The Stress Tensor in Uniaxial Extension

In any elementary textbook on fluid mechanics (see Shames,(8) for example), it is shown that the stress vector f on any surface whose outward-directed unit normal is n, is obtained by matrix multiplication. 

Using symmetry arguments,it can be shown that normal (internal) stresses result from normal strains a shear stress cannot be caused by a normal strain. What this means is that, in a rod-stretching experiment, all the off-diagonal elements of the extra-stress tensor t are zero. Furthermore, if jCj is the direction of stretching, symmetry again requires that T22 equal T33. As a result there are only three nonzero terms in the total stress tensor a, and these can be written as 

Physically, a22 is the ambient (atmospheric) pressure and, without loss of generality, one can set it equal to zero. (Recall that one frequently uses the gauge pressure in elementary fluid mechanics.) Then, the difference between Eq. (8a) and Eq. (8b) yields 

Equation (9) relates to the single material function, Tu - T22, in uniaxial extension. If stretching occurs under the influence of a constant stretch rate, must be a unique function of , since both Tu and T22 depend uniquely on . 

---