Dislocation Density Evolution Research Articles

Deformation mechanism of a metastable medium entropy alloy strengthened by the synergy of heterostructure design and cryo-pre-straining

Face-centered cubic (FCC) medium entropy alloys (MEAs) have received considerable attention due to their impressive mechanical properties and responses. However, their practical application is limited by their modest yield strengths. The potential enhancement of the mechanical properties of single-phase MEAs was explored in this study through a synergistic approach combining heterogeneous structure design with subsequent cryo-pre-straining. A heterogeneous lamella structure was produced in a single-phase Fe55Mn20Cr15Ni10 MEA via two-step rolling and annealing. Cryo-pre-straining at varying degrees (6, 12, 21, and 36%) introduced hexagonal close-packed (HCP) phase, high-density dislocations, twins, and stacking faults, leveraging the reduced stacking fault energy at cryogenic temperatures. This process enhanced the alloy's yield strength from 353 MPa to 1.2 GPa (compared to the baseline uniform coarse-grained structure), while maintaining an acceptable total elongation of 8.4%. The impact of cryo-pre-straining on the microstructure and mechanical properties of the MEA was assessed using in-situ synchrotron X-ray diffraction analysis. Cryo-pre-straining (36%) achieved a higher dislocation density (6.1 × 1015 m−2) compared to room-temperature straining (2.5 × 1015 m−2). The stress contribution from HCP-martensite and the evolution of dislocation density during loading were quantified, along with observations of negative stacking fault probability and strain-induced HCP→FCC reverse transformation in cryo-pre-strained samples under loading conditions. Furthermore, the contributions of regulated microstructures to the enhancement of yield strength were quantitatively assessed.

Tailoring Strength and Ductility in Dual-Phase High-Entropy Alloys: Insights from Deep Learning Molecular Dynamics Simulation on FCC/BCC Thickness Ratios

This study investigates the influence of FCC/BCC thickness ratios on the mechanical properties of AlCoCrFeNi dual-phase high-entropy alloys (DP-HEAs) using deep learning-enhanced molecular dynamics simulations. The results demonstrate that varying the thickness ratio significantly affects the stress-strain behavior, dislocation density evolution, and local structural transformations during tensile deformation. DP_0.5, with a thinner BCC phase, exhibits higher dislocation densities and enhanced strain hardening, resulting in increased strength but reduced ductility. In contrast, DP_3.0, with a thicker BCC phase, shows lower dislocation densities, leading to improved ductility but lower strength. The phase transformation from BCC to HCP structures is a key mechanism contributing to plastic deformation, with the BCC/FCC interface playing a critical role in dislocation nucleation and propagation. These findings provide valuable insights into optimizing the microstructural design of DP-HEAs to achieve a tailored balance of strength and ductility, offering the potential for advanced structural applications.

Electroplasticity constitutive modeling of aluminum alloys based on dislocation density evolution

Electrical current can effectively improve the plasticity of metallic materials. The tensile deformation behavior of Al alloys under the pulsed electrical current assisted quasi-static unidirectional tension (EAT) has been investigated. Materials under the EAT exhibits periodic electro-softening and strain-hardening behaviors, i.e., a ratchet shape mechanical response. However, establishing a constitutive model to accurately predict the ratchet shape mechanical behavior, especially during the EAT interval, and accurately predicting the strain-hardening behavior of materials are critical issues that need to be solved urgently. In this study, based on the Taylor polycrystalline model, thermal activation theory and dislocation density evolution theory, a two-parameter dislocation density electroplasticity constitutive model with forward and reverse dislocation density evolution was developed to describe the periodic coupling effect of the electro-thermal-mechanical fields during EAT. The tensile deformation behaviors of AA 6061-T6 and AA 7075-T6 under the effect of a pulsed electrical current were quantitatively predicted using the proposed constitutive model. The results show that the correlation coefficient between the predicted and experimental results of the constitutive model can reach 0.84–0.99, implying that the proposed constitutive model can accurately predict the complex electroplasticity behavior of Al alloys during EAT.

Mechanism of Fatigue-Life Extension Due to Dynamic Strain Aging in Low-Carbon Steel at High Temperature.

An enhancement in fatigue life for ferrite-pearlite low-carbon steel (LCS) at high temperature (HT) has been discovered, where it increased from 190,873 cycles at room temperature (RT) to 10,000,000 cycles at 400 °C under the same stress conditions. To understand the mechanism behind this phenomenon, the evolution of microstructure and dislocation density during fatigue tests was comprehensively investigated. High-power X-ray diffraction (XRD) was employed to analyze the evolution of total dislocation density, while Electron Backscatter Diffraction (EBSD) and High-Resolution EBSD (HR-EBSD) were conducted to reveal the evolutions of kernel average misorientation (KAM), geometrically necessary dislocations (GND) and elastic strains. Results indicate that the enhancement was attributed to the dynamic strain aging (DSA) effect above the upper temperature limit, where serration and jerky flow disappeared but hindrance of dislocations persisted. Due to the DSA effect, periods of increase and decrease in the total dislocations were observed during HT fatigue tests, and the fraction of screw dislocations increased continuously, caused by viscous movement of the screw dislocations. Furthermore, the increased fraction of screw dislocations resulted in a lower energy configuration, reducing slip traces on sample surfaces and preventing fatigue-crack initiation.

A framework toward fatigue life modeling of machining process verified in burnishing

In the present work a multiscale model was developed to predict fatigue lives of Inconel 718 after a machining process. The model captures the effects of surface integrity like residual stress, microstructure, hardness and roughness. Within this holistic predictive model, firstly burnishing process factors (viz static force, feed rate, spindle speed and pass number) are correlated to surface integrity aspects. Herein, residual stress is modeled using theory of incremental plasticity; also grain size and hardness were correlated to process factors through dislocation density evolution. In order to model the surface roughness, the Z-map approach considering the work hardening of the surface layers was utilized. The foresaid developed models of surface integrity aspects were utilized to predict the fatigue life using Navaro-Rios crack propagation together with Chen’s crack initiation models. Thus, a novel holistic model correlating machining parameters directly to fatigue life has been developed through this hybrid approach. The developed models of fatigue life and surface integrity aspects were then confirmed by series of burnishing experiments on Inconel 718. Later, the confirmed model was utilized to identify the parametric influence of burnishing process on variation of fatigue life where the results were justified using variation of surface integrity factors. Through this developed model, it was found that burnishing setting of 1000 N static force, 0.05 mm/rev feed rate, 300RPM spindle speed and 2 pass number results in improvement of HCF, MCF and LCF of 242 %, 184 % and 212 % where the obtained results from the developed framework were compatible well with experimental values.

Subgrain Size Modeling and Substructure Evolution in an AA1050 Aluminum Alloy during High-Temperature Compression.

For materials with high stacking fault energy (SFE), such as aluminum alloys, dynamic recovery (DRV) and dynamic recrystallization (DRX) are essential softening mechanisms during plastic deformation, which lead to the continuous generation and refinement of newborn subgrains (2° ˂ misorientation angle ˂ 15°). The present work investigates the influence of compression parameters on the evolution of the substructures for a 1050 aluminum alloy at elevated temperatures. The alloy microstructure was investigated under deformation temperatures ranging from 300 °C to 500 °C and strain rates from 0.01 to 0.1 s-1, respectively. A well-defined substructure and subsequent subgrain refinement provided indication of the evolution laws of the substructure under high-temperature compression. Corresponding experimental data on the average subgrain size under various compression conditions were obtained. Two different independent average subgrain size evolution models (empirical and substructure-based) were used and applied with several internal state variables. The substructure model employed physical variables to simulate subgrain refinement and thermal coarsening during deformation, incorporating a corresponding dislocation density evolution model. The correlation coefficient (R) and root mean square error (RMSE) of the substructure-based model were calculated to be 0.98 and 5.7%, respectively. These models can provide good estimates of the average subgrain size, with both predictions and experiments reproducing the expected subgrain size evolution using physically meaningful variables during continuous deformation.

Exploring grain size influence on tensile behavior of 316 H austenitic stainless steel at high temperature: A phenomenological dislocation model

The high temperature mechanical behavior of 316 H austenitic stainless steel with different grain size was investigated in this study through the use of a novel phenomenological dislocation model finite element method. The study was conducted by producing steel specimens with varying grain sizes through adjustments in annealing temperature and time, followed by high temperature tensile tests at 550°C. The finite element method, when integrated with a phenomenological dislocation model that considers grain size parameters, effectively captured the mechanical response under different grain size conditions. Additionally, the two-variable Kocks-Mecking(KM) model accurately captured the grain size effect in 316 H austenitic steel and effectively depicted its tensile flow behavior. The model’s predictions were based on the evolution of forest dislocation density and mobile dislocation density, proving to be a reliable tool for analyzing the microstructural evolution of 316 H austenitic steel specimens with varying grain sizes. This study provides insight into the effect of grain size on the high temperature strength of austenitic stainless steels and demonstrates the utility of a novel phenomenological model finite element method for predicting the mechanical behavior of polycrystalline materials.

Comparison and evaluation of different constitutive models for predicting the hot deformation behavior of Mg-Gd-Y-Zr alloy

The popular constitutive models used in the field of hot forming of magnesium alloys can be divided into phenomenological models, machine learning models, and internal state variables (ISV) models based on physical mechanisms. Currently, there is a lack of comparison and evaluation regarding the suitability of different types of models. In this study, Mg-Gd-Y-Zr alloy is taken as the research object. The hot deformation behavior of the alloy was studied systematically. Subsequently, Arrhenius model with strain compensation, artificial neural network (ANN) model, and ISV model involving dynamic recrystallization (DRX), dislocation density and grain size evolution were established. ANN model demonstrates a higher level of accuracy in fitting the original stress-strain curves compared to both ISV model and modified Arrhenius model, but ANN model is not suitable for predicting the experimental results outside of the initial database. ISV model considers the impact of microstructure evolution history on stress, making it highly effective in reflecting the mechanical responses under complex loading condition. The established ISV model is embedded in the ABAQUS software, which shows good ability in calculating the mechanical response, dimension, and microstructure evolution information of the component during hot forming.

Strain-hardening resilience via the cooperation of geometrically necessary dislocations and deformation twins in a strong and ductile lightweight high-entropy steel

In the net-zero era, the burgeoning demands within engineering applications require materials that are not only lighter and stronger but also ductile. A robust strain hardening, crucial for achieving high strength and ductility, is challenging due to limited dislocation density evolution. This study discovers a cooperative strain-hardening strategy in a newly designed high-entropy steel (HES) with a density of 6.82 g/cm3. In this lightweight HES, the duplex microstructure of compositionally complex austenite and D03 intermetallic compounds facilitates the interplay between geometrically necessary dislocations (GNDs) and deformation twins (DTs) during plastic deformation. It generates a strain-hardening resilience during deformation and yields a very high dislocation density of 6.62 × 1015 m-2, contributing to strain hardening of over 900 MPa and a large elongation of 47 %. The resilient strain hardening achieved by the GND-DT cooperative strategy can be applied to various heterostructured alloys, offering a pathway for strong and ductile lightweight materials.

An internal state variables constitutive model for Ni-based superalloy considering the influence of second phase and its application in flexible skew rolling

Microstructure prediction and control of Ni-based alloy during plastic forming is crucial for obtaining high-performance products. Taking GH4169 alloy as an example for study, firstly, hot compression experiments were conducted using both solution treated (ST) and aging treated (AT) alloys at different true strains (0.223–0.916), temperatures (900℃-1100℃), and strain rates (0.1 s−1-10 s−1). Then, an internal state variable constitutive model was established based on experimental data. In the developed model, the evolution of dislocation density, dynamic recovery, and dynamic recrystallization (DRX) behavior are reasonably described. Specifically, by introducing the volume fraction and size of the initial δ-phase, the interaction between δ-phase and dislocations, the influence of δ-phase on DRX behavior and critical strain, and its pinning effect on grain boundaries are considered. The calculation results indicate that accurate predictions can be achieved within a large parameter range. Subsequently, the model was applied to flexible skew rolling (FSR) process, a novel plastic forming method for forming shaft parts and bars. The finite element (FE) simulations and rolling experiments were carried out. The results indicate that the FE model embedded with the constitutive model can effectively predict the forming dimensions and microstructure distribution of GH4169 alloy. The established constitutive model can provide reference for the high-temperature plastic forming process of alloys containing the second phase.

Strain Hardening in Dilute Binary Al–Cu, Al–Zn, and Al–Mn Alloys: Experiment and Modeling

During thermo-mechanical processing, dissolved alloying elements have a huge impact on the microstructure evolution by influencing the overall dislocation storage rate. Especially, for non-heat treatable Al alloys, the effects of strain-hardening and solid solution strengthening are of significant practical interest. In the present work, a detailed study of the room temperature work-hardening behavior of binary Al–Cu, Al–Zn, and Al–Mn alloys with varying solute concentrations is carried out. Stress–strain curves at different strain rates are recorded and computationally analyzed by an advanced 3-Internal-Variables-Model (3IVM) approach for the dislocation density evolution. The initial strengthening rate is examined as a function of the solute concentration.

Effect of loading modes on uniaxial creep-fatigue deformation: A dislocation based viscoplastic constitutive model

A comprehensive investigation was conducted on the stress-strain responses and microstructural evolutions of 9Cr3Co3WCu martensitic steel at 650 ℃ subjected to low cycle fatigue tests, strain-controlled creep-fatigue tests (SNCFTs), and hybrid stress-strain-controlled creep-fatigue tests (HSSCFTs). The creep strain accumulation per cycle in HSSCFTs exhibited three stages: an initial decrease in the early cycles, followed by a prolonged period of steady increase, culminating in a rapid increase prior to fracture. In contrast, the creep strain accumulation in SNCFTs showed a consistent decreasing trend throughout the test. Through the employment of advanced characterization techniques, the evolution of dislocation density exhibited a decreasing rate in all cases, and a fracture morphology characterized by a large size of creep voids was observed in HSSCFTs. Consequently, a dislocation-based viscoplastic constitutive model was developed, incorporating a relaxation parameter, a slip deformation resistance model and a dislocation density evolution model interacting with the grain boundary. The proposed model demonstrated excellent congruences under fatigue, creep, creep-fatigue, and multi-step creep- fatigue tests between experimental and simulated results, thereby confirming its capability to comprehensively capture cyclic deformation under various loading modes.

Crystal Quality and Efficiency Engineering of InGaN‐Based Red Light‐Emitting Diodes

This article is aimed at understanding of the complex design of metalorganic chemical vapour deposition ‐grown InGaN‐based red light‐emitting diode (LED) structure. The contribution of different elements of red LED structure to the stress distribution and threading dislocation density (TDD) evolution is theoretically investigated. For this purpose a self‐consistent modeling of the structure growth process is used, taking into account stress‐modulated indium incorporation, mismatch stress relaxation by threading dislocations and V‐pits, and nucleation of new threading dislocations. The simulation results, consisting of composition, stress, and TDD profiles, are then utilized for modeling of device operation, which allows to analyze contribution of different elements to the heterostructure operation.

In-situ synchrotron diffraction analysis of deformation mechanisms in an AA5083 sheet metal processed by modified equal-channel angular pressing

Severe plastic deformation (SPD) is a process to significantly improve the mechanical properties of materials by grain refinement. For forming at room temperature (RT), higher strengths can thus be achieved; for increased temperatures (HT), greater elongations can be achieved. This great potential has been chiefly applied only on laboratory scale due to a lack of industrial process design. Equal-channel angular pressing (ECAP) offers the possibility to integrate these advantages in industrial processes. This paper investigates a modified ECAP process for aluminum AA5083 sheet metal. Two passes of ECAP for sheet metal and an additional heat treatment were used. To study the material deformation on a micro-scale, in-situ synchrotron measurements at DESY (Hamburg) were performed to analyze the evolution of lattice strains and dislocation densities during tensile testing. The mechanical properties of the ECAP-processed sheet material reveal an increase in yield strength and a decrease in elongation at RT. Higher strains can be measured for the processed material at elevated test temperatures. In-situ diffraction results show a changed behavior of the ECAP sheet material compared to the reference material for both temperature regimes. These changes are based on the increased defect density in the material, which leads to increased strength at RT. At HT, dynamic recrystallization occurs during tensile testing, and the prevailing deformation mechanisms reconfigure. Electron backscatter diffraction (EBSD) and micrographs are used to interpret and supplement the observations. The findings provide the basis for the in-depth understanding of the deformation behavior of the material and help to apply the ECAP process on an industrial scale.

Study of the microstructure evolution of alloy structural steel and inhomogeneity effect of the microscale pulsed currents during current-assisted plane strain compressions by modeling a novel cellular automata method

The current-assisted forming process is considered to be a novel process that utilizes the electroplasticity effect to reduce the deformation resistance of difficult-to-deform metals and to refine the grain size while improving the mechanical properties of the part. However, the lack of understanding of the mechanisms by which pulsed current affects grain refinement still makes it difficult to experimentally develop analytical or empirical models of the relationship between grain size, pulse current parameters, and deformation amount. It will seriously hinder the further development and application of the current-assisted forming process. To reveal the mechanism of grain refinement coupling with electroplasticity, a series of Electron Back Scattering Diffraction observation tests were carried out after current-assisted plane strain compressions. The results show that the grain orientation spread value of the pulse current condition is lower than that of the non-current condition, while the refined grain area percentage is higher than that of the non-current condition. Based on the microstructure observation results, a cellular automata model for grain refinement is proposed. This consists of a dislocation density evolution, sub-grains generation, grain fragmentation, and a grain orientation rotation algorithm. To study the electroplasticity of the grain refinement process, the cellular automata model is also embedded with a finite difference numerical algorithm for solving the microscopic current density distribution of the model. The cellular automata model simulation results show that the error of grain size accuracy of this cellular automata model under non-current and pulse current conditions is 10.48% and 8.55%, respectively. It shows that the model can accurately characterize the grain refinement behavior of the current-assisted plane strain compression. The simulation results reveal the deep mechanism of pulsed current promotion on grain refinement, i.e., an inhomogeneous microstructure leads to uneven distribution of pulse current density. The inhomogeneous current distribution will further increase the grain refinement rate in the coarse-grained regions, thus increasing the proportion of the refined grain regions in the microstructure and leading to a relatively more homogeneous microstructure under pulse current conditions. The cellular automata model accurately reveals the mechanism of electroplasticity on the grain refinement. The model will serve as a crucial theoretical reference in designing current-assisted forming process routes to ensure excellent microstructure and properties in manufactured parts. It will enhance the widespread utilization of current-assisted forming process.

Study of the Dynamic Recrystallization Behavior of Mg-Gd-Y-Zn-Zr Alloy Based on Experiments and Cellular Automaton Simulation

The exploration of the relationship between process parameters and grain evolution during the thermal deformation of rare-earth magnesium alloys using simulation software has significant implications for enhancing research and development efficiency and advancing the large-scale engineering application of high-performance rare-earth magnesium alloys. Through single-pass hot compression experiments, this study obtained high-temperature flow stress curves for rare-earth magnesium alloys, analyzing the variation patterns of these curves and the softening mechanism of the materials. Drawing on physical metallurgical theories, such as the evolution of dislocation density during dynamic recrystallization, recrystallization nucleation, and grain growth, the authors of this paper establish a cellular automaton model to simulate the dynamic recrystallization process by tracking the sole internal variable—the evolution of dislocation density within cells. This model was developed through the secondary development of the DEFORM-3D finite element software. The results indicate that the model established in this study accurately simulates the evolution process of grain growth during heat treatment and the dynamic recrystallization microstructure during the thermal deformation of rare-earth magnesium alloys. The simulated results align well with relevant theories and metallographic experimental results, enabling the simulation of the dynamic recrystallization microstructure and grain size prediction during the deformation process of rare-earth magnesium alloys.

How are microstructural defect interactions linked to simultaneous intergranular and transgranular fracture modes in polycrystalline systems?

The major objective of this investigation is to fundamentally understand and predict how intergranular (IG) and transgranular (TG) fracture modes nucleate and propagate in f.c.c. polycrystalline systems due to defects, such as total and partial dislocation densities and grain boundary (GB) structures and misorientations. A dislocation density crystalline plasticity (DCP) formulation based on the evolution and interaction of total and partial dislocation densities was integrated with a recently developed fracture approach to investigate the fracture nucleation and propagation of simultaneous multiple fracture events, including both IG and TG fracture events. The aggregate grains and GB orientations and morphologies are based on EBSD measurements that are representative of polycrystalline copper aggregates with a broad range of random high angle and low angle GBs. The predictions indicate that dislocation density pileups induce IG fracture due to interrelated stress, slip, and total and partial dislocation density accumulations and interactions for both high angle GBs (HAGBs) and low angle GBs (LAGBs). TG fracture nucleation and propagation occurred due to normal stress accumulations, which exceeds the fracture stress, along cleavage planes. Furthermore, it is shown how IG cracks transition from the GB plane to the cleavage planes as cracks nucleate and propagate. Accumulated plastic zones due to different slip system activities can impede and blunt crack propagation and fronts. These plastic zones form due to high Lomer and Shockley partial dislocation densities. These predictions, which are consistent with experimental observations, provide a fundamental understanding of how simultaneous failure modes initiate and propagate for physically representative microstructures.

Thermal ratcheting of uranium simulated with a thermo-elasto-visco-plastic self-consistent polycrystal model

Uranium polycrystal aggregates accumulate large plastic deformation during thermal cycling in a process known as thermal ratcheting. This process is induced by highly anisotropic single crystal properties, texture, and a complex coupling between thermal, elastic, and plastic mechanisms of deformation. In this study, a thermo-elasto-visco-plastic polycrystal model is developed and applied to the prediction of thermal ratcheting in uranium. The model is based on the evolution of dislocation density on individual slip systems and accounts for static recovery effects. The model captures the plastic relaxation of intergranular stresses induced by thermal cycling and leads to quantitative understanding of the coupling between thermal, elastic and plastic mechanisms, and the important role played by static dislocation recovery in uranium. Results are compared with thermal ratcheting measurements.

A dislocation density-based crystal plasticity constitutive model: comparison of VPSC effective medium predictions with ρ-CP finite element predictions

This work presents a dislocation density-based crystal plasticity constitutive model for glide kinetics, strengthening and dislocation density evolution, implemented in the effective medium-based visco-plastic self consistent (VPSC) framework and the spatially resolved, ρ-CP crystal plasticity finite element framework. Additionally, a distribution of intragranular stresses is introduced in the VPSC framework, instead of the conventionally used mean value of grain stress for effective medium calculations. The ρ-CP model is first calibrated to predict the mechanical response of a bcc ferritic steel with an initial rolled texture. The same set of constitutive model parameters are then used in VPSC to predict the aggregate stress–strain response and total dislocation densities. For these VPSC simulations, the interaction parameter governing the interaction between the grain and the effective medium in the Eshelby inclusion formalism, and a scalar parameter representative of the distribution of intragranular stresses within a grain, are used to calibrate the VPSC predictions in order to match the predictions of the ρ-CP model. A parametric study is performed to understand the effect of these two parameters on the VPSC predictions. Further, simulations are also performed for a random untextured polycrystal to identify the corresponding VPSC simulation parameters for predicting a similar response as the ρ-CP model. The novelty of the work is in the same set of constitutive models and associated parameters have been implemented in VPSC and ρ-CP to predict similar aggregate stress–strain response and total dislocation densities. This finite element-calibrated effective medium crystal plasticity approach reduces the computational time by at least two orders of magnitude and represents an advance towards the development of multiscale crystal plasticity modeling tools.

Assessment of phenomenological models for Indentation Size Effect (ISE) through physically based dislocation density evolution

The depth dependent hardness, referred to as indentation size effect (ISE) has gained considerable attention. ISE is attributed to the strain gradient caused by the inhomogeneous strain distribution beneath the indenter tip. The depth dependence of hardness is phenomenologically accounted using an appropriate length scale parameter. The Unified model is one such model that is built on Nix–Gao’s model and focuses on integrating various length scale parameters resulting from both microstructural and geometrical effects. One aspect of the current work is to assess the applicability of the Unified model to a material like bulk Nickel with a history of cold work. Also, the physics of ISE in metals can be related to the dislocation mean free path influenced by strengthening mechanisms. Past attempts are scarce to quantify the ISE through the microstructure variables. In the present work, an attempt has been made to simulate the nanoindentation test using a dislocation density based constitutive model. The objective of this study is to establish a correlation between the equivalent plastic strain resulting from deep depth indentation (which does not exhibit a size effect) and an equivalent dislocation density. The dislocation density here serves as a state variable that describes the microstructure-dependent mechanical behaviour. The modified KME, a dislocation density-based model, is incorporated as a user subroutine in a commercial finite element solver. It has been proven to accurately predict the indentation behaviour at substantially large depths, which aligns well with the predictions of the Unified model or experimental observations. The good correlation between the value of effective dislocation density and equivalent plastic strain in the plastic zone underneath the indenter assures to extend the use of dislocation density as a state variable to predict ISE.