This is a report of some recent progress and challenges we have made and encountered in modelling and numerical simulation of materially nonlinear beam structures with applications in micro-electrical-mechanical systems. For simplicity, the fully nonlinear DE’s and the associated initial/boundary value problems arising from modelling Hollomon’s power-law material structures are presented as our representative mathematical models. While for linear elastic materials, the principal operator in the equations appears to be the Laplacian or the bi-Laplacian operator, for the Hollomon’s materials, the principal operator is the p-Laplacian or the bi-p-Laplacian. The main results presented are centered around approximations of solutions to the nonlinear wave equation by lumped-parameter models and numerical methods. Similar, but more challenging models are also introduced for further investigations.

 We will discuss examples of singularity formation from smooth data for linear and quasilinear uniformly parabolic systems in the plane.

In 1934, Wilhelm Blaschke’s attention focused on a recent construction in metric geometry proposed by Dan Barbilian as a generalization of various models of hyperbolic geometry. It was the year when S.-S. Chern started his doctoral program under Blaschke’s supervision in Hamburg and when in several academic centers in Europe scholars were interested in generalizations of Riemannian geometry. Introduced originally in 1934, Barbilian’s metrization procedure induces a distance on a planar domain through a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to more general domains. In our discussion we plan to show that these spaces are naturally related to Gromov hyperbolic spaces. In several works written with W.G. Boskoff, we explore this connection. We conclude our talk by stating several open problems related to this content.