Contact Analysis

CONTACT CHANGES THE LOAD PATH

Contact is the mechanism by which load transfers between two surfaces that touch but are not bonded. When two surfaces are in contact, the load passes from one to the other through the contact pressure. When they separate, the load path is interrupted. When they slide, the load transfer changes direction (from normal to shear, governed by friction). This means that the load path in a structure with contact is not fixed — it depends on the contact status, which depends on the load, which depends on the deformation. This circular dependency is what makes contact non-linear and what makes it so important to model correctly. A bonded contact assumption — gluing the surfaces together as if they never separate — locks the load path into a single configuration and can hide the real behaviour entirely.

contact-interface

Before assuming bonded contact, ask: do these surfaces really never separate, never slide, and never change their load transfer under any load case? If the answer is no, bonded contact is hiding the real load path.

Contact States: Open, Closed, Sliding, Sticking

A contact pair can be in one of several states at any point on the interface. Open: the surfaces are separated, no load is transferred. Closed and sticking: the surfaces are in contact and not sliding — normal load and shear (up to the friction limit) are transferred. Closed and sliding: the surfaces are in contact and sliding relative to each other — normal load and friction-limited shear are transferred. The transition between these states is what makes contact non-linear. A surface that is open at the start of the analysis may close under load; a surface that is sticking may begin to slide if the shear exceeds the friction limit; a surface that is closed may separate if the load reverses.

• Open: surfaces separated, no load transfer
• Closed, sticking: surfaces in contact, no relative sliding — normal and shear load transferred
• Closed, sliding: surfaces in contact, relative sliding — normal load and friction-limited shear transferred
• Separation: a previously closed contact opens under load reversal
• State transitions: open→closed, closed→sliding, closed→open — each changes the load path and the stiffness matrix

Friction

Friction is the shear resistance between two contacting surfaces. The Coulomb friction model is the most common: the maximum shear stress that can be transferred is μ·p, where μ is the friction coefficient and p is the contact pressure. If the applied shear is below this limit, the surfaces stick. If it exceeds the limit, the surfaces slide, and the shear is capped at μ·p. The friction coefficient is one of the most sensitive inputs in a contact analysis. A small change in μ can shift a contact from sticking to sliding, which changes the load transfer and the entire stress field. The friction coefficient is also difficult to know precisely — it depends on surface finish, lubrication, contamination, wear, and temperature. A range of values should be considered, and the sensitivity of the result to μ should be assessed.

    Coulomb friction:
      |τ|  ≤  μ · p    (sticking — no sliding)
      |τ|  =  μ · p    (sliding — shear capped at friction limit)

    where  τ  = shear stress at contact
           μ  = friction coefficient
           p  = contact pressure

The friction coefficient is rarely known to better than ±0.1, and the result can change dramatically across that range. Always run a sensitivity study on friction if the contact is load-path-critical.

Penetration and Contact Stiffness

Penetration is the amount by which one surface "passes through" the other in the FEA model. In reality, surfaces do not interpenetrate — they are impenetrable. In FEA, contact is enforced by a penalty method: a stiff spring is inserted between the surfaces when they are in contact, resisting penetration. The stiffness of this spring — the contact stiffness or penalty stiffness — determines how much penetration occurs. Too low, and the surfaces interpenetrate excessively, giving an incorrect contact area and pressure. Too high, and the contact spring dominates the stiffness matrix, causing convergence problems and oscillation. The contact stiffness is a numerical parameter, not a physical one — it should be high enough to limit penetration to an acceptable level (typically a small fraction of the element size) but low enough to allow convergence, and the result should be insensitive to the specific value chosen.

• Penetration: surfaces interpenetrate in the model — controlled by contact stiffness
• Contact (penalty) stiffness: a numerical spring between contacting surfaces
• Too low: excessive penetration, incorrect contact area and pressure
• Too high: convergence problems, contact chatter, oscillation
• The result should be insensitive to the contact stiffness value — if it is not, the stiffness is too low or the mesh is too coarse

Penalty Method and Augmented Methods

The penalty method enforces contact by adding a stiff spring between penetrating surfaces. The contact force is proportional to the penetration depth: F_contact = k_penalty · δ, where k_penalty is the penalty stiffness and δ is the penetration. The penalty method is simple and efficient, but it allows some penetration — the amount depends on the stiffness. The augmented Lagrangian method combines the penalty method with Lagrange multipliers: it iterates on the contact pressure to reduce penetration further, without increasing the penalty stiffness. This gives better contact accuracy (less penetration) with better convergence than a pure penalty method with high stiffness. Most solvers offer both; the augmented Lagrangian method is the default for many problems because it balances accuracy and convergence.

    Penalty method:
      F_contact  =  k_penalty · δ
      where  k_penalty = penalty stiffness (numerical)
             δ         = penetration depth

    Augmented Lagrangian:
      Iterates on contact pressure to reduce penetration
      Better accuracy, better convergence than pure penalty

Master and Slave Surfaces

In a contact pair, one surface is designated the master (or target) and the other the slave (or source or contact). The contact constraint is enforced at the slave nodes against the master surface. The choice of master and slave affects the contact detection and the result. The general rule is: the stiffer surface, or the surface with the coarser mesh, should be the master. If the slave mesh is much finer than the master, a slave node may "fall through" a large master element without being detected. If the slave is much coarser, the contact pressure distribution on the master is poorly resolved. For self-contact (a surface that contacts itself, e.g., a thin shell that folds), both surfaces are the same, and the solver handles the detection internally.

• Master (target): the surface against which contact is detected
• Slave (source/contact): the surface whose nodes are checked for penetration
• Rule: stiffer or coarser surface should be master
• Mesh mismatch: slave much finer than master — node may fall through large master element
• Self-contact: a surface contacting itself (folding, buckling) — handled internally by the solver

Contact Applications

Contact analysis is used in a wide range of structural problems. Each application has its own modelling considerations.

• Bearing: pin in hole, shaft in bearing — contact pressure distribution, bearing stress, clearance effects
• Bolted joints: clamped interfaces, bolt-to-hole contact, friction load transfer, preload and separation
• Pinned joints: pin-to-lug contact, clearance, friction, load transfer through the pin
• Press fits: interference between shaft and bore — contact pressure, retention force, assembly stress
• Clamped interfaces: two surfaces clamped together by bolt preload — friction load transfer, interface opening
• Mechanical stops: a component that hits a stop at end of travel — contact prevents further motion, generates impact load

Hertzian Contact

Hertzian contact theory provides closed-form solutions for the contact pressure and contact area between two elastic bodies with simple geometries: a sphere on a flat surface, two cylinders in contact, an ellipse on a flat surface. The theory assumes: elastic material, small contact area relative to the body dimensions, frictionless contact, and perfectly smooth surfaces. Under these assumptions, the contact pressure distribution is elliptical (Hertzian), and the maximum pressure, contact radius, and approach distance can be calculated analytically. Hertzian theory is useful for two purposes: it provides a quick estimate for simple contact problems without running FEA, and it provides a validation benchmark for FEA contact models. If an FEA contact model of a sphere-on-flat gives a very different contact pressure to Hertz, the FEA model has a problem — mesh, contact stiffness, or element type.

    Hertzian contact (sphere on flat, radius R, load F, elastic modulus E*):

      Contact radius:    a  =  (3·F·R / (4·E*))^(1/3)
      Maximum pressure:  p₀ =  (6·F / (π·a²))  /  π   =  1.5 · F / (π·a²)
      Approach:          δ  =  a² / R  =  (9·F² / (16·R·E*²))^(1/3)

    where  E* = combined elastic modulus:  1/E* = (1-ν₁²)/E₁ + (1-ν₂²)/E₂

Hertzian contact is a powerful validation tool. If your FEA contact model of a simple geometry gives a contact pressure or contact area that disagrees with Hertz, the FEA model has a mesh, contact stiffness, or element type problem. Fix the model before trusting the result.

Contact Convergence

Contact convergence is the most common source of non-linear solver failure. The three main culprits are: contact chatter (a contact pair oscillates between open and closed at each iteration, preventing equilibrium), excessive penetration (the contact stiffness is too low, the surfaces interpenetrate, and the solver cannot find equilibrium), and friction transition (the contact transitions between sticking and sliding at each iteration, changing the shear load and preventing convergence). The remedies include: increasing contact stiffness (to reduce penetration), decreasing contact stiffness (to reduce chatter — the remedy depends on the cause), using a softer contact override during initial iterations, applying the load in smaller increments, and using augmented Lagrangian methods. Reading the solver log — which contact pair is failing, what the penetration is, what the contact status is at each iteration — is essential for diagnosis.

• Chatter: contact pair oscillates open-closed at each iteration — decrease stiffness or use augmented Lagrangian
• Excessive penetration: stiffness too low — increase stiffness or refine mesh
• Friction transition: contact oscillates stick-slip — smaller increments or regularised friction
• Diagnosis: read the solver log — which pair, what penetration, what status, what residual

Why Bonded Contact Is Not Always Acceptable

Bonded contact — gluing two surfaces together as if they are perfectly joined — is the simplest and most computationally efficient contact type. It is appropriate when the two surfaces are genuinely bonded (welded, adhesively bonded, or integrally machined) and never separate. It is not appropriate when the surfaces are in mechanical contact — bolted, pinned, bearing, press-fit — and can separate, slide, or change their load transfer. Bonded contact locks the load path: it forces the load to transfer through the interface regardless of whether the physical interface would actually carry that load. This can hide interface opening, load redistribution, and friction-dependent behaviour. The result looks plausible — a smooth stress contour across the interface — but it may represent a load path that the real structure does not follow.

Bonded contact is a modelling assumption, not a physical reality. If the real interface can open, slide, or change its load transfer, bonded contact gives a single, fixed load path that may be entirely wrong. Always justify a bonded contact assumption against the real interface behaviour.

Contact Modelling: A Practical Checklist

Contact analysis requires careful setup. The following checklist covers the most common pitfalls.

• Is bonded contact justified, or should the interface be modelled with frictional contact? — If the interface can open or slide, bonded contact is wrong
• Is the friction coefficient realistic, and has a sensitivity study been run? — μ is rarely known to better than ±0.1
• Is penetration under control? — Check penetration is a small fraction of element size; adjust contact stiffness if not
• Is the result insensitive to contact stiffness? — If not, increase stiffness or refine mesh
• Is the mesh fine enough to resolve the contact area? — At least 3-4 elements across the contact half-width
• Has the contact model been validated against Hertz or a hand calculation? — For simple geometries, compare to Hertzian theory
• Has the solver log been checked for contact convergence problems? — Read the residual, penetration, and contact status at each iteration

Key takeaways

• Contact changes the load path — where surfaces touch, separate, or slide, the way load transfers through the structure changes fundamentally.
• Bonded contact is not always an acceptable substitute for a physical interface — it locks the load path and can hide the real behaviour.
• Friction coefficient is one of the most sensitive inputs in a contact analysis — a small change can shift the load from sticking to sliding and change the entire stress field.
• Penetration (one surface "passing through" the other) is controlled by contact stiffness — too low and penetration is excessive, too high and convergence is poor.
• Hertzian contact provides closed-form solutions for simple geometries (sphere on flat, cylinder on cylinder) — useful for validation and for problems where a full contact analysis is unnecessary.
• Contact convergence is the most common source of non-linear solver failure — contact chatter, oscillating status, and excessive penetration all prevent equilibrium.
• Contact stiffness (penalty) is a numerical parameter, not a physical one — it must be high enough to limit penetration but low enough to allow convergence, and the result should be insensitive to the value chosen.