The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org.
Bibliographical noteFunding Information:
This work was partly supported by the National Science Foundation under Awards no. PHY-0941493 and DMR-1408211 . Dr. Admal would like to gratefully acknowledge the support provided by the Institute for Pure and Applied Mathematics at the University of California Los Angeles where a part of this work was carried out. The authors would like to thank Amit Acharya for sowing the seed for this problem by suggesting a possible relationship between the decomposition of the atomistic stress and the Helmholtz decomposition for symmetric tensors. Additionally, the authors would like to thank Roger Fosdick, Richard D. James and Ryan S. Elliott for their valuable comments.
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- Constitutive behavior
- Residual stress
- Stress concentration
- Stress relaxation