Mathematical modelling of graphene-based membranes

该项目的目标是开发一种弹性膜压力驱动变形的理论/计算模型，弹性膜包括石墨烯层，可以是单独的，也可以是安装在更厚的聚合物衬底上的。这种薄膜在AV的小组中进行了实验研究，以期将其应用于微型压力传感器。多尺度均匀化技术将被用来表示基于连续介质的模型中的石墨烯层，该模型将服从有限元离散。将特别注意边界条件的制定，以捕捉膜如何附着在刚性支撑件上的细节-这一特征已被初步联合工作(通过暑期实习进行)确定为特别重要，即使在膜的硬度由聚合物衬底主导的情况下也是如此。需要研究的特定场景(并与现有和正在进行的实验研究的结果进行仔细的交叉验证)包括准稳定膨胀的研究(从小(线性)变形到大(非线性)变形)；预应力的影响和膜安装在其上的空腔的形状。(这一点很重要，因为非圆形空腔可能会导致起皱。)计算模型将在MH和AH开发和维护的面向对象的多物理有限元库OOMPH-LIB中实现，该模型将在与植入式麦克风的应用相关的参数区域中使用一种由可压缩的周围介质产生的声强迫而产生的非定常变形进行探索。OOMP-LIB是由MH和AH开发和维护的面向对象的多物理有限元库(参见http://www.oomph-lib.org).将该模型纳入这个开源图书馆将有助于其在反病毒小组中的使用，并有助于向更广泛的科学界传播。

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.