Three-dimensional third medium contact model for hyperelastic contact and pneumatically actuated systems

TL;DR

This paper develops a fully three-dimensional third-medium contact framework to model hyperelastic contact and pneumatically actuated systems without explicit contact surface tracking. It introduces a soft third medium with a regularization term to stabilize distorted elements and a pneumatic term to capture inflation or suction, enabling unified treatment of contact and actuation. The authors derive the governing equations, weak forms, and high-order tangent tensors, and implement the method within a finite element scheme requiring second-order shape functions. Through multiple 3D benchmarks and soft-robot–like applications, the approach demonstrates accuracy, robustness, and versatility, with potential extensions to isogeometric and meshless discretizations.

Abstract

This work presents a comprehensive three-dimensional third-medium contact framework for modeling complex contact interactions in hyperelastic solids and pneumatically actuated systems. The proposed third-medium formulation embeds a fictitious medium (or third medium) between potentially interacting bodies, enabling a unified and robust treatment of hyperelastic contact and self-contact without the need for discretization of the contact interface. Unlike the widely studied two-dimensional problem, this paper extends the new regularization term given in Reference \cite{TMCWriggers2} to three-dimensional problems and ensures element quality in a third medium. Due to the need for higher-order elements for the regularization term, this paper details the linearization process of this problem within the finite element framework. In addition, pneumatically actuated systems are considered by introducing a pneumatic term to represent pneumatic loading (pressure or suction) and inducing contact caused by internal inflation. This approach is suitable for complex hyperelastic contact and self-contact, and has potential applications in the fields of soft robotics and flexible mechanisms. The framework is developed in a fully three-dimensional setting, making it also suitable for isogeometric methods and meshless methods. Several benchmark and application-level simulations demonstrate the accuracy, robustness, and versatility of the proposed approach. The results highlight the capability of the three-dimensional third-medium model to handle challenging nonlinear contact scenarios relevant to soft materials, soft actuators, and emerging multifunctional structures.

Figures (21)

• Figure 1: Contact of two bodies with a third medium, where $\varphi$ is a map between the initial and current configuration.
• Figure 2: Discretization in the finite element method.
• Figure 3: Self-contact within a 3D box, (a) geometry model and boundary condition, (b) front view of the 3D box.
• Figure 4: Self-contact within a 3D box, (a) geometry model and the third medium, (b) finite element mesh.
• Figure 5: Gap $g$ between the surfaces of the upper and lower flange for different parameters, (a) different $\gamma$ and $\alpha_r = 10$, (b) different $\alpha_r$ and $\gamma = 1\times 10^{-6}$.