  In the field of nonlinear mechanics, many challenging problems (e.g.
plasticity, contact, masonry structures, nonlinear membranes) turn out to be
expressible as conic programs. In general, such problems are non-smooth in
nature (plasticity condition, unilateral condition, etc.), which makes their
numerical resolution through standard Newton methods quite difficult. Their
formulation as conic programs alleviates this difficulty since large-scale
conic optimization problems can now be solved in a very robust and efficient
manner, thanks to the development of dedicated interior-point algorithms. In
this contribution, we review old and novel formulations of various non-smooth
mechanics problems including associated plasticity with nonlinear hardening,
nonlinear membranes, minimal crack surfaces and visco-plastic fluid flows.
