Progress and puzzles in nonlinear elasticity -- Quasiconvex envelopes in nonlinear elasticity -- Anisotropie polyconvex energies -- Construction of polyconvex energies for non-trivial anisotropy classes -- Applications of anisotropic polyconvex energies: thin shells and biomechanics of arterial walls -- Phase transitions with interfacial energy: convexity conditions and the existence of minimizers -- Nematic elastomers: modelling, analysis, and numerical simulations -- Applications of polyconvexity and strong ellipticity to nonlinear elasticity and elastic plate theory -- ?-convergene e for a geometrically exact Cosserat shell-model of defective elastic crystals.

Generalized convexity conditions play a major role in many modern mechanical applications. They serve as the basis for existence proofs and allow for the design of advanced algorithms. Moreover, understanding these convexity conditions helps in deriving reliable mechanical models. The book summarizes the well established as well as the newest results in the field of poly-, quasi and rank-one convexity. Special emphasis is put on the construction of anisotropic polyconvex energy functions with applications to biomechanics and thin shells. In addition, phase transitions with interfacial energy and the relaxation of nematic elastomers are discussed.