分野別セミナー

When deforming a material body (e.g. heating, bending, stretching or otherwise stressing), physics (basically the principle of least action) informs us that the final deformation rearranges itself so as to minimise some energy or action functional. Modelling is often about trying to find the correct functional from other physical first principles. The theory of nonlinear elasticity has been developed over a century to try and say important and generic things about the structure and regularity of minimisers or minimal energy configurations for various classes of functionals. Of course minimisers of functionals satisfy differential equations (Euler-Lagrange). There have been significant advances in solving these differential equations over the years, particularly when they are nice (the technical term is elliptic). But to model more interesting phenomena, like critical phase and transitions, supersonically moving objects and so forth, the equations develop singularities and nonlinear terms can't be ignored. In dealing with these ugly equations (the technical term is nonlinear degenerate elliptic) it's sometimes easier to go back to the functionals themselves.

In this talk I will discuss some recent work with others about a special interesting case modelling nonlinear phenomena in elastic media by minimising a scale invariant measure of the anisotropic properties of the material in the simplest 2D case (with 3D applications). Surprisingly this is connected with a conjecture from J.C.C. Nitsche in 1962 (solved by Iwaniec, Koskela and Onninen) concerning harmonic mappings and minimal surfaces. There is a wonderful dichotomy in the solutions to these equations as one passes through a critical phase when one can identify conformal invariants of the material (= geometric quantities derived from infinitesimal information). This dichotomy shows, for instance, that materials can only be stretched so far before breaking or tearing. There appear to be other applications in modelling cellular structures, foam physics and tissues as well.