Flexoelectricity, the linear coupling of strain gradient and electric polarization, is inherently a size-dependent phenomenon. The energy storage function for a flexoelectric material depends not only on polarization and strain, but also strain-gradient. Thus, conventional finite-element methods formulated solely on displacement are inadequate to treat flexoelectric solids since gradients raise the order of the governing differential equations. Here, we introduce a computational framework based on a mixed formulation developed previously by one of the present authors and a colleague. This formulation uses displacement and displacement-gradient as separate variables which are constrained in a ‘weighted integral sense’ to enforce their known relation. We derive a variational formulation for boundary-value problems for piezo- and/or flexoelectric solids. We validate this computational framework against available exact solutions. Our new computational method is applied to more complex problems, including a plate with an elliptical hole, stationary cracks, as well as tension and shear of solids with a repeating unit cell. Our results address several issues of theoretical interest, generate predictions of experimental merit and reveal interesting flexoelectric phenomena with potential for application.

Biofilm properties change drastically from one point to another inside the matrix, and from one minute to the next, bringing about similar variations in biofilm mechanical properties, both in time and space. In this article, we present a theory that quantifies deformation-dependent changes in the mechanical properties of a composite porous material that undergoes compression. Such changes are a result of the pores either closing (when the biofilm is under compression) or opening (when under tension). The theory borrows well-established principles of continuum mechanics and is modified to represent a biofilm composed of four different phases, three different solid biomass materials (active biomass, extracellular polymers and inert biomass) and pores. We see that, when the evolution of the volume fractions of the different phases in a uniaxial compression test is taken into account, the material “hardens” or becomes stiffer as the deformation increases, due to void closure. Once complete void closure is achieved, the material reaches its maximum stiffness. Different homogenisation schemes are presented and comparisons are performed with stress-strain calculations for all of them.