The discrete origin of configurational forces

Abstract

We develop a discrete formulation of configurational forces for open subsystems of a finite particle system. Membership in an open subsystem may change by particle admission or deletion. We first use zero-one membership functions to formulate standard balances for mass, standard momentum, angular momentum, and kinetic energy: continuous terms are evaluated on fixed-membership subintervals, but admission and deletion enter through jump-transfer sums. We then introduce a configurational momentum balance for open subsystems and configurational contact-force resultants associated with the ordered separation between a subsystem and its complement. To describe membership changes geometrically, we introduce an auxiliary smooth description based on a scalar membership field on the Euclidean space of particle placements. For each particle, the level set through the current particle position is used as a local membership cut, and the gradient at that position defines a normal to the cut. We assign an observed velocity to these level sets; its tangential component can be changed without altering the smooth membership value or its rate. We require the working associated with membership transfer to be invariant under such tangential changes, and we refer to this invariance as intrinsicity. If this invariance is imposed on a working involving only standard contact-force assignments and standard momentum transfer, then an artificial normality condition is imposed on the standard contact-force assignment. We therefore add configurational contact-force assignments. For a particle with assigned mass, standard momentum, standard velocity, and specific configurational momentum, we impose the intrinsicity requirement and obtain that the specific configurational momentum is the negative of the standard velocity. For nondissipative membership transfer, we use a mechanical energy imbalance to identify the particlewise membership tension: the combined standard-plus-configurational cut-force assignment equals the difference between the membership interaction-energy increment and the kinetic energy, multiplied by the cut normal. After imposing localization hypotheses for contact-force assignments, current representative volumes, and energy increments, we obtain the continuum relation $\bu=-\bv$ for the specific configurational momentum and the spatial inertial Eshelby relation $\C=\rho(\psi-\tfrac12|\bv|^2)\sidem-\T$, where $\C$ is the spatial configurational stress, $\T$ is the Cauchy stress, $\rho$ is the mass density, $\psi$ is the specific free energy, $\bv$ is the standard velocity, and $\sidem$ is the identity tensor.