Unilateral contact complementarity ($g\ge 0,r\ge 0,gr=0$)

For a nonadhesive obstacle, $g$ is the signed separation and $r$ its outward normal contact force. Nonpenetration gives $g\ge 0$, absence of adhesion gives $r\ge 0$, and reaction is possible only at contact, giving $gr=0$. In a small-slope taut-string model, $g=w-f$ and force balance gives $r=p-T{w}^{\prime \prime }$ for downward applied load $p$. The three conditions then become $(w-f)(T{w}^{\prime \prime }-p)=0$, $w\ge f$, $T{w}^{\prime \prime }-p\le 0$. Multiplication by admissible displacement differences and integration gives the associated variational inequality, connecting a local contact law to constrained energy minimization.