Third medium finite element contact formulation for pneumatically actuated systems

Active mechanical metamaterials are artificially engineered microstructures that can be externally controlled to exhibit novel and switchable mechanical behavior on the macroscopic scale. Pneumatically actuated variants of these metamaterials can then change their mechanical, acoustic, or other types of effective behavior in response to applied pressure with possible applications ranging from soft robotic actuators to phononic crystals. Such materials often undergo large deformations, leading to self-contact, which makes their modeling and generative design challenging. We propose to use the third medium method to simulate both pneumatic actuation and contact behavior. In contrast to existing formulations, we (i) combine together contact with pneumatic actuation in a single material model for the third medium, (ii) include more permissive regularization scheme for our model, penalizing gradient of pseudo-rotation and gradient of volume change only, and (iii) introduce pneumatic actuation as an exact value of Cauchy stress. Our formulation is energetically consistent and admits advanced finite element solvers, such as the modified Cholesky decomposition to detect instabilities. We demonstrate the behavior of the proposed formulation on several examples of traditional contact benchmarks, including a standard patch test, and validate it with experimental measurement.