Home Math Teaching About Me

≪ Seminars and Talks

Variational Methods in Fracture Mechanics

Speaker: Kerrek Stinson
Date of Talk: May 20, 2025
Upstream link: UCLA Analysis and PDE Seminar

Given an object \(\Omega \subseteq \mathbb{R}^2\), one can describe a deformation of the object via maps \(y, u : \Omega \to \mathbb{R}^2\) (the deformation and displacement, respectively), subject to the relation \(y = \operatorname{Id} + u \).

These maps \(y\) and \(u\) are allowed to be discontinuous; these points of discontinuity represent where the object breaks or cracks. Let \(K\) be this set of discontinuities.

One can model the energy of a deformation \(u\) with a “crack set” \(K\); the following is the Griffith energy: \[E \left[ u, K \right] = \int _{\Omega \setminus K }\left\lvert e(u) \right\rvert^2 dx + \mathcal{H}^1(K).\] Here, \(e\) is the symmetric gradient \(e(u) = \frac{1}{2} \left( \nabla u + \nabla u^\top \right)\). This shows up because it’s invariant under the action of \(\operatorname{SO}(2)\), reflecting the fact that the energy of a deformation should not depend on which perspective you view this deformation from. There are other energies to study, and one wants to understand minimisers of these.

1. The Principle of Concentration Compactness

The classic approach is to take a minimising sequence and appeal to compactness and lower semicontinuity, but the energies above are insensitive to independent translations of disconnected pieces of the object. This makes sense: break an object (with an associated energy cost), then move the pieces around for free, and the latter motion should not affect the energy of the crack itself.

The principle of concentration compactness is a paradigm or technique that allows one to address this failure of compactness. It looks like a common idea used throughout variational approaches…

2. The Mumford-Shah Conjecture

In 1985, Mumford and Shah proposed the following conjecture:

The triple junction part is reasonable: junctions of many, many cracks can be perturbed slightly to reduce the lengths of the cracks. This seems very difficult! The speaker and his co-authors proved instead that, up to a codimension \(> 1\) singular set, minimisers of the slightly different Griffith energy are \(C ^{1, \frac{1}{2}}\) manifolds.

3. Korn’s Inequality

One important ingredient in the speaker’s work is Korn’s inequality, which he described as a variant of Poincaré’s inequality for the symmetric gradient! (This is helpful for identifying chunk of the object that are being “deformed” by more or less a rigid motion.)

Finally, the speaker mentions that the structure of the singularities of crack sets is still not understood. He expects that triple junctions of arcs should not be difficult, but that “crack tips” would be much more difficult to understand. Moreover, everything is being done in \(\mathbb{R}^2\) at the moment, and close to nothing is known for higher dimensions.