Computational Modelling of the Mechanical Properties of Elastic Fibre Networks

From everyday items such as paper, felt and nappies, to sophisticated biological structures such as mammalian cytoskeletons and the collagen of extracellular matrices, many materials are made up of complex disordered fibre networks with varying microstructures. It is often important to tune the mechanical properties of such networks for their specific application. The macroscopic network response to an applied load can be controlled by modifying the properties of component fibres at the microscopic level. A model relating the properties at these two scales is desirable for the design and fabrication of better materials. We developed a numerical code predicting the mechanical properties of 2D and 3D elastic fibre networks. Specifically, we find an efficient solution of the linear matrix system obtained from a large system of equations, representing a given random fibre network in mechanical equilibrium with an applied linear shear at the network boundary. This global system is assembled by considering the individual contributions of fibres modelled as cross-linked slender elastic rods, and then solved to obtain a prediction of the network displacement and total elastic energy for the applied shear. To study various network architectures, we also developed another code for network generation and visualisation. Using our software, we analysed the numerical performance of preconditioners for the iterative methods used to solve the linear system. This was applied to systems representing established 2D networks, and investigated the mechanical properties of layered 3D and fully 3D disordered systems. Our choices of preconditioners were motivated by exploiting the distinct block structure of our assembled matrix. Drawing from the application of needlepunched nonwoven fabrics, we designed a series of novel networks consisting of random 2D Mikado cross-linked layers. Applying our numerical model, we were able to explore the effects of material anisotropy on shear response, and provide a first analysis of how the macroscopic mechanics are driven by variations in microscopic properties. Evidence was also presented that our software can be used to model the mechanical properties of fully 3D random fibre networks under imposed macroscopic shear. This work can be used to direct future research, and offers the opportunity to model 3D fibrous materials using a geometry generated differently to many related works - in terms of both cross-linking procedure and cross-link type variability. The final outcome of this work is a reusable piece of software for modelling the mechanical properties of elastic fibre networks with various geometries under a macroscopically applied shear. Through use of numerical techniques and integration with the PETSc library [4], we solve the resulting systems of these networks effectively and within reasonable time scales, with the opportunity of additional optimisation if further work were to be carried out.