A cohesive interface approach to model the inter-lamellar behaviour of the intervertebral disc annulus fibrosus

INTRODUCTION - The method of representing the inter-lamellar behaviour in Finite Element (FE) models of the disc has been shown to affect the gross disc mechanics [1]. This study aim was to develop an approach to produce inter-lamellar models of the Annulus Fibrosus, i.e. constitutive models of interaction between lamellae. METHODS (Figure 1) - Mature ovine intervertebral discs were used in the present study as a model for the discs of the human lumbar spine. The annulus fibrosus was dissected and radial slices were subjected to mechanical tests. Images were taken under conventional and differential interference contrast (DIC) microscopy. The images before and during loading were used to produce 2D computational models of the annulus structural behaviour. The DIC images were analysed (ImageJ, U.S. N.I.H., USA) to extract the edges between lamellae and converted into quadrilateral FE meshes (ScanIP 5.1, Simpleware Ltd, UK, and Abaqus 6.12, Dassault Systèmes). Each lamella was modelled with Holzapfel’s anisotropic hyperelastic constitutive behaviour, with one fibre orientation per lamella. For this, model parameters were identified to fit to experimental data on the behaviour of tissue components (the proteoglycan matrix and the collagen fibres) [1]. Several modelling hypotheses were tested for the inter-lamellar behaviour: fully bonded conditions, simple frictionless behaviour, friction behaviour with a Coulomb contact, and delamination behaviour, with or without friction, through the use of a cohesive interface model. Displacement boundary conditions were applied reproducing the measured external displacement on the conventional microscopy images. Adequacy of the inter-lamellar behaviour model was assessed by comparing computational and experimental deformed geometries, specifically the change in lamellar interfaces

Spatial and temporal patterns of root distribution in developing stands of four woody crop species grown with drip irrigation and fertilization.

Abstract In forest trees, roots mediate such significant carbon fluxes as primary production and soil C02 efflux. Despite the central role of roots in these critical processes, information on root distribution during stand establishment is limited, yet must be described to accurately predict how various forest types, which are growing with a range of resource limitations, might respond to environmental change. This study reports root length density and biomass development in young stands of eastern cottonwood (Populus deltoidies Bartr.) and American sycamore (Platanus occidentalis L.) that have narrow, high resource site requirements, and compares them with sweetgum (Liquidambar styraczj7ua L.) and loblolly pine (Pinus taeda L.), which have more robust site requirements. Fine roots (5 mm) were sampled to determine spatial distribu-tion in response to fertilizer and irrigation treatments delivered through drip irrigation tubes. Root length density and biomass were predominately controlled by stand development, depth and proximity to drip tubes. After accounting for this spatial and temporal variation, there was a significant increase in RLD with fertilization and irrigation for all genotypes. The response to fertilization was greater than that of irrigation. Both fine and coarse roots responded positively to resources delivered through the drip tube, indicating a wholeroot- system response to resource enrichment and not just a feeder root response. The plastic response to drip tube water and nutrient enrichment demonstmte the capability of root systems to respond to supply heterogeneity by increasing acquisition surface. Fineroot biomass, root density and specific root length were greater for broadleaved species than pine. Roots of all genotypes explored the rooting volume within 2 years, but this occurred faster and to higher root length densities in broadleaved species, indicating they had greater initial opportunity for resource acquisition than pine. Sweetgum's root characteristics and its response to resource availability were similar to the other broadleaved species, despite its hnctional resemblance to pine regarding robust site requirements. It was concluded that genotypes, irrigation arid fertilization significantly influenced tree root system development, which varied spatially in response to resource-supply heterogeneity created by dnp tubes. Knowledge of spatial and temporal patterns of root distribution in these stands will be used to interpret nutrient acquisition and soil respiration measurements

Derivation of inter-lamellar behaviour of the intervertebral disc annulus

The inter-lamellar connectivity of the annulus fibrosus in the intervertebral disc has been shown to affect the prediction of the overall disc behaviour in computational models. Using a combined experimental and computational approach, the inter-lamellar mechanical behaviour of the disc annulus was investigated under conditions of radial loading. Twenty-seven specimens of anterior annulus fibrosus were dissected from twelve discs taken from four frozen ovine thoracolumbar spines. Specimens were grouped depending on their radial provenance within the annulus fibrosus. Standard tensile tests were performed. In addition, micro-tensile tests under microscopy were used to observe the displacement of the lamellae and inter-lamellar connections. Finite elements models matching the experimental protocols were developed with specimen-specific geometries and boundary conditions assuming a known lamellar behaviour. An optimisation process was used to derive the interface stiffness values for each group. The assumption of a linear cohesive interface was used to model the behaviour of the inter-lamellar connectivity. The interface stiffness values derived from the optimisation process were consistently higher than the corresponding lamellar values. The interface stiffness values of the outer annulus were 43% to 75% higher than those of the inner annulus. Tangential stiffness values for the interface were 6% to 39% higher than normal stiffness values within each group and similar to values reported by other investigators. These results reflect the intricate fibrous nature of the inter-lamellar connectivity and provide values for the representation of the inter-lamellar behaviour at a continuum level