  This work performs the convergence analysis of the polytopal nodal
discretisation of contact-mechanics (with Tresca friction) recently introduced
in [18] in the framework of poro-elastic models in fractured porous media. The
scheme is based on a mixed formulation, using face-wise constant approximations
of the Lagrange multipliers along the fracture network and a fully discrete
first order nodal approximation of the displacement field. The displacement
field is enriched with additional bubble degrees of freedom along the fractures
to ensure the inf-sup stability with the Lagrange multiplier space. It is
presented in a fully discrete formulation, which makes its study more
straightforward, but also has a Virtual Element interpretation. The analysis
establishes an abstract error estimate accounting for the fully discrete
framework and the non-conformity of the discretisation. A first order error
estimate is deduced for sufficiently smooth solutions both for the gradient of
the displacement field and the Lagrange multiplier. A key difficulty of the
numerical analysis is the proof of a discrete inf-sup condition, which is based
on a non-standard $H^{-1/2}$-norm (to deal with fracture networks) and involves
the jump of the displacements, not their traces. The analysis also requires the
proof of a discrete Korn inequality for the discrete displacement field which
takes into account fracture networks. Numerical experiments based on analytical
solutions confirm our theoretical findings
