On the Computational Derivation of Bond-Based Peridynamic Stress Tensor
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On the Computational Derivation of Bond-Based Peridynamic Stress Tensor Arash S. Fallah 1,2 & Ilias N. Giannakeas 3 & Rizgar Mella 4 & Mark R. Wenman 5 & Yasser Safa 2 & Hamid Bahai 1 Received: 19 September 2019 / Accepted: 5 June 2020/ # The Author(s) 2020
Abstract
The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bondbased formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain
* Arash S. Fallah [email protected]
1
Department of Mechanical and Aerospace Engineering, Brunel University London, Uxbridge UB8 3PH, UK
2
ICP Institute of Computational Physics, ZHAW Zürich University of Applied Sciences, Technikumstrasse 9, CH-8401 Winterthur, Switzerland
3
Department of Civil and Environmental Engineering, Brunel University London, Uxbridge UB8 3PH, UK
4
Nuclear Faculty, Defence Academy of the UK, HMS Sultan, Military Road, Gosport PO12 3BY, UK
5
Centre for Nuclear Engineering, Department of Materials, Bessemer Building, South Kensington Campus, Imperial College London, London SW7 2AZ, UK
Journal of Peridynamics and Nonlocal Modeling
method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip. Keywords Peridynamics . Stress tensor . J-integral . Discontinuity . Singularity
1 Introduction Peridynamics is a rather recent reformulation of nonlocal continuum mechanics [1] which tries to unify, and incorporate within a single framework, the mathematical modelling of continuous media, dis
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