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**Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials.**
*(English)*
Zbl 0974.58019

Summary: We consider a class of evolutionary variational inequalities arising in quasistatic frictional contact problems for linear elastic materials. We indicate sufficient conditions in order to have the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data, respectively. The existence of the solution is obtained using a time-discretization method, compactness and lower semicontinuity arguments. In the study of the discrete problems we use a recent result obtained by the authors [Adv. Math. Sci. Appl. 10, 103-118 (2000; Zbl 0977.47057)]. Further, we apply the abstract results in the study of a number of mechanical problems modeling the frictional contact between a deformable body and a foundation. The material is assumed to have linear elastic behavior and the processes are quasistatic. The first problem concerns a model with normal compliance and a version of Coulomb’s law of dry friction, for which we prove the existence of a weak solution. We then consider a problem of bilateral contact with Tresca’s friction law and a problem involving a simplified version of Coulomb’s friction law. For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.

### MSC:

58E35 | Variational inequalities (global problems) in infinite-dimensional spaces |

35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |

74M15 | Contact in solid mechanics |

74M10 | Friction in solid mechanics |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

49J40 | Variational inequalities |