A differentiable variational model for structural self-contact and fracture

Numerical modelling of structural self-contact and crack propagation presents significant challenges due to the inherently discontinuous and non-differentiable nature of the underlying physical phenomena. Traditional contact models demand explicit definition and tracking of contact points, while fracture models often rely on predefined crack initiation sites, sharp interfaces, and re-meshing. This study introduces a novel framework that overcomes these limitations within a unified and numerically stable variational formulation. The contact phenomenon is described through the hyperelastic third medium contact model and fracture is represented by a phase field. Structures are embedded in a third medium that stiffens under compression, enabling the transfer of forces between structural members. Crack propagation occurs in regions in which it is energetically favourable for the system to evolve toward a fully damaged state, specifically where the critical energy release rate is exceeded. Careful treatment is required when coupling the two phenomena, particularly concerning the void material behaviour. This work presents an efficient and differentiable numerical model that captures both nonlinear phenomena within a unified framework. This framework will allow designers and engineers to efficiently analyse complex nonlinear structural behaviours, previously requiring separate models that involved pre-defined crack initiation sites and contact points. Lastly, the differentiable nature of the model facilitates straightforward future integration into topology optimisation pipelines, providing designers the ability to intentionally design for and leverage self-contact interactions and material failure as functional, performance-enhancing features.