## Rubber Fatigue ≠ Metal Fatigue Part 2: Linear Superposition

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

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

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

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.

Endurica EIETM: load space discretization and interpolation for nonlinear cases

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

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

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

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

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

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

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

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.

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

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

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

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.

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

Endurica EIETMvalidation for a six-channel non-linear interpolation

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

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.

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

Conclusion

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

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

Rubber and metal are very different materials that exhibit very different behaviors.  Consider the effect of mean strain or stress on the fatigue performance of these materials.  Figure 1 illustrates a few typical constant amplitude strain cycles, each at a different level of mean strain.  If the stress amplitude is equal to the mean stress, we say that we have pulsating tension or fully relaxing tension.  If the mean stress is zero, we say that we have fully reversed tension/compression.  If the minimum stress is always positive, then we have nonrelaxing tension (i.e. always under load).  Nonrelaxing cycles are quite common in applications.  Examples include: pre-loads applied during installation; swaging of a bushing to induce compressive pre-stresses, interference fits, self-stresses occurring due to thermal expansion/contraction; and in tires, shape-memory effects of textile cords.

In metal fatigue analysis, it is customary to define the effect in terms of stress amplitude σa and mean stress σm, relative to the yield stress σy and the ultimate stress σu, as shown in Figure 2.  Below the fatigue threshold stress σ0, indefinite life is predicted. The Haigh (or Goodman)

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.

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

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

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

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.

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.

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 Technology82(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

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

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

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

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

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

Top 10 Code Developments:

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

Testing Hardware

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

Applications, Case Studies, Webinars

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

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

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

## Integrated Durability Solutions for Elastomers

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 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.

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.

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

1. 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.
2. 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.
3. 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.
4. 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.

## Fatigue Life Analysis of Free Surfaces

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.

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