Abstract The Reynolds-Hertz equations for elastohydrodynamic lubrication with an undetermined boundary of the contact region are formulated in the framework of a nonlinear varational inequality due to the constraint that the pressure must be nonnegative. The present study shows that the operator of the variational inequality is bounded, coercive, pseudomonotone and continuous. Then the existence of solutions to the variational inequality is proved and it is shown that the solutions are the weak solutions of the Reynolds-Hertz equations. The variational inequality is regularized by a penalty method and the penalty method is proved to be convergent. A detailed study shows that there exists classical solutions of one-dimensional line contact problems. To show the potential of penalty method, an additional study shows the convergence of sequences of finite-dimensional approximations. A priori error estimates for finite element solutions are derived. Numerical experiments confirm the predicted rates of convergence for a range of applied loads.
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