A unified variational framework for phase-field fracture and third-medium contact in finite deformation hyperelasticity

Abstract

This paper presents a unified variational framework that integrates phase-field fracture (PFF) and third-medium contact (TMC) within finite deformation hyperelasticity. The key idea is that both crack and contact are treated through regularization: the sharp crack topology is regularized into a diffuse damage field, while the discrete contact interface is regularized by a compliant fictitious medium with auxiliary fields. This strategy eliminates the need for explicit contact detection or crack tracking algorithms. The framework is validated through two-dimensional three-point bending and three-dimensional Brazilian disk test simulations, demonstrating the interplay between contact-induced stress concentration and crack nucleation/propagation. In particular, the Brazilian disk simulation naturally reproduces secondary crushing-type fracture zones near the contact regions -- a phenomenon consistently observed in experiments yet inaccessible to simplified loading models. These results pave the way for predictive simulation of coupled contact-fracture phenomena without recourse to explicit interface tracking.

Figures (12)

• Figure 1: Schematic drawing for the unified PFF-TMC framework. (a) Reference configuration showing the substrate $\Omega_1$, indenter $\Omega_2$, and third-medium domain $\Omega_3$ with their respective field variables. (b) Deformed configuration after the mapping $\boldsymbol{\varphi}$, illustrating the compressed third medium ($J \to 0$), contact pressure transmission, stress concentration beneath the indenter, and phase-field damage evolution ($d \to 1$) in the substrate.
• Figure 2: Phase-field regularization of fracture. A body subjected to boundary conditions ($S_u$: Dirichlet, $S_t$: Neumann) containing a sharp crack (center) is approximated by a diffuse crack band (right) with characteristic width $2\ell$, where the damage field $d$ transitions continuously from $d=0$ (intact) to $d=1$ (fully broken).
• Figure 3: 2D C-box benchmark for third-medium contact. (a) Geometry and boundary conditions: the left boundary is clamped, and a prescribed vertical displacement $u_y$ is applied at the top-right corner. (b) Finite element mesh showing the solid domain $\Omega_1$ (blue) and the third-medium domain $\Omega_3$ (red).
• Figure 4: Deformed configurations of the C-box with auxiliary-field regularization: (a) at the onset of contact (corresponding to step 60 in Fig. \ref{['fig:cbox_gap']}) and (b) post-contact stage. The solid domain $\Omega_1$ (blue) and the third medium $\Omega_3$ (red) are shown. The auxiliary fields $p$ and $q$ maintain mesh quality in the severely compressed third medium.
• Figure 5: Tip displacement and contact gap as functions of the loading step for the 2D C-box benchmark. The contact gap decreases monotonically until the inner surfaces come into contact through the third medium. The theoretical contact point is indicated by the vertical dashed line.

Theorems & Definitions (5)

• Remark 1: Notation
• Remark 2: Mixed large/small-strain formulation
• Remark 3: Eigenvalue computation in 2D and 3D
• Remark 4: Numerical regularization
• Remark 5: Phase-field length scale