Representation and Structural Change in Physical Modelling

This paper examines how different physical modelling frameworks represent structural change and asks when the organisation through which physical interactions occur should itself form part of the model state. The literature spans classical continuum mechanics and internal-variable theories, configurational mechanics, phase-field and cohesive fracture, enriched and unfitted methods, nonlocal and particle descriptions, quasicontinuum strategies, adaptive and higher-order networks, graph-based learning, and discrete differential and combinatorial formulations. These approaches provide different mechanisms for accommodating evolving geometry, interfaces, interactions, and topology and therefore reveal that the central issue cannot be reduced to a continuum-discrete distinction. The comparison is organised around the objects that carry physical quantities, the relations that determine admissible interactions, the transformations included in the state evolution, and the information about history retained after structural change. From this review, a relational synthesis is developed in which physical entities, their attributed relations, and the quantities carried by both may enter the state of an evolving system. Four concepts are introduced to make this synthesis operational: accumulative and reorganisational change, representational closure, structural memory, and representational transition. A minimal event-map formulation is proposed to identify the requirements for transferring physical quantities, constitutive information, and history when structural carriers change. Fatigue, environment-assisted degradation, transport, plasticity, and fracture are used as comparative tests of representational adequacy. The resulting framework does not prescribe a universal modelling formalism. It provides criteria for determining when structural information can be reduced to effective fields and internal variables, when explicit organisation becomes consequential, and how relational and continuum descriptions might be connected in a controlled manner.