Logarithmic divergence for dislocations when the Burgers vector dominates the thickness

Establish a lower bound of order \(h^2\log(1/\varepsilon)\) for the elastic energy of a thin sheet with an edge dislocation in the regime \(\varepsilon=h^\beta\) with \(0\leq\beta<1\), thereby matching the existing upper bound and proving the predicted logarithmic divergence beyond the \(h^2\) scaling.

Background

For edge dislocations, the paper proves lower and upper energy bounds that coincide in several parameter regimes. When ε=hβ\varepsilon=h^\beta with β<1\beta<1, however, the lower bound is only of order h2h^2, while the upper bound is of order h2log⁡(1/ε)h^2\log(1/\varepsilon).

The authors expect the logarithmic factor to be genuine, consistent with the iterated thin-sheet limit in which the Kirchhoff bending model diverges logarithmically. They note that their lower-bound method, based on summing independent contributions from annuli, cannot capture the required propagation of bending energy from the inner boundary, so a substantially new argument is needed.

References

For $\beta<1$, however, we have a lower bound of $h2$ and an upper bound of $h2\log(1/\e)$. We believe that the upper bound is tight, i.e., that in this case there is a logarithmic divergence on top of the $h2$ scaling. Indeed, this is the case in the iterated limit of taking $h\to 0$ while fixing $\e$, and then taking $\e\to 0$.

— Energy scaling laws for thin elastic sheets with topological defects  (2608.23134 - Kupferman et al., 24 Aug 2026) in Section 1, paragraph “Dislocations” in the discussion following Theorem 2