Local Reference Frame

Three coordinate systems are introduced in the formulation:

• Global Cartesian fixed system $X=\left(X\overrightarrow{i}+Y\overrightarrow{j}+Z\overrightarrow{k}\right)$
• Natural system $\left(\xi ,\eta ,\zeta \right)$ , covariant axes x,y
• Local systems (x, y, z) defined by an orthogonal set of unit base vectors ( $t_1$ , $t_2$ , $n$ ). $n$ is taken to be normal to the mid-surface coinciding with $\zeta$ , and ( $t_1$ , $t_2$ ) are taken in the tangent plane of the mid-surface.

Figure 1. Local Reference Frame

The vector normal to the plane of the element at the mid point is defined as:(1)

$$n=\frac{x\times y}{\Vert x\times y\Vert }$$

The vector defining the local direction is:(2)

$$t_1=\frac{x}{\Vert x\Vert }$$

Hence, the vector defining the local direction is found from the cross product of the two previous vectors:(3)

$$t_2=n\times t_1$$