The peridynamic stress tensors and the non-local to local passage

Motivated by earlier works by S.A. Silling and R.B. I,ehoucq, we re-examine the notion of stress and force flux in peridynamics - a useful connection to measurable quantities and classical view of continuum mechanics. Based on the idea of traction we define two new peridynamic stress tensors P-y and P which stand, respectively, for analogues of the Cauchy and 1st Piola-Kirchhoff stress tensors from classical elasticity, We show that the tensor P differs from the earlier defined peridynainic stress tensor v; though their divergence is equal. We address the question of symmetry of the tensor P-y which proves to be symmetric in case of bond-based peridynaniics; as opposed to the inverse Piola transform of v (corresponding to the analogue of Cauchy stress tensor) which fails to be symmetric in general. We also derive a gen eral formula of the force-flux in peridynamics and compute the limit of P for vanishing non-locality, denoted by P-0. For the sake of brevity we stick to bond-based peridynamic in our calculations. We show that this tensor P-0 surprisingly coincides with the collapsed tensor vo, the limit of the original tensor v. At the end, using this flux formula, we suggest an explanation why the collapsed tensor P-0 (and hence vo) can be indeed identified with' the 1st Piola-Kirchhoff stress tensor. Throughout the whole paper we suppose that the deformation is sufficiently regular.