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# Rigidity theory meets homogenization: How periodic surfaces bend

In differential geometry, rigidity theory investigates whether a surface can deform by pure bending without stretching. The central problem is to find or disprove the existence of *isometric* deformations.

Classical examples in the "discrete" category include:

- The finite rigidity of convex polyhedra by Cauchy
- The infinitesimal rigidity of convex polyhedra by Dehn
- The existence of flexible (non-convex) polyhedra by Connelly

In the "smooth" category, we have:

- The infinitesimal rigidity of (smooth) convex compact surfaces by Liebmann
- The finite rigidity of (smooth) convex compact surfaces by Cohn-Vossen
- The existence of
*infinitesimally*flexible (non-convex) surfaces again by Cohn-Vossen - various positive results on other particular classes of surfaces: developable, of translation, of revolution, Weingarten, etc.

Here, "smooth" means at least C2. Class C1 is the work of the devil.

Rigidity theory is useful in mechanics, structural engineering in particular, because it allows to characterize the behavior of a shell as *flexure-dominant* or *membrane-dominant*. For instance, it is well-known that thin-walled closed sections resist torsion better than open sections. It is because open sections twist isometrically and closed sections do not. The difference in rigidity is driven by geometry, not material properties. Historically, the nature of deformations in thin shells was first debated by Rayleigh and Love. No disrespect to Love.

The classical literature overlooks a notion that turned out to be important for modern trends (architectural geometry, deployable membranes, origami, etc.). The notion is that of an *effective *isometry: A surface can *effectively *stretch without stretching pointwise. This is the central concept in theory of homogenization (theory of composites) whereby a heterogeneous material has effective properties that can be very different from the local, pointwise, properties.

The theorem I'd like to share with you does just that: it characterizes the *effective membrane and bending strains *produced by (infinitesimal) isometric deformations. Specifically, call **E** and **X** the effective membrane and bending strains produced by a pair of infinitesimally isometric deformations, then

*Theorem: E*11*X*22 + *E*22*X*11 - 2*E*12*X*12 = 0.

The theorem applies to any *periodic piecewise smooth simply connected* surface.

- Periodicity: this is invariance by a 2D lattice of translations. It is needed to define what
*effective*means. - Piecewise smooth: again, at least C2 (piecewise) needed to write compatibility of deformations. But beyond that, polyhedral, "curved-crease" and smooth surfaces are all ok.
- Simply connected: meaning, in particular, no cut-outs. Cut-outs introduce uncontrolled boundaries that are inconvenient for a purely geometric approach.

Note that the theorem does not say what **E** and **X** are possible. Instead, it says: should a certain effective membrane strain **E** be possible, then here is how that limits the options for effective bending strains **X**, and conversely. For instance, it says:

- If a surface can (effectively) stretch in direction 1, then it cannot (effectively) bend in direction 2.
- If it can twist then it cannot shear, and if it can shear then it cannot twist, and so on.

The proof of the theorem relies on a set of tools almost identical to those used in homogenization theory:

- Showing that a certain differential operator is symmetric, similar to how div(
**C**grad) is a symmetric operator in elasticity. - Proving a continuity condition to handle jumps across "crease" lines, similar to the continuity of normal stresses across material heterogeneities.
- Using test functions and the divergence theorem under periodic boundary conditions, as in the proof of, say, the Hill-Mandel condition.

For examples, see the video above. For proofs, see references below.

References for the curious:

For a brief, uninformed, review of rigidity theory, see the introduction of:

For case studies, see:

For the paper that pushed me down the rabbit hole, see:

- nassarh's blog
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