Critical State Model Research Articles

CASM-U: a unified critical state model for unsaturated clays and sands

CASM-U: a unified critical state model for unsaturated clays and sands

An analytical model for predicting the magnetization loss in HTS sector-shaped conductors for fusion

Abstract Within the framework of magnetic confinement fusion, several projects worldwide are demonstrating the possibility of integrating high-temperature superconductors (HTS) in the coil systems. HTS-based technologies are highly attractive for practical applications because they can extend the operating margins of fusion coils in terms of higher temperatures, transport currents and magnetic fields. Based on the results achieved with the twisted-stacked tape cable, we have designed a novel low-loss HTS sector cable-in-conduit conductor, with a target of 60 kA at 4.5 k, 18 T, which is presently of interest for the DEMO Central Solenoid coil. In HTS cables, the AC losses can represent a significant limiting factor, therefore they must be taken into consideration both in the design phase and in the assessment of the overall magnet thermal budget. In this work, to assess the loss behavior and to optimize the cable design, we have explored different aspect ratios and arrangements of the stacked tapes within the cable layout. The magnetization losses are calculated with a 2D finite-element model based on the T-A formulation and analytical approximations based on the Brandt-Halse critical state model. Specifically, we have developed an analytical formulation that allows for the calculation of the instantaneous power losses in HTS stacked cables with a limited number of tapes per stack, achieving sufficient accuracy at high fields. The analytical model enables a sufficiently accurate assessment of the heat deposited on the conductor during those particular instants of a plasma scenario where the variation of the field is very high, such as during the critical initial discharge period of the plasma initiation.

Vortex Phase Diagram and Pinning Mechanisms of Air‐Annealed FeTe0.8S0.2 Single Crystals

The vortex phase diagram and superconducting properties of air‐annealed FeTe0.8S0.2 single crystals are investigated. The annealed samples exhibit a superconducting transition, as confirmed by temperature‐dependent resistivity measurements. The thermally activated energy is calculated from the magnetotransport measurements, and the analysis shows that as the magnetic field increases, there is a crossover from single to collective vortex pinning. The vortex phase diagrams have been determined by analyzing magnetic field‐dependent resistivity, revealing the transitions from an unpinned vortex liquid region to a pinned vortex liquid state and further transition from this pinned state to a vortex glass state. Temperature‐dependent magnetization measurement determines the superconducting volume fraction. The critical current densities, as a function of the magnetic field (JC(H)), have been estimated from the magnetization versus magnetic field loops measured at various temperatures. Bean's critical state model estimates the JC values, and for an optimized annealed crystal, the self‐field JC at 2 K is . The normalized pinning force density in the samples is determined using the Dew Hughes model to identify the pinning mechanisms. The temperature‐dependent JC data analysis indicates the presence of ‐pinning in the samples. Theoretical models analyze experimental observations to understand the vortex pinning mechanisms.

On the trapped magnetic moment in type-II superconductors

Abstract Measurements of the trapped (remanent) magnetic moment, M trap ( H ) , when a small magnetic field H is turned off after cooling below the superconducting transition temperature, Tc , or ramping a magnetic field up and down after cooling in a zero magnetic field, offer substantial advantages in difficult cases of small samples and large field-dependent backgrounds, which is relevant for hydrogen-based ultra-high-T c superconductors (UHTS). Until recently, there was no need for a separate paper on the trapped magnetic flux for well-known critical-state models due to the simplicity of the physics involved. However, recent publications showed the need for such an analysis. This note summarizes the expectations for the Bean model with constant critical current density and the Kim model with field-dependent critical currents. It is shown that if the trapped moment is fitted to the power law, M trap ∝ H α , the fixed exponent α = 2 is exact for the Bean model, while Kim models show a wide interval of possible values, 2 ⩽ α ⩽ 4 . Furthermore, accounting for reversible magnetization expands the range of possible exponents to 1 ⩽ α ⩽ 4 . In addition, demagnetizing factors are essential and make the trapped moment orientation dependent even in isotropic materials. As a concrete application, it is shown that flux trapping experiments performed on H3S UHTS compounds can be well described using this generalized approach, lending further support to the type-II superconducting nature of H3S under ultra-high pressure.

Revealing the pinning landscape and related vortex pattern evolution in granular superconducting films

Most superconducting electronics based on films exhibit granular structures. It has been suggested that grain boundaries form a network with relatively weak superconductivity, potentially acting as pinning centers. Yet, so far, detailed microscopic studies of the pinning landscape and its relation to vortex behavior remain scarce. Here, we imaged the vortex lattices (VL) in granular Nb films using magnetic force microscopy over large scanning areas at various magnetic fields. A non-monotonic evolution in the degree of vortex lattice ordering was observed with increasing vortex density, driven by a combination of vortex-vortex interactions and pinning effects. The spatial distribution of pinning potential within the film was directly mapped using a recently developed scanning quantum vortex microscope (SQVM). Instead of the network formed by grain boundaries, the pinning landscape presents a network-like structure, yet with domains significantly larger than the individual grains. The results of numerical simulations based on pinning landscape revealed by SQVM well reproduce our experiments. The pinning force per unit length at low magnetic fields was calculated. The critical current density, estimated from the relative positions of vortices, aligns well with the critical state model. Our work illustrates the relationship between the evolution of the vortex lattice with magnetic field and the structural features of granular Nb film, providing new insights into the design of high-performance superconducting electronic devices.

The effect of the non-uniform critical current density on the magnetoelastic behavior of bulk superconductors: Case of a long cylindrical superconductor

In this paper, the effect of non-uniform critical current density on bulk superconductors is studied, the case of a long cylindrical superconductor with transport current is chose to discussed. The critical current density is distributed non-uniformly along the radius of the cylinder. Based on the Bean critical state model, the distributions of trapped magnetic flux and shielding current in the cylinder are investigated. Combined with the plane strain approach, the analytical expressions of magnetic flux pinning force and stress are obtained. The magnetostriction of the cylinder is also discussed. Results show that the non-uniform critical current density changes the distribution law of the shielding current and trapped magnetic flux in the cylinder. The increase of non-uniform parameters n leads to an obvious increase in the flux pinning force. Thus, a larger extreme value of the pinning stress is obtained, a bigger structure deformation is produced inside the superconducting cylinder. All those conclusions will provide a helpful guide for engineering application.

The use of a nonlocal critical state model in modelling triaxial and plane strain tests on overconsolidated clays

When modelling the phenomenon of strain localisation in strain-softening soils with the finite element (FE) method, nonlocal approaches have been commonly employed to avoid mesh dependency and numerical instability. This paper first presents the FE formulation of a critical state model for highly overconsolidated clays incorporating a nonlocal method. The performance of the nonlocal strain regularisation is subsequently assessed through a series of coupled hydro-mechanical (HM) analyses of undrained and drained triaxial compression tests on London clay. The mechanism behind the evolution of strain localisation in triaxial tests is investigated and a comparison with equivalent plane strain analyses is discussed. Finally, a comprehensive sensitivity study is presented, investigating the influence of the two nonlocal parameters, in the adopted nonlocal algorithm, on the predicted stress–strain responses. A key outcome is the derived linear relationship between the two parameters, which enables a unique stress–strain response to be achieved in either axisymmetric or plane strain analyses with multiple combinations of the two parameters. Such a modelling capability is essential in applications of the proposed nonlocal strain regularisation in large scale boundary value problems in which restrictions on element size exist.

Superconductivity in atomically thin films: Two-dimensional critical state model

The comprehensive understanding of superconductivity is a multiscale task that involves several levels, starting from the electronic scale determining the microscopic mechanism, going to the phenomenological scale describing vortices and the continuum-elastic scale describing vortex matter, to the macroscopic scale relevant in technological applications. The prime example for such a macro-phenomenological description is the Bean model that is hugely successful in describing the magnetic and transport properties of bulk superconducting devices. Motivated by the development of novel devices based on superconductivity in atomically thin films, such as twisted-layer graphene, here, we present a simple macro-phenomenological description of the critical state in such two-dimensional (2D) thin films. While transverse screening and demagnetization can be neglected in these systems, thereby simplifying the task in comparison with usual film- and platelet-shaped samples, surface and bulk pinning are important elements to be included. We use our 2D critical state model to describe the transport and magnetic properties of 2D thin-film devices, including the phenomenon of nonreciprocal transport in devices with asymmetric boundaries and the superconducting diode effect. Published by the American Physical Society 2024

Electromagnetic response analysis of high temperature superconducting coated conductor tape with local debonding: Case of transporting current in an external magnetic field

In this article, the electromagnetic mechanical behavior of a long superconducting coating tape within a local internal cracking was analyzed when it transport current in a perpendicular magnetic field. Based on the superconducting critical state model, the distribution of current and magnetic field within the structure were calculated. Then, the control equation for the correlation between the flux pinning force inside the superconducting thin film and the shear force at the substrate interface was given using the plane strain method of linear elasticity theory. Combined with the Chebyshev polynomial solution, the distribution patterns of normal stress inside the superconducting thin film and shear stress at the substrate interface were finally obtained. The results indicate that, unlike the phenomenon observed under a single field excitation, the stress distribution in the structure exhibits an asymmetric distribution when both transport current and external magnetic field are applied. The normal stress inside the superconducting thin film is significantly larger on the right side of the structure than that on the left side, and the same distribution law is found in the shear stress at the substrate interface. The stress concentration occurs at the separation point between the film and the substrate. When the Young's modulus of the substrate material is high, the normal stress inside the superconducting film is low, while the shear stress at the interface is high, and vice versa. All these results provide theoretical guidance for the application of superconducting tapes.

Unified modeling for clay and sand with a hybrid-driven fabric evolution law

This paper presents a novel critical state model for both clay and sand considering the inherent and induced anisotropy, named as CASM-h. The CASM-h model is developed based on the well-calibrated CASM model and incorporates the concepts of subloading surface and fabric anisotropy. To describe the induced anisotropy, a hybrid-driven fabric evolution law is proposed, defining fabric evolution as driven jointly by stress and strain rates. Subsequently, the fabric tensor together with its evolution law is appropriately incorporated into constitutive relation by employing the anisotropic transformed stress approach, without introducing additional mechanical mechanisms. The CASM-h model is validated under a series of stress paths, including triaxial tests, simple shear tests, shear tests with fixed principal stress direction as well as principal stress rotation (PSR) tests. The simulation results show that CASM-h model can satisfactorily capture the anisotropic behaviours of both clay and sand, such as the loading direction dependency of strength, non-coaxial behaviour, PSR effects and so on.

Effects of gamma-irradiation on the superconducting properties of FeTe0.55Se0.45 single crystals grown by self-flux method

We have investigated the effect of gamma (γ)-irradiation on the structural and superconducting properties of FeTe0.55Se0.45 single crystals grown by the self-flux method. The impact of γ-irradiation on the superconducting transition temperature (TC), critical current density (JC), and vortex pinning mechanism has been systematically studied. The x-ray diffraction study reveals the growth of single crystals along the c-axis. The superconductivity has been confirmed in pristine and γ-irradiated samples through temperature-dependent resistivity (ρ(T)) and magnetization [M(T)] measurements. After irradiation, a slight improvement is observed in the upper critical field Hc2(0) values. The values of thermally activated energy have been calculated and a crossover from a single to collective vortex pinning regime is observed. Additionally, we have analyzed the vortex phase diagrams, revealing a transition from vortex liquid to vortex glass state. Furthermore, the presence of second magnetization peak (SMP) or fishtail effect has been noticed in the M(H) loops, and with increasing temperature, the position of SMP (Hsp) shifts toward lower magnetic field regions. The critical current density has been estimated by Bean's critical state model at different magnetic fields [JC(H)] and temperatures [JC(T)]. The defects through gamma-irradiation lead to a significant threefold increase in JC compared to pristine samples in self-field and at 2 K. The pinning mechanisms have been explained using collective pinning theory and the Dew-Hughes model by analyzing the normalized pinning force density. Our analysis indicates that δl-pinning is dominant and point defects are present in all the samples.

COMPLETE AND SEMI-COMPLETE EXPLICIT ALGORITHMS OF A UNIFIED CRITICAL STATE MODEL FOR OVER-CONSOLIDATED SOILS

This paper presents a comparison of the performance of explicit algorithm and semi-complete explicit algorithm in the numerical implementations of an unconventional plastic model for soils. The new model, named CASM-S, is developed by incorporating the sub-loading surface theory into the standard unified clay and sand model (i.e., CASM), to enhance the prediction ability for the mechanical behavior of over-consolidated soils. The complete explicit algorithm of CASM-S is based on the sub-stepping method with the technique of automatic error control (SUBM), while the semi-complete explicit algorithm adopts the cutting-plane integration procedure (CPM). The complete implementation process of this model is performed, and the stability, accuracy, and efficiency of these two algorithms are compared through a series of numerical simulations, such as fluid-structure coupling problem, over-consolidation problem, and square-footing problem. These simulations demonstrate that CASM-S implemented by both the SUBM and CPM can obtain a reliable solution under appropriate size of increments. For the sub-loading surface model with highly nonlinear characteristics used in this paper, the CPM has faster local convergence rate, but the SUBM shows a higher efficiency and accuracy at global level.

Exploration of the Character Representation of DNA Chiral Conformations and Deformations via a Curved Surface Discrete Frenet Frame.

While undergoing structural deformation, DNA experiences changes in the interactions between its internal base pairs, presenting challenges to conventional elastic methods. To address this, we propose the Discrete Critical State (DCS) model in this paper. This model combines surface discrete frame theory with gauge theory and Landau phase transition theory to investigate DNA's structural deformation, phase transitions, and chirality. Notably, the DCS model considers both the internal interactions within DNA and formulates an overall equation using unified physical and geometric parameters. By employing the discrete frame, we derive the evolution of physical quantities along the helical axis of DNA, including geodesic curvature, geodesic torsion, and others. Our findings indicate that B-DNA has a significantly lower free energy density compared to Z-DNA, which is in agreement with experimental observations. This research reveals that the direction of base pairs is primarily governed by the geodesic curve within the helical plane, aligning closely with the orientation of the base pairs. Moreover, the geodesic curve has a profound influence on the arrangement of base pairs at the microscopic level and effectively regulates the configuration and geometry of DNA through macroscopic-level free energy considerations.

CEM–DSR-based explicit solution of limit expansion pressure in unsaturated soils and its application

In this study, an explicit solution for the limit expansion pressure of cavity expansion in unsaturated soil, which follows a hydraulic coupling constitutive model, was proposed based on the cavity expansion method (CEM) and deep symbolic regression (DSR). In detail, a semi-analytical elastic–plastic solution for cavity expansion in unsaturated soils was obtained based on the unsaturated critical state model; hence, a database was created. To obtain an explicit solution for the limit expansion pressure, DSR, which is a gradient-based approach to symbolic regression based on reinforcement learning, was applied. Subsequently, 13- and 4-parameter regression analyses were conducted, with satisfactory results being obtained. Finally, based on the proposed explicit expression, a theoretical calculation model for the jacked-pile penetration resistance in unsaturated soils was established. The study presented in this paper opens up possibilities for obtaining theoretical solutions to issues such as predicting pile driving performance in unsaturated soils.

Constraint strategies for estimating in-situ stress from borehole measurements

In this study, we describe strategies to reduce uncertainties in the estimation of in-situ stresses. The strategies are based on the commonly used poro-elastic model with strain corrections for in-situ stress estimation. In addition to the calibration of the minimum horizontal stress magnitudes from measurement data, we further constrain the magnitude of maximum horizontal stress quantitatively with (i) a critical stress state model, (ii) (non)-occurrences of breakout and/or drilling-induced fracture observed from image logs, and (iii) qualitatively with elliptical borehole shapes observed from multi-arm caliper logging data. The methodology is demonstrated using wells near Dawson Creek, Northeast British Columbia and Northwest Alberta, in Western Canada Sedimentary Basin (WCSB). The uncertainty of the estimated maximum horizontal stress magnitude was reduced and the range of the maximum horizontal stress was narrowed after the application of these constraint strategies. It is also observed that in the east of the study area, the in-situ stress regime is a normal fault stress regime. The presence of a normal fault stress regime is unexpected since a strike–slip fault stress regime is typically considered for the entire region. Yet, analysis of caliper data in two horizontal wells has confirmed its presence.

Shear strength reduction factor used in critical state models with hardening

The slope stability analyses are important to predict environmental, financial, and human life impacts. The Limit Equilibrium Method (LEM) is commonly used to estimate a slope’s Safety Factor (SF). However, the Finite Element Method (FEM) is increasingly applied to slope stability analyses, using different approaches, among which the technique of Shear Strength Reduction (SSR) is commonly used in perfectly plastic elastic models. The objective of this study is to present a discussion about these two methodologies, using critical state models with or without hardening will be used to model the stress-strain behavior of the soil mass. The results obtained in the case study presented, using LEM and FEM considering critical state models with and without hardening are consistent and allowed verifying the stability condition of the slope. Also, the reduction factors are smaller when compared to the results using perfectly plastic elastic models.

Critical State Characterization of New Zealand East Coast Sand for Numerical Modeling

This paper outlines the findings of a laboratory-based and numerical study to investigate the undrained flow failure behavior of East Coast Sand (ECS). ECS is a commonly encountered coastal deposit from the upper North Island of New Zealand. The study focused on establishing the undrained strength characteristics of ECS under static, triaxial compressive loading conditions, and at confining pressures in the range of typical engineering interest and for a range of soil densities considered in loosely deposited sands. The research objectives of establishing the basic soil properties and the intrinsic advanced geomechanical properties specific to ECS from Auckland were achieved through laboratory experiments and matching numerical simulations with an advanced critical-state compatible soil constitutive model (Norsand). The current work examined five different aspects of the ECS undrained behavior under static loads. It was shown that loosely deposited ECS within mean effective stresses ranging between 50 and 200 kPa was highly susceptible to expensive flow failures of structures built on or with them. The obtained approximate peak undrained shear strengths before failure and critical states were 29 kPa, 84 kPa, 130 kPa, and 200 kPa for test confining stresses of 50 kPa, 100 kPa, 200 kPa, and 300 kPa, respectively. Similarly, the corresponding excess pore water pressures were 48 kPa, 98 kPa, 200 kPa, and 240 kPa, respectively. The above results proved that the soil’s effective and confining stress are key determinants of the soil’s undrained shear strength characteristics which was consistent with the existing literature.

A Simplified Silty Sand Model

A unified critical state model has been developed for both clean sand and silty sand using the modified Cam-clay model (MCC). The main feature of the proposed model is a new critical state line equation in the e-ln(p) plane that is capable of handling both straight and curved test results. With this feature, the error in calculating plastic volumetric strain is, in theory, eliminated. Another crucial feature of the model is the transformed stress tensor based on the SMP (spatially mobilized plane) criterion, which takes into account the proper shear yield and failure of soil under three-dimensional stresses. Additionally, the proposed model applies the intergranular void ratio with the fines influence factor for silty sand. Only eight soil parameters are required for clean sand, and a total number of twelve soil parameters are needed for silty sand. This model not only enhances the predictive accuracy for granular soils but also broadens the applicability of the model to encompass silty sand in both drained and undrained analyses.

Critical states of a superconducting slab subjected to tension and bending

This work is an extension of a previous critical-state model proposed by the same author [Li and Gao, SuST (2015)]. We want to show the evolution of critical states for a rather long time, and the physics behind the evolution is also desired. It is another purpose to make clear the role of mechanical deformation played in the evolution process of critical state. We thus develop the model with two new features: (a) mechanical loads are applied to both ends of the slab, instead of no external force applied in the previous model; (b) the critical current density is now a function of position, rather than a constant over the slab volume as in the previous model. The new results by the model consist of: (a) the evolution with time for a given strain value is generally the same as the strain-free case; (b) stretching accelerates the development of the critical state, and bending has no significant effect on the critical state; (c) the detailed critical-state profiles may provide how to regulate the electromagnetic field by strain in a superconducting device with slab geometry for practical purpose.

A thermal elastic visco-plastic model for soft clayey soils

It is frequently observed that the stress–strain behaviour of soft clayey soils is affected by temperature changes. Development and verification of a reliable constitutive model with consideration of variable temperature conditions are necessary. Due to the significant rheological and other nonlinear properties of clayey soils, the coupled effects of temperature, time dependency, structuration, nonlinear creep, and anisotropy should be considered in the constitutive model. In this study, a new three-dimensional (3D) thermal elastic visco-plastic model is established and verified for the time-dependent stress–strain behaviour of clayey soils considering temperature changes. The model is developed based on the existing elastic visco-plastic models with the equivalent time concept, the overstress theory, and the critical state model. The thermal elastic line and virgin heating line are introduced and generalized to construct constitutive equations for both thermal elastic and thermal visco-plastic behaviour of clayey soils in general stress conditions. After establishing the 3D basic model, further refinement is introduced to consider the nonlinear creep behaviour and structuration for natural and reconstituted clayey soils. Finally, the model is successfully validated by a series of laboratory test data on different clayey soils under variable temperature paths with reasonably good accuracy.