Abstract
After a brief review on the ability of continuum gradient elasticity (GradEla) to eliminate singularities from dislocation lines and crack tips, we present an extension to its fractional counterpart by replacing the classical Laplacian in the gradient-enhanced Hooke’s Law by a fractional one. Then, a discussion on implications of fractional gradient elasticity to eliminate stress/strain singularities from a screw dislocation is given, followed by the derivation of the fundamental solution of the governing fractional Helmholtz equation, for addressing more general problems. Finally, an elaboration is provided on using these ideas to revisit interatomic potentials used in materials science simulations.
Keywords
- Gradient elasticity
- Interatomic potentials
- London modification
- Fractional laplacian
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Notes
- 1.
The usage of the Caputo fractional derivative \(_0^CD^{\,\alpha }_{k}\) is introduced for convenience as a consequence of the assumed spherical symmetry of the interaction kernel.
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Acknowledgements
This work was benefited from research interactions with the beneficiaries and partners of the RISE European projects FRAMED no. 734485 (https://cordis.europa.eu/project/rcn/207050/factsheet/en), and ATM2BT no. 824022 (https://cordis.europa.eu/project/rcn/219192/factsheet/en). The work was initiated during a visit of E.C. Aifantis to the University of Florida which was supported in part by FRAMED and in part by the following regional grants of NSRF 2014–2020: MIS 5005134 “Nano-chemomechanics in Deformation and Fracture: Theory and Applications in LIBs and SGS”, and MIS 5005454 “Material Instabilities, Size Effects and Morphogenesis: Nanomaterials and Brain”. An extended version of this work is included in the review [6], whereas the special London’s non-fractional gradient case was discussed in [25]. In fact, the article in [25] was advanced in order to realize the aforementioned research collaboration between Aristotle University (E. C. Aifantis team) and the University of Floride (K. E. Aifantis team).
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Parisis, K., Aifantis, E.C. (2021). Gradients, Singularities and Interatomic Potentials. In: TMS 2021 150th Annual Meeting & Exhibition Supplemental Proceedings. The Minerals, Metals & Materials Series. Springer, Cham. https://doi.org/10.1007/978-3-030-65261-6_71
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DOI: https://doi.org/10.1007/978-3-030-65261-6_71
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