Page 60 - Introduction to Continuum Mechanics

Part B Principal Scalar Invariants of a Tensor 45

Without loss of generality, let

2
2
2
then noting that ct +^ +y = 1, we have
i.e.,

Also,

i.e.,

2B20 Principal Scalar Invariants of a Tensor

The characteristic equation of a tensor T, 17^— A<5,y | =0 is a cubic equation in A. It can be
written as

where

Since by definition, the eigenvalues of T do not depend on the choices of the base vectors,
therefore the coefficients of Eq. (2B20.1) will not depend on any particular choice of basis.
They are called the principal scalar invariants of T.
We note that, in terms of the eigenvalues of T which are the roots of Eq.(2B20.1), the //'s
take the simpler form