How To Deal with a Nasty, Nonlinear Problem

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

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.

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.

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.

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?