Page 67 - Introduction to Continuum Mechanics

52 Tensors Calculus

where K is a tensor known as thermal conductivity tensor.
(a)What tensor K corresponds to the Fourier heat conduction law mentioned in the previous
example?
(b)If it is known that K is symmetric, show that there are at least three directions in which heat
flow is normal to the surface of constant temperature.

(c)If 0-2^+3*2 and

find q.

Solution.
(a)Clearly, K=W, so that q= -klVO= -kVO
(b)For symmetric K, we know from Section 2B.18 that there exist at least three principal
directions n^i*-. and n 3 such that
Knj = fcjnj

Fig. 2C.2

n
=
^"2 ^2 2
Kn 3 = %n 3
where k\ki and £3 are eigenvalues of K. Thus, for V0 in the direction of n b
qi = -KV0 = -K|V0|n 1 = -|Vd|Kn! = -ki\VO\ni