on Cartesian coordinates, this is written in the shorter form x=xkik where a summation over values k = 1, 2, 3 is implied when a suffix is repeated (k in this example). Any vector v may be written as v=v1i1+v2i2+v3i3, or as v=vkik when using tensor notation. The scalar quantities vk, k = 1, 2, 3, are the components of the vector v with values depending on the choice of coordinates. The magnitude of the vector v is specified by
and its value is independent of the system of coordinates that is selected. The magnitude is, thus, an invariant of the vector.
The unit vector in the direction of the vector v is specified by v/|v|. Examples of vectors that occur in the physical world are forces, displacements, velocities and tractions.
2.3 Tensors
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:
or, more compactly, using tensor notation in the form
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
A fourth-order tensor T is expressed in terms of the unit vectors of the coordinate system as follows
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.
2.3.1 Fourth-order Elasticity Tensors
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).
It is clear that
which may be written as
where
The fourth-order tensor Iijmn can be defined by
where δij denotes the Kronecker delta symbol which has the value unity when i = j and the value zero otherwise. Clearly
The identity tensor defined by (2.11) does not exhibit the same symmetry as the stiffness and compliance tensors, which are such that
It is noted that
indicating that Iijmn≠Ijimn and Iijmn≠Ijinm.
A symmetric fourth-order identity tensor may be defined by
so that
