The derivative of a parameterized mechanical contact problem with a Tresca's friction law involves Signorini unilateral conditions

Samir Adly (1) , Loïc Bourdin (1) , Fabien Caubet (2)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a mechanical contact problem involving a Tresca's friction law. The weak formulation of this problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space $H^1(\Omega)$ and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability of the parameterized Tresca's solution and that its derivative satisfies Signorini unilateral conditions. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.

Mots clés: mechanical contact problems; Tresca's friction law; Signorini unilateral conditions; variational inequality; convex subdifferential; proximal operator; sensitivity analysis; twice epi-differentiability

Domaines: Optimisation et contrôle [math.OC]

Samir Adly, Loïc Bourdin, Fabien Caubet. The derivative of a parameterized mechanical contact problem with a Tresca's friction law involves Signorini unilateral conditions. 2019. ⟨hal-02368180⟩