**Atomistic two-, three- and four-body potentials. Spatial and material
settings**

P Steinmann and A Smith and E Birang and A McBride and A Javili, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, 154, 104507 (2021).

DOI: 10.1016/j.jmps.2021.104507

In molecular dynamics or molecular statics (MD/MS) multi-body potentials empirically capture the energetic interactions in atomistic systems enabling the computation of the corresponding atomistic forces as energetic conjugates to the atomistic positions. We distinguish here between spatial and material atomistic positions and consequently between the corresponding spatial and material atomistic forces. In quasi-statics, i.e. MS, the former, also denoted as deformational atomistic forces, contribute to the classical deformational mechanics (i.e., equilibrium) problem that seeks to minimise the total potential energy of an atomistic system with respect to the atomistic positions relative to the ambient space. The latter, also denoted as configurational atomistic forces, contribute to the configurational mechanics (i.e., non-equilibrium) problem that determines the release of total potential energy of an atomistic system upon variation of the atomistic positions relative to the ambient material, i.e., due to perturbations of the material (initial) atomistic configuration. The importance of material atomistic forces is that they drive energetically favourable re-organisations of the material atomistic configuration, thereby characterising the tendency of generic atomistic defects to propagate. In this contribution we focus on two-, three-, and four-body potentials, whereby we distinguish between novel stretch-and classical angle-based potentials for the two latter cases. Taken together, as the main contribution, we derive expressions for the corresponding spatial and, for the first time, material atomistic forces and highlight their striking formal similarity. The derivations are detailed but the final expression compact and well-suited for numerical implementation.

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