  From the perspective of physical properties, the cell membrane is an exotic
two-dimensional material that has a dual nature: it exhibits characteristics of
fluids, i.e., lipid molecules show lateral diffusion, while also demonstrating
properties of solids, evidenced by a non-zero shear modulus. We construct a
model for such a $\textit{semi-solid}$ $\textit{membrane}$. Our model is a
fluctuating randomly triangulated mesh with two different kinds of nodes. The
solid nodes never change their neighbors, while the fluid nodes do. As the area
fraction occupied by the solid nodes ($\Phi$) is increased the motion of fluid
nodes transition from diffusion to localization via subdiffusion. Next, the
solid nodes are pinned to mimic the pinning of the plasma membrane to the
cytoskeleton. For the pinned membrane, there exists a range of $\Phi$ over
which the model has both a non-zero shear modulus and a non-zero lateral
diffusivity. The bending modulus, measured through the spectrum of height
fluctuations remains unchanged.
