. --- We consider two-dimensional triangular networks of beads connected by Hookean tethers under isotropic compression. We determine both the compression and the shear modulus as a function of temperature and compression within simple approximations and by a Monte Carlo simulation. At low temperature, this network undergoes a collapse transition with increasing compression. In the two phase region, collapsed and non-collapsed triangles coexist. While the compression modulus vanishes in the two phase region, the shear modulus shows only a small anomaly at the transition. With increasing temperature, this transition disappears in our simulation. Anharmonic shear fluctuations invalidate a harmonic analysis in large regions of the phase space. In application to the red blood cell membrane, we obtain good agreement with more microscopic models for the shear modulus. Our results also indicate that strong compression will lead to non-trivial elastic behavior of the cell membrane. 1. Introdu..
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