Transversely Isotropic Elasticity

Assuming the 1–2 plane to be the plane of isotropy at every point, transverse isotropy requires that $E_1=E_2=E_p$ , ${\nu }_{31}={\nu }_{32}={\nu }_{tp}$ , ${\nu }_{13}={\nu }_{23}={\nu }_{pt}$ , and $G_{13}=G_{23}=G_t$ , where $p$ and $t$ stand for “in-plane” and “transverse,” respectively. Thus, while ${\nu }_{tp}$ has the physical interpretation of the Poisson's ratio that characterizes the strain in the plane of isotropy resulting from stress normal to it, ${\nu }_{pt}$ characterizes the transverse strain in the direction normal to the plane of isotropy resulting from stress in the plane of isotropy. In general, the quantities ${\nu }_{tp}$ and ${\nu }_{pt}$ are not equal and are related by ${\nu }_{tp}/E_t={\nu }_{pt}/E_p$ . The stress-strain laws reduce to

where $G_p=E_p/2(1+{\nu }_p)$ and the total number of independent constants is only five.

The restrictions on the elastic constants due to material stability are