Homogenization of surface energy and elasticity for highly rough surfaces

Surface energy plays a central role in several phenomena pertaining to nearly all aspects of materials science. This includes phenomena such as self-assembly, catalysis, fracture, void growth, and microstructural evolution among others. In particular, due to the large surface-to-volume ratio, the impact of surface energy on the physical response of nanostructures is nothing short of dramatic. How does the roughness of a surface renormalize the surface energy and associated quantities such as surface stress and surface elasticity? In this work, we attempt to address this question by using a multi-scale asymptotic homogenization approach. In particular, the novelty of our work is that we consider highly rough surfaces, reminiscent of experimental observations, as opposed to gentle roughness that is often treated by using a perturbation approach. We find that softening of a rough surface is significantly underestimated by conventional approaches. In addition, our approach naturally permits the consideration of bending resistance of a surface, consistent with the Steigmann-Ogden theory, in sharp contrast to the surfaces in the Gurtin-Murdoch surface elasticity theory that do not offer flexural resistance.

On Some Thermoelastic Problem of a Nonhomogeneous Long Pipe

The paper deals with the thermoelastic problem of a multilayered pipe subjected to normal loadings on its inner surface and temperature differences on its internal and external surfaces. Two types of nonhomogeneous pipe materials of pipe are considered: (1) a ring-layered composite composed of two repeated thermoelastic solids with varying thickness and (2) a functionally graded ring layer. The ring-layered pipe with periodic structure is investigated by using the homogenized model with microlocal parameters. A homogenization approach is proposed for the modelling of the FGM pipe. The analysis of obtained circumferential, radial and axial stress is presented in the form of figures and discussed in detail. It was shown that the proposed approach to the homogenization allows us to correctly calculate the averaged characteristics in the representative cell (the macro-characteristics) and also the characteristics dependent on the choice of the component in the representative cell (the micro-characteristics) for both microperiodic composites and functionally graded materials.

Shallow water theory for structured bathymetry

A shallow water theory is developed which applies to surface wave propagation over structured bathymetry comprising rapid abrupt fluctuations in depth between two smoothly varying levels. Using a homogenization approach coupled to the depth-averaging process which underpins shallow water modelling, governing equations for the wave elevation are derived which explicitly relate local spatially varying anisotropy of wave speeds to properties of the microstructured bed. The model is applied to two water wave scattering problems both to demonstrate the complex wave propagation characteristics exhibited by structured beds and to provide examples of how to use structured beds to engineer bespoke wave propagation. This includes propagating waves with practically zero reflection and loss of form through circular bends in channels.

Homogenization-based design of microstructured membranes: wake flows past permeable shells

A formal framework to characterize and control/optimize the flow past permeable membranes by means of a homogenization approach is proposed and applied to the wake flow past a permeable cylindrical shell. From a macroscopic viewpoint, a Navier-like effective stress jump condition is employed to model the presence of the membrane, in which the normal and tangential velocities at the membrane are respectively proportional to the so-called filtrability and slip numbers multiplied by the stresses. Regarding the particular geometry considered here, a characterization of the steady flow for several combinations of constant filtrability and slip numbers shows that the flow morphology is dominantly influenced by the filtrability and exhibits a recirculation region that moves downstream of the body and eventually disappears as this number increases. A linear stability analysis further shows the suppression of vortex shedding as long as large values of the filtrability number are employed. In the control/optimization phase, specific objectives for the macroscopic flow are formulated by adjoint methods. A homogenization-based inverse procedure is proposed to obtain the optimal constrained microscopic geometry from macroscopic objectives, which accounts for fast variations of the filtrability and slip profiles along the membrane. As a test case for the proposed design methodology, a cylindrical membrane is designed to maximize the resulting drag coefficient.