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.