Gradient elasticity: non-singular dislocation theory and regularization

Markus Lazar (TU Darmstadt)
Vortrag am 06.05.2019

Abstract

A three-dimensional theory of non-singular dislocations in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters is presented [1-4]. The theory is systematically developed as a generalization
of the classical anisotropic theory in the  framework of 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.

From the mathematical point of view, gradient elasticity provides a regularization based on higher order partial differential equations and the length scale parameters are regularization parameters. 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 tensor with the non-singular anisotropic Green tensor derived in [2]. The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green 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 function which is used as the spreading function of the dislocation core. 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.

References:

[1] M. Lazar,
    On gradient field theories: gradient magnetostatics and gradient elasticity,
    Philosophical Magazine 94 (2014), 2840-2874.

[2] M. Lazar, G. Po,
    The non-singular Green tensor of Mindlin's anisotropic gradient
    elasticity with separable weak non-locality,
    Physics Letters A 379 (2015), 1538-1543.

[3] G. Po, M. Lazar, N.C. Admal, N. Ghoniem,
    A non-singular theory of dislocations in anisotropic crystals,
    International Journal of Plasticity 103 (2018), 1-22.

[4] M. Lazar, E. Agiasofitou, G. Po,
    Three-dimensional nonlocal anisotropic elasticity:
    a generalized continuum theory of Angström-mechanics,
    submitted for publication (2019).