Title: The Peridynamic Stress Tensors and the Non-local to Local Passage,
Abstract: We re-examine the notion of stress in peridynamics. Based on the idea of
traction we define two new peridynamic stress tensors $\mathbf{P}^{\mathbf{y}}$
and $\mathbf{P}$ which stand, respectively, for analogues of the Cauchy and 1st
Piola-Kirchhoff stress tensors from classical elasticity. We show that the
tensor $\mathbf{P}$ differs from the earlier defined peridynamic stress tensor
$\nu$; though their divergence is equal. We address the question of symmetry of
the tensor $\mathbf{P}^{\mathbf{y}}$ which proves to be symmetric in case of
bond-based peridynamics; as opposed to the inverse Piola transform of $\nu$
(corresponding to the analogue of Cauchy stress tensor) which fails to be
symmetric in general. We also derive a general formula of the force-flux in
peridynamics and compute the limit of $\mathbf{P}$ for vanishing non-locality,
denoted by $\mathbf{P}_0$. We show that this tensor $\mathbf{P}_0$ surprisingly
coincides with the collapsed tensor $\nu_0$, a limit of the original tensor
$\nu$. At the end, using this flux-formula, we suggest an explanation why the
collapsed tensor $\mathbf{P}_0$ (and hence $\nu_0$) can be indeed identified
with the 1st Piola-Kirchhoff stress tensor.