Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 371

In this section the mass moment of inertia I has been referred to as a tensor. A short
calculation will demonstrate that the terminology is correct. From (A.123) is easy
to see that I may be written relative to the Latin and Greek coordinate systems as

ð
O fð

I ð L Þ ¼ x ð L Þ x ð L Þ Þ x ð L Þ x ð L Þ Þgrð x ð L Þ ; 1 ð t Þ d v ; (A.126a)

and

ð
O fðx ð G Þ

I ð G Þ ¼ x ð G Þ Þ1 ðx ð G Þ x ð G Þ Þgrðx ð G Þ ;

t Þ d v ; (A.126b)

respectively. The transformation law for the open product of x with itself can be
calculated by twice using the transformation law for vectors (A.77) applied to x ,
thus

x ð L Þ x ð L Þ ¼ x ð G Þ x ð G Þ ¼ x ð G Þ x ð G Þ Þ Q T Q Q Q ð : (A.127)

The occurrence of the transpose in the last equality of the last equation may be
more easily perceived by recasting the expression in the indicial notation:

x ð L i x ð L Þ Q j b x ð G Þ x ð G Þ

b

Q i a x ð G Þ Q i a x ð G Þ ¼

b ¼

Q j b :

(A.128)

j a a

Now, contracting the open product of vectors in (A.127) above to the scalar
product, it follows that since Q Q T Q T ¼ Q ¼ 1 ð Q i a Q i b ¼ d ab Þ ,

x ð L Þ x ð L Þ ¼ x ð G Þ x ð G Þ :

(A.129)

Combining the results (A.127) and (A.129) it follows that the non-scalar

portions of the integrands in (A.126a) and (A.126b) are related by

x ð L Þ x ð L Þ Þ x ð L Þ x ð L Þ Þg ¼ x ð G Þ x ð G Þ Þ x ð G Þ x ð G Þ Þg Q T fð 1 ð Q fð 1 ð :

Thus from this result and (A.126a) and (A.126b) the transformation law for
second order tensors is obtained,

I ð L Þ ¼ I ð G Þ Q T Q ; (A.130)

and it follows that tensor terminology is correct in describing the mass moment of
inertia.