Mathematical theory of uniform elastic structures

aus der Einleitung: The theory of continuous distributions of material imperfections, dislocations in particular, the origin of which can be traced back to the period of 1950-1967, has been approached from at least two different points of view, i.e., structural dynamics and continuum mechanics. While the pioneering works of Bilby, Eshelby, Kröner, Kondo (see e.g., [B], [Kr]) and others represent a structural point of view the mathematical theory of materially uniform simple elastic bodies of Noll and Wang, [N], [W], [Bl], is firmly based on continuum mechanics notions. Seen as a natural generalization of the structural approach, this theory takes as its fundamental assumption that the presence of imperfections does not modify the general constitutive nature of the elastic material and that the information required to identify and describe smooth distributions of defects can be found in the material, response functional of a given uniform body without introducing any extra parameters or a priori geometries. Following this line of thought, imperfections are seen as being responsible for a breakdown of homogeneity of these constitutive functionals. Geometrie periodicity of the underlying atomic lattice corresponds, on the other hand, to material uniformity and the form of the material symmetry group. Using the language of modern differential geometry the theory shows that for a materially uniform simple elastic body a linear connection can be defined in a manner consistent with the given constitutive relations but not necessarily in a unique way.[...