Curl from Compression
This is a hyper-specialised addition to the Fragmentation test, added from intellectual curiosity and the knowledge that at least a few people on the planet find it interesting.
Often rigid barrier coatings are applied to a polymer substrate that is heated by the deposition process. The thermal expansion coefficient of the coating is often much less than that of the substrate so it becomes compressed by the shrinkage of the substrate. This produces a curl in the film for which the radius R can be easily measured. If the coating is further annealed (as is often the case with, say, ITO) there is a further compression from the annealing process, so the curl is stronger and the radius is smaller.
This compression can be helpful if the substrate ever gets into tension - rigid barrier films are strong in compression but easily cracked under tension. In the Fragmentation test the compression gives an apparent improvement in the COS - Crack Onset Strain. The Fragmentation app allows you to enter a compression strain to adjust for this fact.
The compressive stress, σ, in a coating can be readily calculated from knowledge of the moduli of the coating and substrate Ec, Es, their thicknesses hc and hs and the Poisson ratio of the substrate νs, here assumed to be 0.3.
σ = Eshs2/(6R(1-vs)hc)[1+hc/hs(4Ec/Es-1)]
The compressive strain γ is calculated as γ = σ(1-νc)/Ec, again with the Poisson ratio assumed to be 0.3. The L input is for those who don't have a Radius measurement but instead have a displacement δ over a sample length L. After entering the value for L, change R till you get the measured value of δ and the strain is calculated correctly. R = L2/2δ.
It sounds easy to "measure the curl". In fact it can be complicated. The first problem is that some substrates are pre-curled so the radius, R should be calculated from the radii before and after via 1/R=1/Rafter-1/Rbefore, where for an originally flat film Rbefore is infinite so that R=Rafter. Unfortunately different parts of a film (especially PET) show different curls so some careful experimentation is required before deciding, for example, only to take samples from the centre of a roll which tends to have less curl. Measuring the radius is also not always trivial. Some use some sort of image analysis from a video image, others create templates of known radius with which to compare the actual sample which, if long enough, can be placed on its edge for the comparisons.
The analysis here and its application to the "correction" to the calculations in the Fragmentation app is described by Y. Leterrier1 and his group at the Ecole Polytechnique Fédérale de Lausanne and helpful discussions are warmly acknowledged. Extra input from Dr Charles Bishop is also warmly acknowledged.