A third medium approach for thermo-mechanical contact based on low order ansatz spaces

The third medium contact approach has been successfully employed in structural applications and extended to various optimization problems. This discretization technique replaces classical contact formulations and algorithms by introducing a compliant interfacial layer – referred to as the third medium – between the contacting bodies. Unlike traditional contact methods, this formulation naturally accommodates finite deformations at the interface. As the two bodies approach each other, the third medium undergoes compression and effectively acts as a deformable barrier, preventing interpenetration and transmitting contact forces in a smooth and numerically stable manner. In thermo-mechanical problems, heat conduction must be incorporated into the model, which typically requires specialized interface laws when using classical contact formulations. These laws aim to capture the complex thermal behavior at the contact interface, including discontinuities and varying conductance. In contrast, the third medium approach offers a significant advantage: the thermo-mechanical formulation inherently accounts for the interface behavior without the need for additional interface conditions. This includes the gradual heat transfer through the surrounding gas when the bodies are near each other, as well as the localized heat conduction that occurs upon physical contact. As a result, the third medium naturally captures both non-contact and contact-phase thermal conduction within a unified framework. In this paper, we propose a new thermo-mechanical model based on a continuum formulation for finite strains and show by means of examples the behavior of the associated finite element formulation based on linear ansatz functions.