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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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Let's consider a point $\renewcommand{\vec}[1]{\mathbf{#1}}\vec{x}$ of the undeformed material, which moves to $\vec{x}'$ after deformation. We define the displacement vector

$$\vec{u} = \vec{x}'-\vec{x}.$$

The variation of $dl^2=d\vec{x}^2$ is then given by (implicit summation on repeated indices everywhere)

$$dl'^2 = dl^2 + 2u_{ik}dx_idx_k,$$

where $u_{ik}$ is the strain tensor,

$$\newcommand{\partialder}[2]{\frac{\partial{#1}}{\partial{#2}}}u_{ik}=\frac{1}{2}\left(\partialder{u_i}{x_k}\partialder{u_k}{u_i}+\partialder{u_l}{x_i}\partialder{u_l}{x_k}\right).$$

Then there is a stress tensor $\sigma_{ik}$ such that the force on a volume $V$ is given by

$$F_i = \int_{\partial V} \sigma_{ik}ds_k,$$

where $ds_k$ are the component of the infinitesimal element of surface. The key postulate to derive this formula is that an element of material exert a force on its immediate neighbourhood only, which precludes the existence of macroscopic electric field, as generated in piezoelectric materials for example, when the material is deformed.

Finally, a widely used model is Hooke's law, which gives the stress tensor knowing the strain tensor,

$$\sigma_{ik} = Ku_{ll}\delta_{ik}+2\mu(u_{ik}-\frac{1}{3}u_{ll}\delta_{ik}),$$

where $K$ is called the bulk modulus and $\mu$ is called the shear modulus.

Reference (shamelessly plundered above!): L.D. Landau and E.M. Lifshitz. Theory of Elasticity, volume 7 of Course of Theoretical Physics. Pergamon Press, 1970.

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    $\begingroup$ +1: funnily enough I was just looking at this text a couple of days ago! $\endgroup$ Sep 5 '17 at 17:55
  • $\begingroup$ Thanks a lot for the answer! If I may ask, is it true that $F_i$ calculated with the integral in your answer represents only surface forces (i.e. forces that are not exerted by external fields but that comes from the action of the rest of the medium)? Is this somehow what you meant by saying that "an element of material exert a force on its immediate neighbourhood only"? Therefore if there are volume forces (e.g. an external gravitational field) the calculation of $F_i$ with that integral is not possible anymore? $\endgroup$
    – Sørën
    Sep 7 '17 at 13:45
  • $\begingroup$ Yes, everything I wrote is for surface forces only. An external gravitational field would not be an issue: just add another term, which would be an integral over a volume. As long as that field is external, i.e. does not depend on the deformation of the material, it would work. But if the deformation induces a field felt throughout the volume, then that would be a different story: I mean if that field depends on the deformation. $\endgroup$
    – user154997
    Sep 8 '17 at 14:44
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The classical-physics analog of the Maxwell stress tensor is the Cauchy stress tensor. It is a linear operator that inputs a unit vector at a point and outputs the stress density vector across an infinitesimal area normal to the unit vector at that point.

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