  Mechanical metamaterials are artificially engineered microstructures that
exhibit novel mechanical behavior on the macroscopic scale. Active
metamaterials can be externally controlled. Pneumatically actuated
metamaterials can 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. To facilitate the
design of such pneumatically actuated metamaterials and structures by topology
optimization, a robust way of their computational modeling, capturing both
pneumatic actuation of internal voids and internal contact, is needed. Since
voids in topology optimization are often modeled using a soft material model,
the third medium contact formulation lends itself as a suitable stepping stone.
We propose a single hyperelastic material model capable of maintaining a
prescribed hydrostatic Cauchy stress within a void in the pre-contact phase
while simultaneously acting as a third medium to enforce frictionless contact,
contrasting existing third medium approaches focused solely on contact. We
split the overall third-medium energy density into contact, regularization, and
pneumatic pressure contributions, all of which can be individually controlled
and tuned. To prevent distortions of the compliant third medium, we include
curvature penalization in our model. This improves on existing formulations in
terms of compliant third medium behavior, leading ultimately to better
numerical stability of the solution. Since our formulation is energetically
consistent, we are able to employ more advanced finite element solvers, such as
the modified Cholesky algorithm 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.
