Critical State Model Research Articles

Below 1 µV cm−1: determining the geometrically-saturated critical transport current of a superconducting tape

In the critical state model, the critical current, , of a superconductor defines the upper limit of dissipation-free current flow. However conventional transport measurements of practical superconductors rely on the detection of a voltage drop along the length of the conductor. This requires that the superconductor has entered the dissipative regime, and hence inherently over-estimates the current at which geometric saturation occurs. Nonetheless, the convenience of the 1 µV cm−1 criterion means that it has become the widely adopted definition of transport . Here, we present an alternative definition for the transport critical current of a superconducting tape under self-field conditions, which is based on the concept of geometric current saturation across the full cross-sectional area. This saturation threshold can be experimentally determined through simple Hall sensor measurements of the evolving perpendicular magnetic field at the tape surface. The surface field exhibits a signature transition when the transport current increases beyond the current-saturation threshold. We present an analytical model which describes this effect and defines the critical saturated current as a function of . This definition is first validated using finite element (FE) modelling, and then experimentally demonstrated through measurements on a variety of commercial REBCO and Bi-2223 tapes of differing widths. It is found that the 1 µV cm−1 criterion consistently leads to an overestimation of the saturated critical current by ∼15% in REBCO tapes, and up to 30% in a Bi-2223 tape. FE modelling indicates that this overestimate is most prominent in superconductors exhibiting a low n-value (i.e. n ≲ 20). A key advantage of the measurement approach presented here is that it allows the unambiguous measurement of a transport value to be completed at lower currents, without entering the dissipative region in which sample damage can occur. There are also implications as to the correct choice of value which should be employed within the well-known Norris and Brandt equations for AC loss.

Numerical integration of an elasto-plastic critical state model for soils under unsaturated conditions

This paper presents the complete set of incremental equations for the numerical integration of the Glasgow Coupled Model (GCM) and a comprehensive algorithm for its numerical integration. The incremental formulation proposed is expressed in terms of strain and suction increments (i.e. strain-driven) and defines an initial value problem (IVP) that can be solved once the initial state and the pair of increments of the driven variables are known. The numerical integration of this IVP is carried out by extending to unsaturated condition, the well-known explicit substepping formulation with automatic error control widely used for saturated soils. A notable feature of the substepping integration scheme presented is that it integrates simultaneously the model equations for both mechanical and water retention responses. Hence, the estimate of the local truncation error to automatically adjust the size of the integration step is not only affected by the local error in stresses and mechanical hardening parameter (as in a saturated soil model) but, additionally, by the local error incurred in the integration of the water retention relations (i.e. degree of saturation and water retention hardening parameter). The correctness of the integration scheme is then verified by comparison of computational outcomes against analytical/reference solutions.

Critical State Model of Sand-Tire Derived Aggregate Mixtures Based on Triaxial Tests

This study is part of an environmental experimental program on the use of scrap automobile tires for geotechnical applications. Different types of laboratory tests were conducted to determine the elastic, plastic, and creep parameters of tire derived aggregate (TDA)-sand granulated mixtures. However, this paper emphasizes the plasticity parameters via the development of a critical state model based on the results of triaxial tests. This was attained by considering loose sand specimens, at a predetermined TDA volumetric content, subject to three different confining pressures under a constant axial displacement rate. The calculated deviatoric stress versus axial strain curves, obtained via the modified Cam Clay model, captured the non-linear elastoplastic response obtained in the tests. Results indicated that the level of the shear strength is highly dependent on critical state friction angle which in turn depends on the TDA content. For the loose TDA-sand mixtures used in the present study, the effect of the TDA content demonstrates a reinforcement of the sand matrix. However this reinforcement diminishes as the TDA content increases.

Nonlocal regularization of an anisotropic critical state model for sand

Many advanced constitutive models which can capture the strain-softening and state-dependent dilatancy response of sand have been developed. These models can give good prediction of the single soil element behaviour under various loading conditions. But the solution will be highly mesh-dependent when they are used in real boundary value problems due to the strain-softening. They can give mesh-dependent strain localization pattern and bearing capacity of foundations on sand. Nonlocal regularization of an anisotropic critical state sand model is presented. The evolution of void ratio which has a significant influence on strain-softening is assumed to depend on the volumetric strain increment of both the local and neighbouring integration points. The regularization method has been implemented using the explicit stress integration method. The nonlocal model has been used in simulating both drained plane strain compression and the response of a strip footing on dry sand. In plane strain compression, mesh-independent results for the force–displacement relationship and shear band thickness can be obtained when the mesh size is smaller than the internal length. The force–displacement relationship of strip footings predicted by the nonlocal model is much less mesh-sensitive than the local model prediction. The strain localization under the strip footing predicted by the nonlocal model is mesh independent. The regularization method is thus proper for application in practical geotechnical engineering problems.

Numerical analysis of slope collapse using SPH and the SIMSAND critical state model

Geological disasters such as slope failure and landslides can cause loss of life and property. Therefore, reproducing their evolution process is of great importance for risk assessment and mitigation. The recently developed SIMSAND critical state sand model combined with the smoothed particle hydrodynamics (SPH) method is adopted in this work to study slope failure under large deformations. To illustrate the efficiency and accuracy of the SIMSAND-SPH approach, a series of slope collapse studies using the discrete element method (DEM) considering various particle shapes (i.e. spherical, tetrahedral and elongated) is adopted as benchmarks. The parameters of the SIMSAND model are calibrated using DEM triaxial tests. In comparison to the DEM simulations, the runout distance and final slope height are well characterized with the SIMSAND-SPH approach with less computational cost. All comparisons show that the SIMSAND-SPH approach is highly efficient and accurate, which can be an alternative numerical tool to simulate real scale granular flow.

Imaging of Strong Nanoscale Vortex Pinning in GdBaCuO High-Temperature Superconducting Tapes.

The high critical current density of second-generation high-temperature superconducting (2G-HTS) tapes is the result of the systematic optimisation of the pinning landscape for superconducting vortices through careful engineering of the size and density of defects and non-superconducting second phases. Here, we use scanning Hall probe microscopy to conduct a vortex-resolved study of commercial GdBaCuO tapes in low fields for the first time and complement this work with “local” magnetisation and transport measurements. Magnetic imaging reveals highly disordered vortex patterns reflecting the presence of strong pinning from a dense distribution of nanoscale Gd2O3 second-phase inclusions in the superconducting film. However, we find that the measured vortex profiles are unexpectedly broad, with full-width-half-maxima typically of 6 μm, and exhibit almost no temperature dependence in the range 10–85 K. Since the lateral displacements of pinned vortex cores are not expected to exceed the superconducting layer thickness, this suggests that the observed broadening is caused by the disruption of the circulating supercurrents due to the high density of nanoscale pinning sites. Deviations of our local magnetisation data from an accepted 2D Bean critical state model also indicate that critical state profiles relax quite rapidly by flux creep. Our measurements provide important information about the role second-phase defects play in enhancing the critical current in these tapes and demonstrate the power of magnetic imaging as a complementary tool in the optimisation of vortex pinning phenomena in 2G-HTS tapes.

Electromagnetic Analysis of HTS DC Cables Based on Critical State Model

High-temperature superconducting (HTS) DC cables have the advantages of low transmission loss, high current density and large transmission capacity, which have gained more and more attentions in recent years. The typical structures of HTS DC cables include unipolar and bipolar coaxial that doubles the cable's transmission capacity. Based on the critical state model, this article analyses and contrasts the critical current density distributions and degradations due to magnetic field in the HTS tapes of the unipolar and coaxial bipolar DC cables. The influences of the bipolar cables with different voltage levels are also compared. The results show that the critical current density degradations due to magnetic field are almost the same in both unipolar and coaxial bipolar structures, which are mainly caused by the self-field. For the bipolar cables, the current density distribution in the outer layer HTS tapes changes with the voltage level. To evaluate the influence of the axial magnetic field, a 3-D model of the unipolar cable is established with a helical wound pitch of the HTS tapes. The results show that the axial magnetic field is relatively low and the critical current density would not degrade if the helical angle of the HTS tapes is reasonably small.

The Magnetization of HTSC Tapes Stack in the Flux Pump Regime

In the present paper, the possibility of magnetization of HTSC tape by means of pumping by a local magnetic field is considered. Within the framework of the critical state model, the calculation of the density of the induced currents and the magnetization of a square (12 × 12 mm) fragment of a second-generation HTS tape (YBCO) was made. The model considers real dependence of the critical current density on the magnetic field and has been successfully tested previously. In the sample, the currents were induced by the impact of a local magnetic field. The size of the localized area of the external magnetic field to be considered was much smaller than the size of the HTS sample. It is showed that the repeated cyclical impact of the local outer field leads to the accumulation of the total magnetic moment in the sample up to a magnitude of 94% of the theoretical magnetization limit for this sample. The results obtained make it possible to optimize the mode of magnetization of a square fragment of a tape (or a stack of tapes) when implementing a magnetic flux pump. The experimental studies showed results comprisable well with theoretical ones.

Elastoplastic solution of drained expansion of a cylindrical cavity in structured soils considering structure degradation

An elastoplastic solution of drained cylindrical cavity expansion is presented based on a well-known critical state model for soils exhibiting a natural structure. The model referred to as the structured Cam-clay (SCC) model is capable of describing the properties of the soil structure and its subsequent stress-induced degradation. The asymmetric stiffness matrix of the elastoplastic constitutive equation for structured soils is derived due to the assumption of the non-associated plastic flow rule. The solution procedure is formulated by transforming the boundary value problem of cylindrical cavity expansion into the problem of solving four ordinary differential equations with three stress components and the specific volume as the basic unknown variables. The theoretical solution is numerically calculated and validated against the existing solution and finite element simulation results. Furthermore, the significance of the soil structure and its stress-induced degradation in the expansion response, and the interaction between the soil structure and overconsolidation are emphasized by extensive parametric analyses. The results demonstrate that the soil structure has a substantial influence on the cavity expansion response, and its degradation is closely related to overconsolidation. Particularly, among the solutions considered, the solution considering the soil structure provides higher effective stress values near the cavity surface due to the additional strength conveyed by the natural structure.

Magnetostriction-induced stress in an orthotropic mechanical thin high-temperature superconducting disk based on the exponential model of critical state

In this paper, the magnetostriction and stress of an orthotropic mechanical high-temperature superconducting disk with a concentric small hole are investigated based on the exponential model of the critical state. Approximate solutions of the displacement and stress for the superconducting disk during zero-field cooling (ZFC) and field-cooling (FC) processes are obtained. The results show that tensile stresses will appear in the disk during the decreasing field of ZFC, while the stresses are always compressive during the increasing field of ZFC and FC processes. This indicates that the superconducting disk is easily damaged during the decreasing field of ZFC and FC processes. The critical point where tensile stress is largest is provided. The value of magnetostriction depends on the size of the disk, the applied field and the magnetization process. This research is helpful in the reliability design of high-temperature superconducting disks and cables with high strength for space solar power stations.

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An express magnetometric technique is proposed, which makes it possible to determine the inhomogeneity of the change in the superconducting properties of a thin superconductor layer due to radiation exposure. A numerical simulation of this technique is carried out using the example of a superconducting GdBa2Cu3O7-x layer irradiated with hydrogen and helium ions. The profile of the concentration of radiation defects was calculated using the SRIM software package. The calculation of the density distribution of superconducting currents induced by an external magnetic field was carried out within the framework of the critical state model. Model calculations have shown the effectiveness of the proposed method.

Magnetocaloric effect and critical behavior of Ni1−xMnxCr2O4 (x = 0, 0.10, and 0.50) compounds

Single-phase polycrystalline samples of Ni1−xMnxCr2O4 (x = 0.0, 0.10, and 0.50) were prepared by the sol–gel route, and their structural, magnetic, and magnetocaloric properties and critical behavior were studied. Structural transformation from a tetragonal (I41/amd) to a cubic (Fd3¯m) phase is observed at room temperature due to the substitution of Jahn–Teller inactive Mn2+ ions. Mn substitution gives rise to a reduction in ferrimagnetic transition temperature. A significant increase in magnetic entropy change (−ΔSM) and relative cooling power (RCP) is observed upon Mn substitution. For x = 0.50, the estimated value of −ΔSM = 2.9 J kg−1 K−1 and the relative of cooling power RCP = 58.2 J kg−1 for an applied field of H = 9 T. The ferrimagnetic transition is studied by recording isothermal magnetization data in a close temperature interval in the vicinity of the transition for the maximum applied field up to 9 T. These data were analyzed based on a modified Arrot plot method. The estimated values of critical exponents do not fall under any specific critical state model and they are discussed in detail.

Critical State Models in Non-Cohesive Media Mechanics (Review)

In this review, we analyze the basic equations and assumptions for the design of critical state models used in the mechanics of non-cohesive media. The relationship of the modified cam clay model with related models of the plasticity theory with isotropic hardening described by closed plasticity surfaces is noted. The state equations of modified cam clay models in the elastic zone are analyzed; the studies, in which the elastic state is described by the hyperelasticity equations with exponential potential, are noted. Generalizations of modified cam clay models to the case of finite deformations are considered. It is important to note that additional research on kinematic combined loading in the spherical and deviatoric regions is needed.

Magnetization Current Simulation of High-Temperature Bulk Superconductors Using the ANSYS Iterative Algorithm Method

The A-V formulation based iterative algorithm method (IAM), developed at Paul Scherrer Institute, shows excellent performance in modeling the critical state magnetization-current in the ReBCO tape stacks after field-cooling magnetization. This article upgrades the function of the IAM, making it realistic to take into account the widely used E-J power law, as well as to model the magnetization-current in bulk high-temperature superconductors under a zero-field-cooling cycle, having both the magnetization and demagnetization stages. The simulation results obtained from the ANSYS-IAM are validated with the well-known H-formulation method for both the critical state model and the flux creep model. The computation times from using the two different numerical methods are compared and discussed. The influence factors, including the iteration steps, the initial superconductor resistivity, the ramping time, and the reservation coefficient, are studied in detail.

Critical fluid density obtained from the theory of generalized charges in accordance with the hypothesis on the first coordination number

In this article, a model of hard spheres for critical fluid, based on the hypothesis on the coordination number of molecules in minimally bound infinite clusters, is considered. Variants of homogeneous (quasi-crystalline lattice) and heterogeneous (“Cayley tree forest”) molecular packings at the vertices of a uniform graph of order 3 are described. Using the theory of generalized charges and taking into account the found structures, the dependences of the critical molar volume on electronic structure parameters, which have the character of non-empirical laws, are derived. A priori calculations of molar volumes by the critical state model are in good agreement with the reference data for simple gases and linear hydrocarbons.

Numerical Modeling of Lateral Dynamic Compaction on the Slope in Dry Sand

Dynamic compaction is a low-cost soil improvement method that is most commonly applied in sandy soils. Free fall compaction is not applicable to compact the soil in the slope, but it is possible to perform lateral dynamic compaction in the slope. In this method, the angle of impact of the tamper with the inclined surface is vertical. In this dynamic compaction method, the tamper is forced to move in a circular path by a cable with specific radius. In this study, ABACUS 6.14 software was used to simulate three-dimensional model of lateral dynamic compaction in the slope. The critical state model that used in numerical simulation is cap plasticity criteria. In this study, the effects of impact position, impact velocity and tamper weight on soil improvement for three different slope were investigated. The analysis focuses on crater depth and improvement region which are compared to the state of flat ground. It was found that on steep slopes to increase the tamper energy, the effect of increasing the impact velocity was greater than the effect of increasing the tamper mass to soil improvement. The crater depth due to lateral dynamic compaction is lower than the horizontal layer as well as the depth of soil improvement.

Pinning mediated field penetration and complex susceptibility in inhomogeneous superconductors

The penetration of the magnetic field in a network consisting of the superconducting grains and intergranular region depends not only on the applied external magnetic field but also on the underlying pinning force density and fraction of the superconducting grains in the sample. Considering the underlying pinning force density in the form of the repetitive Gaussian (RG) nature we have solved the critical state model to get the spatial internal field profile at a particular time instant. The internal magnetic field as a function of the position inside the superconductor exhibits nonlinear, step-like nature. The complicated nature of the penetration of the field gets strongly affected by the size of the granular region and the total pinning force density in the superconductors. We have numerically calculated the real and imaginary parts of the complex susceptibility. The variation of the imaginary part with the temperature has a peak at a temperature which shifts towards the higher temperature with the reduction of the superconducting fraction.

The effect of Ni–Zn ferrite doping on the superconductivity of Y3Ba5Cu8O18 nanocomposite materials

Nanocomposite materials of the Y3Ba5Cu8O18/(Ni0.5Zn0.5Fe2O4)x type, where x = 0.00 wt. %, 0.03 wt. %, 0.10 wt. %, and 0.50 wt. %, are synthesized by the improvement of the solid-state reaction technique. The prepared samples have been characterized by x-ray diffraction (XRD), scanning electron microscopy, x-ray energy dispersion, infra-red spectroscopy, thermo-gravimetric analysis, four probe DC resistance, and current–voltage (I–V) measurements. It was found that the prepared samples have an orthorhombic structure with unit cell parameters (a = 3.9557 Å, b = 3.8197 Å, and c = 30.2460 Å) for pure Y358, and the unit cell volume decreases by the addition of (Ni0.5Zn0.5Fe2O4) nanoparticles into the Y358 compound. The crystallite size (D) was evaluated from XRD patterns by Scherrer’s method. The crystallite size (D) is decreased from 40 nm for pure Y358 to 18 nm for Y358 doped by the Ni–Zn nanoferrite content of 0.50 wt. %. The critical current density Jc and I–V characteristics for superconductor/nanoferrite (Y358/Ni–Zn) composite materials are explained according to Bean’s critical state model and the charge–vortex duality in the Y358 high-Tc mixed-state.

A state-dependent constitutive model for coarse-grained gassy soil and its application in slope instability modelling

Free gas in sandy marine sediments is a common occurrence worldwide. A distinct feature of gassy sand is that, under undrained shearing, presence of occluded gas bubbles in the pore fluid can increase the undrained strength of sand at a relatively loose state, while reduce the strength of a relatively dense sand. Previous theoretical analyses have primarily focused on modelling the ‘beneficial’ effect of free gas on loose sand in migrating static liquefaction, with few attempts to describe the ‘detrimental’ effect of gas on dense sand under undrained loading. This study presents a state-dependent critical state model, which describes the distinct behavior of gassy sand with various states in a unified way. Comparison between the model predictions and test data of three gassy sands shows that the new model can capture the constitutive behavior of gassy marine sand over a wide range of initial states and degrees of saturation (typically between 85% and 100% for unsaturated marine sediments) using a single set of parameters. Parametric studies were performed to quantify the effects of gas (either ‘detrimental’ or ‘beneficial’) on sand with various initial states. The new model has been implemented in ABAQUS and used to simulate the stability of submarine slopes under undrained loading condition. It is found that free gas can improve and weaken the slope stability for loose and dense sand, respectively.

Fast and efficient critical state modelling of field-cooled bulk high-temperature superconductors using a backward computation method

A backward computation method has been developed to accelerate modelling of the critical state magnetization current in a staggered-array bulk high-temperature superconducting (HTS) undulator. The key concept is as follows: (i) a large magnetization current is first generated on the surface of the HTS bulks after rapid field-cooling (FC) magnetization; (ii) the magnetization current then relaxes inwards step-by-step obeying the critical state model; (iii) after tens of backward iterations the magnetization current reaches a steady state. The simulation results show excellent agreement with the H -formulation method for both the electromagnetic and electromagnetic-mechanical coupled analyses, but with significantly faster computation speed. The simulation results using the backward computation method are further validated by the recent experimental results of a five-period Gd–Ba–Cu–O (GdBCO) bulk undulator. Solving the finite element analysis (FEA) model with 1.8 million degrees of freedom (DOFs), the backward computation method takes less than 1.4 h, an order of magnitude or higher faster than other state-of-the-art numerical methods. Finally, the models are used to investigate the influence of the mechanical stress on the distribution of the critical state magnetization current and the undulator field along the central axis.