There are several possible explanations for this discrepancy. One of them is an assumption in the theory which may not be justified, which would account for the direction of the discrepancy, and which can readily be tested by a small extension of the theory. The theory attributes the work of crumpling, and hence the force, to the creation of stretching ridges in the sheet. The energy of these ridges has been pretty well established, eg. in "Properties of Ridges in Elastic Membranes" Alexander E. Lobkovsky and T. A. Witten, Physical Review E 55 1577-1589 (1997). This paper verifies that the energy of a ridge scales as the 1/3 power of its length, and finds the proportionality factor in terms of the Young's modulus of the material, and numerical constants found by simulations of regular ridge-containing structures.
In accounting for the experimental crumpling force, we made the assumption that all the ridges in the crumpled sheet were about the same size. But real crumpled sheets contain ridges of various sizes. We would like to know how important this effect is. More fundamentally, we would like to know whether the system would gain or lose elastic energy if we allowed its ridges to have different sizes.
There is a simple way to test the effect of various ridge sizes. We start using the model with a single ridge size, as described below. Then we modify the model to have two different ridge sizes, and see whether the ridge energy increases or decreases. If the energy decreases, it suggests, that crumpled sheets have many ridge sizes because this allows them to be in a lower-energy state. If the energy increases instead, it suggests that the different ridge sizes occur for some other reason despite the extra energy that it costs.
Here are the explicit assumptions of the model. We model the crumpled sheet a bunch of tetrahedra, which have the same surface area and the same volume that the real crumpled sheet has. The given sheet area is divided up into the largest tetrahedra possible that do not intersect within the allotted volume. This constraint determines the size of the tetrahedra, and hence their ridge length X. In practice, the confined volume is flat and wide. One way to solve the packing problem is to make a monolayer of tetrahedra sitting on the bottom surface. This construction tells the order of magnitude of the size necessary for packing. If we decrease the height by a factor of 2, we then must reduce X by a factor 2. The surface area of each tetrahedron must become 1/4 its original size. On the other hand, 4 times as many tetrahedra are needed to cover the given area. Thus this construction preserves the total area as the height is compressed. The next step is to generalize this construction so that two different sizes are present, and see what is the resulting change in energy in this bimodal packing.
Simple bimodal packing: One simple way to incorporate two different sizes is to imagine that the system consists of two separate layers. One layer contains larger tetrahedra; the other contains smaller ones. If the two layers have equal thickness, their tetrahedra are the same size, and we have our original single-size packing. But if one of the layers is made thicker at the expense of the other, different sizes of tetrahedra are required for each, as shown here:
This version is too simple, though. You can readily convince yourself that the energy must go up when the layers or the tetrahedra become unequal.
Interpenetrating tetrahedra: The presumed reason that unequal sizes are favorable is that they allow small crumpled structures to pack into the spaces between large ones. A simple version of such a structure can be made of tetrahedra: each tetrahedron has a smaller one inscribed within it, like this:
This structure isn't meant to be a realistic packing. It is only meant to provide a case where extra surface can be accomodated by using the small spaces between (or within) the original objects. If one has a given area of material formed into this bimodal structure, within a given volume, will its ridge energy be larger or smaller than with the uniform-sized tetrahedra? Answering this question is the goal of the project.
You may be able to find a more realistic bimodal packing that provides a better test of whether heterogeneity of size is good. Alternatively, you may discover a very simple approach that shows that heterogeneity is favorable or unfavorable in general.