1
$\begingroup$

Short version: Does anyone know of good references for using functional derivatives to obtain equilibrium (or dynamic) equations in elastic theories?

More background:

I've frequently encountered complex elastic problems that might involve strain gradients, nonlinearities, etc. As a physicist, the approach that seems natural to me when I want to find the equilibrium positions $\mathbf{r}$ of bits of matter initially at $\mathbf{R}$ is to take the functional derivative of the energy functional with respect to $\mathbf{r}(\mathbf{R})$ and set it equal to zero. This works out pretty well because the nature of elasticity is that the energy density is semilocal and invariant under rotations and translations, so you get some nice simplifications even for complex solids.

I'm confident this is correct but I don't want to claim it as a new technique and I've found only passing references to it. Does anyone know anything of the history of using functional derivatives in elasticity (or solid mechanics more generally)?

Improve this question
$\endgroup$
4
  • $\begingroup$ Do you mean Fréchet/Gateaux derivatives of functionals in elasticity? $\endgroup$ Commented Jul 16, 2021 at 15:57
  • $\begingroup$ @nicoguaro Maybe! I haven't heard of them, but Wikipedia says the Frechet derivative is used to define a functional derivative. Do you have a reference/explanation for how they're used in elasticity? Thanks. $\endgroup$ Commented Jul 16, 2021 at 16:10
  • $\begingroup$ I have plenty. Do you have a context to define a little bit more the answer? $\endgroup$ Commented Jul 16, 2021 at 16:12
  • 1
    $\begingroup$ Well what we've done is write down an energy as $E = \int d^d \mathbf{R} \left[e(\mathbf{C}) - f_m r_m\right]$, where the energy density is some semilocal functional of the target metric, integrated over the reference coordinate and subject to a force-field $f_m$ and then differentiated with respect to the target position field, $\mathbf{r}(\mathbf{R})$, so something like that again would be nice. But basically, we're looking for a good, authoritative source for generally using functional derivatives to obtain elastic equilibria (or more generally equations of motion). $\endgroup$ Commented Jul 17, 2021 at 1:55

1 Answer 1

Reset to default
1
$\begingroup$

A general class of materials that are considered elastic is hyperelastic materials. Here the stress-strain relationships can be derived from a strain energy function (or functional).

For your specific question, I would suggest that you look into variational methods for elasticity. I suggest the following books:

  • Lanczos, C. (1970). The variational principles of mechanics. Fourth edition, Courier Corporation. This book makes a connection between particle mechanics and continuum mechanics.

  • Reddy, J. N. (2017). Energy principles and variational methods in applied mechanics. John Wiley & Sons. This book presents energy principles for mechanics. It also shows how these principles are used in direct variational methods and concludes with the finite element method.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Awesome, thanks! I've had a quick look through the relevant sections and those seem like just what I was looking for. $\endgroup$ Commented Jul 19, 2021 at 17:39

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.