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In engineering, a steady state equilibrium force balance is typically expressed as $$\nabla\cdot \sigma=0$$ where $\sigma$ is the stress (force per unit area) applied at a point. It seems more intuitive to me to express the force balance as $$\sum F = 0$$. What does the first equation capture that the second equation doesn't? That is, what advantages does $\nabla\cdot \sigma=0$ have over $\sum F = 0$ to represent this steady state?

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up vote 1 down vote accepted

The stress tensor may be continuous rather than due to a discrete number of forces e.g. in fluid mechanics. For example the Cauchy momentum equation is naturally written using the divergence of the stress tensor.

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The divergence form of the equilibrium equation is obtained by summing the forces over a small cube that has the same area on opposite faces. So, in effect, they both represent force equilibrium. General principle: equilibrium applies to forces, not stresses.

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