Variable-order particle dynamics: formulation and application to the simulation of edge dislocations

This study presents the application of variable-order (VO) fractional operators to modelling the dynamics of edge dislocations under the effect of a static state of shear stress. More specifically, a particle dynamic approach is used to simulate the microscopic structure of a material where the constitutive atoms or molecules are modelled via discrete masses and their interaction via inter-particle forces. VO operators are introduced in the formulation in order to capture the complex linear-to-nonlinear dynamic transitions following the translation of dislocations as well as the creation and annihilation of bonds between particles. Remarkably, the motion of the dislocation does not require any a priori assumption in terms of either possible trajectory or sections of the model that could undergo the nonlinear transition associated with the creation and annihilation of bonds. The model only requires the definition of the initial location of the dislocations. Results will show that the VO formulation is fully evolutionary and capable of capturing both the sliding and the coalescence of edge dislocations by simply exploiting the instantaneous response of the system and the state of stress. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.