Tom Shearer

Employing pre-stress to generate finite cloaks for antiplane elastic waves

It is shown that nonlinear elastic pre-stress of neo-Hookean hyperelastic materials can be used as a mechanism to generate finite cloaks and thus render objects near-invisible to incoming antiplane elastic waves. This approach appears to negate the requirement for special cloaking metamaterials with inhomogeneous and anisotropic material properties in this case. These properties are induced naturally by virtue of the pre-stress. This appears to provide a mechanism for broadband cloaking since dispersive effects due to metamaterial microstructure will not arise.

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on the geometrical arrangement of their fibrils is derived. The distribution of the crimp angles of the fibrils is used to determine the stress-strain response of a single fascicle, and this stress-strain response is used to determine the form of the strain energy function, the parameters of which can all potentially be directly measured via experiments - unlike those of commonly used strain energy functions such as the Holzapfel-Gasser-Ogden (HGO) model, whose parameters are phenomenological. We compare the new model with the HGO model and show that the new model gives a better match to existing stress-strain data for human patellar tendon than the HGO model, with the average relative error in matching this data when using the new model being 0.053 (compared with 0.57 when using the HGO model), and the average absolute error when using the new model being 0.12 MPa (compared with 0.31 MPa when using the HGO model).

Three-dimensional visualisation of soft biological structures by X-ray computed micro-tomography.

ABSTRACT Whereas the two-dimensional (2D) visualisation of biological samples is routine, three-dimensional (3D) imaging remains a time-consuming and relatively specialised pursuit. Current commonly adopted techniques for characterising the 3D structure of non-calcified tissues and biomaterials include optical and electron microscopy of serial sections and sectioned block faces, and the visualisation of intact samples by confocal microscopy or electron tomography. As an alternative to these approaches, X-ray computed micro-tomography (microCT) can both rapidly image the internal 3D structure of macroscopic volumes at sub-micron resolutions and visualise dynamic changes in living tissues at a microsecond scale. In this Commentary, we discuss the history and current capabilities of microCT. To that end, we present four case studies to illustrate the ability of microCT to visualise and quantify: (1) pressure-induced changes in the internal structure of unstained rat arteries, (2) the differential morphology of stained collagen fascicles in tendon and ligament, (3) the development of Vanessa cardui chrysalises, and (4) the distribution of cells within a tissue-engineering construct. Future developments in detector design and the use of synchrotron X-ray sources might enable real-time 3D imaging of dynamically remodelling biological samples. Summary: MicroCT is a technology that can rapidly image the internal 3D structure of macroscopic biological specimens (including living systems) at sub-micron resolutions. This Commentary discusses four case studies to illustrate its capabilities.

A new strain energy function for modelling ligaments and tendons whose fascicles have a helical arrangement of fibrils.

A new strain energy function for the hyperelastic modelling of ligaments and tendons whose fascicles have a helical arrangement of fibrils is derived. The stress-strain response of a single fascicle whose fibrils exhibit varying levels of crimp throughout its radius is calculated and used to determine the form of the strain energy function. The new constitutive law is used to model uniaxial extension test data for human patellar tendon and is shown to provide an excellent fit, with the average relative error being 9.8%. It is then used to model shear and predicts that the stresses required to shear a tendon are much smaller than those required to uniaxially stretch it to the same strain level. Finally, the strain energy function is used to model ligaments and tendons whose fascicles are helical, and the relative effects of the fibril helix angle, the fascicle helix angle and the fibril crimp variable are compared. It is shown that they all have a significant effect; the fibril crimp variable governs the non-linearity of the stress-strain curve, whereas the helix angles primarily affect its stiffness. Smaller values of the helix angles lead to stiffer tendons; therefore, the model predicts that one would expect to see fewer helical sub-structures in stiff positional tendons, and more in those that are required to be more flexible.

Optimal Contrast Agent Staining of Ligaments and Tendons for X-Ray Computed Tomography

X-ray computed tomography has become an important tool for studying the microstructures of biological soft tissues, such as ligaments and tendons. Due to the low X-ray attenuation of such tissues, chemical contrast agents are often necessary to enhance contrast during scanning. In this article, the effects of using three different contrast agents—iodine potassium iodide solution, phosphotungstic acid and phosphomolybdic acid—are evaluated and compared. Porcine anterior cruciate ligaments, patellar tendons, medial collateral ligaments and lateral collateral ligaments were used as the basis of the study. Three samples of each of the four ligament/tendon types were each assigned a different contrast agent (giving a total of twelve samples), and the progression of that agent through the tissue was monitored by performing a scan every day for a total period of five days (giving a total of sixty scans). Since the samples were unstained on day one, they had been stained for a total of four days by the time of the final scans. The relative contrast enhancement and tissue deformation were measured. It was observed that the iodine potassium iodide solution penetrated the samples fastest and caused the least sample shrinkage on average (although significant deformation was observed by the time of the final scans), whereas the phosphomolybdic acid caused the greatest sample shrinkage. Equations describing the observed behaviour of the contrast agents, which can be used to predict optimal staining times for ligament and tendon X-ray computed tomography, are presented.

Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media.

The effect of a longitudinal stretch and a pressure-induced inhomogeneous radial deformation on the scattering of antiplane elastic waves from a cylindrical cavity is determined. Three popular nonlinear strain energy functions are considered: the neo-Hookean, the Mooney–Rivlin and a two-term Arruda–Boyce model. A new method is developed to analyse and solve the governing wave equations. It exploits their properties to determine an asymptotic solution in the far-field, which is then used to derive a boundary condition to numerically evaluate the equations local to the cavity. This method could be applied to any linear ordinary differential equation whose inhomogeneous coefficients tend to a constant as its independent variable tends to infinity. The effect of the pre-stress is evaluated by considering the scattering cross section. A longitudinal stretch is found to decrease the scattered power emanating from the cavity, whereas a compression increases it. The effect of the pressure difference depends on the strain energy function employed. For a Mooney–Rivlin material, a cavity inflation increases the scattered power and a deflation decreases it; for a neo-Hookean material, the scattering cross section is unaffected by the radial deformation; and for a two-term Arruda–Boyce material, both inflation and deflation are found to decrease the scattered power.

The relative compliance of energy-storing tendons may be due to the helical fibril arrangement of their fascicles

A nonlinear elastic microstructural model is used to investigate the relationship between structure and function in energy-storing and positional tendons. The model is used to fit mechanical tension test data from the equine common digital extensor tendon (CDET) and superficial digital flexor tendon (SDFT), which are used as archetypes of positional and energy-storing tendons, respectively. The fibril crimp and fascicle helix angles of the two tendon types are used as fitting parameters in the mathematical model to predict their values. The outer fibril crimp angles were predicted to be 15.1° ± 2.3° in the CDET and 15.8° ± 4.1° in the SDFT, and the average crimp angles were predicted to be 10.0° ± 1.5° in the CDET and 10.5° ± 2.7° in the SDFT. The crimp angles were not found to be statistically significantly different between the two tendon types (p = 0.572). By contrast, the fascicle helix angles were predicted to be 7.9° ± 9.3° in the CDET and 29.1° ± 10.3° in the SDFT and were found to be statistically highly significantly different between the two tendon types (p < 0.001). This supports previous qualitative observations that helical substructures are more likely to be found in energy-storing tendons than in positional tendons and suggests that the relative compliance of energy-storing tendons may be directly caused by these helical substructures.

A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.

Correction to ‘Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media’

[This corrects the article DOI: 10.1098/rspa.2015.0450.].

Extracellular matrix fragmentation in young, healthy cartilaginous tissues.

Although the composition and structure of cartilaginous tissues is complex, collagen II fibrils and aggrecan are the most abundant assemblies in both articular cartilage (AC) and the nucleus pulposus (NP) of the intervertebral disc (IVD). Whilst structural heterogeneity of intact aggrecan ( containing three globular domains) is well characterised, the extent of aggrecan fragmentation in healthy tissues is poorly defined. Using young, yet skeletally mature (18-30 months), bovine AC and NP tissues, it was shown that, whilst the ultrastructure of intact aggrecan was tissue-dependent, most molecules (AC: 95 %; NP: 99.5 %) were fragmented (lacking one or more globular domains). Fragments were significantly smaller and more structurally heterogeneous in the NP compared with the AC (molecular area; AC: 8543 nm2; NP: 4625 nm2; p < 0.0001). In contrast, fibrillar collagen appeared structurally intact and tissue-invariant. Molecular fragmentation is considered indicative of a pathology; however, these young, skeletally mature tissues were histologically and mechanically (reduced modulus: AC: ≈ 500 kPa; NP: ≈ 80 kPa) comparable to healthy tissues and devoid of notable gelatinase activity (compared with rat dermis). As aggrecan fragmentation was prevalent in neonatal bovine AC (99.5 % fragmented, molecular area: 5137 nm2) as compared with mature AC (95.0 % fragmented, molecular area: 8667 nm2), it was hypothesised that targeted proteolysis might be an adaptive process that modified aggrecan packing (as simulated computationally) and, hence, tissue charge density, mechanical properties and porosity. These observations provided a baseline against which pathological and/or age-related fragmentation of aggrecan could be assessed and suggested that new strategies might be required to engineer constructs that mimic the mechanical properties of native cartilaginous tissues.