Anwendungen von Tensorfunktionen in der Kontinuumsmechanik anisotroper Materialien

TLDR: A short survey of recent advances in the mathematical modelling of materials behavior including anisotropy and damage can be found in this paper, where the authors present some principles, methods, and recent successful applications of tensor functions in solid mechanics.

Abstract: During the last three decades much effort has been devoted to the elaboration of phenomenological theories describing the relation between force and deformation in bodies of materials which do not obey either the linear laws of the classical theories of elasticity or the hydrodynamics of viscous fluids. Such problems will play a central role for mathematicians, physicists, and engineers also in the future [1]. - Material laws and constitutive theories are the fundamental bases for describing the mechanical behaviour of materials under multi-axial states of stress involving actual boundary conditions. In solving such complex problems, the tensor function theory has become a powerful tool. This paper will provide a short survey of some recent advances in the mathematical modelling of materials behaviour including anisotropy and damage. The mechanical behaviour of anisotropic solids (materials with orientated internal structures, produced by forming processes and manufacturing procedures, or induced by permanent deformation) requires a suitable mathematical modelling. The properties of tensor functions with several argument tensors constitute a rational basis for a consistent mathematical modelling of complex material behaviour, This paper presents certain principles, methods, and recent successful applications of tensor functions in solid mechanics. The rules of specifying irreducible sets of tensor invariants, and tensor generators of material tensors of rank two and four are also discussed. Furthermore, it is very important to determine the scalar coefficients in constitutive and evolutional equations as functions of the integrity basis and experimental data. It is explained in detail that these coefficients can be determinded by using tensorial interpolation methods. Some examples for practical use are discussed. Finally, we have carried out our own experiments in order to examine the validity of the mathematical modelling. - Like applications in solid mechanics, tensor functions also play a significant role in mathematical modelling in fluid mechanics. This paper, however, is restricted to the mechanical behaviour of solids.