In one of my textbooks the following notation is used to describe strain components in a displacement field:
$$\begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix} = \begin{bmatrix} \frac{\partial u_x}{\partial x} \\ \frac{\partial u_y}{\partial y} \\ \frac{\partial u_z}{\partial z} \\ \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y} \\ \frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x} \\ \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \end{bmatrix}$$
or
$$S_I = \nabla_{Ij}u_j$$
I am a little bit confused as to how exactly I should interpret the subscripts in the expression $S_I = \nabla_{Ij}u_j$. Take for instance $S_4$. We are supposed to have:
$$S_4 = \nabla_{4j}u_j = \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y}$$
But I can't see how logically the subscripts indicate this. If anyone can explain this to me, I would really be grateful!
Best Answer
I thought it made non sense as well.
But after looking it up it seems that in this context, $\nabla$ is a symbol that stands for
$$ \nabla = \left[ \begin{array}{ccc} \frac{\partial}{\partial x} & 0 & 0 \\ 0 & \frac{\partial}{\partial y} & 0 \\ 0 & 0 & \frac{\partial}{\partial z} \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \\ 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x} \end{array} \right] $$ Don't ask me why, I find it very weird and I am a total stranger to mechanics :)