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.

翻译：机械超材料是经过人工设计的微结构，在宏观尺度上展现出新颖的力学行为。主动超材料可通过外部手段进行控制。气动驱动超材料能够通过施加压力改变其力学、声学或其他类型的有效行为，潜在应用涵盖软体机器人致动器到声子晶体等。为通过拓扑优化设计此类气动驱动超材料与结构，需要建立稳健的计算建模方法，既能捕捉内部空腔的气动驱动效应，也能处理内部接触问题。由于拓扑优化中的空腔常采用软材料模型建模，第三介质接触公式成为理想的过渡方案。我们提出单一超弹性材料模型：该模型在接触前阶段既能维持空腔内设定的静水柯西应力，同时能作为强制执行无摩擦接触的第三介质——这与仅关注接触的现有第三介质方法形成对比。我们将第三介质总能量密度分解为接触项、正则化项和气动压力贡献项，这些均可独立控制和调节。为抑制柔顺第三介质的畸变，我们在模型中引入曲率惩罚项。相较于现有公式，该方法在柔顺第三介质行为方面实现改进，最终提升了数值解的稳定性。由于本公式满足能量一致性要求，我们能够采用更先进的有限元求解器（如修正Cholesky算法）检测失稳现象。我们通过多个传统接触基准算例（包括标准贴片测试）验证了所提公式的性能，并与实验测量结果进行了对比验证。