A kinematic approach to material deformation: Giving more structure to a swarm considering agents’ local rotations

This paper proposes a discrete kinematic model inspired by the principles of cellular automata and swarm-based models, aimed at simulating deformation phenomena in complex materials. Each point in the system represents a material unit that interacts locally with its neighbors according to rules inspired by solid mechanics, without resorting to global differential equations or force balances. This approach is part of a broader tradition of geometric and kinematic models, from ancient cosmologies to modern computational structures, and is particularly effective in the study of disordered or non-linear systems, where continuous methods are inapplicable or computationally burdensome. Kinematic modeling has historically favored representations of points without orientation, as evidenced in the models of Eudoxo and Descartes. However, such approaches are limited in describing phenomena in which directional orientation is crucial, as in the case of anisotropic deformations of materials. In this study, we introduce a kinematic model based on points with intrinsic direction, extending the traditional representation to a full vector description. Each point in the system possesses a position and a direction vector, allowing the analysis of local interactions that include both translations and rotations. The proposed model is distinguished by its ability to represent emergent phenomena arising from local interactions, offering a new perspective in the simulation of complex mechanical behavior. The intrinsic direction of the points makes it possible to capture anisotropic effects and to more accurately model the dynamics of real material systems. This extension of the kinematics-discrete paradigm opens up new possibilities in the computational modeling of materials, providing a versatile tool for the analysis of complex systems in engineering and materials physics. The potential of the proposed model is discussed in relation to its ability to describe, with reduced computational cost, both reversible elastic phenomena and irreversible processes such as plasticization and damage propagation. The presented framework thus emerges as a promising tool for the qualitative and quantitative analysis of heterogeneous materials, with possible applications in the physical, engineering, and computational fields.