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The biomechanics of the optic nerve head is assumed to play an important role in ganglion cell loss in glaucoma. Organized collagen fibrils form complex networks that introduce strong anisotropic and nonlinear attributes into the constitutive response of the peripapillary sclera (PPS) and lamina cribrosa (LC) dominating the biomechanics of the optic nerve head. The recently presented computational remodeling approach (Grytz and Meschke in Biomech Model Mechanobiol 9:225-235, 2010) was used to predict the micro-architecture in the LC and PPS, and to investigate its impact on intraocular pressure-related deformations. The mechanical properties of the LC and PPS were derived from a microstructure-oriented constitutive model that included the stretch-dependent stiffening and the statistically distributed orientations of the collagen fibrils. Biomechanically induced adaptation of the local micro-architecture was captured by allowing collagen fibrils to be reoriented in response to the intraocular pressure-related loading conditions. In agreement with experimental observations, the remodeling algorithm predicted the existence of an annulus of fibrils around the scleral canal in the PPS, and a predominant radial orientation of fibrils in the periphery of the LC. The peripapillary annulus significantly reduced the intraocular pressure-related expansion of the scleral canal and shielded the LC from high tensile stresses. The radial oriented fibrils in the LC periphery reinforced the LC against transversal shear stresses and reduced LC bending deformations. The numerical approach presents a novel and reasonable biomechanical explanation of the spatial orientation of fibrillar collagen in the optic nerve head.

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Glaucoma is among the leading causes of blindness worldwide. The ocular disease is characterized by irreversible damage of the retinal ganglion cell axons at the level of the lamina cribrosa (LC). The LC is a porous, connective tissue structure whose function is believed to provide mechanical support to the axons as they exit the eye on their path from the retina to the brain. Early experimental glaucoma studies have shown that the LC remodels into a thicker, more posterior structure which incorporates more connective tissue after intraocular pressure (IOP) elevation. The process by which this occurs is unknown. Here we present a microstructure motivated growth and remodeling (G&R) formulation to explore a potential mechanism of these structural changes. We hypothesize that the mechanical strain experienced by the collagen fibrils in the LC stimulates the G&R response at the micro-scale. The proposed G&R algorithm controls collagen fibril synthesis/degradation and adapts the residual strains between collagen fibrils and the surrounding tissue to achieve biomechanical homeostasis. The G&R algorithm was applied to a generic finite element model of the human eye subjected to normal and elevated IOP. The G&R simulation underscores the biomechanical need for a LC at normal IOP. The numerical results suggest that IOP elevation leads to LC thickening due to an increase in collagen fibril mass, which is in good agreement with experimental observations in early glaucoma monkey eyes. This is the first study to demonstrate that a biomechanically-driven G&R mechanism can lead to the LC thickening observed in early experimental glaucoma.

SUMMARYThis paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12:277 -296) mesh objective 2D and 3D ÿnite element formulations are developed using linear and quadratic 2D elements as well as 8-noded 3D elements. In the formulation of the ÿnite-element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter deÿning the amplitude of the displacement jump within the ÿnite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non-linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of ÿxed discontinuities. It is shown, that the rotating localization approach reduces locking e ects observed in analyses based on ÿxed localization directions. The applicability of the proposed SDA ÿnite-element model as well as its numerical performance is investigated by means of a three-dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force.

SUMMARYThe extended finite element method (X-FEM) has proven to be capable of simulating cracking and crack propagation in quasi-brittle materials, such as cement paste or concrete, without the need for re-meshing. In the framework of the X-FEM cracks are represented as surfaces of discontinuous displacements continuously propagating through finite elements. Since crack path continuity is required in X-FEM-based analyses, the reliability of numerical analyses of cracked structures crucially depends on the correct prediction of the crack path and, consequently, on the criterion used for the determination of the crack propagation direction. In this paper four different crack propagation criteria proposed in the literature are investigated including two local and two global criteria. The two local criteria include an averaged stress criterion and the maximum circumferential stress criterion based on the linear elastic fracture mechanics.

A material model for plain concrete formulated within the framework of multisurface elastoplasticity-damage theory is proposed in this paper. Anisotropic sti ness degradation as well as inelastic deformations are taken into account. The applicability of the model encompasses cracking as well as the non-linear response of concrete in compression. The e ect of di erent softening laws on the stress-strain relationship and on the dissipation is investigated in the context of a 1D model problem. The integration of the evolution laws is based on the standard return map scheme. Further computational issues include the stability of the local iteration procedure and the treatment of the apex region of the damage surface. The model is employed for re-analyses of a cylinder splitting test and of a notched concrete beam. Results from the composite elastoplastic-damage model are compared with test results and results from other material models for concrete, respectively. ? 1998 John Wiley & Sons, Ltd.KEY WORDS: damage; plasticity; concrete; cracking; Rankine criterion; ÿnite element analysis INTRODUCTORY REMARKSBrittle materials such as geomaterials and concrete exhibit distributed as well as localized degradation of the mechanical properties with increasing loading. The phenomenological response of plain concrete subjected to predominantly tensile stresses is characterized by a more or less linear ascending branch of the stress-displacement curve followed by a progressively decreasing residual strength resulting in the formation of macrodefects in the form of discrete cracks. When unloaded in the post-peak regime, non-recoverable deformations as well as a degradation of the sti ness of the unloading branch is observed. From a microstructural point of view, the progressive degradation of the elastic moduli, commonly referred to as damage, is the result of growth and coalescence of existing microcracks and microvoids along the interfaces of the cement paste and the aggregates. This deterioration process prevents a complete closure of microcracks in unloading processes. As a consequence, permanent strains develop. On the phenomenological level, this e ect is often modelled by means of classical plasticity theory. Depending on the level of hydrostatic

Organized collagen fibrils form complex networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of human eye tissues. Physiological adaptation of the collagen network and the mechanical condition within biological tissues are complex and mutually dependent phenomena. In this contribution, a computational model is presented to investigate the interaction between the collagen fibril architecture and mechanical loading conditions in the corneo-scleral shell. The biomechanical properties of eye tissues are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. Biomechanically induced remodeling of the collagen network is captured on the meso-scale by allowing for a continuous re-orientation of preferred fibril orientations and a continuous adaptation of the fibril dispersion. The presented approach is applied to a numerical human eye model considering the cornea and sclera. The predicted fibril morphology correlates well with experimental observations from X-ray scattering data.