Vector and Tensor Notation

In this book, both component and compact form are used for representation of vectors and tensors. Component notation is standard, since cartesian reference axes are used in general. For the compact notation that is used in several chapters to simplify the expressions, the following conventions are used:

1. Geometric vectors are bold faced lower case roman latin letters: e. g., n, t, m.

2. Microplane or, in general, microscopic vectors are denoted by a superimposed arrow, thus n, є, d.

3. Except for a few greek boldmath for classical stresses and strains (<r and є), second-order tensors are represented as bold face upper case roman latin letters, such as E, N, M, A, etc.

4. Fourth-order tensors are represented as bold faced upper case italic latin letters, such as E, С, В, etc.

5. The transformation of a vector by a second-order tensor into another vector is represented by simple juxtaposition: t = trn or t = Tn.

6. The transformation of a second-order tensor by a fourth-order tensor into another second-order tensor is represented by simple juxtaposition as well: cr = Еє, є — Ccr or H = I3N, etc.

7. The inner-product of two vectors or two second-rank tensors is represented by adot, e. g., n-m, n-m, cf ■ 6є, a ■ 6є, T • F, etc. Accordingly, the expression T ■ CS represents the inner product of the second order tensors T and E = CS, the latter being the transformed by the fourth-order tensor C of the second-order tensor S.