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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 [65] and [34].

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 104). 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.

If the independent surface consists of quadratic faces (6-node triangles or 8-node quads) its underlying elements are automatically remeshed into linear elements (C3D20(R) into C3D8I, C3D15 into C3D6 and C3D10 into C3D4) such that linear faces result. This remeshing is also done on the dependent surface if it is defined based on faces. If the dependent surface is defined as a nodal surface no remeshing of the dependent side is performed. In the latter case the middle nodes of the quadratic elements belonging to the slave 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). Furthermore, it can result in an irregular contact stress distribution.

Figure 104: Definition of the dependent nodal surface and the independent element face surface

In CalculiX, contact is modeled by the generation of nonlinear spring elements. To this end, 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 (since the independent faces are linear, either because the unterlying volume elements are linear or because of automatic remeshing). 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 and the *FRICTION 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.

The *SURFACE BEHAVIOR card defines the linear (actually quasi bilinear as illustrated by Figure 106) or exponential normal (i.e. locally perpendicular to the master surface) behavior of the spring element. The pressure $ p$ exerted on the independent face of a contact spring element with exponential behavior is given by

$\displaystyle p=p_0 \exp(\beta d),$ (43)

Figure 105: Exponential pressure-overclosure relationship

Figure 106: Linear pressure-overclosure relationship

where $ p_0$ is the pressure at zero clearance, $ \beta$ is a coefficient and $ d$ is the overclosure (penetration of the slave node into the master side; a negative penetration is a clearance). Instead of having to specify $ \beta$, which lacks an immediate physical significance, the user is expected to specify $ c_0$ which is the clearance at which the pressure is 1 % of $ p_0$. From this $ \beta$ can be calculated:

$\displaystyle \beta=\frac{\ln 100}{c_0}.$ (44)

The pressure curve for $ p_0 =1$ and $ c_0=0.5$ looks like in Figure 105. A large value of $ c_0$ leads to soft contact, i.e. large penetrations can occur, hard contact is modeled by a small value of $ c_0$. Hard contact leads to slower convergence than soft contact. If the distance of the slave node to the master surface exceeds $ c_0$ no contact spring element is generated.

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

$\displaystyle p= K d \left[ \frac{1}{2} + \frac{1}{\pi} \tan^{-1} \left( \frac{d}{\epsilon} \right) \right],$ (45)

were $ \epsilon$ is a small number (in CalculiX the default value is $ 10/K$). The term in square brackets makes sure that the value of p is very small for $ d \le 0$. In general, a linear contact spring formulation will converge more easily than an exponential behavior. The pressure curve for $ K =10^{3}$ and $ \epsilon=10^{-2}$ looks like in Figure 106. A large value of $ K$ leads to hard contact. To obtain good results $ K$ should typically exceed 50 times the E-modulus of the adjacent materials. Notice that for negative overclosure a tensile pressure applies (for $ d \rightarrow - \infty$ ) equal to $ K \epsilon / \pi$. For the default value of $ \epsilon$ this amounts to $ 10/\pi [F]/[L]^2$. For a linear contact spring the distance $ c_0$ beyond which no contact spring element is generated is defined by $ 10^{-3} \sqrt{\text{spring area}}$ if the spring area exceeds zero, and $ 10^{-10}$ otherwise.

The normal spring force is defined as the pressure multiplied by the spring area. The spring area is assigned to the slave nodes and defined by 1/4 (rectangular faces) or 1/3 (triangular faces) of the external faces the slave node belongs to.

Figure 107: Shear stress versus relative tangential displacement

The tangential spring force is defined as the shear stress multiplied by the spring area. The shear stress is a function of the relative displacement $ \Vert
\boldsymbol{t} \Vert$ between the slave node and the master face. This function is shown in Figure 107. It consists of a stick range, in which the shear stress is a linear function of the relative tangential displacement, and a slip range, for which the shear stress is a function of the local pressure only. User input consists of the friction coefficient $ \mu$ which is dimensionless and usually takes values between 0.1 and 0.5 and the tangent of the stick range $ \lambda$ which has the dimension of force per unit of volume and should be chosen about 100 times smaller than the spring constant.

The friction can be redefined in all but the first step by the *CHANGE FRICTION keyword card. In the same way contact pairs can be activated or deactivated in all but the first step by using the *MODEL CHANGE card.

If CalculiX detects an overlap of the contacting surfaces at the start of a step, the overlap is completely taken into account at the start of the step for a dynamic calculation (*DYNAMIC or *MODEL DYNAMIC) whereas it is linearly ramped for a static calculation (*STATIC).

Finally a few useful rules if you experience convergence problems:

next up previous contents
Next: Materials Up: Boundary conditions Previous: Multiple point constraints (MPC)   Contents
guido dhondt 2012-10-06