  In this manuscript, an original numerical procedure for the nonlinear
peridynamics on arbitrarily--shaped two-dimensional (2D) closed manifolds is
proposed. When dealing with non parameterized 2D manifolds at the discrete
scale, the problem of computing geodesic distances between two non-adjacent
points arise. Here, a routing procedure is implemented for computing geodesic
distances by re-interpreting the triangular computational mesh as a
non-oriented graph; thus returning a suitable and general method. Moreover, the
time integration of the peridynamics equation is demanded to a P-(EC)$^k$
formulation of the implicit $\beta$-Newmark scheme. The convergence of the
overall proposed procedure is questioned and rigorously proved. Its abilities
and limitations are analyzed by simulating the evolution of a two-dimensional
sphere. The performed numerical investigations are mainly motivated by the
issues related to the insurgence of singularities in the evolution problem. The
obtained results return an interesting picture of the role played by the
nonlocal character of the integrodifferential equation in the intricate
processes leading to the spontaneous formation of singularities in real
materials.
