For geometric non-linearity, we know that as the geometry changes during the analysis, the global stiffness matrix changes since it already is a function of geometry of the structure.

What I want to ask is that how does including a frictional contact within my FEA causes a change in global stiffness matrix so that the analysis becomes non-linear?

According to the basic FEA courses I have taken, the global stiffness matrix doesn't appear to be a function of the contact. So is the global stiffness matrix change different from the condition where geometric non-linearity is present, as compared to where the contact non-linearity (like frictional) is present?

  • $\begingroup$ Friction is a contact "force". It is not a geometric parameter, nor depends on the geometry of an element but the contact surface. $\endgroup$
    – r13
    Jul 12, 2021 at 15:59
  • $\begingroup$ Thank you telling me that, but that is not what I was asking for. $\endgroup$ Jul 12, 2021 at 16:43
  • $\begingroup$ The reason I said that is because I don't think friction force has anything to do with the global stiffness matrix, which stays the same whether the force is linear or non-linear. I could be wrong by misunderstanding the context or the question though. $\endgroup$
    – r13
    Jul 12, 2021 at 16:55
  • $\begingroup$ @r13 it depends how you model "friction". In more complex models than Coulomb friction, there is some "stiffness" or "flexibility" between the two surfaces rather than an instantaneous change from "stuck" to "slipping". $\endgroup$
    – alephzero
    Jul 12, 2021 at 19:13
  • $\begingroup$ It is not about the frictional force, it is about the contact's nature. I can even apply a frictionless contact, or rough contact (where the two faces cannot slide relative to each other), but at the same time the nature of these contacts still makes the analysis non-linear. Whereas Bonded contacts are considered as Linear. As far as my knowledge goes, the non-linearity basically means that the stiffness matrix has changed. So I am pretty much sure that the contact non-linearity has definitely something to do with the stiffness matrix. $\endgroup$ Jul 12, 2021 at 19:13

2 Answers 2


It does not directly have anything to do with the stiffness matrix. But the geometric constraint caused by the friction is a external force that is affected by the deformation of the object.*

Since the contact changes as the object deforms then the frictional force changes by the deformation. Again it becomes nonlinear as the problem to be solved changes over the span of the solution. In fact there need not be a friction at all just a changing contact causes the same problem.

Overall, the answer is the same whatever you add to the FEA. If it isnt a simple, small angles approximation case with simple constant forces and constrains then it is nonlinear. Linear is the special case. Most if, not all, nontrivial realword cases are nonlinear. But it can still be approximately good to do a linear FEM analysis, since nonlinear cases are hard to model correctly.

* this does not mean i can not model the a change to the stifness matrix that approximates this in some cases. But its more a varying load

  • $\begingroup$ Excellent answer. I didn't think about deformation caused friction or vice versa. $\endgroup$
    – r13
    Jul 12, 2021 at 18:45

"Geometric nonlinearity" is a confusing term, because it can include two separate effects.

In general you can write the global stiffness matrix of the structure as the sum of three terms: $K = K_e + K_\sigma + K_L$.

$K_e$ is the elastic stiffness, resulting from the stress-strain relationships in the material. For a linear elastic material it constant.

$K_\sigma$ is the stress stiffness which models the fact that the internal forces in the body (arising from the stress distribution) can do work when the body deforms. A simple example is a stretched string. If you pull a guitar string sideways to pluck it, the deformed shape is (approximately) two straight segments which are not collinear, each with the same tension as in the straight string. The resultant force of the two non-collinear tension forces opposes the sideways deflection. That is equivalent to a stiffness term which is proportional to the tension in the string.

$K_L$ is the "load stiffness" which is similar to the stress stiffness, but arises because the external loads on the structure may depend on the deformations. For example a pressure load acting on structure acts normal to the deformed shape of the surface, not to its original shape. Another example is a structure rotating at constant angular velocity about a fixed axis, where the centripetal force acting on each particle changes both direction and magnitude as the structure moves (the magnitude depends on the distance from the axis of rotation, and the direction always points towards the axis).

An example of $K_L$ is a model of a simple pendulum. The only stiffness term is $K_L$ caused by the fact the the external force (weight) acts at a different position when the structure moves.

A linear analysis with small strains and small displacements ignores $K_\sigma$ and $K_L$.

A "geometric nonlinear" analysis always includes $K_\sigma$ but usually ignores $K_L$. That corresponds to assuming the the strains are (infinitesimally) small, but the rigid body motion (specifically, the rigid body rotation) of the structure can be arbitrarily large.

I don't know any common term used for "including or ignoring $K_L$" In some situations (e.g the dynamics of rotating machinery) including it is essential. In other situations it is negligible and ignored.

A friction force between two surfaces is therefore a (nonlinear) load stiffness. If the surfaces move relative to each other, the friction forces do work and the direction of the friction force depends on the relative motion. But the magnitude of the friction force also depends on the stress component normal to the surface, and when the surfaces are not sliding there is no work done.

So it isn't very practical to include these terms in the global stiffness matrix somehow. In practice you would apply a nonlinear constraint to the model of the structure to create the correct friction forces in the model.

  • $\begingroup$ Changing loads are typically called follower forces in contexts other than contact. $\endgroup$ Jul 12, 2021 at 20:37
  • $\begingroup$ Good explanations. However, my impression is all the stiffness terms are related to the applied force, stress, and displacement, which is the basis of defining the stiffness matrix. Am I missing something that is directly related to the change in material geometric properties (A, I)? $\endgroup$
    – r13
    Jul 12, 2021 at 20:43

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