Page 338 - Introduction to Continuum Mechanics

322 The Elastic Solid

Thus, in order that Eq. (5.32.6) be acceptable as a constitutive law, it must satisfy the condition
given by Eq. (5.32.8). Now, in matrix form, the equation

becomes

and the equation

becomes

Now, if we view the above two matrix equations, Eqs. (5.33.9) and (5.33.11), as those cor-
responding to a change of rectangular Cartesian basis, then we come to the conclusion that
the constitutive equation given by Eq. (5.33.6) describes an isotropic material because both
Eqs. (5.33.9) and (5.33.11) have the same function f.
We note that the special case

where a is a constant, is called a Hookean Solid.

5,34 Constitutive Equation for an Isotropic Elastic Medium

From the above example, we see that the assumption that T is a function of B alone leads
to the constitutive equation for an isotropic elastic medium under large deformation.
A function such as the function f, which satisfies the condition Eq. (5.33.8) is called an
isotropic function. Thus for an isotropic elastic solid, the Cauchy stress tensor is an isotropic
function of the left Cauchy-Green tensor B.
It can be proved that in three dimensional space, the most general isotropic function f(B)
can be represented by the following equation

where a 0, a\ and ai are scalar functions of the scalar invariants of the tensor B, so that the
general constitutive equation for an isotropic elastic solid under large deformation is given by