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

Abstract

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’.

Variable-order fracture mechanics and its application to dynamic fracture

“…In other terms, the governing equations describing a system can be implicitly reformulated via a change in the order α(t), following a change in the underlying physical mechanisms dominating the response of the system. This characteristic was illustrated to formulate evolutionary equations to model contact dynamics, hysteretic behavior 38 , and motion of edge-dislocations in lattice structures 43 . In the present study, we extend this unique behavior of the VO-RL operator to simulate the initiation and propagation of cracks in solids.…”

“…The development of this framework involved the introduction of VO differential operators into the classical elastodynamic equations, to allow them to evolve in a nonlinear fashion while accounting for the growth (or propagation) of nonlinearities and discontinuities typical of dynamic fracture. This reformulation of classical elastodynamics using VO-FC builds upon the mathematical structure presented in 43 which focused on the modeling of the propagation of dislocations through lattices of particles using physics-informed order variations. More specifically, the VO model introduced in 43 leveraged an order variation law based on the relative displacements of particles within the lattice in order to capture the formation and annihilation of pairwise bonds.…”

“…This reformulation of classical elastodynamics using VO-FC builds upon the mathematical structure presented in 43 which focused on the modeling of the propagation of dislocations through lattices of particles using physics-informed order variations. More specifically, the VO model introduced in 43 leveraged an order variation law based on the relative displacements of particles within the lattice in order to capture the formation and annihilation of pairwise bonds. The general strategy followed that outlined in 38,44 for physics-driven VO laws for discrete systems.…”

Advanced materials modelling via fractional calculus: challenges and perspectives

“…In the theme issue, Patnaik & Semperlotti [104] propose a variable-order formulation to capture the evolution of edge dislocations through lattice structures under static shear stress. The authors simulate the microscopic structure of a material by a particle dynamic approach where constitutive atoms or molecules are represented via discrete masses and interparticle forces are represented by variable-order Riemann-Liouville operators depending on a quadratic potential field.…”

Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions

“…The comprehensive overview summarizing state-of-the-art practical applications of FC has been recently published by The Royal Society Publishing. The sixteen-paper issue entitled "Advanced materials modeling via fractional calculus: challenges and perspectives" [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] covers applications of constant-order (CO) and variable-order (VO) fractional differential operators to several fundamental phenomena. These include anomalous diffusion, [13,16] heat conduction [14,27], fractional viscoelasticity of fluids [19], and materials [12,18,22].…”