Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

A path-independent measure in order parameter space is introduced such that, when integrated along any closed contour in a three-dimensional nematic phase, it yields the topological charge of any line defects encircled by the contour. A related measure, when integrated over either closed or open surfaces, reduces to known results for the charge associated with point defects (hedgehogs) or Skyrmions. We further define a tensor density, the disclination density tensor $�{\displaystyle ���\mathbf{D}���}�$, from which the location of a disclination line can be determined. This tensor density has a dyadic decomposition near the line into its tangent and its rotation vector, allowing a convenient determination of both. The tensor $�{\displaystyle ���\mathbf{D}���}�$ may be non-zero in special configurations in which there are no defects (double-splay or double-twist configurations), and its behaviour there is provided. The special cases of Skyrmions and hedgehog defects are also examined, including the computation of their topological charge from $�{\displaystyle ���\mathbf{D}���}�$.

Open cavities are often an essential component in the design of ultra-thin subwavelength metasurfaces and a typical requirement is that cavities have precise, often low frequency, resonances while simultaneously being physically compact. To aid this design challenge, we develop a methodology to allow isospectral twinning of reference cavities with either smaller or larger ones, enforcing their spectra to coincide so that open resonators are identical in terms of their complex eigenfrequencies. For open systems, the spectrum is not purely discrete and real, and we pay special attention to the accurate twinning of leaky modes associated with complex-valued eigenfrequencies with an imaginary part orders of magnitude lower than the real part. We further consider twinning of two-dimensional gratings, and model these with Floquet–Bloch conditions along one direction and perfectly matched layers in the other one; complex eigenfrequencies of special interest are located in the vicinity of the positive real line and further depend upon the Bloch wavenumber. The isospectral behaviour is illustrated, and quantified, throughout by numerical simulation using finite-element analysis.

Gravitational waves are being shown to derive directly from Newtonian dynamics for a continuous mass distribution, e.g. compressible fluids or equivalent. It is shown that the equations governing a continuous mass distribution, i.e. the inviscid Navier–Stokes equations for a general variable gravitational field $�{\displaystyle ���\mathit{g}���\left(��t�,��\mathit{x}���\right)���}�$, are equivalent to a form identical to the Maxwell equations from electromagnetism, subject to a specified condition. The consequence of this equivalence is the creation of gravity waves that propagate at a finite speed. The latter implies that Newtonian gravitation as presented in this paper is not ‘spooky action at a distance’ but rather is similar to electromagnetic waves propagating at finite speed, despite the apparent form appearing in the integrated field formula. In addition, this proves that, in analogy to the Maxwell equations, the Newtonian gravitation equations are Lorentz invariant for waves propagating at the speed of light. Since gravitational waves were so far derived only from Einstein’s general relativity theory, it becomes appealing to check if there is a connection between the Newtonian waves presented in this paper and the general relativity type of waves at least in a certain limit of overlapping validity, i.e. as a flat-space approximation. The latter is left for follow-up research.

We introduce a mathematical model for the mechanical behaviour of the eukaryotic cell cytoskeleton. This discrete model involves a regular array of pre-stressed protein filaments that exhibit resistance to enthalpic stretching, joined at cross-links to form a network. Assuming that the inter-cross-link distance is much shorter than the length scale of the cell, we upscale the discrete force balance to form a continuum system of governing equations and deduce the corresponding macroscopic stress tensor. We use these discrete and continuum models to analyse the imposed displacement of a bead placed in the domain, characterizing the cell rheology through the force–displacement curve. We further derive an analytical approximation to the stress and strain fields in the limit of small bead radius, predicting the net force required to generate a given deformation and elucidating the dependency on the microscale properties of the filaments. We apply these models to networks of the intermediate filament vimentin and demonstrate good agreement between predictions of the discrete, continuum and analytical approaches. In particular, our model predicts that the network stiffness increases sublinearly with the filament pre-stress and scales logarithmically with the bead size.

Laminated media with material properties modulated in space and time in the form of travelling waves have long been known to exhibit non-reciprocity. However, when using the method of low-frequency homogenization, it was so far only possible to obtain non-reciprocal effective media when both material properties are modulated in time, in the form of a Willis-coupling (or bi-anisotropy in electromagnetism) model. If only one of the two properties is modulated in time, while the other is kept constant, it was thought impossible for the method of homogenization to recover the expected non-reciprocity since this Willis-coupling coefficient then vanishes. Contrary to this belief, we show that effective media with a single time-modulated parameter are non-reciprocal, provided homogenization is pushed to the second order. This is illustrated by numerical experiments (dispersion diagrams and time-domain simulations) for a bilayered modulated medium.

We consider wave scattering from a system of highly contrasting resonators with time-modulated material parameters. In this setting, the wave equation reduces to a system of coupled Helmholtz equations that models the scattering problem. We consider the one-dimensional setting. In order to understand the energy of the system, we prove a novel higher-order discrete, capacitance matrix approximation of the subwavelength resonant quasi-frequencies. Further, we perform numerical experiments to support and illustrate our analytical results and show how periodically time-dependent material parameters affect the scattered wave field.

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo methods. Assessment of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.

Over the last half-century, linear viscoelastic models for crack growth in soft solids have flourished but their predictions have rarely been compared to experiments. In fact, most available models are either very approximate or cast in forms which are not quite suitable for the analysis of actual data. Here, we propose a linear viscoelastic approach which consistently exploits the dynamic mechanical analysis data. We apply this method to four sets of results documenting fracture or adhesion rupture in soft solids with various degrees of viscoelasticity. For elastomers, the results reproduce the well-known inconsistency of the process zone size. In more viscoelastic systems, however, the present approach is able to match the measured velocity dependence of the rupture energy with physically acceptable process zone sizes. Moreover, our predictions agree with the damage zone sizes measured by mechanoluminescence. Building on these results, we discuss various issues arising when evaluating the linear viscoelastic contribution to the rupture/adhesion energy in soft solids: data quality, physical interpretation of the parameters, validity of simpler approximations and limitations of the present approach.