We consider two-dimensional triangular networks of beads connected by
Hookean tethers under isotropic compression. We determine both the com
pression and the shear modulus as a function of temperature and compre
ssion within simple approximations and by a Monte Carlo simulation. At
low temperature, this network undergoes a collapse transition with in
creasing compression. In the two phase region, collapsed and non-colla
psed 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 a
nalysis in large regions of the phase space. In application to the red
blood cell membrane, we obtain good agreement with more microscopic m
odels for the shear modulus. Our results also indicate that strong com
pression will lead to non-trivial elastic behavior of the cell membran
e.