The computational framework for continuum-kinematics-inspired peridynamics
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ORIGINAL PAPER
The computational framework for continuum-kinematics-inspired peridynamics A. Javili1 · S. Firooz1 · A. T. McBride2 · P. Steinmann2,3 Received: 25 April 2020 / Accepted: 15 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Peridynamics (PD) is a non-local continuum formulation. The original version of PD was restricted to bond-based interactions. Bond-based PD is geometrically exact and its kinematics are similar to classical continuum mechanics (CCM). However, it cannot capture the Poisson effect correctly. This shortcoming was addressed via state-based PD, but the kinematics are not accurately preserved. Continuum-kinematics-inspired peridynamics (CPD) provides a geometrically exact framework whose underlying kinematics coincide with that of CCM and captures the Poisson effect correctly. In CPD, one distinguishes between one-, two- and three-neighbour interactions. One-neighbour interactions are equivalent to the bond-based interactions of the original PD formalism. However, two- and three-neighbour interactions are fundamentally different from state-based interactions as the basic elements of continuum kinematics are preserved precisely. The objective of this contribution is to elaborate on computational aspects of CPD and present detailed derivations that are essential for its implementation. Key features of the resulting computational CPD are elucidated via a series of numerical examples. These include three-dimensional problems at large deformations. The proposed strategy is robust and the quadratic rate of convergence associated with the Newton–Raphson scheme is observed. Keywords Peridynamics · Continuum kinematics · Computational implementation
1 Introduction Peridynamics is an alternative approach to formulate nonlocal continuum mechanics [1]; its roots can be traced back to the pioneering works of Piola [2,3]. However, it is fundamentally different from established non-local elasticity [see [4,5], among others] as the concepts of stress and strain are absent. As a non-local theory, the behaviour of each material point is influenced by interactions with other material points in their finite vicinity. In contrast to CCM, the governing equations of PD are integro-differential equations appropriate for problems involving discontinuities such as
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A. Javili [email protected]
1
Department of Mechanical Engineering, Bilkent University, 06800 Ankara, Turkey
2
Glasgow Computational Engineering Centre, James Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ, UK
3
Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerland Str. 5, 91058 Erlangen, Germany
cracks and interfaces. Given that PD inherently accounts for geometrical discontinuities, it provides a suitable framework for fracture mechanics and related problems [6–17]. However, the range of PD applications is broad and not limited to fracture. PD has experienced prolific growth as an area of research, with a significant number of contributions in multiple disciplines. V
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