Page 518 - Introduction to Continuum Mechanics

502 Special Single Integral Type Nonlinear Constitutive Equations

Example 15.3

Show that any polynomial function of a real symmetric tensor A can be represented by

where// are real valued functions of the scalar invariants of the symmetric tensor A.

Solution. Let

Since A satisfies its own characteristic equation:

therefore,

» r /-k
etc., Thus, every A for N>3 can be expressed as a sum of A, A and I with coefficients being
functions of the scalar invariants of A. Substituting these expressions in Eq. (i), one obtains

Now, from Eq. (iii), we can obtain

therefore, Eq. (v) can also be written as

which is Eq. (8.15.16). Actually the representation of F(A) given in this example can be shown
to be true under the more general condition that the symmetric function F of the symmetric
tensor A is an isotropic function (of which the polynomial function of A is a special case). An
isotropic function F is a function which satisfies the condition

for any orthogonal tensor Q. Now, let us identify A with C t and /, with the scalar invariants of
C ( (note however that /3 = 1 for incompressible fluid), then the most general representation
of F(C,) is [ we recall that F(C,) is required to satisfy Eq. (viii) for frame indifference, see Eq.
(8.14.5) also],