Fabric Tensor Research Articles

Evolution of structure of cohesive granular ensembles in compression

A small amount of cementation between particles drastically changes the mechanical response of a granular ensemble. Through a series of interrupted uniaxial compression tests inside the x-ray computed tomograph (XCT) on fine and coarse grained cemented sand specimens, a quantification of the behavior at both the continuum and particle scales is made. At the continuum length scale, displacements and porosity fields are examined. While at the particle scale, the coordination number, particle and contact fabric tensors, directional distribution of particles and contact normals are extracted and analyzed. Particle level phenomenology—at different stages of loading—such as breakage of bonds, formation of new contacts, manifest themselves at the ensemble scale in the form of bifurcation zones, which coalesce into split zones through the specimen. A new paradigm for the fabric tensor is also provided using principal component analysis. The results of the continuum and particle scale analysis are used to provide insights on the compression behavior of cemented sands.

Influence of aggregates properties on microstructural properties and mechanical performance of asphalt mixtures

Asphalt mixtures can perform suitably in road and airfield pavements. However, with rapid increases in traffic volumes, harsh climate environments, and heavier loads, there is an increased interest in improving the mechanical properties of asphalt materials through a suitable design of the microstructure of the asphalt mixture. Aggregates’ physical and chemical properties significantly influence the interface interaction between aggregates and binders and achieve strong, durable and cost-effective asphalt mixtures. This study prepared the asphalt mixtures from three different aggregate sources, i.e. Granite 1 (GD 1), Granite 2 (GD 2) and limestone (GD 3) and captured the microstructural properties using a charge-coupled device (CCD) digital camera and analysed them by iPAS software. The effect of the daily temperature variation of the environment to the corresponding temperature gradient through the depth of all asphalt mixtures layers was assessed. The results indicate that sodium oxide (Na2O) + potassium oxide (K2O) + calcium oxide (CaO) + silicon (SiO2) has a significant influence on the electrical conductivity of granite samples at high temperatures and pressure. Marshall, wheel tracking, three-point bending beam, double tension–compression fatigue tests at two periods were conducted to investigate any relationship between the structural performance of the asphalt mixtures and their microstructural properties. Additionally, the normal contact fabric tensor (F) was developed to t evaluate the interlocking of aggregates in asphalt mixtures, and the correlation with the mechanical performance was performed. The results show that the higher amount of silicon and aluminium in aggregates, the better the aggregate texture and the high contact points weight. In addition, high contact points weight has a notable impact on the Marshall stability, rutting depth, dynamic stability, stiffness modulus and complex Poisson’s ratio at high and low temperatures. All samples showed good mechanical performance at all temperatures because of their high aggregate particles F.

A fabric tensor based small strain constitutive law for the elastoplastic behavior of snow

The mechanical behavior of snow depends on its density and microstructure. The anisotropy in the microstructure is expressed in terms of fabric tensor that leads to an anisotropic stress-strain relation. Lately, fabric-based relations have successfully estimated the elastic properties of snow. Motivated by this, we propose a fabric-based macroscopic elasto-plastic constitutive law for snow, which can be used to study avalanche initiation. Mean Intercept Length tensor used as a measure of material fabric is determined by X-ray tomography. Fabric tensor and density-dependent yield surface with a provision for isotropic hardening/softening are used in this process. Beyond the initial yield, the yield function grows till the strength of the snow is reached and then softens. Since snow exhibits tension and compression behavior asymmetry, a piece-wise quadratic yield function is used. Stress-strain curves needed for determining the equation of the hardening/softening law and other parameters of the proposed macroscopic constitutive law are obtained through micro-Finite Element (μ-FE) simulations. The macroscopic constitutive law has been implemented as a user subroutine in a FE code and can predict the snow's multiaxial behavior.

Increased muscle fiber fractional anisotropy value using diffusion tensor imaging after compression without fiber injury.

Recent studies have indicated that injuries such as muscle tears modify the microstructural integrity of muscle, leading to substantial alterations in measured diffusion parameters. Therefore, the fractional anisotropy (FA) value decreases. However, we hypothesized that soft tissue, such as muscle tissue, undergoes reversible changes under conditions of compression without fiber injury. To evaluate the FA change due to compression in muscle tissue without fiber injury. Diffusion tensor imaging (DTI) was performed on both feet of 10 healthy volunteers (mean age = 35.0 ± 10.39 years; age range = 23-52 years) using a 3.0-T magnetic resonance imaging (MRI) scanner with an eight-channel phased array knee coil. An MRI-compatible sphygmomanometer was applied to the individuals' lower legs and individuals were placed in a compressed state. Then, rest intervals of 5 min were set in re-rest state after compression. The FA value, apparent diffusion coefficient (ADC), and eigenvalues (λ1, λ2, λ3) of the gastrocnemius and soleus muscle were measured at each state. The mean FA values increased in all muscles in a compressed state, while the mean λ3 decreased. In all muscles, significant differences were found between the rest and compressed states in terms of mean FA and λ3 (P < 0.0001). We confirmed the reversibility of the DTI metrics, which suggests that there was no muscle injury during this study. In cases of compression without fiber injury, the FA value increases, because fibers are strongly aligned in the longitudinal direction.

Thermodynamic Constitutive Model for Sands with Consideration of Distributed Fabric

A micromechanics-inspired fabric-dependent thermodynamic model is developed to simulate the critical-state and phase-transformation behavior of sands. A generalized fabric tensor that considers the statistical distribution of granular fabric is proposed. The fabric tensor is the sum of three families of a transversely isotropic tensor, which is assumed to be characterized by the probability density function of the normal distribution. The new tensor variable is incorporated into the reversible and irreversible thermodynamic models, and the fabric-dependent hyperelastic and plastic constitutive relations are then derived. The hyperelastic instability leads to a yield criterion that depends on fabric distribution and its evolution. Plastic strain development is linked to fabric distribution by a concept of accumulative yielding probability transitions, which in turn provides smooth transition from macroscopic elasticity to plasticity and finally to the critical state upon shearing. Using this theoretical framework, the physical mechanism underlying the phase-transformation and critical-state behavior of sands are interpreted from the evolution of fabric distribution. The performance of the model is verified by simulating various shear tests of Toyoura sand and a discrete element method (DEM)-modeled sand. The effects of b-value and principal stress direction on the stress-strain behavior are related to the evolution of fabric distribution. The noncoaxiality among stress, strain, and fabric is also captured by the model.

Determination of anisotropic elastic parameters from morphological parameters of cancellous bone for osteoporotic lumbar spine

In biomechanics, large finite element models with macroscopic representation of several bones or joints are necessary to analyze implant failure mechanisms. In order to handle large simulation models of human bone, it is crucial to homogenize the trabecular structure regarding the mechanical behavior without losing information about the realistic material properties. Accordingly, morphology and fabric measurements of 60 vertebral cancellous bone samples from three osteoporotic lumbar spines were performed on the basis of X-ray microtomography (μCT) images to determine anisotropic elastic parameters as a function of bone density in the area of pedicle screw anchorage. The fabric tensor was mapped in cubic bone volumes by a 3D mean-intercept-length method. Fabric measurements resulted in a high degree of anisotropy (DA = 0.554). For the Young’s and shear moduli as a function of bone volume fraction (BV/TV, bone volume/total volume), an individually fit function was determined and high correlations were found (97.3 ≤ R2 ≤ 99.1,p < 0.005). The results suggest that the mathematical formulation for the relationship between anisotropic elastic constants and BV/TV is applicable to current μCT data of cancellous bone in the osteoporotic lumbar spine. In combination with the obtained results and findings, the developed routine allows determination of elastic constants of osteoporotic lumbar spine. Based on this, the elastic constants determined using homogenization theory can enable efficient investigation of human bone using finite element analysis (FEA).Graphical Cancellous Bone with Fabric Tensor Ellipsoid representing anisotropy and principal axis (colored coordinate system) of given trabecular structure

Bulk contributions to the Casimir interaction of Dirac materials

Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk dielectric functions for Dirac materials in the presence of a mass gap, chemical potential, and finite temperature. Using these results (and neglecting eventual boundary effects), we study the Casimir interaction of Dirac materials. We describe in detail the characteristic features of the dielectric functions and their influence on the Casimir pressure.

Quantitative Description for Sand Void Fabric with the Principle of Stereology

Based on the principle of stereology to describe void fabric, the fabric tensor is redefined by the idea of normalization, and a novel quantitative description method for the orthotropic fabric of granular materials is presented. The scan line is described by two independent angles in the stereo space, and the projection of the scan line on three orthogonal planes is used to determine the plane tensor. The second-order plane tensor can be described equivalently by two invariants, which describe the degree and direction of anisotropy of the material, respectively. In the three-dimensional orthogonal space, there are three measurable amplitude parameters on the three orthogonal planes. Due to the normalized definition of tensor in this paper, there are only two independent variations of the three amplitude parameters, and any two amplitude parameters can be used to derive the three-dimensional orthotropic fabric tensor. Therefore, the same orthorhombic anisotropy structure can be described by three fabrics, which enriches the theoretical description of orthotropy greatly. As the geometric relationship of the stereoscopic space scan line changes, the three sets of orthotropic fabrics degenerate into different forms of transversely isotropic and isotropic fabrics naturally and have a clear physical meaning. The novel fabric tensor is quantitatively determined based on mathematical probability and statistics. The discrete distribution of voids in space is projected as a scalar measurable parameter on a plane. This parameter is related to the macroscopic constitutive relationship directly and can be used to describe the effect of microscopic voids on the macroscopic phenomenon of materials.

The influence of anisotropy on compaction bands: The case of coaxiality between stress and fabric anisotropy tensors

AbstractCompaction bands are localized failure patterns that appear in highly porous rock material under the effect of relatively high confining pressure. Being affected mainly by volumetric compression, these bands appear to be almost perpendicular to the most compressive principal stress of a stress state at the so‐called “cap” of the yield surface (YS). In this study, we focus on the mechanism that leads to the onset of compaction bands by using a viscoplasticity model able to describe the post‐localization response of these materials. The proposed constitutive framework is based on the overstress theory of Perzyna (1966) and the anisotropic clay plasticity model of Dafalias (1986), which provides not only the necessary “cap” of the YS, but introduces a rotational hardening (RH) mechanism, thus, accounting for the effect of fabric anisotropy. Following the analysis of Veveakis and Regenauer‐Lieb (2015), we identify the compaction bands as “static” cnoidal wave formations in the medium that occur at a post‐yield regime, and we study the effect of rotational and isotropic hardening on their onset. Moreover, we determine a theoretical range of confining pressures in triaxial compression tests for the compaction bands to develop. Under the assumption of coaxiality between stress and anisotropy tensors, the results show that the isotropic hardening promotes compaction localization, whereas the RH has a slightly negative effect on the onset of compaction localization.

Light behaviors outside a black hole surrounded by dark matter

Here we show the light behaviors outside a Schwarzschild black hole that is surrounded by a dark matter model proposed by Konoplya (see Konoplya R., Phys. Lett. B, 795 (2019) 1). It is worth mentioning that dark matter is studied for the first time in an optical system by transformation optics theory. Different from other works of mimicking curved spaces with geodesic equations, we calculate ray trajectories using the equivalent anisotropic material tensor, which can also be used to simulate field patterns for wave optics. The results from geometric optics and wave optics agree with each other perfectly, showing the gravitational lensing effects and light trapping properties, revealing the influences of dark matter on the black hole shadow and gravity, which can give some hints for future observations.

CONSTITUTIVE RELATIONS OF GRANULAR MATERIALS BY INTEGRATING MICROMECHANICAL KNOWLEDGE WITH DEEP LEARNING

Constitutive relations of granular materials are of great significance to many fields, such as geotechnical engineering. Different from traditional phenomenological constitutive theory, this study explores a micromechanics-informed data-driven constitutive modelling approach for granular materials via machine learning models. On the basis of Vogit’s homogenization assumption, an analytical small-strain stress-strain relation is established. This relation uniquely determines a group of micromechanical fabric variables associated with the constitutive behavior of granular materials. These recognized variables, together with principal strain and stress sequence pairs reflecting macroscopic properties of granular materials, are obtained via a series of discrete element models of triaxial compression tests. Considering the fact that these microscopic fabric tensors are internal variables, which cannot be directly used as inputs of a material constitutive model, a directed graph is introduced to incorporate microstructural information implicitly in the prediction of stress-strain responses. The gated recurrent unit (GRU) based recurrent neural networks are used as basic deep learning models to describe the mapping relation between nodes in the designed directed graph. In this study, the entire stress-strain prediction model can be assembled with two neural networks that are trained separately, after unfolding the directed graph from the target node to the source node. By testing the trained deep learning model based on brand new datasets, the results demonstrate that the proposed training approach can satisfactorily capture the multi-directional stress-strain responses with reversal loadings, such as conventional triaxial compression with unloading-reloading cycles, true-triaxial compression with constant intermediate principal stress (constant-b), and constant mean effective effective stress (constant-p) conditions with unloading-reloading cycles. The prediction results also show that the trained model possesses satisfactory interpolation and extrapolation capability. Considering the excellent ability of deep learning in terms of capturing the mechanical responses of granular materials and the unique features of open learning when new data is available, integrating a data-driven paradigm with theoretical constitutive models may be one of the important directions for constitutive research of granular materials.

The Multipole Structure and Symmetry Classification of Even-Type Deviators Decomposed from the Material Tensor.

The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.

Can DXA image-based deep learning model predict the anisotropic elastic behavior of trabecular bone?

3D image-based finite element (FE) and bone volume fraction (BV/TV)/fabric tensor modeling techniques are currently used to determine the apparent stiffness tensor of trabecular bone for assessing its anisotropic elastic behavior. Inspired by the recent success of deep learning (DL) techniques, we hypothesized that DL modeling techniques could be used to predict the apparent stiffness tensor of trabecular bone directly using dual-energy X-ray absorptiometry (DXA) images. To test the hypothesis, a convolutional neural network (CNN) model was trained and validated to predict the apparent stiffness tensor of trabecular bone cubes using their DXA images. Trabecular bone cubes obtained from human cadaver proximal femurs were used to obtain simulated DXA images as input, and the apparent stiffness tensor of the trabecular cubes determined by using micro-CT based FE simulations was used as output (ground truth) to train the DL model. The prediction accuracy of the DL model was evaluated by comparing it with the micro-CT based FE models, histomorphometric parameter based multiple linear regression models, and BV/TV/fabric tensor based multiple linear regression models. The results showed that DXA image-based DL model achieved high fidelity in predicting the apparent stiffness tensor of trabecular bone cubes (R2 = 0.905–0.973), comparable to or better than the histomorphometric parameter based multiple linear regression and BV/TV/fabric tensor based multiple linear regression models, thus supporting the hypothesis of this study. The outcome of this study could be used to help develop DXA image-based DL techniques for clinical assessment of bone fracture risk.

Non‐coaxial hypoplastic model for sand with evolving fabric anisotropy including non‐proportional loading

AbstractNon‐coaxial response refers to the deviation between the directions of the principal stress and plastic strain increment. In this paper, an extended hypoplastic model is proposed based on the anisotropic critical state theory, to describe the non‐coaxial and anisotropic response of sand subjected to both monotonic loading and rotation of principal stress axis. A fabric tensor that characterizes the internal structure of sand is introduced into a hypoplastic model, to reflect the effect of fabric anisotropy on the dilatancy and strength of sand. A Lode‐angle‐dependent hypoplastic potential surface is employed, rendering the flow direction no longer co‐directional with the stress tensor in the deviatoric plane. The fabric tensor is further incorporated into the flow direction, enabling the model to generate a non‐coaxial response. Upon shearing, the fabric evolves toward the loading direction following a properly defined evolution law, and the model response becomes purely coaxial at the critical state when the fabric becomes co‐directional with the loading direction. The tangential loading effect is further introduced to simulate the stiffness degradation of sand during undrained rotational shearing. The model is demonstrated to be capable of simulating the prismatic yet complex anisotropic behavior of sand under both proportional and non‐proportional loading conditions.

Surface orientation tensor to predict preferred contact orientation and characterise the form of individual particles

The characterisation and classification of particle form are typically based on the consideration of the main particle dimensions, for the derivation of which no method has been unanimously accepted or proven to be representative of its morphology or load-bearing capabilities. This study proposes a weighted fabric tensor, named “surface orientation tensor”, that characterises the form of an individual particle. Using the eigenvalues of this tensor, efficient measures of compactness, flakiness and elongation are proposed. In comparison to the traditional oriented bounding box approaches, it has the advantage that it is based directly on the orientations of the normal vectors of the faces forming the surface of the particle, i.e. those directions along which the particle can best transmit contact forces to its neighbours. The advantages of the proposed approach are pointed out with discrete element simulations on assemblies of polyhedral particles.

Theoretical solution for drained cylindrical cavity expansion in clays with fabric anisotropy and structure

This paper presents a novel, exact, semi-analytical solution for the quasi-static drained expansion of a cylindrical cavity in soft soils with fabric anisotropy and structure. The assumed constitutive model is the S-CLAY1S model, which is a Cam clay-type model that considers fabric anisotropy that evolves with plastic strains, structure and gradual degradation of bonding (destructuration) due to plastic straining. The solution involves the numerical integration of a system of eight first-order ordinary differential equations, three of them corresponding to the effective stresses in cylindrical coordinates, other three corresponding to the components of the fabric tensor and one corresponding to the amount of bonding and another corresponding to the specific volume. The solution is validated against finite element analyses. When destructuration is considered, the solution provides slightly lower values of the effective radial and mean stresses near the cavity wall. Besides, the specific volume is further reduced due to loss of bonding. Parametric analyses and discussion of the influence of soil overconsolidation, expansion of the cavity and initial amount of bonding are presented.

Discrete element simulations of drained granular material response under multidirectional rotational shear

Multidirectional shearing tests on soils in the laboratory suffer from drawbacks such as non-uniformity of stress and unknown horizontal stress of the specimen. As a promising numerical technique, the discrete element method (DEM) enables investigation of the responses of virtual granular samples under general loading conditions and provides ideal numerical results to overcome the aforementioned limitations. In this study, three different drained cyclic tests, including conventional rotational shear, bidirectional circular rotational shear, and bidirectional elliptical rotational shear, were conducted on both isotropic and transversely isotropic samples under various stress ratios and density conditions. In all these tests, the magnitudes of the three principal stresses were kept constant during the entire loading process. Principal stress rotation occurred in one deviatoric stress plane in a conventional rotational shearing test but spatially (not necessarily in one plane) in the other two bidirectional shearing tests. The normalized deviatoric strain increment was decomposed into (1) coaxial and proportional, (2) coaxial and orthogonal, and (3) totally non-coaxial parts in reference to the deviatoric stress tensor. Their evolutions are presented and discussed in detail herein. The internal structure quantified by the contact-normal-based fabric tensor was tracked, and its correlation with the imposed stress path is illustrated.

Optical Chirality Determined from Mueller Matrices

Optical chirality, in terms of circular birefringence and circular dichroism, is described by its electromagnetic and magnetoelectric material tensors, and the corresponding optical activity contributes to the Mueller matrix. Here, spectroscopic ellipsometry in the spectral range 210–1690 nm is used to address chiral phenomena by measuring Mueller matrices in transmission. Three approaches to determine chirality parameters are discussed. In the first approach, applicable in the absence of linear polarization effects, circular birefringence and circular dichroism are evaluated directly from elements of a Mueller matrix. In the second method, differential decomposition is employed, which allows for the unique separation of chirality parameters from linear anisotropic parameters as well as from depolarization provided that the sample is homogeneous along the optical path. Finally, electromagnetic modeling using the Tellegen constitutive relations is presented. The last method also allows structural effects to be included. The three methods to quantify optical chirality are demonstrated for selected materials, including sugar solutions, α-quartz, liquid crystals, beetle cuticle, and films of cellulose nanocrystals.

Micromechanical modelling of the anisotropic creep behaviour of granular medium as a fourth-order fabric tensor

The main objective of the present work is to develop a micromechanics approach to predict the macroscopic anisotropic creep behaviour of granular media. To this end, the linear viscoelastic behaviour of the inter-particle interaction at contact is adopted, and the contact distribution is characterized by a fourth-order fabric tensor in the local scale. Then, fourth-order tensor fabric-based micromechanical approaches based on Voigt and Reuss localization assumptions are applied to granular media in the Laplace–Carson space. With help of the inverse Laplace–Carson transformation of these obtained models, the macroscopic anisotropic creep behaviour of granular media submitted to a constant external loading is examined. Finally, the obtained results by specializing the Burgers model into the obtained models are compared with the numerical simulations in the particle flow code (PFC2D) to illustrate the validation and the accuracy of the analytical models for the macroscopic anisotropic creep behaviour of granular media.

Anisotropic microstructural evolution and coarsening in free sintering and constrained sintering of metal film by using FIB-SEM tomography

The anisotropic microstructure develops during powder processing and constrained sintering. FIB-SEM tomography was used to investigate the microstructural evolutions of Au submicron particles during free sintering and constrained sintering. The decrease in total surface area and the coarsening, i.e. the increase of mean intercept length of solid phase, were observed during the densification process. The anisotropic packing structure of spherical particles induced by casting process was characterized by area weighted fabric tensor, which represented the bond orientation and the anisotropy of contact area. In free sintering, the initial anisotropy in packing structure decreased with densification. In constrained sintering, the initial anisotropy observed by the mean intercept length of pore was reversed in the later stage. The microstructure evolved so as to form the elongated pores along the thickness direction. An open pore structure model was presented to explain the microstructural evolution in the intermediate stage.