Page 49 - Introduction to Continuum Mechanics

34 Tensors

MIJU ~ TijTu=Q miQ njr mnQ rkQslT rs = QmiQnjQr1<QslTmnTrs
i.e.,
Mijkl = QmiQnjQrkQslMmnrs
which is the transformation law for a fourth order tensor.
It is quite clear from the proof given above that the order of the tensor whose components
are obtained from the multiplication of components of tensors is determined by the number
of free indices; no free index corresponds to a scalar, one free index corresponds to a vector,
two free indices correspond a second-order tensor, etc.
(c) The quotient rule:
If a,- are components of an arbitrary vector and 7^- are components of an arbitrary tensor
and a,- = 7^6y for all coordinates, then £>/ are components of a vector. To prove this, we note
that since a,- are components of a vector, and T)y are components of a second-order tensor,
therefore,

and

Now, substituting Eqs. (i) and (ii) into the equation a,- = Tybj, we have

But, the equation a,- = Tqbj is true for all coordinates, thus, we also have

Thus, Eq. (iii) becomes

we
Multiplying the above equation with Q ik and noting that Q^Qi m ~ <5fcm» 8 et

i.e.,

Since the above equation is to be true for any tensor T, therefore, the parenthesis must be
identically zero. Thus,