Page 339 - Introduction to Continuum Mechanics

Constitutive Equation for an Isotropic Elastic Medium 323

Since a tensor satisfies its own characteristic equation [See Example 5.34.1 below], there-
fore we have

or,

Substituting Eq. (5.34.4) into Eq. (5.34.2), we obtain

where <p 0, <p\ and <P2 and <P2 are scalar functions of the scalar invariants of B. This is the
alternate form of the constitutive equation for an isotropic elastic solid under large deforma-
tions.

Example 5.34.1

Derive the Cayley-Hamilton Theorem, Eq. (5.34.3).
Solution. Since B is real and symmetric, there always exists three eigenvalues correspond-
ing to three mutually perpendicular eigenvector directions.[See Section 2B18]. The
eigenvalues A/ satisfies the characteristic equation

The above three equations can be written in a matrix form as

Now, the matrix in this equation is the matrix for the tensor B using its eigenvectors as the
Cartesian rectangular basis. Thus, Eq. (5.34.7) has the invariant form

Equation (5.34.2) or equivalently, Eq. (5.34.5) is the most general constitutive equation for
an isotropic elastic solid under large deformation.
If the material is incompressible, then the constitutive equation is indeterminate to an
arbitrary hydrostatic pressure and the constitutive equation becomes