The peridynamic stress tensors and the non-local to local passage
| Type of publication: | Article |
| Citation: | |
| Publication status: | Published |
| Journal: | ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik |
| Volume: | 99 |
| Number: | 6 |
| Year: | 2020 |
| Pages: | e201800010 |
| ISSN: | 0044-2267 |
| URL: | https://doi.org/10.1002/zamm.2... |
| DOI: | 10.1002/zamm.201800010 |
| Abstract: | Motivated by earlier works by S.A. Silling and R.B. Lehoucq, 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 and which stand, respectively, for analogues of the Cauchy and 1st Piola-Kirchhoff stress tensors from classical elasticity. We show that the tensor differs from the earlier defined peridynamic stress tensor ; though their divergence is equal. We address the question of symmetry of the tensor which proves to be symmetric in case of bond-based peridynamics; as opposed to the inverse Piola transform of (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 for vanishing non-locality, denoted by . For the sake of brevity we stick to bond-based peridynamic in our calculations. We show that this tensor surprisingly coincides with the collapsed tensor , the limit of the original tensor . At the end, using this flux-formula, we suggest an explanation why the collapsed tensor (and hence ) can be indeed identified with the 1st Piola-Kirchhoff stress tensor. Throughout the whole paper we suppose that the deformation is sufficiently regular. |
| Keywords: | continuum mechanics, flux, non-local theory, peridynamics, stress |
| Authors | |
| Added by: | [VP] |
| Total mark: | 0 |
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