A Phase Field Formulation of Frictional Sliding Contact for 3D Fully Eulerian Fluid Structure Interactions

TL;DR

This work addresses the challenge of modeling frictional sliding contact in a fully Eulerian multiphase FSI setting, motivated by ship-ice interactions. It develops a phase-field overlap-based contact formulation that yields normal and frictional forces within a single momentum balance and uses phase-averaged velocities to define slip direction without separate velocity fields. Validation across Hertzian contact, sliding block, and ironing problems demonstrates accurate traction, dynamic friction response, and stability under large deformation, while a ship-ice application showcases 3D capability with free-surface effects. The approach offers a scalable, surface-tracking-free framework for complex multiphase contact in engineering systems, with potential extensions to stick–slip behavior and advanced ice mechanics.

Abstract

Frictional sliding contact in hydrodynamic environments can be found in a range of engineering applications. Accurate modeling requires an integrated numerical framework capable of resolving large relative motions, multiphase interactions, and nonlinear contact responses. Building on our previously developed fully Eulerian fluid structure formulation, we introduce a phase field based formulation for dynamic frictional contact in 3D. Contact detection is achieved via the overlap of diffuse interfaces of colliding solids. The normal contact response is defined as a volumetric body force proportional to the overlap parameter, while the tangential response is computed using the Coulomb friction model. The direction of the friction forces are derived by projecting phase-averaged relative velocities onto the local tangent plane of colliding bodies. This proposed unified treatment enables the computation of both normal and frictional forces within a single momentum balance equation, avoiding separate velocity fields for individual solids. We present several test cases with increasing complexity to verify and demonstrate our proposed frictional contact model. Verification against the Hertzian contact problem shows excellent agreement with the analytical solution, with errors below $3\%$ in the traction profile. In the sliding block benchmark, the computed displacement profiles closely follow the analytical solution for point-mass systems across multiple friction coefficients. The ironing problem demonstrates stable force predictions under finite deformation, with normal and tangential forces matching kinetic friction laws. The robustness and scalability of the proposed formulation are further demonstrated through a representative ship ice interaction scenario with free surface and frictional sliding effects.

Figures (15)

• Figure 1: Schematic of two colliding bodies with diffuse interfaces (top). The shaded regions indicate the contact volumes. Insets (bottom) illustrate: phase-averaged velocities (left), normal contact forces (center), and frictional contact forces (right). Solid lines denote iso-contours of $\phi=0$, while dashed lines denote $\phi=\pm 0.9$. In the formulation, the contact volume is not restricted to the $\phi=\pm 0.9$ contours but decays smoothly into the interior; the bounded shaded regions are shown here only for clarity.
• Figure 2: (a) Schematic of the Hertzian contact problem with a deformable cylinder in contact with an elastic plane, and (b) single mesh layer for the same problem. The right plane wall approaches the fixed cylinder on the left with a constant velocity boundary condition of $u_0 = R/100$. The domain is extruded by 0.1 units in the out-of-plane direction to enable 3D computation. The mesh is refined in the contact zone and progressively coarsened away from it.
• Figure 3: Hertz contact problem: (a) pressure contour for the contact problem and (b) comparison of the numerical force profiles with the analytical Hertz profile for different mesh resolutions. The interface resolution is considered to be same for all cases $\frac{\varepsilon}{h}=10$ in the contact zone.
• Figure 4: Sliding block problem: (a) Schematic representation of the computational domain and (b) single layer of the mesh. Two components of gravity act on the elastic block, thus replicating a block sliding down an inclined plane. The domain is extruded by 0.1 units into the page to solve in 3D. The mesh is refined around the expected sliding zone and gradually coarsened elsewhere.
• Figure 5: Sliding block problem: Comparison of the sliding displacement of the center of mass between the numerical and analytical solutions.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3