  We numerically study the phase structure of two types of triangulated
spherical surface models, which includes an in-plane shear energy in the
Hamiltonian, and we found that the phase structure of the models is
considerably influenced by the presence of the in-plane shear elasticity. The
models undergo a first-order collapsing transition and a first-order (or
second-order) transition of surface fluctuations; the latter transition was
reported to be of second-order in the first model without the in-plane shear
energy. This leads us to conclude that the in-plane elasticity strengthens the
transition of surface fluctuations. We also found that the in-plane elasticity
decreases the variety of phases in the second model without the in-plane
energy. The Hamiltonian of the first model is given by a linear combination of
the Gaussian bond potential, a one-dimensional bending energy, and the in-plane
shear energy. The second model is obtained from the first model by replacing
the Gaussian bond potential with the Nambu-Goto potential, which is defined by
the summation over the area of triangles.
