Theoretical and practical issues in nonlinear contact Finite Element Analysis (FEA): an overview

Sources of nonlinear contact

Nonlinear contact in Finite Element Analysis (FEA) involves complex interactions between surfaces, where stiffness and boundary conditions change during the FEA simulation. Key aspects include:

    >  Geometric nonlinearity, as surfaces may deform significantly;                                                          >  Material nonlinearity, where materials exhibit behaviors like plasticity or hyperelasticity;               >  Contact constraints, such as gap closure or separation.

Nonlinear contact due to geometric nonlinearities: occurs when large deformations, rotations, or changes in the shape of the interacting bodies significantly affect the contact behavior. Unlike small-deformation problems, where the geometry remains essentially unchanged, large deformations alter the contact area, load distribution, and stiffness of the system.

For example, when a thin shell or membrane undergoes significant bending or stretching, the surface curvature changes, leading to variations in the contact pressure and frictional forces. Similarly, large rotations can cause surfaces to lose or gain contact dynamically, introducing additional challenges. Buckling or wrinkling in slender structures, such as beams or plates, also exemplifies geometric nonlinearity, where the interaction depends on the deformed geometry rather than the initial shape.

Geometric nonlinearities require the FEA solver to continuously update the contact conditions and stiffness matrix during each iteration. This is done by considering the deformed configuration rather than the original one. Iterative techniques, like the Newton-Raphson method, are used to achieve convergence. Such nonlinearities are critical for accurately modeling problems like metal forming, crash analysis, or soft tissue interactions, where large deformations are inherent.

Nonlinear contact due to material nonlinearities: occurs when the material properties of the interacting bodies change under applied loads, affecting how they deform and interact at the contact interface. Unlike linear materials, which have a constant relationship between stress and strain, nonlinear materials exhibit behaviors such as plasticity, hyperelasticity, or viscoelasticity.

For example, in plasticity, a material deforms permanently after exceeding its yield point, which can alter the contact pressure distribution and the shape of the contact area. In hyperelastic materials (like rubber), large deformations and non-linear stress-strain relationships result in continuously changing contact forces. Viscoelastic materials add time-dependent behavior, where contact forces vary with the rate of deformation.

Nonlinear material properties create challenges in FEA because the stiffness matrix changes dynamically during the analysis. This requires iterative numerical methods to solve, such as Newton-Raphson, to ensure accurate convergence. Such contact is crucial to model real-world applications, such as elastomer seals, indentation problems, or crash simulations.

Nonlinear contact due to contact constraints: arises when the interaction between surfaces is governed by conditions that change dynamically during the simulation. These constraints include gap closure, frictional behavior, and contact separation, all of which depend on the relative motion and forces between the surfaces. 

For example, when two bodies initially separated by a gap come into contact under an applied load, the contact force is zero until the gap closes, introducing a nonlinearity. Similarly, when friction is present, the tangential force depends on the normal force and whether sliding or sticking occurs. Transitioning between sticking (static friction) and sliding (kinetic friction) creates a nonlinear relationship between forces and motion. In cases of separation, surfaces may lose contact entirely, altering the boundary conditions of the system. 

This type of nonlinearity occurs because contact constraints are inherently conditional—they activate or deactivate based on the geometry and forces. FEA solvers handle this using algorithms like the penalty method or Lagrange multipliers, which iteratively adjust the solution to enforce contact conditions. Nonlinear contact due to constraints is critical in applications like bolted joints, gear meshing, and impact simulations, where the contact state evolves during the analysis.

As well, friction plays a critical role, introducing nonlinear behavior due to varying resistance during sliding. Iterative solvers, like Newton-Raphson, are commonly used to handle the nonlinearity, requiring careful convergence controls. Nonlinear contact is essential for accurately modeling real-world scenarios, such as bolted joints, gear meshing, or soft material interactions.