Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 16

Elastic stress-strain equations are often written in the following form (see, for example, (2.153) and (2.154) given later in the chapter which includes thermal terms).

(2.7)

It is clear that

(2.8)

which may be written as

(2.9)

where

(2.10)

The fourth-order tensor Iijmn can be defined by

(2.11)

where δij denotes the Kronecker delta symbol which has the value unity when i = j and the value zero otherwise. Clearly

(2.12)

The identity tensor defined by (2.11) does not exhibit the same symmetry as the stiffness and compliance tensors, which are such that

(2.13)

It is noted that

(2.14)

indicating that Iijmn≠Ijimn and Iijmn≠Ijinm.

A symmetric fourth-order identity tensor may be defined by

(2.15)

so that

(2.16)

The definition (2.15) is used to define the fourth-order identity tensors used in this book which are denoted by I or I.