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Contact Analysis

When two bodies come into contact, a contact force \(t_c\) is transmitted through the contact surface. The principle of virtual work is rewritten as follows.

\[\begin{equation} \int^{t'}_{^{t'}v}\,^{t'} \sigma : \delta^{t'} A_{(L)} d^{t'}v = \int^{t'}_{^{t'}S_{t}}\,^{t'} \overline{t} \cdot \delta u d^{t'}s+ \int^{t'}_{V} \overline{b} \cdot \delta u d^{t'}v + \int^{t'}_{^{t'} S _{c}}t_{c}[\delta u^{(1)} - u^{(2)}] \label{eq:2.2.70} \end{equation}\]

Here, \(S_c\) denotes the contact area, and \(u^{(1)}\) and \(u^{(2)}\) denote the displacements of contact bodies 1 and 2, respectively.

In contact analysis, surfaces that may come into contact are specified as a pair. One surface in the pair is designated the master surface and the other the slave surface. In this master-slave analysis method, the contact constraint conditions are assumed as follows.

  1. A slave node does not penetrate the master surface.
  2. When contact occurs, the slave node is located at the contact position, and the master and slave surfaces transmit contact force and friction force to each other through this contact point.

Discretizing the last term of Eq.\(\eqref{eq:2.2.70}\) by the finite element method gives the following equation.

\[\begin{equation} \int^{t'}_{^{t'}S_{c}} t_c [\delta u^{(1)} - \delta u^{(2)}] \approx \delta UK_c \Delta U + \delta UF_c \label{eq:2.2.71} \end{equation}\]

Here, \(K_c\) and \(F_c\) denote the contact stiffness matrix and contact force, respectively. The finite element formulations of the total Lagrangian and updated Lagrangian methods including contact constraints are as follows.

\[\begin{equation} \delta U^T ( ^t_0 K_L + ^t_0 K_{NL} + K_c ) \Delta U = \delta U^T {_0^{t'} F} - \partial U^T {^t_0 Q} + \delta UF_c \label{eq:2.2.72} \end{equation}\]
\[\begin{equation} \delta U^T (^t_t K_L + ^t_t K_{NL} + K_c ) \Delta U = \delta U^T {_{t} ^{t'} F} - \partial U^T {^t_t Q}+ \delta UF_c \label{eq:2.2.73} \end{equation}\]