Mass Tensor Research Articles

A semi-theoretical method for determining the permeability tensor of fractured rock masses in three-dimensional space

The permeability tensor is a critical parameter for analyzing the hydraulic behavior of anisotropic permeability in fractured rock masses. However, determining this tensor for three-dimensional (3D) fractured rock masses has proven to be challenging and resource-intensive. Both field tests, requiring numerous costly in situ tests, and numerical experiments, hindered by complex discrete fracture networks with a high fracture density, present difficulties in obtaining accurate results. In response, this study proposes a semi-theoretical method for determining the permeability tensor of 3D fractured rock masses, significantly reducing labor and economic costs. The proposed method focuses on establishing the theoretical relationship of directional permeabilities in a 3D space, with emphasis on the properties of the permeability tensor and the influence of fractures' geometry on the flow rate. To facilitate the construction of the method, anisotropic ellipse and ellipsoid are introduced, providing a description of permeability anisotropy. With this innovative approach, engineers can calculate the permeability tensor even when only one value of permeability is available along any flow direction. The utilization of the anisotropic ellipse and ellipsoid concepts helps simplify the determination process. Through numerical experiments, the method is validated and its accuracy demonstrated, making it a valuable tool for analyzing the hydraulic behavior of 3D fractured rock masses.

Mass moments of functionally graded 2D domains and axisymmetric solids

We present a general methodology to evaluate the mass moments of two-dimensional domains and axisymmetric solids made of functionally graded materials. The approach developed in the paper is based on the sequence of two steps. First, the original domain integrals are converted to integrals extended to the relevant boundary by exploiting Gauss theorem. Second, for domains having a polygonal or circular shape, the boundary integrals are evaluated analytically by providing algebraic expressions that depend upon the parameters defining the density distribution, the position vectors of the vertices of the polygonal domain or the initial and ending points of an arbitrary circular sector, respectively. While the first step refers to moments of arbitrary order, the second step is limited to the most useful quantities for engineering applications, i.e. generalised mass, static moment and inertia tensor. The formulas derived in the paper are validated by means of examples retrieved from the specialised literature for which analytical results are available or have been specifically derived by the authors. Finally, in order to ascertain the computational savings entailed by the use of the proposed analytical formulas with respect to numerical techniques, the mass moments of a longitudinal section of a human femure, made of a functionally graded material and characterised by a linear density distribution, have been computed.

Predicting two-dimensional semiconductors using conductivity effective mass.

In this paper we investigate the relationship between the conductivity effective mass and exfoliation energy of materials to assess whether automatic sampling of the electron band structure can predict the presence of and ease of separating chemically bonded layers. We assess 22 976 materials from the Materials Project database, screen for only those that are thermodynamically stable and identify the 1000 materials with the highest standard deviation for p-type and the 1000 materials with the highest standard deviation for n-type internal conductivity effective mass tensors. We calculate the exfoliation energy of these 2000 materials and report on the correlation between effective mass and exfoliation energy. A relationship is found which is used to identify a previously unconsidered two-dimensional material and could streamline the modelling of other two-dimensional materials in the future.

Plasmons in a Strip with an Anisotropic Two-Dimensional Electron Gas Fully Screened by a Metal Gate

Interest in anisotropic two-dimensional electron systems and in plasma oscillations in them has been growing recently. Plasmons in a strip with a two-dimensional electron gas with an elliptic Fermi surface that is located near a metal gate, which screens the fields of the two-dimensional gas, have been theoretically analyzed. The plasma eigenmodes in this system have been found analytically in the limit of strong screening and the frequencies and damping rates of these modes have been determined taking into account anisotropy, magnetic field, and electromagnetic retardation effects. It has been shown that the fundamental mode in this limit is an edge magnetoplasmon with a linear dispersion relation. The frequency, damping rate, and velocity of this magnetoplasmon are independent of the magnetic field, and the localization length near the edge is proportional to the magnetic field. The square of the frequency of any other mode is the sum of the square of the frequency of this plasma mode without magnetic field and the square of the cyclotron frequency with a coefficient, which is independent of the orientation of the conductivity tensor with respect to the edges of the strip but depends on the principal values of the effective mass tensor when electromagnetic retardation effects are taken into account.

Study on the influence of statistical geometrical characteristics on the permeability tensor of fractured rock masses by the equivalent pipe network method

The permeability of the host rock of the nuclear waste underground repository is one of the most keys for estimating the potential risk for nuclide waste to escape. Therefore, a three-dimensional discrete fracture network (DFN) model using the Monte Carlo method is generated based on the statistical geometrical characteristics of fracture network in Beishan I area which is selected as the site for constructing the nuclear waste repository in China. And the permeability of Beishan I fractured rock masses is calculated through the equivalent pipe network (EPN) method and representative elementary volume (REV) of it is determined. A rotating algorithm is developed to calculate different direction permeability. Furthermore, the influence of fracture statistical geometrical characteristics on REV size is explored by creating a series of DFN models with varying log-normal distribution of side lengths. The results show that the REV size of the Beishan Ⅰ area is 1.66 × 105 m3 and the corresponding principal permeability direction is 111.29°/75.36°. The REV size and permeability both increase with the increment of the mean and variance of the log-normal distribution increases. And with the increase of mean and variance, the growth rate gradually accelerated.

Wave Polarization Control in Anisotropic Locally Resonant Materials

Elastic wave propagation in solids can be controlled and manipulated by properly designed metamaterials. In particular, polarization conversion can be obtained by using anisotropic materials. In this paper, we propose a three-component locally resonant material with non-symmetrically coated inclusions, and we study the effect of the anisotropic equivalent mass on band gap formation and the polarization conversion of elastic waves. The equivalent frequency-dependent mass tensor is obtained through the two-scale homogenization approach. The study of the eigenvalues of the mass tensor enables to predict band gaps and polarization bands, as well as identifying a priori the effect of different geometric and material parameters, thus opening the way to metamaterial optimization.

Colossal transverse magnetoresistance due to nematic superconducting phase fluctuations in a copper oxide.

Electronic anisotropy ("nematicity") has been detected in cuprate superconductors by various experimental techniques. Using angle-resolved transverse resistance (ARTR) measurements, a very sensitive and background-free technique that can detect 0.5% anisotropy in transport, we have observed it also in La2-xSrxCuO4 (LSCO) for 0.02 ≤ x ≤ 0.25. A central enigma in LSCO is the rotation of the nematic director (orientation of the largest longitudinal resistance) with temperature; this has not been seen before in any material. Here, we address this puzzle by measuring the angle-resolved transverse magnetoresistance (ARTMR) in LSCO. We report the discovery of colossal transverse magnetoresistance (CTMR)-an order-of-magnitude drop in the transverse resistivity in the magnetic field of 6 T. We show that the apparent rotation of the nematic director is caused by anisotropic superconducting fluctuations, which are not aligned with the normal electron fluid, consistent with coexisting bond-aligned and diagonal nematic orders. We quantify this by modeling the (magneto-)conductivity as a sum of normal (Drude) and paraconducting (Aslamazov-Larkin) channels but extended to contain anisotropic Drude and Cooper-pair effective mass tensors. Strikingly, the anisotropy of Cooper-pair stiffness is much larger than that of the normal electrons. It grows dramatically on the underdoped side, where the fluctuations become quasi-one-dimensional. Our analysis is general rather than model dependent. Still, we discuss some candidate microscopic models, including coupled strongly-correlated ladders where the transverse (interladder) phase stiffness is low compared with the longitudinal intraladder stiffness, as well as the anisotropic superconducting fluctuations expected close to the transition to a pair-density wave state.

Determination of heat transfer representative element volume and three-dimensional thermal conductivity tensor of fractured rock masses

The natural fracture system exerts a significant influence on the physical and mechanical properties of rock masses. The heat transfer in fractured rock masses is much more complicated than that of intact rock due to the thermal contact resistance induced by natural fractures with varying size, aperture, roughness, and orientation. This work describes a method for determining the representative element volume (REV) of heat transfer and the 3D thermal conductivity tensor of fractured rock masses. The method is based on the element-wise averaging of temperature and heat flux, which can be obtained from a finite element solution to a steady state heat transfer problem with specific boundary conditions. A case study was conducted based on the geological data of a potential area for high level radioactive waste disposal in China. The results indicate that the size of heat transfer REV of the rock masses in the study area is approximately 18 m × 18 m × 18 m. 3D thermal conductivity tensor, the magnitudes and directions of the three principal thermal conductivities, and anisotropy of the fractured rock masses were obtained. Influences of main fracture parameters including density, disc diameter, and thermal contact resistance on the heat transfer characteristics were investigated. The research results in this work can provide a better understanding of the discontinuous heat transfer processes in fractured rock masses and their homogenization methods.

Singularity resolving in solution of the boundary integral equation in two-dimensional vortex methods

The problem of 2D incompressible flow simulation around airfoils with sharp edges and corner point is considered. The solution of the boundary integral equation with respect to vortex sheet intensity arising in Lagrangian vortex method has weak singularity that cannot be resolved correctly in the framework of the existing Galerkin-type numerical schemes. It is shown that for piecewise-smooth bounded solutions the known schemes allow for solution reconstruction with high quality and provide the 2-nd order of accuracy, while for singular solution their order of accuracy goes down to the 1-st. A numerical scheme is suggested that allows for solution singularity resolving and provides the 2-nd order of accuracy. As a model problem, the added mass tensor components computation is considered, since its exact value is known for the Joukowsky wing airfoil with sharp edge (cusp point).

Pseudo-spin order of Wigner crystals in multi-valley electron gases

We study multi-valley electron gases in the low density (rs ≫ 1) limit. Here the ground-state is always a Wigner crystal (WC), with additional pseudo-spin order where the pseudo-spins are related to valley occupancies. Depending on the symmetries of the host semiconductor and the values of the parameters such as the anisotropy of the effective mass tensors, we find a striped or chiral pseudo-spin antiferromagnet, or a time-reversal symmetry breaking orbital loop-current ordered pseudo-spin ferromagnet. Our theory applies to the recently-discovered WC states in AlAs and in mono and bilayer transition metal dichalcogenides. We identify a set of interesting electronic liquid crystalline phases that could arise by continuous quantum melting of such WCs.

Вычисление тензора эффективной массы в ε- и ξ-фосфорене методом теории функционала плотности

In this work, the effective mass tensor was obtained from the law of dispersion of electrons in the conduction band of various allotropic modifications of phosphorene, calculated using the density functional theory. For calculations, we used the quantum chemical modeling package, OpenMX, which significantly reduces the calculation time for systems consisting of hundreds and thousands of atoms. Comparison of the obtained results of tensors for black and blue phosphorene with other works showed the correctness of the methods used, which were subsequently applied to other less studied allotropic modifications of phosphorene. In particular, ε- and ξ-phosphorene were studied, which are characterized by a non-hexagonal crystal lattice with an indirect band gap. This may be of interest from the point of view of discovering interesting acoustic-electronic properties in these materials, for example, resonant absorption of acoustic waves. Parallels between allotropes are drawn. From the point of view of effective mass tensors, ε-phosphorene is isotropic, like blue phosphorene, while ξ-phosphorene, on the contrary, has a non-isotropic structure, like black phosphorene. The results obtained during the study can be used in further study of the physical properties of materials, such as conductivity and photovoltaic effects.

Photoferroic prospective of multiferroic HoMnO3 compound, evaluated on the base of the DFT study of its magnetic, electronic, and optical properties

In this work, non-collinear spin density functional theory based all-electron full-potential linearized augmented plane wave electronic structure method have been employed to study magnetic, electronic, and optical properties of multiferroic HoMnO3 compound in order to evaluate its potential to be used in photovoltaic applications. Exchange and correlation effects of the Ho 4f- and Mn 3d-electrons were treated on the base of the local spin density approximation including effective Hubbard (Ueff) correction. We concluded that the ground state magnetic structure should be described by Γ3 irreducible representation of the P63cm space group, for both the Mn and the Ho magnetic sub-lattices. The calculated indirect and direct band gaps, corresponding to the transition from the O 2p-to the Mn 3d-states, are found to be 1.25 eV and 1.4 eV, respectively, in very good agreement with the experimental findings. Calculated dielectric tensor spectra are found to be in satisfactory agreement with the experiment as well. On the base of these results, we investigated photoferroic properties of the HoMnO3 by calculating its absorption coefficient spectrum and the effective mass tensor of the photogenerated electrons and holes. By comparing results with those of isostructural LuMnO3 and the YMnO3 multiferroics, we concluded that the HoMnO3 exhibits similar photoferroic properties, a fact that classifies it as a material suitable for photovoltaic applications.

A theoretical calculation of the cosmological constant based on a mechanical model of vacuum

Lord Kelvin believed that the electromagnetic aether must also generate gravity. Presently, we have no methods to determine the density of the electromagnetic aether, or we say the Ω(1) substratum. Thus, we also suppose that vacuum is filled with another kind of continuously distributed substance, which may be called the Ω(2) substratum. Based on a theorem of V. Fock on the mass tensor of a fluid, the contravariant energy-momentum tensors of the Ω(1) and Ω(2) substrata are established. Quasi-static solutions of the gravitational field equations in vacuum are obtained. Based on an assumption, relationships between the contravariant energy-momentum tensors of the Ω(1) and Ω(2) substrata and the contravariant metric tensor are obtained. Thus, the cosmological constant is calculated theoretically. The Ω(1) and Ω(2) substrata may be a possible candidate of the dark energy. According to the theory of vacuum mechanics, only those energy-momentum tensors of discrete or continuously distributed sinks in the Ω(0) substratum are permitted to act as the source terms in the generalized Einstein's equations . Thus, the zero-point energy of electromagnetic fields is not qualified for a source term in the generalized Einstein's equations. Some people believed that all kinds of energies should act as source terms in the Einstein's equations. It may be this unwarranted belief that leads to the cosmological constant problem. The mass density of theΩ(1) and Ω(2) substrata is equivalent to that of around 3 protons contained in a box with a volume of 1 cubic metre.

The Space-Time Metric Outside a Pulsating Charged Sphere

We consider the problem of determining the dynamics of the electromagnetic field generated outside a ball whose charge changes depending on time. We are in conditions of perfect symmetry and the electric field is considered to be radial. This is not a simplification since, under such a hypothesis, the magnetic field does not develop. Thus, it is first necessary to find out the appropriate modeling equations. These are obtained by writing a suitable energy tensor that combines the classical electromagnetic stress-energy tensor with a special kind of mass tensor. The next step is to show that it is possible to solve Einstein’s equations by plugging the new tensor on the right-hand side. Interesting connections with some classical solutions related to black holes are finally established.

Numerical investigation of geometrically nonlinear clamped uniform rods and rods with sections varying exponentially free vibration

The present paper is intended to investigate the problem of linear and non-linear longitudinal free vibration of uniform rods and rods whose cross-sections vary exponentially at large vibration amplitudes. The method adopted consists in discretizing the energy term on linear kij and non-linear rigidity tensor bijkl, as well as the mass tensor mij. Therefore, the formulation of this structure is based on Lagrange equations and the harmonic balance method so as to obtain the nonlinear algebraic equations. These latter are solved numerically and analytically through the explicit and linearized method. The response of Clamped-Clamped uniform and non-uniform rods on our structure are highlighted in the amplitude frequency and associated first three mode shapes. Moreover, this research leads to study the influence of the exponential slope on the maximum displacement, thus emphasizing the non-uniform bars usefulness. The obtained results are then compared with the available literature with a view to validating this theory. As a perspective, the method used in this paper would be pushed to study the FDM material, taking into account other parameters related to additive manufacturing, and later to be validated experimentally. Longitudinal vibrations are important in mechanical structures; therefore, the determination of their dynamic behaviour needs to be understood. In the present study, the effect of the displacement amplitude on the exponential slope of the structure was analysed, which led to the determination of the reduction range of the vibration amplitude under resonance. However, this should be taken into account in the design process. Besides, the usefulness of the non-linearity geometric effects was demonstrated to examine these structures by considering all the parameters involved. A linearized procedure is used to solve a nonlinear algebra equation. The use of this method leads to reduce calculation time contrary to iterative methods.

Electronic, Thermal, and Thermoelectric Transport Properties of ε-Ga2O3 from First Principles.

The electronic, thermal, and thermoelectric transport properties of ε-Ga2O3 have been obtained from first-principles calculation. The band structure and electron effective mass tensor of ε-Ga2O3 were investigated by density functional theory. The Born effective charge and dielectric tensor were calculated by density perturbation functional theory. The thermal properties, including the heat capacity, thermal expansion coefficient, bulk modulus, and mode Grüneisen parameters, were obtained using the finite displacement method together with the quasi-harmonic approximation. The results for the relationship between the Seebeck coefficient and the temperature and carrier concentration of ε-Ga2O3 are presented according to the ab initio band energies and maximally localized Wannier function. When the carrier concentration of ε-Ga2O3 increases, the electrical conductivity increases but the Seebeck coefficient decreases. However, the figure of merit of thermoelectric application can still increase with the carrier concentration.

Riemannian Formulation of Pontryagin’s Maximum Principle for the Optimal Control of Robotic Manipulators

In this work, we consider robotic systems for which the mass tensor is identified to be the metric in a Riemannian manifold. Cost functional invariance is achieved by constructing it with the identified metric. Optimal control evolution is revealed in the form of a covariant second-order ordinary differential equation featuring the Riemann curvature tensor that constrains the control variable. In Pontryagin’s framework of the maximum principle, the cost functional has a direct impact on the system Hamiltonian. It is regarded as the performance index, and optimal control variables are affected by this fundamental choice. In the present context of cost functional invariance, we show that the adjoint variables are the first-order representation of the second-order control variable evolution equation. It is also shown that adding supplementary invariant terms to the cost functional does not modify the basic structure of the optimal control covariant evolution equation. Numerical trials show that the proposed invariant cost functionals, as compared to their non-invariant versions, lead to lower joint power consumption and narrower joint angular amplitudes during motion. With our formulation, the differential equations solver gains stability and operates dramatically faster when compared to examples where cost functional invariance is not considered.

Residual Gauge Invariance in a Massive Lorentz-Violating Extension of QED

We reassess an alternative CPT-odd electrodynamics obtained from a Palatini-like procedure. Starting from a more general situation, we analyze the physical consistency of the model for different values of the parameter introduced in the mass tensor. We show that there is a residual gauge invariance in the model if the local transformation is taken to vary only in the direction of the Lorentz-breaking vector. This residual gauge invariance can be extended to all models whose only source of gauge symmetry breaking is such a mass term.

Anisotropy and quench dynamics of quasiholes in fractional quantum Hall liquids

We present a microscopic study of quasiholes in bosonic fractional quantum Hall (FQH) liquids at filling factor $\nu=1/2$ in the lowest Landau level with anisotropic band mass tensors. We use the spatial density profile to characterize the shape of a quasihole and analyze its anisotropy. We then compare the quasihole's anisotropy with the intrinsic geometric metric of the system that is extracted from the maximal overlap between the numerically obtained quasihole ground state and a set of model wave functions of anisotropic quasiholes. For a static system, we find that the quasihole's anisotropy, similar to the intrinsic metric, grows with the anisotropy of the band mass tensor. When the quasihole develops well, we observe a correspondence between the anisotropy of the quasihole and the intrinsic metric of the underlying anisotropic FQH state. We also drive the system out of equilibrium by suddenly changing the band mass tensor. In this case, the shape of the quasihole evolves with time and shows similar dynamics with the intrinsic metric of the postquench state. The evolving frequency matches the energy of a spin-$2$ quadrupole degree of freedom in the system. Our results suggest that the density profile of a quasihole is a useful tool to estimate the intrinsic metric and capture the dynamics of an FQH system.

Quantifying avian inertial properties using calibrated computed tomography.

ABSTRACTEstimating centre of mass and mass moments of inertia is an important aspect of many studies in biomechanics. Characterising these parameters accurately in three dimensions is challenging with traditional methods requiring dissection or suspension of cadavers. Here, we present a method to quantify the three-dimensional centre of mass and inertia tensor of birds of prey using calibrated computed tomography (CT) scans. The technique was validated using several independent methods, providing body segment mass estimates within approximately 1% of physical dissection measurements and moment of inertia measurements with a 0.993 R2 correlation with conventional trifilar pendulum measurements. Calibrated CT offers a relatively straightforward, non-destructive approach that yields highly detailed mass distribution data that can be used for three-dimensional dynamics modelling in biomechanics. Although demonstrated here with birds, this approach should work equally well with any animal or appendage capable of being CT scanned.