CONTINUUM MECHANICS for ENGINEERS

( )
eˆ eˆ
nˆ
( ) − +
=
2
2 1 2
3
3 1 2
Then ˆn is constructed from nˆ = nˆ × nˆ
, so that
Thus, the transformation matrix A is given by Eq 2.6-11 as
In concluding this section, we mention several interesting properties of
symmetric second-order tensors. (1) The principal values and principal directions
of T and T T are the same. (2) The principal values of T –1 are reciprocals
of the principal values of T, and both have the same principal directions. (3)
The product tensors TQ and QT have the same principal values. (4) A
symmetric tensor is said to be positive (negative) definite if all of its principal
values are positive (negative); and positive (negative) semi-definite if one principal
value is zero and the others positive (negative).
2.7 Tensor Fields, Tensor Calculus
A tensor field assigns to every location x, at every instant of time t, a tensor
Tij…k(x,t), for which x ranges over a finite region of space, and t varies over
some interval of time. The field is continuous and hence differentiable if the
components Tij…k(x,t) are continuous functions of x and t. Tensor fields may
be of any order. For example, we may denote typical scalar, vector, and tensor
fields by the notations φ(x,t), vi(x,t), and Tij(x,t), respectively.
Partial differentiation of a tensor field with respect to the variable t is
symbolized by the operator ∂/ ∂tand
follows the usual rules of calculus.
On the other hand, partial differentiation with respect to the coordinate xq will be indicated by the operator ∂/ ∂x,
which may be abbreviated as
q
2
simply . Likewise, the second partial ∂ / ∂x ∂x
may be written ,
∂ q
and so on. As an additional measure in notational compactness it is customary
in continuum mechanics to introduce the subscript comma to denote
partial differentiation with respect to the coordinate variables. For example,
= ˆ
( ) ( ) ( )
e *
2
nˆ eˆ eˆ eˆ e* ˆ
3 ( ) 1
= ( − 1− 2 + 2 3)= 3
2
⎢
[ aij] = ⎢−
⎡ 1/ 2 1/ 2 1/ 2⎤
⎥
1/ 2 1/ 2 0 ⎥
⎢
⎣
−1/ 2 −1/
2 1/ 2⎥
⎦
q m ∂ qm