Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 15

The tensors to be used in the book are either second order or fourth order. Tensors are usually physical quantities that are independent of the coordinate system that is used to describe their properties. For the given coordinate system having unit vectors i1, i2and i3, a second-order tensor t is expressed in terms of the unit vectors as follows:

(2.3)

or, more compactly, using tensor notation in the form

(2.4)

where summation over values 1, 2, 3 is implied by the repeated suffices j and k. The quantities tjk are known as the components of a second-order tensor with values depending on the choice of coordinates. There are three independent invariants of second-order tensors which can be expressed in a variety of forms, the simplest being

(2.5)

A fourth-order tensor T is expressed in terms of the unit vectors of the coordinate system as follows

(2.6)

where summation over values 1, 2, 3, is implied by the repeated suffices i, j, k and l. The quantities Tijkl are known as the components of a fourth-order tensor with values depending on the choice of coordinates.