss points are not located on the free surface, and hydrostatic pressure profile does not conform adequately to element shape function). Fortunately, the problem is not hard to resolve: free surfaces can easily be skinned with membrane elements. Membrane elements are specially formulated to produce an exact state of plane stress.

Let’s look at fatigue life predictions that have been computed with a skin of membrane elements, and compare them with predictions computed from the underlying 3D solid elements.

To study the differences in fatigue life calculations, three simple loading cases were used: simple tension, planar tension, and bending. For each case the fatigue life is calculated for both the surface and solid elements. The results are shown in the table below.

The fatigue life results show that the shortest life always occurred on the free surface. The life for the solid elements varied from 16% to 25% longer than the surface elements. In each case, the critical failure location was on the surface of the part and in the same location for both the solid and surface calculations. The colored contours of fatigue life are shown below for each of the cases.

Figure 1. Fatigue life on simple tension specimen. Isometric view.

Figure 2. Fatigue life on planar tension specimen. Cross-section view through the center of the specimen.

Figure 3. Fatigue life on bending specimen. Cross-section view through the center of the specimen.

Mesh refinement affects the fatigue life results. A mesh refinement study was performed on the bending case. The mesh refinement study consists of the standard mesh model shown above, a coarse mesh model and a fine mesh model. The number of elements in each model triples with each increase in mesh density. The results are shown below.

Figure 4. Mesh Density Analysis on bending specimen.

This mesh density analysis shows that as mesh density increases, the difference in the bulk and surface results decreases. The bulk and surface results converge to a single value. The amount that solid elements on the surface of the part extend into the interior of the part decreases as smaller elements are used. Since the smaller solid elements have a strain history closer to the surface they more closely match the surface element strains and the life results converge to a single value.

Bottom Line: if you have free surfaces, skin your model with membrane elements for high accuracy results. Refining your mesh at the surface may help somewhat, but skinning with membranes is far more reliable.