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### Contact

Contact is a strongly nonlinear kind of boundary condition, preventing bodies to penetrate each other. The contact definition implemented in CalculiX is a node-to-surface penalty method based on a pairwise interaction of surfaces. For details on the penalty method the reader is referred to [61] and [33].

Each pair of interacting surfaces consists of a dependent surface and an independent surface. The dependent surface may be defined based on nodes or element faces, the independent surface must consist of element faces (Figure 84). Usually, the mesh on the dependent side is finer than on the independent side. As many pairs can be defined as needed. A contact pair is defined by the keyword card *CONTACT PAIR.

Figure 84: Definition of the dependent nodal surface and the independent element face surface

For each node on the dependent surface, a face on the independent surface is localized such that a perpendicular line on a point within the face contains the node. If such is face is found a nonlinear spring element is generated consisting of the dependent node and all vertex nodes belonging to the independent face. Depending of the kind of face the contact spring element contains 4 or 5 nodes. The properties of the spring are defined by a *SURFACE INTERACTION definition, whose name must be specified on the *CONTACT PAIR card.

The user can determine how often during the calculation the pairing of the dependent nodes with the independent faces takes place. If the user specifies the parameter SMALL SLIDING on the *CONTACT PAIR card, the pairing is done once per increment. if this parameter is not selected, the pairing is checked every iteration. This is useful if the sliding is particularly large.

The *SURFACE INTERACTION keyword card is very similar to the *MATERIAL card: it starts the definition of interaction properties in the same way a *MATERIAL card starts the definition of material properties. Whereas material properties are characterized by cards such as *DENSITY or *ELASTIC, interaction properties are denoted by the *SURFACE BEHAVIOR card. All cards beneath a *SURFACE INTERACTION card are interpreted as belonging to the surface interaction definition until a keyword card is encountered which is not a surface interaction description card. At that point, the surface interaction description is considered to be finished. Consequently, an interaction description is a closed block in the same way as a material description, Figure 3.

Right now, the *SURFACE BEHAVIOR card is the only surface interaction description card. It defines the linear (actually quasi bilinear as illustrated by Figure 86) or exponential behavior of the spring element. The pressure exerted on the independent face of a contact spring element with exponential behavior is given by

| (43)

Figure 85: Exponential pressure-overclosure relationship

Figure 86: Linear pressure-overclosure relationship

where is the pressure at zero clearance, is a coefficient and is the overclosure (penetration of the slave node into the master side; a negative penetration is a clearance). Instead of having to specify , which lacks an immediate physical significance, the user is expected to specify which is the clearance at which the pressure is 1 % of . From this can be calculated:

| (44)

The pressure curve for and looks like in Figure 85. A large value of leads to soft contact, i.e. large penetrations can occur, hard contact is modeled by a small value of . Hard contact leads to slower convergence than soft contact.

In case of a linear contact spring the pressure-clearance relationship is given by

| (45)

were is a small number (in CalculiX the default value is ). The term in square brackets makes sure that the value of p is very small for . In general, a linear contact spring formulation will converge more easily than an exponential behavior. The pressure curve for and looks like in Figure 86. A large value of leads to hard contact. To obtain good results should typically exceed 50 times the E-modulus of the adjacent materials. Notice that for negative overclosure a tensile pressure applies equal to . For the default value of this amounts to .

Finally a few useful rules if you experience convergence problems:

- Try SMALL SLIDING first, and then large sliding, if applicable.
- Try a linear pressure-overclosure relationship first, with a stiffness constant about 50 to 100 times Young's modulus of the adjacent materials.
- Use linear elements in the contact area. Contact problems with quadratic elements do not converge as well. You may connect the linear elements with underlying quadratic elements with multiple point constraints.

Notice that in CalculiX, middle nodes of quadratic elements belonging to the slave and master contact surfaces are internally connected to their neighboring vertex nodes by means of multiple point constraints (i.e. their displacements are the mean of the displacements of the neighboring end nodes). This makes the contact area stiffer (similar to using linear elements for bending).

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