  We study a surface model with a self-avoiding (SA) interaction using the
canonical Monte Carlo simulation technique on fixed-connectivity (FC)
triangulated lattices of sphere topology. The model is defined by an area
energy, a deficit angle energy, and the SA potential. A pressure term is also
included in the Hamiltonian. The volume enclosed by the surface is well defined
because of the self-avoidance. We focus on whether or not the interaction
influences the phase structure of the FC model under two different conditions
of pressure ${\it \Delta} p$; zero and small negative. The results are compared
with the previous results of the self-intersecting model, which has a rich
variety of phases; the smooth spherical phase, the tubular phase, the linear
phase, and the collapsed phase. We find that the influence of the SA
interaction on the multitude of phases is almost negligible except for the
evidence that no crumpled surface appears under ${\it \Delta} p\=\0$ at least
even in the limit of zero bending rigidity $\alpha\to \0$. The Hausdorff
dimension is obtained in the limit of $\alpha\to \0$ and compared with previous
results of SA models, which are different from the one in this paper.
