For microscale heterogeneous partial differential equations (PDEs), this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. A key innovation herein is to analyse the ensemble of all phase-shifts of the heterogeneity. Dynamical systems theory then frames the homogenization as a slow manifold of the ensemble. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of an exact homogenization at finite scale separation, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology complements existing well-established results by providing a new rigorous and flexible approach to homogenization that potentially also provides correct macroscale initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.

The self-interaction force of dislocation curves in metals depends on the local arrangement of the atoms and on the non-local interaction between dislocation curve segments. While these non-local segment–segment interactions can be accurately described by linear elasticity when the segments are further apart than the atomic scale of size $\varepsilon$ , this model breaks down and blows up when the segments are $O(\varepsilon)$ apart. To separate the non-local interactions from the local contribution, various models depending on $\varepsilon$ have been constructed to account for the non-local term. However, there are no quantitative comparisons available between these models. This paper makes such comparisons possible by expanding the self-interaction force in these models in $\varepsilon$ beyond the O(1)-term. Our derivation of these expansions relies on asymptotic analysis. The practical use of these expansions is demonstrated by developing numerical schemes for them, and by – for the first time – bounding the corresponding discretisation error.

The Kirchhoff approximation is widely used to describe the scatter of elastodynamic waves. It simulates the scattered field as the convolution of the free-space Green’s tensor with the geometrical elastodynamics approximation to the total field on the scatterer surface and, therefore, cannot be used to describe nongeometrical phenomena, such as head waves. The aim of this paper is to demonstrate that an alternative approximation, the convolution of the far-field asymptotics of the Lamb’s Green’s tensor with incident surface tractions, has no such limitation. This is done by simulating the scatter of a critical Gaussian beam of transverse motions from an infinite plane. The results are of interest in ultrasonic nondestructive testing.

An infinite Bernoulli-Euler beam (representing the “combined rail” consisting of the rail and longitudinal sleeper) mounted on periodic flexible point supports (representing the railpads) has already proven to be a suitable mathematical model for the floating ladder track (FLT), to define its natural vibrations and its forced response due to a moving load. Adopting deliberately conservative parameters for the existing FLT design, we present further results for the response to a steadily (uniformly) moving load when the periodic supports are assumed to be elastic, and then introduce the mass and viscous damping of the periodic supports. Typical support damping significantly moderates the resulting steady deflexion at any load speed, and in particular substantially reduces the magnitude of the resonant response at the critical speed. The linear mathematical analysis is then extended to include the inertia of the load that otherwise moves uniformly along the beam, generating overstability at supercritical speeds – i.e. at load speeds notably above the critical speed predicted for the resonant response when the load inertia is neglected. Neither the resonance nor the overstability should prevent the safe implementation of the FLT design in modern high speed rail systems.