A non-singular theory of dislocations in anisotropic crystals

Journal Article · Sat Oct 21 00:00:00 EDT 2017 · International Journal of Plasticity

DOI:https://doi.org/10.1016/j.ijplas.2017.10.003· OSTI ID:1538364

Po, Giacomo ^[1]; Lazar, Markus ^[2]; Admal, Nikhil Chandra  ^[1]; Ghoniem, Nasr ^[1]

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1. University of California, Los Angeles, CA (United States)
2. Darmstadt University of Technology (Germany)

Here we develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by Lazar and Po (2015b). The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures. Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants. The anisotropic non-singular theory is shown to be in good agreement with molecular statics without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. Compared to the isotropic solution, the difference in the energy density per unit length between edge and screw dislocations is more pronounced.

• Research Organization: Univ. of California, Los Angeles, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Fusion Energy Sciences (FES); US Air Force Office of Scientific Research (AFOSR); National Science Foundation (NSF); German Research Foundation (DFG)
• Grant/Contract Number: FG02-03ER54708; FA9550-11-1-0282; CMMI-1563427; La1974/2-2; La1974/3-1; La1974/4-1
• OSTI ID: 1538364
• Alternate ID(s): OSTI ID: 1549430
• Journal Information: International Journal of Plasticity, Vol. 103, Issue C; ISSN 0749-6419
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
dislocation loops
anisotropy
gradient elasticity
singularity
Green's functions