  Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in
a second order in time partial integro-differential equation. In this paper, we
consider a nonlinear model of peridynamics in a two-dimensional spatial domain.
We implement a spectral method for the space discretization based on the
Fourier expansion of the solution while we consider the Newmark-$\beta$ method
for the time marching. This computational approach takes advantages from the
convolutional form of the peridynamic operator and from the use of the discrete
Fourier transform. We show a convergence result for the fully discrete
approximation and study the stability of the method applied to the linear
peridynamic model. Finally, we perform several numerical tests and comparisons
to validate our results and provide simulations implementing a volume
penalization technique to avoid the limitation of periodic boundary conditions
due to the spectral approach.
