Cauchy Momentum Equation Derivation

[Physics_5]

From "Cauchy Momentum Equation" on Wikipedia,
The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes F_i

This is exactly what I am having trouble grasping. It's probably something simple and fundamental, but I'll go ahead and provide my thoughts below.

My latex code also wasn't compiling here, so I just attached a pdf.

Thanks in advance.

 

[Orodruin]

The directions of your force vectors do not comply with the one you need in order to make your sigmas be the components of the stress tensor. Remember that the components of the force density are $f_i = \sigma_{ij}n_j$, where n is the normal vector. The direction of the normal matters!

The easiest way of doing this is to write down the surface force on an arbitrary volume using an integral of the stress tensor over its surface and then applying the divergence theorem.

 

[Chestermiller]

I agree with Orodruin. The direction that they have shown in their figure for $\sigma_1$ is not the tensile stress exerted by the adjacent material on the mass under consideration. It is the tensile stress exerted by the mass under consideration on the adjacent material. As Orodruin also points out, the best way to be sure to get the signs right is to let the mathematics do the work for you by using the Cauchy stress relationship.