Stress Tensor Research Articles

Stress‒dilatancy behaviour of calcareous sand

Calcareous sands are a special geomaterial primarily composed of calcium carbonate or other insoluble carbonate materials susceptible to breakage. Therefore, breakage has a crucial impact on the change of calcareous sand granulation and the shape of individual grains during shear. By analysing some triaxial tests results of some calcareous sands presented in the literature in the light of the Frictional State Concept (FSC), it can be shown that the stress‒dilatancy relationship in different shear phases can be approximated with straight lines. These lines are defined by a critical frictional state angle and two material parameters of the FSC. These parameters express the deviation between the stress‒dilatancy relationship for tested soil and the stress‒dilatancy relationship for isotropic granular material shear deformed without non-coaxiality of the stress tensor and strain increment tensor, breakage, and other effects affecting energy dissipation during shear. It will also be shown that the points representing failure states in the η - D plane lie on a straight line with a much higher slope than for silica sands due to the breakage effect. Using the FSC, the stress‒dilatancy relationship can be simply described in different shear phases and used to build new elasto-plastic models of calcareous sand in the future.

Relativistic dissipative magnetohydrodynamics from the Boltzmann equation for 2-particle species gas

We derive the equations of motion of relativistic magnetohydrodynamics from the Boltzmann equation using the method of moments. We consider a locally electrically neutral system composed of two particle species with opposite charges, with vanishing dipole moment or spin, so that the fluid has vanishing magnetization and polarization. We find that the dynamics of this fluid changes dramatically in the presence of a magnetic field. The shear stress tensor no longer adheres to a single differential equation; instead, it splits into three non-degenerate components, each evolving according to distinct dynamical equations. Exploring these equations in a Bjorken flow scenario, we find that for large magnetic fields, our theory predicts oscillatory behavior beyond the scope of an Israel-Stewart-like theory.

Stress ratio effect on fatigue crack growth assessment via thermoelastic stress analysis

The fatigue performance of welded joint is strongly dependent on welding residual stresses as well as the local geometry of the weld toes. To ensure the structural integrity over time of such components, fine predictive models must be proposed together with efficient experimental methods for their assessment.This study introduces a model based on linear elastic fracture mechanics to forecast fatigue life in service of high strength steel welded joints. This model considers the effect of the local stress ratio (dependent on stabilized residual stresses) on the crack growth rate through the definition of an effective stress range. The use of Thermoelastic Stress Analysis (TSA) is used to perform in-situ monitoring of fatigue cracks on welded joints to experimentally characterize the crack-closure phenomenon at different load ratios. Indeed, under adiabatic conditions, the temperature's first harmonic at the surface of a structure submitted to cyclic loading is proportional to the stress tensor's first invariant's amplitude. As a result, the presence of a surface crack and the non-linearity induced by the alterative opening and closing within one loading cycle can be observed in the immediate temperature response. This phenomenon is used to evaluate a crack-opening rate that is related to the fatigue life of the welded joints.

A New Method to Invert for Interseismic Deep Slip Along Closely Spaced Faults Using Surface Velocities and Subsurface Stressing‐Rate Tensors

AbstractInversions of interseismic geodetic surface velocities often cannot uniquely resolve the three‐dimensional slip‐rate distribution along closely spaced faults. Microseismic focal mechanisms reveal stress information at depth and may provide additional constraints for inversions that estimate slip rates. Here, we present a new inverse approach that utilizes both surface velocities and subsurface stressing‐rate tensors to constrain interseismic slip rates and activity of closely spaced faults. We assess the ability of the inverse approach to recover slip rate distributions from stressing‐rate tensors and surface velocities generated by two forward models: (a) a single strike‐slip fault model and (b) a complex southern San Andreas fault system (SAFS) model. The single fault model inversions reveal that a sparse array of regularly spaced stressing‐rate tensors can recover the forward model slip distribution better than surface velocity inversions alone. Because focal mechanism inversions currently provide normalized deviatoric stress tensors, we perform inversions for slip rate using full, deviatoric or normalized deviatoric forward‐model‐generated stressing‐rate tensors to assess the impact of removing stress magnitude from the constraining data. All the inversions, except for those that use normalized deviatoric stressing‐rate tensors, recover the forward model slip‐rate distribution well, even for the SAFS model. Jointly inverting stressing rate and velocity data best recovers the forward model slip‐rate distribution and may improve estimates of interseismic deep slip rates in regions of complex faulting, such as the southern SAFS; however, successful inversions of crustal data will require methods to estimate stressing‐rate magnitudes.

FORMATION OF THE NEAR-FAULT DAMAGE ZONE DURING DYNAMIC RUPTURE IN A CRYSTALLINE ROCK MASS

Many models for analyzing the dynamic propagation of seismogenic ruptures are based on solving classical problems of fracture mechanics. It is assumed that the fault is a shear fracture mode with uniformly distributed friction and stresses at the crack tip. Known fracture mechanics theories do not describe the formation of laterally oriented damage zones, i.e., in the direction perpendicular to the crack plane. Observational data indicate the presence of a fairly extensive zone of damaged material in the vicinity of the fault. This is the zone of dynamic influence where the material has an increased number of fractures, higher permeability and lower elastic wave velocities. A correct assessment of the properties and sizes of zones of dynamic influence is crucial for constructing adequate earthquake preparation models. This paper analyzes the development patterns and quantitative characteristics of the damage zone during dynamic earthquake rupture and quasi-static evolution of the fault. To estimate the size and mechanical characteristics of the near-fault damage zone caused by movement along the slip surface, it is convenient to use the value of the second invariant of the deviatoric stress tensor (shear intensity). Matching of the calculated value with one or another degree of rock mass damage can be done using measurement data from large-scale explosions, by comparing them with the calculation results. It is shown that coseismic movement along the fault leads to insignificant changes in the properties of the host rock mass. Although the longitudinal wave velocity near the fault decreases markedly by 30-35%, the permeability increases only by approximately a factor of three, and the increase in the degree of fracturing is almost unnoticeable. This means that the properties of the rock mass change due to the opening of preexisting cracks. Repeated movements do not radically change the characteristic dimensions and properties of the damage zone. It is concluded that the fault-affected zone is formed mainly at the quasi-static stage of the formation of a single main fault zone through the coalescence of individual macrofractures, and future seismogenic movements add to the already existing amount of fractures.

Force and stress calculations with a neural-network wave function for solids.

Accurate ab initio calculations of real solids are of fundamental importance in fields such as chemistry, phases and materials science. Recently, variational Monte Carlo (VMC) based on neural-network wave functions has been developed as a promising option to solve the existing challenges in ab initio calculations. In this study, we discuss the calculation of interatomic forces and stress tensors of real solids with a neural-network-based VMC method. A new scheme for computing forces is proposed based on the space-warp coordination transformation method, which achieves better accuracy, efficiency and robustness than existing methods. In addition, we also designed new periodic features of the neural network to further improve the robustness of force calculations for different lattices. This work paves the way for further extending the application of machine-learning quantum Monte Carlo methods in materials modelling.

Space-time decay rate of the 3D diffusive and inviscid Oldroyd-B system

<abstract><p>We investigate the space-time decay rates of solutions to the 3D Cauchy problem of the compressible Oldroyd-B system with diffusive properties and without viscous dissipation. The main novelties of this paper involve two aspects: On the one hand, we prove that the weighted rate of $ k $-th order spatial derivative (where $ 0\leq k\leq3 $) of the global solution $ (\rho, u, \eta, \tau) $ is $ t^{-\frac{3}{4}+\frac{k}{2}+\gamma} $ in the weighted Lebesgue space $ L^2_{\gamma} $. On the other hand, we show that the space-time decay rate of the $ m $-th order spatial derivative (where $ m \in\left [0, 2\right] $) of the extra stress tensor of the field in $ L^2_{\gamma } $ is $ (1+t)^{-\frac{5}{4}-\frac{m}{2}+\gamma} $, which is faster than that of the velocity. The proofs are based on delicate weighted energy methods and interpolation tricks.</p></abstract>

Testing the assumptions of the Effective Field Theory of Large-Scale Structure

The Effective Field Theory of Large-Scale Structure (EFTofLSS) attempts to amend some of the shortcomings of the traditional perturbative methods used in cosmology. It models the evolution of long-wavelength perturbations above a cutoff scale without the need for a detailed description of the short-wavelength ones. Short-scale physics is encoded in the coefficients of a series of operators composed of the long-wavelength fields, and ordered in a systematic expansion. As applied in the literature, the EFTofLSS corrects a summary statistic (such as the power spectrum) calculated from standard perturbation theory by matching it to N-body simulations or observations. This `bottom-up' construction is remarkably successful in extending the range of validity of perturbation theory. In this work, we compare this framework to a `top-down' approach, which estimates the EFT coefficients from the stress tensor of an N-body simulation, and propagates the corrections to the summary statistic. We consider simple initial conditions, viz. two sinusoidal, plane-parallel density perturbations with substantially different frequencies and amplitudes. We find that the leading EFT correction to the power spectrum in the top-down model is in excellent agreement with that inferred from the bottom-up approach which, by construction, provides an exact match to the numerical data. This result is robust to changes in the wavelength separation between the two linear perturbations. However, in our setup, the leading EFT coefficient does not always grow linearly with the cosmic expansion factor as assumed in the literature based on perturbative considerations. Instead, it decreases after orbit crossing takes place.

Numerical viscosity control in Godunov-like smoothed particle hydrodynamics for realistic flows modeling

In this study, we introduce a way to control the viscosity of the numerical approximation in the Godunov-like smoothed particle hydrodynamics (SPH) methods. This group of SPH methods includes momentum and energy fluxes in the right-hand sides of the equations, which are calculated by the solution of the Riemann problem between each pair of neighboring particles within the support radius of the smoothing kernel, which is similar to the procedure for the calculation of fluxes across cell boundaries in Godunov schemes. Such SPH methods do not require the use of artificial viscosity since the significant numerical viscosity is already introduced by a Riemann problem solution. We demonstrate that such a numerical viscosity may be measured and obtain the explicit expression for it depending on smoothed particle properties. In particular, we have found that Godunov-like SPH method with interparticle contact algorithms produces numerical viscosity several orders of magnitude higher than physical viscosity in materials. Modern approaches, such as SPH with monotonic upstream-centered scheme for conservation laws or weighted essentially non-oscillatory reconstruction techniques, have not only lower numerical viscosity but also too large for modeling real-world viscous flows. By constructing a correcting viscous stress tensor based on the analytical solution for discontinuous viscous flow, it is possible to reduce the viscous stresses of numerical origin. The use of such a correction makes it possible to improve the agreement with experiments in the simulation of viscous flows without using schemes of higher order reconstruction.

CREEPING FLOW OF COUPLE STRESS FLUID OVER A SPHERICAL FIELD ON A SATURATED BIPOROUS MEDIUM

This problem emphasizes the dynamic interaction between a biporous medium and a couple stress fluid of laminar flow. The flow around a permeable field engulfed in a couple stress fluid is examined. When examining the motion of an oil droplet in a porous collector that is surrounded by an aqueous medium (oil-in-water emulsion) and is subject to an external pressure drop, this formulation of the problem is typical. A similar issue arises when lymph enters the tissues of humans or animals: the inside permeable spherical field saturated with viscous fluid and outside region saturated with couple stress fluid. The Brinkman equations are utilized to characterize the couple stress fluid flow in a saturated biporous medium. The couple stress tensor and velocity fields are expressed using Gegenbauer polynomials and Macdonald functions. For the axially symmetric motion, both pressure distribution and the stream function solution are explicitly solved. The method of variable separation is used to investigate an analytical resoluteness for the flow field. The drag force on a saturated biporous medium and the drag coefficient <i>D<sub>N</sub></i> are calculated, and the impacts of the permeability κ, the ratio of viscosity (γ<sup>2</sup> = μ<sub>1</sub> /μ<sub>2</sub>), the couple stress viscosity ratio (τ = η'/η), and the parameter of couple stress (λ = √μ/η). The appropriate dependencies are graphically delineated and reviewed, including the permeability κ, couple stress parameter λ, viscosity ratio γ<sup>2</sup>, and couple stress viscosities (η, η'). According to the findings, increasing permeability gradually raises the drag coefficient, which is used to describe a spherical field’s surface with a high level resistance of flow. Limits statements are used to illustrate specific cases that are well-known. The current study is significant primarily in the course through a layer formed by penetrable particles and has very important and compelling applications in both nature and innovation, with a variety of potential outcomes.

Momentum Transfer in Triboelectric Nanogenerators

AbstractThe triboelectric nanogenerator (TENG) reflects a prospering field, that uses Maxwell's displacement current as the driving force to transform mechanical energy into electricity. Based on the law of energy conservation, many theoretical models of TENGs are proposed to provide a detailed insight into how energy flows and transformation in the energy harvesting system, while ignoring a hidden but extremely important point about TENG's momentum transfer and conservation. Here a series of analysis is presented for the momentum transfer and conservation in TENGs based on Maxwell equations and stress tensor. Using a time‐dependent 3D mathematical model, it is elaborated that how the time‐ and spatial‐dependent momentum current is influenced by the field and the dielectric materials, demonstrating that momentum is overall conserved for a TENG. In other words, the TENG device can not only convert mechanical energy into electricity, but it is also able to transfer momentum. Momentum transfer is another important characteristic of TENGs, and finally, the essential differences and similarities among the momentum transfer, energy transfer, and energy transformation in TENGs are systematically discussed. This study will certainly serve as a new starting point for exploring momentum transfer and conservation in the TENG momentum transfer system.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of mathbb {R}^2 into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.

Stress Regimes and Stress Fields Analysis Using Fault-Slip Data; Eastern Iran

In this paper direction of stress tensors and stress ellipsoids shape in Neogene, Paleogene, and older units were investigated. Moreover, relationship between shape of stress ellipsoid and exhumation of igneous units were determined using analysis of stress tensors. Analysis of relationships between stress regime and uplifting indicate that in the areas we observed the uplifting of the igneous units the shape of stress ellipsoid has been displaced locally from prolate to oblate which is due to local change of stress regime. Hence, local variation of stress regime in the eastern part of Shekarab Mountains caused boarder outcrops of igneous units. In the western part of study area shape of stress ellipsoid is prolate and it's due to existing reverse fault with strike-slip component. Comparison shape of stress ellipsoid in Eocene units indicate that shape of stress ellipsoid in the western, middle and eastern Shekarab Mountains is prolate, and as local there is oblate stress ellipsoid shape in eastern part. Results of this research indicate that direction of major stress axis had clockwise rotation from Cretaceous times to the present.

Physics-informed neural network reconciles Australian displacements and tectonic stresses

Stress orientation information is invaluable to evaluate active tectonic forces within the Earth’s crust. The global dataset provided by the World Stress Map offers a rich resource of stress indicators, facilitating the calibration of mechanical models to extract complete stress and displacement fields. However, traditional inversion processes are hampered by the manual tuning of geomechanical properties and boundary conditions to reconcile simulations with observations. In this study, we introduce ML-SEISMIC (machine learning for stress estimation integrating satellite image and computational modelling), a physics-informed deep neural network approach to autonomously align stress orientation data with an elastic model. It nearly completely bypasses the need for explicit boundary condition inputs and yields comprehensive distributions of material properties, displacements, and stress tensors. Application of this methodology to Australia, coupled with precise global navigation satellite systems observations, unveils a robust and scale-independent interpolation framework. Additionally, it pinpoints regions where stress orientation reinterpretation is warranted. Our results present a streamlined yet powerful process, offering a substantial leap forward in geodynamic investigations. This approach promises to unify velocity and stress orientation observations with physical models, ushering in a new era of insights into Earth’s dynamic processes.

Novel method for the accurate calculation of Drucker–Prager cap model parameters and reduction of experimental time and effort

The Drucker–Prager cap (DPC) model is useful for estimating stress and strain distributions in compacted powders; however, it requires numerous stress state parameters, which are obtained from various tests, thus requiring significant time and effort, and a large number of samples. In addition, there is room for improvement in the reproducibility of DPC model simulations with respect to realistic behavior. This study develops a simple method for estimating DPC model parameters, and it can accurately reproduce the pressures measured during tableting. This objective is achieved by estimating the onset of particle deformation using the powder compaction equation and including the shear stress that results from die wall friction in the stress tensor.

Meshless numerical manifold method with novel subspace tracking and CSS locating techniques for slope stability analysis

The meshless numerical manifold method (MNMM) inherits the advantages of handling continuous and discontinuous problems uniformly, improving interpolation precision, and avoiding linear dependence issues. This paper refines the MNMM method for slope stability analysis to overcome the dependence and sensitivity of the strength reduction method of the finite element method on the number of finite element meshes and the algorithm for solving the nonlinear equations when calculating slopes with soft and weak interlayers. First, the strength reduction method (SRM) was incorporated into MNMM to create the numerical model called MNMM-SRM. Based on the MNMM-SRM, a complementary Mohr-Coulomb (M−C) control equation compliant with Koiter's rule of multi-yielding surfaces was established. In the principal stress space, the subspace tracking method was used to track the eigenvalues and eigenvectors of the stress tensor, thereby addressing the corner problem of M−C multi-yield surfaces. Combining the k-means clustering algorithm with wavelet transformation and employing the slope's total nodal displacement field as an input variable, an intelligent and efficient critical slip surface (CSS) automatic extraction method was developed. Three typical cases were used to validate the accuracy of the numerical model. This numerical model has a wide range of potential applications in slope stability research.

Sensitivity analysis for acoustic-driven gas bubble dynamics in tangent hyperbolic fluid

Ultrasound imaging, or sonography, utilizes high-frequency sound waves to create images of the internal structures of the human body. It is widely applied in medical diagnostics due to its non-invasive nature and real-time capabilities. Ultrasound waves are emitted into the body, and their reflections are captured to generate detailed images of organs, tissues, fetuses during pregnancy, and more. The interaction between sound waves and biological tissues and issues related to propagation, reflection, and absorption of ultrasound are crucial considerations for improving image quality and safety. It is frequently used to keep an eye on the health and development of the fetus throughout pregnancy. The examination of several organs, including the heart, liver, kidneys, and blood arteries, is also done to look for anomalies, tumors, and other diseases. The dynamics of spherical gas bubbles within a non-Newtonian fluid exposed to an external sonic field are examined in this paper. The mathematical model offers a useful foundation for examining bubble behavior since it incorporates the additional stress tensor of a tangent hyperbolic fluid. The generalized RP problem may be transformed into an ordinary differential equation that can be solved using numerical methods, allowing a large range of parameter values to be investigated and resolving convergence difficulties. Particularly noteworthy are the numerical results for long acoustic cycles. The link between input and output variables is also investigated using an experimental methodology closely related to sensitivity analysis, which is important for possible device development. This innovative approach offers a fresh perspective on the subject.

Warning! The negative divergence of the stress-tensor does not always yield the Ehrenfest force.

It has been assumed that the negative divergence of all stress tensors in common use yields the same force. This work finds that this is untrue, and, in fact, can vary wildly. We demonstrate this for the hydrogen atom, the one-particle isotropic harmonic oscillator, and a particle in an infinite spherical well where the exact density, pair-density, and the first order reduced density matrix are known for ground and excited states without any approximation. The Ehrenfest stress-tensor is introduced as any stress-tensor whose negative divergence will yield the corresponding Ehrenfest force for the same system when the exact wave-function is utilized. Stress-tensors within the literature are examined to show those that are Ehrenfest stress-tensors. Those that differ are demonstrated by how they differ within an exact formulation. The proof that the negative divergence of an Ehrenfest stress-tensor yields the Ehrenfest force is summarized.

Force-chain finder: A software tool for the recursive detection of force-chains in granular materials via minor principal stress

Force transmission in granular media occurs through an inhomogeneous network of inter-particle contacts referred to as force-chains. A thorough understanding of the structure of these chains is indispensable for a better comprehension of the macroscopic signatures they generate. This paper introduces Force-Chain Finder (FCF), an open-source software tool designed for detecting force-chains in granular materials. Leveraging the stress tensor computed for each particle based on its interactions with neighbouring particles, the tool effectively identifies the magnitude and direction of the most compressive principal stress. Through a recursive traversal of particles and their neighbours, force-chains are robustly detected based on the alignment of the principal stress directions, which is decided by a parameter α (an angle in radians). The software provides a comprehensive suite of post-processing features, including the exportation of results in different formats, enabling detailed analysis of specific regions and dynamic phenomena. Additionally, the software facilitates the computation of statistical measures pertaining to chain size and population. By streamlining the identification and characterization of force-chains within discrete element method (DEM) simulations, this tool significantly enhances the efficiency and accuracy of force-chain analysis. Thus, the software promotes deeper insights into the behaviour of granular materials by enabling researchers to effortlessly detect and analyse force-chains. PROGRAM SUMMARYProgram Title: Force Chain Finder (FCF)CPC Library link to program files:https://doi.org/10.17632/33pkc4f63t.1Developer's repository link:https://github.com/Particles-Research/Force_Chain_Finder_Public/tree/mainLicensing provisions: GNU General Public License 3Programming Language: C++, PythonSupplementary material: Illustrative examplesNature of the Problem: FCF has been developed with the primary objective of efficiently and comprehensively detecting force networks that arise in Discrete Element Modelling (DEM) simulations of granular flow.Solution Method: The method is based on the work of “J. Peters, M. Muthuswamy, J. Wibowo, and A. Tordesillas, Characterization of force chains in granular material, Physical Review E 72, 041307 (2005)”. It utilizes the concept of minor principal stress to identify quasilinear chains, each consisting of at least three particles. However, significant algorithmic modifications have been implemented to enhance the accuracy of the procedure, enabling the detection of chain branching and merging.