Interpretation of the atomic formulae for stress and stiffness coefficients

Abstract

The atomic formulae for stress {δ_αβ(i)} and stiffness coefficients {C*_αβγλ(i)} are carefully examined particularly with their application to the results of computer simulation for imperfect lattices or amorphous matter. It is found that: (1) The formulae are valid irrespective of whether the force balance condition is satisfied for each atom or not. (2) When it is satisfied, the relation holds, where v(i) denotes the volume of the Wigner-Seitz cell surrounding the ith atom and {f^ex _α(i)} the external forces.

The physical meanings of the formulae are investigated, and it is found that: (1) 2v(i)δ_αβ(i) may be interpreted as the sum of the work done by the counterbalance force (CBF), which is imagined to be exerted on each atom surrounding the ith atom to balance the interaction force exerted by the ith atom, upon the deformation described by the unit value of the strain component ϵ_αβ under the supposition that the ith atom is fixed. (2) v(i)C*_αβγλ(i) may be interpreted as the sum of the work done by the increment of the CBF, which balances the increment of the interaction force exerted by the ith atom due to a deformation described by the unit value of u _{γ, λ,} upon the deformation described by the unit value of u _{α, β} under the simultaneous superposition of u _{α β} and u _{γ, λ} with the ith atom fixed. The roles of u _{α β} and u _{γ, λ} may be interchanged in this interpretation. (3) When the force balance condition is satisfied for each atom, the interpretation of the stress components given above predicts that the outermost atoms at a free surface are to possess positive values for v(i)δ_αα(i), where the x _α-axis is taken to be perpendicular to the free surface. This prediction is verified by the results of a computer simulation for a two-dimensional triangular lattice.