Aleksey V. Pichugin

High-frequency homogenization for periodic media

An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.

Four simplified gradient elasticity models for the simulation of dispersive wave propagation

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial-temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems.

Eigenvalue of a semi-infinite elastic strip

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (Rayleigh–Lamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Grams determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν=0, there exists another value of ν≈0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism.

Investigation of shock waves in explosive blasts using fibre optic pressure sensors

We describe miniature all-optical pressure sensors, fabricated by wafer etching techniques, less than 1 mm2 in overall cross-section with rise times in the µs regime and pressure ranges typically 900 kPa (9 bar). Their performance is suitable for experimental studies of the pressure–time history for test models exposed to shocks initiated by an explosive charge. The small size and fast response of the sensors promises higher quality data than has been previously available from conventional electrical sensors, with potential improvements to numerical models of blast effects. Results from blast tests are presented in which up to six sensors were multiplexed, embedded within test models in a range of orientations relative to the shock front.

An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

An asymptotically consistent two–dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long–wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading–order and refined higher–order equations for the mid–surface deflection are derived. In the case of zero normal initial static stress and in–plane tension, the leading–order equation reduces to the classical membrane equation, with its refined counterpart also being obtained. The theory is applied to a one–dimensional edge loading problem for a semi–infinite plate. In doing so, the leading– and higher–order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi–front. A solution is derived by using the method of matched asymptotic expansions.

Optimum structure to carry a uniform load between pinned supports: exact analytical solution

Recent numerical evidence indicates that a parabolic funicular is not necessarily the optimal structural form to carry a uniform load between pinned supports. When the constituent material is capable of resisting equal limiting tensile and compressive stresses, a more efficient structure can be identified, comprising a central parabolic section and networks of truss bars emerging from the supports. In the current article, a precise geometry for this latter structure is identified, avoiding the inconsistencies that render the parabolic form non-optimal. Explicit analytical expressions for the geometry, stress and virtual-displacement fields within and above the structure are presented. Furthermore, a suitable displacement field below the structure is computed numerically and shown to satisfy the Michell–Hemp optimality criteria, hence formally establishing the global optimality of this new structural form.

Extensional edge waves in pre-stressed incompressible plates

Consider a semi-infinite layer of an incompressible elastic material subjected to a finite static homogeneous pre-deformation. This paper describes the long-wave low-frequency limiting behaviour of extensional edge waves that may propagate along the free edge of such a layer. The analysis is done by constructing an asymptotic plate theory that describes the extensional motion of thin pre-stressed incompressible plates. The secular equation for edge waves propagating along one of the principal axes of the pre-stress is studied for bi-axial deformations and several specific material models. We show that certain combinations of material models and configurations of pre-stresses do not support the propagation of edge waves, which mirrors the situation with surface waves in pre-stressed media. More unusually, we present a seemingly first explicit example of the non-unique edge wave solutions existing in thin pre-stressed plates.

The transition between Neumann and Dirichlet boundary conditions in isotropic elastic plates

The transition from Neumann (traction-free) to Dirichlet (fixed-face) boundary conditions is investigated in respect of wave propagation in a linear isotropic elastic layer. Attention is focused on the implications of such a transition on the dispersion curve branches within the long-wave region. The formation of low-frequency band gap that is expected to exist in layers with Dirichlet boundary condition is shown to be caused by different mechanisms in anti-symmetric and symmetric cases. Certain implications to shortwave propagation in the layer are also investigated. The study includes both a numerical investigation and a multi-parameter asymptotic analysis.

Extensional edge modes in elastic plates and shells

The recently discovered undamped localized mode at the end of an elastic strip is demonstrated to be particularly relevant in the plane stress setting, where it exists for the Poisson ratio 0.29. This paper also emphasizes the difference between low-frequency edge modes, typically characterized by low variation across the plate (or shell) thickness, and high-frequency edge modes, whose natural frequencies are of the order of thickness resonance frequencies.

Anti-symmetric motion of a pre-stressed incompressible elastic layer near shear resonance

A two-dimensional model is derived for anti-symmetric motion in the vicinity of the shear resonance frequencies in a pre-stressed incompressible elastic plate. The method of asymptotic integration is used and a second-order solution, for infinitesimal displacement components and incremental pressure, is obtained in terms of the long-wave amplitude. The leading-order hyperbolic governing equation for the long-wave amplitude is observed to be not wave-like for certain pre-stressed states, with time and one of the in-plane spatial variables swapping roles. This phenomenon is shown to be intimately related to the possible existence of negative group velocity at low wave number, i.e. in the vicinity of shear resonance frequencies.