# How to interpret a differential stress element with differing stress magnitudes on opposite faces?

The element is in equilibrium, but with different magnitudes of stress on opposing faces. What meaning does this have physically?

See attached image.

Thanks

• would you like an example of a real situation? Commented Mar 17, 2022 at 0:28
• Sure... taking part of Chemomechanic’s answer, what might be a situation where it’s valid to have a finite difference in stress across an element? Now I’m thinking it doesn’t even make sense to have a finite difference across an infinitesimal element. (?) Commented Mar 18, 2022 at 8:53

If the differences from side to side are infinitesimal, then we have the common situation of a stress state varying throughout a material. We can build up constitutive equations by summing the forces in the three Cartesian directions (in the general case) and summing the moments around the corresponding axes. Then we'd apply Newton's second law (linear and rotational).

In the $$x$$ direction, for instance, we can sum forces as

$$$$-\sigma_{xx}\,dy\,dz+\left(\sigma_{xx}+\frac{\partial\sigma_{xx}}{\partial x}dx\right)dy\,dz+-\sigma_{xy}\,dx\,dz+\left(\sigma_{xy}+\frac{\partial\sigma_{xy}}{\partial y}dy\right)dx\,dz\\ =\rho\,dx\,dy\,dz\,\ddot x,$$$$

where $$\rho$$ is the density and where we've Taylor-expanded the variation in stress throughout the element and retained only the first term because that variation is infinitesimal. This reduces to the equilibrium equation

$$\frac{\partial\sigma_{xx}}{\partial x}+\frac{\partial\sigma_{xy}}{\partial y}=\rho\ddot x.$$

If the differences from side to side are finite, however, then the differential element shown above may not be useful and may even be invalid depending on the framework it's used under. In those cases, it should be divided into smaller elements with infinitesimal variation only.

• Thank you. I guess I’m still confused though. Generally a differential element describing stress at a point is shown in equilibrium with equal and opposite stresses on opposing sides... so is that simply ignoring infinitesimal variation across the element? Commented Mar 18, 2022 at 8:48
• Or maybe there is no variation—maybe the image is meant to represent a uniform stress state. Commented Mar 18, 2022 at 21:34