The tangential behavior of a contact interface, due to friction, can be divided into two states, the stick state and the slip state. The stick state is simply defined by the constraint that there can be no tangential movement between the two contacting boundaries. When the contact enters the slip state, the tangential condition enforces the direction of the relative tangential motion between points xdst and xsrc on the respective side of the interface. On the destination boundary Γdst, the tangential contact condition can be written as a weak equation (the contribution of tangential contact to the virtual work)

where Tt is the friction force. The stick and slip conditions are then enforced by the constitutive model used to define Tt.

Using the same concepts as for the normal contact, and alternative to Equation 3-185 is to use the following weak equation

where Fslip is the slip function (analogue to the yield function in plasticity theory), and λ is friction multiplier. The critical friction force Tt,crit determines when slip occurs, and is defined as

where Tcohe is the cohesion, and Tt,max defines the maximum friction force that is admissible. The friction coefficient μ can either be defined as an arbitrary constant expression or through an exponential decay model in time-dependent studies. The latter model is defined as

where μstat is the static friction coefficient, μdyn is the dynamic friction coefficient, vs is the slip velocity, and αdcf is a decay coefficient.

It is possible to also define the critical friction force through an arbitrary expression. In principle, Tt,crit, or any variable described above, may be defined as a function of any other variable or field. However, the implementation described in the following is only valid as long as it does not dependent on Tt. In other words, the implementation of the friction constitutive model does not allow for hardening behavior of the friction force.

In COMSOL Multiphysics, implementation of Equation 3-187 is made using a backward Euler integration. Together with a penalty regularization, this results in the following set of algebraic equations to solve at increment n+1:

where Tt,n is the friction force at the previous increment. The set of algebraic equations is solved for unknowns Tt,n+1 and λ by introducing a trial state and a return mapping. In doing so, it is assumed that Tt,crit, n+1 is independent of Tt, n+1, which leads to an explicit set of equations for finding Tt, n+1. The trial state is first computed as

The weak contribution added for tangential contact when using the penalty method follows from Equation 3-185, and is given by

or from Equation 3-186 as

where Ttp is the penalized (or augmented) friction, and Tt is a Lagrange multiplier that can be identified as a traction. Alternatively, Equation 3-186 can be augmented, giving the following result

The Lagrange multiplier Tt is defined as the dependent variable of the contact problem and is typically discretized using Lagrange shape functions. There are two methods for solving the above coupled system of equations, Segregated and Fully coupled, as selected in the parent Contact node.

As for the penalty method, Equation 3-187 is implemented using backward Euler integration of the evolution equations, and a trial state and return mapping to solve the nonlinear algebraic equations.

When the Segregated solution method is selected in the parent Contact node, the trial state for outer iteration j+1 is computed as

This scheme then fits in to the algorithmic format described for normal contact where the displacement field u and the Lagrange multipliers associated with contact are solved in a segregated way. Including friction, this algorithm is schematically described as

The segregated algorithm is also used when friction is used together with the Augmented Lagrangian, dynamic formulation.

When the Fully coupled solution method is selected in the parent Contact node, the trial friction force for iteration i+1 is computed as

The weak contribution added for tangential contact when using the Fully coupled solution method follows from Equation 3-189, and is

or similarly from the alternative augmentation in Equation 3-190.

The derivation of Nitsche’s formulation for tangential contact with friction follows similar steps as for the normal contact in The Nitsche’s Method. Replacing the contact pressure vector -Tnn in Equation 3-184 with the friction force Tt, the weak contribution becomes

The definition of Tt is done in a similar way as for the penalty method following Equation 3-188. However, for Nitsche’s formulation of frictional contact, the trial friction force is given as

where Tat is the tangential part of the nominal traction vector. Note that the same penalty factor, pn, is always used for the normal and tangential parts of the contact problem when using the Nitsche method.

In the Slip Velocity subnode to Contact, it is assumed the tangential contact is in a slip state when in contact, and that the relative motion between the source and destination boundaries is known beforehand. That is

where vslip is the prescribed velocity on the destination boundary relative to the source boundary. Given that the rate of the slip is known, the constitutive model for friction in Equation 3-187 is simplified, and it follows that

since the slip velocity and friction force are co-linear. The weak contribution added for tangential contact by the slip velocity follows from Equation 3-185 and is given by

or from Equation 3-186 as

Friction is a dissipative process and the accumulated dissipated energy Wfric can be computed by solving a distributed ODE on the destination boundary: