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Contact Inequality Constraints

Figure (3.1) shows schematically the problem under consideration. Two generic bodies are shown and denoted as contactor and target. The contactor contains the finite element boundary nodes that comes into contact with the target elements. The target and contactor could be within a single body that folds during deformation in a fashion that invokes contact between parts of its boundaries, see Figure (3.2). The displacement compatibility between the two bodies requires that no material overlap can occur along the region of contact. As a result, contact forces are developed that act along the region of contact upon the target and the contactor. The force transmissibility condition requires that these forces to be equal and opposite. In addition, it requires that normal contact forces can only exert a compressive action, and the tangential contact forces have to satisfy a law of frictional resistance. Coulomb's law of friction is used in the current formulation to regulate the friction forces. It requires that the friction force has to be greater than $\nu_s F_n$ for a slippage to occur. Once slippage has occurred, the friction force is set equal to $\nu_s F_n$, where $\nu_s$ and $\nu_d$ are the static and dynamic coefficients of friction, respectively. These two conditions impose inequality constraints on the system that mathematically represent the stress analysis model of the contact bodies. These constraints are given as

$\displaystyle \mbox{No penetration condition:}$$\textstyle {\bf P}_i.{\bf n}\geq 0$ (102)
$\displaystyle \mbox{No tension condition:}$Fn>0 (103)
$\displaystyle \mbox{Coulomb's law of friction condition:}$$\textstyle F_t \leq \nu_s F_n$$\displaystyle \mbox{for no slippage}$(104)
 $\textstyle F_t=\nu_d F_n$$\displaystyle \mbox{for slippage}$(105)

where ${\bf P}_i$ is the penetration vector of contactor node number i into the target's element, ${\bf n}$ is the normal to the target element, Fn is the compressive normal contact force and Ft is the tangential contact force. It should be noted that Equation (3.1) ensures that the contactor nodes can not be within the region of the target body, but the target node can be inside or outside the contactor body as shown in Figure (3.3). This introduces some discretization error which may be reduced by using a finer mesh. In addition, by using Coulomb's law of friction, the elasticity between the particles in contact is neglected and a rigid plastic contact behavior is assumed.
Figure 3.1: Geometry of a General Contact Problem

Figure 3.2: Single Body Contact

Figure 3.3: Discretization Error in the Modeling of Some Contact Problems

next up previous contents
Next: Treatment of Inequality Constraints Up: Contact Mechanics Previous: Contact Mechanics
A. Zeiny