Tangential Contact with Friction
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)
(3-203)
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 3-203 is to use the following weak equation
(3-204)
Constitutive Model for Friction
The constitutive model for friction, including stick and slip, is based on the Coulomb law. In a general framework similar to plasticity theory, the model can be expressed as
(3-205)
where Fslip is the slip function (analogue to the yield function in plasticity theory), and λ is a 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.
The Penalty Method
In COMSOL Multiphysics, implementation of Equation 3-205 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
and the return mapping gives the current value of the friction force
(3-206)
The weak contribution added for tangential contact when using the penalty method follows from Equation 3-203, and is given by
or from Equation 3-204 as
The Augmented Lagrangian Method
The augmented Lagrangian formulation for tangential contact in COMSOL Multiphysics is based on an integral formulation that in a weak sense enforces the stick and slip conditions exactly over the contacting boundaries. It is based on the following augmentation of Equation 3-203
(3-207)
where Ttp is the penalized (or augmented) friction, and Tt is a Lagrange multiplier that can be identified as a traction. Alternatively, Equation 3-204 can be augmented, giving the following result
(3-208)
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 3-205 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
and the return mapping gives the current value of the penalized friction force
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
1
Initialize increment n + 1 by setting the outer iteration counter j = 0, and the initial values for dependent variables , , and
2
Compute the displacement uj+1 by solving for example
3
Compute the Lagrange multipliers Tn,j+1 and Tt,j+1 by solving
4
If convergence, continue to the next increment, else return to step 2
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 important difference here lies in that for the coupled method, the trial state is computed as the change from the previous nonlinear iteration every time the coupled problem is solved. Knowing the trial state, the return mapping is performed similar to the other methods.
The weak contribution added for tangential contact when using the Fully coupled solution method follows from Equation 3-207, and is
or similarly from the alternative augmentation in Equation 3-208.
The Nitsche’s Method
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 3-202 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 3-206. 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.
Prescribed Slip Velocity
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 3-205 is simplified, and it follows that
since the slip velocity and friction force are collinear. The weak contribution added for tangential contact by the slip velocity follows from Equation 3-203 and is given by
or from Equation 3-204 as
Dissipation
Friction is a dissipative process and the accumulated dissipated energy Wfric can be computed by solving a distributed ODE on the destination boundary: