A numerical procedure is developed for the sensitivity analysis with respect to problem parameters or design variables for contact problems with the Coulomb friction. Using the sensitivity results with respect to a loading scale parameter, an efficient incremental analysis technique for the frictional contact is presented. Also, a shape design sensitivity analysis for the frictional contact problem is proposed using the material derivative concept and the direct differentiation method. The design of initial separation for minimizing the maximum contact stress is considered.
For completeness, a general kinematic description with an updated Lagrangian formulation is adopted and finite element method is used for discretization. Kinematic variables such as contact gap and slip have complementarity relations with contact force and friction force. The resulting complementarity becomes linear for a two dimensional contact with the Coulomb friction. Since the linear complementarity problem(LCP) can be transformed to an equivalent minimization problem, sensitivity formulas abtained from the sensitivity analysis for a parametric optimal design(POD) are applicable. It is shown that the sensitivity analysis results of the basic solutions are obtained by pivoting sensitivity vectors of active constraints in the equivalent minimization problem according to the modified simplex solution scheme of LCP. The variations of the non-basic solutions are zero because of the fact that the set of active constraints is not changed for small perturbations of problem parameters. For a degenerate case, different sensitivity matrices are obtained which denote directional derivatives at the solution point.
In frictional contact problem, an incremental analysis is inevitable because of the nonlinearity and the non-uniqueness of the solution to a given level of external load. Since the physical state is unique for a given path of loading, the problem is how to guarantee the correct path numerically. An incremental analysis is proposed, which determines the size of an incremental step using the sensitivity analysis with respect to loading scale parameter. Several illustrative numerical examples have been tested. From a problem with loading and unloading, it is shown that a non-uniqueness solution can be obtained when incremental steps selected differently. A layer pressed against a half-plane by a uniform pressure and subjected to a tangential force varying periodically in time is simulated and compared with the analytical solution. The computational results are in good agreements with the analytical ones and residual frictional stress effects such as shake down phenomena have been shown. As an application of the method, a valve-cotter system of a motorcycle engine subject to a periodic loading has been analyzed as an axisymmetric three body contact problem.
The sensitivity of contact stresses with the change of the contact contour is also derived. For numerical examples, initial separation between a piston rod and a pin has been obtained to minimize the maximum contact pressure. And, initial separation between a valve and a cotter has been designed to reduce the stress concentration considering element stress functionals. The contact stress is found highly sensitive to the change of initial gap. The stiffness matrices are assumed not changing even with the contour variation due to its smallness.
With the correct determination of the step size, the present incremental analysis technique has been shown to be efficient and capable of following complex response histories. The step size determination in the case of material and other geometric nonlinearities such as large elasto-plastic deformation remains as a future topic.