Tensor Analysis, Invariants, and Representations

Abstract

In this chapter, the concepts necessary to formulate the vector and tensor representations needed in the construction of viscoelastic and turbulent constitutive equations are introduced. The material is generally adapted from the review article of Spencer (Theory of invariants, pp. 239–353, 1971) who summarized the research work of Ronald Rivlin and colleagues, and with the goal to present to the reader a description more familiar to a fluid dynamicist. Probably the most general presentation of the theory leading to the appropriate form of constitutive equations resulting from invariance conditions is given in the book by Smith (Constitutive equations for anisotropic and isotropic materials, Mechanics and physics of discrete systems, vol. 3, 1994); however such a general presentation is not needed here and would be more mathematically intense than desired. Nevertheless, it is assumed at the outset that the reader, at a minimum, is familiar with elements of tensor and matrix algebra. The numerous citations within this chapter will guide the reader to the relevant publications.