This paper presents a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche's method which was initially designed for Dirichlet's condition. A main interesting characteristic is that this approximation produces well-posed space semi-discretizations contrary to standard finite element discretizations. This paper is then mainly devoted to present an analysis of the semi-discrete problem in terms of consistency, well-posedness and energy conservation, and also to study the well-posedness of some time-marching schemes (theta-scheme, Newmark and a new hybrid scheme). The stability properties of the schemes and the corresponding numerical experiments can be found in a second paper.
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