Equivalent continuum for dynamically deforming atomistic particle systems

Abstract

An equivalent continuum is defined for dynamically deforming atomistic particle systems treated with concepts of molecular dynamics. The discrete particle systems considered exhibit micropolar interatomic interactions which involve both central interatomic forces and interatomic moments. The equivalence of the continuum to discrete atomic systems includes, firstly, preservation of linear and angular momenta, secondly, conservation of internal, external and inertial work rates and, thirdly, conservation of mass. This equivalence is achieved through the definition of, firstly, continuum stress and couple stress fields that make the same contribution to motion and deformation as internal interatomic forces and couples, secondly, continuum fields of body force, body moment, surface traction and surface moment that make the same contribution to motion and deformation as external forces and moments on the atoms, thirdly, a continuum deformation field that is work conjugate to the continuum kinetic fields and consistent with the atomic deformation field and, fourthly, continuum distributions of mass and moment of inertia that preserve the linear and angular momenta as well as kinetic energy. This equivalence holds for the entire system and for volume elements defined by any subset of particles in the system; therefore, averaging and characterization across different length scales are possible and size-scale effects can be explicitly analysed. The framework of analysis provides an explicit account of arbitrary atom arrangement, admitting applications to both crystalline and amorphous structures. The analysis also applies to both homogeneous materials with identical atoms and heterogeneous materials with dissimilar atoms. For non-polar atomic systems with only central interatomic forces, the fields of couple stress, body moment and surface moment vanish. This demonstrates that, on the interatomic level, interatomic moments give rise to couple stresses of dynamically equivalent nature.