Anwendungen der getrennt konvexen Funktionale in der Mechanik

• Applications of separately convex functionals in mechanics

Ban, Paul Michael; Weichert, Dieter (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2007, 2008)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2007

Abstract

In the present work, the properties of separately convex functionals are investigated and some applications to mechanics are given. Several propositions concerning the theoretical basis of separately convex functionals are formulated and proved and the relation to the class of convex functionals is explored. Characterization theorems for the bipotential are given. Relations to other convexity notions, as well as consistent definitions of separately convex hulls are studied in the sequel. The biaffine hull is introduced as a natural generalization of the affine hull from convex analysis. Sufficient conditions for the uniqueness and typical existence theorems are analyzed. Fundamental differences between convex and separately convex functionals concerning the number and the kind of local minima are displayed. The role of separately convex functionals in the formulation of material models, especially of the ISM model, is explored in chapter 8. Shakedown theorems for a certain class of materials, satisfying the hypotheses of the ISM model, are proved. Further material models, belonging to the class of ISM models, are presented in chapter 10. Finally, algorithms for the numerical treatment of the problem are deduced.