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The metric-restricted inverse design problem

Amit Acharya's picture

Amit Acharya         Marta Lewicka         Mohammad Reza Pakzad

In Nonlinearity, 29, 1769-1797

We study a class of design problems in solid mechanics, leading to a variation on the
classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new
context, we derive a necessary and sufficient existence condition, given through a system of total
differential equations, and discuss its integrability. In the classical context, the same approach
yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor.
In the present situation, the equations do not close in a straightforward manner, and successive
differentiation of the compatibility conditions leads to a more sophisticated algebraic description
of integrability. We also recast the problem in a variational setting and analyze the infimum value
of the appropriate incompatibility energy, resembling "non-Euclidean elasticity".  We then derive a
Γ-convergence result for the dimension reduction from 3d to 2d in the Kirchhoff energy scaling
regime. A practical implementation of the algebraic conditions of integrability is also discussed.

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Comments

Arash_Yavari's picture

Hi Amit,

This paper is very nice. Just a couple of minor comments. 1) On page 1, you mention “2-dimensional shell”. I would use “2-dimensional membrane” as you are not considering bending stresses. In the case of a shell, in addition to metric (first fundamental form) you would need the second fundamental form as well and also need to worry about some extra compatibility equations. 2) On page 6, you define the Christoffel symbols (of “second kind”). I know this has been used in old books but, in my opinion, “first” and “second” kind are misleading or, at best, unnecessary. Christoffel symbols are simply coordinate representations of the Levi-Civita connection.

Regards,
Arash

Amit Acharya's picture

Hi Arash - Glad you liked it and thanks for your comments.

On your comment 1) indeed, it would have been better to say on page 1 that we are primarily concerned with a membrane, as is mentioned in the paper in the first bullet on page 2.

If, on the other hand, the material of the shell is such that given a prestrain metric field it can generate as a 'prebending strain' exactly the second fundamental form of a deformation whose first fundamental form 'annihilates' the given prestrain, then the question posed is good enough even for a shell to attain the given shape S as a consequence of energy minimization. The definition of annihilation will depend on the form in which the bending strain of the shell and the prebending strain of the shell appear in the constitutive equation for the energy. I just made this up, and I agree it is a bit artificial and it would be better just to stick with a membrane.

Now to your comment about the extra compatibility conditions. To keep things simple, let's just think of the forward problem rather than the inverse problem. Then, I believe the extra conditions you are thinking of are the Codazzi-Mainardi eqns coupled to the Gauss equations.

However, there is an interesting twist here.

If, in fact, the question was posed as given the first and second fundamental forms, find conditions on these fields such that a deformation of the shell exists, this is 'not such a difficult question' in the sense that it is a creative application of the Thomas/Froebenius theorem for a creatively defined problem. And the answer is the satisfaction of the Gauss + Codazzi-Mainardi equations.

However, if you ask the same question with only the metric field specified, then on first glance it seems like this is a far less constrained problem than the previous one and should be solvable for all metric fields - basically, one is mapping from 2 to 3 dimensions, and there are three equations for three variables so it does not look overconstrained and some familiarity with the Gauss-Codazzi-Mainardi equations makes one think that the second fundamental form is free to choose with the metric and Christoffel symbols as data (generated from the given metric field), so this should be eminently doable. Guess what, even in the local case, away from the analytic situation that was done by Janet and Cartan, existence of a solution is not known for arbitrary, smooth metric fields. One can appreciate what the issue is by inspecting the Gauss Codazzi-Mainardi equations, seen as a system for defining the second fundamental form in terms of the metric field. Then the C-M equations become a first-order, linear pde system for the 2nd FF  and the Gauss equations become quadratic algebraic constraint equations with the Riemann Curvature of the first fundamental form as data. This, apparently, is not easy to solve as it involves equations that change type depending on the details of the smooth metric field.

So, the pure metric specified problem may be considered in some sense more interesting than the more constrained case involving additional compatibility conditions.

Finally, as for your comment on the Christoffel symbols of first and second kinds. Of course, they are components of the connection, but the first and second kinds are different arrays of numbers and they are needed to be identified as different beasts in our derivation so why you say they are 'misleading' and 'unnecessary' is not clear to me (I think of myself as lazy enough not to use anything unnecessary). I hope we can all agree on freedom of expression :), so what's in a name(s)?.... Especially in mathematics where the only thing you are 'free' to do is to make definitions and set up the basic rules of the game you want to play..... All the best.

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