Durability of 3D Printed Elastomer Structures

If you are involved in 3D printing with elastomers, can you predict the fatigue behavior?

How is product lifetime affected by complex lattice designs with multiaxial stresses, and what is the impact of printing defects?

Scientific literature and social media are abound with amazing examples of the potential for 3D printed articles made from metals, plastics and elastomers for use in many fields including the biomedical area. Researchers at ETH Zürich recently 3D printed a functioning artificial heart made from a silicone material. A picture of the device is shown below, and the story can be viewed elsewhere.1,2 This pioneering work represents a very noteworthy achievement. This research also highlights the importance of understanding elastomer durability in these cutting edge applications, as the silicone heart only survived 3,000 beats or about 30 minutes.

One of the key differences between 3D printing (additive manufacturing) and conventional manufacturing is the ability of 3D printing processes to create complex structures containing open spaces, often lattice-like in nature. Perhaps the most innovative and high profile example of a 3D printed product with lattice construction is the midsole for the Adidas Futurecraft 4D shoe that is created using the Carbon 3D technology.3

Overall stresses that are relatively modest and unidirectional translate into much higher stress, multiaxial conditions within the struts of a lattice structure like the shoe sole example above. The finite element simulation below illustrates this for a lattice structure undergoing simple compression (thanks to Mark Bauman, engineering analyst at Endurica).

Multiaxial load cases, crack closure considerations, and other complexities that arise in lattice designs and make it impossible to predict fatigue behavior using simplistic approaches such as Wohler / stress(S)-lifetime(N) curves, can be readily handled using the Endurica CL elastomer fatigue solver for Abaqus, MSC Marc, and ANSYS finite element analysis to predict when and where cracks will show up in the structure.

Cracks in an elastomer start out as microscopic precursors that grow due to applied cyclic loading according to a characteristic crack growth rate law for the material.4 In combination with critical plane analysis, this rubber fracture mechanics approach is the cornerstone of our Endurica CL software. The crack precursors – also called intrinsic defects or flaws – are especially important to pay attention to in the additive manufacturing of products in which voids or defects can be introduced by the printing process. The Core Module of our Fatigue Property Mapping testing services includes quantification of crack precursor size, and our new Reliability Module characterizes its distribution. The figure below illustrates the clear influence of crack precursor size on tensile strength in a study wherein we intentionally introduced glass microspheres as flaws in the rubber compound.5 Fatigue lifetime shows the same strong dependence on flaw size.

Endurica has the software, testing solutions, and expertise to help you understand and improve the durability of your 3D printed elastomer applications, so contact us to see how we can help you #GetDurabilityRight in the additive manufacturing world.

References

  1. https://www.sciencealert.com/this-3d-printed-soft-artificial-heart-beats-just-like-a-real-one
  2. https://www.youtube.com/watch?v=YUYNXeHfTdQ
  3. https://www.youtube.com/watch?v=qlomslovAnI
  4. W. V. Mars, “Fatigue life prediction for elastomeric structures”, Rubber Chemistry and Technology 80, 481 (2007), https://doi.org/10.5254/1.3548175.
  5. C. G. Robertson, L. B. Tunnicliffe, L. Maciag, M. A. Bauman, K. Miller, C. R. Herd, and W. V. Mars, “Characterizing Tensile Strength Distribution to Evaluate Filler Dispersion Effects and Reliability of Rubber”, paper presented at the Fall 196th Technical Meeting of the Rubber Division, American Chemical Society (International Elastomer Conference), Cleveland, OH, October 8-10, 2019.

 

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Calibrating Crack Precursor Size in Endurica CL

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, Tc, 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, rc, 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.

 

 

 

 

 


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.

 

 

 

 

 

 



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.

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EIE – Effect of Map Discretization on Interpolation Accuracy

Overview

The accuracy of the interpolated results performed by EIE is dependent on the discretization of the map. Specifically, the results will become more accurate as the map’s point density increases. This study uses a simple 2D model to quantify the accuracy of results interpolated from maps with different densities.

Model

A 1 mm x 1 mm rubber 2D plane strain model with two channels is used. The square’s bottom edge is fixed and the top edge is displaced in the x and y directions as shown below. The x displacement corresponds to channel 1 and the y displacement corresponds to channel 2. The working space of the model is defined by the x displacement ranging from 0 mm to 0.8 mm and the y displacement ranging from -0.08 mm to 0.8 mm.

Plane strain model with two channels

The model is meshed with 100 8-node, quadrilateral, plane strain, hybrid, reduced integration elements (shown below).

100 element mesh

History

We define as the benchmark reference solution a history that covers the model’s entire working space with a high density of points. An evenly spaced grid of 128×128 points for a total of 16384 points is used as the history (shown below). It is important that this history is more refined than the maps that we will create to ensure that we are testing all regions of our maps.

128×128 history points

These points are used to drive the finite element model and the results are recorded. For this study, we record the three non-zero strain components and the hydrostatic pressure (NE11, NE22, NE12, and HP) for each element at each time point. In summary, there are 4 result components, 100 elements, and 16384 time increments. This set of results is the reference solution since it is solved directly by the finite element model. We will compare this solution to our interpolated results to measure our interpolation accuracy.

Maps

Six maps with different levels of refinement are used to compute interpolated results for our history points. All of the maps structure their points as an evenly spaced grid. The first map starts with two points along each edge. With each additional map, the number of points along each edge is doubled so that the sixth and final map has 64 edge points. The map points for the six maps are shown below.

Six maps with increasing levels of refinement

The map points for these six maps are used to drive the finite element model’s two channels. The strain and hydrostatic pressure results from the FEA solutions are recorded at each map point in a similar way to how the results were recorded for the FEA solution that was driven by the history points. Next, EIE is used six times to interpolate the map point results at each resolution onto the high resolution reference history points.

We now have seven sets of history results: the true set of results and six interpolated sets of results.

Results

To compare our results, we look at the absolute difference between the sets of results. The absolute error is used, opposed to a relative error, since some regions of the model’s working space will give near zero strain and hydrostatic pressure. Division by these near zero values would cause the relative error to spike in those regions.

Since we have 100 elements and 4 components per element, there are a lot of results that could be compared. To focus our investigation, we look at the element and component that gave the maximum error. The figure below shows contour plots for each of the six maps for this worst-case element and component. The component that gave the maximum error was NE12. The title of each of the contour plot also shows the maximum error found for each of the plots.

Error contours for the worst-case element and component. Titles report the maximum log10 error.

You can see that the error decreases as the map density increases. Also, you can identify the grid pattern in the contour plots since the error gets smaller near the map points.

Plotting the maximum error for each of the maps against the number of map points on a log scale is shown below. The slope of this line is approximately equal to 1 which is expected since a linear local interpolation was used to compute the results.

Maximum error vs the number of points for each of the six maps
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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.

We want to help you #GetDurabilityRight, so please contact me at cgrobertson@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.twitterlinkedinmail

Specifying Strain Crystallization Effects for Fatigue Analysis

Endurica CL and fe-safe/Rubber provide several material models for defining cyclic crack growth under nonrelaxing conditions.  Nonrelaxing cycles occur when the ratio R is greater than zero.  R is defined as

where T is the energy release rate (note that T will always be greater than or equal to zero).

The crack growth rate under nonrelaxing conditions is, in general, a function of both Tmax and R. For purposes of calculation, it is convenient to define an “equivalent” energy release rate Teq that gives the same steady state rate of crack growth as the operating condition on the nonrelaxing crack growth curve, but which is instead on the fully relaxing crack growth curve.  In other words,

Using this scheme, you can set up models for both amorphous and strain-crystallizing rubbers, depending on your definition of Teq.  Amorphous rubbers follow the well-known Paris model, and strain-crystallizing rubbers follow the Mars-Fatemi model (or you can define a lookup table).

Paris Model (Amorphous):

The Paris model is the simplest to derive, as it does not involve any material parameters.  It defines the equivalent energy release rate as

This definition is only suitable for rubbers that do not strain-crystallize.

For strain-crystallizing rubbers, one of the other two models should be used.

Mars-Fatemi Model (Strain-crystallizing):

The Mars-Fatemi model accounts for strain crystallization by treating the power-law slope, F, of the Thomas fatigue crack growth rate law   as a function of R, where

or

The exponential version is more compact, but the polynomial version is more flexible.

By substituting F(R) into the fatigue crack growth rate equations for relaxing and nonrelaxing cases, and doing a bit of algebra, the following relationship is obtained

 

Lookup Table (Strain-crystallizing):

The most flexible and accurate way to define strain crystallization is via a lookup table.  The lookup table takes R as an input and returns x(R) as an output.  This function can be defined as the fraction x(R) by which the nonrelaxing crack growth curve is shifted between the fully relaxing crack growth curve (x=0), and the vertical asymptote at Tc (x=1), at a given R.

This can be rearranged into the desired Teq (Tmax,R) form, as follows

Comparisons:

Visualizing the differences between the models helps gain a better understanding of how strain crystallization can affect fatigue performance.  Since all of these models can be represented in the same form of Teq(Tmax,R), we show 2-D contour plots of Teq with R on the x-axis and ∆T on the y-axis.  ∆T is used instead of Tmax to make it easier to compare back to the simple Paris model.

From the figures above, we see that for the Paris model, the equivalent energy release rate depends only on ∆T.  When using this model, changes in R will have no effect on fatigue performance (when ∆T is also held constant).

For strain-crystallizing rubbers, changes in R should influence fatigue performance.  This is seen in the figures for the Mars-Fatemi and lookup table models.

The Mars-Fatemi example uses the following parameters:

The lookup table example uses Tc=10.0 kJ/m2 and Lindley’s data for unfilled natural rubber (P. B. Lindley, Int. J. Fracture 9, 449 (1973)).

For these models, there is a significant decline in Teq as R increases.  This effect is most pronounced when Tmax is much smaller than the critical energy release rate Tc.  Also, there is a point where the effect is reversed (around R=0.8 in these examples) and the high R-ratio starts to have a negative effect on fatigue performance.

Implications:

A material’s strain crystallization properties’ impact on fatigue performance under non-relaxing conditions should not be ignored.  Whether you are seeking to take advantage of strain-crystallization effects or simply comparing the results of different materials/geometries/loadings, strain-crystallization should be accurately represented in your simulations.

Follow these tips to take advantage of strain crystallization and help ensure your fatigue performance is the best it can be.

  • Take advantage of Endurica’s material characterization service (the FPM-NR Nonrelaxing Module generates the strain crystallization curve) or use your own in-house testing to create an accurate strain crystallization model of your material (the nonrelaxing procedure is available for the Coesfeld Tear and Fatigue Analyser).
  • Use output requests like DAMAGE_SPHERE, CEDMINMAX and CEDRAINFLOW to observe R-ratios for your duty cycles.

 

References

  1. B. Lindley, Int. J. Fracture 9, 449 (1973)

Mars, W. V. “Fatigue life prediction for elastomeric structures.” Rubber chemistry and technology 80, no. 3 (2007): 481-503.

Mars, W. V. “Computed dependence of rubber’s fatigue behavior on strain crystallization.” Rubber Chemistry and Technology 82, no. 1 (2009): 51-61.

Barbash, Kevin P., and William V. Mars. Critical Plane Analysis of Rubber Bushing Durability under Road Loads. No. 2016-01-0393. SAE Technical Paper, 2016.

 

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Durability Analysis in CAE: panel discussion of metals vs. polymers at the SAE World Congress

The relationship between crack nucleation and fracture mechanics experiments for polymers was first documented in 1964 by Gent, Lindley and Thomas (Journal of Applied Polymer Science, 8, 455, 1964.) 

Some weeks ago, I attended the WCX 2017 SAE World Congress and Exhibition, where a Technical Expert Panel Discussion on the topic of Durability Analysis in CAE was held.  The panel was moderated by Yung-Li Lee (FCA US LLC), and included topic experts Abolhassan Khosrovaneh (General Motors LLC), Xuming Su (Ford Motor Co., Ltd.), and Efthimio Duni (FCA EMEA). The discussion was excellent and wide ranging, owing both to the panelists, and also to the audience, which (judging by the high engagement) was very well versed with the core of the topic, as well as its frontiers. I will not attempt to give a complete summary of the event, but I do want to highlight a memorable discussion thread, and to offer a few thoughts.

I do not know who raised the topic.  It could have been a doctoral student or young professional.  Clearly, it was a person wanting to align his own efforts well relative to larger industry trends.  He started out with the observation that the classical crack nucleation methods (in which fatigue behavior is defined by a stress-life or strain-life curve) are quite popular in the automotive sector for analyzing fatigue of metals.  He also observed that modern tools for rubber take a different approach based upon a fracture mechanics method (in which fatigue behavior is defined by a crack growth rate curve). He then asked (I’m paraphrasing from memory here):

  • Which method (nucleation vs. fracture mechanics) is preferred for analysis of polymers?
  • Should we try to unify all testing and analysis efforts for metals and polymers under the same method?

The panelists made several points in responding to this prompt. They started with the point that differences in methodology may be hard to avoid, if only because metals and polymers are so different in composition, molecular structure, and microstructure.  Of course, it is possible to use fracture mechanical methods with metals, although there are some limitations implied by the granular crystalline structure of metals when cracks are very small.  Likewise, it is also possible to use stress-life methods with polymers, although certain aspects of the material behavior may be incompatible with the usual procedures, leading to questionable results.  From a practical standpoint, it would be quite difficult to change the methods used by the industry for metal fatigue analysis – the methods are quite mature at this point, and they have been implemented and validated across so many codes and projects that it is hard to imagine what could be gained by making a change.  For polymers, CAE durability methods are newer, and we should use what works.

There is a final point that I believe will ultimately define how this all plays out.  It is that 1) fatigue analysis for polymers is usually driven by multiple “special effects”, and that 2) the economics of the testing required to characterize these effects scales very differently between the two approaches.

Let me illustrate with a typical example:  we have a Natural Rubber compound used in a high temperature application, for an extended time, under nonrelaxing loads.  Let’s compare our options:

 

Option 1

Stress-Life Method

Option 2

Fracture Mechanics + Critical Plane Method

To use the stress-life method, we will need to develop curves that give the effect of 4 parameters on the fatigue life: 1) strain amplitude, 2) mean strain, 3) temperature, and 4) ageing.  The experiment is a simple cycle-until-rupture procedure, with one test specimen consumed per operating condition tested.

 

Let’s assume that we measure each of the four parameters at only 3 levels, and that we will require 3 replicates of each experiment.  The total number of fatigue experiments we need is therefore:

 

N = 3 amplitudes x 3 means x 3 temperatures x 3 ageing conditions x 3 replicates = 35 = 243 fatigue to failure tests

 

With the fracture mechanics method, a single run of the experiment solicits the crack at many different operating conditions, enabling observation of the crack growth rate at each condition.  Using Endurica’s standard testing modules, the example testing program (including replication) would require the following procedures:

 

Core module: 9 experiments (amplitude effect)

Nonrelaxing module: 3 experiments (mean effect)

Thermal module: 12 experiments (temperature effect)

Ageing module: 30 experiments (ageing effect)

 

243 tests required 54 tests required

 

In this example, the fracture mechanics method is almost 243/54 = 4.5x more efficient than the stress-life method!  If you need more than 3 levels, or if you have more than 4 key operating parameters, the experimental cost for the stress-life method quickly becomes completely impractical, relative to the fracture mechanics method. Based on these scaling rules, and on the fact that polymers exhibit so many special effects, you can now appreciate why the fracture mechanics method must prevail for polymers.  For metals, the case is less compelling: there aren’t as so many special effects, and the industry testing norms are already well established.

Bottom line: for fatigue of polymers, the economics of testing for ‘special effects’ strongly favors a fracture mechanics approach.  This fact is certain to shape the future development of fatigue life prediction methods for polymers.twitterlinkedinmail

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