Third Medium Finite Element Contact Formulation for Pneumatically Actuated Systems

TL;DR

The paper addresses robust computational modeling of pneumatically actuated metamaterials that combine internal void actuation with contact, a key capability for topology optimization and soft robotics. It introduces a tripartite, energetically consistent third-medium formulation with $W = \psi_p W_p + \psi_c W_c + \psi_r W_r$, where $W_p$ enforces hydrostatic stress via $W_p = p J(F)$, $W_c$ provides a contact response through a neo-Hookean-like term $W_c = \ln^2 J + (J^{-2/3} I_1 - 3)$, and $W_r$ regularizes curvature with $W_r = \tfrac{1}{2} c (\nabla \ln \bm{Q} : \nabla \ln \bm{Q} + \nabla J \cdot \nabla J)$. The model achieves exact pneumatic follower loading, improves numerical stability with a novel rotation-based regularization via $\nabla \ln \bm{Q}$, and preserves energy consistency to enable advanced solvers. Validation includes a patch test, a self-contact C-shape benchmark, and buckling of a four-void metamaterial, with experimental comparison showing good agreement for critical pressure and deformation patterns, demonstrating practical applicability to design and optimization of pneumatically actuated metamaterials.

Abstract

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.

Figures (17)

• Figure 1: Behavior of the neo-Hookean strain energy density terms in response to in-plane biaxial and full triaxial compression. Under in-plane compression, due to the $J^{-2/3}$ term, the isochoric energy term tends to infinity more rapidly than the volumetric term. The isochoric term alone can therefore act as the contact term for plane strain geometries.
• Figure 1: Uniaxial compression of silicone rubber cylinders, various stages of loading (compressive strain increases left to right). The samples are fixed in the loading machine between lubricated plexiglass plates.
• Figure 2: Pneumatic box subjected to suction within the internal void: geometry and boundary conditions.
• Figure 2: Stress-strain diagrams for uniaxial compression tests on cylindrical silicone rubber samples shown in Figure \ref{['fig:axialtests']}. Experimental data from three independent tests and a response curve of a near incompressible neo-Hookean material model with parameters determined from a least-square fit procedure.
• Figure 3: Simulation of monotone suction in a single pneumatic cell. (a) Mesh and deformed configuration for $\gamma = 10$ (left), for $\gamma = 1$ (center), where only the last converged configuration is shown, with contact penetration and distorted third medium, and for $\gamma = 1$ with the use of additional regularizing terms (right). (b) Comparison of pressure-gap diagrams for different values of $\gamma$ and regularization (only relevant area shown).