ABSTRACT
CHAPTER 1
TENSOR PRELIMINARIES
1.1. Vectors
An orthonormal basis for the three-dimensional Euclidean vector space is a
set of three orthogonal unit vectors. The scalar product of any two of these
vectors is
e
i
e
j
= Æ
ij
=
(
1; if i = j;
0; if i 6= j;
(1.1.1)
Æ
ij
being the Kronecker delta symbol. An arbitrary vector a can be decom-
posed in the introduced basis as
a = a
i
e
i
; a
i
= a e
i
: (1.1.2)
The summation convention is assumed over the repeated indices. The scalar
product of the vectors a and b is
a b = a
i
b
i
: (1.1.3)
The vector product of two base vectors is dened by
e
i
e
j
=
ijk
e
k
; (1.1.4)
where
ijk
is the permutation symbol
ijk
=
>
<
>
:
1; if ijk is an even permutation of 123;
1; if ijk is an odd permutation of 123;
0; otherwise:
(1.1.5)
The vector product of the vectors a and b can consequently be written as
a b =
ijk
a
i
b
j
e
k
: (1.1.6)
The triple scalar product of the base vectors is
(e
i
e
j
) e
k
=
ijk
; (1.1.7)
so that
(a b) c =
ijk
a
i
b
j
c
k
=
a
b
c
a
b
c
a
b
c
: (1.1.8)
In view of the vector relationship
(e
e
) (e
k
e
) = (e
e
)(e
e
) (e
e
)(e
e
); (1.1.9)
ijm
klm
= Æ
ik
Æ
jl
Æ
il
Æ
jk
: (1.1.10)
In particular,
ikl
jkl
= 2Æ
ij
;
ijk
ijk
= 6: (1.1.11)
The triple vector product of the base vectors is
(e
i
e
j
) e
k
=
ijm
klm
e
l
= Æ
ik
e
j
Æ
jk
e
i
: (1.1.12)
Thus,
(a b) c = a
i
b
j
(c
i
e
j
c
j
e
i
); (1.1.13)
which conrms the vector identity
(a b) c = (a c)b (b c)a: (1.1.14)
1.2. Second-Order Tensors
A dyadic product of two base vectors is the second-order tensor e
i
e
j
, such
that
(e
i
e
j
) e
k
= e
k
(e
j
e
i
) = Æ
jk
e
i
: (1.2.1)
For arbitrary vectors a, b and c, it follows that
(a b) e
k
= b
k
a; (a b) c = (b c)a: (1.2.2)
The tensors e
i
e
j
serve as base tensors for the representation of an
arbitrary second-order tensor,
A = A
ij
e
i
e
j
; A
ij
= e
i
A e
j
: (1.2.3)
A dot product of the second-order tensor A and the vector a is the vector
b = A a = b
i
e
i
; b
i
= A
ij
a
j
: (1.2.4)
Similarly, a dot product of two second-order tensors A and B is the second-
order tensor
C = A B = C
ij
e
i
e
j
; C
ij
= A
ik
B
kj
: (1.2.5)
The unit (identity) second-order tensor is
I = Æ
ij
e
i
e
j
; (1.2.6)
which satises
A I = I A = A; I a = a: (1.2.7)
The transpose of the tensor A is the tensor A
T
, which, for any vectors a
and b, meets
A a = a A
T
; b A a = a A
T
b: (1.2.8)
Thus, if A = A
ij
e
i
e
j
, then
A
T
= A
e
e
: (1.2.9)
symmetric) if A
T
= A. If A is nonsingular (detA 6= 0), there is a unique
inverse tensor A
such that
A A
= A
A = I: (1.2.10)
In this case, b = A a implies a = A
b. For an orthogonal tensor
A
T
= A
, so that detA = 1. The plus sign corresponds to proper and
minus to improper orthogonal tensors.
