APPLICATIONS OF TENSOR FUNCTIONS IN THE CONTINUUM-MECHANICS OF ANISOTROPIC MATERIALS

During the last three decades much effort has been devoted to the elab oration of phenomenological theories describing the relation between f orce and deformation in bodies of materials which do not obey either t he linear laws of the classical theories of elasticity or the hydrodyn amics of viscous fluids. Such problems will play a central role for ma thematicians, physicists, and engineers also in, the future [1]. - Mat erial laws and constitutive theories are the fundamental bases for des cribing the mechanical behaviour of materials under multi-axial states of stress involving actual boundary conditions. In solving such compl ex problems, the tensor function theory has become a powerful tool. Th is paper will provide a short survey of some recent advances in the ma thematical modelling of materials behaviour including anisotropy and d amage. The mechanical behaviour of anisotropic solids (materials with orientated internal structures, produced by forming processes and manu facturing procedures, or induced by permanent deformation) requires a suitable mathematical modelling. The properties of tensor functions wi th several argument tensors constitute a rational basis for a consiste nt mathematical modelling of complex material behaviour. This paper pr esents certain principles, methods, and recent successful applications of tensor functions in solid mechanics. The rules of specifying irred ucible sets of tensor invariants, and tensor generators of material te nsors of rank two and Sour are also discussed. Furthermore, it is very important to determine the scalar coefficients in constitutive and ev olutional equations as functions of the integrity basis and experiment al data. A is explained in detail that these coefficients can be deter minded by using tensorial interpolation methods. Some examples for pra ctical use are discussed. Finally, we have carried out our own experim ents in order to examine the validity of the mathematical modelling. - Like applications in solid mechanics, tensor functions also play a si gnificant role in mathematical modelling in fluid mechanics. This pape r, however, is restricted to the mechanical behaviour of solids.