Page 48 - Introduction to Continuum Mechanics

Part B Defining Tensors by Transformation Laws 33

completely characterizes a vector or a tensor. Thus, it is perfectly meaningful to use a statement
such as "consider a tensor T/,-" meaning consider the tensor T whose components with respect
to some set of {e,-} are 7)y. In fact, an alternative way of defining a tensor is through the use of
transformation laws relating the components of a tensor with respect to different bases.
Confining ourselves to only rectangular Cartesian coordinate systems and using unit vectors
along positive coordinate directions as base vectors, we now define Cartesian components of
tensors of different orders in terms of their transformation laws in the following where the
primed quantities are referred to basis {e/ } and unprimed quantities to basis {e,-}, the e/ and
e, are related by e,'-Qe/, Q being an orthogonal transformation

a' = a zeroth-order tensor(or scalar)
first-order tensor (or vector)
a- ~ Q mia m
TJJ = QmiQnjTmn second-order tensor(or tensor)
T/jk = QmiQnjQrkTmnr third-order tensor
etc.

Using the above transformation laws, one can easily establish the following three rules
(a)the addition rule (b) the multiplication rule and (c) the quotient rule.
(a)The addition rule:
If TJ; and Sy are components of any two tensors, then TJJ+SJJ are components of a tensor.
Similarly if TpandS,-^ are components of any two third order tensors, then Tp.-1-Sp. are
components of a third order tensor.
To prove this rule, we note that since Tl jk=Q miQ njQ rkT mnr and S; jk=Q miQ njQ rkS mnr we
have,
+
+
*ijk Sijk = QmiQnjQrk*mnr+QmiQnjQrkTmnr ~ QmiQn}Qrk(^mnr ^nmr)
Letting W- jk = T^+S^ and W mnr=T mnr+S mnr, we have,
™ijk — QmiQnjQrkTmnr
i.e, Wfjff are components of a third order tensor.

(b)The multiplication rule:
Let a/ be components of any vector and Tjy be components of any tensor. We can form many
kinds of products from these components. Examples are (a)a/a,« (b)a/(3ya^ (c) TijT kl, etc. It can
be proved that each of these products are components of a tensor, whose order is equal to the
number of the free indices. For example, a/a/ is a scalar (zeroth order tensor), a^ija k are
components of a third order tensor, 7]y7]y are components of a fourth order tensor.

To prove that T^Tjy are components of a fourth-order tensor, let M /y W=^r w, then