Nonsmooth mechanics and convex optimization The principal subject of this book is to discuss how to make use of theory and algorithms of optimization for treating problems in applied mechanics in a comprehensive way. Particular emphasis however is to be put on the two terms involved in the title \\nonsmooth and \\convex which distinguish the methodology of the present work from the conventional methods in applied and computational mechanics. This book consists of four parts dealing with the abstract framework of convex analysis for comprehensive treatment of nonsmooth mechanics (Chapters 1 3) demonstration of our methodology through in depth study of a selected class of structures (Chapters 4 5) numerical algorithms for solving the problems in nonsmooth mechanics (Chapters 6 7) and the application of theoretical and numerical methodologies to the problems covering many topics in nonsmooth mechanics (Chapters 8 11). After more than three decades since the work by Duvaut Lions the author hopes that the present work serves as a new bridge between nonsmooth mechanics of deformable bodies and modern convex optimization. Although this book is primarily aimed at mechanicians it also provides applied mathematicians with a successful case study in which achievements of modern mathematical engineering are fully applied to real world problems. Basic and detailed exposition of the notion of complementarity and its links with convex analysis including many examples taken from applied mechanics may open a new door for the communities of applied and computational mechanics to a comprehensive treatment of nonsmoothness properties This book presents a methodology for comprehensive treatment of nonsmooth laws in mechanics in accordance with contemporary theory and algorithms of optimization. The author deals with theory and numeiral algorithms comprehensively providing a new perspective n nonsmooth mechanics based on contemporary optimization. Covering linear programs; semidefinite programs; second order cone programs; complementarity problems; optimality conditions; Fenchel and Lagrangian dualities; algorithms of operations research and treating cable networks; membranes; masonry structures; contact problems; plasticity this is an ideal guide of nonsmooth mechanics for graduate students and researchers in civil and mechanical engineering and applied mathematics