文档介绍：Chapter 1
Tensor Notation
A Working Knowledge in Tensor Analysis
This chapter is not meant as a replacement for a course in tensor analysis, but it will
provide a working background to tensor notation and algebra.
Cartesian Frame of Reference
Physical quantities encountered are either scalars (., time, temperature, pres-
sure, volume, density), or vectors (., displacement, velocity, acceleration, force,
torque, or tensors (., stress, displacement gradient, velocity gradient, alternating
tensors—we deal mostly with second-order tensors). These quantities are distin-
guished by the following generic notation:
s denotes a scalar (lightface italic)
u denotes a vector (boldface)
F denotes a tensor (boldface)
The distinction between vector and tensor is usually clear from the context. When
they are functions of points in a three-dimensional Euclidean space E,theyare
called fields. The set of all vectors (or tensors) form a normed vector space U.
Distances and time are measured in the Cartesian frame of reference, or simply
frame of reference, F ={O; e1, e2, e3}, which consists of an origin O, a clock, and
an orthonormal basis {e1, e2, e3}, see Fig. ,
ei · ej = δij ,i,j= 1, 2, 3 ()
where the Kronecker delta is defined as
1,i= j,
δij = ()
0,i= j.
We only deal with right-handed frames of reference (applying the right-hand rule:
the thumb is in direction 1, and the forefinger in direction 2, the middle finger lies
in direction 3), where (e1 × e2) · e3 = 1.
N. Phan-Thien, Understanding Viscoelasticity, Graduate Texts in Physics, 1
DOI -3-642-32958-6_1, © Springer-Verlag Berlin Heidelberg 2013
2 1 Tensor Notation
Fig. Cartesian frame of
referen