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

TL;DR

The paper develops a continuum, finite‑strain model for thermo‑mechanical contact based on a third, compliant interfacial medium that surrounds the contacting bodies. It introduces a regularization strategy enabling linear low‑order finite elements to accurately capture deformation while preventing element distortion, and couples mechanical and thermal fields through a pseudo‑potential formulation. The framework unifies non‑contact and contact heat transfer within a single formulation by modeling the third medium as a gas‑like conductor whose conductivity increases during contact. Numerical tests in 2D and 3D validate accurate gap closure, smooth stress and heat flux transitions, and robust performance across regularizations, with explicit guidance on parameter choices and mesh layering. The approach offers a computationally efficient, robust alternative to classical interface laws with potential extensions to frictional heating and more complex multiphysics scenarios.

Abstract

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 behaviour of the associated finite element formulation based on linear ansatz functions.

Figures (25)

• Figure 1: General idea of third medium contact: the third medium $\Omega^{m}$ is placed around the two physical bodies $\Omega^{(1)}$ and $\Omega^{(2)}$ (left). During the loading the solids undergo the deformation $\boldsymbol{\varphi}$ and the two bodies come into contact. The third medium is severely deformed (right).
• Figure 2: Block under displacement and thermal load
• Figure 3: Block: mesh and deformed solid with temperature distribution.
• Figure 4: Block: Load-gap curve.
• Figure 5: Block: progression of the temperature just before and after contact.