Mathematical modelling of graphene-based membranes

The aims of the project are to develop a theoretical/computational model of the pressure-driven deformation of elastic membranes which comprise a graphene layer, either by itself or mounted on a thicker polymer backing. Such membranes are studied experimentally in AV's group with a view towards employing them in miniaturized pressure sensors. Multi-scale homogenization techniques will be employed to represent the graphene layer in a continuum-based model which will be amenable to a finite-element discretization. Particular attention will be paid to the formulation of boundary conditions which capture the details of how the membrane is attached the rigid support - a feature that preliminary joint work (performed via summer internships) has already identified as being particularly important, even for cases where the membrane's stiffness is dominated by the polymer backing.

Specific scenarios to be studied (and carefully cross-validated against the results of existing and ongoing experimental studies) include the study of quasi-steady inflation (from small (linear) to large (nonlinear) deformations); the effect of pre-stress and the shape of the cavity the membrane is mounted on. (This is important because non-circular cavities can cause wrinkling.) Unsteady deformations in response to acoustic forcing from a compressible surrounding medium will be explored using a "small-on-large"-deformation approach in parameter regimes that are relevant for the application to an implantable microphone that is being developed in a project funded by the NIHR Manchester Biomedical Research Centre.

The computational model will be implemented in oomph-lib, the object-oriented multi-physics finite-element library, developed and maintained by MH and AH (see http://www.oomph-lib.org). The incorporation of the model into this open-source library will facilitate its use in AV's group and also aid dissemination to the wider scientific community.