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How To Deal with a Nasty, Nonlinear Problem

How To Deal with a Nasty, Nonlinear Problem

My engineering team was recently confronted by one of those really difficult analysis problems where there appeared to be few if any simplifying assumptions and no classical approaches that would fit. It is the kind of problem that has become increasingly typical, reflecting the complex nature of many of today's real-world analysis challenges.

We were given the task of improving the design of an inflatable containment system supported by a polymeric nonwoven envelope. The envelope material was highly flexible like paper, but much stronger and more versatile in this particular application. Our mission: To pick, for a given envelope size, the optimum grade of this nonwoven material such that the containment system could withstand 14 psi internal pressurization without bursting.

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Having tackled several other highly nonlinear problems like reducing paper jams in copiers, improving cell phone robustness to severe shocks like being dropped, analyzing a storm room capable of withstanding 100 mph debris like flying tree limbs, I figured we had a shot at solving this problem too.

Step 1: Figure Out What You Know

First, we started with what we knew. The facts were scant. We knew the flat, uninflated dimensions for several envelope sizes. We had some material data for several grades of the candidate material, including sheet thickness and uniaxial stress-strain curves measured to specimen failure. We observed from some scouting tests that these envelopes stretched significantly, in the neighborhood of 20 percent, as the pressure approached burst values. We were puzzled by this result, since ultimate failure data from uniaxial tests indicated that we should have observed noticeably lower burst strengths. Additionally, we found that reported Mullen burst pressures (TAPPI-industry standard for measuring failure behavior in sheet structure) for the material showed around 100 percent variability, yet the variability that we observed when we inflated actual envelopes to burst was much less. Finally, we knew that classical pressure vessel design equations were too simplistic for the analysis-underestimating stresses and strains by as much as 300 percent!

Step 2: Decide if Modeling Can Help

Un-inflated Shape Low Pressure Inflation (~ 1 psi)

CAN IT BE MODELED? A pragmatic scouting experiment, inflation of a sample envelope using a straw in an engineer's office, was compared to some initial FEA predictions. These early proof-of-concept models clearly showed that the explicit dynamics FEA approach could predict the envelope inflation, including the complex wrinkling.(above)

Using a drink straw and manually blowing into a sample envelope provided crude insight into the challenges that lay ahead. The complex inflated shape, including wrinkles, indicated that our best chance for analysis success was via Explicit Dynamics, a particular style of nonlinear FEA popular among automotive engineers to perform simulations of car crashes and airbag deployment.

First up: Determine whether we could make an FEA model that inflated like the envelope in our straw exercise. Using the uniaxial material data, we built an ABAQUS/Explicit FEA model of a flat envelope and then analyzed it as the internal pressure was raised to 1 psi (our estimate of the lung pressure we could place into the test envelope). The FEA model inflated much like the real envelope, including all the complex wrinkles and bulges. This initial scouting effort took a few days and told us, "Yep, it could be modeled."

Step 3: Characterize the Material

We needed to address the two puzzling inconsistencies related to the material behavior:

  1. Failure strains from uniaxial tension tests were much lower than failure strains measured in envelope burst tests.

  2. Mullen burst data indicated that the material should exhibit much larger variation in envelope burst pressures than was being observed in actual envelope burst tests.

  1. Visual inspection of the nonwoven material indicated that its surface homogeneity was irregular; causing the material to have weak and strong spots. During uniaxial tests, a long thin test strip would always break at a weak spot, like the proverbial weak link in a chain. However, the 2D nature of loading a sheet during inflation allowed for multiple load paths to exist over the surface of the sheet, making the structure less vulnerable to the "weakest link problem." In the form of an envelope, weak areas in the material were supported by stronger neighboring areas.
    This explained why uniaxial failure strain data underestimated the material's actual failure strain limits during envelope inflation. We determined the problem with the Mullen burst data to be with the test protocol, which requires a one-inch-diameter sample. For our materials, this equated to around the size of the surface variations (weak and strong spots). Thus, in a Mullen burst test, we could easily get a fully weak or fully strong sample; creating large variations in observed Mullen burst values. However, when used in a typical envelope, say 10 x 10 inches, the envelope was large enough to allow stronger sections to support the load of weaker sections, thereby significantly reducing the observed variation of burst pressures.

    Improving Characterization of Material FailureBubble test data was combined with a nonlinear FEA model to obtain improved estimates of a material's ultimate stress limit, including statistical variations. Numerous samples of a material were tested in the bubble test, recording the applied burst pressure for each sample. A nonlinear FEA model of the bubble test was computed to obtain the general relationship between applied bubble pressure and material stress. The discrete FEA data was imported into Mathcad engineering software that used splines and curve-fitting to compute a nonlinear transfer function relating inflation pressure to material stress. Lastly, all the burst pressure values from the physical bubble test were converted to estimates of ultimate material stress (one for each sample tested) using the nonlinear transfer function. In the example shown, 12 samples were tested and 12 values of ultimate stress were determined, which were then used for statistical estimates of the variation in ultimate stress.&?xml:namespace prefix = o />&o:p>&/o:p>

    These observations led us to replace the Mullen burst data with data measured using our own "bubble tester"-a super-sized version of the Mullen tester. The diameter of the bubble tester was set to six inches, allowing sufficient "neighboring support" in the material similar to the range of envelope sizes of interest. Unfortunately, it was not possible for the bubble test (nor Mullen test) to report the ultimate stress of the material directly. It could only record the strains in the bubble and applied bubble pressures, including those at burst.
    In both the Mullen burst test and our bubble test, a flat circular sheet is clamped around its circumference and inflated with pressure into the shape of a bubble. The nature of such a deformation is that at any given applied pressure, the strain (and stress) state will vary over the surface of the bubble (as opposed to an ideal uniaxial tension test or an ideal sphere-inflation test which have a uniform stress state at a given load). For a uniform material, the maximum strain in the bubble will occur at the top center of the bubble. For the nonwoven material with surface variations, it will occur in the vicinity of the top center of the bubble.
    Analyzing both the uniaxial data and bubble test data together, in conjunction with nonlinear FEA models of the bubble test, we were able to determine a practical approach to model the statistical variability of the material behavior. We used the uniaxial data to determine the shape of the nonlinear stress/strain curve and the burst results from the bubble test to determine the material's failure strain (and stress) limits. Since the failure limits observed in the bubble test went beyond the uniaxial data, we simply extrapolated (using 2nd order curve fits) the uniaxial data. The nonlinear FEA models of the actual bubble tests were then used to validate the approach.
    We performed the characterization for a given material grade as follows: A single nonlinear FEA model of the bubble inflation test was run using the uniaxial material data with extrapolation, but without the ability to fail. This produced the general relationship between bubble inflation pressure and predicted material strain/stress at the center of the bubble.
    From the model, discrete FEA data pairs of strain and stress at the center of the bubble vs applied pressure were then imported into Mathcad engineering calculation software. Within Mathcad, a pseudo analytical representation of this nonlinear transfer function was easily defined using a combination of cubic spline interpolation and curve-fitting. Lastly, burst pressure values for a number of experimental samples were physically measured with the bubble test and entered into the Mathcad worksheet where the nonlinear transfer function was used to map burst pressures to estimates of ultimate stresses. Using this approach, actual physically observed values of material failure behavior (and their variations) were converted to individual sample values of ultimate stress under a loading state similar to the actual envelope material.
    A key feature to this methodology was that all the material variability was lumped into the failure behavior; the shape of the underlying stress/strain curve was an average representation. This allowed significant simplification and efficiency in the analysis, both for material characterization as well as envelope analysis. Compared to the idea of making numerous FEA models with random patches of weak and strong material and performing Monte Carlo analysis, our nonlinear transfer function method using a single FEA model per material grade was a viable pragmatic approach.

    Comparing Envelope Shape
    Modest pressure

    Just prior to burst

    Comparing Envelope Strain

    Blow Them Up 'Til They Burst

    We inflated envelopes until they burst, and compared both the shape of the envelope and the development of strain within the envelope to an FEA model. At low and modest inflation pressures, wrinkles around the outer edges were visible. At higher pressures, these wrinkles disappear. A non-contact laser extensometer aids in quantitative model validation by recording strain throughout the inflation test for numerous samples. The FEA model's prediction of this quantity was excellent. Note, this was a prediction from the FEA model, not a curve fit to the measured envelope strain data.

    Step 4: Validate the Model
    To validate our envelope inflation model further, we compared experiments of envelope inflation to our FEA predictions that now utilized our improved material representations. We looked at envelope shape during inflation, local strains measured at the center of the envelope, and burst pressures.
    We obtained statistical estimates of burst pressures from the FEA envelope models using a nonlinear transfer function technique similar to that used in the bubble analysis. In this case though, we created nonlinear transfer functions using stress/strain data from the envelope models (one model for a given material grade and envelope size) to map the material's ultimate stress limits (obtained from the bubble test) to envelope inflation pressures at burst. The validation experiments demonstrated that the analysis approach characterized the system well and could be efficiently used to assess a variety of material grades and envelope shapes.
    In the end, we pulled another rabbit out of the hat. How cool is that?


    When using cubic splines and curve-fits to create pseudo-analytical functions from discrete data, we must be careful. The image shows how extrapolation beyond the original data set can produce terrible results with cubic splines. Extrapolation with a 2nd order curve-fit using only that last few data points to compute the curve-fit coefficients produced a viable representation. The two representations were easily combined using "if" logic within Mathcad.

    Web Resources
    To see an animation of Ted's exploding envelope:
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