An energy-conserving contact theory for discrete element modelling of arbitrarily shaped particles: Basic framework and general contact model

The first paper of this series establishes a unified theoretical framework that lays a solid foundation for developing energy-conserving normal contact models for arbitrarily shaped bodies in the discrete element method. It is derived based solely on the requirement that the potential energy must be conserved for an elastic impact of two shapes under any condition. The resulting general energy-conserving contact model states that the normal force as a vector must be the gradient of a contact potential field. When such a contact potential or energy function is specified, a complete normal contact model for a pair of arbitrarily shaped particles, including the contact normal direction, contact point/line and force magnitude, will be automatically followed without introducing any additional assumptions. In this framework, the contact geometry and contact force are indispensably related and are evaluated in a consistent manner. Due to the paramount role that the energy function plays in the current theory, its fundamental properties are discussed, which serve as general guidance for choosing a valid energy function. In addition, both single and multiple contacts and their evolution can be handled in a seamless way. Some symmetric properties of particle shapes can also be utilised to simplify the contact models. Within the proposed theoretical framework, different choices or combinations of geometric features as variables for the contact energy function can give rise to unique types of energy-conserving contact models with distinct characteristics and features. Two such functions using only one primary feature, which lead to two specialised energy-conserving contact models, will be presented in the subsequent papers of this series.