Fabric Tensor Research Articles

Magnetic Anomalies Caused by 3D Polyhedral Structures With Arbitrary Polynomial Magnetization

AbstractMagnetization is a natural property of magnetic materials which is widely used to analyze paramagnetic data, locate unexploded ordnance, explore mineral deposits and gas resources, and image anomalous magnetic structures in the Earth's crust. An essential and challenging task is to compute accurate magnetic anomalies caused by realistic magnetic 3D structures. In this study, for the first time, we derive the analytical expressions of magnetic potential, magnetic field and magnetic gradient tensor for magnetic materials with 3D polyhedral shapes and magnetization vectors that may vary in the space following polynomial trends. The order of polynomials can vary from one to high positive integers in both horizontal and vertical directions. Synthetic and realistic deposit models are used to validate our analytical solutions. We release the open‐source code in C++.

A unified structure-based constitutive model for both tension and shear of passive myocardium

This paper presents a hyperelastic model for replicating the passive response of ventricular myocardium and pericardium using fourth-order structural tensors. Models that can accurately predict mechanical response of cardiac samples are useful in understanding diseased tissues or postulating potential biomaterials for the replacement of damaged tissue in surgical applications. An orthotropic model is postulated here and is demonstrated to perform well in recreating experimental results in a variety of planar biaxial testing protocols and six modes of simple shear. The work outlines the fourth-order structural tensor (FOST) approach to representations of hyperelasticity for prescribing individual microstructural components (or interaction between them) that contribute to the overall elastic response of a biological tissue, through selection of specific FOSTs and constructing fourth-order material tensors possessing the constitutive properties and directional biases of that constituent. A fourth-order scalar product is then performed between this and an appropriately chosen fourth-order finite strain measure that returns a physically meaningful contribution to the strain energy density function for the overall tissue. A structure-based model for ventricular myocardium and pericardium is presented in this work using this approach. In the case of human myocardium, a consistent set of seven material constants simultaneously replicates five planar biaxial tension protocols and six modes of simple shear response.

A micromechanical analysis of strain concentration tensor for elastoplastic medium containing aligned and misaligned pores

In this paper, making use of Eshelby tensor for the elastoplastic medium, will be presented an approach for the strain concentration tensor for materials that present spherical or spheroidal pores. The pores are embedded in a homogeneous isotropic matrix which presents an elastoplastic behavior, governed by the von Mises yield criterion and isotropic hardening. In this context, it will also be analyzed the behavior of a porous material through the influence of the extraction of the isotropic part of the anisotropic operator. The effective properties of the material in each incremental step are obtained through the micromechanical Mori-Tanaka's model. Through elastoplastic strain concentration tensor for aligned or misaligned pores, it was observed that the pore direction represents a greater influence on the elastoplastic behavior of the material studied than aspect ratio (length divided by pore diameter). This approach has the advantage of being easily incorporated into conventional mean-field homogenization (MFH) schemes for the case where inclusion stiffness is neglected and presents different geometries or directions in 3D space.

P11.24.A The anisotropic component of the decomposed diffusion tensor predicts overall survival in patients with glioblastoma

Abstract Background The diffusion tensor can be decomposed into isotropic (DTI-p) and anisotropic (DTI-q) components (Peña et al., 2006). Regions of DTI-q abnormality have a high tumour burden and increased surgical resection of abnormal DTI-q positively correlates with overall survival (OS) (Yan et al., 2017). We aimed to establish if median voxel DTI-q (MVQ) or a distribution measure of DTI-q (DMQ) could act as a neuro-imaging biomarker, to predict OS in patients with glioblastoma. Material and Methods Diffusion tensor decomposition was used to create DTI-p and DTI-q maps, using FSL software (FMRIB Software Library, Oxford). MVQ and DMQ (the 95th centile minus the 5th centile of the DTI-q distribution, divided by the MVQ) were calculated from the preoperative whole brain (WB), contrast-enhancing (CE), and non-contrast-enhancing (NCE) hemisphere DTI-q maps, using fslstats, for 33 patients with glioblastoma. Using R Studio, multiple linear regression (MLR) models were computed to establish if MVQ or DMQ of the WB, CE and NCE hemispheres or age, significantly predicts OS. The Breusch-Pagan Test, on package “lmtest” in R, was calculated for all MLR models, to determine if heteroscedasticity was present and, if so, bootstrapped multiple regression (BMR) models were computed using package “boot” in R. Results Evidence for heteroscedasticity was present in MLRs that modelled the relationship between DMQ of WB, age, and OS (BP = 6.032, p = 0.014) and DMQ of CE hemisphere, age, and OS (BP = 7.163, p = 0.028). In the BMR of WB DMQ, age, and OS, the 95% bias-corrected accelerated confidence intervals (BCa-CI) for the WB DMQ regression coefficient was 133.5 - 1851.4 and included the WB DMQ estimated coefficient of 803.9. The BMR of CE hemisphere DMQ, age, and OS, computed a 95% BCa-CI for the CE hemisphere DMQ coefficient of 101.8 - 1579.6, containing the CE hemisphere DMQ coefficient estimate of 612.414. For both BMRs, the 95% BCa-CI for age coefficients crossed a value of 0. From computed MLR models, WB MVQ (t = -2.569, p = 0.015), CE hemisphere MVQ (t = -2.143, p = 0.040), NCE hemisphere MVQ (t = -2.567, p = 0.015) and NCE hemisphere DMQ (t = 2.39, p = 0.024) were significant predictors of OS. Age did not significantly predict OS in any models and was not significantly related to WB, CE and NCE hemisphere MVQ or DMQ. Conclusion WB, CE and NCE hemisphere MVQ and DMQ predict OS in our tested subgroup of patients with glioblastoma. Age is not a significant predictor of OS and does not significantly correlate with MVQ or DMQ. The median value and distribution spread of DTI-q may act as a prognostic biomarker in glioblastoma, facilitating patient stratification.

3D Mechanical Characters and Their Fabric Evolutions of Granular Materials by DEM Simulation

To quantitatively describe the influence of the contact characteristics of granular materials on their mechanical response, the true-triaxial tests with different particle shapes are simulated by the discrete element method (DEM), and the connection between the evolutions of particle contact fabric and the anisotropic mechanical responses is studied. The contact normal vector of the particle in 3D space is described by two independent angles, by which the contact fabric tensor is defined. The amplitude parameters in three orthogonal directions are defined by the invariants of the plane fabric tensor, which are scalars and describe the degree of anisotropy of the contact fabric in each plane. The expression of orthotropic fabric tensor is derived from the amplitude parameters, with the change of geometric space of contact normal vector, which is reduced to the different tensor of transverse isotropic naturally. The fabric tensor can be directly applied to the constitutive equation to describe the effects of the particle contact on the mechanical response. For verifying the rationality of contact characteristics described by fabric tensor, four particle shapes are clumped by PFC3D. The mechanical properties of specimens with different particles are simulated under the true-triaxial loading path, and the data of contact normal vector is extracted in real time. The simulation results showed that the particle shapes have a significant effect on the 3D stress-strain relationship and strength, which showed apparent anisotropy, and the invariants of fabric tensor can be used to describe the evolution of particle contact in the loading process.

Upgrading SANISAND model to address continuous stress principal axis rotation: A conservative approach

AbstractThe SANISAND models are known for their ability to simulate sands behavior with various properties under a wide range of loadings. Nevertheless, there is much room for improvement in simulating stress principal axis (PA) rotation effects. This study tried to address the continuous stress PA rotation problem following a conservative approach in preserving the SANISAND basic formulation. The inelastic response of the material under the stress rotation is regulated by a plastic intensity index as a function of the stress rotation rate with respect to the deviatoric fabric tensor and the non‐coaxiality between the stress and the plastic strain rate. The model performance is verified compared to the experimental data as well as other constitutive models. Supplementary discussions are provided to shed light on how fabric evolution results in a lower degree of anisotropy under continuous stress PA rotation. Based on continuum bounding surface plasticity, the proposed model addresses the fabric anisotropy effects on the severity of plasticity under continuous stress PA rotation.

Behavior of Sheared Granular Materials at Micro-Scale during the Cyclic Loading

The aim of this paper is to explore the evolution of different micro-scale quantities during the cyclic loading using the discrete element method (DEM) for a granulate system such as sand. The numerical samples comprising of 9826 spheres were generated and consolidated isotropically using the periodic boundaries. These numerical samples were subjected to the cyclic loading for different maximum applied strains. The simulated stress-strain behavior was validated with the experiment and found an excellent agreement between them during loading and unloading. The evolutions of different micro-scale quantities were investigated in detail considering the variation of the maximum applied strain and the density of sample. It is noted that the evolution of the coordination number and the slip coordination number is a function of the maximum applied strain and the density of sample during the cyclic loading. The change of the slip coordination number is larger at the end of unloading than that at end of loading during the cyclic loading regardless of the values of the maximum applied strain and the density of sample. The ratio of strong contacts to all the contacts increases abruptly when the load is reversed, which is opposite to what is observed for the coordination number and the slip coordination number. The deviatoric fabric computed by the fabric tensor considering the strong contacts mimics the deviatoric stress irrespective of the values of the maximum applied strain and the density of sample during the cyclic loading. Moreover, a linear correlation between the macro and micro quantities exists regardless of the variation of the maximum applied strains or the variation of the density of the sample during the cyclic loading. The slopes of the lines of these correlations are almost same.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Propagation Characteristics of Rotation Waves in Transversely Isotropic Granular Media Considering Microstructure Effect

The purpose of this study is to develop a micromechanical-based microstructure model for transversely isotropic granular media and then use it to investigate the propagation characteristics of particle rotation waves. In this paper, the particle translation and rotation are selected as basic independent variables and the particle displacement at contact due to particle rotation is ignored. The relative deformation tensors are introduced to describe the local deformational fluctuation because of their discrete nature and microstructure effect. Based on micro–macro deformation energy conservation, the constitutive relations are derived through transferring the summation into an integral and introducing the contact fabric tensor. The governing equations and corresponding boundary conditions can then be obtained based on Hamilton’s principle. Subsequently, the dispersion characteristics and bandgap features of particle rotation waves in transversely isotropic granular media are analyzed based on the present model. The research shows that: the present microstructure model can predict 12 particle rotation waves and reflect 8 dispersion relations; the effect of the change in fabric on the dispersion relation of particle rotation waves can be mainly attributed to the effect of equivalent stiffness on frequency; and the degree of anisotropy has significant effects on the width of frequency bandgap of longitudinal waves, while it has little effect on the width of frequency bandgap of transverse and in-plane shear waves.

An Efficient Model For Scaffold Mediated Bone Regeneration

We present a three dimensional, time dependent model for bone regeneration in the presence of porous scaffolds to bridge critical size bone defects. Our approach uses homogenized quantities, thus drastically reducing computational cost compared to models resolving the microstructural scale of the scaffold. Using abstract functional relationships instead of concrete effective material properties, our model can incorporate the homogenized material tensors for a large class of scaffold microstructure designs. We prove an existence and uniqueness theorem for solutions based on a fixed point argument. We include the cases of mixed boundary conditions and multiple, interacting signalling molecules, both being important for application. Furthermore we present numerical simulations showing good agreement with experimental findings.

Quantitative correlation between rafting microstructure and anisotropic mechanical behavior in dual-phase materials

Continuum descriptions of the microstructure in dual-phase materials are introduced by fabric tensors computed from the mean intercept length (MIL) method. The fabric tensor can be decomposed into the isotropic and deviatoric components to represent the dilatation and orientation preferred properties, respectively. Systematic numerical simulations on micromechanics models and simplified microstructures are conducted. It is confirmed that the anisotropic elastic behavior can be quantitatively predicted with the help of deviatoric components. This component can be further implemented into the constitutive model to describe the time- and load-dependent evolution of the material state variable. Anisotropy induced by morphology and anisotropic constituents is discussed. The results match well with computational experiments, showing the promising application potential of the tensorial representation method and the microstructure-based mechanical property prediction model.

Microstructure Characterization of High Explosives by Wavelet Transform

The key macro properties of high explosives including sensitivity to shock, the possibility of initiation, and the subsequent chemical reaction are known to be controlled by processes occurring at their microstructure level. However, there is a lack of an easy, effective, and accurate method to quantify the microstructure, termed as fabric, of high explosives despite an abundance of evidence regarding its importance. This study proposes a rotational Haar wavelet transform (RHWT) method to characterize the fabric of high explosives from two-dimensional images, yielding key fabric parameters including rose diagram, fabric direction, and degree of fabric anisotropy. The fabric tensor commonly used in numerical simulations and constitutive models can also be determined by RHWT. The RHWT was implemented on microscopic images of six high explosives captured by various imaging techniques including scanning electrical microscopy, polarized light microscopy, and micro X-ray computed tomography. Despite these variables, the proposed RHWT successfully identifies fabric in these images, demonstrating robustness and validity of RHWT.

Fabric-enriched continuum breakage mechanics (F-CBM)

A fabric-enriched continuum breakage mechanics (F-CBM) framework is developed to examine the relation between elastic and inelastic anisotropy in granular materials subjected to high pressure. For this purpose, emphasis is given to the influence of the anisotropic fabric on the plastic cap regime – that is, a macroscopic signature of the inelasticity of granular materials widely dependent on the particle crushability. Anisotropic behaviour prior to yielding is introduced through a symmetric second-order fabric tensor embedded in the expression of the elastic energy potential. It is shown that, thanks to its energy-based formulation, the F-CBM framework provides a platform to predict the rotation and distortion of the yield cap of granular materials as an outcome of anisotropic strain energy storage prior to grain crushing. Parametric analyses indicate that both the degree of anisotropy and the orientation of the fabric have an impact on the energy release resulting from grain crushing and contribute to the resulting stress–strain behaviour. It is shown that, despite the use of very few parameters, the proposed framework can accurately predict the anisotropic yielding and inelastic stress–strain response solely on the basis of the directional properties inferred from measurements of elastic anisotropy.

Fabric response to stress probing in granular materials: Two-dimensional, anisotropic systems

The microstructure of granular materials has a significant influence on their macroscopic quasi-static strength and deformational behaviour. This microstructure is often quantified by a second-order fabric tensor that describes the primary orientational statistics of interparticle contacts. Here, it is investigated how the fabric tensor changes when samples are subjected to small (strain) loadings with different ‘directions’, i.e. probes. This is accomplished by the analysis of extensive sets of Discrete Element Method (DEM) simulations for various anisotropic, pre-peak two-dimensional samples, where both in-plane (i.e. coaxial with the current stress and fabric tensor) and out-of-plane, noncoaxial probes are considered. The results of DEM simulations show that the in-plane and out-of-plane fabric responses are effectively decoupled, i.e. they are only dependent on the in-plane and out-of-plane strain increment, respectively. The out-of-plane fabric increment is proportional to the out-of-plane strain increment whereas the in-plane fabric increment is linearly dependent on the in-plane strain increment. An accurate theoretical description (with a modest number of model parameters) has been developed that describes the fabric response to the imposed, in-plane as well out-of-plane, strain increments for the considered systems.

Predicting the trabecular bone apparent stiffness tensor with spherical convolutional neural networks

The apparent stiffness tensor is relevant for characterizing trabecular bone quality. Previous studies have used morphology-stiffness relationships for estimating the apparent stiffness tensor. In this paper, we propose to train spherical convolutional neural networks (SphCNNs) to estimate this tensor. Information of the edges, trabecular thickness, and spacing are summarized in functions on the unitary sphere used as inputs for the SphCNNs. The concomitant dimensionality reduction makes it possible to train neural networks on relatively small datasets. The predicted tensors were compared to the stiffness tensors computed by using the micro-finite element method (μFE), which was considered as the gold standard, and models based on fourth-order fabric tensors. Combining edges and trabecular thickness yields significant improvements in the accuracy compared to the methods based on fourth-order fabric tensors. From the results, SphCNNs are promising for replacing the more expensive μFE stiffness estimations.

On the mechanical anisotropy of argillaceous Cobourg limestone: fabric tensor approach

The mechanical response of Cobourg limestone is anisotropic, which stems from heterogeneity of its fabric. In this work, a new fabric-dependent approach is developed for the description of strength and deformation characteristics of this rock. The quantification of fabric is carried out by employing the basic principles of stereology and the specific fabric descriptor used is the so-called mean intercept length. The results demonstrate that the material may be perceived as transversely isotropic and the mechanical response is affected by the volume fraction of argillaceous partings. The latter are spatially variable, so that the notion of Representative Elementary Volume (REV) is not, in general, applicable. A numerical study is included involving simulation of triaxial tests conducted on differently oriented samples of Cobourg limestone.

Simulation of the flow past random arrays of spherical particles: Microstructure-based tensor quantities as a tool to predict fluid–particle forces

A recent challenge in the modelling of particle flows is to build microstructure-informed drag models to overcome the average description of the fluid–particle force in the drag force correlations currently used in Euler–Lagrange and Euler–Euler models. To that end, we study through particle-resolved direct numerical simulations (PR-DNS) the flow past random assemblies of mono-dispersed spherical particles at three particle Reynolds numbers (10, 50, 100) and four solid volume fractions (0.10, 0.20, 0.30, 0.40). The present methodology is validated against theoretical and numerical results for the mean drag force in Stokes flows and finite Reynolds numbers flows. PR-DNS results are then used to characterize in details the statistics of the force distribution over the particle array, highlighting the substantial dispersion of the fluid force along the streamwise and transverse directions. The microstructure formed by the solid phase is described by means of a limited number of tensor quantities inspired from the fabric tensor used in granular media. Significant correlations are identified between the force experienced by a given particle immersed in a random array and a few key quantities that describe the anisotropy of its neighbourhood. A microstructure-based multi-linear model is proposed and validated against independent test cases. The model appears to perform best in the viscous and dense regimes. The addition of a stochastic contribution to the model allows to recover the correct level of force fluctuations at the cost of a lower correlation between the model and the data.

Evidence of a unique critical fabric surface for granular soils

The critical state of granular soils needs to make proper reference to the fabric that develops at critical state. This study substantialises the concept of critical fabric surface (CFS), which attracts the fabric state of granular soils upon continuous shearing. Numerical experiments using discrete-element method (DEM) modelling are conducted under drained and undrained conditions with varying Lode angles. Fabric tensors are defined based on the normals of all contacts and of the strong force contacts only. Both tensors have their spherical component preserved such that the information of coordination number can be carried. A separate series of low-confining-pressure undrained tests is conducted to probe the fabric states of soils in the post-liquefaction regime. Finally, a single CFS spanning across a wide range of coordination numbers is established based on the DEM results. The CFS concept provides an important reference state for soils sheared to large strains, complementary to the traditionally defined critical state. This provides a new perspective to interpret and model the mechanics of granular soils in both pre- and post-liquefied regimes. The evolution of fabric shows that the normalised strong-contact fabric evolves linearly with the stress ratio even for liquefied or anisotropically consolidated soils.

On the dual effect of obstacles in preventing silo clogging in 2D

Placing an obstacle in front of a bottleneck has been proposed as a sound alternative to improve the flow of discrete materials in a wide variety of scenarios. Nevertheless, the physical reasons behind this behavior are not fully understood and the suitability of this practice has been recently challenged for pedestrian evacuations. In this work, we experimentally demonstrate that for the case of inert grains discharging from a silo, an obstacle above the exit leads to a reduction of clog formation via two different mechanisms: i) an alteration of the kinematic properties in the outlet proximities that prevents the stabilization of arches; and ii) an introduction of a clear anisotropy in the contact fabric tensor that becomes relevant when working at a quasi-static regime. Then, both mechanisms are encompassed using a single formulation that could be inspiring for other, more complex, systems.

Form-function relationship between trabecular bone architecture and biomechanical function in the horse humerus

In mechanical design, it is a common challenge to design parts with high strength-to-weight ratio. Throughout evolution, natural organisms have faced the same challenge and developed a variety of solutions. One of these is the endoskeleton, composed of bones linked by joints. In long bones, near joints, Nature uses an anisotropic porous material, trabecular bone. This architecture is adapted to the mechanical stresses it undergoes. The aim of this work is to study the relationship between a given trabecular architecture and its biomechanical function. Understanding this relationship allows to predict the natural trabecular architecture that would be optimal to support a given stress field. Trabecular architecture is often characterised through a Fabric Tensor (FT) whose eigenvectors and eigenvalues respectively describe trabecular orientation and anisotropy. A first FT (Mmes) was measured using BoneJ2 in 20 trabecular Regions Of Interest (ROI) identified in a microtomographied horse humerus. In parallel, the stress field in this humerus was calculated using a Finite Elements Analysis (FEA) based on a musculoskeletal model of the horse forelimb during trotting. This FEA was used to obtain a second FT (Mest) for each ROI through an equation from the literature relating FT to stress in the human femur. Difference percentages between Mmes and Mest were calculated and showed that the proposed law is accurate to predict FT from stress in the horse humerus. This law is interesting for the design of mechanical parts bio-inspired from bones, which reproduce the adaptation of trabecular architecture to the mechanical stress it undergoes.

Shear–induced anisotropy in granular materials under various saturation states

This study presents a micromechanical study on the characteristics of shear-induced anisotropy in granular media under various saturation states. A series of unsaturated triaxial compression tests under the full range of degree of saturation are performed by using Discrete Element Method (DEM). The effect of the suctions on the characteristics of anisotropy is illustrated and discussed. Good agreements between anisotropy and stress ratio are observed for saturated and unsaturated states if skeleton contact stress is adopted. With the increase of the suctions, the weight of fabric tensor anisotropy decreases while the weight of anisotropy related to normal forces increases. The mechanical anisotropy plays a dominant role in providing shear resistance for the full saturation range.