We present a framework for generating convex approximations of complex contact models, incorporating experimentally validated models like Hunt & Crossley coupled with Coulomb's law of friction alongside the principle of maximum dissipation. Our approach is robust across a wide range of stiffness values, making it suitable for both compliant surfaces and rigid approximations. We evaluate these approximations across a wide variety of test cases, detailing properties and limitations. We implement a fully differentiable solution in the open-source robotics toolkit, Drake. Our novel hybrid approach enables computation of gradients for complex geometric models while reusing factorizations from contact resolution. We demonstrate robust simulation of robotic tasks at interactive rates, with accurately resolved stiction and contact transitions, supporting effective sim-to-real transfer.