  This paper analyzes a new self-avoiding (SA) meshwork model using the
canonical Monte Carlo simulation technique on lattices that consist of
connection-fixed triangles. The Hamiltonian of this model includes a
self-avoiding potential and a pressure term. The model identifies a
crumpled-to-tubule (CT) transition between the crumpled and tubular phases.
This is a second-order transition, which occurs when the pressure difference
between the inner and outer sides of the surface is close to zero. We obtain
the Flory swelling exponents $\nu_{{\rm R}^2}(=\!D_f/2)$ and $\bar{\nu}_{\rm
v}$ corresponding to the mean square radius of gyration $R_g^2$ and enclosed
volume $V$, where $D_f$ is the fractal dimension. The analysis shows that
$\bar{\nu}_{\rm v}$ at the transition is almost identical to the one of the
smooth phase of previously reported SA model which has no crumpled phase.
