Frictional contact models with local compliance: semismooth formulation

A complementarity‐based, 3‐dimensional frictional contact model with local compliance and damping was introduced in the Ph.D. thesis [45] and the paper [47] and was subsequently studied extensively in [31, 49]. In this paper, we examine a variant of this model where there is no damping in the normal contact forces but there is coupled stiffness between the normal and tangential forces via body deformations. We show that this frictional contact model admits a formulation as an ordinary differential equation with a boundedly Lipschitz continuous, albeit implicitly defined, semismooth right‐hand side with global linear growth. Several major consequences follow from such a formulation: (a) existence and uniqueness of a continuously differentiable solution trajectory originated from an arbitrary initial state, (b) finite contact forces that are semismooth functions of the system state, (c) semismooth dependence of the trajectory on the initial state, and (d) convergence of a shooting method for solving two‐point boundary problems. The derived results are valid for both a dynamic model and a quasistatic model, the latter being one in which inertia effects are ignored, and for a broad class of friction cones that include the well‐known quadratic Coulomb cone and its polygonal approximations. Part II of this work establishes the absence of Zeno states in such a frictional contact model.