Objectivity of State-Based Peridynamic Models for Elasticity
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We verify the objectivity (invariance to rigid body rotations) ordinary state-based peridynamic models published in the literature that differ in their formulas. We find and explain the sources for the differences between these published formulas. We demonstrate that a primary cause leading to these differences is the way in which the peridynamic volume dilatation is defined in the different formulations. We show that the equations of motion derived from one approach apply to deformations with small or large rotations and is objective. The other approach is valid only for deformations with zero/infinitesimal rotations, thus it is not objective. We also show that both state-based models recover the correct bond-based formulations with the appropriate Poisson ratio values when the term containing the volume dilatation vanishes.
KeywordsPeridynamics Elasticity Objectivity Ordinary state-based formulation Volume dilatation Large rotations
Mathematics Subject Classification74B05 74A45 74G15 74G65 74G70
This work has been supported by AFOSR MURI Center for Materials Failure Prediction through Peridynamics (program managers Drs. James Fillerup, Ali Sayir, David Stargel, and Fariba Fahroo), by a grant from ARO (program managers Dr. Larry Russell), and by ONR (program manager William Nickerson). We would like to thank Dr. Stewart Silling, Prof. Erdogan Madenci and Prof. Erkan Oterkus for very helpful discussions. We also thank the anonymous reviewer who pointed out a possible derivation for the nonlocal dilatation formula in Sect. 2.2.2 and gave us permission to include it here.
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