Keywords: homogenization, thin sheets, micro-macro approach, contact, friction, continuum mechanics.

In the present paper, the mechanical modeling and the numerical simulation of a pile of thin sheets under compressive and in-plane forces is presented. Such a pile may consist of thousands of sheets with thickness below one millimeter and length of several meters. These sheets are not laminated or glued, but interact through frictional contact only, where Coulomb's friction law is assumed at the interfaces of adjacent sheets. In the case of sufficient compressive forces normal to the sheet plane, the pile of sheets can withstand in-plane forces. In practical applications, several stacks of sheets interfinger within overlapping zones.

The challenge of the present investigation is the large number of very thin sheets, which can not be modeled by three dimensional finite elements directly on conventional computers, because of the large number of unknowns. The sheets cannot be modeled by means of classical shell or plate elements, because of the important influence of thickness deformation to the distribution of compressive forces. The solution to the problem is based on the smoothening of the pile of sheets in such a way that the frictional contact behavior of the thin sheets under compressive and in-plane forces is replaced by a macroscopic material model, which captures the essential deformation state and stress distribution within the loaded pile of sheets. In the macroscopic material, which can be considered as a continuous and homogenous material, the sheets are considered to have negligibly small bending stiffness. The discrete contact conditions of each sheet are transformed to a continuous constitutive model in a phenomenological way, which is very similar to elasto-plasticity. The discrete normal contact behavior is replaced by a condition which states that only compressive stresses might occur in a transverse direction normal to the sheet plane. Further, in order to include frictional contact, a restriction is added to the shear stresses within the pile. As a third condition, the overlapping of many sheets which interfinger each other, is modeled by a limitation of the in-plane normal stresses. In the case of planar sheets and small deformations, the formulation can be based on one global coordinate system, in which the in-plane and the transverse coordinates are specified at once. However, for more complex problems, such a s cylindrical sheets, the macroscopic material model needs to be built upon a local coordinate system.

While the phenomenological considerations to the material model seem to be somewhat simplistic, the comparison of results with the macroscopic material and results of detailed and computationally expensive contact computations shows that the approach covers the relevant deformation quantities and leads to good agreement of stresses. In laboratory measurements, the numerical results have been verified, which agree well within the measured deformation and stress quantities. As a challenging problem, a model has been evaluated where several cases of transverse pressure and in-plane tensional forces have been applied to a pile of sheets with overlapping zones. It turned out, that the macroscopic model is able to retain the theoretically expected behavior of the pile of sheets as well as the results agree well within the laboratory measurements.

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