Electrical analogy for stress and strain It feels like the relation between stress and strain (and other mechanical properties) is analogous to that of some electrical properties (voltage and current?).
I'm comfortable with electrical engineering and trying to read up on mechanics and viscoelasticity. Could someone help my understanding by drawing some analogies or pointing me to such a description?
 A: The analogies may be built in various ways – similar simple mathematical relationships like $U=RI$ are among many of them – but I would choose the analogy consistent with the Czech language where "napětí" [nuh-pyeh-tyea] means both "voltage" and "tension". I guess that even English speakers must sometimes say "electric tension" instead of "voltage".
In the analogy inspired by this (not quite) coincidence, the voltage and analogously the stress (or tension) may be considered as the reason, and the current and analogously the deformation (strain, the relative change of the length etc.) is the consequence. For a resistor, we have Ohm's law
$$ I = \frac{U}{R} $$
and for springs, we have the analogous Hooke's law
$$ X = \frac{F}{k} $$
where $R,k$ are the resistance and the spring constant, respectively. Hooke's law for the continuous media generalizes the $F=kX$ Hooke's law above:
$$ \sigma_{ij} = -\sum_{k,\ell=1}^{3} c_{ijkl} \epsilon_{kl} $$
where $c$ is the stiffness/elasticity tensor generalizing $k$, the stress tensor $\sigma$ generalizes $F$, the $\epsilon$ is the strain tensor generalizing $X$. There are various symmetries over the indices etc. This tensor equation is the counterpart of the local version of Ohm's law
$$ \vec \jmath = \sigma \vec E $$
where the new $\sigma$ is the conductivity generalizing $1/R$ for a wire, $\vec \jmath$ is the current density, and $\vec E$ is the electric field. Sometimes, the conductivity may also be replaced by a tensor to get
$$ j_i = \sum_{k=1}^3 \sigma_{ik} E_k $$
The number and structure of the tensor indices is different in the electric/resistor and mechanical/stress examples but there is an analogy. To some extent, one could even find the mechanical counterparts of the coils or capacitors in this dictionary.
A: Another interesting analogy, pointed out by Prof. Graeme Milton, can be seen when you examine Maxwell's equations in media at a fixed frequency $\omega$:
$$
  \boldsymbol{\nabla} \times \mathbf{E} = i\omega\boldsymbol{\mu}(\mathbf{x})\cdot\mathbf{H}(\mathbf{x}) ~;~~
  \boldsymbol{\nabla} \times \mathbf{H} = -i\omega\boldsymbol{\epsilon}(\mathbf{x})\cdot\mathbf{E}(\mathbf{x}) ~.
$$
Therefore,
$$
  \omega^2~\boldsymbol{\epsilon}\cdot\mathbf{E} = i\omega~\boldsymbol{\nabla} \times \mathbf{H} = 
    \boldsymbol{\nabla} \times [\boldsymbol{\mu}^{-1}\cdot(\boldsymbol{\nabla} \times \mathbf{E})] ~.
$$
In index notation (wrt an orthonormal basis)
$$
  [\boldsymbol{\nabla} \times \mathbf{a}]_i = \mathcal{E}_{ijk}~\frac{\partial a_k}{\partial x_j}
$$
where $\mathcal{E}_{ijk}$ is the permutation tensor defined as
$$
  \mathcal{E}_{ijk} = 
   \begin{cases}
     1 & \text{for even permutations, i.e., 123, 231, 312 } \\
     -1 & \text{for odd permutations, i.e., 132, 321, 213 } \\
     0 & \text{otherwise}.
   \end{cases}
$$
Working through the algebra,
$$
  \begin{align}
  \left[\omega^2~\boldsymbol{\epsilon}\cdot\mathbf{E}\right]_j & = 
    \mathcal{E}_{jim}~\frac{\partial }{\partial x_i}[\boldsymbol{\mu}^{-1}\cdot(\boldsymbol{\nabla} \times \mathbf{E})]_m \\
    & = 
    \mathcal{E}_{jim}~\frac{\partial }{\partial x_i}[(\boldsymbol{\mu}^{-1})_{mn}~(\boldsymbol{\nabla} \times \mathbf{E})_n] \\
    & = 
    \mathcal{E}_{jim}~\frac{\partial }{\partial x_i}[(\boldsymbol{\mu}^{-1})_{mn}~\mathcal{E}_{nkl}~
       \frac{\partial E_l}{\partial x_k}] \\
    & = 
     \frac{\partial }{\partial x_i}[C_{ijkl} \frac{\partial E_l}{\partial x_k}] \quad \text{where} \quad
       C_{ijkl} := \mathcal{E}_{jim}~\mathcal{E}_{nkl}~[\boldsymbol{\mu}^{-1}]_{mn}
  \end{align}
$$
or, 
$$
  {
    \omega^2~\boldsymbol{\epsilon}\cdot\mathbf{E} = \boldsymbol{\nabla} \cdot (\boldsymbol{\mathsf{C}}:\boldsymbol{\nabla} \mathbf{E}) ~.
  }
$$
This is very similar to the linear elasticity equation at fixed frequency
$$
    -\omega^2~\rho~\mathbf{u} = \boldsymbol{\nabla} \cdot (\boldsymbol{\mathsf{C}}:\boldsymbol{\nabla} \mathbf{u}) ~.
$$
The permittivity is similar to a negative density and the electric field
is similar to the displacement.  The equations also hint at a tensorial
density.  However, continuity conditions are different for the two equations. At an interface the elastic displacement is continuous while only the tangential component of the electric field is continuous.
Also, the elastic stiffness tensor $\boldsymbol{\mathsf{C}}$ has different symmetries for the two situations.
