Poly-, Quasi- and Rank-One Convexity in Applied Mechanics
September 24, 2007 — September 28, 2007
Coordinators:
- Joerg Schroeder (Univ. Duisburg-Essen, Essen, Germany)
- Patrizio Neff (Techn. Univ. Darmstadt, Darmstadt, Germany)
A variety of mechanical applications is associated to “generalized” convexity conditions. This includes aspects of the modeling of fracture and self contact, the status of elasticity with respect to atomistic models, the understanding of microstructure induced by phase transformations, the passage from three-dimensional elasticity to models of rods and shells, mechanical applications in the field of biomechanics, the interpretation of various convexity conditions for fibrous materials including networks and fabrics, mechanical applications in carbon nanotube modeling, and finite-element formulation of nematic liquid crystal elastomers.
Related to the above mentioned problems, the conditions of polyconvexity (Ball 1977), quasiconvexity (Morrey 1952) and rank-one convexity (Legendre-Hadamard ellipticity) play a major role. In contrast to some well-known isotropic models the construction of anisotropic polyconvex functions remains an open field of research and is treated in the course. Some well-known material models do not fulfill the quasiconvexity inequality. In these cases the construction of quasiconvex hulls is an appropriate venue to take. It may lead to some physical restrictions limiting the applicability of the underlying models. Applications of this concept are discussed in the course for the St. Venant-Kirchhoff model and for nematic liquid crystals. Furthermore, if we focus on material models satisfying the Legendre-Hadamard condition, the construction of rank-one-convex functions is another important strategy. Thus, general construction rules to do this are proposed.
The course is addressed to master students, doctoral students, post docs and experienced researchers in engineering, applied mathematics and science who wish to broaden their knowledge in generalized convexity conditions and their impact in applied mechanics, particularly with regard to the constitutive modeling of complex material behavior as well as on the consequences of “validity” (existence) of solutions obtained within direct variational methods.