An overview of elastic waveguides with dynamic sub-structures.

The dynamic response of elastic waveguides is important for a wide range of applications that involve dispersive waves as well as wave localization. In particular, a case of special interest relates to waveguides subjected to moving loads. In the case where the elongated structure includes a sequence of built-in resonators, the range of resonance regimes may be extended accordingly. The present paper gives an overview of several mathematical formulations that connect Floquet theory to the dynamic response of multi-scale waveguides, which include inertial sub-structures subjected to external forces. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions.

We present formal asymptotic approximations of fields representing the in-plane dynamic response of elastic solids containing clusters of closely interacting small rigid inclusions. For finite densely perforated bodies, the asymptotic scheme is developed to approximate the eigenfrequencies and the associated eigenmodes of the elastic medium with clamped boundaries. The asymptotic algorithm is also adapted to address the scattering of in-plane waves in infinite elastic media containing dense clusters. The results are accompanied by numerical simulations that illustrate the accuracy of the asymptotic approach. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

Modelling of seismic assessment for large geological systems.

A new approach to seismic analysis has been introduced and demonstrated for a sequence of recent seismic events recorded in the Blackpool region of Lancashire, UK. The seismic activity, induced by an industrial hydraulic fracturing at a depth exceeding 2 km, had the extent of registered surface elastic vibrations reaching a distance exceeding 15 km. The analysis is based on the study of elastic fields, three-dimensional extrapolations of the landscape and the novel reconstruction of a three-dimensional digital model of seismic map boundaries and vertical profiles. The verification of the proposed approach is carried out via the comparison with published data of the Blackpool seismic events, combined with the new spectral analysis linked to the identified regions of seismic activity. The latter was accompanied by a finite-element simulation of vibrations for an elastic layer of variable thickness, approximating the test region. The analysis and numerical modelling have demonstrated consistency with the dynamic nature of structural stratification of the geological systems, and in addition, the predictive nature of the modelling work was demonstrated by the comparison of the model eigenmodes with the published parameters of registered earthquakes in the Blackpool area. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Fibril density reduction in keratoconic corneas.

This study aims to estimate the reduction in collagen fibril density within the central 6 mm radius of keratoconic corneas through the processing of microstructure and videokeratography data. Collagen fibril distribution maps and topography maps were obtained for seven keratoconic and six healthy corneas, and topographic features were assessed to detect and calculate the area of the cone in each keratoconic eye. The reduction in collagen fibril density within the cone area was estimated with reference to the same region in the characteristic collagen fibril maps of healthy corneas. Together with minimum thickness and mean central corneal refractive power, the cone area was correlated with the reduction in the cone collagen fibrils. For the corneas considered, the mean area of keratoconic cones was 3.30 ± 1.90 mm2. Compared with healthy corneas, fibril density in the cones of keratoconic corneas was lower by as much as 35%, and the mean reduction was 17 ± 10%. A linear approximation was developed to relate the magnitude of reduction to the refractive power, minimum corneal thickness and cone area (R2 = 0.95, p < 0.001). Outside the cone area, there was no significant difference between fibril arrangement in healthy and keratoconic corneas. The presented method can predict the mean fibril density in the keratoconic eye's cone area. The technique can be applied in microstructure-based finite-element models of the eye to regulate its stiffness level and the stiffness distribution within the areas affected by keratoconus.

Waves in elastic bodies with discrete and continuous dynamic microstructure.

This paper presents a unified approach to the modelling of elastic solids with embedded dynamic microstructures. General dependences are derived based on Green's kernel formulations. Specifically, we consider systems consisting of a master structure and continuously or discretely distributed oscillators. Several classes of connections between oscillators are studied. We examine how the microstructure affects the dispersion relations and determine the energy distribution between the master structure and microstructures, including the vibration shield phenomenon. Special attention is given to the comparative analysis of discrete and continuous distributions of the oscillators, and to the effects of non-locality and trapped vibrations. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

Numerical Simulation of Corneal Fibril Reorientation in Response to External Loading.

PURPOSE: To simulate numerically the collagen fibril reorientation observed experimentally in the cornea. METHODS: Fibril distribution in corneal strip specimens was monitored using X-ray scattering while under gradually increasing axial loading. The data were analysed at each strain level in order to quantify the changes in the angular distribution of fibrils with strain growth. The resulting relationship between stain and fibril reorientation was adopted in a constitutive model to control the mechanical anisotropy of the tissue material. The outcome of the model was validated against the experimental measurements before using the model in simplified representations of two surgical procedures. RESULTS: The numerical model was able to reproduce the experimental measurements of specimen deformation and fibril reorientation under uniaxial loading with errors below 8.0%. With tissue removal simulated in a full eye numerical model, fibril reorientation could be predicted around the affected area, and this change both increased with larger tissue removal and reduced gradually away from that area. CONCLUSION: The presented method can successfully simulate fibril reorientation with changes in the strain regime affecting cornea tissue. Analyses based on this method showed that fibrils tend to align parallel to the tissue cut following keratoplasty operations. With the ability to simulate fibril reorientation, numerical modelling can have a greater potential in modelling the behaviour following surgery and injury to the cornea.

Microstructure-based numerical simulation of the mechanical behaviour of ocular tissue.

This paper aims to present a novel full-eye biomechanical material model that incorporates the characteristics of ocular tissues at microstructural level, and use the model to analyse the age-related stiffening in tissue behaviour. The collagen content in ocular tissues, as obtained using X-ray scattering measurements, was represented by sets of Zernike polynomials that covered both the cornea and sclera, then used to reconstruct maps of collagen fibril magnitude and orientation on the three-dimensional geometry of the eye globe. Fine-mesh finite-element (FE) models with eye-specific geometry were built and supported by a user-defined material model (UMAT), which considered the regional variation of fibril density and orientation. The models were then used in an iterative inverse modelling study to derive the material parameters that represent the experimental behaviour of ocular tissues from donors aged between 50 and 90 years obtained in earlier ex vivo studies. Sensitivity analysis showed that reducing the number of directions that represented the anisotropy of collagen fibril orientation at each X-ray scattering measurement point from 180 to 16 would have limited and insignificant effect on the FE solution (0.08%). Inverse analysis resulted in material parameters that provided a close match with experimental intraocular pressure-deformation behaviour with a root mean square of error between 3.6% and 4.3%. The results also demonstrated a steady increase in mechanical stiffness in all ocular regions with age. A constitutive material model based on distributions of collagen fibril density and orientation has been developed to enable the accurate representation of the biomechanical behaviour of ocular tissues. The model offers a high level of control of stiffness and anisotropy across ocular globe, and therefore has the potential for use in planning surgical and medical procedures.

Analysis of X-ray scattering microstructure data for implementation in numerical simulations of ocular biomechanical behaviour.

This study aimed to analyse microstructure data on the density and orientation of collagen fibrils in whole eye globes and to propose an effective method for the preparation of data for use in numerical simulations of the eye's biomechanical performance. Wide-angle X-ray scattering was applied to seven healthy ex-vivo human eyes. Each eye was dissected into an anterior and a posterior cup, and radial incisions were used to flatten the tissue before microstructure characterisation. A method was developed to use the microstructure data obtained for the dissected tissue to build realistic 3D maps of fibril density and orientation covering the whole eye globe. At the central cornea, 61.5±2.3% of fibrils were aligned within 45° sectors surrounding the two orthogonal directions. In contrast, more than one-third of the total fibril content was concentrated along the circumferential direction at the limbus (37.0±2.4%) and around the optic nerve head (34.8±2.1%). The insertion locations of the four recti muscles exhibited a preference in the meridional direction near the equator (38.6±3.9%). There was also a significant difference in fibril density between the limbus and other regions (ratio = 1.91±0.45, p <0.01 at the central cornea and ratio = 0.80±0.21, p <0.01 at the posterior pole). Characterisation of collagen fibril density and orientation across the whole ocular surface has been possible but required the use of a technique that involved tissue dissection and hence caused tissue damage. The method presented in this paper aimed to minimise the effect of dissection on the quality of obtained data and was successful in identifying fibril distribution trends that were compatible with earlier studies, which concentrated on localised areas of the ocular globe.