Contact normal behaviour
The penalty regularisation approach used to enforce the normal contact constraints requires the penalty factor \(\epsilon\) between the two contact surfaces. The amount of penetration between the two surfaces depends on the magnitude of \(\epsilon\). Higher values decrease the amount of penetration but can lead to ill-conditioning of the global stiffness matrix and to convergence issues. Typically, the stiffness is given by
where \(K^e\) is a representative underlying element stiffness and \(s\) is a scaling factor. Only few open literature present formulas to estimate \(K^e\). In numgeo we approximate \(K^e\) based on the constitutive Jacobian \(\textbf{J}\) of the adjacent element:
The default value of the stiffness factor is \(s=20\).
Normal contact stress law
In the absence of adhesion in the normal direction, the contact normal stress follows a standard penalty formulation:
This means that compressive contact \((g_n<0)\) generates a repulsive contact stress proportional to the penetration, while no stress is transmitted when the surfaces are separated.
If a normal adhesion \(a_n > 0\) is prescribed, limited tensile stresses can be transmitted across the interface. The contact law then becomes:
In this case, the interface can sustain tensile stresses up to the adhesion limit \(a_n\). Once the tensile stress reaches this limit \((t_n = -a_n)\), the contact is lost and the stress drops to zero.
A graphical representation of this is given in Figure 1.
Note
Adhesion is only activated after the surfaces have been in compressive contact at least once. This ensures that tensile forces are not transmitted across interfaces that have never been in physical contact.