Chapter 2 - Mathematical Preliminaries

Abstract

As mentioned in Section 1.2, continuum mechanics must be formulated in terms of quantities that are independent of the coordinate system used to describe the problem. This requires the use of a tensor format to express all variables and relations. The main purpose of this chapter is to present the tensor language and theory necessary for the development of our study. The focus is mainly on Cartesian tensors, but will explore non-Cartesian curvilinear coordinates in the last two sections. Both tensor algebra and tensor calculus for field variables will be presented. Considerable time will be devoted to definitions, proper language, and notational issues that will appear many times in the later chapters. Additional related matrix and tensor mathematical topics will also be discussed. Further reading on the general topics of tensors can be found in Goodbody, A.M., 1982. Cartesian Tensors: With Applications to Mechanics, Fluid Mechanics and Elasticity. Ellis Horwood, New York; Simmons, J.G., 1994. A Brief on Tensor Analysis. Springer, New York; Itskov, M., 2015. Tensor Algebra and Tensor Analysis for Engineers, Fourth ed., Springer, New York.