Marta Lewicka

Problems from Materials Science provide a rich source of challenges in research directions traditionally associated with pure mathematics, such as: Mathematical Analysis, Geometry, Differential Equations, or Calculus of Variations. These problems often have a multiscale character; for example, the atomic structure of a material influences its macroscopic properties. Purely continuum models also raise multiscale issues, involving the formation and evolution of patterns as a consequence of loading or phase transformation.

Recently, there has been sustained interest in the growth-induced morphogenesis, particularly of the low-dimensional structures such as fillaments, laminae and their assemblies, arising routinely in biological systems and their artificial mimics. The physical basis for morphogenesis can be presented in terms of a simple principle: differential growth in a body leads to residual strains that generically result in changes of the body’s shape. Eventually, the growth patterns are expected to be regulated by these strains, so that this principle might well be the basis for the physical self-organization of the tissues.

The proposed lectures will be concerned with the analysis of elastic films exhibiting residual stress at free equilibria, i.e. in the absence of any boundary conditions and external forces. Examples of such structures include, in particular, growing tissues such as leaves, flowers or marine invertebrates, as well as specifically engineered gels. There, it is conjectured that the growth process results in the formation of non-Euclidean, Riemannian metrics, leading to complicated morphogenesis of the tissue while it strives to attain a configuration closest possible to an isometric immersion of the metric.

This phenomenon can be studied through a variational model, pertaining to the non-Euclidean version of the nonlinear elasticity. For metrics with non-zero Riemann curvature (i.e. metrics which have no orientation-preserving isometric immersion), the infimum of the energy turns out to be positive at free equilibria. Further analysis of scaling of the energy minimizers in terms of the reference plate’s thickness leads to the rigorous derivation of the corresponding limiting theories. As we shall see, these theories are differentiated by the embeddability properties of the target metrics – in the same spirit as different scalings of external forces lead to a hierarchy of nonlinear plate theories in classical elasticity. There are also many interesting experimental results allowing to print a metric into a gellic disk, and then to compare their compatibility with our findings.

The lectures will be self-contained and will require only basic knowledge of modern analysis and differential geometry in