Mixed double-hemivariational analysis of piezoelectric contact with multiplier-coupled nonmonotone friction

We introduce and analyze a static piezoelectric contact model for Hencky-type materials in which the nonsmooth interface response depends on the tangential displacement, the electric potential jump, and a regularized representation of the normal contact reaction through the Lagrange multiplier. This yields a genuinely multiplier-dependent double nonsmooth coupling on the contact boundary. The model is formulated as a mixed problem in which a Lagrange multiplier is associated with the unilateral Signorini-type contact constraint and, through its regularized representation, also enters the nonsmooth interface response. The resulting boundary functional is expressed in hemivariational form through Clarke’s generalized directional derivative with respect to both the primal interface variables and the regularized multiplier variable. Under standard assumptions on the constitutive operators and suitable growth, compactness, and upper semicontinuity conditions on the coupled superpotential, we prove the existence of weak solutions. The proof combines a truncation procedure, a Fan–KKM argument with a Minty-type reformulation, a uniform multiplier estimate based on an inf–sup property for the normal trace, and a weak compactness argument. We also establish a sequential stability result showing the robustness of weak solutions under perturbations of the loadings and of the foundation data. Finally, a numerical illustration is provided to highlight the influence of the regularized normal reaction on the effective frictional and electrical interface responses.