Dr Olga Barrera

This work presents a theoretical and numerical study of the flow of the interstitial fluid saturating a porous medium, principally aimed at modeling a bio-mimetic material and assumed to experience a dynamic regime different from the Darcian one, as is typically hypothesized in biomechanical scenarios. The main aspect of our research is the conjecture according to which, for a particular mechanical state of the porous medium, the fluid exhibits two types of deviation from Darcy’s law. One is due to the inertial forces characterizing the pore scale dynamics of the fluid. This aspect can be resolved by turning to the Forchheimer correction to Darcy’s law, which introduces non-linearities in the relationship between the fluid filtration velocity and the dissipative forces describing the interactions between the fluid and the solid matrix. The second source of discrepancies from classical Darcy’s law emerges, for example, when pore scale disturbances to the flow, such as obstructions of the fluid path or clogging of the pores, result in a time delay between drag force and filtration velocity. Recently, models have been proposed in which such delay is described through constitutive laws featuring fractional operators. Whereas, to the best of our knowledge, the aforementioned behaviors have been studied separately or in small deformations, we present a model of fluid flow in a deformable porous medium undergoing large deformations in which the fluid motion obeys a fractionalized Forchheimer’s correction. After reviewing Forchheimer’s formulation, we present a fractionalization of the Darcy–Forchheimer law, and we explain the numerical procedure adopted to solve the highly non-linear boundary value problem resulting from the presence of the two considered deviations from the Darcian regime. We complete our study by highlighting the way in which the fractional order of the model tunes the magnitude of the pore pressure and fluid filtration velocity.

The meniscal tissue is a layered material with varying properties influenced by collagen content and arrangement. Understanding the relationship between structure and properties is crucial for disease management, treatment development, and biomaterial design. The internal layer of the meniscus is softer and more deformable than the outer layers, thanks to interconnected collagen channels that guide fluid flow. To investigate these relationships, we propose an integrated approach that combines Computational Fluid Dynamics (CFD) with Image Analysis (CFD-IA). We analyze fluid flow in the internal architecture of the human meniscus across a range of inlet velocities (0.1 mm/s to 1.6 m/s) using high-resolution 3D micro-computed tomography scans. Statistical correlations are observed between architectural parameters (tortuosity, connectivity, porosity, pore size) and fluid flow parameters (Re number distribution, permeability). Some channels exhibit Re values of 1400 at an inlet velocity of 1.6 m/s, and a transition from Darcy’s regime to a non-Darcian regime occurs around an inlet velocity of 0.02 m/s. Location-dependent permeability ranges from 20-32 Darcy. Regression modelling reveals a strong correlation between fluid velocity and tortuosity at high inlet velocities, as well as with channel diameter at low inlet velocities. At higher inlet velocities, flow paths deviate more from the preferential direction, resulting in a decrease in the concentration parameter by an average of 0.4. This research provides valuable insights into the fluid flow behaviour within the meniscus and its structural influences. 3D models and image stack are available to download at https://doi.org/10.5281/zenodo.10401592

This paper focuses on the origin of the poroelastic anisotropic behaviour of the meniscal tissue and its spatially varying properties. We present confined compression creep test results on samples extracted from three parts of the tissue (Central body, Anterior horn and Posterior horn) in three orientations (Circumferential, Radial and Vertical). We show that a poroelastic model in which the fluid flow evolution is ruled by non-integer order operators (fractional Darcy’s law) provides accurate agreement with the experimental creep data. The model is validated against two additional sets of experimental data: stress relaxation and fluid loss during the consolidation process measured as weight reduction. Results show that the meniscus can be considered as a transversely isotropic poroelastic material. This behaviour is due to the fluid flow rate being about three times higher in the circumferential direction than in the radial and vertical directions in the body region of the meniscus. The 3D fractional poroelastic model is implemented in the finite element software to estimate the weight loss during the confined compression tests.

Introduction: Complex soft tissues, such as knee meniscus, play a crucial role in mobility and joint health but are incredibly difficult to repair and replace when damaged. This difficulty is due to the highly hierarchical and porous nature of the tissues, which, in turn, leads to their unique mechanical properties that provide joint stability, load redistribution, and friction reduction. To design tissue substitutes, the internal architecture of the native tissue needs to be understood and replicated.

Methods: We explore a combined audiovisual approach, a so-called transperceptual approach, to generate artificial architectures mimicking the native architectures. The proposed methodology uses both traditional imagery and sound generated from each image to rapidly compare and contrast the porosity and pore size within the samples. We have trained and tested a generative adversarial network (GAN) on 2D image stacks of a knee meniscus. To understand how the resolution of the set of training images impacts the similarity of the artificial dataset to the original, we have trained the GAN with two datasets. The first consists of 478 pairs of audio and image files for which the images were downsampled to 64 × 64 pixels. The second dataset contains 7,640 pairs of audio and image files for which the full resolution of 256 × 256 pixels is retained, but each image is divided into 16 square sections to maintain the limit of 64 × 64 pixels required by the GAN.

Results: We reconstructed the 2D stacks of artificially generated datasets into 3D objects and ran image analysis algorithms to characterize the architectural parameters statistically (pore size, tortuosity, and pore connectivity). Comparison with the original dataset showed that the artificially generated dataset based on the downsampled images performs best in terms of parameter matching, achieving between 4% and 8% of the mean of grayscale values of the pixels, mean porosity, and pore size of the native dataset.

Discussion: Our audiovisual approach has the potential to be extended to larger datasets to explore how similarities and differences can be audibly recognized across multiple samples.

Mathematical constitutive models are crucially important for the real viscoelastic solid materials in academic investigation and engineering application. The viscoelastic solid materials and their composites are extensively applied in practice engineering, including biomechanics, aviation, aerospace, and geology, and it has captured more and more attention in recent years. Proper application and selection for the constitutive models of the solid viscoelastic materials will be directly influenced by the investigation accuracy in analysis. The aim of this monograph is to essentially describe the development and evolution of the constitutive equations for viscoelastic solid materials in terms of their mechanics and physical behaviors. For isotropic viscoelastic material, its physical behaviors and mechanical properties are influenced by the environmental effects and external exaction factors. In this report, more than 14 classical and fundamental types/categories of the constitutive models for the real viscoelastic solid material are discussed and reviewed. Physical and mechanics behaviors in the time-, temperature-, moisture-, and frequency-domain are provided. Moreover, the development of the time-temperature-, frequency-temperature-, temperature-moisture-, and pressure-time-dependent constitutive models are also presented. We intend to illustrate and emphasize the mathematical and graphical aspects related to the constitutive models for the real viscoelastic solid material, serving as a foundation reference in future academic investigation and practical application.

We develop a novel a posteriori error estimator for the L2 error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using rational approximations which allow to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank–Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. In addition we propose an algorithm to adapt both the finite element mesh and the rational scheme in order to balance the discretization errors. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy

This work focuses on the constitutive modelling of damage development in particle strengthened materials in the presence of hydrogen. We apply the model to an area at the interface of a dissimilar weld of 8630 steel/IN625 nickel alloy which is known as the ’featureless region’. This region contains an array of M7C3 carbides each measuring about 40nm. Cleavage-like fracture has been observed only in the presence of hydrogen and it is attributed to a combination of two types of hydrogen embrittlement mechanisms: hydrogen-induced decohesion (HID) along the M7C3-matrix interface of and a ductile-type fracture (Hydrogen Enhanced Local Plasticity, HELP). Modelling the constitutive behaviour of this region at a continuum level is not appropriate as the major role in the material response is the interaction of the carbide particles with dislocations which is captured at the mesoscopic level. Here we propose a constitutive model of the ”featureless zone” that accurately represents the effect of hydrogen on the constitutive response of the M7C3 region at the mesoscopic scale. Hydrogen enhances the evolution of dislocations around the M7C3 carbides, therefore the stress locally increases affecting the interfacial strength. The result is that, in the region of high hydrogen concentration, the material exhibits softening

In this study, we observe that the poromechanical parameters in human meniscus vary spatially throughout the tissue. The response is anisotropic and the porosity is functionally graded. To draw these conclusions, we measured the anisotropic permeability and the “aggregate modulus” of the tissue, i.e., the stiffness of the material at equilibrium, after the interstitial fluid has ceased flowing. We estimated those parameters within the central portion of the meniscus in three directions (i.e., vertical, radial and circumferential) by fitting an enhanced model on stress relation confined compression tests. We noticed that a classical biphasic model was not sufficient to reproduce the observed experimental behaviour. We propose a poroelastic model based on the assumption that the fluid flow inside the human meniscus is described by a fractional porous medium equation analogous to Darcy’s law, which involves fractional operators. The fluid flux is then time-dependent for a constant applied pressure gradient (in contrast with the classical Darcy’s law, which describes a time independent fluid flux relation). We show that a fractional poroelastic model is well-suited to describe the flow within the meniscus and to identify the associated parameters (i.e., the order of the time derivative and the permeability). The results indicate that mean values of λβ, β in the central body are λβ = 5.5443 × 10−10 m4 Ns1−β , β = 0.0434, while, in the posterior and anterior regions, are λβ = 2.851 × 10−10 m4 Ns 1−β , β = 0.0326 and λβ = 1.2636 × 10−10 m4 Ns 1−β , β = 0.0232, respectively. Furthermore, numerical simulations show that the fluid flux diffusion is facilitated in the central part of the meniscus and hindered in the posterior and anterior regions.

This paper presents an unified mathematical and computational framework for mechanics-coupled “anomalous” transport phenomena in porous media. The anomalous diffusion is mainly due to variable fluid flow rates caused by spatially and temporally varying permeability. This type of behaviour is described by a fractional pore pressure diffusion equation. The diffusion transient phenomena is significantly affected by the order of the fractional operators. In order to solve 3D consolidation problems of large scale structures, the fractional pore pressure diffusion equation is implemented in a finite element framework adopting the discretised formulation of fractional derivatives given by Grunwald–Letnikov (GL). Here the fractional pore pressure diffusion equation is implemented in the commercial software Abaqus through an open-source UMATHT subroutine. The similarity between pore pressure, heat and hydrogen transport is also discussed in order to show that it is possible to use the coupled thermal-stress analysis to solve fractional consolidation problems.

An innovative technique, called conversion, is introduced to model circumferential cracks in thin cylindrical shells. The semi-analytical finite element method is applied to investigate the modal deformation of the cylinder. An element including the crack is divided into three sub-elements with four nodes in which the stiffness matrix is enriched. The crack characteristics are included in the finite element method relations through conversion matrices and a rotational spring corresponding to the crack. Conversion matrices obtained by applying continuity conditions at the crack tip are used to transform displacements of the middle nodes to those of the main nodes. Moreover, another technique, called spring set, is represented based on a set of springs to model the crack as a separated element. Components of the stiffness matrix related to the separated element are incorporated while the geometric boundary conditions at the crack tip are satisfied. The effects of the circumferential mode number, the crack depth and the length of the cylinder on the critical buckling load are investigated. Experimental tests, ABAQUS modeling and results from literature are used to verify and validate the results and derived relations. In addition, the crack effect on the natural frequency is examined using the vibration analysis based on the conversion technique.

The knee meniscus is a highly porous structure which exhibits a grading architecture through the depth of the tissue. The superficial layers on both femoral and tibial sides are constituted by a fine mesh of randomly distributed collagen fibers while the internal layer is constituted by a network of collagen channels of a mean size of 22.14 μμm aligned at a 30∘30∘ inclination with respect to the vertical. Horizontal dog-bone samples extracted from different depths of the tissue were mechanically tested in uniaxial tension to examine the variation of elastic and viscoelastic properties across the meniscus. The tests show that a random alignment of the collagen fibers in the superficial layers leads to stiffer mechanical responses (E = 105 and 189 MPa) in comparison to the internal regions (E = 34 MPa). All regions exhibit two modes of relaxation at a constant strain (τ1=6.4τ1=6.4 to 7.7 s, τ2τ2 = 49.9 to 59.7 s).

The meniscus is an integral part of the human knee, preventing joint degradation by distributing load from the femoral condyles to the tibial plateau. Recent qualitative studies suggested that the meniscus is constituted by an intricate net of collagen channels inside which the fluid flows during loading. The aim of this study is to describe in detail the structure in which this fluid flows by quantifying the orientation and morphology of the collagen channels of the meniscal tissue. A 7 mm cylindrical sample, extracted vertically from the central part of a lateral porcine meniscus was freeze-dried and scanned using the highest-to-date resolution Microscopic Computed Tomography. The orientation of the collagen channels, their size and distribution was calculated. Comparisons with confocal multi-photon microscopy imaging performed on portions of fresh tissue have shown that the freeze-dried procedure adopted here ensures that the native architecture of the tissue is maintained. Sections of the probe at different heights were examined to determine differences in composition and structure along the sample from the superficial to the internal layers. Results reveal a different arrangement of the collagen channels in the superficial layers with respect to the internal layers with the internal layers showing a more ordered structure of the channels oriented at 30∘∘ with respect to the vertical, a porosity of 66.28% and the mean size of the channels of 22.14 μmμm.

This paper presents a slicing technique useful to prepare precise and repeatable samples from organs which are irregularly shaped and highly heterogeneous for mechanical testing and advanced microscopy observation. The suggested technique does not seem to influence the internal microstructure and it is employed here for testing specimens cut from different region of the meniscal tissue. Fast Fourier Transform analysis is used to quantify characteristic features of the microstructure (collagen fibers orientation, pore size) after slicing prior mechanical testing. Uniaxial (relaxation) tests are performed on dog-bone meniscal samples. Stress relaxation testing results on samples cut from different regions of the meniscus highlights a significant scatter in the maximum stress even though the preferential collagen fibers orientation is not too dissimilar. The proposed slicing method, which requires inexpensive tools and allows to minimize sample preparation, is proved to be applicable for a range of applications from macroscopic/nano mechanical testing to microscopy observations to accurately characterize location-dependent microstructural features and mechanical properties.

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In this paper we present the results from a recent micromechanical investigation aimed at developing methodologies for testing and understanding the fundamental behaviour of meniscal tissue. To achieve this, we employed two distinctly different, but equally relevant mechanical testing platforms – uniaxial tensile testing and Dynamic Mechanical Analysis. The results from the tensile tests revealed that the studied material exhibits non-linear stress-strain behaviour and that its viscoelastic properties are timedependent. Furthermore, by using DMA it was possible to perform walking and running simulations, which provided furtherinformation of the strain=time response of the meniscal samples. The importance of accurate specimen preparation and actual method development are also presented and discussed in detail.

The complex inhomogeneous architecture of the human meniscal tissue at the micro and nano scale in the absence of artefacts introduced by sample treatments has not yet been fully revealed. The knowledge of the internal structure organization is essential to understand the mechanical functionality of the meniscus and its relationship with the tissue’s complex structure. In this work, we investigated human meniscal tissue structure using up-to-date non-invasive imaging techniques, based on multiphoton fluorescence and quantitative second harmonic generation microscopy complemented with Environmental Scanning Electron Microscopy measurements. Observations on 50 meniscal samples extracted from 6 human menisci (3 lateral and 3 medial) revealed fundamental features of structural morphology and allowed us to quantitatively describe the 3D organisation of elastin and collagen fibres bundles. 3D regular waves of collagen bundles are arranged in “honeycomb-like” cells that are comprised of pores surrounded by the collagen and elastin network at the micro-scale. This type of arrangement propagates from macro to the nanoscale.

Finite element analysis of linear-elastic structures with spatially varying uncertain properties is addressed within the framework of the interval model of uncertainty. Resorting to a recently proposed interval field model, the uncertain properties are expressed as the superposition of deterministic basis functions weighted by particular unitary intervals. An Interval Finite Element Method (IFEM) incorporating the interval field representation of uncertainties is formulated by applying an interval extension in conjunction with the standard energy approach. Uncertainty propagation analysis is performed by adopting a response surface approach which provides approximate explicit expressions of response bounds requiring only a few deterministic analyses. Then, the whole procedure is implemented in ABAQUS’ environment by coding User Subroutines and Python scripts.

2D plane stress and bending problems involving uncertain Young's modulus of the material are analyzed. The accuracy of the proposed IFEM as well as response variability under spatially dependent uncertainty are investigated.