# I Continuum mechanical analogous of Maxwell stress tensor

#### crick

Maxwell stress tensor $\bar{\bar{\mathbf{T}}}$ in the static case can be used to determine the total force $\mathbf{f}$ acting on a system of charges contanined in the volume bounded by $S$

$$\int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf {{Q_{mech}}}\tag{1}$$

Where $\mathbf {{Q_{mech}}}$ is the (mechanical) momentum of the system of charges.

What theorem/relation is formally analogous to $(1)$ in continuum mechanics? I've read that also in continuum mechanic one can introduce a tensor such that the value of its components on a surface $S$ enclosing a system of masses determines the forces acting on the masses completely.

I could not find this analogy on Jackson or Griffiths, so what is the tensor that is similar to Maxwell stress tensor in mechanics? Is it the stress tensor? By which theorem does it determine the forces on a system of masses?

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#### Orodruin

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Is it the stress tensor?
Partially, the stress tensor is the spatial-spatial part. The stress energy tensor also incorporates the momentum and energy densities.

Edit: So forget that. You were only asking about the 3D case, not the spacetime 4D case. It is just the stress tensor. The Maxwell stress tensor is the stress tensor of a continuum - the EM field.

"Continuum mechanical analogous of Maxwell stress tensor"

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